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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>One-loop 𝑁-point correlators in pure gravity</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2411.07939v2/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S1" title="In One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">I </span>Introduction</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S2" title="In One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II </span>Equations of motion</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S3" title="In One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">III </span>Loop recursions</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S4" title="In One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">IV </span>Some examples</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S5" title="In One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">V </span>Final remarks</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line" lang="en"> <h1 class="ltx_title ltx_title_document">One-loop <math alttext="N" class="ltx_Math" display="inline" id="id1.m1.1"><semantics id="id1.m1.1b"><mi id="id1.m1.1.1" xref="id1.m1.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="id1.m1.1c"><ci id="id1.m1.1.1.cmml" xref="id1.m1.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="id1.m1.1d">N</annotation><annotation encoding="application/x-llamapun" id="id1.m1.1e">italic_N</annotation></semantics></math>-point correlators in pure gravity</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Humberto Gomez<sup class="ltx_sup" id="id13.2.id1"><span class="ltx_text ltx_font_italic" id="id13.2.id1.1">a,b</span></sup> </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Renann Lipinski Jusinskas<sup class="ltx_sup" id="id14.2.id1"><span class="ltx_text ltx_font_italic" id="id14.2.id1.1">b</span></sup> </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Cristhiam Lopez-Arcos<sup class="ltx_sup" id="id15.2.id1"><span class="ltx_text ltx_font_italic" id="id15.2.id1.1">c,d</span></sup> </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Alexander Quintero Vélez<sup class="ltx_sup" id="id16.8.id1"><span class="ltx_text ltx_font_italic" id="id16.8.id1.1">c</span></sup> </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation"><sup class="ltx_sup" id="id17.9.id1"><span class="ltx_text ltx_font_italic" id="id17.9.id1.1">a</span></sup> Facultad de Ciencias Basicas, Universidad Santiago de Cali, <br class="ltx_break"/>Calle 5 <math alttext="N^{\circ}" class="ltx_Math" display="inline" id="id7.3.m2.1"><semantics id="id7.3.m2.1a"><msup id="id7.3.m2.1.1" xref="id7.3.m2.1.1.cmml"><mi id="id7.3.m2.1.1.2" xref="id7.3.m2.1.1.2.cmml">N</mi><mo id="id7.3.m2.1.1.3" xref="id7.3.m2.1.1.3.cmml">∘</mo></msup><annotation-xml encoding="MathML-Content" id="id7.3.m2.1b"><apply id="id7.3.m2.1.1.cmml" xref="id7.3.m2.1.1"><csymbol cd="ambiguous" id="id7.3.m2.1.1.1.cmml" xref="id7.3.m2.1.1">superscript</csymbol><ci id="id7.3.m2.1.1.2.cmml" xref="id7.3.m2.1.1.2">𝑁</ci><compose id="id7.3.m2.1.1.3.cmml" xref="id7.3.m2.1.1.3"></compose></apply></annotation-xml><annotation encoding="application/x-tex" id="id7.3.m2.1c">N^{\circ}</annotation><annotation encoding="application/x-llamapun" id="id7.3.m2.1d">italic_N start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT</annotation></semantics></math> 62-00 Barrio Pampalinda, Cali, Valle, Colombia </span> <span class="ltx_contact ltx_role_affiliation"><sup class="ltx_sup" id="id18.10.id1"><span class="ltx_text ltx_font_italic" id="id18.10.id1.1">b</span></sup> Institute of Physics of the Czech Academy of Sciences & CEICO <br class="ltx_break"/>Na Slovance 2, 18221 Prague, Czech Republic </span> <span class="ltx_contact ltx_role_affiliation"><sup class="ltx_sup" id="id19.11.id1"><span class="ltx_text ltx_font_italic" id="id19.11.id1.1">c</span></sup> Escuela de Matemáticas, Universidad Nacional de Colombia, <br class="ltx_break"/>Sede Medellín, Carrera 65 <math alttext="\#" class="ltx_Math" display="inline" id="id10.6.m2.1"><semantics id="id10.6.m2.1a"><mi id="id10.6.m2.1.1" mathvariant="normal" xref="id10.6.m2.1.1.cmml">#</mi><annotation-xml encoding="MathML-Content" id="id10.6.m2.1b"><ci id="id10.6.m2.1.1.cmml" xref="id10.6.m2.1.1">#</ci></annotation-xml><annotation encoding="application/x-tex" id="id10.6.m2.1c">\#</annotation><annotation encoding="application/x-llamapun" id="id10.6.m2.1d">#</annotation></semantics></math> 59A–110, Medellín, Colombia </span> <span class="ltx_contact ltx_role_affiliation"><sup class="ltx_sup" id="id20.12.id1"><span class="ltx_text ltx_font_italic" id="id20.12.id1.1">d</span></sup> Universidad EIA, C.P. 055428, Envigado, Colombia </span></span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id12.1"><span class="ltx_text" id="id12.1.1">In this work we propose a simple algebraic recursion for the complete one-loop integrands of <math alttext="N" class="ltx_Math" display="inline" id="id12.1.1.m1.1"><semantics id="id12.1.1.m1.1a"><mi id="id12.1.1.m1.1.1" xref="id12.1.1.m1.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="id12.1.1.m1.1b"><ci id="id12.1.1.m1.1.1.cmml" xref="id12.1.1.m1.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="id12.1.1.m1.1c">N</annotation><annotation encoding="application/x-llamapun" id="id12.1.1.m1.1d">italic_N</annotation></semantics></math>-graviton correlators. This formula automatically yields the correct symmetry factors of individual diagrams, taking into account both the graviton and the ghost loop, and seamlessly controlling the related combinatorics.</span></p> </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">I </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">Achieving a satisfactory quantum description of gravity has long been a major aspiration in theoretical physics. As a quantum field theory, earlier research has introduced seminal works (e.g. <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib1" title="">Feynman:1996kb </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib2" title="">Weinberg:1964kqu </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib3" title="">Weinberg:1964ew </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib4" title="">Weinberg:1965rz </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib5" title="">Weinberg:1965nx </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib6" title="">DeWitt:1967yk </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib7" title="">DeWitt:1967ub </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib8" title="">DeWitt:1967uc </a></cite>), but we were soon confronted with an inconvenient observation: gravity is not renormalizable <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib9" title="">tHooft:1974toh </a></cite>. From an effective field theory (EFT) perspective, this is not an obstacle but rather a source of valuable lessons using perturbation theory (see <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib10" title="">Donoghue:2022eay </a></cite> and references therein).</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">In ordinary gauge theories, perturbative computations using a traditional diagrammatic approach are incumbered by gauge redundancies. This is even more pronounced in gravity, which in addition contains an infinite number of field theory vertices and non-trivial field redefinitions (e.g. <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib11" title="">Ananth:2007zy </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib12" title="">Knorr:2023usb </a></cite>). While these redundancies appear to be unavoidable in a Lagrangian formulation, modern scattering amplitudes techniques have long abandoned them. The so-called on-shell methods (see e.g. <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib13" title="">Travaglini:2022uwo </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib14" title="">Brandhuber:2022qbk </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib15" title="">Bern:2022wqg </a></cite>) have revolutionized the area, enabling the systematic computation of S-matrices based on just a few ingredients (particle content, global symmetries, factorization, unitarity). A paradigmatic example is the double copy <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib16" title="">Bern:2008qj </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib17" title="">Bern:2010ue </a></cite>.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">Beyond on-shell results, there is increased interest on graviton correlators in different contexts: renormalization group analysis, form factors, curved backgrounds (including AdS/CFT), and gravitational EFTs, to name a few. Since the role of gravitons is firmly established in modern theoretical physics, going off the mass-shell is an important step. This is especially relevant in the exciting new era of gravitational waves and black hole observations. An off-shell formulation would naturally lead to a sounder description of the quanta of gravity, whether or not they can be detected <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib18" title="">Rothman:2006fp </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib19" title="">Dyson:2013hbl </a></cite>. At the same time, it could further expose the shortcomings of a particle/field description in lieu of a more overarching model such as string theory <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib20" title="">Weinberg:2021exr </a></cite>.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">Current off-shell methods are few and limited. The string-inspired world-line formalism <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib21" title="">Schubert:2001he </a></cite>, for instance, is presently well-suited to compute only the irreducible part of one-loop correlators <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib22" title="">Bastianelli:2013tsa </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib25" title="">Ahmadiniaz:2012xp </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib23" title="">Ahmadiniaz:2020jgo </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib24" title="">Ahmadiniaz:2023vrk </a></cite>. In spite of ongoing interest (e.g. <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib26" title="">Bastianelli:2022pqq </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib27" title="">Brandt:2022und </a></cite>), state-of-the-art loop computations with off-shell gravitons are still inneficiently based on diagram expansions, as in <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib28" title="">Goroff:1985sz </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib29" title="">Goroff:1985th </a></cite>. Some time ago, we established a recursive technique to compute one-loop correlators in colored theories <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib30" title="">Gomez:2022dzk </a></cite> via an extension of the perturbiner method <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib31" title="">Rosly:1996vr </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib32" title="">Rosly:1997ap </a></cite>. The underlying idea is to generate scattering trees from field equations. This has been long known at tree level <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib33" title="">Boulware:1968zz </a></cite> (also <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib34" title="">Berends:1987me </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib35" title="">Bardeen:1995gk </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib36" title="">Cangemi:1996rx </a></cite>), and the perturbiner streamlined the procedure (see <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib37" title="">Mafra:2015gia </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib38" title="">Lee:2015upy </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib39" title="">Mafra:2015vca </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib40" title="">Mafra:2016ltu </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib41" title="">Mafra:2016mcc </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib42" title="">Mizera:2018jbh </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib43" title="">Garozzo:2018uzj </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib44" title="">Lopez-Arcos:2019hvg </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib45" title="">Gomez:2020vat </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib46" title="">Guillen:2021mwp </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib47" title="">Gomez:2021shh </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib48" title="">Ben-Shahar:2021doh </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib49" title="">Cho:2021nim </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib50" title="">Escudero:2022zdz </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib51" title="">Lee:2022aiu </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib52" title="">Cho:2022faq </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib53" title="">Cho:2023kux </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib54" title="">Damgaard:2024fqj </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib55" title="">Chen:2023bji </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib56" title="">Tao:2023yxy </a></cite> for a series of recent developments and applications). The quantum (off-shell) step of <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib30" title="">Gomez:2022dzk </a></cite> involved a controlled sewing procedure to recursively generate one-loop integrands. These results relied heavily on the color ordering (like in bi-adjoint scalar and Yang-Mills theories), and could not be applied to colorless fields.</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.1">In this work we finally report a breakthrough in the recursive construction of one-loop <math alttext="N" class="ltx_Math" display="inline" id="S1.p5.1.m1.1"><semantics id="S1.p5.1.m1.1a"><mi id="S1.p5.1.m1.1.1" xref="S1.p5.1.m1.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S1.p5.1.m1.1b"><ci id="S1.p5.1.m1.1.1.cmml" xref="S1.p5.1.m1.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.1.m1.1c">N</annotation><annotation encoding="application/x-llamapun" id="S1.p5.1.m1.1d">italic_N</annotation></semantics></math>-graviton correlators. The perturbiner offers a natural solution to the problem of defining a recursion encompassing the infinite number of graviton vertices <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib47" title="">Gomez:2021shh </a></cite>. Our proposal is based on an elegant solution to the expected overcounting of Feynman diagrams generated by the sewing procedure, ultimately taming their combinatorial nature. We show that our formula automatically generates the fully off-shell integrands, with both graviton and ghost loops. As a side result, we also propose a formula for one-loop amplitudes, automatically removing tadpoles and external-leg bubbles.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">II </span>Equations of motion</h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.1">The Einstein-Hilbert action in <math alttext="d" class="ltx_Math" display="inline" id="S2.p1.1.m1.1"><semantics id="S2.p1.1.m1.1a"><mi id="S2.p1.1.m1.1.1" xref="S2.p1.1.m1.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S2.p1.1.m1.1b"><ci id="S2.p1.1.m1.1.1.cmml" xref="S2.p1.1.m1.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.1.m1.1c">d</annotation><annotation encoding="application/x-llamapun" id="S2.p1.1.m1.1d">italic_d</annotation></semantics></math> dimensions is</p> <table class="ltx_equation ltx_eqn_table" id="S2.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="S_{\text{EH}}=\frac{1}{2\kappa}\int d^{d}x\sqrt{-g}R," class="ltx_Math" display="block" id="S2.E1.m1.1"><semantics id="S2.E1.m1.1a"><mrow id="S2.E1.m1.1.1.1" xref="S2.E1.m1.1.1.1.1.cmml"><mrow id="S2.E1.m1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.cmml"><msub id="S2.E1.m1.1.1.1.1.2" xref="S2.E1.m1.1.1.1.1.2.cmml"><mi id="S2.E1.m1.1.1.1.1.2.2" xref="S2.E1.m1.1.1.1.1.2.2.cmml">S</mi><mtext id="S2.E1.m1.1.1.1.1.2.3" xref="S2.E1.m1.1.1.1.1.2.3a.cmml">EH</mtext></msub><mo id="S2.E1.m1.1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.1.cmml">=</mo><mrow id="S2.E1.m1.1.1.1.1.3" xref="S2.E1.m1.1.1.1.1.3.cmml"><mfrac id="S2.E1.m1.1.1.1.1.3.2" xref="S2.E1.m1.1.1.1.1.3.2.cmml"><mn id="S2.E1.m1.1.1.1.1.3.2.2" xref="S2.E1.m1.1.1.1.1.3.2.2.cmml">1</mn><mrow id="S2.E1.m1.1.1.1.1.3.2.3" xref="S2.E1.m1.1.1.1.1.3.2.3.cmml"><mn id="S2.E1.m1.1.1.1.1.3.2.3.2" xref="S2.E1.m1.1.1.1.1.3.2.3.2.cmml">2</mn><mo id="S2.E1.m1.1.1.1.1.3.2.3.1" xref="S2.E1.m1.1.1.1.1.3.2.3.1.cmml">⁢</mo><mi id="S2.E1.m1.1.1.1.1.3.2.3.3" xref="S2.E1.m1.1.1.1.1.3.2.3.3.cmml">κ</mi></mrow></mfrac><mo id="S2.E1.m1.1.1.1.1.3.1" xref="S2.E1.m1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S2.E1.m1.1.1.1.1.3.3" xref="S2.E1.m1.1.1.1.1.3.3.cmml"><mo id="S2.E1.m1.1.1.1.1.3.3.1" xref="S2.E1.m1.1.1.1.1.3.3.1.cmml">∫</mo><mrow id="S2.E1.m1.1.1.1.1.3.3.2" xref="S2.E1.m1.1.1.1.1.3.3.2.cmml"><msup id="S2.E1.m1.1.1.1.1.3.3.2.2" xref="S2.E1.m1.1.1.1.1.3.3.2.2.cmml"><mi id="S2.E1.m1.1.1.1.1.3.3.2.2.2" xref="S2.E1.m1.1.1.1.1.3.3.2.2.2.cmml">d</mi><mi id="S2.E1.m1.1.1.1.1.3.3.2.2.3" xref="S2.E1.m1.1.1.1.1.3.3.2.2.3.cmml">d</mi></msup><mo id="S2.E1.m1.1.1.1.1.3.3.2.1" xref="S2.E1.m1.1.1.1.1.3.3.2.1.cmml">⁢</mo><mi id="S2.E1.m1.1.1.1.1.3.3.2.3" xref="S2.E1.m1.1.1.1.1.3.3.2.3.cmml">x</mi><mo id="S2.E1.m1.1.1.1.1.3.3.2.1a" xref="S2.E1.m1.1.1.1.1.3.3.2.1.cmml">⁢</mo><msqrt id="S2.E1.m1.1.1.1.1.3.3.2.4" 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xref="S2.E1.m1.1.1.1.1.3.3.2.4.2.2">𝑔</ci></apply></apply><ci id="S2.E1.m1.1.1.1.1.3.3.2.5.cmml" xref="S2.E1.m1.1.1.1.1.3.3.2.5">𝑅</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E1.m1.1c">S_{\text{EH}}=\frac{1}{2\kappa}\int d^{d}x\sqrt{-g}R,</annotation><annotation encoding="application/x-llamapun" id="S2.E1.m1.1d">italic_S start_POSTSUBSCRIPT EH end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 italic_κ end_ARG ∫ italic_d start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT italic_x square-root start_ARG - italic_g end_ARG italic_R ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p1.6">where <math alttext="\kappa" class="ltx_Math" display="inline" id="S2.p1.2.m1.1"><semantics id="S2.p1.2.m1.1a"><mi id="S2.p1.2.m1.1.1" xref="S2.p1.2.m1.1.1.cmml">κ</mi><annotation-xml encoding="MathML-Content" id="S2.p1.2.m1.1b"><ci id="S2.p1.2.m1.1.1.cmml" xref="S2.p1.2.m1.1.1">𝜅</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.2.m1.1c">\kappa</annotation><annotation encoding="application/x-llamapun" id="S2.p1.2.m1.1d">italic_κ</annotation></semantics></math> is the gravitational constant, <math alttext="g" class="ltx_Math" display="inline" id="S2.p1.3.m2.1"><semantics id="S2.p1.3.m2.1a"><mi id="S2.p1.3.m2.1.1" xref="S2.p1.3.m2.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S2.p1.3.m2.1b"><ci id="S2.p1.3.m2.1.1.cmml" xref="S2.p1.3.m2.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.3.m2.1c">g</annotation><annotation encoding="application/x-llamapun" id="S2.p1.3.m2.1d">italic_g</annotation></semantics></math> is the determinat of the inverse metric <math alttext="g_{\mu\nu}" class="ltx_Math" display="inline" id="S2.p1.4.m3.1"><semantics id="S2.p1.4.m3.1a"><msub id="S2.p1.4.m3.1.1" xref="S2.p1.4.m3.1.1.cmml"><mi id="S2.p1.4.m3.1.1.2" xref="S2.p1.4.m3.1.1.2.cmml">g</mi><mrow id="S2.p1.4.m3.1.1.3" xref="S2.p1.4.m3.1.1.3.cmml"><mi id="S2.p1.4.m3.1.1.3.2" xref="S2.p1.4.m3.1.1.3.2.cmml">μ</mi><mo id="S2.p1.4.m3.1.1.3.1" xref="S2.p1.4.m3.1.1.3.1.cmml">⁢</mo><mi id="S2.p1.4.m3.1.1.3.3" xref="S2.p1.4.m3.1.1.3.3.cmml">ν</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.p1.4.m3.1b"><apply id="S2.p1.4.m3.1.1.cmml" xref="S2.p1.4.m3.1.1"><csymbol cd="ambiguous" id="S2.p1.4.m3.1.1.1.cmml" xref="S2.p1.4.m3.1.1">subscript</csymbol><ci id="S2.p1.4.m3.1.1.2.cmml" xref="S2.p1.4.m3.1.1.2">𝑔</ci><apply id="S2.p1.4.m3.1.1.3.cmml" xref="S2.p1.4.m3.1.1.3"><times id="S2.p1.4.m3.1.1.3.1.cmml" xref="S2.p1.4.m3.1.1.3.1"></times><ci id="S2.p1.4.m3.1.1.3.2.cmml" xref="S2.p1.4.m3.1.1.3.2">𝜇</ci><ci id="S2.p1.4.m3.1.1.3.3.cmml" xref="S2.p1.4.m3.1.1.3.3">𝜈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.4.m3.1c">g_{\mu\nu}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.4.m3.1d">italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="R=g^{\mu\nu}R_{\mu\nu}" class="ltx_Math" display="inline" id="S2.p1.5.m4.1"><semantics id="S2.p1.5.m4.1a"><mrow id="S2.p1.5.m4.1.1" xref="S2.p1.5.m4.1.1.cmml"><mi id="S2.p1.5.m4.1.1.2" xref="S2.p1.5.m4.1.1.2.cmml">R</mi><mo id="S2.p1.5.m4.1.1.1" xref="S2.p1.5.m4.1.1.1.cmml">=</mo><mrow id="S2.p1.5.m4.1.1.3" xref="S2.p1.5.m4.1.1.3.cmml"><msup id="S2.p1.5.m4.1.1.3.2" xref="S2.p1.5.m4.1.1.3.2.cmml"><mi id="S2.p1.5.m4.1.1.3.2.2" xref="S2.p1.5.m4.1.1.3.2.2.cmml">g</mi><mrow id="S2.p1.5.m4.1.1.3.2.3" xref="S2.p1.5.m4.1.1.3.2.3.cmml"><mi id="S2.p1.5.m4.1.1.3.2.3.2" xref="S2.p1.5.m4.1.1.3.2.3.2.cmml">μ</mi><mo id="S2.p1.5.m4.1.1.3.2.3.1" xref="S2.p1.5.m4.1.1.3.2.3.1.cmml">⁢</mo><mi id="S2.p1.5.m4.1.1.3.2.3.3" xref="S2.p1.5.m4.1.1.3.2.3.3.cmml">ν</mi></mrow></msup><mo id="S2.p1.5.m4.1.1.3.1" xref="S2.p1.5.m4.1.1.3.1.cmml">⁢</mo><msub id="S2.p1.5.m4.1.1.3.3" xref="S2.p1.5.m4.1.1.3.3.cmml"><mi id="S2.p1.5.m4.1.1.3.3.2" xref="S2.p1.5.m4.1.1.3.3.2.cmml">R</mi><mrow id="S2.p1.5.m4.1.1.3.3.3" xref="S2.p1.5.m4.1.1.3.3.3.cmml"><mi id="S2.p1.5.m4.1.1.3.3.3.2" xref="S2.p1.5.m4.1.1.3.3.3.2.cmml">μ</mi><mo id="S2.p1.5.m4.1.1.3.3.3.1" xref="S2.p1.5.m4.1.1.3.3.3.1.cmml">⁢</mo><mi id="S2.p1.5.m4.1.1.3.3.3.3" xref="S2.p1.5.m4.1.1.3.3.3.3.cmml">ν</mi></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.5.m4.1b"><apply id="S2.p1.5.m4.1.1.cmml" xref="S2.p1.5.m4.1.1"><eq id="S2.p1.5.m4.1.1.1.cmml" xref="S2.p1.5.m4.1.1.1"></eq><ci id="S2.p1.5.m4.1.1.2.cmml" xref="S2.p1.5.m4.1.1.2">𝑅</ci><apply id="S2.p1.5.m4.1.1.3.cmml" xref="S2.p1.5.m4.1.1.3"><times id="S2.p1.5.m4.1.1.3.1.cmml" xref="S2.p1.5.m4.1.1.3.1"></times><apply id="S2.p1.5.m4.1.1.3.2.cmml" xref="S2.p1.5.m4.1.1.3.2"><csymbol cd="ambiguous" id="S2.p1.5.m4.1.1.3.2.1.cmml" xref="S2.p1.5.m4.1.1.3.2">superscript</csymbol><ci id="S2.p1.5.m4.1.1.3.2.2.cmml" xref="S2.p1.5.m4.1.1.3.2.2">𝑔</ci><apply id="S2.p1.5.m4.1.1.3.2.3.cmml" xref="S2.p1.5.m4.1.1.3.2.3"><times id="S2.p1.5.m4.1.1.3.2.3.1.cmml" xref="S2.p1.5.m4.1.1.3.2.3.1"></times><ci id="S2.p1.5.m4.1.1.3.2.3.2.cmml" xref="S2.p1.5.m4.1.1.3.2.3.2">𝜇</ci><ci id="S2.p1.5.m4.1.1.3.2.3.3.cmml" xref="S2.p1.5.m4.1.1.3.2.3.3">𝜈</ci></apply></apply><apply id="S2.p1.5.m4.1.1.3.3.cmml" xref="S2.p1.5.m4.1.1.3.3"><csymbol cd="ambiguous" id="S2.p1.5.m4.1.1.3.3.1.cmml" xref="S2.p1.5.m4.1.1.3.3">subscript</csymbol><ci id="S2.p1.5.m4.1.1.3.3.2.cmml" xref="S2.p1.5.m4.1.1.3.3.2">𝑅</ci><apply id="S2.p1.5.m4.1.1.3.3.3.cmml" xref="S2.p1.5.m4.1.1.3.3.3"><times id="S2.p1.5.m4.1.1.3.3.3.1.cmml" xref="S2.p1.5.m4.1.1.3.3.3.1"></times><ci id="S2.p1.5.m4.1.1.3.3.3.2.cmml" xref="S2.p1.5.m4.1.1.3.3.3.2">𝜇</ci><ci id="S2.p1.5.m4.1.1.3.3.3.3.cmml" xref="S2.p1.5.m4.1.1.3.3.3.3">𝜈</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.5.m4.1c">R=g^{\mu\nu}R_{\mu\nu}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.5.m4.1d">italic_R = italic_g start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT italic_R start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT</annotation></semantics></math> is the scalar curvature. The Ricci tensor is expressed as <math alttext="R_{\mu\nu}=R_{\mu\rho\nu}^{\rho}" class="ltx_Math" display="inline" id="S2.p1.6.m5.1"><semantics id="S2.p1.6.m5.1a"><mrow id="S2.p1.6.m5.1.1" xref="S2.p1.6.m5.1.1.cmml"><msub id="S2.p1.6.m5.1.1.2" xref="S2.p1.6.m5.1.1.2.cmml"><mi id="S2.p1.6.m5.1.1.2.2" xref="S2.p1.6.m5.1.1.2.2.cmml">R</mi><mrow id="S2.p1.6.m5.1.1.2.3" xref="S2.p1.6.m5.1.1.2.3.cmml"><mi id="S2.p1.6.m5.1.1.2.3.2" xref="S2.p1.6.m5.1.1.2.3.2.cmml">μ</mi><mo id="S2.p1.6.m5.1.1.2.3.1" xref="S2.p1.6.m5.1.1.2.3.1.cmml">⁢</mo><mi id="S2.p1.6.m5.1.1.2.3.3" xref="S2.p1.6.m5.1.1.2.3.3.cmml">ν</mi></mrow></msub><mo id="S2.p1.6.m5.1.1.1" xref="S2.p1.6.m5.1.1.1.cmml">=</mo><msubsup id="S2.p1.6.m5.1.1.3" xref="S2.p1.6.m5.1.1.3.cmml"><mi id="S2.p1.6.m5.1.1.3.2.2" xref="S2.p1.6.m5.1.1.3.2.2.cmml">R</mi><mrow id="S2.p1.6.m5.1.1.3.2.3" xref="S2.p1.6.m5.1.1.3.2.3.cmml"><mi id="S2.p1.6.m5.1.1.3.2.3.2" xref="S2.p1.6.m5.1.1.3.2.3.2.cmml">μ</mi><mo id="S2.p1.6.m5.1.1.3.2.3.1" xref="S2.p1.6.m5.1.1.3.2.3.1.cmml">⁢</mo><mi id="S2.p1.6.m5.1.1.3.2.3.3" xref="S2.p1.6.m5.1.1.3.2.3.3.cmml">ρ</mi><mo id="S2.p1.6.m5.1.1.3.2.3.1a" xref="S2.p1.6.m5.1.1.3.2.3.1.cmml">⁢</mo><mi id="S2.p1.6.m5.1.1.3.2.3.4" xref="S2.p1.6.m5.1.1.3.2.3.4.cmml">ν</mi></mrow><mi id="S2.p1.6.m5.1.1.3.3" xref="S2.p1.6.m5.1.1.3.3.cmml">ρ</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.p1.6.m5.1b"><apply id="S2.p1.6.m5.1.1.cmml" xref="S2.p1.6.m5.1.1"><eq id="S2.p1.6.m5.1.1.1.cmml" xref="S2.p1.6.m5.1.1.1"></eq><apply id="S2.p1.6.m5.1.1.2.cmml" xref="S2.p1.6.m5.1.1.2"><csymbol cd="ambiguous" id="S2.p1.6.m5.1.1.2.1.cmml" xref="S2.p1.6.m5.1.1.2">subscript</csymbol><ci id="S2.p1.6.m5.1.1.2.2.cmml" xref="S2.p1.6.m5.1.1.2.2">𝑅</ci><apply id="S2.p1.6.m5.1.1.2.3.cmml" xref="S2.p1.6.m5.1.1.2.3"><times id="S2.p1.6.m5.1.1.2.3.1.cmml" xref="S2.p1.6.m5.1.1.2.3.1"></times><ci id="S2.p1.6.m5.1.1.2.3.2.cmml" xref="S2.p1.6.m5.1.1.2.3.2">𝜇</ci><ci id="S2.p1.6.m5.1.1.2.3.3.cmml" xref="S2.p1.6.m5.1.1.2.3.3">𝜈</ci></apply></apply><apply id="S2.p1.6.m5.1.1.3.cmml" xref="S2.p1.6.m5.1.1.3"><csymbol cd="ambiguous" id="S2.p1.6.m5.1.1.3.1.cmml" xref="S2.p1.6.m5.1.1.3">superscript</csymbol><apply id="S2.p1.6.m5.1.1.3.2.cmml" xref="S2.p1.6.m5.1.1.3"><csymbol cd="ambiguous" id="S2.p1.6.m5.1.1.3.2.1.cmml" xref="S2.p1.6.m5.1.1.3">subscript</csymbol><ci id="S2.p1.6.m5.1.1.3.2.2.cmml" xref="S2.p1.6.m5.1.1.3.2.2">𝑅</ci><apply id="S2.p1.6.m5.1.1.3.2.3.cmml" xref="S2.p1.6.m5.1.1.3.2.3"><times id="S2.p1.6.m5.1.1.3.2.3.1.cmml" xref="S2.p1.6.m5.1.1.3.2.3.1"></times><ci id="S2.p1.6.m5.1.1.3.2.3.2.cmml" xref="S2.p1.6.m5.1.1.3.2.3.2">𝜇</ci><ci id="S2.p1.6.m5.1.1.3.2.3.3.cmml" xref="S2.p1.6.m5.1.1.3.2.3.3">𝜌</ci><ci id="S2.p1.6.m5.1.1.3.2.3.4.cmml" xref="S2.p1.6.m5.1.1.3.2.3.4">𝜈</ci></apply></apply><ci id="S2.p1.6.m5.1.1.3.3.cmml" xref="S2.p1.6.m5.1.1.3.3">𝜌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p1.6.m5.1c">R_{\mu\nu}=R_{\mu\rho\nu}^{\rho}</annotation><annotation encoding="application/x-llamapun" id="S2.p1.6.m5.1d">italic_R start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT = italic_R start_POSTSUBSCRIPT italic_μ italic_ρ italic_ν end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ρ end_POSTSUPERSCRIPT</annotation></semantics></math>, with Riemann tensor given by</p> <table class="ltx_equation ltx_eqn_table" id="S2.E2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="R_{\mu\rho\nu}^{\sigma}=\partial_{\rho}\Gamma_{\mu\nu}^{\sigma}-\partial_{\nu}% \Gamma_{\rho\mu}^{\sigma}+\Gamma_{\rho\gamma}^{\sigma}\Gamma_{\mu\nu}^{\gamma}% -\Gamma_{\nu\gamma}^{\sigma}\Gamma_{\mu\rho}^{\gamma}," class="ltx_Math" display="block" id="S2.E2.m1.1"><semantics id="S2.E2.m1.1a"><mrow id="S2.E2.m1.1.1.1" xref="S2.E2.m1.1.1.1.1.cmml"><mrow id="S2.E2.m1.1.1.1.1" xref="S2.E2.m1.1.1.1.1.cmml"><msubsup id="S2.E2.m1.1.1.1.1.2" xref="S2.E2.m1.1.1.1.1.2.cmml"><mi id="S2.E2.m1.1.1.1.1.2.2.2" xref="S2.E2.m1.1.1.1.1.2.2.2.cmml">R</mi><mrow id="S2.E2.m1.1.1.1.1.2.2.3" xref="S2.E2.m1.1.1.1.1.2.2.3.cmml"><mi id="S2.E2.m1.1.1.1.1.2.2.3.2" xref="S2.E2.m1.1.1.1.1.2.2.3.2.cmml">μ</mi><mo id="S2.E2.m1.1.1.1.1.2.2.3.1" xref="S2.E2.m1.1.1.1.1.2.2.3.1.cmml">⁢</mo><mi id="S2.E2.m1.1.1.1.1.2.2.3.3" xref="S2.E2.m1.1.1.1.1.2.2.3.3.cmml">ρ</mi><mo id="S2.E2.m1.1.1.1.1.2.2.3.1a" xref="S2.E2.m1.1.1.1.1.2.2.3.1.cmml">⁢</mo><mi id="S2.E2.m1.1.1.1.1.2.2.3.4" xref="S2.E2.m1.1.1.1.1.2.2.3.4.cmml">ν</mi></mrow><mi id="S2.E2.m1.1.1.1.1.2.3" xref="S2.E2.m1.1.1.1.1.2.3.cmml">σ</mi></msubsup><mo id="S2.E2.m1.1.1.1.1.1" rspace="0.1389em" xref="S2.E2.m1.1.1.1.1.1.cmml">=</mo><mrow id="S2.E2.m1.1.1.1.1.3" xref="S2.E2.m1.1.1.1.1.3.cmml"><mrow id="S2.E2.m1.1.1.1.1.3.2" xref="S2.E2.m1.1.1.1.1.3.2.cmml"><mrow id="S2.E2.m1.1.1.1.1.3.2.2" xref="S2.E2.m1.1.1.1.1.3.2.2.cmml"><mrow id="S2.E2.m1.1.1.1.1.3.2.2.2" xref="S2.E2.m1.1.1.1.1.3.2.2.2.cmml"><msub id="S2.E2.m1.1.1.1.1.3.2.2.2.1" xref="S2.E2.m1.1.1.1.1.3.2.2.2.1.cmml"><mo id="S2.E2.m1.1.1.1.1.3.2.2.2.1.2" lspace="0.1389em" rspace="0em" xref="S2.E2.m1.1.1.1.1.3.2.2.2.1.2.cmml">∂</mo><mi id="S2.E2.m1.1.1.1.1.3.2.2.2.1.3" xref="S2.E2.m1.1.1.1.1.3.2.2.2.1.3.cmml">ρ</mi></msub><msubsup id="S2.E2.m1.1.1.1.1.3.2.2.2.2" xref="S2.E2.m1.1.1.1.1.3.2.2.2.2.cmml"><mi id="S2.E2.m1.1.1.1.1.3.2.2.2.2.2.2" mathvariant="normal" xref="S2.E2.m1.1.1.1.1.3.2.2.2.2.2.2.cmml">Γ</mi><mrow id="S2.E2.m1.1.1.1.1.3.2.2.2.2.2.3" xref="S2.E2.m1.1.1.1.1.3.2.2.2.2.2.3.cmml"><mi id="S2.E2.m1.1.1.1.1.3.2.2.2.2.2.3.2" xref="S2.E2.m1.1.1.1.1.3.2.2.2.2.2.3.2.cmml">μ</mi><mo id="S2.E2.m1.1.1.1.1.3.2.2.2.2.2.3.1" xref="S2.E2.m1.1.1.1.1.3.2.2.2.2.2.3.1.cmml">⁢</mo><mi id="S2.E2.m1.1.1.1.1.3.2.2.2.2.2.3.3" xref="S2.E2.m1.1.1.1.1.3.2.2.2.2.2.3.3.cmml">ν</mi></mrow><mi id="S2.E2.m1.1.1.1.1.3.2.2.2.2.3" xref="S2.E2.m1.1.1.1.1.3.2.2.2.2.3.cmml">σ</mi></msubsup></mrow><mo id="S2.E2.m1.1.1.1.1.3.2.2.1" xref="S2.E2.m1.1.1.1.1.3.2.2.1.cmml">−</mo><mrow id="S2.E2.m1.1.1.1.1.3.2.2.3" xref="S2.E2.m1.1.1.1.1.3.2.2.3.cmml"><msub id="S2.E2.m1.1.1.1.1.3.2.2.3.1" xref="S2.E2.m1.1.1.1.1.3.2.2.3.1.cmml"><mo id="S2.E2.m1.1.1.1.1.3.2.2.3.1.2" lspace="0em" rspace="0em" xref="S2.E2.m1.1.1.1.1.3.2.2.3.1.2.cmml">∂</mo><mi id="S2.E2.m1.1.1.1.1.3.2.2.3.1.3" xref="S2.E2.m1.1.1.1.1.3.2.2.3.1.3.cmml">ν</mi></msub><msubsup id="S2.E2.m1.1.1.1.1.3.2.2.3.2" xref="S2.E2.m1.1.1.1.1.3.2.2.3.2.cmml"><mi id="S2.E2.m1.1.1.1.1.3.2.2.3.2.2.2" mathvariant="normal" xref="S2.E2.m1.1.1.1.1.3.2.2.3.2.2.2.cmml">Γ</mi><mrow id="S2.E2.m1.1.1.1.1.3.2.2.3.2.2.3" xref="S2.E2.m1.1.1.1.1.3.2.2.3.2.2.3.cmml"><mi id="S2.E2.m1.1.1.1.1.3.2.2.3.2.2.3.2" xref="S2.E2.m1.1.1.1.1.3.2.2.3.2.2.3.2.cmml">ρ</mi><mo id="S2.E2.m1.1.1.1.1.3.2.2.3.2.2.3.1" xref="S2.E2.m1.1.1.1.1.3.2.2.3.2.2.3.1.cmml">⁢</mo><mi id="S2.E2.m1.1.1.1.1.3.2.2.3.2.2.3.3" xref="S2.E2.m1.1.1.1.1.3.2.2.3.2.2.3.3.cmml">μ</mi></mrow><mi id="S2.E2.m1.1.1.1.1.3.2.2.3.2.3" xref="S2.E2.m1.1.1.1.1.3.2.2.3.2.3.cmml">σ</mi></msubsup></mrow></mrow><mo id="S2.E2.m1.1.1.1.1.3.2.1" xref="S2.E2.m1.1.1.1.1.3.2.1.cmml">+</mo><mrow id="S2.E2.m1.1.1.1.1.3.2.3" xref="S2.E2.m1.1.1.1.1.3.2.3.cmml"><msubsup id="S2.E2.m1.1.1.1.1.3.2.3.2" xref="S2.E2.m1.1.1.1.1.3.2.3.2.cmml"><mi id="S2.E2.m1.1.1.1.1.3.2.3.2.2.2" mathvariant="normal" xref="S2.E2.m1.1.1.1.1.3.2.3.2.2.2.cmml">Γ</mi><mrow id="S2.E2.m1.1.1.1.1.3.2.3.2.2.3" xref="S2.E2.m1.1.1.1.1.3.2.3.2.2.3.cmml"><mi id="S2.E2.m1.1.1.1.1.3.2.3.2.2.3.2" xref="S2.E2.m1.1.1.1.1.3.2.3.2.2.3.2.cmml">ρ</mi><mo id="S2.E2.m1.1.1.1.1.3.2.3.2.2.3.1" xref="S2.E2.m1.1.1.1.1.3.2.3.2.2.3.1.cmml">⁢</mo><mi id="S2.E2.m1.1.1.1.1.3.2.3.2.2.3.3" xref="S2.E2.m1.1.1.1.1.3.2.3.2.2.3.3.cmml">γ</mi></mrow><mi id="S2.E2.m1.1.1.1.1.3.2.3.2.3" xref="S2.E2.m1.1.1.1.1.3.2.3.2.3.cmml">σ</mi></msubsup><mo id="S2.E2.m1.1.1.1.1.3.2.3.1" 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\Gamma_{\rho\mu}^{\sigma}+\Gamma_{\rho\gamma}^{\sigma}\Gamma_{\mu\nu}^{\gamma}% -\Gamma_{\nu\gamma}^{\sigma}\Gamma_{\mu\rho}^{\gamma},</annotation><annotation encoding="application/x-llamapun" id="S2.E2.m1.1d">italic_R start_POSTSUBSCRIPT italic_μ italic_ρ italic_ν end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT = ∂ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT roman_Γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT - ∂ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT roman_Γ start_POSTSUBSCRIPT italic_ρ italic_μ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT + roman_Γ start_POSTSUBSCRIPT italic_ρ italic_γ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT roman_Γ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_γ end_POSTSUPERSCRIPT - roman_Γ start_POSTSUBSCRIPT italic_ν italic_γ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT roman_Γ start_POSTSUBSCRIPT italic_μ italic_ρ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_γ end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p1.7">and <math alttext="\Gamma^{\rho}_{\mu\nu}=g^{\rho\sigma}(\partial_{\mu}g_{\nu\sigma}+\partial_{% \nu}g_{\mu\sigma}-\partial_{\sigma}g_{\mu\nu})/2" class="ltx_Math" display="inline" id="S2.p1.7.m1.1"><semantics id="S2.p1.7.m1.1a"><mrow id="S2.p1.7.m1.1.1" xref="S2.p1.7.m1.1.1.cmml"><msubsup id="S2.p1.7.m1.1.1.3" xref="S2.p1.7.m1.1.1.3.cmml"><mi id="S2.p1.7.m1.1.1.3.2.2" mathvariant="normal" xref="S2.p1.7.m1.1.1.3.2.2.cmml">Γ</mi><mrow id="S2.p1.7.m1.1.1.3.3" xref="S2.p1.7.m1.1.1.3.3.cmml"><mi id="S2.p1.7.m1.1.1.3.3.2" 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start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT = italic_g start_POSTSUPERSCRIPT italic_ρ italic_σ end_POSTSUPERSCRIPT ( ∂ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_ν italic_σ end_POSTSUBSCRIPT + ∂ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_μ italic_σ end_POSTSUBSCRIPT - ∂ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ) / 2</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.p2"> <p class="ltx_p" id="S2.p2.1">Regardless of the natural geometrical interpretation, the action (<a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S2.E1" title="In II Equations of motion ‣ One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_tag">1</span></a>) and the metric <math alttext="g_{\mu\nu}" class="ltx_Math" display="inline" id="S2.p2.1.m1.1"><semantics id="S2.p2.1.m1.1a"><msub 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id="S2.p2.1.m1.1c">g_{\mu\nu}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.1.m1.1d">italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT</annotation></semantics></math> often make the analysis of graviton scattering overly involved. We will work instead with the metric density <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib57" title="">Landau:1975pou </a></cite></p> <table class="ltx_equation ltx_eqn_table" id="S2.E3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathfrak{g}^{\mu\nu}=\sqrt{-g}g^{\mu\nu}," class="ltx_Math" display="block" id="S2.E3.m1.1"><semantics id="S2.E3.m1.1a"><mrow id="S2.E3.m1.1.1.1" xref="S2.E3.m1.1.1.1.1.cmml"><mrow id="S2.E3.m1.1.1.1.1" xref="S2.E3.m1.1.1.1.1.cmml"><msup id="S2.E3.m1.1.1.1.1.2" xref="S2.E3.m1.1.1.1.1.2.cmml"><mi id="S2.E3.m1.1.1.1.1.2.2" xref="S2.E3.m1.1.1.1.1.2.2.cmml">𝔤</mi><mrow id="S2.E3.m1.1.1.1.1.2.3" xref="S2.E3.m1.1.1.1.1.2.3.cmml"><mi id="S2.E3.m1.1.1.1.1.2.3.2" xref="S2.E3.m1.1.1.1.1.2.3.2.cmml">μ</mi><mo id="S2.E3.m1.1.1.1.1.2.3.1" xref="S2.E3.m1.1.1.1.1.2.3.1.cmml">⁢</mo><mi id="S2.E3.m1.1.1.1.1.2.3.3" xref="S2.E3.m1.1.1.1.1.2.3.3.cmml">ν</mi></mrow></msup><mo id="S2.E3.m1.1.1.1.1.1" xref="S2.E3.m1.1.1.1.1.1.cmml">=</mo><mrow id="S2.E3.m1.1.1.1.1.3" xref="S2.E3.m1.1.1.1.1.3.cmml"><msqrt id="S2.E3.m1.1.1.1.1.3.2" xref="S2.E3.m1.1.1.1.1.3.2.cmml"><mrow id="S2.E3.m1.1.1.1.1.3.2.2" xref="S2.E3.m1.1.1.1.1.3.2.2.cmml"><mo id="S2.E3.m1.1.1.1.1.3.2.2a" xref="S2.E3.m1.1.1.1.1.3.2.2.cmml">−</mo><mi id="S2.E3.m1.1.1.1.1.3.2.2.2" xref="S2.E3.m1.1.1.1.1.3.2.2.2.cmml">g</mi></mrow></msqrt><mo id="S2.E3.m1.1.1.1.1.3.1" xref="S2.E3.m1.1.1.1.1.3.1.cmml">⁢</mo><msup id="S2.E3.m1.1.1.1.1.3.3" xref="S2.E3.m1.1.1.1.1.3.3.cmml"><mi id="S2.E3.m1.1.1.1.1.3.3.2" xref="S2.E3.m1.1.1.1.1.3.3.2.cmml">g</mi><mrow id="S2.E3.m1.1.1.1.1.3.3.3" xref="S2.E3.m1.1.1.1.1.3.3.3.cmml"><mi id="S2.E3.m1.1.1.1.1.3.3.3.2" xref="S2.E3.m1.1.1.1.1.3.3.3.2.cmml">μ</mi><mo id="S2.E3.m1.1.1.1.1.3.3.3.1" xref="S2.E3.m1.1.1.1.1.3.3.3.1.cmml">⁢</mo><mi id="S2.E3.m1.1.1.1.1.3.3.3.3" xref="S2.E3.m1.1.1.1.1.3.3.3.3.cmml">ν</mi></mrow></msup></mrow></mrow><mo id="S2.E3.m1.1.1.1.2" xref="S2.E3.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E3.m1.1b"><apply id="S2.E3.m1.1.1.1.1.cmml" xref="S2.E3.m1.1.1.1"><eq id="S2.E3.m1.1.1.1.1.1.cmml" xref="S2.E3.m1.1.1.1.1.1"></eq><apply id="S2.E3.m1.1.1.1.1.2.cmml" xref="S2.E3.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.E3.m1.1.1.1.1.2.1.cmml" xref="S2.E3.m1.1.1.1.1.2">superscript</csymbol><ci id="S2.E3.m1.1.1.1.1.2.2.cmml" xref="S2.E3.m1.1.1.1.1.2.2">𝔤</ci><apply id="S2.E3.m1.1.1.1.1.2.3.cmml" xref="S2.E3.m1.1.1.1.1.2.3"><times id="S2.E3.m1.1.1.1.1.2.3.1.cmml" xref="S2.E3.m1.1.1.1.1.2.3.1"></times><ci id="S2.E3.m1.1.1.1.1.2.3.2.cmml" xref="S2.E3.m1.1.1.1.1.2.3.2">𝜇</ci><ci id="S2.E3.m1.1.1.1.1.2.3.3.cmml" xref="S2.E3.m1.1.1.1.1.2.3.3">𝜈</ci></apply></apply><apply id="S2.E3.m1.1.1.1.1.3.cmml" xref="S2.E3.m1.1.1.1.1.3"><times id="S2.E3.m1.1.1.1.1.3.1.cmml" xref="S2.E3.m1.1.1.1.1.3.1"></times><apply id="S2.E3.m1.1.1.1.1.3.2.cmml" xref="S2.E3.m1.1.1.1.1.3.2"><root id="S2.E3.m1.1.1.1.1.3.2a.cmml" xref="S2.E3.m1.1.1.1.1.3.2"></root><apply id="S2.E3.m1.1.1.1.1.3.2.2.cmml" xref="S2.E3.m1.1.1.1.1.3.2.2"><minus id="S2.E3.m1.1.1.1.1.3.2.2.1.cmml" xref="S2.E3.m1.1.1.1.1.3.2.2"></minus><ci id="S2.E3.m1.1.1.1.1.3.2.2.2.cmml" xref="S2.E3.m1.1.1.1.1.3.2.2.2">𝑔</ci></apply></apply><apply id="S2.E3.m1.1.1.1.1.3.3.cmml" xref="S2.E3.m1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S2.E3.m1.1.1.1.1.3.3.1.cmml" xref="S2.E3.m1.1.1.1.1.3.3">superscript</csymbol><ci id="S2.E3.m1.1.1.1.1.3.3.2.cmml" xref="S2.E3.m1.1.1.1.1.3.3.2">𝑔</ci><apply id="S2.E3.m1.1.1.1.1.3.3.3.cmml" xref="S2.E3.m1.1.1.1.1.3.3.3"><times id="S2.E3.m1.1.1.1.1.3.3.3.1.cmml" xref="S2.E3.m1.1.1.1.1.3.3.3.1"></times><ci id="S2.E3.m1.1.1.1.1.3.3.3.2.cmml" xref="S2.E3.m1.1.1.1.1.3.3.3.2">𝜇</ci><ci id="S2.E3.m1.1.1.1.1.3.3.3.3.cmml" xref="S2.E3.m1.1.1.1.1.3.3.3.3">𝜈</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E3.m1.1c">\mathfrak{g}^{\mu\nu}=\sqrt{-g}g^{\mu\nu},</annotation><annotation encoding="application/x-llamapun" id="S2.E3.m1.1d">fraktur_g start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT = square-root start_ARG - italic_g end_ARG italic_g start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p2.4">with inverse <math alttext="\mathfrak{g}_{\mu\nu}" class="ltx_Math" display="inline" id="S2.p2.2.m1.1"><semantics id="S2.p2.2.m1.1a"><msub id="S2.p2.2.m1.1.1" xref="S2.p2.2.m1.1.1.cmml"><mi id="S2.p2.2.m1.1.1.2" xref="S2.p2.2.m1.1.1.2.cmml">𝔤</mi><mrow id="S2.p2.2.m1.1.1.3" xref="S2.p2.2.m1.1.1.3.cmml"><mi id="S2.p2.2.m1.1.1.3.2" xref="S2.p2.2.m1.1.1.3.2.cmml">μ</mi><mo id="S2.p2.2.m1.1.1.3.1" xref="S2.p2.2.m1.1.1.3.1.cmml">⁢</mo><mi id="S2.p2.2.m1.1.1.3.3" xref="S2.p2.2.m1.1.1.3.3.cmml">ν</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.p2.2.m1.1b"><apply id="S2.p2.2.m1.1.1.cmml" xref="S2.p2.2.m1.1.1"><csymbol cd="ambiguous" id="S2.p2.2.m1.1.1.1.cmml" xref="S2.p2.2.m1.1.1">subscript</csymbol><ci id="S2.p2.2.m1.1.1.2.cmml" xref="S2.p2.2.m1.1.1.2">𝔤</ci><apply id="S2.p2.2.m1.1.1.3.cmml" xref="S2.p2.2.m1.1.1.3"><times id="S2.p2.2.m1.1.1.3.1.cmml" xref="S2.p2.2.m1.1.1.3.1"></times><ci id="S2.p2.2.m1.1.1.3.2.cmml" xref="S2.p2.2.m1.1.1.3.2">𝜇</ci><ci id="S2.p2.2.m1.1.1.3.3.cmml" xref="S2.p2.2.m1.1.1.3.3">𝜈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.2.m1.1c">\mathfrak{g}_{\mu\nu}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.2.m1.1d">fraktur_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT</annotation></semantics></math>. Under diffeomorphisms (<math alttext="\delta x^{\mu}=\tilde{c}^{\mu}" class="ltx_Math" display="inline" id="S2.p2.3.m2.1"><semantics id="S2.p2.3.m2.1a"><mrow id="S2.p2.3.m2.1.1" xref="S2.p2.3.m2.1.1.cmml"><mrow id="S2.p2.3.m2.1.1.2" xref="S2.p2.3.m2.1.1.2.cmml"><mi id="S2.p2.3.m2.1.1.2.2" xref="S2.p2.3.m2.1.1.2.2.cmml">δ</mi><mo id="S2.p2.3.m2.1.1.2.1" xref="S2.p2.3.m2.1.1.2.1.cmml">⁢</mo><msup id="S2.p2.3.m2.1.1.2.3" xref="S2.p2.3.m2.1.1.2.3.cmml"><mi id="S2.p2.3.m2.1.1.2.3.2" xref="S2.p2.3.m2.1.1.2.3.2.cmml">x</mi><mi id="S2.p2.3.m2.1.1.2.3.3" xref="S2.p2.3.m2.1.1.2.3.3.cmml">μ</mi></msup></mrow><mo id="S2.p2.3.m2.1.1.1" xref="S2.p2.3.m2.1.1.1.cmml">=</mo><msup id="S2.p2.3.m2.1.1.3" xref="S2.p2.3.m2.1.1.3.cmml"><mover accent="true" id="S2.p2.3.m2.1.1.3.2" xref="S2.p2.3.m2.1.1.3.2.cmml"><mi id="S2.p2.3.m2.1.1.3.2.2" xref="S2.p2.3.m2.1.1.3.2.2.cmml">c</mi><mo id="S2.p2.3.m2.1.1.3.2.1" xref="S2.p2.3.m2.1.1.3.2.1.cmml">~</mo></mover><mi id="S2.p2.3.m2.1.1.3.3" xref="S2.p2.3.m2.1.1.3.3.cmml">μ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.p2.3.m2.1b"><apply id="S2.p2.3.m2.1.1.cmml" xref="S2.p2.3.m2.1.1"><eq id="S2.p2.3.m2.1.1.1.cmml" xref="S2.p2.3.m2.1.1.1"></eq><apply id="S2.p2.3.m2.1.1.2.cmml" xref="S2.p2.3.m2.1.1.2"><times id="S2.p2.3.m2.1.1.2.1.cmml" xref="S2.p2.3.m2.1.1.2.1"></times><ci id="S2.p2.3.m2.1.1.2.2.cmml" xref="S2.p2.3.m2.1.1.2.2">𝛿</ci><apply id="S2.p2.3.m2.1.1.2.3.cmml" xref="S2.p2.3.m2.1.1.2.3"><csymbol cd="ambiguous" id="S2.p2.3.m2.1.1.2.3.1.cmml" xref="S2.p2.3.m2.1.1.2.3">superscript</csymbol><ci id="S2.p2.3.m2.1.1.2.3.2.cmml" xref="S2.p2.3.m2.1.1.2.3.2">𝑥</ci><ci id="S2.p2.3.m2.1.1.2.3.3.cmml" xref="S2.p2.3.m2.1.1.2.3.3">𝜇</ci></apply></apply><apply id="S2.p2.3.m2.1.1.3.cmml" xref="S2.p2.3.m2.1.1.3"><csymbol cd="ambiguous" id="S2.p2.3.m2.1.1.3.1.cmml" xref="S2.p2.3.m2.1.1.3">superscript</csymbol><apply id="S2.p2.3.m2.1.1.3.2.cmml" xref="S2.p2.3.m2.1.1.3.2"><ci id="S2.p2.3.m2.1.1.3.2.1.cmml" xref="S2.p2.3.m2.1.1.3.2.1">~</ci><ci id="S2.p2.3.m2.1.1.3.2.2.cmml" xref="S2.p2.3.m2.1.1.3.2.2">𝑐</ci></apply><ci id="S2.p2.3.m2.1.1.3.3.cmml" xref="S2.p2.3.m2.1.1.3.3">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.3.m2.1c">\delta x^{\mu}=\tilde{c}^{\mu}</annotation><annotation encoding="application/x-llamapun" id="S2.p2.3.m2.1d">italic_δ italic_x start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT = over~ start_ARG italic_c end_ARG start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT</annotation></semantics></math>), it transforms as <math alttext="\delta\mathfrak{g}^{\mu\nu}=\mathfrak{g}^{\mu\rho}\partial_{\rho}\tilde{c}^{% \nu}+\mathfrak{g}^{\nu\rho}\partial_{\rho}\tilde{c}^{\mu}-\partial_{\rho}(% \mathfrak{g}^{\mu\nu}\tilde{c}^{\rho})" class="ltx_Math" display="inline" id="S2.p2.4.m3.1"><semantics id="S2.p2.4.m3.1a"><mrow id="S2.p2.4.m3.1.1" xref="S2.p2.4.m3.1.1.cmml"><mrow id="S2.p2.4.m3.1.1.3" xref="S2.p2.4.m3.1.1.3.cmml"><mi 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id="S2.p2.4.m3.1.1.1.1.1.1.1.3.3.cmml" xref="S2.p2.4.m3.1.1.1.1.1.1.1.3.3">𝜌</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p2.4.m3.1c">\delta\mathfrak{g}^{\mu\nu}=\mathfrak{g}^{\mu\rho}\partial_{\rho}\tilde{c}^{% \nu}+\mathfrak{g}^{\nu\rho}\partial_{\rho}\tilde{c}^{\mu}-\partial_{\rho}(% \mathfrak{g}^{\mu\nu}\tilde{c}^{\rho})</annotation><annotation encoding="application/x-llamapun" id="S2.p2.4.m3.1d">italic_δ fraktur_g start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT = fraktur_g start_POSTSUPERSCRIPT italic_μ italic_ρ end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT over~ start_ARG italic_c end_ARG start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT + fraktur_g start_POSTSUPERSCRIPT italic_ν italic_ρ end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT over~ start_ARG italic_c end_ARG start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT - ∂ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT ( fraktur_g start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT over~ start_ARG italic_c end_ARG start_POSTSUPERSCRIPT italic_ρ end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.p3"> <p class="ltx_p" id="S2.p3.3">Towards gauge-fixing, we introduce the Faddeev–Popov ghost <math alttext="c^{\mu}" class="ltx_Math" display="inline" id="S2.p3.1.m1.1"><semantics id="S2.p3.1.m1.1a"><msup id="S2.p3.1.m1.1.1" xref="S2.p3.1.m1.1.1.cmml"><mi id="S2.p3.1.m1.1.1.2" xref="S2.p3.1.m1.1.1.2.cmml">c</mi><mi id="S2.p3.1.m1.1.1.3" xref="S2.p3.1.m1.1.1.3.cmml">μ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.p3.1.m1.1b"><apply id="S2.p3.1.m1.1.1.cmml" xref="S2.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S2.p3.1.m1.1.1.1.cmml" xref="S2.p3.1.m1.1.1">superscript</csymbol><ci id="S2.p3.1.m1.1.1.2.cmml" xref="S2.p3.1.m1.1.1.2">𝑐</ci><ci id="S2.p3.1.m1.1.1.3.cmml" xref="S2.p3.1.m1.1.1.3">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.1.m1.1c">c^{\mu}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.1.m1.1d">italic_c start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT</annotation></semantics></math>, with antighost <math alttext="b_{\mu}" class="ltx_Math" display="inline" id="S2.p3.2.m2.1"><semantics id="S2.p3.2.m2.1a"><msub id="S2.p3.2.m2.1.1" xref="S2.p3.2.m2.1.1.cmml"><mi id="S2.p3.2.m2.1.1.2" xref="S2.p3.2.m2.1.1.2.cmml">b</mi><mi id="S2.p3.2.m2.1.1.3" xref="S2.p3.2.m2.1.1.3.cmml">μ</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p3.2.m2.1b"><apply id="S2.p3.2.m2.1.1.cmml" xref="S2.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S2.p3.2.m2.1.1.1.cmml" xref="S2.p3.2.m2.1.1">subscript</csymbol><ci id="S2.p3.2.m2.1.1.2.cmml" xref="S2.p3.2.m2.1.1.2">𝑏</ci><ci id="S2.p3.2.m2.1.1.3.cmml" xref="S2.p3.2.m2.1.1.3">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.2.m2.1c">b_{\mu}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.2.m2.1d">italic_b start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT</annotation></semantics></math>. The usual harmonic gauge reads <math alttext="g^{\nu\rho}\Gamma_{\nu\rho}^{\mu}=\partial_{\nu}\mathfrak{g}^{\mu\nu}=0" class="ltx_Math" display="inline" id="S2.p3.3.m3.1"><semantics id="S2.p3.3.m3.1a"><mrow id="S2.p3.3.m3.1.1" xref="S2.p3.3.m3.1.1.cmml"><mrow id="S2.p3.3.m3.1.1.2" xref="S2.p3.3.m3.1.1.2.cmml"><msup id="S2.p3.3.m3.1.1.2.2" xref="S2.p3.3.m3.1.1.2.2.cmml"><mi id="S2.p3.3.m3.1.1.2.2.2" xref="S2.p3.3.m3.1.1.2.2.2.cmml">g</mi><mrow id="S2.p3.3.m3.1.1.2.2.3" xref="S2.p3.3.m3.1.1.2.2.3.cmml"><mi id="S2.p3.3.m3.1.1.2.2.3.2" xref="S2.p3.3.m3.1.1.2.2.3.2.cmml">ν</mi><mo id="S2.p3.3.m3.1.1.2.2.3.1" xref="S2.p3.3.m3.1.1.2.2.3.1.cmml">⁢</mo><mi id="S2.p3.3.m3.1.1.2.2.3.3" xref="S2.p3.3.m3.1.1.2.2.3.3.cmml">ρ</mi></mrow></msup><mo id="S2.p3.3.m3.1.1.2.1" xref="S2.p3.3.m3.1.1.2.1.cmml">⁢</mo><msubsup id="S2.p3.3.m3.1.1.2.3" xref="S2.p3.3.m3.1.1.2.3.cmml"><mi id="S2.p3.3.m3.1.1.2.3.2.2" mathvariant="normal" xref="S2.p3.3.m3.1.1.2.3.2.2.cmml">Γ</mi><mrow id="S2.p3.3.m3.1.1.2.3.2.3" 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id="S2.p3.3.m3.1.1.4.2.3.3.cmml" xref="S2.p3.3.m3.1.1.4.2.3.3">𝜈</ci></apply></apply></apply></apply><apply id="S2.p3.3.m3.1.1c.cmml" xref="S2.p3.3.m3.1.1"><eq id="S2.p3.3.m3.1.1.5.cmml" xref="S2.p3.3.m3.1.1.5"></eq><share href="https://arxiv.org/html/2411.07939v2#S2.p3.3.m3.1.1.4.cmml" id="S2.p3.3.m3.1.1d.cmml" xref="S2.p3.3.m3.1.1"></share><cn id="S2.p3.3.m3.1.1.6.cmml" type="integer" xref="S2.p3.3.m3.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.3.m3.1c">g^{\nu\rho}\Gamma_{\nu\rho}^{\mu}=\partial_{\nu}\mathfrak{g}^{\mu\nu}=0</annotation><annotation encoding="application/x-llamapun" id="S2.p3.3.m3.1d">italic_g start_POSTSUPERSCRIPT italic_ν italic_ρ end_POSTSUPERSCRIPT roman_Γ start_POSTSUBSCRIPT italic_ν italic_ρ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT = ∂ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT fraktur_g start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT = 0</annotation></semantics></math>, and the gauge-fixed action is given by</p> <table class="ltx_equation ltx_eqn_table" id="S2.E4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="S=\frac{1}{8\kappa}\int d^{d}x\,\big{\{}\mathfrak{g}^{\mu\nu}\partial_{\mu}% \mathfrak{g}^{\rho\sigma}\partial_{\nu}\mathfrak{g}_{\rho\sigma}-2\mathfrak{g}% ^{\mu\nu}\partial_{\mu}\mathfrak{g}^{\rho\sigma}\partial_{\rho}\mathfrak{g}_{% \nu\sigma}\\ -\frac{1}{(2-d)}\mathfrak{g}^{\mu\nu}\mathfrak{g}^{\rho\sigma}\partial_{\mu}% \mathfrak{g}_{\rho\sigma}\mathfrak{g}^{\gamma\lambda}\partial_{\nu}\mathfrak{g% }_{\gamma\lambda}-2\eta_{\mu\nu}\partial_{\rho}\mathfrak{g}^{\mu\rho}\partial_% {\sigma}\mathfrak{g}^{\nu\sigma}\\ \vphantom{\frac{1}{(2-d)}}+4[\mathfrak{g}^{\mu\rho}\partial_{\rho}c^{\nu}+% \mathfrak{g}^{\nu\rho}\partial_{\rho}c^{\mu}-\partial_{\rho}(\mathfrak{g}^{\mu% \nu}c^{\rho})]\partial_{\mu}b_{\nu}\big{\}}," class="ltx_Math" display="block" id="S2.E4.m1.91"><semantics id="S2.E4.m1.91a"><mtable displaystyle="true" id="S2.E4.m1.90.90" rowspacing="0pt"><mtr id="S2.E4.m1.90.90a"><mtd class="ltx_align_left" columnalign="left" id="S2.E4.m1.90.90b"><mrow id="S2.E4.m1.30.30.30.30.30"><mi id="S2.E4.m1.1.1.1.1.1.1" xref="S2.E4.m1.1.1.1.1.1.1.cmml">S</mi><mo id="S2.E4.m1.2.2.2.2.2.2" xref="S2.E4.m1.2.2.2.2.2.2.cmml">=</mo><mfrac id="S2.E4.m1.3.3.3.3.3.3" xref="S2.E4.m1.3.3.3.3.3.3.cmml"><mn id="S2.E4.m1.3.3.3.3.3.3.2" xref="S2.E4.m1.3.3.3.3.3.3.2.cmml">1</mn><mrow id="S2.E4.m1.3.3.3.3.3.3.3" xref="S2.E4.m1.3.3.3.3.3.3.3.cmml"><mn id="S2.E4.m1.3.3.3.3.3.3.3.2" xref="S2.E4.m1.3.3.3.3.3.3.3.2.cmml">8</mn><mo id="S2.E4.m1.3.3.3.3.3.3.3.1" xref="S2.E4.m1.3.3.3.3.3.3.3.1.cmml">⁢</mo><mi id="S2.E4.m1.3.3.3.3.3.3.3.3" xref="S2.E4.m1.3.3.3.3.3.3.3.3.cmml">κ</mi></mrow></mfrac><mo id="S2.E4.m1.4.4.4.4.4.4" xref="S2.E4.m1.4.4.4.4.4.4.cmml">∫</mo><msup id="S2.E4.m1.30.30.30.30.30.31"><mi id="S2.E4.m1.5.5.5.5.5.5" xref="S2.E4.m1.5.5.5.5.5.5.cmml">d</mi><mi id="S2.E4.m1.6.6.6.6.6.6.1" xref="S2.E4.m1.6.6.6.6.6.6.1.cmml">d</mi></msup><mi id="S2.E4.m1.7.7.7.7.7.7" xref="S2.E4.m1.7.7.7.7.7.7.cmml">x</mi><mrow id="S2.E4.m1.30.30.30.30.30.32"><mo id="S2.E4.m1.8.8.8.8.8.8" lspace="0.170em" maxsize="120%" minsize="120%" xref="S2.E4.m1.91.91.1.1.1.cmml">{</mo><msup id="S2.E4.m1.30.30.30.30.30.32.1"><mi id="S2.E4.m1.9.9.9.9.9.9" xref="S2.E4.m1.9.9.9.9.9.9.cmml">𝔤</mi><mrow id="S2.E4.m1.10.10.10.10.10.10.1" xref="S2.E4.m1.10.10.10.10.10.10.1.cmml"><mi id="S2.E4.m1.10.10.10.10.10.10.1.2" xref="S2.E4.m1.10.10.10.10.10.10.1.2.cmml">μ</mi><mo id="S2.E4.m1.10.10.10.10.10.10.1.1" xref="S2.E4.m1.10.10.10.10.10.10.1.1.cmml">⁢</mo><mi id="S2.E4.m1.10.10.10.10.10.10.1.3" xref="S2.E4.m1.10.10.10.10.10.10.1.3.cmml">ν</mi></mrow></msup><msub id="S2.E4.m1.30.30.30.30.30.32.2"><mo id="S2.E4.m1.11.11.11.11.11.11" lspace="0em" rspace="0em" xref="S2.E4.m1.11.11.11.11.11.11.cmml">∂</mo><mi id="S2.E4.m1.12.12.12.12.12.12.1" xref="S2.E4.m1.12.12.12.12.12.12.1.cmml">μ</mi></msub><msup id="S2.E4.m1.30.30.30.30.30.32.3"><mi id="S2.E4.m1.13.13.13.13.13.13" xref="S2.E4.m1.13.13.13.13.13.13.cmml">𝔤</mi><mrow id="S2.E4.m1.14.14.14.14.14.14.1" xref="S2.E4.m1.14.14.14.14.14.14.1.cmml"><mi id="S2.E4.m1.14.14.14.14.14.14.1.2" xref="S2.E4.m1.14.14.14.14.14.14.1.2.cmml">ρ</mi><mo id="S2.E4.m1.14.14.14.14.14.14.1.1" xref="S2.E4.m1.14.14.14.14.14.14.1.1.cmml">⁢</mo><mi id="S2.E4.m1.14.14.14.14.14.14.1.3" xref="S2.E4.m1.14.14.14.14.14.14.1.3.cmml">σ</mi></mrow></msup><msub id="S2.E4.m1.30.30.30.30.30.32.4"><mo id="S2.E4.m1.15.15.15.15.15.15" lspace="0em" rspace="0em" xref="S2.E4.m1.15.15.15.15.15.15.cmml">∂</mo><mi id="S2.E4.m1.16.16.16.16.16.16.1" xref="S2.E4.m1.16.16.16.16.16.16.1.cmml">ν</mi></msub><msub id="S2.E4.m1.30.30.30.30.30.32.5"><mi id="S2.E4.m1.17.17.17.17.17.17" xref="S2.E4.m1.17.17.17.17.17.17.cmml">𝔤</mi><mrow id="S2.E4.m1.18.18.18.18.18.18.1" xref="S2.E4.m1.18.18.18.18.18.18.1.cmml"><mi id="S2.E4.m1.18.18.18.18.18.18.1.2" xref="S2.E4.m1.18.18.18.18.18.18.1.2.cmml">ρ</mi><mo id="S2.E4.m1.18.18.18.18.18.18.1.1" xref="S2.E4.m1.18.18.18.18.18.18.1.1.cmml">⁢</mo><mi id="S2.E4.m1.18.18.18.18.18.18.1.3" xref="S2.E4.m1.18.18.18.18.18.18.1.3.cmml">σ</mi></mrow></msub><mo id="S2.E4.m1.19.19.19.19.19.19" xref="S2.E4.m1.19.19.19.19.19.19.cmml">−</mo><mn id="S2.E4.m1.20.20.20.20.20.20" xref="S2.E4.m1.20.20.20.20.20.20.cmml">2</mn><msup id="S2.E4.m1.30.30.30.30.30.32.6"><mi id="S2.E4.m1.21.21.21.21.21.21" xref="S2.E4.m1.21.21.21.21.21.21.cmml">𝔤</mi><mrow id="S2.E4.m1.22.22.22.22.22.22.1" xref="S2.E4.m1.22.22.22.22.22.22.1.cmml"><mi id="S2.E4.m1.22.22.22.22.22.22.1.2" xref="S2.E4.m1.22.22.22.22.22.22.1.2.cmml">μ</mi><mo id="S2.E4.m1.22.22.22.22.22.22.1.1" xref="S2.E4.m1.22.22.22.22.22.22.1.1.cmml">⁢</mo><mi id="S2.E4.m1.22.22.22.22.22.22.1.3" xref="S2.E4.m1.22.22.22.22.22.22.1.3.cmml">ν</mi></mrow></msup><msub id="S2.E4.m1.30.30.30.30.30.32.7"><mo id="S2.E4.m1.23.23.23.23.23.23" lspace="0em" rspace="0em" xref="S2.E4.m1.23.23.23.23.23.23.cmml">∂</mo><mi id="S2.E4.m1.24.24.24.24.24.24.1" xref="S2.E4.m1.24.24.24.24.24.24.1.cmml">μ</mi></msub><msup id="S2.E4.m1.30.30.30.30.30.32.8"><mi id="S2.E4.m1.25.25.25.25.25.25" xref="S2.E4.m1.25.25.25.25.25.25.cmml">𝔤</mi><mrow id="S2.E4.m1.26.26.26.26.26.26.1" xref="S2.E4.m1.26.26.26.26.26.26.1.cmml"><mi id="S2.E4.m1.26.26.26.26.26.26.1.2" xref="S2.E4.m1.26.26.26.26.26.26.1.2.cmml">ρ</mi><mo id="S2.E4.m1.26.26.26.26.26.26.1.1" xref="S2.E4.m1.26.26.26.26.26.26.1.1.cmml">⁢</mo><mi id="S2.E4.m1.26.26.26.26.26.26.1.3" xref="S2.E4.m1.26.26.26.26.26.26.1.3.cmml">σ</mi></mrow></msup><msub id="S2.E4.m1.30.30.30.30.30.32.9"><mo id="S2.E4.m1.27.27.27.27.27.27" lspace="0em" rspace="0em" xref="S2.E4.m1.27.27.27.27.27.27.cmml">∂</mo><mi id="S2.E4.m1.28.28.28.28.28.28.1" xref="S2.E4.m1.28.28.28.28.28.28.1.cmml">ρ</mi></msub><msub id="S2.E4.m1.30.30.30.30.30.32.10"><mi id="S2.E4.m1.29.29.29.29.29.29" xref="S2.E4.m1.29.29.29.29.29.29.cmml">𝔤</mi><mrow id="S2.E4.m1.30.30.30.30.30.30.1" xref="S2.E4.m1.30.30.30.30.30.30.1.cmml"><mi id="S2.E4.m1.30.30.30.30.30.30.1.2" xref="S2.E4.m1.30.30.30.30.30.30.1.2.cmml">ν</mi><mo id="S2.E4.m1.30.30.30.30.30.30.1.1" xref="S2.E4.m1.30.30.30.30.30.30.1.1.cmml">⁢</mo><mi id="S2.E4.m1.30.30.30.30.30.30.1.3" xref="S2.E4.m1.30.30.30.30.30.30.1.3.cmml">σ</mi></mrow></msub></mrow></mrow></mtd></mtr><mtr id="S2.E4.m1.90.90c"><mtd class="ltx_align_right" columnalign="right" id="S2.E4.m1.90.90d"><mrow id="S2.E4.m1.58.58.58.28.28"><mrow id="S2.E4.m1.58.58.58.28.28.29"><mo id="S2.E4.m1.58.58.58.28.28.29a" xref="S2.E4.m1.91.91.1.1.1.cmml">−</mo><mrow id="S2.E4.m1.58.58.58.28.28.29.1"><mfrac id="S2.E4.m1.32.32.32.2.2.2" xref="S2.E4.m1.32.32.32.2.2.2.cmml"><mn id="S2.E4.m1.32.32.32.2.2.2.3" xref="S2.E4.m1.32.32.32.2.2.2.3.cmml">1</mn><mrow id="S2.E4.m1.32.32.32.2.2.2.1.1" xref="S2.E4.m1.32.32.32.2.2.2.1.1.1.cmml"><mo id="S2.E4.m1.32.32.32.2.2.2.1.1.2" stretchy="false" xref="S2.E4.m1.32.32.32.2.2.2.1.1.1.cmml">(</mo><mrow id="S2.E4.m1.32.32.32.2.2.2.1.1.1" xref="S2.E4.m1.32.32.32.2.2.2.1.1.1.cmml"><mn id="S2.E4.m1.32.32.32.2.2.2.1.1.1.2" xref="S2.E4.m1.32.32.32.2.2.2.1.1.1.2.cmml">2</mn><mo id="S2.E4.m1.32.32.32.2.2.2.1.1.1.1" xref="S2.E4.m1.32.32.32.2.2.2.1.1.1.1.cmml">−</mo><mi id="S2.E4.m1.32.32.32.2.2.2.1.1.1.3" xref="S2.E4.m1.32.32.32.2.2.2.1.1.1.3.cmml">d</mi></mrow><mo id="S2.E4.m1.32.32.32.2.2.2.1.1.3" stretchy="false" xref="S2.E4.m1.32.32.32.2.2.2.1.1.1.cmml">)</mo></mrow></mfrac><mo id="S2.E4.m1.58.58.58.28.28.29.1.1" xref="S2.E4.m1.91.91.1.1.1.cmml">⁢</mo><msup id="S2.E4.m1.58.58.58.28.28.29.1.2"><mi id="S2.E4.m1.33.33.33.3.3.3" xref="S2.E4.m1.33.33.33.3.3.3.cmml">𝔤</mi><mrow id="S2.E4.m1.34.34.34.4.4.4.1" xref="S2.E4.m1.34.34.34.4.4.4.1.cmml"><mi id="S2.E4.m1.34.34.34.4.4.4.1.2" xref="S2.E4.m1.34.34.34.4.4.4.1.2.cmml">μ</mi><mo id="S2.E4.m1.34.34.34.4.4.4.1.1" xref="S2.E4.m1.34.34.34.4.4.4.1.1.cmml">⁢</mo><mi id="S2.E4.m1.34.34.34.4.4.4.1.3" xref="S2.E4.m1.34.34.34.4.4.4.1.3.cmml">ν</mi></mrow></msup><mo id="S2.E4.m1.58.58.58.28.28.29.1.1a" xref="S2.E4.m1.91.91.1.1.1.cmml">⁢</mo><msup id="S2.E4.m1.58.58.58.28.28.29.1.3"><mi id="S2.E4.m1.35.35.35.5.5.5" xref="S2.E4.m1.35.35.35.5.5.5.cmml">𝔤</mi><mrow id="S2.E4.m1.36.36.36.6.6.6.1" xref="S2.E4.m1.36.36.36.6.6.6.1.cmml"><mi id="S2.E4.m1.36.36.36.6.6.6.1.2" xref="S2.E4.m1.36.36.36.6.6.6.1.2.cmml">ρ</mi><mo id="S2.E4.m1.36.36.36.6.6.6.1.1" xref="S2.E4.m1.36.36.36.6.6.6.1.1.cmml">⁢</mo><mi id="S2.E4.m1.36.36.36.6.6.6.1.3" xref="S2.E4.m1.36.36.36.6.6.6.1.3.cmml">σ</mi></mrow></msup><mo id="S2.E4.m1.58.58.58.28.28.29.1.1b" lspace="0em" xref="S2.E4.m1.91.91.1.1.1.cmml">⁢</mo><mrow id="S2.E4.m1.58.58.58.28.28.29.1.4"><msub id="S2.E4.m1.58.58.58.28.28.29.1.4.1"><mo id="S2.E4.m1.37.37.37.7.7.7" rspace="0em" xref="S2.E4.m1.37.37.37.7.7.7.cmml">∂</mo><mi id="S2.E4.m1.38.38.38.8.8.8.1" xref="S2.E4.m1.38.38.38.8.8.8.1.cmml">μ</mi></msub><mrow id="S2.E4.m1.58.58.58.28.28.29.1.4.2"><msub id="S2.E4.m1.58.58.58.28.28.29.1.4.2.2"><mi id="S2.E4.m1.39.39.39.9.9.9" xref="S2.E4.m1.39.39.39.9.9.9.cmml">𝔤</mi><mrow id="S2.E4.m1.40.40.40.10.10.10.1" xref="S2.E4.m1.40.40.40.10.10.10.1.cmml"><mi id="S2.E4.m1.40.40.40.10.10.10.1.2" xref="S2.E4.m1.40.40.40.10.10.10.1.2.cmml">ρ</mi><mo id="S2.E4.m1.40.40.40.10.10.10.1.1" xref="S2.E4.m1.40.40.40.10.10.10.1.1.cmml">⁢</mo><mi id="S2.E4.m1.40.40.40.10.10.10.1.3" xref="S2.E4.m1.40.40.40.10.10.10.1.3.cmml">σ</mi></mrow></msub><mo id="S2.E4.m1.58.58.58.28.28.29.1.4.2.1" xref="S2.E4.m1.91.91.1.1.1.cmml">⁢</mo><msup id="S2.E4.m1.58.58.58.28.28.29.1.4.2.3"><mi id="S2.E4.m1.41.41.41.11.11.11" xref="S2.E4.m1.41.41.41.11.11.11.cmml">𝔤</mi><mrow id="S2.E4.m1.42.42.42.12.12.12.1" xref="S2.E4.m1.42.42.42.12.12.12.1.cmml"><mi id="S2.E4.m1.42.42.42.12.12.12.1.2" xref="S2.E4.m1.42.42.42.12.12.12.1.2.cmml">γ</mi><mo id="S2.E4.m1.42.42.42.12.12.12.1.1" xref="S2.E4.m1.42.42.42.12.12.12.1.1.cmml">⁢</mo><mi id="S2.E4.m1.42.42.42.12.12.12.1.3" xref="S2.E4.m1.42.42.42.12.12.12.1.3.cmml">λ</mi></mrow></msup><mo id="S2.E4.m1.58.58.58.28.28.29.1.4.2.1a" lspace="0em" xref="S2.E4.m1.91.91.1.1.1.cmml">⁢</mo><mrow id="S2.E4.m1.58.58.58.28.28.29.1.4.2.4"><msub id="S2.E4.m1.58.58.58.28.28.29.1.4.2.4.1"><mo id="S2.E4.m1.43.43.43.13.13.13" rspace="0em" xref="S2.E4.m1.43.43.43.13.13.13.cmml">∂</mo><mi id="S2.E4.m1.44.44.44.14.14.14.1" xref="S2.E4.m1.44.44.44.14.14.14.1.cmml">ν</mi></msub><msub id="S2.E4.m1.58.58.58.28.28.29.1.4.2.4.2"><mi id="S2.E4.m1.45.45.45.15.15.15" xref="S2.E4.m1.45.45.45.15.15.15.cmml">𝔤</mi><mrow id="S2.E4.m1.46.46.46.16.16.16.1" xref="S2.E4.m1.46.46.46.16.16.16.1.cmml"><mi id="S2.E4.m1.46.46.46.16.16.16.1.2" xref="S2.E4.m1.46.46.46.16.16.16.1.2.cmml">γ</mi><mo id="S2.E4.m1.46.46.46.16.16.16.1.1" xref="S2.E4.m1.46.46.46.16.16.16.1.1.cmml">⁢</mo><mi id="S2.E4.m1.46.46.46.16.16.16.1.3" xref="S2.E4.m1.46.46.46.16.16.16.1.3.cmml">λ</mi></mrow></msub></mrow></mrow></mrow></mrow></mrow><mo id="S2.E4.m1.47.47.47.17.17.17" xref="S2.E4.m1.91.91.1.1.1.cmml">−</mo><mrow id="S2.E4.m1.58.58.58.28.28.30"><mn id="S2.E4.m1.48.48.48.18.18.18" xref="S2.E4.m1.48.48.48.18.18.18.cmml">2</mn><mo id="S2.E4.m1.58.58.58.28.28.30.1" xref="S2.E4.m1.91.91.1.1.1.cmml">⁢</mo><msub id="S2.E4.m1.58.58.58.28.28.30.2"><mi id="S2.E4.m1.49.49.49.19.19.19" xref="S2.E4.m1.49.49.49.19.19.19.cmml">η</mi><mrow id="S2.E4.m1.50.50.50.20.20.20.1" xref="S2.E4.m1.50.50.50.20.20.20.1.cmml"><mi id="S2.E4.m1.50.50.50.20.20.20.1.2" xref="S2.E4.m1.50.50.50.20.20.20.1.2.cmml">μ</mi><mo id="S2.E4.m1.50.50.50.20.20.20.1.1" xref="S2.E4.m1.50.50.50.20.20.20.1.1.cmml">⁢</mo><mi id="S2.E4.m1.50.50.50.20.20.20.1.3" xref="S2.E4.m1.50.50.50.20.20.20.1.3.cmml">ν</mi></mrow></msub><mo id="S2.E4.m1.58.58.58.28.28.30.1a" lspace="0em" xref="S2.E4.m1.91.91.1.1.1.cmml">⁢</mo><mrow id="S2.E4.m1.58.58.58.28.28.30.3"><msub id="S2.E4.m1.58.58.58.28.28.30.3.1"><mo id="S2.E4.m1.51.51.51.21.21.21" rspace="0em" xref="S2.E4.m1.51.51.51.21.21.21.cmml">∂</mo><mi id="S2.E4.m1.52.52.52.22.22.22.1" xref="S2.E4.m1.52.52.52.22.22.22.1.cmml">ρ</mi></msub><mrow id="S2.E4.m1.58.58.58.28.28.30.3.2"><msup id="S2.E4.m1.58.58.58.28.28.30.3.2.2"><mi id="S2.E4.m1.53.53.53.23.23.23" xref="S2.E4.m1.53.53.53.23.23.23.cmml">𝔤</mi><mrow id="S2.E4.m1.54.54.54.24.24.24.1" xref="S2.E4.m1.54.54.54.24.24.24.1.cmml"><mi id="S2.E4.m1.54.54.54.24.24.24.1.2" xref="S2.E4.m1.54.54.54.24.24.24.1.2.cmml">μ</mi><mo id="S2.E4.m1.54.54.54.24.24.24.1.1" xref="S2.E4.m1.54.54.54.24.24.24.1.1.cmml">⁢</mo><mi id="S2.E4.m1.54.54.54.24.24.24.1.3" xref="S2.E4.m1.54.54.54.24.24.24.1.3.cmml">ρ</mi></mrow></msup><mo id="S2.E4.m1.58.58.58.28.28.30.3.2.1" lspace="0em" xref="S2.E4.m1.91.91.1.1.1.cmml">⁢</mo><mrow id="S2.E4.m1.58.58.58.28.28.30.3.2.3"><msub id="S2.E4.m1.58.58.58.28.28.30.3.2.3.1"><mo id="S2.E4.m1.55.55.55.25.25.25" rspace="0em" xref="S2.E4.m1.55.55.55.25.25.25.cmml">∂</mo><mi id="S2.E4.m1.56.56.56.26.26.26.1" xref="S2.E4.m1.56.56.56.26.26.26.1.cmml">σ</mi></msub><msup id="S2.E4.m1.58.58.58.28.28.30.3.2.3.2"><mi id="S2.E4.m1.57.57.57.27.27.27" xref="S2.E4.m1.57.57.57.27.27.27.cmml">𝔤</mi><mrow id="S2.E4.m1.58.58.58.28.28.28.1" xref="S2.E4.m1.58.58.58.28.28.28.1.cmml"><mi id="S2.E4.m1.58.58.58.28.28.28.1.2" xref="S2.E4.m1.58.58.58.28.28.28.1.2.cmml">ν</mi><mo id="S2.E4.m1.58.58.58.28.28.28.1.1" xref="S2.E4.m1.58.58.58.28.28.28.1.1.cmml">⁢</mo><mi id="S2.E4.m1.58.58.58.28.28.28.1.3" xref="S2.E4.m1.58.58.58.28.28.28.1.3.cmml">σ</mi></mrow></msup></mrow></mrow></mrow></mrow></mrow></mtd></mtr><mtr id="S2.E4.m1.90.90e"><mtd class="ltx_align_right" columnalign="right" id="S2.E4.m1.90.90f"><mrow id="S2.E4.m1.90.90.90.32.32"><mo id="S2.E4.m1.59.59.59.1.1.1" xref="S2.E4.m1.59.59.59.1.1.1.cmml">+</mo><mn id="S2.E4.m1.60.60.60.2.2.2" xref="S2.E4.m1.60.60.60.2.2.2.cmml">4</mn><mrow id="S2.E4.m1.90.90.90.32.32.33"><mo id="S2.E4.m1.61.61.61.3.3.3" stretchy="false" xref="S2.E4.m1.91.91.1.1.1.cmml">[</mo><msup id="S2.E4.m1.90.90.90.32.32.33.1"><mi id="S2.E4.m1.62.62.62.4.4.4" xref="S2.E4.m1.62.62.62.4.4.4.cmml">𝔤</mi><mrow id="S2.E4.m1.63.63.63.5.5.5.1" xref="S2.E4.m1.63.63.63.5.5.5.1.cmml"><mi id="S2.E4.m1.63.63.63.5.5.5.1.2" xref="S2.E4.m1.63.63.63.5.5.5.1.2.cmml">μ</mi><mo id="S2.E4.m1.63.63.63.5.5.5.1.1" xref="S2.E4.m1.63.63.63.5.5.5.1.1.cmml">⁢</mo><mi id="S2.E4.m1.63.63.63.5.5.5.1.3" xref="S2.E4.m1.63.63.63.5.5.5.1.3.cmml">ρ</mi></mrow></msup><msub id="S2.E4.m1.90.90.90.32.32.33.2"><mo id="S2.E4.m1.64.64.64.6.6.6" lspace="0em" rspace="0em" xref="S2.E4.m1.64.64.64.6.6.6.cmml">∂</mo><mi id="S2.E4.m1.65.65.65.7.7.7.1" xref="S2.E4.m1.65.65.65.7.7.7.1.cmml">ρ</mi></msub><msup id="S2.E4.m1.90.90.90.32.32.33.3"><mi id="S2.E4.m1.66.66.66.8.8.8" xref="S2.E4.m1.66.66.66.8.8.8.cmml">c</mi><mi id="S2.E4.m1.67.67.67.9.9.9.1" xref="S2.E4.m1.67.67.67.9.9.9.1.cmml">ν</mi></msup><mo id="S2.E4.m1.68.68.68.10.10.10" xref="S2.E4.m1.68.68.68.10.10.10.cmml">+</mo><msup id="S2.E4.m1.90.90.90.32.32.33.4"><mi id="S2.E4.m1.69.69.69.11.11.11" xref="S2.E4.m1.69.69.69.11.11.11.cmml">𝔤</mi><mrow id="S2.E4.m1.70.70.70.12.12.12.1" xref="S2.E4.m1.70.70.70.12.12.12.1.cmml"><mi id="S2.E4.m1.70.70.70.12.12.12.1.2" xref="S2.E4.m1.70.70.70.12.12.12.1.2.cmml">ν</mi><mo id="S2.E4.m1.70.70.70.12.12.12.1.1" xref="S2.E4.m1.70.70.70.12.12.12.1.1.cmml">⁢</mo><mi id="S2.E4.m1.70.70.70.12.12.12.1.3" xref="S2.E4.m1.70.70.70.12.12.12.1.3.cmml">ρ</mi></mrow></msup><msub id="S2.E4.m1.90.90.90.32.32.33.5"><mo id="S2.E4.m1.71.71.71.13.13.13" lspace="0em" rspace="0em" xref="S2.E4.m1.71.71.71.13.13.13.cmml">∂</mo><mi id="S2.E4.m1.72.72.72.14.14.14.1" xref="S2.E4.m1.72.72.72.14.14.14.1.cmml">ρ</mi></msub><msup id="S2.E4.m1.90.90.90.32.32.33.6"><mi id="S2.E4.m1.73.73.73.15.15.15" xref="S2.E4.m1.73.73.73.15.15.15.cmml">c</mi><mi id="S2.E4.m1.74.74.74.16.16.16.1" xref="S2.E4.m1.74.74.74.16.16.16.1.cmml">μ</mi></msup><mo id="S2.E4.m1.75.75.75.17.17.17" xref="S2.E4.m1.75.75.75.17.17.17.cmml">−</mo><msub id="S2.E4.m1.90.90.90.32.32.33.7"><mo id="S2.E4.m1.76.76.76.18.18.18" lspace="0em" rspace="0em" xref="S2.E4.m1.76.76.76.18.18.18.cmml">∂</mo><mi id="S2.E4.m1.77.77.77.19.19.19.1" xref="S2.E4.m1.77.77.77.19.19.19.1.cmml">ρ</mi></msub><mrow id="S2.E4.m1.90.90.90.32.32.33.8"><mo id="S2.E4.m1.78.78.78.20.20.20" stretchy="false" xref="S2.E4.m1.91.91.1.1.1.cmml">(</mo><msup id="S2.E4.m1.90.90.90.32.32.33.8.1"><mi id="S2.E4.m1.79.79.79.21.21.21" xref="S2.E4.m1.79.79.79.21.21.21.cmml">𝔤</mi><mrow id="S2.E4.m1.80.80.80.22.22.22.1" xref="S2.E4.m1.80.80.80.22.22.22.1.cmml"><mi id="S2.E4.m1.80.80.80.22.22.22.1.2" xref="S2.E4.m1.80.80.80.22.22.22.1.2.cmml">μ</mi><mo id="S2.E4.m1.80.80.80.22.22.22.1.1" xref="S2.E4.m1.80.80.80.22.22.22.1.1.cmml">⁢</mo><mi id="S2.E4.m1.80.80.80.22.22.22.1.3" xref="S2.E4.m1.80.80.80.22.22.22.1.3.cmml">ν</mi></mrow></msup><msup id="S2.E4.m1.90.90.90.32.32.33.8.2"><mi id="S2.E4.m1.81.81.81.23.23.23" xref="S2.E4.m1.81.81.81.23.23.23.cmml">c</mi><mi id="S2.E4.m1.82.82.82.24.24.24.1" xref="S2.E4.m1.82.82.82.24.24.24.1.cmml">ρ</mi></msup><mo id="S2.E4.m1.83.83.83.25.25.25" stretchy="false" xref="S2.E4.m1.91.91.1.1.1.cmml">)</mo></mrow><mo id="S2.E4.m1.84.84.84.26.26.26" stretchy="false" xref="S2.E4.m1.91.91.1.1.1.cmml">]</mo></mrow><msub id="S2.E4.m1.90.90.90.32.32.34"><mo id="S2.E4.m1.85.85.85.27.27.27" lspace="0em" rspace="0em" xref="S2.E4.m1.85.85.85.27.27.27.cmml">∂</mo><mi id="S2.E4.m1.86.86.86.28.28.28.1" xref="S2.E4.m1.86.86.86.28.28.28.1.cmml">μ</mi></msub><msub id="S2.E4.m1.90.90.90.32.32.35"><mi id="S2.E4.m1.87.87.87.29.29.29" xref="S2.E4.m1.87.87.87.29.29.29.cmml">b</mi><mi id="S2.E4.m1.88.88.88.30.30.30.1" xref="S2.E4.m1.88.88.88.30.30.30.1.cmml">ν</mi></msub><mo id="S2.E4.m1.89.89.89.31.31.31" maxsize="120%" minsize="120%" 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\vphantom{\frac{1}{(2-d)}}+4[\mathfrak{g}^{\mu\rho}\partial_{\rho}c^{\nu}+% \mathfrak{g}^{\nu\rho}\partial_{\rho}c^{\mu}-\partial_{\rho}(\mathfrak{g}^{\mu% \nu}c^{\rho})]\partial_{\mu}b_{\nu}\big{\}},</annotation><annotation encoding="application/x-llamapun" id="S2.E4.m1.91d">start_ROW start_CELL italic_S = divide start_ARG 1 end_ARG start_ARG 8 italic_κ end_ARG ∫ italic_d start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT italic_x { fraktur_g start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT fraktur_g start_POSTSUPERSCRIPT italic_ρ italic_σ end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT fraktur_g start_POSTSUBSCRIPT italic_ρ italic_σ end_POSTSUBSCRIPT - 2 fraktur_g start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT fraktur_g start_POSTSUPERSCRIPT italic_ρ italic_σ end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT fraktur_g start_POSTSUBSCRIPT italic_ν italic_σ end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL - divide start_ARG 1 end_ARG start_ARG ( 2 - italic_d ) end_ARG fraktur_g start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT fraktur_g start_POSTSUPERSCRIPT italic_ρ italic_σ end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT fraktur_g start_POSTSUBSCRIPT italic_ρ italic_σ end_POSTSUBSCRIPT fraktur_g start_POSTSUPERSCRIPT italic_γ italic_λ end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT fraktur_g start_POSTSUBSCRIPT italic_γ italic_λ end_POSTSUBSCRIPT - 2 italic_η start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ∂ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT fraktur_g start_POSTSUPERSCRIPT italic_μ italic_ρ end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT fraktur_g start_POSTSUPERSCRIPT italic_ν italic_σ end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL + 4 [ fraktur_g start_POSTSUPERSCRIPT italic_μ italic_ρ end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT italic_c start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT + fraktur_g start_POSTSUPERSCRIPT italic_ν italic_ρ end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT italic_c start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT - ∂ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT ( fraktur_g start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT italic_c start_POSTSUPERSCRIPT italic_ρ end_POSTSUPERSCRIPT ) ] ∂ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT } , end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p3.4">where <math alttext="\eta_{\mu\nu}" class="ltx_Math" display="inline" id="S2.p3.4.m1.1"><semantics id="S2.p3.4.m1.1a"><msub id="S2.p3.4.m1.1.1" xref="S2.p3.4.m1.1.1.cmml"><mi id="S2.p3.4.m1.1.1.2" xref="S2.p3.4.m1.1.1.2.cmml">η</mi><mrow id="S2.p3.4.m1.1.1.3" xref="S2.p3.4.m1.1.1.3.cmml"><mi id="S2.p3.4.m1.1.1.3.2" xref="S2.p3.4.m1.1.1.3.2.cmml">μ</mi><mo id="S2.p3.4.m1.1.1.3.1" xref="S2.p3.4.m1.1.1.3.1.cmml">⁢</mo><mi id="S2.p3.4.m1.1.1.3.3" xref="S2.p3.4.m1.1.1.3.3.cmml">ν</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.p3.4.m1.1b"><apply id="S2.p3.4.m1.1.1.cmml" xref="S2.p3.4.m1.1.1"><csymbol cd="ambiguous" id="S2.p3.4.m1.1.1.1.cmml" xref="S2.p3.4.m1.1.1">subscript</csymbol><ci id="S2.p3.4.m1.1.1.2.cmml" xref="S2.p3.4.m1.1.1.2">𝜂</ci><apply id="S2.p3.4.m1.1.1.3.cmml" xref="S2.p3.4.m1.1.1.3"><times id="S2.p3.4.m1.1.1.3.1.cmml" xref="S2.p3.4.m1.1.1.3.1"></times><ci id="S2.p3.4.m1.1.1.3.2.cmml" xref="S2.p3.4.m1.1.1.3.2">𝜇</ci><ci id="S2.p3.4.m1.1.1.3.3.cmml" xref="S2.p3.4.m1.1.1.3.3">𝜈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p3.4.m1.1c">\eta_{\mu\nu}</annotation><annotation encoding="application/x-llamapun" id="S2.p3.4.m1.1d">italic_η start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT</annotation></semantics></math> is the flat space metric.</p> </div> <div class="ltx_para" id="S2.p4"> <p class="ltx_p" id="S2.p4.1">The respective equations of motion are straightforward to derive. For the metric density we obtain</p> <table class="ltx_equation ltx_eqn_table" id="S2.E5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\partial_{\rho}(\mathfrak{g}^{\rho\sigma}\partial_{\sigma}\mathfrak{g}_{\mu\nu% })=\eta_{\mu\sigma}\partial_{\nu}\partial_{\rho}\mathfrak{g}^{\rho\sigma}-% \partial_{\rho}(\mathfrak{g}_{\mu\sigma}\partial_{\nu}\mathfrak{g}^{\rho\sigma% })\\ +\frac{1}{4}\partial_{\mu}\mathfrak{g}^{\rho\sigma}\partial_{\nu}\mathfrak{g}_% {\rho\sigma}+\frac{1}{2}\mathfrak{g}^{\rho\sigma}\mathfrak{g}^{\gamma\lambda}% \partial_{\gamma}\mathfrak{g}_{\mu\rho}(\partial_{\lambda}\mathfrak{g}_{\nu% \sigma}-\partial_{\sigma}\mathfrak{g}_{\nu\lambda})\\ -\frac{1}{4}\frac{1}{(2-d)}[2\mathfrak{g}_{\mu\nu}\partial_{\rho}(\mathfrak{g}% ^{\rho\sigma}\mathfrak{g}^{\gamma\lambda}\partial_{\sigma}\mathfrak{g}_{\gamma% \lambda})+\mathfrak{g}^{\rho\sigma}\partial_{\mu}\mathfrak{g}_{\rho\sigma}% \mathfrak{g}^{\gamma\lambda}\partial_{\nu}\mathfrak{g}_{\gamma\lambda}]\\ +\frac{1}{2}\partial_{\mu}c^{\rho}\partial_{\nu}b_{\rho}+\frac{1}{2}\partial_{% \mu}(c^{\rho}\partial_{\rho}b_{\nu})+(\mu\leftrightarrow\nu)." class="ltx_math_unparsed" display="block" id="S2.E5.m1.127"><semantics id="S2.E5.m1.127a"><mtable displaystyle="true" id="S2.E5.m1.127.127.4" rowspacing="0pt"><mtr id="S2.E5.m1.127.127.4a"><mtd class="ltx_align_left" columnalign="left" id="S2.E5.m1.127.127.4b"><mrow id="S2.E5.m1.125.125.2.125.32.32"><mrow id="S2.E5.m1.124.124.1.124.31.31.31"><msub id="S2.E5.m1.124.124.1.124.31.31.31.2"><mo id="S2.E5.m1.1.1.1.1.1.1">∂</mo><mi id="S2.E5.m1.2.2.2.2.2.2.1">ρ</mi></msub><mrow id="S2.E5.m1.124.124.1.124.31.31.31.1.1"><mo id="S2.E5.m1.3.3.3.3.3.3" lspace="0em" stretchy="false">(</mo><mrow id="S2.E5.m1.124.124.1.124.31.31.31.1.1.1"><msup id="S2.E5.m1.124.124.1.124.31.31.31.1.1.1.2"><mi id="S2.E5.m1.4.4.4.4.4.4">𝔤</mi><mrow id="S2.E5.m1.5.5.5.5.5.5.1"><mi id="S2.E5.m1.5.5.5.5.5.5.1.2">ρ</mi><mo id="S2.E5.m1.5.5.5.5.5.5.1.1">⁢</mo><mi id="S2.E5.m1.5.5.5.5.5.5.1.3">σ</mi></mrow></msup><mo id="S2.E5.m1.124.124.1.124.31.31.31.1.1.1.1" lspace="0em">⁢</mo><mrow id="S2.E5.m1.124.124.1.124.31.31.31.1.1.1.3"><msub id="S2.E5.m1.124.124.1.124.31.31.31.1.1.1.3.1"><mo id="S2.E5.m1.6.6.6.6.6.6" rspace="0em">∂</mo><mi id="S2.E5.m1.7.7.7.7.7.7.1">σ</mi></msub><msub id="S2.E5.m1.124.124.1.124.31.31.31.1.1.1.3.2"><mi id="S2.E5.m1.8.8.8.8.8.8">𝔤</mi><mrow id="S2.E5.m1.9.9.9.9.9.9.1"><mi id="S2.E5.m1.9.9.9.9.9.9.1.2">μ</mi><mo id="S2.E5.m1.9.9.9.9.9.9.1.1">⁢</mo><mi id="S2.E5.m1.9.9.9.9.9.9.1.3">ν</mi></mrow></msub></mrow></mrow><mo id="S2.E5.m1.10.10.10.10.10.10" stretchy="false">)</mo></mrow></mrow><mo id="S2.E5.m1.11.11.11.11.11.11">=</mo><mrow id="S2.E5.m1.125.125.2.125.32.32.32"><mrow id="S2.E5.m1.125.125.2.125.32.32.32.2"><msub id="S2.E5.m1.125.125.2.125.32.32.32.2.2"><mi id="S2.E5.m1.12.12.12.12.12.12">η</mi><mrow id="S2.E5.m1.13.13.13.13.13.13.1"><mi id="S2.E5.m1.13.13.13.13.13.13.1.2">μ</mi><mo id="S2.E5.m1.13.13.13.13.13.13.1.1">⁢</mo><mi id="S2.E5.m1.13.13.13.13.13.13.1.3">σ</mi></mrow></msub><mo id="S2.E5.m1.125.125.2.125.32.32.32.2.1" lspace="0em">⁢</mo><mrow id="S2.E5.m1.125.125.2.125.32.32.32.2.3"><msub id="S2.E5.m1.125.125.2.125.32.32.32.2.3.1"><mo id="S2.E5.m1.14.14.14.14.14.14" rspace="0em">∂</mo><mi id="S2.E5.m1.15.15.15.15.15.15.1">ν</mi></msub><mrow id="S2.E5.m1.125.125.2.125.32.32.32.2.3.2"><msub id="S2.E5.m1.125.125.2.125.32.32.32.2.3.2.1"><mo id="S2.E5.m1.16.16.16.16.16.16" lspace="0em" rspace="0em">∂</mo><mi id="S2.E5.m1.17.17.17.17.17.17.1">ρ</mi></msub><msup id="S2.E5.m1.125.125.2.125.32.32.32.2.3.2.2"><mi id="S2.E5.m1.18.18.18.18.18.18">𝔤</mi><mrow id="S2.E5.m1.19.19.19.19.19.19.1"><mi id="S2.E5.m1.19.19.19.19.19.19.1.2">ρ</mi><mo id="S2.E5.m1.19.19.19.19.19.19.1.1">⁢</mo><mi id="S2.E5.m1.19.19.19.19.19.19.1.3">σ</mi></mrow></msup></mrow></mrow></mrow><mo id="S2.E5.m1.20.20.20.20.20.20">−</mo><mrow id="S2.E5.m1.125.125.2.125.32.32.32.1"><msub id="S2.E5.m1.125.125.2.125.32.32.32.1.2"><mo id="S2.E5.m1.21.21.21.21.21.21" lspace="0em" rspace="0em">∂</mo><mi id="S2.E5.m1.22.22.22.22.22.22.1">ρ</mi></msub><mrow id="S2.E5.m1.125.125.2.125.32.32.32.1.1.1"><mo id="S2.E5.m1.23.23.23.23.23.23" stretchy="false">(</mo><mrow id="S2.E5.m1.125.125.2.125.32.32.32.1.1.1.1"><msub id="S2.E5.m1.125.125.2.125.32.32.32.1.1.1.1.2"><mi id="S2.E5.m1.24.24.24.24.24.24">𝔤</mi><mrow id="S2.E5.m1.25.25.25.25.25.25.1"><mi id="S2.E5.m1.25.25.25.25.25.25.1.2">μ</mi><mo id="S2.E5.m1.25.25.25.25.25.25.1.1">⁢</mo><mi id="S2.E5.m1.25.25.25.25.25.25.1.3">σ</mi></mrow></msub><mo id="S2.E5.m1.125.125.2.125.32.32.32.1.1.1.1.1" lspace="0em">⁢</mo><mrow id="S2.E5.m1.125.125.2.125.32.32.32.1.1.1.1.3"><msub id="S2.E5.m1.125.125.2.125.32.32.32.1.1.1.1.3.1"><mo id="S2.E5.m1.26.26.26.26.26.26" rspace="0em">∂</mo><mi id="S2.E5.m1.27.27.27.27.27.27.1">ν</mi></msub><msup id="S2.E5.m1.125.125.2.125.32.32.32.1.1.1.1.3.2"><mi id="S2.E5.m1.28.28.28.28.28.28">𝔤</mi><mrow id="S2.E5.m1.29.29.29.29.29.29.1"><mi id="S2.E5.m1.29.29.29.29.29.29.1.2">ρ</mi><mo id="S2.E5.m1.29.29.29.29.29.29.1.1">⁢</mo><mi id="S2.E5.m1.29.29.29.29.29.29.1.3">σ</mi></mrow></msup></mrow></mrow><mo id="S2.E5.m1.30.30.30.30.30.30" stretchy="false">)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr id="S2.E5.m1.127.127.4c"><mtd class="ltx_align_right" columnalign="right" id="S2.E5.m1.127.127.4d"><mrow id="S2.E5.m1.126.126.3.126.32.32"><mrow id="S2.E5.m1.126.126.3.126.32.32.33"><mo id="S2.E5.m1.126.126.3.126.32.32.33a">+</mo><mrow id="S2.E5.m1.126.126.3.126.32.32.33.1"><mfrac id="S2.E5.m1.32.32.32.2.2.2"><mn id="S2.E5.m1.32.32.32.2.2.2.2">1</mn><mn id="S2.E5.m1.32.32.32.2.2.2.3">4</mn></mfrac><mo id="S2.E5.m1.126.126.3.126.32.32.33.1.1" lspace="0em">⁢</mo><mrow id="S2.E5.m1.126.126.3.126.32.32.33.1.2"><msub id="S2.E5.m1.126.126.3.126.32.32.33.1.2.1"><mo id="S2.E5.m1.33.33.33.3.3.3" rspace="0em">∂</mo><mi id="S2.E5.m1.34.34.34.4.4.4.1">μ</mi></msub><mrow id="S2.E5.m1.126.126.3.126.32.32.33.1.2.2"><msup id="S2.E5.m1.126.126.3.126.32.32.33.1.2.2.2"><mi id="S2.E5.m1.35.35.35.5.5.5">𝔤</mi><mrow id="S2.E5.m1.36.36.36.6.6.6.1"><mi id="S2.E5.m1.36.36.36.6.6.6.1.2">ρ</mi><mo id="S2.E5.m1.36.36.36.6.6.6.1.1">⁢</mo><mi id="S2.E5.m1.36.36.36.6.6.6.1.3">σ</mi></mrow></msup><mo id="S2.E5.m1.126.126.3.126.32.32.33.1.2.2.1" lspace="0em">⁢</mo><mrow id="S2.E5.m1.126.126.3.126.32.32.33.1.2.2.3"><msub id="S2.E5.m1.126.126.3.126.32.32.33.1.2.2.3.1"><mo id="S2.E5.m1.37.37.37.7.7.7" rspace="0em">∂</mo><mi id="S2.E5.m1.38.38.38.8.8.8.1">ν</mi></msub><msub id="S2.E5.m1.126.126.3.126.32.32.33.1.2.2.3.2"><mi id="S2.E5.m1.39.39.39.9.9.9">𝔤</mi><mrow id="S2.E5.m1.40.40.40.10.10.10.1"><mi id="S2.E5.m1.40.40.40.10.10.10.1.2">ρ</mi><mo id="S2.E5.m1.40.40.40.10.10.10.1.1">⁢</mo><mi id="S2.E5.m1.40.40.40.10.10.10.1.3">σ</mi></mrow></msub></mrow></mrow></mrow></mrow></mrow><mo id="S2.E5.m1.41.41.41.11.11.11">+</mo><mrow id="S2.E5.m1.126.126.3.126.32.32.32"><mfrac id="S2.E5.m1.42.42.42.12.12.12"><mn id="S2.E5.m1.42.42.42.12.12.12.2">1</mn><mn id="S2.E5.m1.42.42.42.12.12.12.3">2</mn></mfrac><mo id="S2.E5.m1.126.126.3.126.32.32.32.2">⁢</mo><msup id="S2.E5.m1.126.126.3.126.32.32.32.3"><mi id="S2.E5.m1.43.43.43.13.13.13">𝔤</mi><mrow id="S2.E5.m1.44.44.44.14.14.14.1"><mi id="S2.E5.m1.44.44.44.14.14.14.1.2">ρ</mi><mo id="S2.E5.m1.44.44.44.14.14.14.1.1">⁢</mo><mi id="S2.E5.m1.44.44.44.14.14.14.1.3">σ</mi></mrow></msup><mo id="S2.E5.m1.126.126.3.126.32.32.32.2a">⁢</mo><msup id="S2.E5.m1.126.126.3.126.32.32.32.4"><mi id="S2.E5.m1.45.45.45.15.15.15">𝔤</mi><mrow id="S2.E5.m1.46.46.46.16.16.16.1"><mi id="S2.E5.m1.46.46.46.16.16.16.1.2">γ</mi><mo id="S2.E5.m1.46.46.46.16.16.16.1.1">⁢</mo><mi id="S2.E5.m1.46.46.46.16.16.16.1.3">λ</mi></mrow></msup><mo id="S2.E5.m1.126.126.3.126.32.32.32.2b" lspace="0em">⁢</mo><mrow id="S2.E5.m1.126.126.3.126.32.32.32.1"><msub id="S2.E5.m1.126.126.3.126.32.32.32.1.2"><mo id="S2.E5.m1.47.47.47.17.17.17" rspace="0em">∂</mo><mi id="S2.E5.m1.48.48.48.18.18.18.1">γ</mi></msub><mrow id="S2.E5.m1.126.126.3.126.32.32.32.1.1"><msub id="S2.E5.m1.126.126.3.126.32.32.32.1.1.3"><mi id="S2.E5.m1.49.49.49.19.19.19">𝔤</mi><mrow id="S2.E5.m1.50.50.50.20.20.20.1"><mi id="S2.E5.m1.50.50.50.20.20.20.1.2">μ</mi><mo id="S2.E5.m1.50.50.50.20.20.20.1.1">⁢</mo><mi id="S2.E5.m1.50.50.50.20.20.20.1.3">ρ</mi></mrow></msub><mo id="S2.E5.m1.126.126.3.126.32.32.32.1.1.2">⁢</mo><mrow id="S2.E5.m1.126.126.3.126.32.32.32.1.1.1.1"><mo id="S2.E5.m1.51.51.51.21.21.21" stretchy="false">(</mo><mrow id="S2.E5.m1.126.126.3.126.32.32.32.1.1.1.1.1"><mrow id="S2.E5.m1.126.126.3.126.32.32.32.1.1.1.1.1.1"><msub id="S2.E5.m1.126.126.3.126.32.32.32.1.1.1.1.1.1.1"><mo id="S2.E5.m1.52.52.52.22.22.22" lspace="0em" rspace="0em">∂</mo><mi id="S2.E5.m1.53.53.53.23.23.23.1">λ</mi></msub><msub id="S2.E5.m1.126.126.3.126.32.32.32.1.1.1.1.1.1.2"><mi id="S2.E5.m1.54.54.54.24.24.24">𝔤</mi><mrow id="S2.E5.m1.55.55.55.25.25.25.1"><mi id="S2.E5.m1.55.55.55.25.25.25.1.2">ν</mi><mo id="S2.E5.m1.55.55.55.25.25.25.1.1">⁢</mo><mi id="S2.E5.m1.55.55.55.25.25.25.1.3">σ</mi></mrow></msub></mrow><mo id="S2.E5.m1.56.56.56.26.26.26">−</mo><mrow id="S2.E5.m1.126.126.3.126.32.32.32.1.1.1.1.1.2"><msub id="S2.E5.m1.126.126.3.126.32.32.32.1.1.1.1.1.2.1"><mo id="S2.E5.m1.57.57.57.27.27.27" lspace="0em" rspace="0em">∂</mo><mi id="S2.E5.m1.58.58.58.28.28.28.1">σ</mi></msub><msub id="S2.E5.m1.126.126.3.126.32.32.32.1.1.1.1.1.2.2"><mi id="S2.E5.m1.59.59.59.29.29.29">𝔤</mi><mrow id="S2.E5.m1.60.60.60.30.30.30.1"><mi id="S2.E5.m1.60.60.60.30.30.30.1.2">ν</mi><mo id="S2.E5.m1.60.60.60.30.30.30.1.1">⁢</mo><mi id="S2.E5.m1.60.60.60.30.30.30.1.3">λ</mi></mrow></msub></mrow></mrow><mo id="S2.E5.m1.61.61.61.31.31.31" stretchy="false">)</mo></mrow></mrow></mrow></mrow></mrow></mtd></mtr><mtr id="S2.E5.m1.127.127.4e"><mtd class="ltx_align_right" columnalign="right" id="S2.E5.m1.127.127.4f"><mrow id="S2.E5.m1.127.127.4.127.34.34"><mo id="S2.E5.m1.127.127.4.127.34.34a">−</mo><mrow id="S2.E5.m1.127.127.4.127.34.34.34"><mfrac id="S2.E5.m1.63.63.63.2.2.2"><mn id="S2.E5.m1.63.63.63.2.2.2.2">1</mn><mn id="S2.E5.m1.63.63.63.2.2.2.3">4</mn></mfrac><mo id="S2.E5.m1.127.127.4.127.34.34.34.2">⁢</mo><mfrac id="S2.E5.m1.64.64.64.3.3.3"><mn id="S2.E5.m1.64.64.64.3.3.3.3">1</mn><mrow id="S2.E5.m1.64.64.64.3.3.3.1.1"><mo id="S2.E5.m1.64.64.64.3.3.3.1.1.2" stretchy="false">(</mo><mrow id="S2.E5.m1.64.64.64.3.3.3.1.1.1"><mn id="S2.E5.m1.64.64.64.3.3.3.1.1.1.2">2</mn><mo id="S2.E5.m1.64.64.64.3.3.3.1.1.1.1">−</mo><mi id="S2.E5.m1.64.64.64.3.3.3.1.1.1.3">d</mi></mrow><mo id="S2.E5.m1.64.64.64.3.3.3.1.1.3" stretchy="false">)</mo></mrow></mfrac><mo id="S2.E5.m1.127.127.4.127.34.34.34.2a">⁢</mo><mrow id="S2.E5.m1.127.127.4.127.34.34.34.1.1"><mo id="S2.E5.m1.65.65.65.4.4.4" stretchy="false">[</mo><mrow id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1"><mrow id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.1"><mn id="S2.E5.m1.66.66.66.5.5.5">2</mn><mo id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.1.2">⁢</mo><msub id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.1.3"><mi id="S2.E5.m1.67.67.67.6.6.6">𝔤</mi><mrow id="S2.E5.m1.68.68.68.7.7.7.1"><mi id="S2.E5.m1.68.68.68.7.7.7.1.2">μ</mi><mo id="S2.E5.m1.68.68.68.7.7.7.1.1">⁢</mo><mi id="S2.E5.m1.68.68.68.7.7.7.1.3">ν</mi></mrow></msub><mo id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.1.2a" lspace="0em">⁢</mo><mrow id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.1.1"><msub id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.1.1.2"><mo id="S2.E5.m1.69.69.69.8.8.8" rspace="0em">∂</mo><mi id="S2.E5.m1.70.70.70.9.9.9.1">ρ</mi></msub><mrow id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.1.1.1.1"><mo id="S2.E5.m1.71.71.71.10.10.10" stretchy="false">(</mo><mrow id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.1.1.1.1.1"><msup id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.1.1.1.1.1.2"><mi id="S2.E5.m1.72.72.72.11.11.11">𝔤</mi><mrow id="S2.E5.m1.73.73.73.12.12.12.1"><mi id="S2.E5.m1.73.73.73.12.12.12.1.2">ρ</mi><mo id="S2.E5.m1.73.73.73.12.12.12.1.1">⁢</mo><mi id="S2.E5.m1.73.73.73.12.12.12.1.3">σ</mi></mrow></msup><mo id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.1.1.1.1.1.1">⁢</mo><msup id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.1.1.1.1.1.3"><mi id="S2.E5.m1.74.74.74.13.13.13">𝔤</mi><mrow id="S2.E5.m1.75.75.75.14.14.14.1"><mi id="S2.E5.m1.75.75.75.14.14.14.1.2">γ</mi><mo id="S2.E5.m1.75.75.75.14.14.14.1.1">⁢</mo><mi id="S2.E5.m1.75.75.75.14.14.14.1.3">λ</mi></mrow></msup><mo id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.1.1.1.1.1.1a" lspace="0em">⁢</mo><mrow id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.1.1.1.1.1.4"><msub id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.1.1.1.1.1.4.1"><mo id="S2.E5.m1.76.76.76.15.15.15" rspace="0em">∂</mo><mi id="S2.E5.m1.77.77.77.16.16.16.1">σ</mi></msub><msub id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.1.1.1.1.1.4.2"><mi id="S2.E5.m1.78.78.78.17.17.17">𝔤</mi><mrow id="S2.E5.m1.79.79.79.18.18.18.1"><mi id="S2.E5.m1.79.79.79.18.18.18.1.2">γ</mi><mo id="S2.E5.m1.79.79.79.18.18.18.1.1">⁢</mo><mi id="S2.E5.m1.79.79.79.18.18.18.1.3">λ</mi></mrow></msub></mrow></mrow><mo id="S2.E5.m1.80.80.80.19.19.19" stretchy="false">)</mo></mrow></mrow></mrow><mo id="S2.E5.m1.81.81.81.20.20.20">+</mo><mrow id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.2"><msup id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.2.2"><mi id="S2.E5.m1.82.82.82.21.21.21">𝔤</mi><mrow id="S2.E5.m1.83.83.83.22.22.22.1"><mi id="S2.E5.m1.83.83.83.22.22.22.1.2">ρ</mi><mo id="S2.E5.m1.83.83.83.22.22.22.1.1">⁢</mo><mi id="S2.E5.m1.83.83.83.22.22.22.1.3">σ</mi></mrow></msup><mo id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.2.1" lspace="0em">⁢</mo><mrow id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.2.3"><msub id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.2.3.1"><mo id="S2.E5.m1.84.84.84.23.23.23" rspace="0em">∂</mo><mi id="S2.E5.m1.85.85.85.24.24.24.1">μ</mi></msub><mrow id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.2.3.2"><msub id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.2.3.2.2"><mi id="S2.E5.m1.86.86.86.25.25.25">𝔤</mi><mrow id="S2.E5.m1.87.87.87.26.26.26.1"><mi id="S2.E5.m1.87.87.87.26.26.26.1.2">ρ</mi><mo id="S2.E5.m1.87.87.87.26.26.26.1.1">⁢</mo><mi id="S2.E5.m1.87.87.87.26.26.26.1.3">σ</mi></mrow></msub><mo id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.2.3.2.1">⁢</mo><msup id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.2.3.2.3"><mi id="S2.E5.m1.88.88.88.27.27.27">𝔤</mi><mrow id="S2.E5.m1.89.89.89.28.28.28.1"><mi id="S2.E5.m1.89.89.89.28.28.28.1.2">γ</mi><mo id="S2.E5.m1.89.89.89.28.28.28.1.1">⁢</mo><mi id="S2.E5.m1.89.89.89.28.28.28.1.3">λ</mi></mrow></msup><mo id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.2.3.2.1a" lspace="0em">⁢</mo><mrow id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.2.3.2.4"><msub id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.2.3.2.4.1"><mo id="S2.E5.m1.90.90.90.29.29.29" rspace="0em">∂</mo><mi id="S2.E5.m1.91.91.91.30.30.30.1">ν</mi></msub><msub id="S2.E5.m1.127.127.4.127.34.34.34.1.1.1.2.3.2.4.2"><mi id="S2.E5.m1.92.92.92.31.31.31">𝔤</mi><mrow id="S2.E5.m1.93.93.93.32.32.32.1"><mi id="S2.E5.m1.93.93.93.32.32.32.1.2">γ</mi><mo id="S2.E5.m1.93.93.93.32.32.32.1.1">⁢</mo><mi id="S2.E5.m1.93.93.93.32.32.32.1.3">λ</mi></mrow></msub></mrow></mrow></mrow></mrow></mrow><mo id="S2.E5.m1.94.94.94.33.33.33" stretchy="false">]</mo></mrow></mrow></mrow></mtd></mtr><mtr id="S2.E5.m1.127.127.4g"><mtd class="ltx_align_right" columnalign="right" id="S2.E5.m1.127.127.4h"><mrow id="S2.E5.m1.123.123.123.29.29"><mo id="S2.E5.m1.95.95.95.1.1.1">+</mo><mfrac id="S2.E5.m1.96.96.96.2.2.2"><mn id="S2.E5.m1.96.96.96.2.2.2.2">1</mn><mn id="S2.E5.m1.96.96.96.2.2.2.3">2</mn></mfrac><msub id="S2.E5.m1.123.123.123.29.29.30"><mo id="S2.E5.m1.97.97.97.3.3.3" lspace="0em" rspace="0em">∂</mo><mi id="S2.E5.m1.98.98.98.4.4.4.1">μ</mi></msub><msup id="S2.E5.m1.123.123.123.29.29.31"><mi id="S2.E5.m1.99.99.99.5.5.5">c</mi><mi id="S2.E5.m1.100.100.100.6.6.6.1">ρ</mi></msup><msub id="S2.E5.m1.123.123.123.29.29.32"><mo id="S2.E5.m1.101.101.101.7.7.7" lspace="0em" rspace="0em">∂</mo><mi id="S2.E5.m1.102.102.102.8.8.8.1">ν</mi></msub><msub id="S2.E5.m1.123.123.123.29.29.33"><mi id="S2.E5.m1.103.103.103.9.9.9">b</mi><mi id="S2.E5.m1.104.104.104.10.10.10.1">ρ</mi></msub><mo id="S2.E5.m1.105.105.105.11.11.11">+</mo><mfrac id="S2.E5.m1.106.106.106.12.12.12"><mn id="S2.E5.m1.106.106.106.12.12.12.2">1</mn><mn id="S2.E5.m1.106.106.106.12.12.12.3">2</mn></mfrac><msub id="S2.E5.m1.123.123.123.29.29.34"><mo id="S2.E5.m1.107.107.107.13.13.13" lspace="0em" rspace="0em">∂</mo><mi id="S2.E5.m1.108.108.108.14.14.14.1">μ</mi></msub><mrow id="S2.E5.m1.123.123.123.29.29.35"><mo id="S2.E5.m1.109.109.109.15.15.15" stretchy="false">(</mo><msup id="S2.E5.m1.123.123.123.29.29.35.1"><mi id="S2.E5.m1.110.110.110.16.16.16">c</mi><mi id="S2.E5.m1.111.111.111.17.17.17.1">ρ</mi></msup><msub id="S2.E5.m1.123.123.123.29.29.35.2"><mo id="S2.E5.m1.112.112.112.18.18.18" lspace="0em" rspace="0em">∂</mo><mi id="S2.E5.m1.113.113.113.19.19.19.1">ρ</mi></msub><msub id="S2.E5.m1.123.123.123.29.29.35.3"><mi id="S2.E5.m1.114.114.114.20.20.20">b</mi><mi id="S2.E5.m1.115.115.115.21.21.21.1">ν</mi></msub><mo id="S2.E5.m1.116.116.116.22.22.22" stretchy="false">)</mo></mrow><mo id="S2.E5.m1.117.117.117.23.23.23">+</mo><mrow id="S2.E5.m1.123.123.123.29.29.36"><mo id="S2.E5.m1.118.118.118.24.24.24" stretchy="false">(</mo><mi id="S2.E5.m1.119.119.119.25.25.25">μ</mi><mo id="S2.E5.m1.120.120.120.26.26.26" stretchy="false">↔</mo><mi id="S2.E5.m1.121.121.121.27.27.27">ν</mi><mo id="S2.E5.m1.122.122.122.28.28.28" stretchy="false">)</mo></mrow><mo id="S2.E5.m1.123.123.123.29.29.29" lspace="0em">.</mo></mrow></mtd></mtr></mtable><annotation encoding="application/x-tex" id="S2.E5.m1.127b">\partial_{\rho}(\mathfrak{g}^{\rho\sigma}\partial_{\sigma}\mathfrak{g}_{\mu\nu% })=\eta_{\mu\sigma}\partial_{\nu}\partial_{\rho}\mathfrak{g}^{\rho\sigma}-% \partial_{\rho}(\mathfrak{g}_{\mu\sigma}\partial_{\nu}\mathfrak{g}^{\rho\sigma% })\\ +\frac{1}{4}\partial_{\mu}\mathfrak{g}^{\rho\sigma}\partial_{\nu}\mathfrak{g}_% {\rho\sigma}+\frac{1}{2}\mathfrak{g}^{\rho\sigma}\mathfrak{g}^{\gamma\lambda}% \partial_{\gamma}\mathfrak{g}_{\mu\rho}(\partial_{\lambda}\mathfrak{g}_{\nu% \sigma}-\partial_{\sigma}\mathfrak{g}_{\nu\lambda})\\ -\frac{1}{4}\frac{1}{(2-d)}[2\mathfrak{g}_{\mu\nu}\partial_{\rho}(\mathfrak{g}% ^{\rho\sigma}\mathfrak{g}^{\gamma\lambda}\partial_{\sigma}\mathfrak{g}_{\gamma% \lambda})+\mathfrak{g}^{\rho\sigma}\partial_{\mu}\mathfrak{g}_{\rho\sigma}% \mathfrak{g}^{\gamma\lambda}\partial_{\nu}\mathfrak{g}_{\gamma\lambda}]\\ +\frac{1}{2}\partial_{\mu}c^{\rho}\partial_{\nu}b_{\rho}+\frac{1}{2}\partial_{% \mu}(c^{\rho}\partial_{\rho}b_{\nu})+(\mu\leftrightarrow\nu).</annotation><annotation encoding="application/x-llamapun" id="S2.E5.m1.127c">start_ROW start_CELL ∂ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT ( fraktur_g start_POSTSUPERSCRIPT italic_ρ italic_σ end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT fraktur_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ) = italic_η start_POSTSUBSCRIPT italic_μ italic_σ end_POSTSUBSCRIPT ∂ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ∂ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT fraktur_g start_POSTSUPERSCRIPT italic_ρ italic_σ end_POSTSUPERSCRIPT - ∂ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT ( fraktur_g start_POSTSUBSCRIPT italic_μ italic_σ end_POSTSUBSCRIPT ∂ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT fraktur_g start_POSTSUPERSCRIPT italic_ρ italic_σ end_POSTSUPERSCRIPT ) end_CELL end_ROW start_ROW start_CELL + divide start_ARG 1 end_ARG start_ARG 4 end_ARG ∂ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT fraktur_g start_POSTSUPERSCRIPT italic_ρ italic_σ end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT fraktur_g start_POSTSUBSCRIPT italic_ρ italic_σ end_POSTSUBSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG fraktur_g start_POSTSUPERSCRIPT italic_ρ italic_σ end_POSTSUPERSCRIPT fraktur_g start_POSTSUPERSCRIPT italic_γ italic_λ end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT fraktur_g start_POSTSUBSCRIPT italic_μ italic_ρ end_POSTSUBSCRIPT ( ∂ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT fraktur_g start_POSTSUBSCRIPT italic_ν italic_σ end_POSTSUBSCRIPT - ∂ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT fraktur_g start_POSTSUBSCRIPT italic_ν italic_λ end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL - divide start_ARG 1 end_ARG start_ARG 4 end_ARG divide start_ARG 1 end_ARG start_ARG ( 2 - italic_d ) end_ARG [ 2 fraktur_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ∂ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT ( fraktur_g start_POSTSUPERSCRIPT italic_ρ italic_σ end_POSTSUPERSCRIPT fraktur_g start_POSTSUPERSCRIPT italic_γ italic_λ end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT fraktur_g start_POSTSUBSCRIPT italic_γ italic_λ end_POSTSUBSCRIPT ) + fraktur_g start_POSTSUPERSCRIPT italic_ρ italic_σ end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT fraktur_g start_POSTSUBSCRIPT italic_ρ italic_σ end_POSTSUBSCRIPT fraktur_g start_POSTSUPERSCRIPT italic_γ italic_λ end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT fraktur_g start_POSTSUBSCRIPT italic_γ italic_λ end_POSTSUBSCRIPT ] end_CELL end_ROW start_ROW start_CELL + divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∂ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_c start_POSTSUPERSCRIPT italic_ρ end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∂ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ( italic_c start_POSTSUPERSCRIPT italic_ρ end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ) + ( italic_μ ↔ italic_ν ) . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p4.2">while the equations of motion for the ghosts are</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S5.EGx1"> <tbody id="S2.E6"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\partial_{\nu}(\mathfrak{g}^{\nu\rho}\partial_{\rho}c^{\mu})-% \partial_{\rho}(\partial_{\nu}\mathfrak{g}^{\mu\nu}c^{\rho})" class="ltx_Math" display="inline" id="S2.E6.m1.2"><semantics id="S2.E6.m1.2a"><mrow id="S2.E6.m1.2.2" xref="S2.E6.m1.2.2.cmml"><mrow id="S2.E6.m1.1.1.1" xref="S2.E6.m1.1.1.1.cmml"><msub id="S2.E6.m1.1.1.1.2" xref="S2.E6.m1.1.1.1.2.cmml"><mo id="S2.E6.m1.1.1.1.2.2" xref="S2.E6.m1.1.1.1.2.2.cmml">∂</mo><mi id="S2.E6.m1.1.1.1.2.3" xref="S2.E6.m1.1.1.1.2.3.cmml">ν</mi></msub><mrow id="S2.E6.m1.1.1.1.1.1" xref="S2.E6.m1.1.1.1.1.1.1.cmml"><mo id="S2.E6.m1.1.1.1.1.1.2" lspace="0em" stretchy="false" xref="S2.E6.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E6.m1.1.1.1.1.1.1" xref="S2.E6.m1.1.1.1.1.1.1.cmml"><msup id="S2.E6.m1.1.1.1.1.1.1.2" xref="S2.E6.m1.1.1.1.1.1.1.2.cmml"><mi id="S2.E6.m1.1.1.1.1.1.1.2.2" xref="S2.E6.m1.1.1.1.1.1.1.2.2.cmml">𝔤</mi><mrow id="S2.E6.m1.1.1.1.1.1.1.2.3" xref="S2.E6.m1.1.1.1.1.1.1.2.3.cmml"><mi id="S2.E6.m1.1.1.1.1.1.1.2.3.2" xref="S2.E6.m1.1.1.1.1.1.1.2.3.2.cmml">ν</mi><mo id="S2.E6.m1.1.1.1.1.1.1.2.3.1" xref="S2.E6.m1.1.1.1.1.1.1.2.3.1.cmml">⁢</mo><mi id="S2.E6.m1.1.1.1.1.1.1.2.3.3" xref="S2.E6.m1.1.1.1.1.1.1.2.3.3.cmml">ρ</mi></mrow></msup><mo id="S2.E6.m1.1.1.1.1.1.1.1" lspace="0em" xref="S2.E6.m1.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.E6.m1.1.1.1.1.1.1.3" xref="S2.E6.m1.1.1.1.1.1.1.3.cmml"><msub id="S2.E6.m1.1.1.1.1.1.1.3.1" xref="S2.E6.m1.1.1.1.1.1.1.3.1.cmml"><mo id="S2.E6.m1.1.1.1.1.1.1.3.1.2" rspace="0em" xref="S2.E6.m1.1.1.1.1.1.1.3.1.2.cmml">∂</mo><mi id="S2.E6.m1.1.1.1.1.1.1.3.1.3" xref="S2.E6.m1.1.1.1.1.1.1.3.1.3.cmml">ρ</mi></msub><msup id="S2.E6.m1.1.1.1.1.1.1.3.2" xref="S2.E6.m1.1.1.1.1.1.1.3.2.cmml"><mi id="S2.E6.m1.1.1.1.1.1.1.3.2.2" xref="S2.E6.m1.1.1.1.1.1.1.3.2.2.cmml">c</mi><mi id="S2.E6.m1.1.1.1.1.1.1.3.2.3" xref="S2.E6.m1.1.1.1.1.1.1.3.2.3.cmml">μ</mi></msup></mrow></mrow><mo id="S2.E6.m1.1.1.1.1.1.3" stretchy="false" xref="S2.E6.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.E6.m1.2.2.3" xref="S2.E6.m1.2.2.3.cmml">−</mo><mrow id="S2.E6.m1.2.2.2" 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id="S2.E6.m1.2c">\displaystyle\partial_{\nu}(\mathfrak{g}^{\nu\rho}\partial_{\rho}c^{\mu})-% \partial_{\rho}(\partial_{\nu}\mathfrak{g}^{\mu\nu}c^{\rho})</annotation><annotation encoding="application/x-llamapun" id="S2.E6.m1.2d">∂ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ( fraktur_g start_POSTSUPERSCRIPT italic_ν italic_ρ end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT italic_c start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT ) - ∂ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT ( ∂ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT fraktur_g start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT italic_c start_POSTSUPERSCRIPT italic_ρ end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S2.E6.m2.1"><semantics id="S2.E6.m2.1a"><mo id="S2.E6.m2.1.1" xref="S2.E6.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S2.E6.m2.1b"><eq id="S2.E6.m2.1.1.cmml" xref="S2.E6.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S2.E6.m2.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S2.E6.m2.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle 0," class="ltx_Math" display="inline" id="S2.E6.m3.1"><semantics id="S2.E6.m3.1a"><mrow id="S2.E6.m3.1.2.2"><mn id="S2.E6.m3.1.1" xref="S2.E6.m3.1.1.cmml">0</mn><mo id="S2.E6.m3.1.2.2.1">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E6.m3.1b"><cn id="S2.E6.m3.1.1.cmml" type="integer" xref="S2.E6.m3.1.1">0</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.E6.m3.1c">\displaystyle 0,</annotation><annotation encoding="application/x-llamapun" id="S2.E6.m3.1d">0 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6)</span></td> </tr></tbody> <tbody id="S2.E7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\partial_{\nu}(\mathfrak{g}^{\nu\rho}\partial_{\rho}b_{\mu})+% \partial_{\nu}\mathfrak{g}^{\nu\rho}\partial_{\mu}b_{\rho}" class="ltx_Math" display="inline" id="S2.E7.m1.1"><semantics id="S2.E7.m1.1a"><mrow id="S2.E7.m1.1.1" xref="S2.E7.m1.1.1.cmml"><mrow id="S2.E7.m1.1.1.1" xref="S2.E7.m1.1.1.1.cmml"><msub id="S2.E7.m1.1.1.1.2" xref="S2.E7.m1.1.1.1.2.cmml"><mo id="S2.E7.m1.1.1.1.2.2" xref="S2.E7.m1.1.1.1.2.2.cmml">∂</mo><mi id="S2.E7.m1.1.1.1.2.3" xref="S2.E7.m1.1.1.1.2.3.cmml">ν</mi></msub><mrow id="S2.E7.m1.1.1.1.1.1" xref="S2.E7.m1.1.1.1.1.1.1.cmml"><mo id="S2.E7.m1.1.1.1.1.1.2" lspace="0em" stretchy="false" xref="S2.E7.m1.1.1.1.1.1.1.cmml">(</mo><mrow 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id="S2.E7.m1.1.1.3.2.2.3.1" xref="S2.E7.m1.1.1.3.2.2.3.1.cmml">⁢</mo><mi id="S2.E7.m1.1.1.3.2.2.3.3" xref="S2.E7.m1.1.1.3.2.2.3.3.cmml">ρ</mi></mrow></msup><mo id="S2.E7.m1.1.1.3.2.1" lspace="0em" xref="S2.E7.m1.1.1.3.2.1.cmml">⁢</mo><mrow id="S2.E7.m1.1.1.3.2.3" xref="S2.E7.m1.1.1.3.2.3.cmml"><msub id="S2.E7.m1.1.1.3.2.3.1" xref="S2.E7.m1.1.1.3.2.3.1.cmml"><mo id="S2.E7.m1.1.1.3.2.3.1.2" rspace="0em" xref="S2.E7.m1.1.1.3.2.3.1.2.cmml">∂</mo><mi id="S2.E7.m1.1.1.3.2.3.1.3" xref="S2.E7.m1.1.1.3.2.3.1.3.cmml">μ</mi></msub><msub id="S2.E7.m1.1.1.3.2.3.2" xref="S2.E7.m1.1.1.3.2.3.2.cmml"><mi id="S2.E7.m1.1.1.3.2.3.2.2" xref="S2.E7.m1.1.1.3.2.3.2.2.cmml">b</mi><mi id="S2.E7.m1.1.1.3.2.3.2.3" xref="S2.E7.m1.1.1.3.2.3.2.3.cmml">ρ</mi></msub></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E7.m1.1b"><apply id="S2.E7.m1.1.1.cmml" xref="S2.E7.m1.1.1"><plus id="S2.E7.m1.1.1.2.cmml" xref="S2.E7.m1.1.1.2"></plus><apply id="S2.E7.m1.1.1.1.cmml" xref="S2.E7.m1.1.1.1"><apply id="S2.E7.m1.1.1.1.2.cmml" xref="S2.E7.m1.1.1.1.2"><csymbol cd="ambiguous" id="S2.E7.m1.1.1.1.2.1.cmml" xref="S2.E7.m1.1.1.1.2">subscript</csymbol><partialdiff id="S2.E7.m1.1.1.1.2.2.cmml" xref="S2.E7.m1.1.1.1.2.2"></partialdiff><ci id="S2.E7.m1.1.1.1.2.3.cmml" xref="S2.E7.m1.1.1.1.2.3">𝜈</ci></apply><apply id="S2.E7.m1.1.1.1.1.1.1.cmml" xref="S2.E7.m1.1.1.1.1.1"><times id="S2.E7.m1.1.1.1.1.1.1.1.cmml" xref="S2.E7.m1.1.1.1.1.1.1.1"></times><apply id="S2.E7.m1.1.1.1.1.1.1.2.cmml" xref="S2.E7.m1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.E7.m1.1.1.1.1.1.1.2.1.cmml" xref="S2.E7.m1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S2.E7.m1.1.1.1.1.1.1.2.2.cmml" xref="S2.E7.m1.1.1.1.1.1.1.2.2">𝔤</ci><apply id="S2.E7.m1.1.1.1.1.1.1.2.3.cmml" xref="S2.E7.m1.1.1.1.1.1.1.2.3"><times id="S2.E7.m1.1.1.1.1.1.1.2.3.1.cmml" xref="S2.E7.m1.1.1.1.1.1.1.2.3.1"></times><ci id="S2.E7.m1.1.1.1.1.1.1.2.3.2.cmml" xref="S2.E7.m1.1.1.1.1.1.1.2.3.2">𝜈</ci><ci 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id="S2.E7.m1.1.1.3.1.cmml" xref="S2.E7.m1.1.1.3.1"><csymbol cd="ambiguous" id="S2.E7.m1.1.1.3.1.1.cmml" xref="S2.E7.m1.1.1.3.1">subscript</csymbol><partialdiff id="S2.E7.m1.1.1.3.1.2.cmml" xref="S2.E7.m1.1.1.3.1.2"></partialdiff><ci id="S2.E7.m1.1.1.3.1.3.cmml" xref="S2.E7.m1.1.1.3.1.3">𝜈</ci></apply><apply id="S2.E7.m1.1.1.3.2.cmml" xref="S2.E7.m1.1.1.3.2"><times id="S2.E7.m1.1.1.3.2.1.cmml" xref="S2.E7.m1.1.1.3.2.1"></times><apply id="S2.E7.m1.1.1.3.2.2.cmml" xref="S2.E7.m1.1.1.3.2.2"><csymbol cd="ambiguous" id="S2.E7.m1.1.1.3.2.2.1.cmml" xref="S2.E7.m1.1.1.3.2.2">superscript</csymbol><ci id="S2.E7.m1.1.1.3.2.2.2.cmml" xref="S2.E7.m1.1.1.3.2.2.2">𝔤</ci><apply id="S2.E7.m1.1.1.3.2.2.3.cmml" xref="S2.E7.m1.1.1.3.2.2.3"><times id="S2.E7.m1.1.1.3.2.2.3.1.cmml" xref="S2.E7.m1.1.1.3.2.2.3.1"></times><ci id="S2.E7.m1.1.1.3.2.2.3.2.cmml" xref="S2.E7.m1.1.1.3.2.2.3.2">𝜈</ci><ci id="S2.E7.m1.1.1.3.2.2.3.3.cmml" xref="S2.E7.m1.1.1.3.2.2.3.3">𝜌</ci></apply></apply><apply id="S2.E7.m1.1.1.3.2.3.cmml" xref="S2.E7.m1.1.1.3.2.3"><apply id="S2.E7.m1.1.1.3.2.3.1.cmml" xref="S2.E7.m1.1.1.3.2.3.1"><csymbol cd="ambiguous" id="S2.E7.m1.1.1.3.2.3.1.1.cmml" xref="S2.E7.m1.1.1.3.2.3.1">subscript</csymbol><partialdiff id="S2.E7.m1.1.1.3.2.3.1.2.cmml" xref="S2.E7.m1.1.1.3.2.3.1.2"></partialdiff><ci id="S2.E7.m1.1.1.3.2.3.1.3.cmml" xref="S2.E7.m1.1.1.3.2.3.1.3">𝜇</ci></apply><apply id="S2.E7.m1.1.1.3.2.3.2.cmml" xref="S2.E7.m1.1.1.3.2.3.2"><csymbol cd="ambiguous" id="S2.E7.m1.1.1.3.2.3.2.1.cmml" xref="S2.E7.m1.1.1.3.2.3.2">subscript</csymbol><ci id="S2.E7.m1.1.1.3.2.3.2.2.cmml" xref="S2.E7.m1.1.1.3.2.3.2.2">𝑏</ci><ci id="S2.E7.m1.1.1.3.2.3.2.3.cmml" xref="S2.E7.m1.1.1.3.2.3.2.3">𝜌</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E7.m1.1c">\displaystyle\partial_{\nu}(\mathfrak{g}^{\nu\rho}\partial_{\rho}b_{\mu})+% \partial_{\nu}\mathfrak{g}^{\nu\rho}\partial_{\mu}b_{\rho}</annotation><annotation encoding="application/x-llamapun" id="S2.E7.m1.1d">∂ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ( fraktur_g start_POSTSUPERSCRIPT italic_ν italic_ρ end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ) + ∂ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT fraktur_g start_POSTSUPERSCRIPT italic_ν italic_ρ end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S2.E7.m2.1"><semantics id="S2.E7.m2.1a"><mo id="S2.E7.m2.1.1" xref="S2.E7.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S2.E7.m2.1b"><eq id="S2.E7.m2.1.1.cmml" xref="S2.E7.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S2.E7.m2.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S2.E7.m2.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle 0." class="ltx_Math" display="inline" id="S2.E7.m3.1"><semantics id="S2.E7.m3.1a"><mrow id="S2.E7.m3.1.2.2"><mn id="S2.E7.m3.1.1" xref="S2.E7.m3.1.1.cmml">0</mn><mo id="S2.E7.m3.1.2.2.1" lspace="0em">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E7.m3.1b"><cn id="S2.E7.m3.1.1.cmml" type="integer" xref="S2.E7.m3.1.1">0</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.E7.m3.1c">\displaystyle 0.</annotation><annotation encoding="application/x-llamapun" id="S2.E7.m3.1d">0 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(7)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.p5"> <p class="ltx_p" id="S2.p5.4">We can use them to conveniently extract scattering trees through the gravitational perturbiner <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib47" title="">Gomez:2021shh </a></cite>. The so-called multiparticle expansions of the fields <math alttext="\mathfrak{g}_{\mu\nu}" class="ltx_Math" display="inline" id="S2.p5.1.m1.1"><semantics id="S2.p5.1.m1.1a"><msub id="S2.p5.1.m1.1.1" xref="S2.p5.1.m1.1.1.cmml"><mi id="S2.p5.1.m1.1.1.2" xref="S2.p5.1.m1.1.1.2.cmml">𝔤</mi><mrow id="S2.p5.1.m1.1.1.3" xref="S2.p5.1.m1.1.1.3.cmml"><mi id="S2.p5.1.m1.1.1.3.2" xref="S2.p5.1.m1.1.1.3.2.cmml">μ</mi><mo id="S2.p5.1.m1.1.1.3.1" xref="S2.p5.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S2.p5.1.m1.1.1.3.3" xref="S2.p5.1.m1.1.1.3.3.cmml">ν</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.p5.1.m1.1b"><apply id="S2.p5.1.m1.1.1.cmml" xref="S2.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S2.p5.1.m1.1.1.1.cmml" xref="S2.p5.1.m1.1.1">subscript</csymbol><ci id="S2.p5.1.m1.1.1.2.cmml" xref="S2.p5.1.m1.1.1.2">𝔤</ci><apply id="S2.p5.1.m1.1.1.3.cmml" xref="S2.p5.1.m1.1.1.3"><times id="S2.p5.1.m1.1.1.3.1.cmml" xref="S2.p5.1.m1.1.1.3.1"></times><ci id="S2.p5.1.m1.1.1.3.2.cmml" xref="S2.p5.1.m1.1.1.3.2">𝜇</ci><ci id="S2.p5.1.m1.1.1.3.3.cmml" xref="S2.p5.1.m1.1.1.3.3">𝜈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.1.m1.1c">\mathfrak{g}_{\mu\nu}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.1.m1.1d">fraktur_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\mathfrak{g}^{\mu\nu}" class="ltx_Math" display="inline" id="S2.p5.2.m2.1"><semantics id="S2.p5.2.m2.1a"><msup id="S2.p5.2.m2.1.1" xref="S2.p5.2.m2.1.1.cmml"><mi id="S2.p5.2.m2.1.1.2" xref="S2.p5.2.m2.1.1.2.cmml">𝔤</mi><mrow id="S2.p5.2.m2.1.1.3" xref="S2.p5.2.m2.1.1.3.cmml"><mi id="S2.p5.2.m2.1.1.3.2" xref="S2.p5.2.m2.1.1.3.2.cmml">μ</mi><mo id="S2.p5.2.m2.1.1.3.1" xref="S2.p5.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S2.p5.2.m2.1.1.3.3" xref="S2.p5.2.m2.1.1.3.3.cmml">ν</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.p5.2.m2.1b"><apply id="S2.p5.2.m2.1.1.cmml" xref="S2.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S2.p5.2.m2.1.1.1.cmml" xref="S2.p5.2.m2.1.1">superscript</csymbol><ci id="S2.p5.2.m2.1.1.2.cmml" xref="S2.p5.2.m2.1.1.2">𝔤</ci><apply id="S2.p5.2.m2.1.1.3.cmml" xref="S2.p5.2.m2.1.1.3"><times id="S2.p5.2.m2.1.1.3.1.cmml" xref="S2.p5.2.m2.1.1.3.1"></times><ci id="S2.p5.2.m2.1.1.3.2.cmml" xref="S2.p5.2.m2.1.1.3.2">𝜇</ci><ci id="S2.p5.2.m2.1.1.3.3.cmml" xref="S2.p5.2.m2.1.1.3.3">𝜈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.2.m2.1c">\mathfrak{g}^{\mu\nu}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.2.m2.1d">fraktur_g start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="b_{\mu}" class="ltx_Math" display="inline" id="S2.p5.3.m3.1"><semantics id="S2.p5.3.m3.1a"><msub id="S2.p5.3.m3.1.1" xref="S2.p5.3.m3.1.1.cmml"><mi id="S2.p5.3.m3.1.1.2" xref="S2.p5.3.m3.1.1.2.cmml">b</mi><mi id="S2.p5.3.m3.1.1.3" xref="S2.p5.3.m3.1.1.3.cmml">μ</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p5.3.m3.1b"><apply id="S2.p5.3.m3.1.1.cmml" xref="S2.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S2.p5.3.m3.1.1.1.cmml" xref="S2.p5.3.m3.1.1">subscript</csymbol><ci id="S2.p5.3.m3.1.1.2.cmml" xref="S2.p5.3.m3.1.1.2">𝑏</ci><ci id="S2.p5.3.m3.1.1.3.cmml" xref="S2.p5.3.m3.1.1.3">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.3.m3.1c">b_{\mu}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.3.m3.1d">italic_b start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="c^{\mu}" class="ltx_Math" display="inline" id="S2.p5.4.m4.1"><semantics id="S2.p5.4.m4.1a"><msup id="S2.p5.4.m4.1.1" xref="S2.p5.4.m4.1.1.cmml"><mi id="S2.p5.4.m4.1.1.2" xref="S2.p5.4.m4.1.1.2.cmml">c</mi><mi id="S2.p5.4.m4.1.1.3" xref="S2.p5.4.m4.1.1.3.cmml">μ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.p5.4.m4.1b"><apply id="S2.p5.4.m4.1.1.cmml" xref="S2.p5.4.m4.1.1"><csymbol cd="ambiguous" id="S2.p5.4.m4.1.1.1.cmml" xref="S2.p5.4.m4.1.1">superscript</csymbol><ci id="S2.p5.4.m4.1.1.2.cmml" xref="S2.p5.4.m4.1.1.2">𝑐</ci><ci id="S2.p5.4.m4.1.1.3.cmml" xref="S2.p5.4.m4.1.1.3">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.4.m4.1c">c^{\mu}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.4.m4.1d">italic_c start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT</annotation></semantics></math> can be cast as</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S5.EGx2"> <tbody id="S2.E8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathfrak{g}_{\mu\nu}(x)" class="ltx_Math" display="inline" id="S2.E8.m1.1"><semantics id="S2.E8.m1.1a"><mrow id="S2.E8.m1.1.2" xref="S2.E8.m1.1.2.cmml"><msub id="S2.E8.m1.1.2.2" xref="S2.E8.m1.1.2.2.cmml"><mi id="S2.E8.m1.1.2.2.2" xref="S2.E8.m1.1.2.2.2.cmml">𝔤</mi><mrow id="S2.E8.m1.1.2.2.3" xref="S2.E8.m1.1.2.2.3.cmml"><mi id="S2.E8.m1.1.2.2.3.2" xref="S2.E8.m1.1.2.2.3.2.cmml">μ</mi><mo id="S2.E8.m1.1.2.2.3.1" xref="S2.E8.m1.1.2.2.3.1.cmml">⁢</mo><mi id="S2.E8.m1.1.2.2.3.3" xref="S2.E8.m1.1.2.2.3.3.cmml">ν</mi></mrow></msub><mo id="S2.E8.m1.1.2.1" xref="S2.E8.m1.1.2.1.cmml">⁢</mo><mrow id="S2.E8.m1.1.2.3.2" xref="S2.E8.m1.1.2.cmml"><mo id="S2.E8.m1.1.2.3.2.1" stretchy="false" xref="S2.E8.m1.1.2.cmml">(</mo><mi id="S2.E8.m1.1.1" xref="S2.E8.m1.1.1.cmml">x</mi><mo id="S2.E8.m1.1.2.3.2.2" stretchy="false" xref="S2.E8.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E8.m1.1b"><apply id="S2.E8.m1.1.2.cmml" xref="S2.E8.m1.1.2"><times id="S2.E8.m1.1.2.1.cmml" xref="S2.E8.m1.1.2.1"></times><apply id="S2.E8.m1.1.2.2.cmml" xref="S2.E8.m1.1.2.2"><csymbol cd="ambiguous" id="S2.E8.m1.1.2.2.1.cmml" xref="S2.E8.m1.1.2.2">subscript</csymbol><ci id="S2.E8.m1.1.2.2.2.cmml" xref="S2.E8.m1.1.2.2.2">𝔤</ci><apply id="S2.E8.m1.1.2.2.3.cmml" xref="S2.E8.m1.1.2.2.3"><times id="S2.E8.m1.1.2.2.3.1.cmml" xref="S2.E8.m1.1.2.2.3.1"></times><ci id="S2.E8.m1.1.2.2.3.2.cmml" xref="S2.E8.m1.1.2.2.3.2">𝜇</ci><ci id="S2.E8.m1.1.2.2.3.3.cmml" xref="S2.E8.m1.1.2.2.3.3">𝜈</ci></apply></apply><ci id="S2.E8.m1.1.1.cmml" xref="S2.E8.m1.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E8.m1.1c">\displaystyle\mathfrak{g}_{\mu\nu}(x)</annotation><annotation encoding="application/x-llamapun" id="S2.E8.m1.1d">fraktur_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S2.E8.m2.1"><semantics id="S2.E8.m2.1a"><mo id="S2.E8.m2.1.1" xref="S2.E8.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S2.E8.m2.1b"><eq id="S2.E8.m2.1.1.cmml" xref="S2.E8.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S2.E8.m2.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S2.E8.m2.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\eta_{\mu\nu}+\sum_{P}H_{P\mu\nu}e^{ik_{P}\cdot x}," class="ltx_Math" display="inline" id="S2.E8.m3.1"><semantics id="S2.E8.m3.1a"><mrow id="S2.E8.m3.1.1.1" xref="S2.E8.m3.1.1.1.1.cmml"><mrow id="S2.E8.m3.1.1.1.1" xref="S2.E8.m3.1.1.1.1.cmml"><msub id="S2.E8.m3.1.1.1.1.2" xref="S2.E8.m3.1.1.1.1.2.cmml"><mi id="S2.E8.m3.1.1.1.1.2.2" xref="S2.E8.m3.1.1.1.1.2.2.cmml">η</mi><mrow id="S2.E8.m3.1.1.1.1.2.3" xref="S2.E8.m3.1.1.1.1.2.3.cmml"><mi id="S2.E8.m3.1.1.1.1.2.3.2" xref="S2.E8.m3.1.1.1.1.2.3.2.cmml">μ</mi><mo id="S2.E8.m3.1.1.1.1.2.3.1" xref="S2.E8.m3.1.1.1.1.2.3.1.cmml">⁢</mo><mi id="S2.E8.m3.1.1.1.1.2.3.3" 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xref="S2.E8.m3.1.1.1.1.3.2.3.3.2.2">𝑖</ci><apply id="S2.E8.m3.1.1.1.1.3.2.3.3.2.3.cmml" xref="S2.E8.m3.1.1.1.1.3.2.3.3.2.3"><csymbol cd="ambiguous" id="S2.E8.m3.1.1.1.1.3.2.3.3.2.3.1.cmml" xref="S2.E8.m3.1.1.1.1.3.2.3.3.2.3">subscript</csymbol><ci id="S2.E8.m3.1.1.1.1.3.2.3.3.2.3.2.cmml" xref="S2.E8.m3.1.1.1.1.3.2.3.3.2.3.2">𝑘</ci><ci id="S2.E8.m3.1.1.1.1.3.2.3.3.2.3.3.cmml" xref="S2.E8.m3.1.1.1.1.3.2.3.3.2.3.3">𝑃</ci></apply></apply><ci id="S2.E8.m3.1.1.1.1.3.2.3.3.3.cmml" xref="S2.E8.m3.1.1.1.1.3.2.3.3.3">𝑥</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E8.m3.1c">\displaystyle\eta_{\mu\nu}+\sum_{P}H_{P\mu\nu}e^{ik_{P}\cdot x},</annotation><annotation encoding="application/x-llamapun" id="S2.E8.m3.1d">italic_η start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT italic_P italic_μ italic_ν end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_i italic_k start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ⋅ italic_x end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(8)</span></td> </tr></tbody> <tbody id="S2.E9"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathfrak{g}^{\mu\nu}(x)" class="ltx_Math" display="inline" id="S2.E9.m1.1"><semantics id="S2.E9.m1.1a"><mrow id="S2.E9.m1.1.2" xref="S2.E9.m1.1.2.cmml"><msup id="S2.E9.m1.1.2.2" xref="S2.E9.m1.1.2.2.cmml"><mi id="S2.E9.m1.1.2.2.2" xref="S2.E9.m1.1.2.2.2.cmml">𝔤</mi><mrow id="S2.E9.m1.1.2.2.3" xref="S2.E9.m1.1.2.2.3.cmml"><mi id="S2.E9.m1.1.2.2.3.2" xref="S2.E9.m1.1.2.2.3.2.cmml">μ</mi><mo id="S2.E9.m1.1.2.2.3.1" 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xref="S2.E9.m1.1.2.2.3.2">𝜇</ci><ci id="S2.E9.m1.1.2.2.3.3.cmml" xref="S2.E9.m1.1.2.2.3.3">𝜈</ci></apply></apply><ci id="S2.E9.m1.1.1.cmml" xref="S2.E9.m1.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E9.m1.1c">\displaystyle\mathfrak{g}^{\mu\nu}(x)</annotation><annotation encoding="application/x-llamapun" id="S2.E9.m1.1d">fraktur_g start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT ( italic_x )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S2.E9.m2.1"><semantics id="S2.E9.m2.1a"><mo id="S2.E9.m2.1.1" xref="S2.E9.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S2.E9.m2.1b"><eq id="S2.E9.m2.1.1.cmml" xref="S2.E9.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S2.E9.m2.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" 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xref="S2.E9.m3.1.1.1.1.3.2.2.3.1"></times><ci id="S2.E9.m3.1.1.1.1.3.2.2.3.2.cmml" xref="S2.E9.m3.1.1.1.1.3.2.2.3.2">𝜇</ci><ci id="S2.E9.m3.1.1.1.1.3.2.2.3.3.cmml" xref="S2.E9.m3.1.1.1.1.3.2.2.3.3">𝜈</ci></apply></apply><apply id="S2.E9.m3.1.1.1.1.3.2.3.cmml" xref="S2.E9.m3.1.1.1.1.3.2.3"><csymbol cd="ambiguous" id="S2.E9.m3.1.1.1.1.3.2.3.1.cmml" xref="S2.E9.m3.1.1.1.1.3.2.3">superscript</csymbol><ci id="S2.E9.m3.1.1.1.1.3.2.3.2.cmml" xref="S2.E9.m3.1.1.1.1.3.2.3.2">𝑒</ci><apply id="S2.E9.m3.1.1.1.1.3.2.3.3.cmml" xref="S2.E9.m3.1.1.1.1.3.2.3.3"><ci id="S2.E9.m3.1.1.1.1.3.2.3.3.1.cmml" xref="S2.E9.m3.1.1.1.1.3.2.3.3.1">⋅</ci><apply id="S2.E9.m3.1.1.1.1.3.2.3.3.2.cmml" xref="S2.E9.m3.1.1.1.1.3.2.3.3.2"><times id="S2.E9.m3.1.1.1.1.3.2.3.3.2.1.cmml" xref="S2.E9.m3.1.1.1.1.3.2.3.3.2.1"></times><ci id="S2.E9.m3.1.1.1.1.3.2.3.3.2.2.cmml" xref="S2.E9.m3.1.1.1.1.3.2.3.3.2.2">𝑖</ci><apply id="S2.E9.m3.1.1.1.1.3.2.3.3.2.3.cmml" xref="S2.E9.m3.1.1.1.1.3.2.3.3.2.3"><csymbol cd="ambiguous" id="S2.E9.m3.1.1.1.1.3.2.3.3.2.3.1.cmml" xref="S2.E9.m3.1.1.1.1.3.2.3.3.2.3">subscript</csymbol><ci id="S2.E9.m3.1.1.1.1.3.2.3.3.2.3.2.cmml" xref="S2.E9.m3.1.1.1.1.3.2.3.3.2.3.2">𝑘</ci><ci id="S2.E9.m3.1.1.1.1.3.2.3.3.2.3.3.cmml" xref="S2.E9.m3.1.1.1.1.3.2.3.3.2.3.3">𝑃</ci></apply></apply><ci id="S2.E9.m3.1.1.1.1.3.2.3.3.3.cmml" xref="S2.E9.m3.1.1.1.1.3.2.3.3.3">𝑥</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E9.m3.1c">\displaystyle\eta^{\mu\nu}-\sum_{P}I_{P}^{\mu\nu}e^{ik_{P}\cdot x},</annotation><annotation encoding="application/x-llamapun" id="S2.E9.m3.1d">italic_η start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT - ∑ start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT italic_i italic_k start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ⋅ italic_x end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(9)</span></td> </tr></tbody> <tbody id="S2.E10"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle c^{\mu}(x)" class="ltx_Math" display="inline" id="S2.E10.m1.1"><semantics id="S2.E10.m1.1a"><mrow id="S2.E10.m1.1.2" xref="S2.E10.m1.1.2.cmml"><msup id="S2.E10.m1.1.2.2" xref="S2.E10.m1.1.2.2.cmml"><mi id="S2.E10.m1.1.2.2.2" xref="S2.E10.m1.1.2.2.2.cmml">c</mi><mi id="S2.E10.m1.1.2.2.3" xref="S2.E10.m1.1.2.2.3.cmml">μ</mi></msup><mo id="S2.E10.m1.1.2.1" xref="S2.E10.m1.1.2.1.cmml">⁢</mo><mrow id="S2.E10.m1.1.2.3.2" xref="S2.E10.m1.1.2.cmml"><mo id="S2.E10.m1.1.2.3.2.1" stretchy="false" xref="S2.E10.m1.1.2.cmml">(</mo><mi id="S2.E10.m1.1.1" xref="S2.E10.m1.1.1.cmml">x</mi><mo id="S2.E10.m1.1.2.3.2.2" stretchy="false" xref="S2.E10.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E10.m1.1b"><apply id="S2.E10.m1.1.2.cmml" xref="S2.E10.m1.1.2"><times id="S2.E10.m1.1.2.1.cmml" xref="S2.E10.m1.1.2.1"></times><apply id="S2.E10.m1.1.2.2.cmml" xref="S2.E10.m1.1.2.2"><csymbol cd="ambiguous" id="S2.E10.m1.1.2.2.1.cmml" xref="S2.E10.m1.1.2.2">superscript</csymbol><ci id="S2.E10.m1.1.2.2.2.cmml" xref="S2.E10.m1.1.2.2.2">𝑐</ci><ci id="S2.E10.m1.1.2.2.3.cmml" xref="S2.E10.m1.1.2.2.3">𝜇</ci></apply><ci id="S2.E10.m1.1.1.cmml" xref="S2.E10.m1.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E10.m1.1c">\displaystyle c^{\mu}(x)</annotation><annotation encoding="application/x-llamapun" id="S2.E10.m1.1d">italic_c start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT ( italic_x )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S2.E10.m2.1"><semantics id="S2.E10.m2.1a"><mo id="S2.E10.m2.1.1" xref="S2.E10.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S2.E10.m2.1b"><eq id="S2.E10.m2.1.1.cmml" xref="S2.E10.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S2.E10.m2.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S2.E10.m2.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\sum_{P}C_{P}^{\mu}e^{ik_{P}\cdot x}," class="ltx_Math" display="inline" id="S2.E10.m3.1"><semantics id="S2.E10.m3.1a"><mrow id="S2.E10.m3.1.1.1" xref="S2.E10.m3.1.1.1.1.cmml"><mrow id="S2.E10.m3.1.1.1.1" xref="S2.E10.m3.1.1.1.1.cmml"><mstyle displaystyle="true" id="S2.E10.m3.1.1.1.1.1" xref="S2.E10.m3.1.1.1.1.1.cmml"><munder id="S2.E10.m3.1.1.1.1.1a" xref="S2.E10.m3.1.1.1.1.1.cmml"><mo id="S2.E10.m3.1.1.1.1.1.2" movablelimits="false" xref="S2.E10.m3.1.1.1.1.1.2.cmml">∑</mo><mi id="S2.E10.m3.1.1.1.1.1.3" xref="S2.E10.m3.1.1.1.1.1.3.cmml">P</mi></munder></mstyle><mrow id="S2.E10.m3.1.1.1.1.2" xref="S2.E10.m3.1.1.1.1.2.cmml"><msubsup id="S2.E10.m3.1.1.1.1.2.2" xref="S2.E10.m3.1.1.1.1.2.2.cmml"><mi id="S2.E10.m3.1.1.1.1.2.2.2.2" xref="S2.E10.m3.1.1.1.1.2.2.2.2.cmml">C</mi><mi id="S2.E10.m3.1.1.1.1.2.2.2.3" xref="S2.E10.m3.1.1.1.1.2.2.2.3.cmml">P</mi><mi id="S2.E10.m3.1.1.1.1.2.2.3" xref="S2.E10.m3.1.1.1.1.2.2.3.cmml">μ</mi></msubsup><mo id="S2.E10.m3.1.1.1.1.2.1" xref="S2.E10.m3.1.1.1.1.2.1.cmml">⁢</mo><msup id="S2.E10.m3.1.1.1.1.2.3" xref="S2.E10.m3.1.1.1.1.2.3.cmml"><mi id="S2.E10.m3.1.1.1.1.2.3.2" xref="S2.E10.m3.1.1.1.1.2.3.2.cmml">e</mi><mrow id="S2.E10.m3.1.1.1.1.2.3.3" xref="S2.E10.m3.1.1.1.1.2.3.3.cmml"><mrow id="S2.E10.m3.1.1.1.1.2.3.3.2" xref="S2.E10.m3.1.1.1.1.2.3.3.2.cmml"><mi id="S2.E10.m3.1.1.1.1.2.3.3.2.2" xref="S2.E10.m3.1.1.1.1.2.3.3.2.2.cmml">i</mi><mo id="S2.E10.m3.1.1.1.1.2.3.3.2.1" xref="S2.E10.m3.1.1.1.1.2.3.3.2.1.cmml">⁢</mo><msub id="S2.E10.m3.1.1.1.1.2.3.3.2.3" xref="S2.E10.m3.1.1.1.1.2.3.3.2.3.cmml"><mi id="S2.E10.m3.1.1.1.1.2.3.3.2.3.2" xref="S2.E10.m3.1.1.1.1.2.3.3.2.3.2.cmml">k</mi><mi id="S2.E10.m3.1.1.1.1.2.3.3.2.3.3" xref="S2.E10.m3.1.1.1.1.2.3.3.2.3.3.cmml">P</mi></msub></mrow><mo id="S2.E10.m3.1.1.1.1.2.3.3.1" lspace="0.222em" rspace="0.222em" xref="S2.E10.m3.1.1.1.1.2.3.3.1.cmml">⋅</mo><mi id="S2.E10.m3.1.1.1.1.2.3.3.3" xref="S2.E10.m3.1.1.1.1.2.3.3.3.cmml">x</mi></mrow></msup></mrow></mrow><mo id="S2.E10.m3.1.1.1.2" xref="S2.E10.m3.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E10.m3.1b"><apply id="S2.E10.m3.1.1.1.1.cmml" xref="S2.E10.m3.1.1.1"><apply id="S2.E10.m3.1.1.1.1.1.cmml" xref="S2.E10.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.E10.m3.1.1.1.1.1.1.cmml" xref="S2.E10.m3.1.1.1.1.1">subscript</csymbol><sum id="S2.E10.m3.1.1.1.1.1.2.cmml" xref="S2.E10.m3.1.1.1.1.1.2"></sum><ci id="S2.E10.m3.1.1.1.1.1.3.cmml" xref="S2.E10.m3.1.1.1.1.1.3">𝑃</ci></apply><apply id="S2.E10.m3.1.1.1.1.2.cmml" xref="S2.E10.m3.1.1.1.1.2"><times id="S2.E10.m3.1.1.1.1.2.1.cmml" xref="S2.E10.m3.1.1.1.1.2.1"></times><apply id="S2.E10.m3.1.1.1.1.2.2.cmml" xref="S2.E10.m3.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S2.E10.m3.1.1.1.1.2.2.1.cmml" xref="S2.E10.m3.1.1.1.1.2.2">superscript</csymbol><apply id="S2.E10.m3.1.1.1.1.2.2.2.cmml" xref="S2.E10.m3.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S2.E10.m3.1.1.1.1.2.2.2.1.cmml" xref="S2.E10.m3.1.1.1.1.2.2">subscript</csymbol><ci id="S2.E10.m3.1.1.1.1.2.2.2.2.cmml" xref="S2.E10.m3.1.1.1.1.2.2.2.2">𝐶</ci><ci id="S2.E10.m3.1.1.1.1.2.2.2.3.cmml" xref="S2.E10.m3.1.1.1.1.2.2.2.3">𝑃</ci></apply><ci id="S2.E10.m3.1.1.1.1.2.2.3.cmml" xref="S2.E10.m3.1.1.1.1.2.2.3">𝜇</ci></apply><apply id="S2.E10.m3.1.1.1.1.2.3.cmml" xref="S2.E10.m3.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S2.E10.m3.1.1.1.1.2.3.1.cmml" xref="S2.E10.m3.1.1.1.1.2.3">superscript</csymbol><ci id="S2.E10.m3.1.1.1.1.2.3.2.cmml" xref="S2.E10.m3.1.1.1.1.2.3.2">𝑒</ci><apply id="S2.E10.m3.1.1.1.1.2.3.3.cmml" xref="S2.E10.m3.1.1.1.1.2.3.3"><ci id="S2.E10.m3.1.1.1.1.2.3.3.1.cmml" xref="S2.E10.m3.1.1.1.1.2.3.3.1">⋅</ci><apply id="S2.E10.m3.1.1.1.1.2.3.3.2.cmml" xref="S2.E10.m3.1.1.1.1.2.3.3.2"><times id="S2.E10.m3.1.1.1.1.2.3.3.2.1.cmml" xref="S2.E10.m3.1.1.1.1.2.3.3.2.1"></times><ci id="S2.E10.m3.1.1.1.1.2.3.3.2.2.cmml" xref="S2.E10.m3.1.1.1.1.2.3.3.2.2">𝑖</ci><apply id="S2.E10.m3.1.1.1.1.2.3.3.2.3.cmml" xref="S2.E10.m3.1.1.1.1.2.3.3.2.3"><csymbol cd="ambiguous" id="S2.E10.m3.1.1.1.1.2.3.3.2.3.1.cmml" xref="S2.E10.m3.1.1.1.1.2.3.3.2.3">subscript</csymbol><ci id="S2.E10.m3.1.1.1.1.2.3.3.2.3.2.cmml" xref="S2.E10.m3.1.1.1.1.2.3.3.2.3.2">𝑘</ci><ci id="S2.E10.m3.1.1.1.1.2.3.3.2.3.3.cmml" xref="S2.E10.m3.1.1.1.1.2.3.3.2.3.3">𝑃</ci></apply></apply><ci id="S2.E10.m3.1.1.1.1.2.3.3.3.cmml" xref="S2.E10.m3.1.1.1.1.2.3.3.3">𝑥</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E10.m3.1c">\displaystyle\sum_{P}C_{P}^{\mu}e^{ik_{P}\cdot x},</annotation><annotation encoding="application/x-llamapun" id="S2.E10.m3.1d">∑ start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT italic_i italic_k start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ⋅ italic_x end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(10)</span></td> </tr></tbody> <tbody id="S2.E11"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle b_{\mu}(x)" class="ltx_Math" display="inline" id="S2.E11.m1.1"><semantics id="S2.E11.m1.1a"><mrow id="S2.E11.m1.1.2" xref="S2.E11.m1.1.2.cmml"><msub id="S2.E11.m1.1.2.2" xref="S2.E11.m1.1.2.2.cmml"><mi id="S2.E11.m1.1.2.2.2" xref="S2.E11.m1.1.2.2.2.cmml">b</mi><mi id="S2.E11.m1.1.2.2.3" xref="S2.E11.m1.1.2.2.3.cmml">μ</mi></msub><mo id="S2.E11.m1.1.2.1" xref="S2.E11.m1.1.2.1.cmml">⁢</mo><mrow id="S2.E11.m1.1.2.3.2" xref="S2.E11.m1.1.2.cmml"><mo id="S2.E11.m1.1.2.3.2.1" stretchy="false" xref="S2.E11.m1.1.2.cmml">(</mo><mi id="S2.E11.m1.1.1" xref="S2.E11.m1.1.1.cmml">x</mi><mo id="S2.E11.m1.1.2.3.2.2" stretchy="false" xref="S2.E11.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E11.m1.1b"><apply id="S2.E11.m1.1.2.cmml" xref="S2.E11.m1.1.2"><times id="S2.E11.m1.1.2.1.cmml" xref="S2.E11.m1.1.2.1"></times><apply id="S2.E11.m1.1.2.2.cmml" xref="S2.E11.m1.1.2.2"><csymbol cd="ambiguous" id="S2.E11.m1.1.2.2.1.cmml" xref="S2.E11.m1.1.2.2">subscript</csymbol><ci id="S2.E11.m1.1.2.2.2.cmml" xref="S2.E11.m1.1.2.2.2">𝑏</ci><ci id="S2.E11.m1.1.2.2.3.cmml" xref="S2.E11.m1.1.2.2.3">𝜇</ci></apply><ci id="S2.E11.m1.1.1.cmml" xref="S2.E11.m1.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E11.m1.1c">\displaystyle b_{\mu}(x)</annotation><annotation encoding="application/x-llamapun" id="S2.E11.m1.1d">italic_b start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ( italic_x )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S2.E11.m2.1"><semantics id="S2.E11.m2.1a"><mo id="S2.E11.m2.1.1" xref="S2.E11.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S2.E11.m2.1b"><eq id="S2.E11.m2.1.1.cmml" xref="S2.E11.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S2.E11.m2.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S2.E11.m2.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\sum_{P}B_{P\mu}e^{ik_{P}\cdot x}," class="ltx_Math" display="inline" id="S2.E11.m3.1"><semantics id="S2.E11.m3.1a"><mrow id="S2.E11.m3.1.1.1" xref="S2.E11.m3.1.1.1.1.cmml"><mrow id="S2.E11.m3.1.1.1.1" xref="S2.E11.m3.1.1.1.1.cmml"><mstyle displaystyle="true" id="S2.E11.m3.1.1.1.1.1" xref="S2.E11.m3.1.1.1.1.1.cmml"><munder id="S2.E11.m3.1.1.1.1.1a" xref="S2.E11.m3.1.1.1.1.1.cmml"><mo id="S2.E11.m3.1.1.1.1.1.2" movablelimits="false" xref="S2.E11.m3.1.1.1.1.1.2.cmml">∑</mo><mi id="S2.E11.m3.1.1.1.1.1.3" xref="S2.E11.m3.1.1.1.1.1.3.cmml">P</mi></munder></mstyle><mrow id="S2.E11.m3.1.1.1.1.2" xref="S2.E11.m3.1.1.1.1.2.cmml"><msub id="S2.E11.m3.1.1.1.1.2.2" xref="S2.E11.m3.1.1.1.1.2.2.cmml"><mi id="S2.E11.m3.1.1.1.1.2.2.2" xref="S2.E11.m3.1.1.1.1.2.2.2.cmml">B</mi><mrow id="S2.E11.m3.1.1.1.1.2.2.3" xref="S2.E11.m3.1.1.1.1.2.2.3.cmml"><mi id="S2.E11.m3.1.1.1.1.2.2.3.2" xref="S2.E11.m3.1.1.1.1.2.2.3.2.cmml">P</mi><mo id="S2.E11.m3.1.1.1.1.2.2.3.1" xref="S2.E11.m3.1.1.1.1.2.2.3.1.cmml">⁢</mo><mi id="S2.E11.m3.1.1.1.1.2.2.3.3" xref="S2.E11.m3.1.1.1.1.2.2.3.3.cmml">μ</mi></mrow></msub><mo id="S2.E11.m3.1.1.1.1.2.1" xref="S2.E11.m3.1.1.1.1.2.1.cmml">⁢</mo><msup id="S2.E11.m3.1.1.1.1.2.3" xref="S2.E11.m3.1.1.1.1.2.3.cmml"><mi id="S2.E11.m3.1.1.1.1.2.3.2" xref="S2.E11.m3.1.1.1.1.2.3.2.cmml">e</mi><mrow id="S2.E11.m3.1.1.1.1.2.3.3" xref="S2.E11.m3.1.1.1.1.2.3.3.cmml"><mrow id="S2.E11.m3.1.1.1.1.2.3.3.2" xref="S2.E11.m3.1.1.1.1.2.3.3.2.cmml"><mi id="S2.E11.m3.1.1.1.1.2.3.3.2.2" xref="S2.E11.m3.1.1.1.1.2.3.3.2.2.cmml">i</mi><mo id="S2.E11.m3.1.1.1.1.2.3.3.2.1" xref="S2.E11.m3.1.1.1.1.2.3.3.2.1.cmml">⁢</mo><msub id="S2.E11.m3.1.1.1.1.2.3.3.2.3" xref="S2.E11.m3.1.1.1.1.2.3.3.2.3.cmml"><mi id="S2.E11.m3.1.1.1.1.2.3.3.2.3.2" xref="S2.E11.m3.1.1.1.1.2.3.3.2.3.2.cmml">k</mi><mi id="S2.E11.m3.1.1.1.1.2.3.3.2.3.3" xref="S2.E11.m3.1.1.1.1.2.3.3.2.3.3.cmml">P</mi></msub></mrow><mo id="S2.E11.m3.1.1.1.1.2.3.3.1" lspace="0.222em" rspace="0.222em" xref="S2.E11.m3.1.1.1.1.2.3.3.1.cmml">⋅</mo><mi id="S2.E11.m3.1.1.1.1.2.3.3.3" xref="S2.E11.m3.1.1.1.1.2.3.3.3.cmml">x</mi></mrow></msup></mrow></mrow><mo id="S2.E11.m3.1.1.1.2" xref="S2.E11.m3.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E11.m3.1b"><apply id="S2.E11.m3.1.1.1.1.cmml" xref="S2.E11.m3.1.1.1"><apply id="S2.E11.m3.1.1.1.1.1.cmml" xref="S2.E11.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.E11.m3.1.1.1.1.1.1.cmml" xref="S2.E11.m3.1.1.1.1.1">subscript</csymbol><sum id="S2.E11.m3.1.1.1.1.1.2.cmml" xref="S2.E11.m3.1.1.1.1.1.2"></sum><ci id="S2.E11.m3.1.1.1.1.1.3.cmml" xref="S2.E11.m3.1.1.1.1.1.3">𝑃</ci></apply><apply id="S2.E11.m3.1.1.1.1.2.cmml" xref="S2.E11.m3.1.1.1.1.2"><times id="S2.E11.m3.1.1.1.1.2.1.cmml" xref="S2.E11.m3.1.1.1.1.2.1"></times><apply id="S2.E11.m3.1.1.1.1.2.2.cmml" xref="S2.E11.m3.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S2.E11.m3.1.1.1.1.2.2.1.cmml" xref="S2.E11.m3.1.1.1.1.2.2">subscript</csymbol><ci 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id="S2.E11.m3.1.1.1.1.2.3.3.2.2.cmml" xref="S2.E11.m3.1.1.1.1.2.3.3.2.2">𝑖</ci><apply id="S2.E11.m3.1.1.1.1.2.3.3.2.3.cmml" xref="S2.E11.m3.1.1.1.1.2.3.3.2.3"><csymbol cd="ambiguous" id="S2.E11.m3.1.1.1.1.2.3.3.2.3.1.cmml" xref="S2.E11.m3.1.1.1.1.2.3.3.2.3">subscript</csymbol><ci id="S2.E11.m3.1.1.1.1.2.3.3.2.3.2.cmml" xref="S2.E11.m3.1.1.1.1.2.3.3.2.3.2">𝑘</ci><ci id="S2.E11.m3.1.1.1.1.2.3.3.2.3.3.cmml" xref="S2.E11.m3.1.1.1.1.2.3.3.2.3.3">𝑃</ci></apply></apply><ci id="S2.E11.m3.1.1.1.1.2.3.3.3.cmml" xref="S2.E11.m3.1.1.1.1.2.3.3.3">𝑥</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E11.m3.1c">\displaystyle\sum_{P}B_{P\mu}e^{ik_{P}\cdot x},</annotation><annotation encoding="application/x-llamapun" id="S2.E11.m3.1d">∑ start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_P italic_μ end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_i italic_k start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ⋅ italic_x end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(11)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p5.16">These expansions are given in terms of plane waves, with single-particle states as building blocks, and the metric density is expanded around flat space. <math alttext="P=p_{1}p_{2}\ldots p_{n}" class="ltx_Math" display="inline" id="S2.p5.5.m1.1"><semantics id="S2.p5.5.m1.1a"><mrow id="S2.p5.5.m1.1.1" xref="S2.p5.5.m1.1.1.cmml"><mi id="S2.p5.5.m1.1.1.2" xref="S2.p5.5.m1.1.1.2.cmml">P</mi><mo id="S2.p5.5.m1.1.1.1" xref="S2.p5.5.m1.1.1.1.cmml">=</mo><mrow id="S2.p5.5.m1.1.1.3" xref="S2.p5.5.m1.1.1.3.cmml"><msub id="S2.p5.5.m1.1.1.3.2" xref="S2.p5.5.m1.1.1.3.2.cmml"><mi id="S2.p5.5.m1.1.1.3.2.2" xref="S2.p5.5.m1.1.1.3.2.2.cmml">p</mi><mn id="S2.p5.5.m1.1.1.3.2.3" xref="S2.p5.5.m1.1.1.3.2.3.cmml">1</mn></msub><mo id="S2.p5.5.m1.1.1.3.1" xref="S2.p5.5.m1.1.1.3.1.cmml">⁢</mo><msub id="S2.p5.5.m1.1.1.3.3" xref="S2.p5.5.m1.1.1.3.3.cmml"><mi id="S2.p5.5.m1.1.1.3.3.2" xref="S2.p5.5.m1.1.1.3.3.2.cmml">p</mi><mn id="S2.p5.5.m1.1.1.3.3.3" xref="S2.p5.5.m1.1.1.3.3.3.cmml">2</mn></msub><mo id="S2.p5.5.m1.1.1.3.1a" xref="S2.p5.5.m1.1.1.3.1.cmml">⁢</mo><mi id="S2.p5.5.m1.1.1.3.4" mathvariant="normal" xref="S2.p5.5.m1.1.1.3.4.cmml">…</mi><mo id="S2.p5.5.m1.1.1.3.1b" xref="S2.p5.5.m1.1.1.3.1.cmml">⁢</mo><msub id="S2.p5.5.m1.1.1.3.5" xref="S2.p5.5.m1.1.1.3.5.cmml"><mi id="S2.p5.5.m1.1.1.3.5.2" xref="S2.p5.5.m1.1.1.3.5.2.cmml">p</mi><mi id="S2.p5.5.m1.1.1.3.5.3" xref="S2.p5.5.m1.1.1.3.5.3.cmml">n</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p5.5.m1.1b"><apply id="S2.p5.5.m1.1.1.cmml" xref="S2.p5.5.m1.1.1"><eq id="S2.p5.5.m1.1.1.1.cmml" xref="S2.p5.5.m1.1.1.1"></eq><ci id="S2.p5.5.m1.1.1.2.cmml" xref="S2.p5.5.m1.1.1.2">𝑃</ci><apply id="S2.p5.5.m1.1.1.3.cmml" xref="S2.p5.5.m1.1.1.3"><times id="S2.p5.5.m1.1.1.3.1.cmml" xref="S2.p5.5.m1.1.1.3.1"></times><apply id="S2.p5.5.m1.1.1.3.2.cmml" xref="S2.p5.5.m1.1.1.3.2"><csymbol cd="ambiguous" id="S2.p5.5.m1.1.1.3.2.1.cmml" xref="S2.p5.5.m1.1.1.3.2">subscript</csymbol><ci id="S2.p5.5.m1.1.1.3.2.2.cmml" xref="S2.p5.5.m1.1.1.3.2.2">𝑝</ci><cn id="S2.p5.5.m1.1.1.3.2.3.cmml" type="integer" xref="S2.p5.5.m1.1.1.3.2.3">1</cn></apply><apply id="S2.p5.5.m1.1.1.3.3.cmml" xref="S2.p5.5.m1.1.1.3.3"><csymbol cd="ambiguous" id="S2.p5.5.m1.1.1.3.3.1.cmml" xref="S2.p5.5.m1.1.1.3.3">subscript</csymbol><ci id="S2.p5.5.m1.1.1.3.3.2.cmml" xref="S2.p5.5.m1.1.1.3.3.2">𝑝</ci><cn id="S2.p5.5.m1.1.1.3.3.3.cmml" type="integer" xref="S2.p5.5.m1.1.1.3.3.3">2</cn></apply><ci id="S2.p5.5.m1.1.1.3.4.cmml" xref="S2.p5.5.m1.1.1.3.4">…</ci><apply id="S2.p5.5.m1.1.1.3.5.cmml" xref="S2.p5.5.m1.1.1.3.5"><csymbol cd="ambiguous" id="S2.p5.5.m1.1.1.3.5.1.cmml" xref="S2.p5.5.m1.1.1.3.5">subscript</csymbol><ci id="S2.p5.5.m1.1.1.3.5.2.cmml" xref="S2.p5.5.m1.1.1.3.5.2">𝑝</ci><ci id="S2.p5.5.m1.1.1.3.5.3.cmml" xref="S2.p5.5.m1.1.1.3.5.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.5.m1.1c">P=p_{1}p_{2}\ldots p_{n}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.5.m1.1d">italic_P = italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> denotes an ordered word composed of <math alttext="n" class="ltx_Math" display="inline" id="S2.p5.6.m2.1"><semantics id="S2.p5.6.m2.1a"><mi id="S2.p5.6.m2.1.1" xref="S2.p5.6.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.p5.6.m2.1b"><ci id="S2.p5.6.m2.1.1.cmml" xref="S2.p5.6.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.6.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.p5.6.m2.1d">italic_n</annotation></semantics></math> single-particle labels (letters) <math alttext="p_{i}" class="ltx_Math" display="inline" id="S2.p5.7.m3.1"><semantics id="S2.p5.7.m3.1a"><msub id="S2.p5.7.m3.1.1" xref="S2.p5.7.m3.1.1.cmml"><mi id="S2.p5.7.m3.1.1.2" xref="S2.p5.7.m3.1.1.2.cmml">p</mi><mi id="S2.p5.7.m3.1.1.3" xref="S2.p5.7.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p5.7.m3.1b"><apply id="S2.p5.7.m3.1.1.cmml" xref="S2.p5.7.m3.1.1"><csymbol cd="ambiguous" id="S2.p5.7.m3.1.1.1.cmml" xref="S2.p5.7.m3.1.1">subscript</csymbol><ci id="S2.p5.7.m3.1.1.2.cmml" xref="S2.p5.7.m3.1.1.2">𝑝</ci><ci id="S2.p5.7.m3.1.1.3.cmml" xref="S2.p5.7.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.7.m3.1c">p_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.7.m3.1d">italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, with <math alttext="p_{i}<p_{i+1}" class="ltx_Math" display="inline" id="S2.p5.8.m4.1"><semantics id="S2.p5.8.m4.1a"><mrow id="S2.p5.8.m4.1.1" xref="S2.p5.8.m4.1.1.cmml"><msub id="S2.p5.8.m4.1.1.2" xref="S2.p5.8.m4.1.1.2.cmml"><mi id="S2.p5.8.m4.1.1.2.2" xref="S2.p5.8.m4.1.1.2.2.cmml">p</mi><mi id="S2.p5.8.m4.1.1.2.3" xref="S2.p5.8.m4.1.1.2.3.cmml">i</mi></msub><mo id="S2.p5.8.m4.1.1.1" xref="S2.p5.8.m4.1.1.1.cmml"><</mo><msub id="S2.p5.8.m4.1.1.3" xref="S2.p5.8.m4.1.1.3.cmml"><mi id="S2.p5.8.m4.1.1.3.2" xref="S2.p5.8.m4.1.1.3.2.cmml">p</mi><mrow id="S2.p5.8.m4.1.1.3.3" xref="S2.p5.8.m4.1.1.3.3.cmml"><mi id="S2.p5.8.m4.1.1.3.3.2" xref="S2.p5.8.m4.1.1.3.3.2.cmml">i</mi><mo id="S2.p5.8.m4.1.1.3.3.1" xref="S2.p5.8.m4.1.1.3.3.1.cmml">+</mo><mn id="S2.p5.8.m4.1.1.3.3.3" xref="S2.p5.8.m4.1.1.3.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.p5.8.m4.1b"><apply id="S2.p5.8.m4.1.1.cmml" xref="S2.p5.8.m4.1.1"><lt id="S2.p5.8.m4.1.1.1.cmml" xref="S2.p5.8.m4.1.1.1"></lt><apply id="S2.p5.8.m4.1.1.2.cmml" xref="S2.p5.8.m4.1.1.2"><csymbol cd="ambiguous" id="S2.p5.8.m4.1.1.2.1.cmml" xref="S2.p5.8.m4.1.1.2">subscript</csymbol><ci id="S2.p5.8.m4.1.1.2.2.cmml" xref="S2.p5.8.m4.1.1.2.2">𝑝</ci><ci id="S2.p5.8.m4.1.1.2.3.cmml" xref="S2.p5.8.m4.1.1.2.3">𝑖</ci></apply><apply id="S2.p5.8.m4.1.1.3.cmml" xref="S2.p5.8.m4.1.1.3"><csymbol cd="ambiguous" id="S2.p5.8.m4.1.1.3.1.cmml" xref="S2.p5.8.m4.1.1.3">subscript</csymbol><ci id="S2.p5.8.m4.1.1.3.2.cmml" xref="S2.p5.8.m4.1.1.3.2">𝑝</ci><apply id="S2.p5.8.m4.1.1.3.3.cmml" xref="S2.p5.8.m4.1.1.3.3"><plus id="S2.p5.8.m4.1.1.3.3.1.cmml" xref="S2.p5.8.m4.1.1.3.3.1"></plus><ci id="S2.p5.8.m4.1.1.3.3.2.cmml" xref="S2.p5.8.m4.1.1.3.3.2">𝑖</ci><cn id="S2.p5.8.m4.1.1.3.3.3.cmml" type="integer" xref="S2.p5.8.m4.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.8.m4.1c">p_{i}<p_{i+1}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.8.m4.1d">italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT < italic_p start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT</annotation></semantics></math>. The sum over words, <math alttext="\sum_{P}" class="ltx_Math" display="inline" id="S2.p5.9.m5.1"><semantics id="S2.p5.9.m5.1a"><msub id="S2.p5.9.m5.1.1" xref="S2.p5.9.m5.1.1.cmml"><mo id="S2.p5.9.m5.1.1.2" xref="S2.p5.9.m5.1.1.2.cmml">∑</mo><mi id="S2.p5.9.m5.1.1.3" xref="S2.p5.9.m5.1.1.3.cmml">P</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p5.9.m5.1b"><apply id="S2.p5.9.m5.1.1.cmml" xref="S2.p5.9.m5.1.1"><csymbol cd="ambiguous" id="S2.p5.9.m5.1.1.1.cmml" xref="S2.p5.9.m5.1.1">subscript</csymbol><sum id="S2.p5.9.m5.1.1.2.cmml" xref="S2.p5.9.m5.1.1.2"></sum><ci id="S2.p5.9.m5.1.1.3.cmml" xref="S2.p5.9.m5.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.9.m5.1c">\sum_{P}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.9.m5.1d">∑ start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math>, goes from single-particle states (one-letter words) to any multiparticle composition. The multiparticle momenta <math alttext="k_{P}" class="ltx_Math" display="inline" id="S2.p5.10.m6.1"><semantics id="S2.p5.10.m6.1a"><msub id="S2.p5.10.m6.1.1" xref="S2.p5.10.m6.1.1.cmml"><mi id="S2.p5.10.m6.1.1.2" xref="S2.p5.10.m6.1.1.2.cmml">k</mi><mi id="S2.p5.10.m6.1.1.3" xref="S2.p5.10.m6.1.1.3.cmml">P</mi></msub><annotation-xml encoding="MathML-Content" id="S2.p5.10.m6.1b"><apply id="S2.p5.10.m6.1.1.cmml" xref="S2.p5.10.m6.1.1"><csymbol cd="ambiguous" id="S2.p5.10.m6.1.1.1.cmml" xref="S2.p5.10.m6.1.1">subscript</csymbol><ci id="S2.p5.10.m6.1.1.2.cmml" xref="S2.p5.10.m6.1.1.2">𝑘</ci><ci id="S2.p5.10.m6.1.1.3.cmml" xref="S2.p5.10.m6.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.10.m6.1c">k_{P}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.10.m6.1d">italic_k start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math> are defined as <math alttext="k_{P}=k_{p_{1}}+...+k_{p_{n}}" class="ltx_Math" display="inline" id="S2.p5.11.m7.1"><semantics id="S2.p5.11.m7.1a"><mrow id="S2.p5.11.m7.1.1" xref="S2.p5.11.m7.1.1.cmml"><msub id="S2.p5.11.m7.1.1.2" xref="S2.p5.11.m7.1.1.2.cmml"><mi id="S2.p5.11.m7.1.1.2.2" xref="S2.p5.11.m7.1.1.2.2.cmml">k</mi><mi id="S2.p5.11.m7.1.1.2.3" xref="S2.p5.11.m7.1.1.2.3.cmml">P</mi></msub><mo id="S2.p5.11.m7.1.1.1" xref="S2.p5.11.m7.1.1.1.cmml">=</mo><mrow id="S2.p5.11.m7.1.1.3" xref="S2.p5.11.m7.1.1.3.cmml"><msub id="S2.p5.11.m7.1.1.3.2" xref="S2.p5.11.m7.1.1.3.2.cmml"><mi id="S2.p5.11.m7.1.1.3.2.2" xref="S2.p5.11.m7.1.1.3.2.2.cmml">k</mi><msub id="S2.p5.11.m7.1.1.3.2.3" xref="S2.p5.11.m7.1.1.3.2.3.cmml"><mi id="S2.p5.11.m7.1.1.3.2.3.2" xref="S2.p5.11.m7.1.1.3.2.3.2.cmml">p</mi><mn id="S2.p5.11.m7.1.1.3.2.3.3" xref="S2.p5.11.m7.1.1.3.2.3.3.cmml">1</mn></msub></msub><mo id="S2.p5.11.m7.1.1.3.1" xref="S2.p5.11.m7.1.1.3.1.cmml">+</mo><mi id="S2.p5.11.m7.1.1.3.3" mathvariant="normal" xref="S2.p5.11.m7.1.1.3.3.cmml">…</mi><mo id="S2.p5.11.m7.1.1.3.1a" xref="S2.p5.11.m7.1.1.3.1.cmml">+</mo><msub id="S2.p5.11.m7.1.1.3.4" xref="S2.p5.11.m7.1.1.3.4.cmml"><mi id="S2.p5.11.m7.1.1.3.4.2" xref="S2.p5.11.m7.1.1.3.4.2.cmml">k</mi><msub id="S2.p5.11.m7.1.1.3.4.3" xref="S2.p5.11.m7.1.1.3.4.3.cmml"><mi id="S2.p5.11.m7.1.1.3.4.3.2" xref="S2.p5.11.m7.1.1.3.4.3.2.cmml">p</mi><mi id="S2.p5.11.m7.1.1.3.4.3.3" xref="S2.p5.11.m7.1.1.3.4.3.3.cmml">n</mi></msub></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p5.11.m7.1b"><apply id="S2.p5.11.m7.1.1.cmml" xref="S2.p5.11.m7.1.1"><eq id="S2.p5.11.m7.1.1.1.cmml" xref="S2.p5.11.m7.1.1.1"></eq><apply id="S2.p5.11.m7.1.1.2.cmml" xref="S2.p5.11.m7.1.1.2"><csymbol cd="ambiguous" id="S2.p5.11.m7.1.1.2.1.cmml" xref="S2.p5.11.m7.1.1.2">subscript</csymbol><ci id="S2.p5.11.m7.1.1.2.2.cmml" xref="S2.p5.11.m7.1.1.2.2">𝑘</ci><ci id="S2.p5.11.m7.1.1.2.3.cmml" xref="S2.p5.11.m7.1.1.2.3">𝑃</ci></apply><apply id="S2.p5.11.m7.1.1.3.cmml" xref="S2.p5.11.m7.1.1.3"><plus id="S2.p5.11.m7.1.1.3.1.cmml" xref="S2.p5.11.m7.1.1.3.1"></plus><apply id="S2.p5.11.m7.1.1.3.2.cmml" xref="S2.p5.11.m7.1.1.3.2"><csymbol cd="ambiguous" id="S2.p5.11.m7.1.1.3.2.1.cmml" xref="S2.p5.11.m7.1.1.3.2">subscript</csymbol><ci id="S2.p5.11.m7.1.1.3.2.2.cmml" xref="S2.p5.11.m7.1.1.3.2.2">𝑘</ci><apply id="S2.p5.11.m7.1.1.3.2.3.cmml" xref="S2.p5.11.m7.1.1.3.2.3"><csymbol cd="ambiguous" id="S2.p5.11.m7.1.1.3.2.3.1.cmml" xref="S2.p5.11.m7.1.1.3.2.3">subscript</csymbol><ci id="S2.p5.11.m7.1.1.3.2.3.2.cmml" xref="S2.p5.11.m7.1.1.3.2.3.2">𝑝</ci><cn id="S2.p5.11.m7.1.1.3.2.3.3.cmml" type="integer" xref="S2.p5.11.m7.1.1.3.2.3.3">1</cn></apply></apply><ci id="S2.p5.11.m7.1.1.3.3.cmml" xref="S2.p5.11.m7.1.1.3.3">…</ci><apply id="S2.p5.11.m7.1.1.3.4.cmml" xref="S2.p5.11.m7.1.1.3.4"><csymbol cd="ambiguous" id="S2.p5.11.m7.1.1.3.4.1.cmml" xref="S2.p5.11.m7.1.1.3.4">subscript</csymbol><ci id="S2.p5.11.m7.1.1.3.4.2.cmml" xref="S2.p5.11.m7.1.1.3.4.2">𝑘</ci><apply id="S2.p5.11.m7.1.1.3.4.3.cmml" xref="S2.p5.11.m7.1.1.3.4.3"><csymbol cd="ambiguous" id="S2.p5.11.m7.1.1.3.4.3.1.cmml" xref="S2.p5.11.m7.1.1.3.4.3">subscript</csymbol><ci id="S2.p5.11.m7.1.1.3.4.3.2.cmml" xref="S2.p5.11.m7.1.1.3.4.3.2">𝑝</ci><ci id="S2.p5.11.m7.1.1.3.4.3.3.cmml" xref="S2.p5.11.m7.1.1.3.4.3.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.11.m7.1c">k_{P}=k_{p_{1}}+...+k_{p_{n}}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.11.m7.1d">italic_k start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT = italic_k start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT + … + italic_k start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. The multiparticle currents (<math alttext="H_{P\mu\nu}" class="ltx_Math" display="inline" id="S2.p5.12.m8.1"><semantics id="S2.p5.12.m8.1a"><msub id="S2.p5.12.m8.1.1" xref="S2.p5.12.m8.1.1.cmml"><mi id="S2.p5.12.m8.1.1.2" xref="S2.p5.12.m8.1.1.2.cmml">H</mi><mrow id="S2.p5.12.m8.1.1.3" xref="S2.p5.12.m8.1.1.3.cmml"><mi id="S2.p5.12.m8.1.1.3.2" xref="S2.p5.12.m8.1.1.3.2.cmml">P</mi><mo id="S2.p5.12.m8.1.1.3.1" xref="S2.p5.12.m8.1.1.3.1.cmml">⁢</mo><mi id="S2.p5.12.m8.1.1.3.3" xref="S2.p5.12.m8.1.1.3.3.cmml">μ</mi><mo id="S2.p5.12.m8.1.1.3.1a" xref="S2.p5.12.m8.1.1.3.1.cmml">⁢</mo><mi id="S2.p5.12.m8.1.1.3.4" xref="S2.p5.12.m8.1.1.3.4.cmml">ν</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.p5.12.m8.1b"><apply id="S2.p5.12.m8.1.1.cmml" xref="S2.p5.12.m8.1.1"><csymbol cd="ambiguous" id="S2.p5.12.m8.1.1.1.cmml" xref="S2.p5.12.m8.1.1">subscript</csymbol><ci id="S2.p5.12.m8.1.1.2.cmml" xref="S2.p5.12.m8.1.1.2">𝐻</ci><apply id="S2.p5.12.m8.1.1.3.cmml" xref="S2.p5.12.m8.1.1.3"><times id="S2.p5.12.m8.1.1.3.1.cmml" xref="S2.p5.12.m8.1.1.3.1"></times><ci id="S2.p5.12.m8.1.1.3.2.cmml" xref="S2.p5.12.m8.1.1.3.2">𝑃</ci><ci id="S2.p5.12.m8.1.1.3.3.cmml" xref="S2.p5.12.m8.1.1.3.3">𝜇</ci><ci id="S2.p5.12.m8.1.1.3.4.cmml" xref="S2.p5.12.m8.1.1.3.4">𝜈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.12.m8.1c">H_{P\mu\nu}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.12.m8.1d">italic_H start_POSTSUBSCRIPT italic_P italic_μ italic_ν end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="I_{P}^{\mu\nu}" class="ltx_Math" display="inline" id="S2.p5.13.m9.1"><semantics id="S2.p5.13.m9.1a"><msubsup id="S2.p5.13.m9.1.1" xref="S2.p5.13.m9.1.1.cmml"><mi id="S2.p5.13.m9.1.1.2.2" xref="S2.p5.13.m9.1.1.2.2.cmml">I</mi><mi id="S2.p5.13.m9.1.1.2.3" xref="S2.p5.13.m9.1.1.2.3.cmml">P</mi><mrow id="S2.p5.13.m9.1.1.3" xref="S2.p5.13.m9.1.1.3.cmml"><mi id="S2.p5.13.m9.1.1.3.2" xref="S2.p5.13.m9.1.1.3.2.cmml">μ</mi><mo id="S2.p5.13.m9.1.1.3.1" xref="S2.p5.13.m9.1.1.3.1.cmml">⁢</mo><mi id="S2.p5.13.m9.1.1.3.3" xref="S2.p5.13.m9.1.1.3.3.cmml">ν</mi></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S2.p5.13.m9.1b"><apply id="S2.p5.13.m9.1.1.cmml" xref="S2.p5.13.m9.1.1"><csymbol cd="ambiguous" id="S2.p5.13.m9.1.1.1.cmml" xref="S2.p5.13.m9.1.1">superscript</csymbol><apply id="S2.p5.13.m9.1.1.2.cmml" xref="S2.p5.13.m9.1.1"><csymbol cd="ambiguous" id="S2.p5.13.m9.1.1.2.1.cmml" xref="S2.p5.13.m9.1.1">subscript</csymbol><ci id="S2.p5.13.m9.1.1.2.2.cmml" xref="S2.p5.13.m9.1.1.2.2">𝐼</ci><ci id="S2.p5.13.m9.1.1.2.3.cmml" xref="S2.p5.13.m9.1.1.2.3">𝑃</ci></apply><apply id="S2.p5.13.m9.1.1.3.cmml" xref="S2.p5.13.m9.1.1.3"><times id="S2.p5.13.m9.1.1.3.1.cmml" xref="S2.p5.13.m9.1.1.3.1"></times><ci id="S2.p5.13.m9.1.1.3.2.cmml" xref="S2.p5.13.m9.1.1.3.2">𝜇</ci><ci id="S2.p5.13.m9.1.1.3.3.cmml" xref="S2.p5.13.m9.1.1.3.3">𝜈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.13.m9.1c">I_{P}^{\mu\nu}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.13.m9.1d">italic_I start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="C_{P}^{\mu}" class="ltx_Math" display="inline" id="S2.p5.14.m10.1"><semantics id="S2.p5.14.m10.1a"><msubsup id="S2.p5.14.m10.1.1" xref="S2.p5.14.m10.1.1.cmml"><mi id="S2.p5.14.m10.1.1.2.2" xref="S2.p5.14.m10.1.1.2.2.cmml">C</mi><mi id="S2.p5.14.m10.1.1.2.3" xref="S2.p5.14.m10.1.1.2.3.cmml">P</mi><mi id="S2.p5.14.m10.1.1.3" xref="S2.p5.14.m10.1.1.3.cmml">μ</mi></msubsup><annotation-xml encoding="MathML-Content" id="S2.p5.14.m10.1b"><apply id="S2.p5.14.m10.1.1.cmml" xref="S2.p5.14.m10.1.1"><csymbol cd="ambiguous" id="S2.p5.14.m10.1.1.1.cmml" xref="S2.p5.14.m10.1.1">superscript</csymbol><apply id="S2.p5.14.m10.1.1.2.cmml" xref="S2.p5.14.m10.1.1"><csymbol cd="ambiguous" id="S2.p5.14.m10.1.1.2.1.cmml" xref="S2.p5.14.m10.1.1">subscript</csymbol><ci id="S2.p5.14.m10.1.1.2.2.cmml" xref="S2.p5.14.m10.1.1.2.2">𝐶</ci><ci id="S2.p5.14.m10.1.1.2.3.cmml" xref="S2.p5.14.m10.1.1.2.3">𝑃</ci></apply><ci id="S2.p5.14.m10.1.1.3.cmml" xref="S2.p5.14.m10.1.1.3">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.14.m10.1c">C_{P}^{\mu}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.14.m10.1d">italic_C start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="B_{P\mu}" class="ltx_Math" display="inline" id="S2.p5.15.m11.1"><semantics id="S2.p5.15.m11.1a"><msub id="S2.p5.15.m11.1.1" xref="S2.p5.15.m11.1.1.cmml"><mi id="S2.p5.15.m11.1.1.2" xref="S2.p5.15.m11.1.1.2.cmml">B</mi><mrow id="S2.p5.15.m11.1.1.3" xref="S2.p5.15.m11.1.1.3.cmml"><mi id="S2.p5.15.m11.1.1.3.2" xref="S2.p5.15.m11.1.1.3.2.cmml">P</mi><mo id="S2.p5.15.m11.1.1.3.1" xref="S2.p5.15.m11.1.1.3.1.cmml">⁢</mo><mi id="S2.p5.15.m11.1.1.3.3" xref="S2.p5.15.m11.1.1.3.3.cmml">μ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.p5.15.m11.1b"><apply id="S2.p5.15.m11.1.1.cmml" xref="S2.p5.15.m11.1.1"><csymbol cd="ambiguous" id="S2.p5.15.m11.1.1.1.cmml" xref="S2.p5.15.m11.1.1">subscript</csymbol><ci id="S2.p5.15.m11.1.1.2.cmml" xref="S2.p5.15.m11.1.1.2">𝐵</ci><apply id="S2.p5.15.m11.1.1.3.cmml" xref="S2.p5.15.m11.1.1.3"><times id="S2.p5.15.m11.1.1.3.1.cmml" xref="S2.p5.15.m11.1.1.3.1"></times><ci id="S2.p5.15.m11.1.1.3.2.cmml" xref="S2.p5.15.m11.1.1.3.2">𝑃</ci><ci id="S2.p5.15.m11.1.1.3.3.cmml" xref="S2.p5.15.m11.1.1.3.3">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.15.m11.1c">B_{P\mu}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.15.m11.1d">italic_B start_POSTSUBSCRIPT italic_P italic_μ end_POSTSUBSCRIPT</annotation></semantics></math>) are recursively determined by the equations of motion. Their single-particle instance corresponds to the respective polarizations. For example, <math alttext="H_{p\mu\nu}=h_{p\mu\nu}" class="ltx_Math" display="inline" id="S2.p5.16.m12.1"><semantics id="S2.p5.16.m12.1a"><mrow id="S2.p5.16.m12.1.1" xref="S2.p5.16.m12.1.1.cmml"><msub id="S2.p5.16.m12.1.1.2" xref="S2.p5.16.m12.1.1.2.cmml"><mi id="S2.p5.16.m12.1.1.2.2" xref="S2.p5.16.m12.1.1.2.2.cmml">H</mi><mrow id="S2.p5.16.m12.1.1.2.3" xref="S2.p5.16.m12.1.1.2.3.cmml"><mi id="S2.p5.16.m12.1.1.2.3.2" xref="S2.p5.16.m12.1.1.2.3.2.cmml">p</mi><mo id="S2.p5.16.m12.1.1.2.3.1" xref="S2.p5.16.m12.1.1.2.3.1.cmml">⁢</mo><mi id="S2.p5.16.m12.1.1.2.3.3" xref="S2.p5.16.m12.1.1.2.3.3.cmml">μ</mi><mo id="S2.p5.16.m12.1.1.2.3.1a" xref="S2.p5.16.m12.1.1.2.3.1.cmml">⁢</mo><mi id="S2.p5.16.m12.1.1.2.3.4" xref="S2.p5.16.m12.1.1.2.3.4.cmml">ν</mi></mrow></msub><mo id="S2.p5.16.m12.1.1.1" xref="S2.p5.16.m12.1.1.1.cmml">=</mo><msub id="S2.p5.16.m12.1.1.3" xref="S2.p5.16.m12.1.1.3.cmml"><mi id="S2.p5.16.m12.1.1.3.2" xref="S2.p5.16.m12.1.1.3.2.cmml">h</mi><mrow id="S2.p5.16.m12.1.1.3.3" xref="S2.p5.16.m12.1.1.3.3.cmml"><mi id="S2.p5.16.m12.1.1.3.3.2" xref="S2.p5.16.m12.1.1.3.3.2.cmml">p</mi><mo id="S2.p5.16.m12.1.1.3.3.1" xref="S2.p5.16.m12.1.1.3.3.1.cmml">⁢</mo><mi id="S2.p5.16.m12.1.1.3.3.3" xref="S2.p5.16.m12.1.1.3.3.3.cmml">μ</mi><mo id="S2.p5.16.m12.1.1.3.3.1a" xref="S2.p5.16.m12.1.1.3.3.1.cmml">⁢</mo><mi id="S2.p5.16.m12.1.1.3.3.4" xref="S2.p5.16.m12.1.1.3.3.4.cmml">ν</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.p5.16.m12.1b"><apply id="S2.p5.16.m12.1.1.cmml" xref="S2.p5.16.m12.1.1"><eq id="S2.p5.16.m12.1.1.1.cmml" xref="S2.p5.16.m12.1.1.1"></eq><apply id="S2.p5.16.m12.1.1.2.cmml" xref="S2.p5.16.m12.1.1.2"><csymbol cd="ambiguous" id="S2.p5.16.m12.1.1.2.1.cmml" xref="S2.p5.16.m12.1.1.2">subscript</csymbol><ci id="S2.p5.16.m12.1.1.2.2.cmml" xref="S2.p5.16.m12.1.1.2.2">𝐻</ci><apply id="S2.p5.16.m12.1.1.2.3.cmml" xref="S2.p5.16.m12.1.1.2.3"><times id="S2.p5.16.m12.1.1.2.3.1.cmml" xref="S2.p5.16.m12.1.1.2.3.1"></times><ci id="S2.p5.16.m12.1.1.2.3.2.cmml" xref="S2.p5.16.m12.1.1.2.3.2">𝑝</ci><ci id="S2.p5.16.m12.1.1.2.3.3.cmml" xref="S2.p5.16.m12.1.1.2.3.3">𝜇</ci><ci id="S2.p5.16.m12.1.1.2.3.4.cmml" xref="S2.p5.16.m12.1.1.2.3.4">𝜈</ci></apply></apply><apply id="S2.p5.16.m12.1.1.3.cmml" xref="S2.p5.16.m12.1.1.3"><csymbol cd="ambiguous" id="S2.p5.16.m12.1.1.3.1.cmml" xref="S2.p5.16.m12.1.1.3">subscript</csymbol><ci id="S2.p5.16.m12.1.1.3.2.cmml" xref="S2.p5.16.m12.1.1.3.2">ℎ</ci><apply id="S2.p5.16.m12.1.1.3.3.cmml" xref="S2.p5.16.m12.1.1.3.3"><times id="S2.p5.16.m12.1.1.3.3.1.cmml" xref="S2.p5.16.m12.1.1.3.3.1"></times><ci id="S2.p5.16.m12.1.1.3.3.2.cmml" xref="S2.p5.16.m12.1.1.3.3.2">𝑝</ci><ci id="S2.p5.16.m12.1.1.3.3.3.cmml" xref="S2.p5.16.m12.1.1.3.3.3">𝜇</ci><ci id="S2.p5.16.m12.1.1.3.3.4.cmml" xref="S2.p5.16.m12.1.1.3.3.4">𝜈</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p5.16.m12.1c">H_{p\mu\nu}=h_{p\mu\nu}</annotation><annotation encoding="application/x-llamapun" id="S2.p5.16.m12.1d">italic_H start_POSTSUBSCRIPT italic_p italic_μ italic_ν end_POSTSUBSCRIPT = italic_h start_POSTSUBSCRIPT italic_p italic_μ italic_ν end_POSTSUBSCRIPT</annotation></semantics></math> describes a graviton. In our construction, single-particle states can also be ghosts.</p> </div> <div class="ltx_para ltx_noindent" id="S2.p6"> <p class="ltx_p" id="S2.p6.1">Using (<a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S2.E5" title="In II Equations of motion ‣ One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_tag">5</span></a>), the recursion for <math alttext="I_{P}^{\mu\nu}" class="ltx_Math" display="inline" id="S2.p6.1.m1.1"><semantics id="S2.p6.1.m1.1a"><msubsup id="S2.p6.1.m1.1.1" xref="S2.p6.1.m1.1.1.cmml"><mi id="S2.p6.1.m1.1.1.2.2" xref="S2.p6.1.m1.1.1.2.2.cmml">I</mi><mi id="S2.p6.1.m1.1.1.2.3" xref="S2.p6.1.m1.1.1.2.3.cmml">P</mi><mrow id="S2.p6.1.m1.1.1.3" xref="S2.p6.1.m1.1.1.3.cmml"><mi id="S2.p6.1.m1.1.1.3.2" xref="S2.p6.1.m1.1.1.3.2.cmml">μ</mi><mo id="S2.p6.1.m1.1.1.3.1" xref="S2.p6.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S2.p6.1.m1.1.1.3.3" xref="S2.p6.1.m1.1.1.3.3.cmml">ν</mi></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S2.p6.1.m1.1b"><apply id="S2.p6.1.m1.1.1.cmml" xref="S2.p6.1.m1.1.1"><csymbol cd="ambiguous" id="S2.p6.1.m1.1.1.1.cmml" xref="S2.p6.1.m1.1.1">superscript</csymbol><apply id="S2.p6.1.m1.1.1.2.cmml" xref="S2.p6.1.m1.1.1"><csymbol cd="ambiguous" id="S2.p6.1.m1.1.1.2.1.cmml" xref="S2.p6.1.m1.1.1">subscript</csymbol><ci id="S2.p6.1.m1.1.1.2.2.cmml" xref="S2.p6.1.m1.1.1.2.2">𝐼</ci><ci id="S2.p6.1.m1.1.1.2.3.cmml" xref="S2.p6.1.m1.1.1.2.3">𝑃</ci></apply><apply id="S2.p6.1.m1.1.1.3.cmml" xref="S2.p6.1.m1.1.1.3"><times id="S2.p6.1.m1.1.1.3.1.cmml" xref="S2.p6.1.m1.1.1.3.1"></times><ci id="S2.p6.1.m1.1.1.3.2.cmml" xref="S2.p6.1.m1.1.1.3.2">𝜇</ci><ci id="S2.p6.1.m1.1.1.3.3.cmml" xref="S2.p6.1.m1.1.1.3.3">𝜈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.1.m1.1c">I_{P}^{\mu\nu}</annotation><annotation encoding="application/x-llamapun" id="S2.p6.1.m1.1d">italic_I start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT</annotation></semantics></math> is shown to be</p> <table class="ltx_equation ltx_eqn_table" id="S2.E12"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="s_{P}\mathbb{P}_{\mu\nu\rho\sigma}I_{P}^{\rho\sigma}=\frac{1}{2}\sum_{P=Q\cup R% }I_{Q}^{\rho\sigma}I_{R}^{\gamma\lambda}\mathcal{V}^{(3)}_{\mu\nu\rho\sigma% \gamma\lambda}(Q,R)\\ +\sum_{P=Q\cup R\cup S}I_{Q}^{\rho\sigma}I_{R}^{\gamma\lambda}H_{S}^{\tau% \delta}V^{(4)}_{\mu\nu\rho\sigma\gamma\lambda\tau\delta}(Q,R,S)\\ +\sum_{P=Q\cup R\cup S\cup T}I_{Q}^{\rho\sigma}I_{R}^{\gamma\lambda}H_{S}^{% \tau\delta}H_{T}^{\alpha\beta}V^{(5)}_{\mu\nu\rho\sigma\gamma\lambda\tau\delta% \alpha\beta}(Q,R,S)\\ +\sum_{P=Q\cup R}B_{Q\rho}C_{R}^{\sigma}\mathcal{V}^{(\textrm{g})\rho}_{\mu\nu% \sigma}(P,Q)," class="ltx_Math" display="block" id="S2.E12.m1.91"><semantics id="S2.E12.m1.91a"><mtable displaystyle="true" id="S2.E12.m1.91.91.2" rowspacing="0pt"><mtr id="S2.E12.m1.91.91.2a"><mtd class="ltx_align_left" columnalign="left" id="S2.E12.m1.91.91.2b"><mrow id="S2.E12.m1.25.25.25.25.25"><mrow id="S2.E12.m1.25.25.25.25.25.26"><msub id="S2.E12.m1.25.25.25.25.25.26.2"><mi id="S2.E12.m1.1.1.1.1.1.1" xref="S2.E12.m1.1.1.1.1.1.1.cmml">s</mi><mi id="S2.E12.m1.2.2.2.2.2.2.1" xref="S2.E12.m1.2.2.2.2.2.2.1.cmml">P</mi></msub><mo id="S2.E12.m1.25.25.25.25.25.26.1" xref="S2.E12.m1.90.90.1.1.1.cmml">⁢</mo><msub id="S2.E12.m1.25.25.25.25.25.26.3"><mi id="S2.E12.m1.3.3.3.3.3.3" xref="S2.E12.m1.3.3.3.3.3.3.cmml">ℙ</mi><mrow id="S2.E12.m1.4.4.4.4.4.4.1" xref="S2.E12.m1.4.4.4.4.4.4.1.cmml"><mi id="S2.E12.m1.4.4.4.4.4.4.1.2" xref="S2.E12.m1.4.4.4.4.4.4.1.2.cmml">μ</mi><mo id="S2.E12.m1.4.4.4.4.4.4.1.1" xref="S2.E12.m1.4.4.4.4.4.4.1.1.cmml">⁢</mo><mi id="S2.E12.m1.4.4.4.4.4.4.1.3" xref="S2.E12.m1.4.4.4.4.4.4.1.3.cmml">ν</mi><mo id="S2.E12.m1.4.4.4.4.4.4.1.1a" xref="S2.E12.m1.4.4.4.4.4.4.1.1.cmml">⁢</mo><mi id="S2.E12.m1.4.4.4.4.4.4.1.4" xref="S2.E12.m1.4.4.4.4.4.4.1.4.cmml">ρ</mi><mo id="S2.E12.m1.4.4.4.4.4.4.1.1b" xref="S2.E12.m1.4.4.4.4.4.4.1.1.cmml">⁢</mo><mi id="S2.E12.m1.4.4.4.4.4.4.1.5" xref="S2.E12.m1.4.4.4.4.4.4.1.5.cmml">σ</mi></mrow></msub><mo id="S2.E12.m1.25.25.25.25.25.26.1a" xref="S2.E12.m1.90.90.1.1.1.cmml">⁢</mo><msubsup id="S2.E12.m1.25.25.25.25.25.26.4"><mi id="S2.E12.m1.5.5.5.5.5.5" xref="S2.E12.m1.5.5.5.5.5.5.cmml">I</mi><mi id="S2.E12.m1.6.6.6.6.6.6.1" xref="S2.E12.m1.6.6.6.6.6.6.1.cmml">P</mi><mrow id="S2.E12.m1.7.7.7.7.7.7.1" xref="S2.E12.m1.7.7.7.7.7.7.1.cmml"><mi id="S2.E12.m1.7.7.7.7.7.7.1.2" xref="S2.E12.m1.7.7.7.7.7.7.1.2.cmml">ρ</mi><mo id="S2.E12.m1.7.7.7.7.7.7.1.1" xref="S2.E12.m1.7.7.7.7.7.7.1.1.cmml">⁢</mo><mi id="S2.E12.m1.7.7.7.7.7.7.1.3" xref="S2.E12.m1.7.7.7.7.7.7.1.3.cmml">σ</mi></mrow></msubsup></mrow><mo id="S2.E12.m1.8.8.8.8.8.8" xref="S2.E12.m1.8.8.8.8.8.8.cmml">=</mo><mrow id="S2.E12.m1.25.25.25.25.25.27"><mfrac id="S2.E12.m1.9.9.9.9.9.9" xref="S2.E12.m1.9.9.9.9.9.9.cmml"><mn id="S2.E12.m1.9.9.9.9.9.9.2" xref="S2.E12.m1.9.9.9.9.9.9.2.cmml">1</mn><mn id="S2.E12.m1.9.9.9.9.9.9.3" xref="S2.E12.m1.9.9.9.9.9.9.3.cmml">2</mn></mfrac><mo id="S2.E12.m1.25.25.25.25.25.27.1" xref="S2.E12.m1.90.90.1.1.1.cmml">⁢</mo><mrow id="S2.E12.m1.25.25.25.25.25.27.2"><munder id="S2.E12.m1.25.25.25.25.25.27.2.1"><mo id="S2.E12.m1.10.10.10.10.10.10" movablelimits="false" xref="S2.E12.m1.10.10.10.10.10.10.cmml">∑</mo><mrow id="S2.E12.m1.11.11.11.11.11.11.1" xref="S2.E12.m1.11.11.11.11.11.11.1.cmml"><mi id="S2.E12.m1.11.11.11.11.11.11.1.2" xref="S2.E12.m1.11.11.11.11.11.11.1.2.cmml">P</mi><mo id="S2.E12.m1.11.11.11.11.11.11.1.1" xref="S2.E12.m1.11.11.11.11.11.11.1.1.cmml">=</mo><mrow id="S2.E12.m1.11.11.11.11.11.11.1.3" xref="S2.E12.m1.11.11.11.11.11.11.1.3.cmml"><mi id="S2.E12.m1.11.11.11.11.11.11.1.3.2" xref="S2.E12.m1.11.11.11.11.11.11.1.3.2.cmml">Q</mi><mo id="S2.E12.m1.11.11.11.11.11.11.1.3.1" xref="S2.E12.m1.11.11.11.11.11.11.1.3.1.cmml">∪</mo><mi id="S2.E12.m1.11.11.11.11.11.11.1.3.3" xref="S2.E12.m1.11.11.11.11.11.11.1.3.3.cmml">R</mi></mrow></mrow></munder><mrow id="S2.E12.m1.25.25.25.25.25.27.2.2"><msubsup id="S2.E12.m1.25.25.25.25.25.27.2.2.2"><mi id="S2.E12.m1.12.12.12.12.12.12" xref="S2.E12.m1.12.12.12.12.12.12.cmml">I</mi><mi id="S2.E12.m1.13.13.13.13.13.13.1" xref="S2.E12.m1.13.13.13.13.13.13.1.cmml">Q</mi><mrow id="S2.E12.m1.14.14.14.14.14.14.1" xref="S2.E12.m1.14.14.14.14.14.14.1.cmml"><mi id="S2.E12.m1.14.14.14.14.14.14.1.2" xref="S2.E12.m1.14.14.14.14.14.14.1.2.cmml">ρ</mi><mo id="S2.E12.m1.14.14.14.14.14.14.1.1" xref="S2.E12.m1.14.14.14.14.14.14.1.1.cmml">⁢</mo><mi id="S2.E12.m1.14.14.14.14.14.14.1.3" xref="S2.E12.m1.14.14.14.14.14.14.1.3.cmml">σ</mi></mrow></msubsup><mo id="S2.E12.m1.25.25.25.25.25.27.2.2.1" xref="S2.E12.m1.90.90.1.1.1.cmml">⁢</mo><msubsup id="S2.E12.m1.25.25.25.25.25.27.2.2.3"><mi id="S2.E12.m1.15.15.15.15.15.15" xref="S2.E12.m1.15.15.15.15.15.15.cmml">I</mi><mi id="S2.E12.m1.16.16.16.16.16.16.1" xref="S2.E12.m1.16.16.16.16.16.16.1.cmml">R</mi><mrow id="S2.E12.m1.17.17.17.17.17.17.1" xref="S2.E12.m1.17.17.17.17.17.17.1.cmml"><mi id="S2.E12.m1.17.17.17.17.17.17.1.2" xref="S2.E12.m1.17.17.17.17.17.17.1.2.cmml">γ</mi><mo id="S2.E12.m1.17.17.17.17.17.17.1.1" xref="S2.E12.m1.17.17.17.17.17.17.1.1.cmml">⁢</mo><mi id="S2.E12.m1.17.17.17.17.17.17.1.3" xref="S2.E12.m1.17.17.17.17.17.17.1.3.cmml">λ</mi></mrow></msubsup><mo id="S2.E12.m1.25.25.25.25.25.27.2.2.1a" xref="S2.E12.m1.90.90.1.1.1.cmml">⁢</mo><msubsup id="S2.E12.m1.25.25.25.25.25.27.2.2.4"><mi class="ltx_font_mathcaligraphic" id="S2.E12.m1.18.18.18.18.18.18" xref="S2.E12.m1.18.18.18.18.18.18.cmml">𝒱</mi><mrow id="S2.E12.m1.20.20.20.20.20.20.1" xref="S2.E12.m1.20.20.20.20.20.20.1.cmml"><mi id="S2.E12.m1.20.20.20.20.20.20.1.2" xref="S2.E12.m1.20.20.20.20.20.20.1.2.cmml">μ</mi><mo id="S2.E12.m1.20.20.20.20.20.20.1.1" xref="S2.E12.m1.20.20.20.20.20.20.1.1.cmml">⁢</mo><mi id="S2.E12.m1.20.20.20.20.20.20.1.3" xref="S2.E12.m1.20.20.20.20.20.20.1.3.cmml">ν</mi><mo id="S2.E12.m1.20.20.20.20.20.20.1.1a" xref="S2.E12.m1.20.20.20.20.20.20.1.1.cmml">⁢</mo><mi id="S2.E12.m1.20.20.20.20.20.20.1.4" xref="S2.E12.m1.20.20.20.20.20.20.1.4.cmml">ρ</mi><mo id="S2.E12.m1.20.20.20.20.20.20.1.1b" xref="S2.E12.m1.20.20.20.20.20.20.1.1.cmml">⁢</mo><mi id="S2.E12.m1.20.20.20.20.20.20.1.5" xref="S2.E12.m1.20.20.20.20.20.20.1.5.cmml">σ</mi><mo id="S2.E12.m1.20.20.20.20.20.20.1.1c" xref="S2.E12.m1.20.20.20.20.20.20.1.1.cmml">⁢</mo><mi id="S2.E12.m1.20.20.20.20.20.20.1.6" xref="S2.E12.m1.20.20.20.20.20.20.1.6.cmml">γ</mi><mo id="S2.E12.m1.20.20.20.20.20.20.1.1d" xref="S2.E12.m1.20.20.20.20.20.20.1.1.cmml">⁢</mo><mi id="S2.E12.m1.20.20.20.20.20.20.1.7" xref="S2.E12.m1.20.20.20.20.20.20.1.7.cmml">λ</mi></mrow><mrow id="S2.E12.m1.19.19.19.19.19.19.1.3"><mo id="S2.E12.m1.19.19.19.19.19.19.1.3.1" stretchy="false">(</mo><mn id="S2.E12.m1.19.19.19.19.19.19.1.1" xref="S2.E12.m1.19.19.19.19.19.19.1.1.cmml">3</mn><mo id="S2.E12.m1.19.19.19.19.19.19.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S2.E12.m1.25.25.25.25.25.27.2.2.1b" xref="S2.E12.m1.90.90.1.1.1.cmml">⁢</mo><mrow id="S2.E12.m1.25.25.25.25.25.27.2.2.5"><mo id="S2.E12.m1.21.21.21.21.21.21" stretchy="false" xref="S2.E12.m1.90.90.1.1.1.cmml">(</mo><mi id="S2.E12.m1.22.22.22.22.22.22" xref="S2.E12.m1.22.22.22.22.22.22.cmml">Q</mi><mo id="S2.E12.m1.23.23.23.23.23.23" xref="S2.E12.m1.90.90.1.1.1.cmml">,</mo><mi id="S2.E12.m1.24.24.24.24.24.24" xref="S2.E12.m1.24.24.24.24.24.24.cmml">R</mi><mo id="S2.E12.m1.25.25.25.25.25.25" stretchy="false" xref="S2.E12.m1.90.90.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow></mrow></mtd></mtr><mtr id="S2.E12.m1.91.91.2c"><mtd class="ltx_align_right" columnalign="right" id="S2.E12.m1.91.91.2d"><mrow id="S2.E12.m1.47.47.47.22.22"><mo id="S2.E12.m1.47.47.47.22.22a" xref="S2.E12.m1.90.90.1.1.1.cmml">+</mo><mrow id="S2.E12.m1.47.47.47.22.22.23"><munder id="S2.E12.m1.47.47.47.22.22.23.1"><mo id="S2.E12.m1.27.27.27.2.2.2" movablelimits="false" xref="S2.E12.m1.27.27.27.2.2.2.cmml">∑</mo><mrow id="S2.E12.m1.28.28.28.3.3.3.1" xref="S2.E12.m1.28.28.28.3.3.3.1.cmml"><mi id="S2.E12.m1.28.28.28.3.3.3.1.2" xref="S2.E12.m1.28.28.28.3.3.3.1.2.cmml">P</mi><mo id="S2.E12.m1.28.28.28.3.3.3.1.1" xref="S2.E12.m1.28.28.28.3.3.3.1.1.cmml">=</mo><mrow id="S2.E12.m1.28.28.28.3.3.3.1.3" xref="S2.E12.m1.28.28.28.3.3.3.1.3.cmml"><mi id="S2.E12.m1.28.28.28.3.3.3.1.3.2" xref="S2.E12.m1.28.28.28.3.3.3.1.3.2.cmml">Q</mi><mo id="S2.E12.m1.28.28.28.3.3.3.1.3.1" xref="S2.E12.m1.28.28.28.3.3.3.1.3.1.cmml">∪</mo><mi id="S2.E12.m1.28.28.28.3.3.3.1.3.3" xref="S2.E12.m1.28.28.28.3.3.3.1.3.3.cmml">R</mi><mo id="S2.E12.m1.28.28.28.3.3.3.1.3.1a" xref="S2.E12.m1.28.28.28.3.3.3.1.3.1.cmml">∪</mo><mi id="S2.E12.m1.28.28.28.3.3.3.1.3.4" xref="S2.E12.m1.28.28.28.3.3.3.1.3.4.cmml">S</mi></mrow></mrow></munder><mrow id="S2.E12.m1.47.47.47.22.22.23.2"><msubsup id="S2.E12.m1.47.47.47.22.22.23.2.2"><mi id="S2.E12.m1.29.29.29.4.4.4" xref="S2.E12.m1.29.29.29.4.4.4.cmml">I</mi><mi id="S2.E12.m1.30.30.30.5.5.5.1" xref="S2.E12.m1.30.30.30.5.5.5.1.cmml">Q</mi><mrow id="S2.E12.m1.31.31.31.6.6.6.1" xref="S2.E12.m1.31.31.31.6.6.6.1.cmml"><mi id="S2.E12.m1.31.31.31.6.6.6.1.2" xref="S2.E12.m1.31.31.31.6.6.6.1.2.cmml">ρ</mi><mo id="S2.E12.m1.31.31.31.6.6.6.1.1" xref="S2.E12.m1.31.31.31.6.6.6.1.1.cmml">⁢</mo><mi id="S2.E12.m1.31.31.31.6.6.6.1.3" xref="S2.E12.m1.31.31.31.6.6.6.1.3.cmml">σ</mi></mrow></msubsup><mo id="S2.E12.m1.47.47.47.22.22.23.2.1" xref="S2.E12.m1.90.90.1.1.1.cmml">⁢</mo><msubsup id="S2.E12.m1.47.47.47.22.22.23.2.3"><mi id="S2.E12.m1.32.32.32.7.7.7" xref="S2.E12.m1.32.32.32.7.7.7.cmml">I</mi><mi id="S2.E12.m1.33.33.33.8.8.8.1" xref="S2.E12.m1.33.33.33.8.8.8.1.cmml">R</mi><mrow id="S2.E12.m1.34.34.34.9.9.9.1" xref="S2.E12.m1.34.34.34.9.9.9.1.cmml"><mi id="S2.E12.m1.34.34.34.9.9.9.1.2" xref="S2.E12.m1.34.34.34.9.9.9.1.2.cmml">γ</mi><mo id="S2.E12.m1.34.34.34.9.9.9.1.1" xref="S2.E12.m1.34.34.34.9.9.9.1.1.cmml">⁢</mo><mi id="S2.E12.m1.34.34.34.9.9.9.1.3" xref="S2.E12.m1.34.34.34.9.9.9.1.3.cmml">λ</mi></mrow></msubsup><mo id="S2.E12.m1.47.47.47.22.22.23.2.1a" xref="S2.E12.m1.90.90.1.1.1.cmml">⁢</mo><msubsup id="S2.E12.m1.47.47.47.22.22.23.2.4"><mi id="S2.E12.m1.35.35.35.10.10.10" xref="S2.E12.m1.35.35.35.10.10.10.cmml">H</mi><mi id="S2.E12.m1.36.36.36.11.11.11.1" xref="S2.E12.m1.36.36.36.11.11.11.1.cmml">S</mi><mrow id="S2.E12.m1.37.37.37.12.12.12.1" xref="S2.E12.m1.37.37.37.12.12.12.1.cmml"><mi id="S2.E12.m1.37.37.37.12.12.12.1.2" xref="S2.E12.m1.37.37.37.12.12.12.1.2.cmml">τ</mi><mo id="S2.E12.m1.37.37.37.12.12.12.1.1" xref="S2.E12.m1.37.37.37.12.12.12.1.1.cmml">⁢</mo><mi id="S2.E12.m1.37.37.37.12.12.12.1.3" xref="S2.E12.m1.37.37.37.12.12.12.1.3.cmml">δ</mi></mrow></msubsup><mo id="S2.E12.m1.47.47.47.22.22.23.2.1b" xref="S2.E12.m1.90.90.1.1.1.cmml">⁢</mo><msubsup id="S2.E12.m1.47.47.47.22.22.23.2.5"><mi id="S2.E12.m1.38.38.38.13.13.13" xref="S2.E12.m1.38.38.38.13.13.13.cmml">V</mi><mrow id="S2.E12.m1.40.40.40.15.15.15.1" xref="S2.E12.m1.40.40.40.15.15.15.1.cmml"><mi id="S2.E12.m1.40.40.40.15.15.15.1.2" xref="S2.E12.m1.40.40.40.15.15.15.1.2.cmml">μ</mi><mo id="S2.E12.m1.40.40.40.15.15.15.1.1" xref="S2.E12.m1.40.40.40.15.15.15.1.1.cmml">⁢</mo><mi id="S2.E12.m1.40.40.40.15.15.15.1.3" xref="S2.E12.m1.40.40.40.15.15.15.1.3.cmml">ν</mi><mo id="S2.E12.m1.40.40.40.15.15.15.1.1a" xref="S2.E12.m1.40.40.40.15.15.15.1.1.cmml">⁢</mo><mi id="S2.E12.m1.40.40.40.15.15.15.1.4" xref="S2.E12.m1.40.40.40.15.15.15.1.4.cmml">ρ</mi><mo id="S2.E12.m1.40.40.40.15.15.15.1.1b" xref="S2.E12.m1.40.40.40.15.15.15.1.1.cmml">⁢</mo><mi id="S2.E12.m1.40.40.40.15.15.15.1.5" xref="S2.E12.m1.40.40.40.15.15.15.1.5.cmml">σ</mi><mo id="S2.E12.m1.40.40.40.15.15.15.1.1c" xref="S2.E12.m1.40.40.40.15.15.15.1.1.cmml">⁢</mo><mi id="S2.E12.m1.40.40.40.15.15.15.1.6" xref="S2.E12.m1.40.40.40.15.15.15.1.6.cmml">γ</mi><mo id="S2.E12.m1.40.40.40.15.15.15.1.1d" xref="S2.E12.m1.40.40.40.15.15.15.1.1.cmml">⁢</mo><mi id="S2.E12.m1.40.40.40.15.15.15.1.7" xref="S2.E12.m1.40.40.40.15.15.15.1.7.cmml">λ</mi><mo id="S2.E12.m1.40.40.40.15.15.15.1.1e" xref="S2.E12.m1.40.40.40.15.15.15.1.1.cmml">⁢</mo><mi id="S2.E12.m1.40.40.40.15.15.15.1.8" xref="S2.E12.m1.40.40.40.15.15.15.1.8.cmml">τ</mi><mo id="S2.E12.m1.40.40.40.15.15.15.1.1f" xref="S2.E12.m1.40.40.40.15.15.15.1.1.cmml">⁢</mo><mi id="S2.E12.m1.40.40.40.15.15.15.1.9" xref="S2.E12.m1.40.40.40.15.15.15.1.9.cmml">δ</mi></mrow><mrow id="S2.E12.m1.39.39.39.14.14.14.1.3"><mo id="S2.E12.m1.39.39.39.14.14.14.1.3.1" stretchy="false">(</mo><mn id="S2.E12.m1.39.39.39.14.14.14.1.1" xref="S2.E12.m1.39.39.39.14.14.14.1.1.cmml">4</mn><mo id="S2.E12.m1.39.39.39.14.14.14.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S2.E12.m1.47.47.47.22.22.23.2.1c" xref="S2.E12.m1.90.90.1.1.1.cmml">⁢</mo><mrow id="S2.E12.m1.47.47.47.22.22.23.2.6"><mo id="S2.E12.m1.41.41.41.16.16.16" stretchy="false" xref="S2.E12.m1.90.90.1.1.1.cmml">(</mo><mi id="S2.E12.m1.42.42.42.17.17.17" xref="S2.E12.m1.42.42.42.17.17.17.cmml">Q</mi><mo id="S2.E12.m1.43.43.43.18.18.18" xref="S2.E12.m1.90.90.1.1.1.cmml">,</mo><mi id="S2.E12.m1.44.44.44.19.19.19" xref="S2.E12.m1.44.44.44.19.19.19.cmml">R</mi><mo id="S2.E12.m1.45.45.45.20.20.20" xref="S2.E12.m1.90.90.1.1.1.cmml">,</mo><mi id="S2.E12.m1.46.46.46.21.21.21" xref="S2.E12.m1.46.46.46.21.21.21.cmml">S</mi><mo id="S2.E12.m1.47.47.47.22.22.22" stretchy="false" xref="S2.E12.m1.90.90.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr id="S2.E12.m1.91.91.2e"><mtd class="ltx_align_right" columnalign="right" id="S2.E12.m1.91.91.2f"><mrow id="S2.E12.m1.72.72.72.25.25"><mo id="S2.E12.m1.72.72.72.25.25a" xref="S2.E12.m1.90.90.1.1.1.cmml">+</mo><mrow id="S2.E12.m1.72.72.72.25.25.26"><munder id="S2.E12.m1.72.72.72.25.25.26.1"><mo id="S2.E12.m1.49.49.49.2.2.2" movablelimits="false" xref="S2.E12.m1.49.49.49.2.2.2.cmml">∑</mo><mrow id="S2.E12.m1.50.50.50.3.3.3.1" xref="S2.E12.m1.50.50.50.3.3.3.1.cmml"><mi id="S2.E12.m1.50.50.50.3.3.3.1.2" xref="S2.E12.m1.50.50.50.3.3.3.1.2.cmml">P</mi><mo id="S2.E12.m1.50.50.50.3.3.3.1.1" xref="S2.E12.m1.50.50.50.3.3.3.1.1.cmml">=</mo><mrow id="S2.E12.m1.50.50.50.3.3.3.1.3" xref="S2.E12.m1.50.50.50.3.3.3.1.3.cmml"><mi id="S2.E12.m1.50.50.50.3.3.3.1.3.2" xref="S2.E12.m1.50.50.50.3.3.3.1.3.2.cmml">Q</mi><mo id="S2.E12.m1.50.50.50.3.3.3.1.3.1" xref="S2.E12.m1.50.50.50.3.3.3.1.3.1.cmml">∪</mo><mi id="S2.E12.m1.50.50.50.3.3.3.1.3.3" xref="S2.E12.m1.50.50.50.3.3.3.1.3.3.cmml">R</mi><mo id="S2.E12.m1.50.50.50.3.3.3.1.3.1a" xref="S2.E12.m1.50.50.50.3.3.3.1.3.1.cmml">∪</mo><mi id="S2.E12.m1.50.50.50.3.3.3.1.3.4" xref="S2.E12.m1.50.50.50.3.3.3.1.3.4.cmml">S</mi><mo id="S2.E12.m1.50.50.50.3.3.3.1.3.1b" xref="S2.E12.m1.50.50.50.3.3.3.1.3.1.cmml">∪</mo><mi id="S2.E12.m1.50.50.50.3.3.3.1.3.5" xref="S2.E12.m1.50.50.50.3.3.3.1.3.5.cmml">T</mi></mrow></mrow></munder><mrow id="S2.E12.m1.72.72.72.25.25.26.2"><msubsup id="S2.E12.m1.72.72.72.25.25.26.2.2"><mi id="S2.E12.m1.51.51.51.4.4.4" xref="S2.E12.m1.51.51.51.4.4.4.cmml">I</mi><mi id="S2.E12.m1.52.52.52.5.5.5.1" xref="S2.E12.m1.52.52.52.5.5.5.1.cmml">Q</mi><mrow id="S2.E12.m1.53.53.53.6.6.6.1" xref="S2.E12.m1.53.53.53.6.6.6.1.cmml"><mi id="S2.E12.m1.53.53.53.6.6.6.1.2" xref="S2.E12.m1.53.53.53.6.6.6.1.2.cmml">ρ</mi><mo id="S2.E12.m1.53.53.53.6.6.6.1.1" xref="S2.E12.m1.53.53.53.6.6.6.1.1.cmml">⁢</mo><mi id="S2.E12.m1.53.53.53.6.6.6.1.3" xref="S2.E12.m1.53.53.53.6.6.6.1.3.cmml">σ</mi></mrow></msubsup><mo id="S2.E12.m1.72.72.72.25.25.26.2.1" xref="S2.E12.m1.90.90.1.1.1.cmml">⁢</mo><msubsup id="S2.E12.m1.72.72.72.25.25.26.2.3"><mi id="S2.E12.m1.54.54.54.7.7.7" xref="S2.E12.m1.54.54.54.7.7.7.cmml">I</mi><mi id="S2.E12.m1.55.55.55.8.8.8.1" xref="S2.E12.m1.55.55.55.8.8.8.1.cmml">R</mi><mrow id="S2.E12.m1.56.56.56.9.9.9.1" xref="S2.E12.m1.56.56.56.9.9.9.1.cmml"><mi id="S2.E12.m1.56.56.56.9.9.9.1.2" xref="S2.E12.m1.56.56.56.9.9.9.1.2.cmml">γ</mi><mo id="S2.E12.m1.56.56.56.9.9.9.1.1" xref="S2.E12.m1.56.56.56.9.9.9.1.1.cmml">⁢</mo><mi id="S2.E12.m1.56.56.56.9.9.9.1.3" xref="S2.E12.m1.56.56.56.9.9.9.1.3.cmml">λ</mi></mrow></msubsup><mo id="S2.E12.m1.72.72.72.25.25.26.2.1a" xref="S2.E12.m1.90.90.1.1.1.cmml">⁢</mo><msubsup id="S2.E12.m1.72.72.72.25.25.26.2.4"><mi id="S2.E12.m1.57.57.57.10.10.10" xref="S2.E12.m1.57.57.57.10.10.10.cmml">H</mi><mi id="S2.E12.m1.58.58.58.11.11.11.1" xref="S2.E12.m1.58.58.58.11.11.11.1.cmml">S</mi><mrow id="S2.E12.m1.59.59.59.12.12.12.1" xref="S2.E12.m1.59.59.59.12.12.12.1.cmml"><mi id="S2.E12.m1.59.59.59.12.12.12.1.2" xref="S2.E12.m1.59.59.59.12.12.12.1.2.cmml">τ</mi><mo id="S2.E12.m1.59.59.59.12.12.12.1.1" xref="S2.E12.m1.59.59.59.12.12.12.1.1.cmml">⁢</mo><mi id="S2.E12.m1.59.59.59.12.12.12.1.3" xref="S2.E12.m1.59.59.59.12.12.12.1.3.cmml">δ</mi></mrow></msubsup><mo id="S2.E12.m1.72.72.72.25.25.26.2.1b" xref="S2.E12.m1.90.90.1.1.1.cmml">⁢</mo><msubsup id="S2.E12.m1.72.72.72.25.25.26.2.5"><mi id="S2.E12.m1.60.60.60.13.13.13" xref="S2.E12.m1.60.60.60.13.13.13.cmml">H</mi><mi id="S2.E12.m1.61.61.61.14.14.14.1" xref="S2.E12.m1.61.61.61.14.14.14.1.cmml">T</mi><mrow id="S2.E12.m1.62.62.62.15.15.15.1" xref="S2.E12.m1.62.62.62.15.15.15.1.cmml"><mi id="S2.E12.m1.62.62.62.15.15.15.1.2" xref="S2.E12.m1.62.62.62.15.15.15.1.2.cmml">α</mi><mo id="S2.E12.m1.62.62.62.15.15.15.1.1" xref="S2.E12.m1.62.62.62.15.15.15.1.1.cmml">⁢</mo><mi id="S2.E12.m1.62.62.62.15.15.15.1.3" xref="S2.E12.m1.62.62.62.15.15.15.1.3.cmml">β</mi></mrow></msubsup><mo id="S2.E12.m1.72.72.72.25.25.26.2.1c" xref="S2.E12.m1.90.90.1.1.1.cmml">⁢</mo><msubsup id="S2.E12.m1.72.72.72.25.25.26.2.6"><mi id="S2.E12.m1.63.63.63.16.16.16" xref="S2.E12.m1.63.63.63.16.16.16.cmml">V</mi><mrow id="S2.E12.m1.65.65.65.18.18.18.1" xref="S2.E12.m1.65.65.65.18.18.18.1.cmml"><mi id="S2.E12.m1.65.65.65.18.18.18.1.2" xref="S2.E12.m1.65.65.65.18.18.18.1.2.cmml">μ</mi><mo id="S2.E12.m1.65.65.65.18.18.18.1.1" xref="S2.E12.m1.65.65.65.18.18.18.1.1.cmml">⁢</mo><mi id="S2.E12.m1.65.65.65.18.18.18.1.3" xref="S2.E12.m1.65.65.65.18.18.18.1.3.cmml">ν</mi><mo id="S2.E12.m1.65.65.65.18.18.18.1.1a" xref="S2.E12.m1.65.65.65.18.18.18.1.1.cmml">⁢</mo><mi id="S2.E12.m1.65.65.65.18.18.18.1.4" xref="S2.E12.m1.65.65.65.18.18.18.1.4.cmml">ρ</mi><mo id="S2.E12.m1.65.65.65.18.18.18.1.1b" xref="S2.E12.m1.65.65.65.18.18.18.1.1.cmml">⁢</mo><mi id="S2.E12.m1.65.65.65.18.18.18.1.5" xref="S2.E12.m1.65.65.65.18.18.18.1.5.cmml">σ</mi><mo id="S2.E12.m1.65.65.65.18.18.18.1.1c" xref="S2.E12.m1.65.65.65.18.18.18.1.1.cmml">⁢</mo><mi id="S2.E12.m1.65.65.65.18.18.18.1.6" xref="S2.E12.m1.65.65.65.18.18.18.1.6.cmml">γ</mi><mo id="S2.E12.m1.65.65.65.18.18.18.1.1d" xref="S2.E12.m1.65.65.65.18.18.18.1.1.cmml">⁢</mo><mi id="S2.E12.m1.65.65.65.18.18.18.1.7" xref="S2.E12.m1.65.65.65.18.18.18.1.7.cmml">λ</mi><mo id="S2.E12.m1.65.65.65.18.18.18.1.1e" xref="S2.E12.m1.65.65.65.18.18.18.1.1.cmml">⁢</mo><mi id="S2.E12.m1.65.65.65.18.18.18.1.8" xref="S2.E12.m1.65.65.65.18.18.18.1.8.cmml">τ</mi><mo id="S2.E12.m1.65.65.65.18.18.18.1.1f" xref="S2.E12.m1.65.65.65.18.18.18.1.1.cmml">⁢</mo><mi id="S2.E12.m1.65.65.65.18.18.18.1.9" xref="S2.E12.m1.65.65.65.18.18.18.1.9.cmml">δ</mi><mo id="S2.E12.m1.65.65.65.18.18.18.1.1g" xref="S2.E12.m1.65.65.65.18.18.18.1.1.cmml">⁢</mo><mi id="S2.E12.m1.65.65.65.18.18.18.1.10" xref="S2.E12.m1.65.65.65.18.18.18.1.10.cmml">α</mi><mo id="S2.E12.m1.65.65.65.18.18.18.1.1h" xref="S2.E12.m1.65.65.65.18.18.18.1.1.cmml">⁢</mo><mi id="S2.E12.m1.65.65.65.18.18.18.1.11" xref="S2.E12.m1.65.65.65.18.18.18.1.11.cmml">β</mi></mrow><mrow id="S2.E12.m1.64.64.64.17.17.17.1.3"><mo id="S2.E12.m1.64.64.64.17.17.17.1.3.1" stretchy="false">(</mo><mn id="S2.E12.m1.64.64.64.17.17.17.1.1" xref="S2.E12.m1.64.64.64.17.17.17.1.1.cmml">5</mn><mo id="S2.E12.m1.64.64.64.17.17.17.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S2.E12.m1.72.72.72.25.25.26.2.1d" xref="S2.E12.m1.90.90.1.1.1.cmml">⁢</mo><mrow id="S2.E12.m1.72.72.72.25.25.26.2.7"><mo id="S2.E12.m1.66.66.66.19.19.19" stretchy="false" xref="S2.E12.m1.90.90.1.1.1.cmml">(</mo><mi id="S2.E12.m1.67.67.67.20.20.20" xref="S2.E12.m1.67.67.67.20.20.20.cmml">Q</mi><mo id="S2.E12.m1.68.68.68.21.21.21" xref="S2.E12.m1.90.90.1.1.1.cmml">,</mo><mi id="S2.E12.m1.69.69.69.22.22.22" xref="S2.E12.m1.69.69.69.22.22.22.cmml">R</mi><mo id="S2.E12.m1.70.70.70.23.23.23" xref="S2.E12.m1.90.90.1.1.1.cmml">,</mo><mi id="S2.E12.m1.71.71.71.24.24.24" xref="S2.E12.m1.71.71.71.24.24.24.cmml">S</mi><mo id="S2.E12.m1.72.72.72.25.25.25" stretchy="false" xref="S2.E12.m1.90.90.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr id="S2.E12.m1.91.91.2g"><mtd class="ltx_align_right" columnalign="right" id="S2.E12.m1.91.91.2h"><mrow id="S2.E12.m1.91.91.2.90.18.18.18"><mrow 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xref="S2.E12.m1.87.87.87.15.15.15">𝑄</ci></interval></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E12.m1.91c">s_{P}\mathbb{P}_{\mu\nu\rho\sigma}I_{P}^{\rho\sigma}=\frac{1}{2}\sum_{P=Q\cup R% }I_{Q}^{\rho\sigma}I_{R}^{\gamma\lambda}\mathcal{V}^{(3)}_{\mu\nu\rho\sigma% \gamma\lambda}(Q,R)\\ +\sum_{P=Q\cup R\cup S}I_{Q}^{\rho\sigma}I_{R}^{\gamma\lambda}H_{S}^{\tau% \delta}V^{(4)}_{\mu\nu\rho\sigma\gamma\lambda\tau\delta}(Q,R,S)\\ +\sum_{P=Q\cup R\cup S\cup T}I_{Q}^{\rho\sigma}I_{R}^{\gamma\lambda}H_{S}^{% \tau\delta}H_{T}^{\alpha\beta}V^{(5)}_{\mu\nu\rho\sigma\gamma\lambda\tau\delta% \alpha\beta}(Q,R,S)\\ +\sum_{P=Q\cup R}B_{Q\rho}C_{R}^{\sigma}\mathcal{V}^{(\textrm{g})\rho}_{\mu\nu% \sigma}(P,Q),</annotation><annotation encoding="application/x-llamapun" id="S2.E12.m1.91d">start_ROW start_CELL italic_s start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT blackboard_P start_POSTSUBSCRIPT italic_μ italic_ν italic_ρ italic_σ end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ρ italic_σ end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∑ start_POSTSUBSCRIPT italic_P = italic_Q ∪ italic_R end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ρ italic_σ end_POSTSUPERSCRIPT italic_I start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_γ italic_λ end_POSTSUPERSCRIPT caligraphic_V start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ italic_ν italic_ρ italic_σ italic_γ italic_λ end_POSTSUBSCRIPT ( italic_Q , italic_R ) end_CELL end_ROW start_ROW start_CELL + ∑ start_POSTSUBSCRIPT italic_P = italic_Q ∪ italic_R ∪ italic_S end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ρ italic_σ end_POSTSUPERSCRIPT italic_I start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_γ italic_λ end_POSTSUPERSCRIPT italic_H start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_τ italic_δ end_POSTSUPERSCRIPT italic_V start_POSTSUPERSCRIPT ( 4 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ italic_ν italic_ρ italic_σ italic_γ italic_λ italic_τ italic_δ end_POSTSUBSCRIPT ( italic_Q , italic_R , italic_S ) end_CELL end_ROW start_ROW start_CELL + ∑ start_POSTSUBSCRIPT italic_P = italic_Q ∪ italic_R ∪ italic_S ∪ italic_T end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ρ italic_σ end_POSTSUPERSCRIPT italic_I start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_γ italic_λ end_POSTSUPERSCRIPT italic_H start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_τ italic_δ end_POSTSUPERSCRIPT italic_H start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_α italic_β end_POSTSUPERSCRIPT italic_V start_POSTSUPERSCRIPT ( 5 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ italic_ν italic_ρ italic_σ italic_γ italic_λ italic_τ italic_δ italic_α italic_β end_POSTSUBSCRIPT ( italic_Q , italic_R , italic_S ) end_CELL end_ROW start_ROW start_CELL + ∑ start_POSTSUBSCRIPT italic_P = italic_Q ∪ italic_R end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_Q italic_ρ end_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT caligraphic_V start_POSTSUPERSCRIPT ( g ) italic_ρ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ italic_ν italic_σ end_POSTSUBSCRIPT ( italic_P , italic_Q ) , end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(12)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p6.2">where we have <math alttext="s_{P}=k^{2}_{P}" class="ltx_Math" display="inline" id="S2.p6.2.m1.1"><semantics id="S2.p6.2.m1.1a"><mrow id="S2.p6.2.m1.1.1" xref="S2.p6.2.m1.1.1.cmml"><msub id="S2.p6.2.m1.1.1.2" xref="S2.p6.2.m1.1.1.2.cmml"><mi id="S2.p6.2.m1.1.1.2.2" xref="S2.p6.2.m1.1.1.2.2.cmml">s</mi><mi id="S2.p6.2.m1.1.1.2.3" xref="S2.p6.2.m1.1.1.2.3.cmml">P</mi></msub><mo id="S2.p6.2.m1.1.1.1" xref="S2.p6.2.m1.1.1.1.cmml">=</mo><msubsup id="S2.p6.2.m1.1.1.3" xref="S2.p6.2.m1.1.1.3.cmml"><mi id="S2.p6.2.m1.1.1.3.2.2" xref="S2.p6.2.m1.1.1.3.2.2.cmml">k</mi><mi id="S2.p6.2.m1.1.1.3.3" xref="S2.p6.2.m1.1.1.3.3.cmml">P</mi><mn id="S2.p6.2.m1.1.1.3.2.3" xref="S2.p6.2.m1.1.1.3.2.3.cmml">2</mn></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.p6.2.m1.1b"><apply id="S2.p6.2.m1.1.1.cmml" xref="S2.p6.2.m1.1.1"><eq id="S2.p6.2.m1.1.1.1.cmml" xref="S2.p6.2.m1.1.1.1"></eq><apply id="S2.p6.2.m1.1.1.2.cmml" xref="S2.p6.2.m1.1.1.2"><csymbol cd="ambiguous" id="S2.p6.2.m1.1.1.2.1.cmml" xref="S2.p6.2.m1.1.1.2">subscript</csymbol><ci id="S2.p6.2.m1.1.1.2.2.cmml" xref="S2.p6.2.m1.1.1.2.2">𝑠</ci><ci id="S2.p6.2.m1.1.1.2.3.cmml" xref="S2.p6.2.m1.1.1.2.3">𝑃</ci></apply><apply id="S2.p6.2.m1.1.1.3.cmml" xref="S2.p6.2.m1.1.1.3"><csymbol cd="ambiguous" id="S2.p6.2.m1.1.1.3.1.cmml" xref="S2.p6.2.m1.1.1.3">subscript</csymbol><apply id="S2.p6.2.m1.1.1.3.2.cmml" xref="S2.p6.2.m1.1.1.3"><csymbol cd="ambiguous" id="S2.p6.2.m1.1.1.3.2.1.cmml" xref="S2.p6.2.m1.1.1.3">superscript</csymbol><ci id="S2.p6.2.m1.1.1.3.2.2.cmml" xref="S2.p6.2.m1.1.1.3.2.2">𝑘</ci><cn id="S2.p6.2.m1.1.1.3.2.3.cmml" type="integer" xref="S2.p6.2.m1.1.1.3.2.3">2</cn></apply><ci id="S2.p6.2.m1.1.1.3.3.cmml" xref="S2.p6.2.m1.1.1.3.3">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.2.m1.1c">s_{P}=k^{2}_{P}</annotation><annotation encoding="application/x-llamapun" id="S2.p6.2.m1.1d">italic_s start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT = italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math>, and</p> <table class="ltx_equation ltx_eqn_table" id="S2.E13"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathbb{P}_{\mu\nu\rho\sigma}=\frac{1}{2}\eta_{\mu\rho}\eta_{\nu\sigma}+\frac{% 1}{2}\eta_{\mu\sigma}\eta_{\nu\rho}+\frac{1}{(2-d)}\eta_{\mu\nu}\eta_{\rho% \sigma}." class="ltx_Math" display="block" id="S2.E13.m1.2"><semantics id="S2.E13.m1.2a"><mrow id="S2.E13.m1.2.2.1" xref="S2.E13.m1.2.2.1.1.cmml"><mrow id="S2.E13.m1.2.2.1.1" xref="S2.E13.m1.2.2.1.1.cmml"><msub id="S2.E13.m1.2.2.1.1.2" xref="S2.E13.m1.2.2.1.1.2.cmml"><mi id="S2.E13.m1.2.2.1.1.2.2" xref="S2.E13.m1.2.2.1.1.2.2.cmml">ℙ</mi><mrow id="S2.E13.m1.2.2.1.1.2.3" xref="S2.E13.m1.2.2.1.1.2.3.cmml"><mi id="S2.E13.m1.2.2.1.1.2.3.2" xref="S2.E13.m1.2.2.1.1.2.3.2.cmml">μ</mi><mo id="S2.E13.m1.2.2.1.1.2.3.1" 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start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_ρ italic_σ end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(13)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p6.3">We define also the inverse of <math alttext="\mathbb{P}_{\mu\nu\rho\sigma}" class="ltx_Math" display="inline" id="S2.p6.3.m1.1"><semantics id="S2.p6.3.m1.1a"><msub id="S2.p6.3.m1.1.1" xref="S2.p6.3.m1.1.1.cmml"><mi id="S2.p6.3.m1.1.1.2" xref="S2.p6.3.m1.1.1.2.cmml">ℙ</mi><mrow id="S2.p6.3.m1.1.1.3" xref="S2.p6.3.m1.1.1.3.cmml"><mi id="S2.p6.3.m1.1.1.3.2" xref="S2.p6.3.m1.1.1.3.2.cmml">μ</mi><mo id="S2.p6.3.m1.1.1.3.1" xref="S2.p6.3.m1.1.1.3.1.cmml">⁢</mo><mi id="S2.p6.3.m1.1.1.3.3" xref="S2.p6.3.m1.1.1.3.3.cmml">ν</mi><mo id="S2.p6.3.m1.1.1.3.1a" 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start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_μ italic_ν italic_ρ italic_σ end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( italic_η start_POSTSUPERSCRIPT italic_μ italic_ρ end_POSTSUPERSCRIPT italic_η start_POSTSUPERSCRIPT italic_ν italic_σ end_POSTSUPERSCRIPT + italic_η start_POSTSUPERSCRIPT italic_μ italic_σ end_POSTSUPERSCRIPT italic_η start_POSTSUPERSCRIPT italic_ν italic_ρ end_POSTSUPERSCRIPT - italic_η start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT italic_η start_POSTSUPERSCRIPT italic_ρ italic_σ end_POSTSUPERSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(14)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p6.8">The recursion sums in (<a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S2.E12" title="In II Equations of motion ‣ One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_tag">12</span></a>) go over the possible disjoint, ordered sub-words <math alttext="\{Q,R,\ldots\}" class="ltx_Math" display="inline" id="S2.p6.4.m1.3"><semantics id="S2.p6.4.m1.3a"><mrow id="S2.p6.4.m1.3.4.2" xref="S2.p6.4.m1.3.4.1.cmml"><mo id="S2.p6.4.m1.3.4.2.1" stretchy="false" xref="S2.p6.4.m1.3.4.1.cmml">{</mo><mi id="S2.p6.4.m1.1.1" xref="S2.p6.4.m1.1.1.cmml">Q</mi><mo id="S2.p6.4.m1.3.4.2.2" xref="S2.p6.4.m1.3.4.1.cmml">,</mo><mi id="S2.p6.4.m1.2.2" xref="S2.p6.4.m1.2.2.cmml">R</mi><mo id="S2.p6.4.m1.3.4.2.3" xref="S2.p6.4.m1.3.4.1.cmml">,</mo><mi id="S2.p6.4.m1.3.3" mathvariant="normal" xref="S2.p6.4.m1.3.3.cmml">…</mi><mo id="S2.p6.4.m1.3.4.2.4" stretchy="false" xref="S2.p6.4.m1.3.4.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p6.4.m1.3b"><set id="S2.p6.4.m1.3.4.1.cmml" xref="S2.p6.4.m1.3.4.2"><ci id="S2.p6.4.m1.1.1.cmml" xref="S2.p6.4.m1.1.1">𝑄</ci><ci id="S2.p6.4.m1.2.2.cmml" xref="S2.p6.4.m1.2.2">𝑅</ci><ci id="S2.p6.4.m1.3.3.cmml" xref="S2.p6.4.m1.3.3">…</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.4.m1.3c">\{Q,R,\ldots\}</annotation><annotation encoding="application/x-llamapun" id="S2.p6.4.m1.3d">{ italic_Q , italic_R , … }</annotation></semantics></math> that form the word <math alttext="P=Q\cup R\cup\ldots" class="ltx_Math" display="inline" id="S2.p6.5.m2.1"><semantics id="S2.p6.5.m2.1a"><mrow id="S2.p6.5.m2.1.1" xref="S2.p6.5.m2.1.1.cmml"><mi id="S2.p6.5.m2.1.1.2" xref="S2.p6.5.m2.1.1.2.cmml">P</mi><mo id="S2.p6.5.m2.1.1.1" xref="S2.p6.5.m2.1.1.1.cmml">=</mo><mrow id="S2.p6.5.m2.1.1.3" xref="S2.p6.5.m2.1.1.3.cmml"><mi id="S2.p6.5.m2.1.1.3.2" xref="S2.p6.5.m2.1.1.3.2.cmml">Q</mi><mo id="S2.p6.5.m2.1.1.3.1" xref="S2.p6.5.m2.1.1.3.1.cmml">∪</mo><mi id="S2.p6.5.m2.1.1.3.3" xref="S2.p6.5.m2.1.1.3.3.cmml">R</mi><mo id="S2.p6.5.m2.1.1.3.1a" xref="S2.p6.5.m2.1.1.3.1.cmml">∪</mo><mi id="S2.p6.5.m2.1.1.3.4" mathvariant="normal" xref="S2.p6.5.m2.1.1.3.4.cmml">…</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p6.5.m2.1b"><apply id="S2.p6.5.m2.1.1.cmml" xref="S2.p6.5.m2.1.1"><eq id="S2.p6.5.m2.1.1.1.cmml" xref="S2.p6.5.m2.1.1.1"></eq><ci id="S2.p6.5.m2.1.1.2.cmml" xref="S2.p6.5.m2.1.1.2">𝑃</ci><apply id="S2.p6.5.m2.1.1.3.cmml" xref="S2.p6.5.m2.1.1.3"><union id="S2.p6.5.m2.1.1.3.1.cmml" xref="S2.p6.5.m2.1.1.3.1"></union><ci id="S2.p6.5.m2.1.1.3.2.cmml" xref="S2.p6.5.m2.1.1.3.2">𝑄</ci><ci id="S2.p6.5.m2.1.1.3.3.cmml" xref="S2.p6.5.m2.1.1.3.3">𝑅</ci><ci id="S2.p6.5.m2.1.1.3.4.cmml" xref="S2.p6.5.m2.1.1.3.4">…</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.5.m2.1c">P=Q\cup R\cup\ldots</annotation><annotation encoding="application/x-llamapun" id="S2.p6.5.m2.1d">italic_P = italic_Q ∪ italic_R ∪ …</annotation></semantics></math>. The current <math alttext="H_{P\mu\nu}" class="ltx_Math" display="inline" id="S2.p6.6.m3.1"><semantics id="S2.p6.6.m3.1a"><msub id="S2.p6.6.m3.1.1" xref="S2.p6.6.m3.1.1.cmml"><mi id="S2.p6.6.m3.1.1.2" xref="S2.p6.6.m3.1.1.2.cmml">H</mi><mrow id="S2.p6.6.m3.1.1.3" xref="S2.p6.6.m3.1.1.3.cmml"><mi id="S2.p6.6.m3.1.1.3.2" xref="S2.p6.6.m3.1.1.3.2.cmml">P</mi><mo id="S2.p6.6.m3.1.1.3.1" xref="S2.p6.6.m3.1.1.3.1.cmml">⁢</mo><mi id="S2.p6.6.m3.1.1.3.3" xref="S2.p6.6.m3.1.1.3.3.cmml">μ</mi><mo id="S2.p6.6.m3.1.1.3.1a" xref="S2.p6.6.m3.1.1.3.1.cmml">⁢</mo><mi id="S2.p6.6.m3.1.1.3.4" xref="S2.p6.6.m3.1.1.3.4.cmml">ν</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.p6.6.m3.1b"><apply id="S2.p6.6.m3.1.1.cmml" xref="S2.p6.6.m3.1.1"><csymbol cd="ambiguous" id="S2.p6.6.m3.1.1.1.cmml" xref="S2.p6.6.m3.1.1">subscript</csymbol><ci id="S2.p6.6.m3.1.1.2.cmml" xref="S2.p6.6.m3.1.1.2">𝐻</ci><apply id="S2.p6.6.m3.1.1.3.cmml" xref="S2.p6.6.m3.1.1.3"><times id="S2.p6.6.m3.1.1.3.1.cmml" xref="S2.p6.6.m3.1.1.3.1"></times><ci id="S2.p6.6.m3.1.1.3.2.cmml" xref="S2.p6.6.m3.1.1.3.2">𝑃</ci><ci id="S2.p6.6.m3.1.1.3.3.cmml" xref="S2.p6.6.m3.1.1.3.3">𝜇</ci><ci id="S2.p6.6.m3.1.1.3.4.cmml" xref="S2.p6.6.m3.1.1.3.4">𝜈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.6.m3.1c">H_{P\mu\nu}</annotation><annotation encoding="application/x-llamapun" id="S2.p6.6.m3.1d">italic_H start_POSTSUBSCRIPT italic_P italic_μ italic_ν end_POSTSUBSCRIPT</annotation></semantics></math> is implicitly determined by <math alttext="\mathfrak{g}^{\mu\rho}\mathfrak{g}_{\nu\rho}=\delta^{\mu}_{\nu}" class="ltx_Math" display="inline" id="S2.p6.7.m4.1"><semantics id="S2.p6.7.m4.1a"><mrow id="S2.p6.7.m4.1.1" xref="S2.p6.7.m4.1.1.cmml"><mrow id="S2.p6.7.m4.1.1.2" xref="S2.p6.7.m4.1.1.2.cmml"><msup id="S2.p6.7.m4.1.1.2.2" xref="S2.p6.7.m4.1.1.2.2.cmml"><mi id="S2.p6.7.m4.1.1.2.2.2" xref="S2.p6.7.m4.1.1.2.2.2.cmml">𝔤</mi><mrow id="S2.p6.7.m4.1.1.2.2.3" xref="S2.p6.7.m4.1.1.2.2.3.cmml"><mi id="S2.p6.7.m4.1.1.2.2.3.2" xref="S2.p6.7.m4.1.1.2.2.3.2.cmml">μ</mi><mo id="S2.p6.7.m4.1.1.2.2.3.1" xref="S2.p6.7.m4.1.1.2.2.3.1.cmml">⁢</mo><mi id="S2.p6.7.m4.1.1.2.2.3.3" xref="S2.p6.7.m4.1.1.2.2.3.3.cmml">ρ</mi></mrow></msup><mo id="S2.p6.7.m4.1.1.2.1" xref="S2.p6.7.m4.1.1.2.1.cmml">⁢</mo><msub id="S2.p6.7.m4.1.1.2.3" xref="S2.p6.7.m4.1.1.2.3.cmml"><mi id="S2.p6.7.m4.1.1.2.3.2" xref="S2.p6.7.m4.1.1.2.3.2.cmml">𝔤</mi><mrow id="S2.p6.7.m4.1.1.2.3.3" xref="S2.p6.7.m4.1.1.2.3.3.cmml"><mi id="S2.p6.7.m4.1.1.2.3.3.2" xref="S2.p6.7.m4.1.1.2.3.3.2.cmml">ν</mi><mo id="S2.p6.7.m4.1.1.2.3.3.1" xref="S2.p6.7.m4.1.1.2.3.3.1.cmml">⁢</mo><mi id="S2.p6.7.m4.1.1.2.3.3.3" xref="S2.p6.7.m4.1.1.2.3.3.3.cmml">ρ</mi></mrow></msub></mrow><mo id="S2.p6.7.m4.1.1.1" xref="S2.p6.7.m4.1.1.1.cmml">=</mo><msubsup id="S2.p6.7.m4.1.1.3" xref="S2.p6.7.m4.1.1.3.cmml"><mi id="S2.p6.7.m4.1.1.3.2.2" xref="S2.p6.7.m4.1.1.3.2.2.cmml">δ</mi><mi id="S2.p6.7.m4.1.1.3.3" xref="S2.p6.7.m4.1.1.3.3.cmml">ν</mi><mi id="S2.p6.7.m4.1.1.3.2.3" xref="S2.p6.7.m4.1.1.3.2.3.cmml">μ</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.p6.7.m4.1b"><apply id="S2.p6.7.m4.1.1.cmml" xref="S2.p6.7.m4.1.1"><eq id="S2.p6.7.m4.1.1.1.cmml" xref="S2.p6.7.m4.1.1.1"></eq><apply id="S2.p6.7.m4.1.1.2.cmml" xref="S2.p6.7.m4.1.1.2"><times id="S2.p6.7.m4.1.1.2.1.cmml" xref="S2.p6.7.m4.1.1.2.1"></times><apply id="S2.p6.7.m4.1.1.2.2.cmml" xref="S2.p6.7.m4.1.1.2.2"><csymbol cd="ambiguous" id="S2.p6.7.m4.1.1.2.2.1.cmml" xref="S2.p6.7.m4.1.1.2.2">superscript</csymbol><ci id="S2.p6.7.m4.1.1.2.2.2.cmml" xref="S2.p6.7.m4.1.1.2.2.2">𝔤</ci><apply id="S2.p6.7.m4.1.1.2.2.3.cmml" xref="S2.p6.7.m4.1.1.2.2.3"><times id="S2.p6.7.m4.1.1.2.2.3.1.cmml" xref="S2.p6.7.m4.1.1.2.2.3.1"></times><ci id="S2.p6.7.m4.1.1.2.2.3.2.cmml" xref="S2.p6.7.m4.1.1.2.2.3.2">𝜇</ci><ci id="S2.p6.7.m4.1.1.2.2.3.3.cmml" xref="S2.p6.7.m4.1.1.2.2.3.3">𝜌</ci></apply></apply><apply id="S2.p6.7.m4.1.1.2.3.cmml" xref="S2.p6.7.m4.1.1.2.3"><csymbol cd="ambiguous" id="S2.p6.7.m4.1.1.2.3.1.cmml" xref="S2.p6.7.m4.1.1.2.3">subscript</csymbol><ci id="S2.p6.7.m4.1.1.2.3.2.cmml" xref="S2.p6.7.m4.1.1.2.3.2">𝔤</ci><apply id="S2.p6.7.m4.1.1.2.3.3.cmml" xref="S2.p6.7.m4.1.1.2.3.3"><times id="S2.p6.7.m4.1.1.2.3.3.1.cmml" xref="S2.p6.7.m4.1.1.2.3.3.1"></times><ci id="S2.p6.7.m4.1.1.2.3.3.2.cmml" xref="S2.p6.7.m4.1.1.2.3.3.2">𝜈</ci><ci id="S2.p6.7.m4.1.1.2.3.3.3.cmml" xref="S2.p6.7.m4.1.1.2.3.3.3">𝜌</ci></apply></apply></apply><apply id="S2.p6.7.m4.1.1.3.cmml" xref="S2.p6.7.m4.1.1.3"><csymbol cd="ambiguous" id="S2.p6.7.m4.1.1.3.1.cmml" xref="S2.p6.7.m4.1.1.3">subscript</csymbol><apply id="S2.p6.7.m4.1.1.3.2.cmml" xref="S2.p6.7.m4.1.1.3"><csymbol cd="ambiguous" id="S2.p6.7.m4.1.1.3.2.1.cmml" xref="S2.p6.7.m4.1.1.3">superscript</csymbol><ci id="S2.p6.7.m4.1.1.3.2.2.cmml" xref="S2.p6.7.m4.1.1.3.2.2">𝛿</ci><ci id="S2.p6.7.m4.1.1.3.2.3.cmml" xref="S2.p6.7.m4.1.1.3.2.3">𝜇</ci></apply><ci id="S2.p6.7.m4.1.1.3.3.cmml" xref="S2.p6.7.m4.1.1.3.3">𝜈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.7.m4.1c">\mathfrak{g}^{\mu\rho}\mathfrak{g}_{\nu\rho}=\delta^{\mu}_{\nu}</annotation><annotation encoding="application/x-llamapun" id="S2.p6.7.m4.1d">fraktur_g start_POSTSUPERSCRIPT italic_μ italic_ρ end_POSTSUPERSCRIPT fraktur_g start_POSTSUBSCRIPT italic_ν italic_ρ end_POSTSUBSCRIPT = italic_δ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT</annotation></semantics></math> in terms of <math alttext="I^{\mu\nu}_{P}" class="ltx_Math" display="inline" id="S2.p6.8.m5.1"><semantics id="S2.p6.8.m5.1a"><msubsup id="S2.p6.8.m5.1.1" xref="S2.p6.8.m5.1.1.cmml"><mi id="S2.p6.8.m5.1.1.2.2" xref="S2.p6.8.m5.1.1.2.2.cmml">I</mi><mi id="S2.p6.8.m5.1.1.3" xref="S2.p6.8.m5.1.1.3.cmml">P</mi><mrow id="S2.p6.8.m5.1.1.2.3" xref="S2.p6.8.m5.1.1.2.3.cmml"><mi id="S2.p6.8.m5.1.1.2.3.2" xref="S2.p6.8.m5.1.1.2.3.2.cmml">μ</mi><mo id="S2.p6.8.m5.1.1.2.3.1" xref="S2.p6.8.m5.1.1.2.3.1.cmml">⁢</mo><mi id="S2.p6.8.m5.1.1.2.3.3" xref="S2.p6.8.m5.1.1.2.3.3.cmml">ν</mi></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S2.p6.8.m5.1b"><apply id="S2.p6.8.m5.1.1.cmml" xref="S2.p6.8.m5.1.1"><csymbol cd="ambiguous" id="S2.p6.8.m5.1.1.1.cmml" xref="S2.p6.8.m5.1.1">subscript</csymbol><apply id="S2.p6.8.m5.1.1.2.cmml" xref="S2.p6.8.m5.1.1"><csymbol cd="ambiguous" id="S2.p6.8.m5.1.1.2.1.cmml" xref="S2.p6.8.m5.1.1">superscript</csymbol><ci id="S2.p6.8.m5.1.1.2.2.cmml" xref="S2.p6.8.m5.1.1.2.2">𝐼</ci><apply id="S2.p6.8.m5.1.1.2.3.cmml" xref="S2.p6.8.m5.1.1.2.3"><times id="S2.p6.8.m5.1.1.2.3.1.cmml" xref="S2.p6.8.m5.1.1.2.3.1"></times><ci id="S2.p6.8.m5.1.1.2.3.2.cmml" xref="S2.p6.8.m5.1.1.2.3.2">𝜇</ci><ci id="S2.p6.8.m5.1.1.2.3.3.cmml" xref="S2.p6.8.m5.1.1.2.3.3">𝜈</ci></apply></apply><ci id="S2.p6.8.m5.1.1.3.cmml" xref="S2.p6.8.m5.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p6.8.m5.1c">I^{\mu\nu}_{P}</annotation><annotation encoding="application/x-llamapun" id="S2.p6.8.m5.1d">italic_I start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math>,</p> <table class="ltx_equation ltx_eqn_table" id="S2.E15"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="H_{P\mu\nu}=\eta_{\mu\rho}I_{P}^{\rho\sigma}\eta_{\sigma\nu}+\eta_{\mu\rho}% \sum_{P=Q\cup R}I_{Q}^{\rho\sigma}H_{R\sigma\nu}," class="ltx_Math" display="block" id="S2.E15.m1.1"><semantics id="S2.E15.m1.1a"><mrow id="S2.E15.m1.1.1.1" xref="S2.E15.m1.1.1.1.1.cmml"><mrow id="S2.E15.m1.1.1.1.1" xref="S2.E15.m1.1.1.1.1.cmml"><msub id="S2.E15.m1.1.1.1.1.2" xref="S2.E15.m1.1.1.1.1.2.cmml"><mi id="S2.E15.m1.1.1.1.1.2.2" xref="S2.E15.m1.1.1.1.1.2.2.cmml">H</mi><mrow id="S2.E15.m1.1.1.1.1.2.3" xref="S2.E15.m1.1.1.1.1.2.3.cmml"><mi id="S2.E15.m1.1.1.1.1.2.3.2" xref="S2.E15.m1.1.1.1.1.2.3.2.cmml">P</mi><mo id="S2.E15.m1.1.1.1.1.2.3.1" xref="S2.E15.m1.1.1.1.1.2.3.1.cmml">⁢</mo><mi id="S2.E15.m1.1.1.1.1.2.3.3" xref="S2.E15.m1.1.1.1.1.2.3.3.cmml">μ</mi><mo id="S2.E15.m1.1.1.1.1.2.3.1a" xref="S2.E15.m1.1.1.1.1.2.3.1.cmml">⁢</mo><mi id="S2.E15.m1.1.1.1.1.2.3.4" xref="S2.E15.m1.1.1.1.1.2.3.4.cmml">ν</mi></mrow></msub><mo id="S2.E15.m1.1.1.1.1.1" xref="S2.E15.m1.1.1.1.1.1.cmml">=</mo><mrow id="S2.E15.m1.1.1.1.1.3" xref="S2.E15.m1.1.1.1.1.3.cmml"><mrow id="S2.E15.m1.1.1.1.1.3.2" xref="S2.E15.m1.1.1.1.1.3.2.cmml"><msub id="S2.E15.m1.1.1.1.1.3.2.2" xref="S2.E15.m1.1.1.1.1.3.2.2.cmml"><mi id="S2.E15.m1.1.1.1.1.3.2.2.2" xref="S2.E15.m1.1.1.1.1.3.2.2.2.cmml">η</mi><mrow id="S2.E15.m1.1.1.1.1.3.2.2.3" xref="S2.E15.m1.1.1.1.1.3.2.2.3.cmml"><mi id="S2.E15.m1.1.1.1.1.3.2.2.3.2" xref="S2.E15.m1.1.1.1.1.3.2.2.3.2.cmml">μ</mi><mo id="S2.E15.m1.1.1.1.1.3.2.2.3.1" xref="S2.E15.m1.1.1.1.1.3.2.2.3.1.cmml">⁢</mo><mi id="S2.E15.m1.1.1.1.1.3.2.2.3.3" xref="S2.E15.m1.1.1.1.1.3.2.2.3.3.cmml">ρ</mi></mrow></msub><mo id="S2.E15.m1.1.1.1.1.3.2.1" xref="S2.E15.m1.1.1.1.1.3.2.1.cmml">⁢</mo><msubsup id="S2.E15.m1.1.1.1.1.3.2.3" xref="S2.E15.m1.1.1.1.1.3.2.3.cmml"><mi id="S2.E15.m1.1.1.1.1.3.2.3.2.2" xref="S2.E15.m1.1.1.1.1.3.2.3.2.2.cmml">I</mi><mi id="S2.E15.m1.1.1.1.1.3.2.3.2.3" xref="S2.E15.m1.1.1.1.1.3.2.3.2.3.cmml">P</mi><mrow id="S2.E15.m1.1.1.1.1.3.2.3.3" xref="S2.E15.m1.1.1.1.1.3.2.3.3.cmml"><mi id="S2.E15.m1.1.1.1.1.3.2.3.3.2" xref="S2.E15.m1.1.1.1.1.3.2.3.3.2.cmml">ρ</mi><mo id="S2.E15.m1.1.1.1.1.3.2.3.3.1" xref="S2.E15.m1.1.1.1.1.3.2.3.3.1.cmml">⁢</mo><mi id="S2.E15.m1.1.1.1.1.3.2.3.3.3" xref="S2.E15.m1.1.1.1.1.3.2.3.3.3.cmml">σ</mi></mrow></msubsup><mo id="S2.E15.m1.1.1.1.1.3.2.1a" xref="S2.E15.m1.1.1.1.1.3.2.1.cmml">⁢</mo><msub id="S2.E15.m1.1.1.1.1.3.2.4" xref="S2.E15.m1.1.1.1.1.3.2.4.cmml"><mi id="S2.E15.m1.1.1.1.1.3.2.4.2" xref="S2.E15.m1.1.1.1.1.3.2.4.2.cmml">η</mi><mrow id="S2.E15.m1.1.1.1.1.3.2.4.3" xref="S2.E15.m1.1.1.1.1.3.2.4.3.cmml"><mi id="S2.E15.m1.1.1.1.1.3.2.4.3.2" xref="S2.E15.m1.1.1.1.1.3.2.4.3.2.cmml">σ</mi><mo id="S2.E15.m1.1.1.1.1.3.2.4.3.1" xref="S2.E15.m1.1.1.1.1.3.2.4.3.1.cmml">⁢</mo><mi id="S2.E15.m1.1.1.1.1.3.2.4.3.3" xref="S2.E15.m1.1.1.1.1.3.2.4.3.3.cmml">ν</mi></mrow></msub></mrow><mo id="S2.E15.m1.1.1.1.1.3.1" xref="S2.E15.m1.1.1.1.1.3.1.cmml">+</mo><mrow id="S2.E15.m1.1.1.1.1.3.3" xref="S2.E15.m1.1.1.1.1.3.3.cmml"><msub id="S2.E15.m1.1.1.1.1.3.3.2" xref="S2.E15.m1.1.1.1.1.3.3.2.cmml"><mi id="S2.E15.m1.1.1.1.1.3.3.2.2" xref="S2.E15.m1.1.1.1.1.3.3.2.2.cmml">η</mi><mrow id="S2.E15.m1.1.1.1.1.3.3.2.3" xref="S2.E15.m1.1.1.1.1.3.3.2.3.cmml"><mi id="S2.E15.m1.1.1.1.1.3.3.2.3.2" xref="S2.E15.m1.1.1.1.1.3.3.2.3.2.cmml">μ</mi><mo id="S2.E15.m1.1.1.1.1.3.3.2.3.1" xref="S2.E15.m1.1.1.1.1.3.3.2.3.1.cmml">⁢</mo><mi 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R}I_{Q}^{\rho\sigma}H_{R\sigma\nu},</annotation><annotation encoding="application/x-llamapun" id="S2.E15.m1.1d">italic_H start_POSTSUBSCRIPT italic_P italic_μ italic_ν end_POSTSUBSCRIPT = italic_η start_POSTSUBSCRIPT italic_μ italic_ρ end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ρ italic_σ end_POSTSUPERSCRIPT italic_η start_POSTSUBSCRIPT italic_σ italic_ν end_POSTSUBSCRIPT + italic_η start_POSTSUBSCRIPT italic_μ italic_ρ end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_P = italic_Q ∪ italic_R end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ρ italic_σ end_POSTSUPERSCRIPT italic_H start_POSTSUBSCRIPT italic_R italic_σ italic_ν end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(15)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p6.9">and the remaining tensor structures in (<a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S2.E12" title="In II Equations of motion ‣ One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_tag">12</span></a>) are</p> <table class="ltx_equation ltx_eqn_table" id="S2.E16"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{V}^{(3)}_{\mu\nu\rho\sigma\gamma\lambda}(Q,R)=k_{QR\gamma}k_{R\nu}% \eta_{\mu\rho}\eta_{\sigma\lambda}-k_{Q\gamma}k_{R\sigma}\eta_{\mu\rho}\eta_{% \nu\lambda}\\ +k_{QR\rho}k_{Q\nu}\eta_{\mu\gamma}\eta_{\sigma\lambda}+\frac{1}{2}\Big{(}k_{% QR\gamma}k_{Q\lambda}\mathbb{P}_{\mu\nu\rho\sigma}\\ \vphantom{\frac{1}{2}}-k_{Q\mu}k_{R\nu}\mathbb{P}_{\rho\sigma\gamma\lambda}+k_% {QR\rho}k_{R\sigma}\mathbb{P}_{\mu\nu\gamma\lambda}-s_{Q}\eta_{\mu\gamma}% \mathbb{P}_{\rho\sigma\nu\lambda}\\ -s_{R}\eta_{\mu\rho}\mathbb{P}_{\gamma\lambda\nu\sigma}-s_{QR}\eta_{\sigma% \lambda}\mathbb{P}_{\mu\nu\rho\gamma}\Big{)}+(\mu\leftrightarrow\nu)," class="ltx_math_unparsed" display="block" id="S2.E16.m1.87"><semantics id="S2.E16.m1.87a"><mtable displaystyle="true" id="S2.E16.m1.87.87" rowspacing="0pt"><mtr id="S2.E16.m1.87.87a"><mtd class="ltx_align_left" columnalign="left" id="S2.E16.m1.87.87b"><mrow id="S2.E16.m1.26.26.26.26.26"><mrow id="S2.E16.m1.26.26.26.26.26.27"><msubsup id="S2.E16.m1.26.26.26.26.26.27.2"><mi class="ltx_font_mathcaligraphic" id="S2.E16.m1.1.1.1.1.1.1">𝒱</mi><mrow id="S2.E16.m1.3.3.3.3.3.3.1"><mi id="S2.E16.m1.3.3.3.3.3.3.1.2">μ</mi><mo id="S2.E16.m1.3.3.3.3.3.3.1.1">⁢</mo><mi id="S2.E16.m1.3.3.3.3.3.3.1.3">ν</mi><mo id="S2.E16.m1.3.3.3.3.3.3.1.1a">⁢</mo><mi id="S2.E16.m1.3.3.3.3.3.3.1.4">ρ</mi><mo id="S2.E16.m1.3.3.3.3.3.3.1.1b">⁢</mo><mi 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id="S2.E16.m1.26.26.26.26.26.28.2.4"><mi id="S2.E16.m1.23.23.23.23.23.23">η</mi><mrow id="S2.E16.m1.24.24.24.24.24.24.1"><mi id="S2.E16.m1.24.24.24.24.24.24.1.2">μ</mi><mo id="S2.E16.m1.24.24.24.24.24.24.1.1">⁢</mo><mi id="S2.E16.m1.24.24.24.24.24.24.1.3">ρ</mi></mrow></msub><mo id="S2.E16.m1.26.26.26.26.26.28.2.1b">⁢</mo><msub id="S2.E16.m1.26.26.26.26.26.28.2.5"><mi id="S2.E16.m1.25.25.25.25.25.25">η</mi><mrow id="S2.E16.m1.26.26.26.26.26.26.1"><mi id="S2.E16.m1.26.26.26.26.26.26.1.2">ν</mi><mo id="S2.E16.m1.26.26.26.26.26.26.1.1">⁢</mo><mi id="S2.E16.m1.26.26.26.26.26.26.1.3">λ</mi></mrow></msub></mrow></mrow></mrow></mtd></mtr><mtr id="S2.E16.m1.87.87c"><mtd class="ltx_align_right" columnalign="right" id="S2.E16.m1.87.87d"><mrow id="S2.E16.m1.44.44.44.18.18"><mo id="S2.E16.m1.27.27.27.1.1.1">+</mo><msub id="S2.E16.m1.44.44.44.18.18.19"><mi id="S2.E16.m1.28.28.28.2.2.2">k</mi><mrow id="S2.E16.m1.29.29.29.3.3.3.1"><mi id="S2.E16.m1.29.29.29.3.3.3.1.2">Q</mi><mo id="S2.E16.m1.29.29.29.3.3.3.1.1">⁢</mo><mi id="S2.E16.m1.29.29.29.3.3.3.1.3">R</mi><mo id="S2.E16.m1.29.29.29.3.3.3.1.1a">⁢</mo><mi id="S2.E16.m1.29.29.29.3.3.3.1.4">ρ</mi></mrow></msub><msub id="S2.E16.m1.44.44.44.18.18.20"><mi id="S2.E16.m1.30.30.30.4.4.4">k</mi><mrow id="S2.E16.m1.31.31.31.5.5.5.1"><mi id="S2.E16.m1.31.31.31.5.5.5.1.2">Q</mi><mo id="S2.E16.m1.31.31.31.5.5.5.1.1">⁢</mo><mi id="S2.E16.m1.31.31.31.5.5.5.1.3">ν</mi></mrow></msub><msub id="S2.E16.m1.44.44.44.18.18.21"><mi id="S2.E16.m1.32.32.32.6.6.6">η</mi><mrow id="S2.E16.m1.33.33.33.7.7.7.1"><mi id="S2.E16.m1.33.33.33.7.7.7.1.2">μ</mi><mo id="S2.E16.m1.33.33.33.7.7.7.1.1">⁢</mo><mi id="S2.E16.m1.33.33.33.7.7.7.1.3">γ</mi></mrow></msub><msub id="S2.E16.m1.44.44.44.18.18.22"><mi id="S2.E16.m1.34.34.34.8.8.8">η</mi><mrow id="S2.E16.m1.35.35.35.9.9.9.1"><mi id="S2.E16.m1.35.35.35.9.9.9.1.2">σ</mi><mo id="S2.E16.m1.35.35.35.9.9.9.1.1">⁢</mo><mi id="S2.E16.m1.35.35.35.9.9.9.1.3">λ</mi></mrow></msub><mo id="S2.E16.m1.36.36.36.10.10.10">+</mo><mfrac id="S2.E16.m1.37.37.37.11.11.11"><mn id="S2.E16.m1.37.37.37.11.11.11.2">1</mn><mn id="S2.E16.m1.37.37.37.11.11.11.3">2</mn></mfrac><mrow id="S2.E16.m1.44.44.44.18.18.23"><mo id="S2.E16.m1.38.38.38.12.12.12" maxsize="160%" minsize="160%">(</mo><msub id="S2.E16.m1.44.44.44.18.18.23.1"><mi id="S2.E16.m1.39.39.39.13.13.13">k</mi><mrow id="S2.E16.m1.40.40.40.14.14.14.1"><mi id="S2.E16.m1.40.40.40.14.14.14.1.2">Q</mi><mo id="S2.E16.m1.40.40.40.14.14.14.1.1">⁢</mo><mi id="S2.E16.m1.40.40.40.14.14.14.1.3">R</mi><mo id="S2.E16.m1.40.40.40.14.14.14.1.1a">⁢</mo><mi id="S2.E16.m1.40.40.40.14.14.14.1.4">γ</mi></mrow></msub><msub id="S2.E16.m1.44.44.44.18.18.23.2"><mi id="S2.E16.m1.41.41.41.15.15.15">k</mi><mrow id="S2.E16.m1.42.42.42.16.16.16.1"><mi id="S2.E16.m1.42.42.42.16.16.16.1.2">Q</mi><mo id="S2.E16.m1.42.42.42.16.16.16.1.1">⁢</mo><mi id="S2.E16.m1.42.42.42.16.16.16.1.3">λ</mi></mrow></msub><msub id="S2.E16.m1.44.44.44.18.18.23.3"><mi id="S2.E16.m1.43.43.43.17.17.17">ℙ</mi><mrow id="S2.E16.m1.44.44.44.18.18.18.1"><mi id="S2.E16.m1.44.44.44.18.18.18.1.2">μ</mi><mo id="S2.E16.m1.44.44.44.18.18.18.1.1">⁢</mo><mi id="S2.E16.m1.44.44.44.18.18.18.1.3">ν</mi><mo id="S2.E16.m1.44.44.44.18.18.18.1.1a">⁢</mo><mi id="S2.E16.m1.44.44.44.18.18.18.1.4">ρ</mi><mo id="S2.E16.m1.44.44.44.18.18.18.1.1b">⁢</mo><mi id="S2.E16.m1.44.44.44.18.18.18.1.5">σ</mi></mrow></msub></mrow></mrow></mtd></mtr><mtr id="S2.E16.m1.87.87e"><mtd class="ltx_align_right" columnalign="right" id="S2.E16.m1.87.87f"><mrow id="S2.E16.m1.65.65.65.21.21"><mrow id="S2.E16.m1.65.65.65.21.21.22"><mrow id="S2.E16.m1.65.65.65.21.21.22.1"><mo id="S2.E16.m1.65.65.65.21.21.22.1a">−</mo><mrow id="S2.E16.m1.65.65.65.21.21.22.1.1"><msub id="S2.E16.m1.65.65.65.21.21.22.1.1.2"><mi id="S2.E16.m1.46.46.46.2.2.2">k</mi><mrow id="S2.E16.m1.47.47.47.3.3.3.1"><mi id="S2.E16.m1.47.47.47.3.3.3.1.2">Q</mi><mo id="S2.E16.m1.47.47.47.3.3.3.1.1">⁢</mo><mi id="S2.E16.m1.47.47.47.3.3.3.1.3">μ</mi></mrow></msub><mo id="S2.E16.m1.65.65.65.21.21.22.1.1.1">⁢</mo><msub id="S2.E16.m1.65.65.65.21.21.22.1.1.3"><mi id="S2.E16.m1.48.48.48.4.4.4">k</mi><mrow id="S2.E16.m1.49.49.49.5.5.5.1"><mi id="S2.E16.m1.49.49.49.5.5.5.1.2">R</mi><mo id="S2.E16.m1.49.49.49.5.5.5.1.1">⁢</mo><mi id="S2.E16.m1.49.49.49.5.5.5.1.3">ν</mi></mrow></msub><mo id="S2.E16.m1.65.65.65.21.21.22.1.1.1a">⁢</mo><msub id="S2.E16.m1.65.65.65.21.21.22.1.1.4"><mi id="S2.E16.m1.50.50.50.6.6.6">ℙ</mi><mrow id="S2.E16.m1.51.51.51.7.7.7.1"><mi id="S2.E16.m1.51.51.51.7.7.7.1.2">ρ</mi><mo id="S2.E16.m1.51.51.51.7.7.7.1.1">⁢</mo><mi id="S2.E16.m1.51.51.51.7.7.7.1.3">σ</mi><mo id="S2.E16.m1.51.51.51.7.7.7.1.1a">⁢</mo><mi id="S2.E16.m1.51.51.51.7.7.7.1.4">γ</mi><mo id="S2.E16.m1.51.51.51.7.7.7.1.1b">⁢</mo><mi id="S2.E16.m1.51.51.51.7.7.7.1.5">λ</mi></mrow></msub></mrow></mrow><mo id="S2.E16.m1.52.52.52.8.8.8">+</mo><mrow id="S2.E16.m1.65.65.65.21.21.22.2"><msub id="S2.E16.m1.65.65.65.21.21.22.2.2"><mi id="S2.E16.m1.53.53.53.9.9.9">k</mi><mrow id="S2.E16.m1.54.54.54.10.10.10.1"><mi id="S2.E16.m1.54.54.54.10.10.10.1.2">Q</mi><mo id="S2.E16.m1.54.54.54.10.10.10.1.1">⁢</mo><mi id="S2.E16.m1.54.54.54.10.10.10.1.3">R</mi><mo id="S2.E16.m1.54.54.54.10.10.10.1.1a">⁢</mo><mi id="S2.E16.m1.54.54.54.10.10.10.1.4">ρ</mi></mrow></msub><mo id="S2.E16.m1.65.65.65.21.21.22.2.1">⁢</mo><msub id="S2.E16.m1.65.65.65.21.21.22.2.3"><mi id="S2.E16.m1.55.55.55.11.11.11">k</mi><mrow id="S2.E16.m1.56.56.56.12.12.12.1"><mi id="S2.E16.m1.56.56.56.12.12.12.1.2">R</mi><mo id="S2.E16.m1.56.56.56.12.12.12.1.1">⁢</mo><mi id="S2.E16.m1.56.56.56.12.12.12.1.3">σ</mi></mrow></msub><mo id="S2.E16.m1.65.65.65.21.21.22.2.1a">⁢</mo><msub id="S2.E16.m1.65.65.65.21.21.22.2.4"><mi id="S2.E16.m1.57.57.57.13.13.13">ℙ</mi><mrow id="S2.E16.m1.58.58.58.14.14.14.1"><mi id="S2.E16.m1.58.58.58.14.14.14.1.2">μ</mi><mo id="S2.E16.m1.58.58.58.14.14.14.1.1">⁢</mo><mi id="S2.E16.m1.58.58.58.14.14.14.1.3">ν</mi><mo id="S2.E16.m1.58.58.58.14.14.14.1.1a">⁢</mo><mi id="S2.E16.m1.58.58.58.14.14.14.1.4">γ</mi><mo id="S2.E16.m1.58.58.58.14.14.14.1.1b">⁢</mo><mi id="S2.E16.m1.58.58.58.14.14.14.1.5">λ</mi></mrow></msub></mrow></mrow><mo id="S2.E16.m1.59.59.59.15.15.15">−</mo><mrow id="S2.E16.m1.65.65.65.21.21.23"><msub id="S2.E16.m1.65.65.65.21.21.23.2"><mi id="S2.E16.m1.60.60.60.16.16.16">s</mi><mi id="S2.E16.m1.61.61.61.17.17.17.1">Q</mi></msub><mo id="S2.E16.m1.65.65.65.21.21.23.1">⁢</mo><msub id="S2.E16.m1.65.65.65.21.21.23.3"><mi id="S2.E16.m1.62.62.62.18.18.18">η</mi><mrow id="S2.E16.m1.63.63.63.19.19.19.1"><mi id="S2.E16.m1.63.63.63.19.19.19.1.2">μ</mi><mo id="S2.E16.m1.63.63.63.19.19.19.1.1">⁢</mo><mi id="S2.E16.m1.63.63.63.19.19.19.1.3">γ</mi></mrow></msub><mo id="S2.E16.m1.65.65.65.21.21.23.1a">⁢</mo><msub id="S2.E16.m1.65.65.65.21.21.23.4"><mi id="S2.E16.m1.64.64.64.20.20.20">ℙ</mi><mrow id="S2.E16.m1.65.65.65.21.21.21.1"><mi id="S2.E16.m1.65.65.65.21.21.21.1.2">ρ</mi><mo id="S2.E16.m1.65.65.65.21.21.21.1.1">⁢</mo><mi id="S2.E16.m1.65.65.65.21.21.21.1.3">σ</mi><mo id="S2.E16.m1.65.65.65.21.21.21.1.1a">⁢</mo><mi id="S2.E16.m1.65.65.65.21.21.21.1.4">ν</mi><mo id="S2.E16.m1.65.65.65.21.21.21.1.1b">⁢</mo><mi id="S2.E16.m1.65.65.65.21.21.21.1.5">λ</mi></mrow></msub></mrow></mrow></mtd></mtr><mtr id="S2.E16.m1.87.87g"><mtd class="ltx_align_right" columnalign="right" id="S2.E16.m1.87.87h"><mrow id="S2.E16.m1.87.87.87.22.22"><mo id="S2.E16.m1.66.66.66.1.1.1">−</mo><msub id="S2.E16.m1.87.87.87.22.22.23"><mi id="S2.E16.m1.67.67.67.2.2.2">s</mi><mi id="S2.E16.m1.68.68.68.3.3.3.1">R</mi></msub><msub id="S2.E16.m1.87.87.87.22.22.24"><mi id="S2.E16.m1.69.69.69.4.4.4">η</mi><mrow id="S2.E16.m1.70.70.70.5.5.5.1"><mi id="S2.E16.m1.70.70.70.5.5.5.1.2">μ</mi><mo id="S2.E16.m1.70.70.70.5.5.5.1.1">⁢</mo><mi id="S2.E16.m1.70.70.70.5.5.5.1.3">ρ</mi></mrow></msub><msub id="S2.E16.m1.87.87.87.22.22.25"><mi id="S2.E16.m1.71.71.71.6.6.6">ℙ</mi><mrow id="S2.E16.m1.72.72.72.7.7.7.1"><mi id="S2.E16.m1.72.72.72.7.7.7.1.2">γ</mi><mo id="S2.E16.m1.72.72.72.7.7.7.1.1">⁢</mo><mi id="S2.E16.m1.72.72.72.7.7.7.1.3">λ</mi><mo id="S2.E16.m1.72.72.72.7.7.7.1.1a">⁢</mo><mi id="S2.E16.m1.72.72.72.7.7.7.1.4">ν</mi><mo id="S2.E16.m1.72.72.72.7.7.7.1.1b">⁢</mo><mi id="S2.E16.m1.72.72.72.7.7.7.1.5">σ</mi></mrow></msub><mo id="S2.E16.m1.73.73.73.8.8.8">−</mo><msub id="S2.E16.m1.87.87.87.22.22.26"><mi id="S2.E16.m1.74.74.74.9.9.9">s</mi><mrow id="S2.E16.m1.75.75.75.10.10.10.1"><mi id="S2.E16.m1.75.75.75.10.10.10.1.2">Q</mi><mo id="S2.E16.m1.75.75.75.10.10.10.1.1">⁢</mo><mi id="S2.E16.m1.75.75.75.10.10.10.1.3">R</mi></mrow></msub><msub id="S2.E16.m1.87.87.87.22.22.27"><mi id="S2.E16.m1.76.76.76.11.11.11">η</mi><mrow id="S2.E16.m1.77.77.77.12.12.12.1"><mi id="S2.E16.m1.77.77.77.12.12.12.1.2">σ</mi><mo id="S2.E16.m1.77.77.77.12.12.12.1.1">⁢</mo><mi id="S2.E16.m1.77.77.77.12.12.12.1.3">λ</mi></mrow></msub><msub id="S2.E16.m1.87.87.87.22.22.28"><mi id="S2.E16.m1.78.78.78.13.13.13">ℙ</mi><mrow id="S2.E16.m1.79.79.79.14.14.14.1"><mi id="S2.E16.m1.79.79.79.14.14.14.1.2">μ</mi><mo id="S2.E16.m1.79.79.79.14.14.14.1.1">⁢</mo><mi id="S2.E16.m1.79.79.79.14.14.14.1.3">ν</mi><mo id="S2.E16.m1.79.79.79.14.14.14.1.1a">⁢</mo><mi id="S2.E16.m1.79.79.79.14.14.14.1.4">ρ</mi><mo id="S2.E16.m1.79.79.79.14.14.14.1.1b">⁢</mo><mi id="S2.E16.m1.79.79.79.14.14.14.1.5">γ</mi></mrow></msub><mo id="S2.E16.m1.80.80.80.15.15.15" maxsize="160%" minsize="160%">)</mo><mo id="S2.E16.m1.81.81.81.16.16.16">+</mo><mo id="S2.E16.m1.82.82.82.17.17.17" stretchy="false">(</mo><mi id="S2.E16.m1.83.83.83.18.18.18">μ</mi><mo id="S2.E16.m1.84.84.84.19.19.19" stretchy="false">↔</mo><mi id="S2.E16.m1.85.85.85.20.20.20">ν</mi><mo id="S2.E16.m1.86.86.86.21.21.21" stretchy="false">)</mo><mo id="S2.E16.m1.87.87.87.22.22.22">,</mo></mrow></mtd></mtr></mtable><annotation encoding="application/x-tex" id="S2.E16.m1.87b">\mathcal{V}^{(3)}_{\mu\nu\rho\sigma\gamma\lambda}(Q,R)=k_{QR\gamma}k_{R\nu}% \eta_{\mu\rho}\eta_{\sigma\lambda}-k_{Q\gamma}k_{R\sigma}\eta_{\mu\rho}\eta_{% \nu\lambda}\\ +k_{QR\rho}k_{Q\nu}\eta_{\mu\gamma}\eta_{\sigma\lambda}+\frac{1}{2}\Big{(}k_{% QR\gamma}k_{Q\lambda}\mathbb{P}_{\mu\nu\rho\sigma}\\ \vphantom{\frac{1}{2}}-k_{Q\mu}k_{R\nu}\mathbb{P}_{\rho\sigma\gamma\lambda}+k_% {QR\rho}k_{R\sigma}\mathbb{P}_{\mu\nu\gamma\lambda}-s_{Q}\eta_{\mu\gamma}% \mathbb{P}_{\rho\sigma\nu\lambda}\\ -s_{R}\eta_{\mu\rho}\mathbb{P}_{\gamma\lambda\nu\sigma}-s_{QR}\eta_{\sigma% \lambda}\mathbb{P}_{\mu\nu\rho\gamma}\Big{)}+(\mu\leftrightarrow\nu),</annotation><annotation encoding="application/x-llamapun" id="S2.E16.m1.87c">start_ROW start_CELL caligraphic_V start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ italic_ν italic_ρ italic_σ italic_γ italic_λ end_POSTSUBSCRIPT ( italic_Q , italic_R ) = italic_k start_POSTSUBSCRIPT italic_Q italic_R italic_γ end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_R italic_ν end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_μ italic_ρ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_σ italic_λ end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT italic_Q italic_γ end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_R italic_σ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_μ italic_ρ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_ν italic_λ end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL + italic_k start_POSTSUBSCRIPT italic_Q italic_R italic_ρ end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_Q italic_ν end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_μ italic_γ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_σ italic_λ end_POSTSUBSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( italic_k start_POSTSUBSCRIPT italic_Q italic_R italic_γ end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_Q italic_λ end_POSTSUBSCRIPT blackboard_P start_POSTSUBSCRIPT italic_μ italic_ν italic_ρ italic_σ end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL - italic_k start_POSTSUBSCRIPT italic_Q italic_μ end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_R italic_ν end_POSTSUBSCRIPT blackboard_P start_POSTSUBSCRIPT italic_ρ italic_σ italic_γ italic_λ end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT italic_Q italic_R italic_ρ end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_R italic_σ end_POSTSUBSCRIPT blackboard_P start_POSTSUBSCRIPT italic_μ italic_ν italic_γ italic_λ end_POSTSUBSCRIPT - italic_s start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_μ italic_γ end_POSTSUBSCRIPT blackboard_P start_POSTSUBSCRIPT italic_ρ italic_σ italic_ν italic_λ end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL - italic_s start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_μ italic_ρ end_POSTSUBSCRIPT blackboard_P start_POSTSUBSCRIPT italic_γ italic_λ italic_ν italic_σ end_POSTSUBSCRIPT - italic_s start_POSTSUBSCRIPT italic_Q italic_R end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_σ italic_λ end_POSTSUBSCRIPT blackboard_P start_POSTSUBSCRIPT italic_μ italic_ν italic_ρ italic_γ end_POSTSUBSCRIPT ) + ( italic_μ ↔ italic_ν ) , end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(16)</span></td> </tr></tbody> </table> <table class="ltx_equation ltx_eqn_table" id="S2.E17"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="V^{(4)}_{\mu\nu\rho\sigma\gamma\lambda\tau\delta}(Q,R,S)=k_{QRS\rho}k_{Q\nu}% \eta_{\mu\lambda}\eta_{\gamma\tau}\eta_{\sigma\delta}-k_{Q\lambda}k_{R\sigma}% \eta_{\mu\rho}\eta_{\nu\delta}\eta_{\gamma\tau}-\frac{1}{4}k_{Q\mu}k_{RS\nu}% \eta_{\rho\lambda}\eta_{\gamma\tau}\eta_{\sigma\delta}\\ +\frac{1}{2}(k_{QRS\rho}k_{RS\sigma}-k_{R\rho}k_{S\sigma})\eta_{\mu\lambda}% \eta_{\nu\delta}\eta_{\gamma\tau}+\frac{1}{2}[(k_{Q}\cdot k_{R})-s_{QRS}]\eta_% {\mu\rho}\eta_{\nu\delta}\eta_{\sigma\lambda}\eta_{\gamma\tau}+\frac{1}{2}(k_{% Q}\cdot k_{RS})\eta_{\mu\rho}\eta_{\nu\lambda}\eta_{\sigma\delta}\eta_{\gamma% \tau}\\ +\frac{1}{2}\frac{1}{(2-d)}[k_{QRS\rho}k_{R\sigma}\eta_{\mu\nu}\eta_{\gamma% \tau}\eta_{\lambda\delta}-k_{Q\mu}k_{R\nu}\eta_{\rho\sigma}\eta_{\gamma\tau}% \eta_{\lambda\delta}-(k_{QR}\cdot k_{Q})\eta_{\mu\tau}\eta_{\nu\delta}\eta_{% \rho\gamma}\eta_{\sigma\lambda}]\\ +\frac{1}{2}\frac{1}{(2-d)}[k_{Q\gamma}k_{QR\lambda}\eta_{\mu\tau}\eta_{\nu% \delta}\eta_{\rho\sigma}-(k_{QRS}\cdot k_{Q})\eta_{\mu\nu}\eta_{\rho\lambda}% \eta_{\gamma\tau}\eta_{\sigma\delta}-s_{Q}\eta_{\mu\lambda}\eta_{\nu\delta}% \eta_{\rho\sigma}\eta_{\gamma\tau}]+(\mu\leftrightarrow\nu)," class="ltx_math_unparsed" display="block" id="S2.E17.m1.197"><semantics id="S2.E17.m1.197a"><mtable displaystyle="true" id="S2.E17.m1.197.197.4" rowspacing="0pt"><mtr id="S2.E17.m1.197.197.4a"><mtd class="ltx_align_left" columnalign="left" id="S2.E17.m1.197.197.4b"><mrow id="S2.E17.m1.44.44.44.44.44"><mrow id="S2.E17.m1.44.44.44.44.44.45"><msubsup id="S2.E17.m1.44.44.44.44.44.45.2"><mi id="S2.E17.m1.1.1.1.1.1.1">V</mi><mrow id="S2.E17.m1.3.3.3.3.3.3.1"><mi id="S2.E17.m1.3.3.3.3.3.3.1.2">μ</mi><mo id="S2.E17.m1.3.3.3.3.3.3.1.1">⁢</mo><mi id="S2.E17.m1.3.3.3.3.3.3.1.3">ν</mi><mo id="S2.E17.m1.3.3.3.3.3.3.1.1a">⁢</mo><mi id="S2.E17.m1.3.3.3.3.3.3.1.4">ρ</mi><mo id="S2.E17.m1.3.3.3.3.3.3.1.1b">⁢</mo><mi id="S2.E17.m1.3.3.3.3.3.3.1.5">σ</mi><mo id="S2.E17.m1.3.3.3.3.3.3.1.1c">⁢</mo><mi id="S2.E17.m1.3.3.3.3.3.3.1.6">γ</mi><mo id="S2.E17.m1.3.3.3.3.3.3.1.1d">⁢</mo><mi id="S2.E17.m1.3.3.3.3.3.3.1.7">λ</mi><mo id="S2.E17.m1.3.3.3.3.3.3.1.1e">⁢</mo><mi id="S2.E17.m1.3.3.3.3.3.3.1.8">τ</mi><mo id="S2.E17.m1.3.3.3.3.3.3.1.1f">⁢</mo><mi id="S2.E17.m1.3.3.3.3.3.3.1.9">δ</mi></mrow><mrow id="S2.E17.m1.2.2.2.2.2.2.1.3"><mo id="S2.E17.m1.2.2.2.2.2.2.1.3.1" stretchy="false">(</mo><mn id="S2.E17.m1.2.2.2.2.2.2.1.1">4</mn><mo id="S2.E17.m1.2.2.2.2.2.2.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S2.E17.m1.44.44.44.44.44.45.1">⁢</mo><mrow id="S2.E17.m1.44.44.44.44.44.45.3"><mo id="S2.E17.m1.4.4.4.4.4.4" stretchy="false">(</mo><mi id="S2.E17.m1.5.5.5.5.5.5">Q</mi><mo id="S2.E17.m1.6.6.6.6.6.6">,</mo><mi id="S2.E17.m1.7.7.7.7.7.7">R</mi><mo id="S2.E17.m1.8.8.8.8.8.8">,</mo><mi id="S2.E17.m1.9.9.9.9.9.9">S</mi><mo id="S2.E17.m1.10.10.10.10.10.10" stretchy="false">)</mo></mrow></mrow><mo id="S2.E17.m1.11.11.11.11.11.11">=</mo><mrow id="S2.E17.m1.44.44.44.44.44.46"><mrow id="S2.E17.m1.44.44.44.44.44.46.1"><msub id="S2.E17.m1.44.44.44.44.44.46.1.2"><mi id="S2.E17.m1.12.12.12.12.12.12">k</mi><mrow id="S2.E17.m1.13.13.13.13.13.13.1"><mi id="S2.E17.m1.13.13.13.13.13.13.1.2">Q</mi><mo id="S2.E17.m1.13.13.13.13.13.13.1.1">⁢</mo><mi id="S2.E17.m1.13.13.13.13.13.13.1.3">R</mi><mo id="S2.E17.m1.13.13.13.13.13.13.1.1a">⁢</mo><mi id="S2.E17.m1.13.13.13.13.13.13.1.4">S</mi><mo id="S2.E17.m1.13.13.13.13.13.13.1.1b">⁢</mo><mi id="S2.E17.m1.13.13.13.13.13.13.1.5">ρ</mi></mrow></msub><mo id="S2.E17.m1.44.44.44.44.44.46.1.1">⁢</mo><msub id="S2.E17.m1.44.44.44.44.44.46.1.3"><mi id="S2.E17.m1.14.14.14.14.14.14">k</mi><mrow id="S2.E17.m1.15.15.15.15.15.15.1"><mi id="S2.E17.m1.15.15.15.15.15.15.1.2">Q</mi><mo id="S2.E17.m1.15.15.15.15.15.15.1.1">⁢</mo><mi id="S2.E17.m1.15.15.15.15.15.15.1.3">ν</mi></mrow></msub><mo id="S2.E17.m1.44.44.44.44.44.46.1.1a">⁢</mo><msub id="S2.E17.m1.44.44.44.44.44.46.1.4"><mi id="S2.E17.m1.16.16.16.16.16.16">η</mi><mrow id="S2.E17.m1.17.17.17.17.17.17.1"><mi id="S2.E17.m1.17.17.17.17.17.17.1.2">μ</mi><mo id="S2.E17.m1.17.17.17.17.17.17.1.1">⁢</mo><mi id="S2.E17.m1.17.17.17.17.17.17.1.3">λ</mi></mrow></msub><mo id="S2.E17.m1.44.44.44.44.44.46.1.1b">⁢</mo><msub id="S2.E17.m1.44.44.44.44.44.46.1.5"><mi id="S2.E17.m1.18.18.18.18.18.18">η</mi><mrow id="S2.E17.m1.19.19.19.19.19.19.1"><mi id="S2.E17.m1.19.19.19.19.19.19.1.2">γ</mi><mo id="S2.E17.m1.19.19.19.19.19.19.1.1">⁢</mo><mi id="S2.E17.m1.19.19.19.19.19.19.1.3">τ</mi></mrow></msub><mo id="S2.E17.m1.44.44.44.44.44.46.1.1c">⁢</mo><msub id="S2.E17.m1.44.44.44.44.44.46.1.6"><mi id="S2.E17.m1.20.20.20.20.20.20">η</mi><mrow id="S2.E17.m1.21.21.21.21.21.21.1"><mi id="S2.E17.m1.21.21.21.21.21.21.1.2">σ</mi><mo id="S2.E17.m1.21.21.21.21.21.21.1.1">⁢</mo><mi id="S2.E17.m1.21.21.21.21.21.21.1.3">δ</mi></mrow></msub></mrow><mo id="S2.E17.m1.22.22.22.22.22.22">−</mo><mrow id="S2.E17.m1.44.44.44.44.44.46.2"><msub id="S2.E17.m1.44.44.44.44.44.46.2.2"><mi id="S2.E17.m1.23.23.23.23.23.23">k</mi><mrow id="S2.E17.m1.24.24.24.24.24.24.1"><mi id="S2.E17.m1.24.24.24.24.24.24.1.2">Q</mi><mo id="S2.E17.m1.24.24.24.24.24.24.1.1">⁢</mo><mi id="S2.E17.m1.24.24.24.24.24.24.1.3">λ</mi></mrow></msub><mo id="S2.E17.m1.44.44.44.44.44.46.2.1">⁢</mo><msub id="S2.E17.m1.44.44.44.44.44.46.2.3"><mi id="S2.E17.m1.25.25.25.25.25.25">k</mi><mrow id="S2.E17.m1.26.26.26.26.26.26.1"><mi id="S2.E17.m1.26.26.26.26.26.26.1.2">R</mi><mo id="S2.E17.m1.26.26.26.26.26.26.1.1">⁢</mo><mi id="S2.E17.m1.26.26.26.26.26.26.1.3">σ</mi></mrow></msub><mo id="S2.E17.m1.44.44.44.44.44.46.2.1a">⁢</mo><msub id="S2.E17.m1.44.44.44.44.44.46.2.4"><mi id="S2.E17.m1.27.27.27.27.27.27">η</mi><mrow id="S2.E17.m1.28.28.28.28.28.28.1"><mi id="S2.E17.m1.28.28.28.28.28.28.1.2">μ</mi><mo id="S2.E17.m1.28.28.28.28.28.28.1.1">⁢</mo><mi id="S2.E17.m1.28.28.28.28.28.28.1.3">ρ</mi></mrow></msub><mo id="S2.E17.m1.44.44.44.44.44.46.2.1b">⁢</mo><msub id="S2.E17.m1.44.44.44.44.44.46.2.5"><mi id="S2.E17.m1.29.29.29.29.29.29">η</mi><mrow id="S2.E17.m1.30.30.30.30.30.30.1"><mi id="S2.E17.m1.30.30.30.30.30.30.1.2">ν</mi><mo id="S2.E17.m1.30.30.30.30.30.30.1.1">⁢</mo><mi id="S2.E17.m1.30.30.30.30.30.30.1.3">δ</mi></mrow></msub><mo id="S2.E17.m1.44.44.44.44.44.46.2.1c">⁢</mo><msub id="S2.E17.m1.44.44.44.44.44.46.2.6"><mi id="S2.E17.m1.31.31.31.31.31.31">η</mi><mrow id="S2.E17.m1.32.32.32.32.32.32.1"><mi id="S2.E17.m1.32.32.32.32.32.32.1.2">γ</mi><mo id="S2.E17.m1.32.32.32.32.32.32.1.1">⁢</mo><mi id="S2.E17.m1.32.32.32.32.32.32.1.3">τ</mi></mrow></msub></mrow><mo id="S2.E17.m1.22.22.22.22.22.22a">−</mo><mrow id="S2.E17.m1.44.44.44.44.44.46.3"><mfrac id="S2.E17.m1.34.34.34.34.34.34"><mn id="S2.E17.m1.34.34.34.34.34.34.2">1</mn><mn id="S2.E17.m1.34.34.34.34.34.34.3">4</mn></mfrac><mo id="S2.E17.m1.44.44.44.44.44.46.3.1">⁢</mo><msub id="S2.E17.m1.44.44.44.44.44.46.3.2"><mi id="S2.E17.m1.35.35.35.35.35.35">k</mi><mrow id="S2.E17.m1.36.36.36.36.36.36.1"><mi id="S2.E17.m1.36.36.36.36.36.36.1.2">Q</mi><mo id="S2.E17.m1.36.36.36.36.36.36.1.1">⁢</mo><mi id="S2.E17.m1.36.36.36.36.36.36.1.3">μ</mi></mrow></msub><mo id="S2.E17.m1.44.44.44.44.44.46.3.1a">⁢</mo><msub id="S2.E17.m1.44.44.44.44.44.46.3.3"><mi id="S2.E17.m1.37.37.37.37.37.37">k</mi><mrow id="S2.E17.m1.38.38.38.38.38.38.1"><mi id="S2.E17.m1.38.38.38.38.38.38.1.2">R</mi><mo id="S2.E17.m1.38.38.38.38.38.38.1.1">⁢</mo><mi id="S2.E17.m1.38.38.38.38.38.38.1.3">S</mi><mo id="S2.E17.m1.38.38.38.38.38.38.1.1a">⁢</mo><mi id="S2.E17.m1.38.38.38.38.38.38.1.4">ν</mi></mrow></msub><mo id="S2.E17.m1.44.44.44.44.44.46.3.1b">⁢</mo><msub id="S2.E17.m1.44.44.44.44.44.46.3.4"><mi id="S2.E17.m1.39.39.39.39.39.39">η</mi><mrow id="S2.E17.m1.40.40.40.40.40.40.1"><mi id="S2.E17.m1.40.40.40.40.40.40.1.2">ρ</mi><mo id="S2.E17.m1.40.40.40.40.40.40.1.1">⁢</mo><mi id="S2.E17.m1.40.40.40.40.40.40.1.3">λ</mi></mrow></msub><mo id="S2.E17.m1.44.44.44.44.44.46.3.1c">⁢</mo><msub id="S2.E17.m1.44.44.44.44.44.46.3.5"><mi id="S2.E17.m1.41.41.41.41.41.41">η</mi><mrow id="S2.E17.m1.42.42.42.42.42.42.1"><mi id="S2.E17.m1.42.42.42.42.42.42.1.2">γ</mi><mo id="S2.E17.m1.42.42.42.42.42.42.1.1">⁢</mo><mi id="S2.E17.m1.42.42.42.42.42.42.1.3">τ</mi></mrow></msub><mo id="S2.E17.m1.44.44.44.44.44.46.3.1d">⁢</mo><msub id="S2.E17.m1.44.44.44.44.44.46.3.6"><mi id="S2.E17.m1.43.43.43.43.43.43">η</mi><mrow id="S2.E17.m1.44.44.44.44.44.44.1"><mi id="S2.E17.m1.44.44.44.44.44.44.1.2">σ</mi><mo id="S2.E17.m1.44.44.44.44.44.44.1.1">⁢</mo><mi id="S2.E17.m1.44.44.44.44.44.44.1.3">δ</mi></mrow></msub></mrow></mrow></mrow></mtd></mtr><mtr id="S2.E17.m1.197.197.4c"><mtd class="ltx_align_right" columnalign="right" id="S2.E17.m1.197.197.4d"><mrow id="S2.E17.m1.196.196.3.196.61.61"><mrow id="S2.E17.m1.194.194.1.194.59.59.59"><mo id="S2.E17.m1.194.194.1.194.59.59.59a">+</mo><mrow id="S2.E17.m1.194.194.1.194.59.59.59.1"><mfrac id="S2.E17.m1.46.46.46.2.2.2"><mn id="S2.E17.m1.46.46.46.2.2.2.2">1</mn><mn id="S2.E17.m1.46.46.46.2.2.2.3">2</mn></mfrac><mo id="S2.E17.m1.194.194.1.194.59.59.59.1.2">⁢</mo><mrow id="S2.E17.m1.194.194.1.194.59.59.59.1.1.1"><mo id="S2.E17.m1.47.47.47.3.3.3" stretchy="false">(</mo><mrow id="S2.E17.m1.194.194.1.194.59.59.59.1.1.1.1"><mrow id="S2.E17.m1.194.194.1.194.59.59.59.1.1.1.1.1"><msub id="S2.E17.m1.194.194.1.194.59.59.59.1.1.1.1.1.2"><mi id="S2.E17.m1.48.48.48.4.4.4">k</mi><mrow id="S2.E17.m1.49.49.49.5.5.5.1"><mi id="S2.E17.m1.49.49.49.5.5.5.1.2">Q</mi><mo id="S2.E17.m1.49.49.49.5.5.5.1.1">⁢</mo><mi id="S2.E17.m1.49.49.49.5.5.5.1.3">R</mi><mo id="S2.E17.m1.49.49.49.5.5.5.1.1a">⁢</mo><mi id="S2.E17.m1.49.49.49.5.5.5.1.4">S</mi><mo id="S2.E17.m1.49.49.49.5.5.5.1.1b">⁢</mo><mi id="S2.E17.m1.49.49.49.5.5.5.1.5">ρ</mi></mrow></msub><mo id="S2.E17.m1.194.194.1.194.59.59.59.1.1.1.1.1.1">⁢</mo><msub id="S2.E17.m1.194.194.1.194.59.59.59.1.1.1.1.1.3"><mi id="S2.E17.m1.50.50.50.6.6.6">k</mi><mrow id="S2.E17.m1.51.51.51.7.7.7.1"><mi id="S2.E17.m1.51.51.51.7.7.7.1.2">R</mi><mo id="S2.E17.m1.51.51.51.7.7.7.1.1">⁢</mo><mi id="S2.E17.m1.51.51.51.7.7.7.1.3">S</mi><mo id="S2.E17.m1.51.51.51.7.7.7.1.1a">⁢</mo><mi id="S2.E17.m1.51.51.51.7.7.7.1.4">σ</mi></mrow></msub></mrow><mo id="S2.E17.m1.52.52.52.8.8.8">−</mo><mrow id="S2.E17.m1.194.194.1.194.59.59.59.1.1.1.1.2"><msub id="S2.E17.m1.194.194.1.194.59.59.59.1.1.1.1.2.2"><mi id="S2.E17.m1.53.53.53.9.9.9">k</mi><mrow id="S2.E17.m1.54.54.54.10.10.10.1"><mi id="S2.E17.m1.54.54.54.10.10.10.1.2">R</mi><mo id="S2.E17.m1.54.54.54.10.10.10.1.1">⁢</mo><mi id="S2.E17.m1.54.54.54.10.10.10.1.3">ρ</mi></mrow></msub><mo id="S2.E17.m1.194.194.1.194.59.59.59.1.1.1.1.2.1">⁢</mo><msub id="S2.E17.m1.194.194.1.194.59.59.59.1.1.1.1.2.3"><mi id="S2.E17.m1.55.55.55.11.11.11">k</mi><mrow id="S2.E17.m1.56.56.56.12.12.12.1"><mi id="S2.E17.m1.56.56.56.12.12.12.1.2">S</mi><mo id="S2.E17.m1.56.56.56.12.12.12.1.1">⁢</mo><mi id="S2.E17.m1.56.56.56.12.12.12.1.3">σ</mi></mrow></msub></mrow></mrow><mo id="S2.E17.m1.57.57.57.13.13.13" stretchy="false">)</mo></mrow><mo id="S2.E17.m1.194.194.1.194.59.59.59.1.2a">⁢</mo><msub id="S2.E17.m1.194.194.1.194.59.59.59.1.3"><mi id="S2.E17.m1.58.58.58.14.14.14">η</mi><mrow id="S2.E17.m1.59.59.59.15.15.15.1"><mi id="S2.E17.m1.59.59.59.15.15.15.1.2">μ</mi><mo id="S2.E17.m1.59.59.59.15.15.15.1.1">⁢</mo><mi id="S2.E17.m1.59.59.59.15.15.15.1.3">λ</mi></mrow></msub><mo id="S2.E17.m1.194.194.1.194.59.59.59.1.2b">⁢</mo><msub id="S2.E17.m1.194.194.1.194.59.59.59.1.4"><mi id="S2.E17.m1.60.60.60.16.16.16">η</mi><mrow id="S2.E17.m1.61.61.61.17.17.17.1"><mi id="S2.E17.m1.61.61.61.17.17.17.1.2">ν</mi><mo id="S2.E17.m1.61.61.61.17.17.17.1.1">⁢</mo><mi id="S2.E17.m1.61.61.61.17.17.17.1.3">δ</mi></mrow></msub><mo id="S2.E17.m1.194.194.1.194.59.59.59.1.2c">⁢</mo><msub id="S2.E17.m1.194.194.1.194.59.59.59.1.5"><mi id="S2.E17.m1.62.62.62.18.18.18">η</mi><mrow id="S2.E17.m1.63.63.63.19.19.19.1"><mi id="S2.E17.m1.63.63.63.19.19.19.1.2">γ</mi><mo id="S2.E17.m1.63.63.63.19.19.19.1.1">⁢</mo><mi id="S2.E17.m1.63.63.63.19.19.19.1.3">τ</mi></mrow></msub></mrow></mrow><mo id="S2.E17.m1.64.64.64.20.20.20">+</mo><mrow id="S2.E17.m1.195.195.2.195.60.60.60"><mfrac id="S2.E17.m1.65.65.65.21.21.21"><mn id="S2.E17.m1.65.65.65.21.21.21.2">1</mn><mn id="S2.E17.m1.65.65.65.21.21.21.3">2</mn></mfrac><mo id="S2.E17.m1.195.195.2.195.60.60.60.2">⁢</mo><mrow id="S2.E17.m1.195.195.2.195.60.60.60.1.1"><mo id="S2.E17.m1.66.66.66.22.22.22" stretchy="false">[</mo><mrow id="S2.E17.m1.195.195.2.195.60.60.60.1.1.1"><mrow id="S2.E17.m1.195.195.2.195.60.60.60.1.1.1.1.1"><mo id="S2.E17.m1.67.67.67.23.23.23" stretchy="false">(</mo><mrow id="S2.E17.m1.195.195.2.195.60.60.60.1.1.1.1.1.1"><msub id="S2.E17.m1.195.195.2.195.60.60.60.1.1.1.1.1.1.1"><mi id="S2.E17.m1.68.68.68.24.24.24">k</mi><mi id="S2.E17.m1.69.69.69.25.25.25.1">Q</mi></msub><mo id="S2.E17.m1.70.70.70.26.26.26" lspace="0.222em" rspace="0.222em">⋅</mo><msub id="S2.E17.m1.195.195.2.195.60.60.60.1.1.1.1.1.1.2"><mi id="S2.E17.m1.71.71.71.27.27.27">k</mi><mi id="S2.E17.m1.72.72.72.28.28.28.1">R</mi></msub></mrow><mo id="S2.E17.m1.73.73.73.29.29.29" stretchy="false">)</mo></mrow><mo id="S2.E17.m1.74.74.74.30.30.30">−</mo><msub id="S2.E17.m1.195.195.2.195.60.60.60.1.1.1.2"><mi id="S2.E17.m1.75.75.75.31.31.31">s</mi><mrow id="S2.E17.m1.76.76.76.32.32.32.1"><mi id="S2.E17.m1.76.76.76.32.32.32.1.2">Q</mi><mo id="S2.E17.m1.76.76.76.32.32.32.1.1">⁢</mo><mi id="S2.E17.m1.76.76.76.32.32.32.1.3">R</mi><mo id="S2.E17.m1.76.76.76.32.32.32.1.1a">⁢</mo><mi id="S2.E17.m1.76.76.76.32.32.32.1.4">S</mi></mrow></msub></mrow><mo id="S2.E17.m1.77.77.77.33.33.33" stretchy="false">]</mo></mrow><mo id="S2.E17.m1.195.195.2.195.60.60.60.2a">⁢</mo><msub id="S2.E17.m1.195.195.2.195.60.60.60.3"><mi id="S2.E17.m1.78.78.78.34.34.34">η</mi><mrow id="S2.E17.m1.79.79.79.35.35.35.1"><mi id="S2.E17.m1.79.79.79.35.35.35.1.2">μ</mi><mo id="S2.E17.m1.79.79.79.35.35.35.1.1">⁢</mo><mi id="S2.E17.m1.79.79.79.35.35.35.1.3">ρ</mi></mrow></msub><mo id="S2.E17.m1.195.195.2.195.60.60.60.2b">⁢</mo><msub id="S2.E17.m1.195.195.2.195.60.60.60.4"><mi id="S2.E17.m1.80.80.80.36.36.36">η</mi><mrow id="S2.E17.m1.81.81.81.37.37.37.1"><mi id="S2.E17.m1.81.81.81.37.37.37.1.2">ν</mi><mo id="S2.E17.m1.81.81.81.37.37.37.1.1">⁢</mo><mi id="S2.E17.m1.81.81.81.37.37.37.1.3">δ</mi></mrow></msub><mo id="S2.E17.m1.195.195.2.195.60.60.60.2c">⁢</mo><msub id="S2.E17.m1.195.195.2.195.60.60.60.5"><mi id="S2.E17.m1.82.82.82.38.38.38">η</mi><mrow id="S2.E17.m1.83.83.83.39.39.39.1"><mi id="S2.E17.m1.83.83.83.39.39.39.1.2">σ</mi><mo id="S2.E17.m1.83.83.83.39.39.39.1.1">⁢</mo><mi id="S2.E17.m1.83.83.83.39.39.39.1.3">λ</mi></mrow></msub><mo id="S2.E17.m1.195.195.2.195.60.60.60.2d">⁢</mo><msub id="S2.E17.m1.195.195.2.195.60.60.60.6"><mi id="S2.E17.m1.84.84.84.40.40.40">η</mi><mrow id="S2.E17.m1.85.85.85.41.41.41.1"><mi id="S2.E17.m1.85.85.85.41.41.41.1.2">γ</mi><mo id="S2.E17.m1.85.85.85.41.41.41.1.1">⁢</mo><mi id="S2.E17.m1.85.85.85.41.41.41.1.3">τ</mi></mrow></msub></mrow><mo id="S2.E17.m1.64.64.64.20.20.20a">+</mo><mrow id="S2.E17.m1.196.196.3.196.61.61.61"><mfrac id="S2.E17.m1.87.87.87.43.43.43"><mn id="S2.E17.m1.87.87.87.43.43.43.2">1</mn><mn id="S2.E17.m1.87.87.87.43.43.43.3">2</mn></mfrac><mo id="S2.E17.m1.196.196.3.196.61.61.61.2">⁢</mo><mrow id="S2.E17.m1.196.196.3.196.61.61.61.1.1"><mo id="S2.E17.m1.88.88.88.44.44.44" stretchy="false">(</mo><mrow id="S2.E17.m1.196.196.3.196.61.61.61.1.1.1"><msub id="S2.E17.m1.196.196.3.196.61.61.61.1.1.1.1"><mi id="S2.E17.m1.89.89.89.45.45.45">k</mi><mi id="S2.E17.m1.90.90.90.46.46.46.1">Q</mi></msub><mo id="S2.E17.m1.91.91.91.47.47.47" lspace="0.222em" rspace="0.222em">⋅</mo><msub id="S2.E17.m1.196.196.3.196.61.61.61.1.1.1.2"><mi id="S2.E17.m1.92.92.92.48.48.48">k</mi><mrow id="S2.E17.m1.93.93.93.49.49.49.1"><mi id="S2.E17.m1.93.93.93.49.49.49.1.2">R</mi><mo id="S2.E17.m1.93.93.93.49.49.49.1.1">⁢</mo><mi id="S2.E17.m1.93.93.93.49.49.49.1.3">S</mi></mrow></msub></mrow><mo id="S2.E17.m1.94.94.94.50.50.50" stretchy="false">)</mo></mrow><mo id="S2.E17.m1.196.196.3.196.61.61.61.2a">⁢</mo><msub id="S2.E17.m1.196.196.3.196.61.61.61.3"><mi id="S2.E17.m1.95.95.95.51.51.51">η</mi><mrow id="S2.E17.m1.96.96.96.52.52.52.1"><mi id="S2.E17.m1.96.96.96.52.52.52.1.2">μ</mi><mo id="S2.E17.m1.96.96.96.52.52.52.1.1">⁢</mo><mi id="S2.E17.m1.96.96.96.52.52.52.1.3">ρ</mi></mrow></msub><mo id="S2.E17.m1.196.196.3.196.61.61.61.2b">⁢</mo><msub id="S2.E17.m1.196.196.3.196.61.61.61.4"><mi id="S2.E17.m1.97.97.97.53.53.53">η</mi><mrow id="S2.E17.m1.98.98.98.54.54.54.1"><mi id="S2.E17.m1.98.98.98.54.54.54.1.2">ν</mi><mo id="S2.E17.m1.98.98.98.54.54.54.1.1">⁢</mo><mi id="S2.E17.m1.98.98.98.54.54.54.1.3">λ</mi></mrow></msub><mo id="S2.E17.m1.196.196.3.196.61.61.61.2c">⁢</mo><msub id="S2.E17.m1.196.196.3.196.61.61.61.5"><mi id="S2.E17.m1.99.99.99.55.55.55">η</mi><mrow id="S2.E17.m1.100.100.100.56.56.56.1"><mi id="S2.E17.m1.100.100.100.56.56.56.1.2">σ</mi><mo id="S2.E17.m1.100.100.100.56.56.56.1.1">⁢</mo><mi id="S2.E17.m1.100.100.100.56.56.56.1.3">δ</mi></mrow></msub><mo id="S2.E17.m1.196.196.3.196.61.61.61.2d">⁢</mo><msub id="S2.E17.m1.196.196.3.196.61.61.61.6"><mi id="S2.E17.m1.101.101.101.57.57.57">η</mi><mrow id="S2.E17.m1.102.102.102.58.58.58.1"><mi id="S2.E17.m1.102.102.102.58.58.58.1.2">γ</mi><mo id="S2.E17.m1.102.102.102.58.58.58.1.1">⁢</mo><mi id="S2.E17.m1.102.102.102.58.58.58.1.3">τ</mi></mrow></msub></mrow></mrow></mtd></mtr><mtr id="S2.E17.m1.197.197.4e"><mtd class="ltx_align_right" columnalign="right" id="S2.E17.m1.197.197.4f"><mrow id="S2.E17.m1.197.197.4.197.43.43"><mo id="S2.E17.m1.197.197.4.197.43.43a">+</mo><mrow id="S2.E17.m1.197.197.4.197.43.43.43"><mfrac id="S2.E17.m1.104.104.104.2.2.2"><mn id="S2.E17.m1.104.104.104.2.2.2.2">1</mn><mn id="S2.E17.m1.104.104.104.2.2.2.3">2</mn></mfrac><mo id="S2.E17.m1.197.197.4.197.43.43.43.2">⁢</mo><mfrac id="S2.E17.m1.105.105.105.3.3.3"><mn id="S2.E17.m1.105.105.105.3.3.3.3">1</mn><mrow id="S2.E17.m1.105.105.105.3.3.3.1.1"><mo id="S2.E17.m1.105.105.105.3.3.3.1.1.2" stretchy="false">(</mo><mrow id="S2.E17.m1.105.105.105.3.3.3.1.1.1"><mn id="S2.E17.m1.105.105.105.3.3.3.1.1.1.2">2</mn><mo id="S2.E17.m1.105.105.105.3.3.3.1.1.1.1">−</mo><mi id="S2.E17.m1.105.105.105.3.3.3.1.1.1.3">d</mi></mrow><mo id="S2.E17.m1.105.105.105.3.3.3.1.1.3" stretchy="false">)</mo></mrow></mfrac><mo id="S2.E17.m1.197.197.4.197.43.43.43.2a">⁢</mo><mrow id="S2.E17.m1.197.197.4.197.43.43.43.1.1"><mo id="S2.E17.m1.106.106.106.4.4.4" stretchy="false">[</mo><mrow id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1"><mrow id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.2"><msub id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.2.2"><mi id="S2.E17.m1.107.107.107.5.5.5">k</mi><mrow id="S2.E17.m1.108.108.108.6.6.6.1"><mi id="S2.E17.m1.108.108.108.6.6.6.1.2">Q</mi><mo id="S2.E17.m1.108.108.108.6.6.6.1.1">⁢</mo><mi id="S2.E17.m1.108.108.108.6.6.6.1.3">R</mi><mo id="S2.E17.m1.108.108.108.6.6.6.1.1a">⁢</mo><mi id="S2.E17.m1.108.108.108.6.6.6.1.4">S</mi><mo id="S2.E17.m1.108.108.108.6.6.6.1.1b">⁢</mo><mi id="S2.E17.m1.108.108.108.6.6.6.1.5">ρ</mi></mrow></msub><mo id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.2.1">⁢</mo><msub id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.2.3"><mi id="S2.E17.m1.109.109.109.7.7.7">k</mi><mrow id="S2.E17.m1.110.110.110.8.8.8.1"><mi id="S2.E17.m1.110.110.110.8.8.8.1.2">R</mi><mo id="S2.E17.m1.110.110.110.8.8.8.1.1">⁢</mo><mi id="S2.E17.m1.110.110.110.8.8.8.1.3">σ</mi></mrow></msub><mo id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.2.1a">⁢</mo><msub id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.2.4"><mi id="S2.E17.m1.111.111.111.9.9.9">η</mi><mrow id="S2.E17.m1.112.112.112.10.10.10.1"><mi id="S2.E17.m1.112.112.112.10.10.10.1.2">μ</mi><mo id="S2.E17.m1.112.112.112.10.10.10.1.1">⁢</mo><mi id="S2.E17.m1.112.112.112.10.10.10.1.3">ν</mi></mrow></msub><mo id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.2.1b">⁢</mo><msub id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.2.5"><mi id="S2.E17.m1.113.113.113.11.11.11">η</mi><mrow id="S2.E17.m1.114.114.114.12.12.12.1"><mi id="S2.E17.m1.114.114.114.12.12.12.1.2">γ</mi><mo id="S2.E17.m1.114.114.114.12.12.12.1.1">⁢</mo><mi id="S2.E17.m1.114.114.114.12.12.12.1.3">τ</mi></mrow></msub><mo id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.2.1c">⁢</mo><msub id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.2.6"><mi id="S2.E17.m1.115.115.115.13.13.13">η</mi><mrow id="S2.E17.m1.116.116.116.14.14.14.1"><mi id="S2.E17.m1.116.116.116.14.14.14.1.2">λ</mi><mo id="S2.E17.m1.116.116.116.14.14.14.1.1">⁢</mo><mi id="S2.E17.m1.116.116.116.14.14.14.1.3">δ</mi></mrow></msub></mrow><mo id="S2.E17.m1.117.117.117.15.15.15">−</mo><mrow id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.3"><msub id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.3.2"><mi id="S2.E17.m1.118.118.118.16.16.16">k</mi><mrow id="S2.E17.m1.119.119.119.17.17.17.1"><mi id="S2.E17.m1.119.119.119.17.17.17.1.2">Q</mi><mo id="S2.E17.m1.119.119.119.17.17.17.1.1">⁢</mo><mi id="S2.E17.m1.119.119.119.17.17.17.1.3">μ</mi></mrow></msub><mo id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.3.1">⁢</mo><msub id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.3.3"><mi id="S2.E17.m1.120.120.120.18.18.18">k</mi><mrow id="S2.E17.m1.121.121.121.19.19.19.1"><mi id="S2.E17.m1.121.121.121.19.19.19.1.2">R</mi><mo id="S2.E17.m1.121.121.121.19.19.19.1.1">⁢</mo><mi id="S2.E17.m1.121.121.121.19.19.19.1.3">ν</mi></mrow></msub><mo id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.3.1a">⁢</mo><msub id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.3.4"><mi id="S2.E17.m1.122.122.122.20.20.20">η</mi><mrow id="S2.E17.m1.123.123.123.21.21.21.1"><mi id="S2.E17.m1.123.123.123.21.21.21.1.2">ρ</mi><mo id="S2.E17.m1.123.123.123.21.21.21.1.1">⁢</mo><mi id="S2.E17.m1.123.123.123.21.21.21.1.3">σ</mi></mrow></msub><mo id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.3.1b">⁢</mo><msub id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.3.5"><mi id="S2.E17.m1.124.124.124.22.22.22">η</mi><mrow id="S2.E17.m1.125.125.125.23.23.23.1"><mi id="S2.E17.m1.125.125.125.23.23.23.1.2">γ</mi><mo id="S2.E17.m1.125.125.125.23.23.23.1.1">⁢</mo><mi id="S2.E17.m1.125.125.125.23.23.23.1.3">τ</mi></mrow></msub><mo id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.3.1c">⁢</mo><msub id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.3.6"><mi id="S2.E17.m1.126.126.126.24.24.24">η</mi><mrow id="S2.E17.m1.127.127.127.25.25.25.1"><mi id="S2.E17.m1.127.127.127.25.25.25.1.2">λ</mi><mo id="S2.E17.m1.127.127.127.25.25.25.1.1">⁢</mo><mi id="S2.E17.m1.127.127.127.25.25.25.1.3">δ</mi></mrow></msub></mrow><mo id="S2.E17.m1.117.117.117.15.15.15a">−</mo><mrow id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.1"><mrow id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.1.1.1"><mo id="S2.E17.m1.129.129.129.27.27.27" stretchy="false">(</mo><mrow id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.1.1.1.1"><msub id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.1.1.1.1.1"><mi id="S2.E17.m1.130.130.130.28.28.28">k</mi><mrow id="S2.E17.m1.131.131.131.29.29.29.1"><mi id="S2.E17.m1.131.131.131.29.29.29.1.2">Q</mi><mo id="S2.E17.m1.131.131.131.29.29.29.1.1">⁢</mo><mi id="S2.E17.m1.131.131.131.29.29.29.1.3">R</mi></mrow></msub><mo id="S2.E17.m1.132.132.132.30.30.30" lspace="0.222em" rspace="0.222em">⋅</mo><msub id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.1.1.1.1.2"><mi id="S2.E17.m1.133.133.133.31.31.31">k</mi><mi id="S2.E17.m1.134.134.134.32.32.32.1">Q</mi></msub></mrow><mo id="S2.E17.m1.135.135.135.33.33.33" stretchy="false">)</mo></mrow><mo id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.1.2">⁢</mo><msub id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.1.3"><mi id="S2.E17.m1.136.136.136.34.34.34">η</mi><mrow id="S2.E17.m1.137.137.137.35.35.35.1"><mi id="S2.E17.m1.137.137.137.35.35.35.1.2">μ</mi><mo id="S2.E17.m1.137.137.137.35.35.35.1.1">⁢</mo><mi id="S2.E17.m1.137.137.137.35.35.35.1.3">τ</mi></mrow></msub><mo id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.1.2a">⁢</mo><msub id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.1.4"><mi id="S2.E17.m1.138.138.138.36.36.36">η</mi><mrow id="S2.E17.m1.139.139.139.37.37.37.1"><mi id="S2.E17.m1.139.139.139.37.37.37.1.2">ν</mi><mo id="S2.E17.m1.139.139.139.37.37.37.1.1">⁢</mo><mi id="S2.E17.m1.139.139.139.37.37.37.1.3">δ</mi></mrow></msub><mo id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.1.2b">⁢</mo><msub id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.1.5"><mi id="S2.E17.m1.140.140.140.38.38.38">η</mi><mrow id="S2.E17.m1.141.141.141.39.39.39.1"><mi id="S2.E17.m1.141.141.141.39.39.39.1.2">ρ</mi><mo id="S2.E17.m1.141.141.141.39.39.39.1.1">⁢</mo><mi id="S2.E17.m1.141.141.141.39.39.39.1.3">γ</mi></mrow></msub><mo id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.1.2c">⁢</mo><msub id="S2.E17.m1.197.197.4.197.43.43.43.1.1.1.1.6"><mi id="S2.E17.m1.142.142.142.40.40.40">η</mi><mrow id="S2.E17.m1.143.143.143.41.41.41.1"><mi id="S2.E17.m1.143.143.143.41.41.41.1.2">σ</mi><mo id="S2.E17.m1.143.143.143.41.41.41.1.1">⁢</mo><mi id="S2.E17.m1.143.143.143.41.41.41.1.3">λ</mi></mrow></msub></mrow></mrow><mo id="S2.E17.m1.144.144.144.42.42.42" stretchy="false">]</mo></mrow></mrow></mrow></mtd></mtr><mtr id="S2.E17.m1.197.197.4g"><mtd class="ltx_align_right" columnalign="right" id="S2.E17.m1.197.197.4h"><mrow id="S2.E17.m1.193.193.193.49.49"><mo id="S2.E17.m1.145.145.145.1.1.1">+</mo><mfrac id="S2.E17.m1.146.146.146.2.2.2"><mn id="S2.E17.m1.146.146.146.2.2.2.2">1</mn><mn id="S2.E17.m1.146.146.146.2.2.2.3">2</mn></mfrac><mfrac id="S2.E17.m1.147.147.147.3.3.3"><mn id="S2.E17.m1.147.147.147.3.3.3.3">1</mn><mrow id="S2.E17.m1.147.147.147.3.3.3.1.1"><mo id="S2.E17.m1.147.147.147.3.3.3.1.1.2" stretchy="false">(</mo><mrow id="S2.E17.m1.147.147.147.3.3.3.1.1.1"><mn id="S2.E17.m1.147.147.147.3.3.3.1.1.1.2">2</mn><mo id="S2.E17.m1.147.147.147.3.3.3.1.1.1.1">−</mo><mi id="S2.E17.m1.147.147.147.3.3.3.1.1.1.3">d</mi></mrow><mo id="S2.E17.m1.147.147.147.3.3.3.1.1.3" stretchy="false">)</mo></mrow></mfrac><mrow id="S2.E17.m1.193.193.193.49.49.50"><mo id="S2.E17.m1.148.148.148.4.4.4" stretchy="false">[</mo><msub id="S2.E17.m1.193.193.193.49.49.50.1"><mi id="S2.E17.m1.149.149.149.5.5.5">k</mi><mrow id="S2.E17.m1.150.150.150.6.6.6.1"><mi id="S2.E17.m1.150.150.150.6.6.6.1.2">Q</mi><mo id="S2.E17.m1.150.150.150.6.6.6.1.1">⁢</mo><mi id="S2.E17.m1.150.150.150.6.6.6.1.3">γ</mi></mrow></msub><msub id="S2.E17.m1.193.193.193.49.49.50.2"><mi id="S2.E17.m1.151.151.151.7.7.7">k</mi><mrow id="S2.E17.m1.152.152.152.8.8.8.1"><mi id="S2.E17.m1.152.152.152.8.8.8.1.2">Q</mi><mo id="S2.E17.m1.152.152.152.8.8.8.1.1">⁢</mo><mi id="S2.E17.m1.152.152.152.8.8.8.1.3">R</mi><mo id="S2.E17.m1.152.152.152.8.8.8.1.1a">⁢</mo><mi id="S2.E17.m1.152.152.152.8.8.8.1.4">λ</mi></mrow></msub><msub id="S2.E17.m1.193.193.193.49.49.50.3"><mi id="S2.E17.m1.153.153.153.9.9.9">η</mi><mrow id="S2.E17.m1.154.154.154.10.10.10.1"><mi id="S2.E17.m1.154.154.154.10.10.10.1.2">μ</mi><mo id="S2.E17.m1.154.154.154.10.10.10.1.1">⁢</mo><mi id="S2.E17.m1.154.154.154.10.10.10.1.3">τ</mi></mrow></msub><msub id="S2.E17.m1.193.193.193.49.49.50.4"><mi id="S2.E17.m1.155.155.155.11.11.11">η</mi><mrow id="S2.E17.m1.156.156.156.12.12.12.1"><mi id="S2.E17.m1.156.156.156.12.12.12.1.2">ν</mi><mo id="S2.E17.m1.156.156.156.12.12.12.1.1">⁢</mo><mi id="S2.E17.m1.156.156.156.12.12.12.1.3">δ</mi></mrow></msub><msub id="S2.E17.m1.193.193.193.49.49.50.5"><mi id="S2.E17.m1.157.157.157.13.13.13">η</mi><mrow id="S2.E17.m1.158.158.158.14.14.14.1"><mi id="S2.E17.m1.158.158.158.14.14.14.1.2">ρ</mi><mo id="S2.E17.m1.158.158.158.14.14.14.1.1">⁢</mo><mi id="S2.E17.m1.158.158.158.14.14.14.1.3">σ</mi></mrow></msub><mo id="S2.E17.m1.159.159.159.15.15.15">−</mo><mrow id="S2.E17.m1.193.193.193.49.49.50.6"><mo id="S2.E17.m1.160.160.160.16.16.16" stretchy="false">(</mo><msub id="S2.E17.m1.193.193.193.49.49.50.6.1"><mi id="S2.E17.m1.161.161.161.17.17.17">k</mi><mrow id="S2.E17.m1.162.162.162.18.18.18.1"><mi id="S2.E17.m1.162.162.162.18.18.18.1.2">Q</mi><mo id="S2.E17.m1.162.162.162.18.18.18.1.1">⁢</mo><mi id="S2.E17.m1.162.162.162.18.18.18.1.3">R</mi><mo id="S2.E17.m1.162.162.162.18.18.18.1.1a">⁢</mo><mi id="S2.E17.m1.162.162.162.18.18.18.1.4">S</mi></mrow></msub><mo id="S2.E17.m1.163.163.163.19.19.19" lspace="0.222em" rspace="0.222em">⋅</mo><msub id="S2.E17.m1.193.193.193.49.49.50.6.2"><mi id="S2.E17.m1.164.164.164.20.20.20">k</mi><mi id="S2.E17.m1.165.165.165.21.21.21.1">Q</mi></msub><mo id="S2.E17.m1.166.166.166.22.22.22" stretchy="false">)</mo></mrow><msub id="S2.E17.m1.193.193.193.49.49.50.7"><mi id="S2.E17.m1.167.167.167.23.23.23">η</mi><mrow id="S2.E17.m1.168.168.168.24.24.24.1"><mi id="S2.E17.m1.168.168.168.24.24.24.1.2">μ</mi><mo id="S2.E17.m1.168.168.168.24.24.24.1.1">⁢</mo><mi id="S2.E17.m1.168.168.168.24.24.24.1.3">ν</mi></mrow></msub><msub id="S2.E17.m1.193.193.193.49.49.50.8"><mi id="S2.E17.m1.169.169.169.25.25.25">η</mi><mrow id="S2.E17.m1.170.170.170.26.26.26.1"><mi id="S2.E17.m1.170.170.170.26.26.26.1.2">ρ</mi><mo id="S2.E17.m1.170.170.170.26.26.26.1.1">⁢</mo><mi id="S2.E17.m1.170.170.170.26.26.26.1.3">λ</mi></mrow></msub><msub id="S2.E17.m1.193.193.193.49.49.50.9"><mi id="S2.E17.m1.171.171.171.27.27.27">η</mi><mrow id="S2.E17.m1.172.172.172.28.28.28.1"><mi id="S2.E17.m1.172.172.172.28.28.28.1.2">γ</mi><mo id="S2.E17.m1.172.172.172.28.28.28.1.1">⁢</mo><mi id="S2.E17.m1.172.172.172.28.28.28.1.3">τ</mi></mrow></msub><msub id="S2.E17.m1.193.193.193.49.49.50.10"><mi id="S2.E17.m1.173.173.173.29.29.29">η</mi><mrow id="S2.E17.m1.174.174.174.30.30.30.1"><mi id="S2.E17.m1.174.174.174.30.30.30.1.2">σ</mi><mo id="S2.E17.m1.174.174.174.30.30.30.1.1">⁢</mo><mi id="S2.E17.m1.174.174.174.30.30.30.1.3">δ</mi></mrow></msub><mo id="S2.E17.m1.175.175.175.31.31.31">−</mo><msub id="S2.E17.m1.193.193.193.49.49.50.11"><mi id="S2.E17.m1.176.176.176.32.32.32">s</mi><mi id="S2.E17.m1.177.177.177.33.33.33.1">Q</mi></msub><msub id="S2.E17.m1.193.193.193.49.49.50.12"><mi id="S2.E17.m1.178.178.178.34.34.34">η</mi><mrow id="S2.E17.m1.179.179.179.35.35.35.1"><mi id="S2.E17.m1.179.179.179.35.35.35.1.2">μ</mi><mo id="S2.E17.m1.179.179.179.35.35.35.1.1">⁢</mo><mi id="S2.E17.m1.179.179.179.35.35.35.1.3">λ</mi></mrow></msub><msub id="S2.E17.m1.193.193.193.49.49.50.13"><mi id="S2.E17.m1.180.180.180.36.36.36">η</mi><mrow id="S2.E17.m1.181.181.181.37.37.37.1"><mi id="S2.E17.m1.181.181.181.37.37.37.1.2">ν</mi><mo id="S2.E17.m1.181.181.181.37.37.37.1.1">⁢</mo><mi id="S2.E17.m1.181.181.181.37.37.37.1.3">δ</mi></mrow></msub><msub id="S2.E17.m1.193.193.193.49.49.50.14"><mi id="S2.E17.m1.182.182.182.38.38.38">η</mi><mrow id="S2.E17.m1.183.183.183.39.39.39.1"><mi id="S2.E17.m1.183.183.183.39.39.39.1.2">ρ</mi><mo id="S2.E17.m1.183.183.183.39.39.39.1.1">⁢</mo><mi id="S2.E17.m1.183.183.183.39.39.39.1.3">σ</mi></mrow></msub><msub id="S2.E17.m1.193.193.193.49.49.50.15"><mi id="S2.E17.m1.184.184.184.40.40.40">η</mi><mrow id="S2.E17.m1.185.185.185.41.41.41.1"><mi id="S2.E17.m1.185.185.185.41.41.41.1.2">γ</mi><mo id="S2.E17.m1.185.185.185.41.41.41.1.1">⁢</mo><mi id="S2.E17.m1.185.185.185.41.41.41.1.3">τ</mi></mrow></msub><mo id="S2.E17.m1.186.186.186.42.42.42" stretchy="false">]</mo></mrow><mo id="S2.E17.m1.187.187.187.43.43.43">+</mo><mrow id="S2.E17.m1.193.193.193.49.49.51"><mo id="S2.E17.m1.188.188.188.44.44.44" stretchy="false">(</mo><mi id="S2.E17.m1.189.189.189.45.45.45">μ</mi><mo id="S2.E17.m1.190.190.190.46.46.46" stretchy="false">↔</mo><mi id="S2.E17.m1.191.191.191.47.47.47">ν</mi><mo id="S2.E17.m1.192.192.192.48.48.48" stretchy="false">)</mo></mrow><mo id="S2.E17.m1.193.193.193.49.49.49">,</mo></mrow></mtd></mtr></mtable><annotation encoding="application/x-tex" id="S2.E17.m1.197b">V^{(4)}_{\mu\nu\rho\sigma\gamma\lambda\tau\delta}(Q,R,S)=k_{QRS\rho}k_{Q\nu}% \eta_{\mu\lambda}\eta_{\gamma\tau}\eta_{\sigma\delta}-k_{Q\lambda}k_{R\sigma}% \eta_{\mu\rho}\eta_{\nu\delta}\eta_{\gamma\tau}-\frac{1}{4}k_{Q\mu}k_{RS\nu}% \eta_{\rho\lambda}\eta_{\gamma\tau}\eta_{\sigma\delta}\\ +\frac{1}{2}(k_{QRS\rho}k_{RS\sigma}-k_{R\rho}k_{S\sigma})\eta_{\mu\lambda}% \eta_{\nu\delta}\eta_{\gamma\tau}+\frac{1}{2}[(k_{Q}\cdot k_{R})-s_{QRS}]\eta_% {\mu\rho}\eta_{\nu\delta}\eta_{\sigma\lambda}\eta_{\gamma\tau}+\frac{1}{2}(k_{% Q}\cdot k_{RS})\eta_{\mu\rho}\eta_{\nu\lambda}\eta_{\sigma\delta}\eta_{\gamma% \tau}\\ +\frac{1}{2}\frac{1}{(2-d)}[k_{QRS\rho}k_{R\sigma}\eta_{\mu\nu}\eta_{\gamma% \tau}\eta_{\lambda\delta}-k_{Q\mu}k_{R\nu}\eta_{\rho\sigma}\eta_{\gamma\tau}% \eta_{\lambda\delta}-(k_{QR}\cdot k_{Q})\eta_{\mu\tau}\eta_{\nu\delta}\eta_{% \rho\gamma}\eta_{\sigma\lambda}]\\ +\frac{1}{2}\frac{1}{(2-d)}[k_{Q\gamma}k_{QR\lambda}\eta_{\mu\tau}\eta_{\nu% \delta}\eta_{\rho\sigma}-(k_{QRS}\cdot k_{Q})\eta_{\mu\nu}\eta_{\rho\lambda}% \eta_{\gamma\tau}\eta_{\sigma\delta}-s_{Q}\eta_{\mu\lambda}\eta_{\nu\delta}% \eta_{\rho\sigma}\eta_{\gamma\tau}]+(\mu\leftrightarrow\nu),</annotation><annotation encoding="application/x-llamapun" id="S2.E17.m1.197c">start_ROW start_CELL italic_V start_POSTSUPERSCRIPT ( 4 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ italic_ν italic_ρ italic_σ italic_γ italic_λ italic_τ italic_δ end_POSTSUBSCRIPT ( italic_Q , italic_R , italic_S ) = italic_k start_POSTSUBSCRIPT italic_Q italic_R italic_S italic_ρ end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_Q italic_ν end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_μ italic_λ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_γ italic_τ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_σ italic_δ end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT italic_Q italic_λ end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_R italic_σ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_μ italic_ρ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_ν italic_δ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_γ italic_τ end_POSTSUBSCRIPT - divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_k start_POSTSUBSCRIPT italic_Q italic_μ end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_R italic_S italic_ν end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_ρ italic_λ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_γ italic_τ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_σ italic_δ end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL + divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( italic_k start_POSTSUBSCRIPT italic_Q italic_R italic_S italic_ρ end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_R italic_S italic_σ end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT italic_R italic_ρ end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_S italic_σ end_POSTSUBSCRIPT ) italic_η start_POSTSUBSCRIPT italic_μ italic_λ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_ν italic_δ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_γ italic_τ end_POSTSUBSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ ( italic_k start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT ⋅ italic_k start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ) - italic_s start_POSTSUBSCRIPT italic_Q italic_R italic_S end_POSTSUBSCRIPT ] italic_η start_POSTSUBSCRIPT italic_μ italic_ρ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_ν italic_δ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_σ italic_λ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_γ italic_τ end_POSTSUBSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( italic_k start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT ⋅ italic_k start_POSTSUBSCRIPT italic_R italic_S end_POSTSUBSCRIPT ) italic_η start_POSTSUBSCRIPT italic_μ italic_ρ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_ν italic_λ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_σ italic_δ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_γ italic_τ end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL + divide start_ARG 1 end_ARG start_ARG 2 end_ARG divide start_ARG 1 end_ARG start_ARG ( 2 - italic_d ) end_ARG [ italic_k start_POSTSUBSCRIPT italic_Q italic_R italic_S italic_ρ end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_R italic_σ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_γ italic_τ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_λ italic_δ end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT italic_Q italic_μ end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_R italic_ν end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_ρ italic_σ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_γ italic_τ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_λ italic_δ end_POSTSUBSCRIPT - ( italic_k start_POSTSUBSCRIPT italic_Q italic_R end_POSTSUBSCRIPT ⋅ italic_k start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT ) italic_η start_POSTSUBSCRIPT italic_μ italic_τ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_ν italic_δ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_ρ italic_γ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_σ italic_λ end_POSTSUBSCRIPT ] end_CELL end_ROW start_ROW start_CELL + divide start_ARG 1 end_ARG start_ARG 2 end_ARG divide start_ARG 1 end_ARG start_ARG ( 2 - italic_d ) end_ARG [ italic_k start_POSTSUBSCRIPT italic_Q italic_γ end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_Q italic_R italic_λ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_μ italic_τ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_ν italic_δ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_ρ italic_σ end_POSTSUBSCRIPT - ( italic_k start_POSTSUBSCRIPT italic_Q italic_R italic_S end_POSTSUBSCRIPT ⋅ italic_k start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT ) italic_η start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_ρ italic_λ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_γ italic_τ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_σ italic_δ end_POSTSUBSCRIPT - italic_s start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_μ italic_λ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_ν italic_δ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_ρ italic_σ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_γ italic_τ end_POSTSUBSCRIPT ] + ( italic_μ ↔ italic_ν ) , end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(17)</span></td> </tr></tbody> </table> <table class="ltx_equation ltx_eqn_table" id="S2.E18"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="V^{(5)}_{\mu\nu\rho\sigma\gamma\lambda\tau\delta\alpha\beta}(Q,R,S)=\frac{1}{2% }\eta_{\mu\alpha}\eta_{\rho\beta}\eta_{\sigma\delta}[(k_{Q}\cdot k_{RS})\eta_{% \nu\gamma}\eta_{\lambda\tau}-k_{Q\gamma}k_{S\lambda}\eta_{\nu\tau}]-\frac{1}{2% }k_{Q\gamma}k_{R\sigma}\eta_{\mu\tau}\eta_{\nu\alpha}\eta_{\rho\delta}\eta_{% \lambda\beta}\\ +\frac{1}{2}\frac{1}{(2-d)}\eta_{\mu\alpha}\eta_{\nu\beta}\eta_{\gamma\tau}[k_% {QRS\rho}k_{R\sigma}\eta_{\lambda\delta}-(k_{QRS}\cdot k_{Q})\eta_{\rho\lambda% }\eta_{\sigma\delta}]-\frac{1}{4}\frac{1}{(2-d)}k_{Q\mu}k_{R\nu}\eta_{\rho\tau% }\eta_{\sigma\delta}\eta_{\gamma\alpha}\eta_{\lambda\beta}+(\mu\leftrightarrow% \nu)," class="ltx_math_unparsed" display="block" id="S2.E18.m1.104"><semantics id="S2.E18.m1.104a"><mtable displaystyle="true" id="S2.E18.m1.104.104.1" rowspacing="0pt"><mtr id="S2.E18.m1.104.104.1a"><mtd class="ltx_align_left" columnalign="left" id="S2.E18.m1.104.104.1b"><mrow id="S2.E18.m1.104.104.1.104.53.53"><mrow id="S2.E18.m1.104.104.1.104.53.53.54"><msubsup id="S2.E18.m1.104.104.1.104.53.53.54.2"><mi id="S2.E18.m1.1.1.1.1.1.1">V</mi><mrow id="S2.E18.m1.3.3.3.3.3.3.1"><mi id="S2.E18.m1.3.3.3.3.3.3.1.2">μ</mi><mo id="S2.E18.m1.3.3.3.3.3.3.1.1">⁢</mo><mi id="S2.E18.m1.3.3.3.3.3.3.1.3">ν</mi><mo id="S2.E18.m1.3.3.3.3.3.3.1.1a">⁢</mo><mi id="S2.E18.m1.3.3.3.3.3.3.1.4">ρ</mi><mo id="S2.E18.m1.3.3.3.3.3.3.1.1b">⁢</mo><mi id="S2.E18.m1.3.3.3.3.3.3.1.5">σ</mi><mo id="S2.E18.m1.3.3.3.3.3.3.1.1c">⁢</mo><mi id="S2.E18.m1.3.3.3.3.3.3.1.6">γ</mi><mo id="S2.E18.m1.3.3.3.3.3.3.1.1d">⁢</mo><mi id="S2.E18.m1.3.3.3.3.3.3.1.7">λ</mi><mo id="S2.E18.m1.3.3.3.3.3.3.1.1e">⁢</mo><mi id="S2.E18.m1.3.3.3.3.3.3.1.8">τ</mi><mo id="S2.E18.m1.3.3.3.3.3.3.1.1f">⁢</mo><mi id="S2.E18.m1.3.3.3.3.3.3.1.9">δ</mi><mo id="S2.E18.m1.3.3.3.3.3.3.1.1g">⁢</mo><mi id="S2.E18.m1.3.3.3.3.3.3.1.10">α</mi><mo id="S2.E18.m1.3.3.3.3.3.3.1.1h">⁢</mo><mi id="S2.E18.m1.3.3.3.3.3.3.1.11">β</mi></mrow><mrow id="S2.E18.m1.2.2.2.2.2.2.1.3"><mo id="S2.E18.m1.2.2.2.2.2.2.1.3.1" stretchy="false">(</mo><mn id="S2.E18.m1.2.2.2.2.2.2.1.1">5</mn><mo id="S2.E18.m1.2.2.2.2.2.2.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S2.E18.m1.104.104.1.104.53.53.54.1">⁢</mo><mrow id="S2.E18.m1.104.104.1.104.53.53.54.3"><mo id="S2.E18.m1.4.4.4.4.4.4" stretchy="false">(</mo><mi id="S2.E18.m1.5.5.5.5.5.5">Q</mi><mo id="S2.E18.m1.6.6.6.6.6.6">,</mo><mi id="S2.E18.m1.7.7.7.7.7.7">R</mi><mo id="S2.E18.m1.8.8.8.8.8.8">,</mo><mi id="S2.E18.m1.9.9.9.9.9.9">S</mi><mo id="S2.E18.m1.10.10.10.10.10.10" stretchy="false">)</mo></mrow></mrow><mo id="S2.E18.m1.11.11.11.11.11.11">=</mo><mrow id="S2.E18.m1.104.104.1.104.53.53.53"><mrow id="S2.E18.m1.104.104.1.104.53.53.53.1"><mfrac id="S2.E18.m1.12.12.12.12.12.12"><mn id="S2.E18.m1.12.12.12.12.12.12.2">1</mn><mn id="S2.E18.m1.12.12.12.12.12.12.3">2</mn></mfrac><mo id="S2.E18.m1.104.104.1.104.53.53.53.1.2">⁢</mo><msub id="S2.E18.m1.104.104.1.104.53.53.53.1.3"><mi id="S2.E18.m1.13.13.13.13.13.13">η</mi><mrow id="S2.E18.m1.14.14.14.14.14.14.1"><mi id="S2.E18.m1.14.14.14.14.14.14.1.2">μ</mi><mo id="S2.E18.m1.14.14.14.14.14.14.1.1">⁢</mo><mi id="S2.E18.m1.14.14.14.14.14.14.1.3">α</mi></mrow></msub><mo id="S2.E18.m1.104.104.1.104.53.53.53.1.2a">⁢</mo><msub id="S2.E18.m1.104.104.1.104.53.53.53.1.4"><mi id="S2.E18.m1.15.15.15.15.15.15">η</mi><mrow id="S2.E18.m1.16.16.16.16.16.16.1"><mi id="S2.E18.m1.16.16.16.16.16.16.1.2">ρ</mi><mo id="S2.E18.m1.16.16.16.16.16.16.1.1">⁢</mo><mi id="S2.E18.m1.16.16.16.16.16.16.1.3">β</mi></mrow></msub><mo id="S2.E18.m1.104.104.1.104.53.53.53.1.2b">⁢</mo><msub id="S2.E18.m1.104.104.1.104.53.53.53.1.5"><mi id="S2.E18.m1.17.17.17.17.17.17">η</mi><mrow id="S2.E18.m1.18.18.18.18.18.18.1"><mi id="S2.E18.m1.18.18.18.18.18.18.1.2">σ</mi><mo id="S2.E18.m1.18.18.18.18.18.18.1.1">⁢</mo><mi id="S2.E18.m1.18.18.18.18.18.18.1.3">δ</mi></mrow></msub><mo id="S2.E18.m1.104.104.1.104.53.53.53.1.2c">⁢</mo><mrow id="S2.E18.m1.104.104.1.104.53.53.53.1.1.1"><mo id="S2.E18.m1.19.19.19.19.19.19" stretchy="false">[</mo><mrow id="S2.E18.m1.104.104.1.104.53.53.53.1.1.1.1"><mrow id="S2.E18.m1.104.104.1.104.53.53.53.1.1.1.1.1"><mrow id="S2.E18.m1.104.104.1.104.53.53.53.1.1.1.1.1.1.1"><mo id="S2.E18.m1.20.20.20.20.20.20" stretchy="false">(</mo><mrow id="S2.E18.m1.104.104.1.104.53.53.53.1.1.1.1.1.1.1.1"><msub id="S2.E18.m1.104.104.1.104.53.53.53.1.1.1.1.1.1.1.1.1"><mi id="S2.E18.m1.21.21.21.21.21.21">k</mi><mi id="S2.E18.m1.22.22.22.22.22.22.1">Q</mi></msub><mo id="S2.E18.m1.23.23.23.23.23.23" lspace="0.222em" rspace="0.222em">⋅</mo><msub id="S2.E18.m1.104.104.1.104.53.53.53.1.1.1.1.1.1.1.1.2"><mi id="S2.E18.m1.24.24.24.24.24.24">k</mi><mrow id="S2.E18.m1.25.25.25.25.25.25.1"><mi id="S2.E18.m1.25.25.25.25.25.25.1.2">R</mi><mo id="S2.E18.m1.25.25.25.25.25.25.1.1">⁢</mo><mi id="S2.E18.m1.25.25.25.25.25.25.1.3">S</mi></mrow></msub></mrow><mo id="S2.E18.m1.26.26.26.26.26.26" stretchy="false">)</mo></mrow><mo id="S2.E18.m1.104.104.1.104.53.53.53.1.1.1.1.1.2">⁢</mo><msub id="S2.E18.m1.104.104.1.104.53.53.53.1.1.1.1.1.3"><mi id="S2.E18.m1.27.27.27.27.27.27">η</mi><mrow id="S2.E18.m1.28.28.28.28.28.28.1"><mi id="S2.E18.m1.28.28.28.28.28.28.1.2">ν</mi><mo id="S2.E18.m1.28.28.28.28.28.28.1.1">⁢</mo><mi id="S2.E18.m1.28.28.28.28.28.28.1.3">γ</mi></mrow></msub><mo id="S2.E18.m1.104.104.1.104.53.53.53.1.1.1.1.1.2a">⁢</mo><msub id="S2.E18.m1.104.104.1.104.53.53.53.1.1.1.1.1.4"><mi id="S2.E18.m1.29.29.29.29.29.29">η</mi><mrow id="S2.E18.m1.30.30.30.30.30.30.1"><mi id="S2.E18.m1.30.30.30.30.30.30.1.2">λ</mi><mo id="S2.E18.m1.30.30.30.30.30.30.1.1">⁢</mo><mi id="S2.E18.m1.30.30.30.30.30.30.1.3">τ</mi></mrow></msub></mrow><mo id="S2.E18.m1.31.31.31.31.31.31">−</mo><mrow id="S2.E18.m1.104.104.1.104.53.53.53.1.1.1.1.2"><msub id="S2.E18.m1.104.104.1.104.53.53.53.1.1.1.1.2.2"><mi id="S2.E18.m1.32.32.32.32.32.32">k</mi><mrow id="S2.E18.m1.33.33.33.33.33.33.1"><mi id="S2.E18.m1.33.33.33.33.33.33.1.2">Q</mi><mo id="S2.E18.m1.33.33.33.33.33.33.1.1">⁢</mo><mi id="S2.E18.m1.33.33.33.33.33.33.1.3">γ</mi></mrow></msub><mo id="S2.E18.m1.104.104.1.104.53.53.53.1.1.1.1.2.1">⁢</mo><msub id="S2.E18.m1.104.104.1.104.53.53.53.1.1.1.1.2.3"><mi id="S2.E18.m1.34.34.34.34.34.34">k</mi><mrow id="S2.E18.m1.35.35.35.35.35.35.1"><mi id="S2.E18.m1.35.35.35.35.35.35.1.2">S</mi><mo id="S2.E18.m1.35.35.35.35.35.35.1.1">⁢</mo><mi id="S2.E18.m1.35.35.35.35.35.35.1.3">λ</mi></mrow></msub><mo id="S2.E18.m1.104.104.1.104.53.53.53.1.1.1.1.2.1a">⁢</mo><msub id="S2.E18.m1.104.104.1.104.53.53.53.1.1.1.1.2.4"><mi id="S2.E18.m1.36.36.36.36.36.36">η</mi><mrow id="S2.E18.m1.37.37.37.37.37.37.1"><mi id="S2.E18.m1.37.37.37.37.37.37.1.2">ν</mi><mo id="S2.E18.m1.37.37.37.37.37.37.1.1">⁢</mo><mi id="S2.E18.m1.37.37.37.37.37.37.1.3">τ</mi></mrow></msub></mrow></mrow><mo id="S2.E18.m1.38.38.38.38.38.38" stretchy="false">]</mo></mrow></mrow><mo id="S2.E18.m1.39.39.39.39.39.39">−</mo><mrow id="S2.E18.m1.104.104.1.104.53.53.53.2"><mfrac id="S2.E18.m1.40.40.40.40.40.40"><mn id="S2.E18.m1.40.40.40.40.40.40.2">1</mn><mn id="S2.E18.m1.40.40.40.40.40.40.3">2</mn></mfrac><mo id="S2.E18.m1.104.104.1.104.53.53.53.2.1">⁢</mo><msub id="S2.E18.m1.104.104.1.104.53.53.53.2.2"><mi id="S2.E18.m1.41.41.41.41.41.41">k</mi><mrow id="S2.E18.m1.42.42.42.42.42.42.1"><mi id="S2.E18.m1.42.42.42.42.42.42.1.2">Q</mi><mo id="S2.E18.m1.42.42.42.42.42.42.1.1">⁢</mo><mi id="S2.E18.m1.42.42.42.42.42.42.1.3">γ</mi></mrow></msub><mo id="S2.E18.m1.104.104.1.104.53.53.53.2.1a">⁢</mo><msub id="S2.E18.m1.104.104.1.104.53.53.53.2.3"><mi id="S2.E18.m1.43.43.43.43.43.43">k</mi><mrow id="S2.E18.m1.44.44.44.44.44.44.1"><mi id="S2.E18.m1.44.44.44.44.44.44.1.2">R</mi><mo id="S2.E18.m1.44.44.44.44.44.44.1.1">⁢</mo><mi id="S2.E18.m1.44.44.44.44.44.44.1.3">σ</mi></mrow></msub><mo id="S2.E18.m1.104.104.1.104.53.53.53.2.1b">⁢</mo><msub id="S2.E18.m1.104.104.1.104.53.53.53.2.4"><mi id="S2.E18.m1.45.45.45.45.45.45">η</mi><mrow id="S2.E18.m1.46.46.46.46.46.46.1"><mi id="S2.E18.m1.46.46.46.46.46.46.1.2">μ</mi><mo id="S2.E18.m1.46.46.46.46.46.46.1.1">⁢</mo><mi id="S2.E18.m1.46.46.46.46.46.46.1.3">τ</mi></mrow></msub><mo id="S2.E18.m1.104.104.1.104.53.53.53.2.1c">⁢</mo><msub id="S2.E18.m1.104.104.1.104.53.53.53.2.5"><mi id="S2.E18.m1.47.47.47.47.47.47">η</mi><mrow id="S2.E18.m1.48.48.48.48.48.48.1"><mi id="S2.E18.m1.48.48.48.48.48.48.1.2">ν</mi><mo id="S2.E18.m1.48.48.48.48.48.48.1.1">⁢</mo><mi id="S2.E18.m1.48.48.48.48.48.48.1.3">α</mi></mrow></msub><mo id="S2.E18.m1.104.104.1.104.53.53.53.2.1d">⁢</mo><msub id="S2.E18.m1.104.104.1.104.53.53.53.2.6"><mi id="S2.E18.m1.49.49.49.49.49.49">η</mi><mrow id="S2.E18.m1.50.50.50.50.50.50.1"><mi id="S2.E18.m1.50.50.50.50.50.50.1.2">ρ</mi><mo id="S2.E18.m1.50.50.50.50.50.50.1.1">⁢</mo><mi id="S2.E18.m1.50.50.50.50.50.50.1.3">δ</mi></mrow></msub><mo id="S2.E18.m1.104.104.1.104.53.53.53.2.1e">⁢</mo><msub id="S2.E18.m1.104.104.1.104.53.53.53.2.7"><mi id="S2.E18.m1.51.51.51.51.51.51">η</mi><mrow id="S2.E18.m1.52.52.52.52.52.52.1"><mi id="S2.E18.m1.52.52.52.52.52.52.1.2">λ</mi><mo id="S2.E18.m1.52.52.52.52.52.52.1.1">⁢</mo><mi id="S2.E18.m1.52.52.52.52.52.52.1.3">β</mi></mrow></msub></mrow></mrow></mrow></mtd></mtr><mtr id="S2.E18.m1.104.104.1c"><mtd class="ltx_align_right" columnalign="right" id="S2.E18.m1.104.104.1d"><mrow id="S2.E18.m1.103.103.103.51.51"><mo id="S2.E18.m1.53.53.53.1.1.1">+</mo><mfrac id="S2.E18.m1.54.54.54.2.2.2"><mn id="S2.E18.m1.54.54.54.2.2.2.2">1</mn><mn id="S2.E18.m1.54.54.54.2.2.2.3">2</mn></mfrac><mfrac id="S2.E18.m1.55.55.55.3.3.3"><mn id="S2.E18.m1.55.55.55.3.3.3.3">1</mn><mrow id="S2.E18.m1.55.55.55.3.3.3.1.1"><mo id="S2.E18.m1.55.55.55.3.3.3.1.1.2" stretchy="false">(</mo><mrow id="S2.E18.m1.55.55.55.3.3.3.1.1.1"><mn id="S2.E18.m1.55.55.55.3.3.3.1.1.1.2">2</mn><mo id="S2.E18.m1.55.55.55.3.3.3.1.1.1.1">−</mo><mi id="S2.E18.m1.55.55.55.3.3.3.1.1.1.3">d</mi></mrow><mo id="S2.E18.m1.55.55.55.3.3.3.1.1.3" stretchy="false">)</mo></mrow></mfrac><msub id="S2.E18.m1.103.103.103.51.51.52"><mi id="S2.E18.m1.56.56.56.4.4.4">η</mi><mrow id="S2.E18.m1.57.57.57.5.5.5.1"><mi id="S2.E18.m1.57.57.57.5.5.5.1.2">μ</mi><mo id="S2.E18.m1.57.57.57.5.5.5.1.1">⁢</mo><mi id="S2.E18.m1.57.57.57.5.5.5.1.3">α</mi></mrow></msub><msub id="S2.E18.m1.103.103.103.51.51.53"><mi id="S2.E18.m1.58.58.58.6.6.6">η</mi><mrow id="S2.E18.m1.59.59.59.7.7.7.1"><mi id="S2.E18.m1.59.59.59.7.7.7.1.2">ν</mi><mo id="S2.E18.m1.59.59.59.7.7.7.1.1">⁢</mo><mi id="S2.E18.m1.59.59.59.7.7.7.1.3">β</mi></mrow></msub><msub id="S2.E18.m1.103.103.103.51.51.54"><mi id="S2.E18.m1.60.60.60.8.8.8">η</mi><mrow id="S2.E18.m1.61.61.61.9.9.9.1"><mi id="S2.E18.m1.61.61.61.9.9.9.1.2">γ</mi><mo id="S2.E18.m1.61.61.61.9.9.9.1.1">⁢</mo><mi id="S2.E18.m1.61.61.61.9.9.9.1.3">τ</mi></mrow></msub><mrow id="S2.E18.m1.103.103.103.51.51.55"><mo id="S2.E18.m1.62.62.62.10.10.10" stretchy="false">[</mo><msub id="S2.E18.m1.103.103.103.51.51.55.1"><mi id="S2.E18.m1.63.63.63.11.11.11">k</mi><mrow id="S2.E18.m1.64.64.64.12.12.12.1"><mi id="S2.E18.m1.64.64.64.12.12.12.1.2">Q</mi><mo id="S2.E18.m1.64.64.64.12.12.12.1.1">⁢</mo><mi id="S2.E18.m1.64.64.64.12.12.12.1.3">R</mi><mo id="S2.E18.m1.64.64.64.12.12.12.1.1a">⁢</mo><mi id="S2.E18.m1.64.64.64.12.12.12.1.4">S</mi><mo id="S2.E18.m1.64.64.64.12.12.12.1.1b">⁢</mo><mi id="S2.E18.m1.64.64.64.12.12.12.1.5">ρ</mi></mrow></msub><msub id="S2.E18.m1.103.103.103.51.51.55.2"><mi id="S2.E18.m1.65.65.65.13.13.13">k</mi><mrow id="S2.E18.m1.66.66.66.14.14.14.1"><mi id="S2.E18.m1.66.66.66.14.14.14.1.2">R</mi><mo id="S2.E18.m1.66.66.66.14.14.14.1.1">⁢</mo><mi id="S2.E18.m1.66.66.66.14.14.14.1.3">σ</mi></mrow></msub><msub id="S2.E18.m1.103.103.103.51.51.55.3"><mi id="S2.E18.m1.67.67.67.15.15.15">η</mi><mrow id="S2.E18.m1.68.68.68.16.16.16.1"><mi id="S2.E18.m1.68.68.68.16.16.16.1.2">λ</mi><mo id="S2.E18.m1.68.68.68.16.16.16.1.1">⁢</mo><mi id="S2.E18.m1.68.68.68.16.16.16.1.3">δ</mi></mrow></msub><mo id="S2.E18.m1.69.69.69.17.17.17">−</mo><mrow id="S2.E18.m1.103.103.103.51.51.55.4"><mo id="S2.E18.m1.70.70.70.18.18.18" stretchy="false">(</mo><msub id="S2.E18.m1.103.103.103.51.51.55.4.1"><mi id="S2.E18.m1.71.71.71.19.19.19">k</mi><mrow id="S2.E18.m1.72.72.72.20.20.20.1"><mi id="S2.E18.m1.72.72.72.20.20.20.1.2">Q</mi><mo id="S2.E18.m1.72.72.72.20.20.20.1.1">⁢</mo><mi id="S2.E18.m1.72.72.72.20.20.20.1.3">R</mi><mo id="S2.E18.m1.72.72.72.20.20.20.1.1a">⁢</mo><mi id="S2.E18.m1.72.72.72.20.20.20.1.4">S</mi></mrow></msub><mo id="S2.E18.m1.73.73.73.21.21.21" lspace="0.222em" rspace="0.222em">⋅</mo><msub id="S2.E18.m1.103.103.103.51.51.55.4.2"><mi id="S2.E18.m1.74.74.74.22.22.22">k</mi><mi id="S2.E18.m1.75.75.75.23.23.23.1">Q</mi></msub><mo id="S2.E18.m1.76.76.76.24.24.24" stretchy="false">)</mo></mrow><msub id="S2.E18.m1.103.103.103.51.51.55.5"><mi id="S2.E18.m1.77.77.77.25.25.25">η</mi><mrow id="S2.E18.m1.78.78.78.26.26.26.1"><mi id="S2.E18.m1.78.78.78.26.26.26.1.2">ρ</mi><mo id="S2.E18.m1.78.78.78.26.26.26.1.1">⁢</mo><mi id="S2.E18.m1.78.78.78.26.26.26.1.3">λ</mi></mrow></msub><msub id="S2.E18.m1.103.103.103.51.51.55.6"><mi id="S2.E18.m1.79.79.79.27.27.27">η</mi><mrow id="S2.E18.m1.80.80.80.28.28.28.1"><mi id="S2.E18.m1.80.80.80.28.28.28.1.2">σ</mi><mo id="S2.E18.m1.80.80.80.28.28.28.1.1">⁢</mo><mi id="S2.E18.m1.80.80.80.28.28.28.1.3">δ</mi></mrow></msub><mo id="S2.E18.m1.81.81.81.29.29.29" stretchy="false">]</mo></mrow><mo id="S2.E18.m1.82.82.82.30.30.30">−</mo><mfrac id="S2.E18.m1.83.83.83.31.31.31"><mn id="S2.E18.m1.83.83.83.31.31.31.2">1</mn><mn id="S2.E18.m1.83.83.83.31.31.31.3">4</mn></mfrac><mfrac id="S2.E18.m1.84.84.84.32.32.32"><mn id="S2.E18.m1.84.84.84.32.32.32.3">1</mn><mrow id="S2.E18.m1.84.84.84.32.32.32.1.1"><mo id="S2.E18.m1.84.84.84.32.32.32.1.1.2" stretchy="false">(</mo><mrow id="S2.E18.m1.84.84.84.32.32.32.1.1.1"><mn id="S2.E18.m1.84.84.84.32.32.32.1.1.1.2">2</mn><mo id="S2.E18.m1.84.84.84.32.32.32.1.1.1.1">−</mo><mi id="S2.E18.m1.84.84.84.32.32.32.1.1.1.3">d</mi></mrow><mo id="S2.E18.m1.84.84.84.32.32.32.1.1.3" stretchy="false">)</mo></mrow></mfrac><msub id="S2.E18.m1.103.103.103.51.51.56"><mi id="S2.E18.m1.85.85.85.33.33.33">k</mi><mrow id="S2.E18.m1.86.86.86.34.34.34.1"><mi id="S2.E18.m1.86.86.86.34.34.34.1.2">Q</mi><mo id="S2.E18.m1.86.86.86.34.34.34.1.1">⁢</mo><mi id="S2.E18.m1.86.86.86.34.34.34.1.3">μ</mi></mrow></msub><msub id="S2.E18.m1.103.103.103.51.51.57"><mi id="S2.E18.m1.87.87.87.35.35.35">k</mi><mrow id="S2.E18.m1.88.88.88.36.36.36.1"><mi id="S2.E18.m1.88.88.88.36.36.36.1.2">R</mi><mo id="S2.E18.m1.88.88.88.36.36.36.1.1">⁢</mo><mi id="S2.E18.m1.88.88.88.36.36.36.1.3">ν</mi></mrow></msub><msub id="S2.E18.m1.103.103.103.51.51.58"><mi id="S2.E18.m1.89.89.89.37.37.37">η</mi><mrow id="S2.E18.m1.90.90.90.38.38.38.1"><mi id="S2.E18.m1.90.90.90.38.38.38.1.2">ρ</mi><mo id="S2.E18.m1.90.90.90.38.38.38.1.1">⁢</mo><mi id="S2.E18.m1.90.90.90.38.38.38.1.3">τ</mi></mrow></msub><msub id="S2.E18.m1.103.103.103.51.51.59"><mi id="S2.E18.m1.91.91.91.39.39.39">η</mi><mrow id="S2.E18.m1.92.92.92.40.40.40.1"><mi id="S2.E18.m1.92.92.92.40.40.40.1.2">σ</mi><mo id="S2.E18.m1.92.92.92.40.40.40.1.1">⁢</mo><mi id="S2.E18.m1.92.92.92.40.40.40.1.3">δ</mi></mrow></msub><msub id="S2.E18.m1.103.103.103.51.51.60"><mi id="S2.E18.m1.93.93.93.41.41.41">η</mi><mrow id="S2.E18.m1.94.94.94.42.42.42.1"><mi id="S2.E18.m1.94.94.94.42.42.42.1.2">γ</mi><mo id="S2.E18.m1.94.94.94.42.42.42.1.1">⁢</mo><mi id="S2.E18.m1.94.94.94.42.42.42.1.3">α</mi></mrow></msub><msub id="S2.E18.m1.103.103.103.51.51.61"><mi id="S2.E18.m1.95.95.95.43.43.43">η</mi><mrow id="S2.E18.m1.96.96.96.44.44.44.1"><mi id="S2.E18.m1.96.96.96.44.44.44.1.2">λ</mi><mo id="S2.E18.m1.96.96.96.44.44.44.1.1">⁢</mo><mi id="S2.E18.m1.96.96.96.44.44.44.1.3">β</mi></mrow></msub><mo id="S2.E18.m1.97.97.97.45.45.45">+</mo><mrow id="S2.E18.m1.103.103.103.51.51.62"><mo id="S2.E18.m1.98.98.98.46.46.46" stretchy="false">(</mo><mi id="S2.E18.m1.99.99.99.47.47.47">μ</mi><mo id="S2.E18.m1.100.100.100.48.48.48" stretchy="false">↔</mo><mi id="S2.E18.m1.101.101.101.49.49.49">ν</mi><mo id="S2.E18.m1.102.102.102.50.50.50" stretchy="false">)</mo></mrow><mo id="S2.E18.m1.103.103.103.51.51.51">,</mo></mrow></mtd></mtr></mtable><annotation encoding="application/x-tex" id="S2.E18.m1.104b">V^{(5)}_{\mu\nu\rho\sigma\gamma\lambda\tau\delta\alpha\beta}(Q,R,S)=\frac{1}{2% }\eta_{\mu\alpha}\eta_{\rho\beta}\eta_{\sigma\delta}[(k_{Q}\cdot k_{RS})\eta_{% \nu\gamma}\eta_{\lambda\tau}-k_{Q\gamma}k_{S\lambda}\eta_{\nu\tau}]-\frac{1}{2% }k_{Q\gamma}k_{R\sigma}\eta_{\mu\tau}\eta_{\nu\alpha}\eta_{\rho\delta}\eta_{% \lambda\beta}\\ +\frac{1}{2}\frac{1}{(2-d)}\eta_{\mu\alpha}\eta_{\nu\beta}\eta_{\gamma\tau}[k_% {QRS\rho}k_{R\sigma}\eta_{\lambda\delta}-(k_{QRS}\cdot k_{Q})\eta_{\rho\lambda% }\eta_{\sigma\delta}]-\frac{1}{4}\frac{1}{(2-d)}k_{Q\mu}k_{R\nu}\eta_{\rho\tau% }\eta_{\sigma\delta}\eta_{\gamma\alpha}\eta_{\lambda\beta}+(\mu\leftrightarrow% \nu),</annotation><annotation encoding="application/x-llamapun" id="S2.E18.m1.104c">start_ROW start_CELL italic_V start_POSTSUPERSCRIPT ( 5 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ italic_ν italic_ρ italic_σ italic_γ italic_λ italic_τ italic_δ italic_α italic_β end_POSTSUBSCRIPT ( italic_Q , italic_R , italic_S ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_η start_POSTSUBSCRIPT italic_μ italic_α end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_ρ italic_β end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_σ italic_δ end_POSTSUBSCRIPT [ ( italic_k start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT ⋅ italic_k start_POSTSUBSCRIPT italic_R italic_S end_POSTSUBSCRIPT ) italic_η start_POSTSUBSCRIPT italic_ν italic_γ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_λ italic_τ end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT italic_Q italic_γ end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_S italic_λ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_ν italic_τ end_POSTSUBSCRIPT ] - divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_k start_POSTSUBSCRIPT italic_Q italic_γ end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_R italic_σ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_μ italic_τ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_ν italic_α end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_ρ italic_δ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_λ italic_β end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL + divide start_ARG 1 end_ARG start_ARG 2 end_ARG divide start_ARG 1 end_ARG start_ARG ( 2 - italic_d ) end_ARG italic_η start_POSTSUBSCRIPT italic_μ italic_α end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_ν italic_β end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_γ italic_τ end_POSTSUBSCRIPT [ italic_k start_POSTSUBSCRIPT italic_Q italic_R italic_S italic_ρ end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_R italic_σ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_λ italic_δ end_POSTSUBSCRIPT - ( italic_k start_POSTSUBSCRIPT italic_Q italic_R italic_S end_POSTSUBSCRIPT ⋅ italic_k start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT ) italic_η start_POSTSUBSCRIPT italic_ρ italic_λ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_σ italic_δ end_POSTSUBSCRIPT ] - divide start_ARG 1 end_ARG start_ARG 4 end_ARG divide start_ARG 1 end_ARG start_ARG ( 2 - italic_d ) end_ARG italic_k start_POSTSUBSCRIPT italic_Q italic_μ end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_R italic_ν end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_ρ italic_τ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_σ italic_δ end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_γ italic_α end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_λ italic_β end_POSTSUBSCRIPT + ( italic_μ ↔ italic_ν ) , end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(18)</span></td> </tr></tbody> </table> <table class="ltx_equation ltx_eqn_table" id="S2.E19"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{V}^{(\textrm{g})\rho}_{\mu\nu\sigma}(P,Q)=\frac{1}{2}(k_{Q\mu}-k_{P% \mu})k_{Q\nu}\delta^{\rho}_{\sigma}\\ -\frac{1}{2}k_{P\mu}k_{Q\sigma}\delta^{\rho}_{\nu}+(\mu\leftrightarrow\nu)." class="ltx_math_unparsed" display="block" id="S2.E19.m1.39"><semantics id="S2.E19.m1.39a"><mtable displaystyle="true" id="S2.E19.m1.39.39.1" rowspacing="0pt"><mtr id="S2.E19.m1.39.39.1a"><mtd class="ltx_align_left" columnalign="left" id="S2.E19.m1.39.39.1b"><mrow id="S2.E19.m1.39.39.1.39.23.23"><mrow id="S2.E19.m1.39.39.1.39.23.23.24"><msubsup id="S2.E19.m1.39.39.1.39.23.23.24.2"><mi class="ltx_font_mathcaligraphic" id="S2.E19.m1.1.1.1.1.1.1">𝒱</mi><mrow id="S2.E19.m1.3.3.3.3.3.3.1"><mi id="S2.E19.m1.3.3.3.3.3.3.1.2">μ</mi><mo id="S2.E19.m1.3.3.3.3.3.3.1.1">⁢</mo><mi id="S2.E19.m1.3.3.3.3.3.3.1.3">ν</mi><mo id="S2.E19.m1.3.3.3.3.3.3.1.1a">⁢</mo><mi id="S2.E19.m1.3.3.3.3.3.3.1.4">σ</mi></mrow><mrow id="S2.E19.m1.2.2.2.2.2.2.1"><mrow id="S2.E19.m1.2.2.2.2.2.2.1.3.2"><mo id="S2.E19.m1.2.2.2.2.2.2.1.3.2.1" stretchy="false">(</mo><mtext id="S2.E19.m1.2.2.2.2.2.2.1.1">g</mtext><mo id="S2.E19.m1.2.2.2.2.2.2.1.3.2.2" stretchy="false">)</mo></mrow><mo id="S2.E19.m1.2.2.2.2.2.2.1.2">⁢</mo><mi id="S2.E19.m1.2.2.2.2.2.2.1.4">ρ</mi></mrow></msubsup><mo id="S2.E19.m1.39.39.1.39.23.23.24.1">⁢</mo><mrow id="S2.E19.m1.39.39.1.39.23.23.24.3"><mo id="S2.E19.m1.4.4.4.4.4.4" stretchy="false">(</mo><mi id="S2.E19.m1.5.5.5.5.5.5">P</mi><mo id="S2.E19.m1.6.6.6.6.6.6">,</mo><mi id="S2.E19.m1.7.7.7.7.7.7">Q</mi><mo id="S2.E19.m1.8.8.8.8.8.8" stretchy="false">)</mo></mrow></mrow><mo id="S2.E19.m1.9.9.9.9.9.9">=</mo><mrow id="S2.E19.m1.39.39.1.39.23.23.23"><mfrac id="S2.E19.m1.10.10.10.10.10.10"><mn id="S2.E19.m1.10.10.10.10.10.10.2">1</mn><mn id="S2.E19.m1.10.10.10.10.10.10.3">2</mn></mfrac><mo id="S2.E19.m1.39.39.1.39.23.23.23.2">⁢</mo><mrow id="S2.E19.m1.39.39.1.39.23.23.23.1.1"><mo id="S2.E19.m1.11.11.11.11.11.11" stretchy="false">(</mo><mrow id="S2.E19.m1.39.39.1.39.23.23.23.1.1.1"><msub id="S2.E19.m1.39.39.1.39.23.23.23.1.1.1.1"><mi id="S2.E19.m1.12.12.12.12.12.12">k</mi><mrow id="S2.E19.m1.13.13.13.13.13.13.1"><mi id="S2.E19.m1.13.13.13.13.13.13.1.2">Q</mi><mo id="S2.E19.m1.13.13.13.13.13.13.1.1">⁢</mo><mi id="S2.E19.m1.13.13.13.13.13.13.1.3">μ</mi></mrow></msub><mo id="S2.E19.m1.14.14.14.14.14.14">−</mo><msub id="S2.E19.m1.39.39.1.39.23.23.23.1.1.1.2"><mi id="S2.E19.m1.15.15.15.15.15.15">k</mi><mrow id="S2.E19.m1.16.16.16.16.16.16.1"><mi id="S2.E19.m1.16.16.16.16.16.16.1.2">P</mi><mo id="S2.E19.m1.16.16.16.16.16.16.1.1">⁢</mo><mi id="S2.E19.m1.16.16.16.16.16.16.1.3">μ</mi></mrow></msub></mrow><mo id="S2.E19.m1.17.17.17.17.17.17" stretchy="false">)</mo></mrow><mo id="S2.E19.m1.39.39.1.39.23.23.23.2a">⁢</mo><msub id="S2.E19.m1.39.39.1.39.23.23.23.3"><mi id="S2.E19.m1.18.18.18.18.18.18">k</mi><mrow id="S2.E19.m1.19.19.19.19.19.19.1"><mi id="S2.E19.m1.19.19.19.19.19.19.1.2">Q</mi><mo id="S2.E19.m1.19.19.19.19.19.19.1.1">⁢</mo><mi id="S2.E19.m1.19.19.19.19.19.19.1.3">ν</mi></mrow></msub><mo id="S2.E19.m1.39.39.1.39.23.23.23.2b">⁢</mo><msubsup id="S2.E19.m1.39.39.1.39.23.23.23.4"><mi id="S2.E19.m1.20.20.20.20.20.20">δ</mi><mi id="S2.E19.m1.22.22.22.22.22.22.1">σ</mi><mi id="S2.E19.m1.21.21.21.21.21.21.1">ρ</mi></msubsup></mrow></mrow></mtd></mtr><mtr id="S2.E19.m1.39.39.1c"><mtd class="ltx_align_right" columnalign="right" id="S2.E19.m1.39.39.1d"><mrow id="S2.E19.m1.38.38.38.16.16"><mo id="S2.E19.m1.23.23.23.1.1.1">−</mo><mfrac id="S2.E19.m1.24.24.24.2.2.2"><mn id="S2.E19.m1.24.24.24.2.2.2.2">1</mn><mn id="S2.E19.m1.24.24.24.2.2.2.3">2</mn></mfrac><msub id="S2.E19.m1.38.38.38.16.16.17"><mi id="S2.E19.m1.25.25.25.3.3.3">k</mi><mrow id="S2.E19.m1.26.26.26.4.4.4.1"><mi id="S2.E19.m1.26.26.26.4.4.4.1.2">P</mi><mo id="S2.E19.m1.26.26.26.4.4.4.1.1">⁢</mo><mi id="S2.E19.m1.26.26.26.4.4.4.1.3">μ</mi></mrow></msub><msub id="S2.E19.m1.38.38.38.16.16.18"><mi id="S2.E19.m1.27.27.27.5.5.5">k</mi><mrow id="S2.E19.m1.28.28.28.6.6.6.1"><mi id="S2.E19.m1.28.28.28.6.6.6.1.2">Q</mi><mo id="S2.E19.m1.28.28.28.6.6.6.1.1">⁢</mo><mi id="S2.E19.m1.28.28.28.6.6.6.1.3">σ</mi></mrow></msub><msubsup id="S2.E19.m1.38.38.38.16.16.19"><mi id="S2.E19.m1.29.29.29.7.7.7">δ</mi><mi id="S2.E19.m1.31.31.31.9.9.9.1">ν</mi><mi id="S2.E19.m1.30.30.30.8.8.8.1">ρ</mi></msubsup><mo id="S2.E19.m1.32.32.32.10.10.10">+</mo><mrow id="S2.E19.m1.38.38.38.16.16.20"><mo id="S2.E19.m1.33.33.33.11.11.11" stretchy="false">(</mo><mi id="S2.E19.m1.34.34.34.12.12.12">μ</mi><mo id="S2.E19.m1.35.35.35.13.13.13" stretchy="false">↔</mo><mi id="S2.E19.m1.36.36.36.14.14.14">ν</mi><mo id="S2.E19.m1.37.37.37.15.15.15" stretchy="false">)</mo></mrow><mo id="S2.E19.m1.38.38.38.16.16.16" lspace="0em">.</mo></mrow></mtd></mtr></mtable><annotation encoding="application/x-tex" id="S2.E19.m1.39b">\mathcal{V}^{(\textrm{g})\rho}_{\mu\nu\sigma}(P,Q)=\frac{1}{2}(k_{Q\mu}-k_{P% \mu})k_{Q\nu}\delta^{\rho}_{\sigma}\\ -\frac{1}{2}k_{P\mu}k_{Q\sigma}\delta^{\rho}_{\nu}+(\mu\leftrightarrow\nu).</annotation><annotation encoding="application/x-llamapun" id="S2.E19.m1.39c">start_ROW start_CELL caligraphic_V start_POSTSUPERSCRIPT ( g ) italic_ρ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ italic_ν italic_σ end_POSTSUBSCRIPT ( italic_P , italic_Q ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( italic_k start_POSTSUBSCRIPT italic_Q italic_μ end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT italic_P italic_μ end_POSTSUBSCRIPT ) italic_k start_POSTSUBSCRIPT italic_Q italic_ν end_POSTSUBSCRIPT italic_δ start_POSTSUPERSCRIPT italic_ρ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL - divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_k start_POSTSUBSCRIPT italic_P italic_μ end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_Q italic_σ end_POSTSUBSCRIPT italic_δ start_POSTSUPERSCRIPT italic_ρ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT + ( italic_μ ↔ italic_ν ) . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(19)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.p7"> <p class="ltx_p" id="S2.p7.8">These objects are clearly connected to the Feynman vertices of the theory. Using (<a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S2.E15" title="In II Equations of motion ‣ One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_tag">15</span></a>), we can make this more explicit. The recursion (<a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S2.E12" title="In II Equations of motion ‣ One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_tag">12</span></a>) is suggestively rewritten as</p> <table class="ltx_equation ltx_eqn_table" id="S2.E20"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="s_{P}\mathbb{P}_{\mu\nu\rho\sigma}I_{P}^{\rho\sigma}=\sum_{n=2}^{\infty}\sum_{% P=P_{1}\cup\ldots\cup P_{n}}\prod_{i=1}^{n}I_{P_{i}}^{\mu_{i}\nu_{i}}\\ \times\frac{1}{n!}\mathcal{V}_{\mu\nu\mu_{1}\nu_{1}\cdots\mu_{n}\nu_{n}}^{(n+1% )}(P_{1},\ldots,P_{n})+\textrm{ghosts}," class="ltx_Math" display="block" id="S2.E20.m1.38"><semantics id="S2.E20.m1.38a"><mtable displaystyle="true" id="S2.E20.m1.38.38.2" rowspacing="0pt"><mtr id="S2.E20.m1.38.38.2a"><mtd class="ltx_align_left" columnalign="left" id="S2.E20.m1.38.38.2b"><mrow id="S2.E20.m1.19.19.19.19.19"><mrow id="S2.E20.m1.19.19.19.19.19.20"><msub id="S2.E20.m1.19.19.19.19.19.20.2"><mi id="S2.E20.m1.1.1.1.1.1.1" xref="S2.E20.m1.1.1.1.1.1.1.cmml">s</mi><mi id="S2.E20.m1.2.2.2.2.2.2.1" xref="S2.E20.m1.2.2.2.2.2.2.1.cmml">P</mi></msub><mo id="S2.E20.m1.19.19.19.19.19.20.1" xref="S2.E20.m1.37.37.1.1.1.cmml">⁢</mo><msub id="S2.E20.m1.19.19.19.19.19.20.3"><mi id="S2.E20.m1.3.3.3.3.3.3" xref="S2.E20.m1.3.3.3.3.3.3.cmml">ℙ</mi><mrow id="S2.E20.m1.4.4.4.4.4.4.1" xref="S2.E20.m1.4.4.4.4.4.4.1.cmml"><mi id="S2.E20.m1.4.4.4.4.4.4.1.2" xref="S2.E20.m1.4.4.4.4.4.4.1.2.cmml">μ</mi><mo id="S2.E20.m1.4.4.4.4.4.4.1.1" xref="S2.E20.m1.4.4.4.4.4.4.1.1.cmml">⁢</mo><mi id="S2.E20.m1.4.4.4.4.4.4.1.3" xref="S2.E20.m1.4.4.4.4.4.4.1.3.cmml">ν</mi><mo id="S2.E20.m1.4.4.4.4.4.4.1.1a" xref="S2.E20.m1.4.4.4.4.4.4.1.1.cmml">⁢</mo><mi id="S2.E20.m1.4.4.4.4.4.4.1.4" xref="S2.E20.m1.4.4.4.4.4.4.1.4.cmml">ρ</mi><mo id="S2.E20.m1.4.4.4.4.4.4.1.1b" xref="S2.E20.m1.4.4.4.4.4.4.1.1.cmml">⁢</mo><mi id="S2.E20.m1.4.4.4.4.4.4.1.5" xref="S2.E20.m1.4.4.4.4.4.4.1.5.cmml">σ</mi></mrow></msub><mo id="S2.E20.m1.19.19.19.19.19.20.1a" xref="S2.E20.m1.37.37.1.1.1.cmml">⁢</mo><msubsup id="S2.E20.m1.19.19.19.19.19.20.4"><mi id="S2.E20.m1.5.5.5.5.5.5" xref="S2.E20.m1.5.5.5.5.5.5.cmml">I</mi><mi id="S2.E20.m1.6.6.6.6.6.6.1" xref="S2.E20.m1.6.6.6.6.6.6.1.cmml">P</mi><mrow id="S2.E20.m1.7.7.7.7.7.7.1" xref="S2.E20.m1.7.7.7.7.7.7.1.cmml"><mi id="S2.E20.m1.7.7.7.7.7.7.1.2" xref="S2.E20.m1.7.7.7.7.7.7.1.2.cmml">ρ</mi><mo id="S2.E20.m1.7.7.7.7.7.7.1.1" xref="S2.E20.m1.7.7.7.7.7.7.1.1.cmml">⁢</mo><mi id="S2.E20.m1.7.7.7.7.7.7.1.3" xref="S2.E20.m1.7.7.7.7.7.7.1.3.cmml">σ</mi></mrow></msubsup></mrow><mo id="S2.E20.m1.8.8.8.8.8.8" rspace="0.111em" xref="S2.E20.m1.8.8.8.8.8.8.cmml">=</mo><mrow id="S2.E20.m1.19.19.19.19.19.21"><munderover id="S2.E20.m1.19.19.19.19.19.21.1"><mo id="S2.E20.m1.9.9.9.9.9.9" movablelimits="false" rspace="0em" xref="S2.E20.m1.9.9.9.9.9.9.cmml">∑</mo><mrow id="S2.E20.m1.10.10.10.10.10.10.1" xref="S2.E20.m1.10.10.10.10.10.10.1.cmml"><mi id="S2.E20.m1.10.10.10.10.10.10.1.2" xref="S2.E20.m1.10.10.10.10.10.10.1.2.cmml">n</mi><mo id="S2.E20.m1.10.10.10.10.10.10.1.1" xref="S2.E20.m1.10.10.10.10.10.10.1.1.cmml">=</mo><mn id="S2.E20.m1.10.10.10.10.10.10.1.3" xref="S2.E20.m1.10.10.10.10.10.10.1.3.cmml">2</mn></mrow><mi id="S2.E20.m1.11.11.11.11.11.11.1" mathvariant="normal" xref="S2.E20.m1.11.11.11.11.11.11.1.cmml">∞</mi></munderover><mrow id="S2.E20.m1.19.19.19.19.19.21.2"><munder id="S2.E20.m1.19.19.19.19.19.21.2.1"><mo id="S2.E20.m1.12.12.12.12.12.12" movablelimits="false" rspace="0em" xref="S2.E20.m1.12.12.12.12.12.12.cmml">∑</mo><mrow id="S2.E20.m1.13.13.13.13.13.13.1" xref="S2.E20.m1.13.13.13.13.13.13.1.cmml"><mi id="S2.E20.m1.13.13.13.13.13.13.1.2" xref="S2.E20.m1.13.13.13.13.13.13.1.2.cmml">P</mi><mo id="S2.E20.m1.13.13.13.13.13.13.1.1" xref="S2.E20.m1.13.13.13.13.13.13.1.1.cmml">=</mo><mrow 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xref="S2.E20.m1.35.35.35.16.16.16">ghosts</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E20.m1.38c">s_{P}\mathbb{P}_{\mu\nu\rho\sigma}I_{P}^{\rho\sigma}=\sum_{n=2}^{\infty}\sum_{% P=P_{1}\cup\ldots\cup P_{n}}\prod_{i=1}^{n}I_{P_{i}}^{\mu_{i}\nu_{i}}\\ \times\frac{1}{n!}\mathcal{V}_{\mu\nu\mu_{1}\nu_{1}\cdots\mu_{n}\nu_{n}}^{(n+1% )}(P_{1},\ldots,P_{n})+\textrm{ghosts},</annotation><annotation encoding="application/x-llamapun" id="S2.E20.m1.38d">start_ROW start_CELL italic_s start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT blackboard_P start_POSTSUBSCRIPT italic_μ italic_ν italic_ρ italic_σ end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ρ italic_σ end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_n = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_P = italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∪ … ∪ italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∏ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_I start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL × divide start_ARG 1 end_ARG start_ARG italic_n ! end_ARG caligraphic_V start_POSTSUBSCRIPT italic_μ italic_ν italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋯ italic_μ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_n + 1 ) end_POSTSUPERSCRIPT ( italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) + ghosts , end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(20)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p7.2">where <math alttext="\mathcal{V}^{(3)}" class="ltx_Math" display="inline" id="S2.p7.1.m1.1"><semantics id="S2.p7.1.m1.1a"><msup id="S2.p7.1.m1.1.2" xref="S2.p7.1.m1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.p7.1.m1.1.2.2" xref="S2.p7.1.m1.1.2.2.cmml">𝒱</mi><mrow id="S2.p7.1.m1.1.1.1.3" xref="S2.p7.1.m1.1.2.cmml"><mo id="S2.p7.1.m1.1.1.1.3.1" stretchy="false" xref="S2.p7.1.m1.1.2.cmml">(</mo><mn id="S2.p7.1.m1.1.1.1.1" xref="S2.p7.1.m1.1.1.1.1.cmml">3</mn><mo id="S2.p7.1.m1.1.1.1.3.2" stretchy="false" xref="S2.p7.1.m1.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.p7.1.m1.1b"><apply id="S2.p7.1.m1.1.2.cmml" xref="S2.p7.1.m1.1.2"><csymbol cd="ambiguous" id="S2.p7.1.m1.1.2.1.cmml" xref="S2.p7.1.m1.1.2">superscript</csymbol><ci id="S2.p7.1.m1.1.2.2.cmml" xref="S2.p7.1.m1.1.2.2">𝒱</ci><cn id="S2.p7.1.m1.1.1.1.1.cmml" type="integer" xref="S2.p7.1.m1.1.1.1.1">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p7.1.m1.1c">\mathcal{V}^{(3)}</annotation><annotation encoding="application/x-llamapun" id="S2.p7.1.m1.1d">caligraphic_V start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT</annotation></semantics></math> is given in (<a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S2.E16" title="In II Equations of motion ‣ One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_tag">16</span></a>) and the general expression for <math alttext="n\geq 3" class="ltx_Math" display="inline" id="S2.p7.2.m2.1"><semantics id="S2.p7.2.m2.1a"><mrow id="S2.p7.2.m2.1.1" xref="S2.p7.2.m2.1.1.cmml"><mi id="S2.p7.2.m2.1.1.2" xref="S2.p7.2.m2.1.1.2.cmml">n</mi><mo id="S2.p7.2.m2.1.1.1" xref="S2.p7.2.m2.1.1.1.cmml">≥</mo><mn id="S2.p7.2.m2.1.1.3" xref="S2.p7.2.m2.1.1.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p7.2.m2.1b"><apply id="S2.p7.2.m2.1.1.cmml" xref="S2.p7.2.m2.1.1"><geq id="S2.p7.2.m2.1.1.1.cmml" xref="S2.p7.2.m2.1.1.1"></geq><ci id="S2.p7.2.m2.1.1.2.cmml" xref="S2.p7.2.m2.1.1.2">𝑛</ci><cn id="S2.p7.2.m2.1.1.3.cmml" type="integer" xref="S2.p7.2.m2.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p7.2.m2.1c">n\geq 3</annotation><annotation encoding="application/x-llamapun" id="S2.p7.2.m2.1d">italic_n ≥ 3</annotation></semantics></math> is given by</p> <table class="ltx_equation ltx_eqn_table" id="S2.E21"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{V}^{(n+1)}_{\mu\nu\mu_{1}\nu_{1}\cdots\mu_{n}\nu_{n}}(P_{1},\ldots,P_% {n})\\ =V^{(4)}_{\mu\nu\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{n}}(P_{1},P_{2},P_{3}% \cup\ldots\cup P_{n})\prod_{i=3}^{n-1}\eta_{\nu_{i}\mu_{i+1}}\\ +\sum_{k=4}^{n}V^{(5)}_{\mu\nu\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{k-1}\mu_% {k}\nu_{n}}(P_{1},P_{2},P_{3}\cup\ldots\cup P_{k-1})\\ \times\prod_{i=4}^{k-1}\eta_{\nu_{i-1}\mu_{i}}\prod_{j=k}^{n-1}\eta_{\nu_{j}% \mu_{j+1}}+S(1,\ldots,n)." class="ltx_Math" display="block" id="S2.E21.m1.88"><semantics id="S2.E21.m1.88a"><mtable displaystyle="true" id="S2.E21.m1.88.88.9" rowspacing="0pt"><mtr id="S2.E21.m1.88.88.9a"><mtd class="ltx_align_left" columnalign="left" id="S2.E21.m1.88.88.9b"><mrow id="S2.E21.m1.82.82.3.81.14.14"><msubsup id="S2.E21.m1.82.82.3.81.14.14.16"><mi class="ltx_font_mathcaligraphic" id="S2.E21.m1.1.1.1.1.1.1" xref="S2.E21.m1.1.1.1.1.1.1.cmml">𝒱</mi><mrow id="S2.E21.m1.3.3.3.3.3.3.1" xref="S2.E21.m1.3.3.3.3.3.3.1.cmml"><mi id="S2.E21.m1.3.3.3.3.3.3.1.2" xref="S2.E21.m1.3.3.3.3.3.3.1.2.cmml">μ</mi><mo 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id="S2.E21.m1.82.82.3.81.14.14.14.2.2"><mi id="S2.E21.m1.10.10.10.10.10.10" xref="S2.E21.m1.10.10.10.10.10.10.cmml">P</mi><mi id="S2.E21.m1.11.11.11.11.11.11.1" xref="S2.E21.m1.11.11.11.11.11.11.1.cmml">n</mi></msub><mo id="S2.E21.m1.12.12.12.12.12.12" stretchy="false" xref="S2.E21.m1.80.80.1.1.1.cmml">)</mo></mrow></mrow></mtd></mtr><mtr id="S2.E21.m1.88.88.9c"><mtd class="ltx_align_right" columnalign="right" id="S2.E21.m1.88.88.9d"><mrow id="S2.E21.m1.85.85.6.84.27.27"><mi id="S2.E21.m1.85.85.6.84.27.27.28" xref="S2.E21.m1.80.80.1.1.1.cmml"></mi><mo id="S2.E21.m1.13.13.13.1.1.1" xref="S2.E21.m1.13.13.13.1.1.1.cmml">=</mo><mrow id="S2.E21.m1.85.85.6.84.27.27.27"><msubsup id="S2.E21.m1.85.85.6.84.27.27.27.5"><mi id="S2.E21.m1.14.14.14.2.2.2" xref="S2.E21.m1.14.14.14.2.2.2.cmml">V</mi><mrow id="S2.E21.m1.16.16.16.4.4.4.1" xref="S2.E21.m1.16.16.16.4.4.4.1.cmml"><mi id="S2.E21.m1.16.16.16.4.4.4.1.2" xref="S2.E21.m1.16.16.16.4.4.4.1.2.cmml">μ</mi><mo id="S2.E21.m1.16.16.16.4.4.4.1.1" xref="S2.E21.m1.16.16.16.4.4.4.1.1.cmml">⁢</mo><mi id="S2.E21.m1.16.16.16.4.4.4.1.3" xref="S2.E21.m1.16.16.16.4.4.4.1.3.cmml">ν</mi><mo id="S2.E21.m1.16.16.16.4.4.4.1.1a" xref="S2.E21.m1.16.16.16.4.4.4.1.1.cmml">⁢</mo><msub id="S2.E21.m1.16.16.16.4.4.4.1.4" xref="S2.E21.m1.16.16.16.4.4.4.1.4.cmml"><mi id="S2.E21.m1.16.16.16.4.4.4.1.4.2" xref="S2.E21.m1.16.16.16.4.4.4.1.4.2.cmml">μ</mi><mn id="S2.E21.m1.16.16.16.4.4.4.1.4.3" xref="S2.E21.m1.16.16.16.4.4.4.1.4.3.cmml">1</mn></msub><mo id="S2.E21.m1.16.16.16.4.4.4.1.1b" xref="S2.E21.m1.16.16.16.4.4.4.1.1.cmml">⁢</mo><msub id="S2.E21.m1.16.16.16.4.4.4.1.5" xref="S2.E21.m1.16.16.16.4.4.4.1.5.cmml"><mi id="S2.E21.m1.16.16.16.4.4.4.1.5.2" xref="S2.E21.m1.16.16.16.4.4.4.1.5.2.cmml">ν</mi><mn id="S2.E21.m1.16.16.16.4.4.4.1.5.3" xref="S2.E21.m1.16.16.16.4.4.4.1.5.3.cmml">1</mn></msub><mo id="S2.E21.m1.16.16.16.4.4.4.1.1c" xref="S2.E21.m1.16.16.16.4.4.4.1.1.cmml">⁢</mo><msub id="S2.E21.m1.16.16.16.4.4.4.1.6" xref="S2.E21.m1.16.16.16.4.4.4.1.6.cmml"><mi id="S2.E21.m1.16.16.16.4.4.4.1.6.2" xref="S2.E21.m1.16.16.16.4.4.4.1.6.2.cmml">μ</mi><mn id="S2.E21.m1.16.16.16.4.4.4.1.6.3" xref="S2.E21.m1.16.16.16.4.4.4.1.6.3.cmml">2</mn></msub><mo id="S2.E21.m1.16.16.16.4.4.4.1.1d" xref="S2.E21.m1.16.16.16.4.4.4.1.1.cmml">⁢</mo><msub id="S2.E21.m1.16.16.16.4.4.4.1.7" xref="S2.E21.m1.16.16.16.4.4.4.1.7.cmml"><mi id="S2.E21.m1.16.16.16.4.4.4.1.7.2" xref="S2.E21.m1.16.16.16.4.4.4.1.7.2.cmml">ν</mi><mn id="S2.E21.m1.16.16.16.4.4.4.1.7.3" xref="S2.E21.m1.16.16.16.4.4.4.1.7.3.cmml">2</mn></msub><mo id="S2.E21.m1.16.16.16.4.4.4.1.1e" xref="S2.E21.m1.16.16.16.4.4.4.1.1.cmml">⁢</mo><msub id="S2.E21.m1.16.16.16.4.4.4.1.8" xref="S2.E21.m1.16.16.16.4.4.4.1.8.cmml"><mi id="S2.E21.m1.16.16.16.4.4.4.1.8.2" xref="S2.E21.m1.16.16.16.4.4.4.1.8.2.cmml">μ</mi><mn id="S2.E21.m1.16.16.16.4.4.4.1.8.3" xref="S2.E21.m1.16.16.16.4.4.4.1.8.3.cmml">3</mn></msub><mo id="S2.E21.m1.16.16.16.4.4.4.1.1f" xref="S2.E21.m1.16.16.16.4.4.4.1.1.cmml">⁢</mo><msub id="S2.E21.m1.16.16.16.4.4.4.1.9" xref="S2.E21.m1.16.16.16.4.4.4.1.9.cmml"><mi id="S2.E21.m1.16.16.16.4.4.4.1.9.2" xref="S2.E21.m1.16.16.16.4.4.4.1.9.2.cmml">ν</mi><mi id="S2.E21.m1.16.16.16.4.4.4.1.9.3" xref="S2.E21.m1.16.16.16.4.4.4.1.9.3.cmml">n</mi></msub></mrow><mrow id="S2.E21.m1.15.15.15.3.3.3.1.3"><mo id="S2.E21.m1.15.15.15.3.3.3.1.3.1" stretchy="false">(</mo><mn id="S2.E21.m1.15.15.15.3.3.3.1.1" xref="S2.E21.m1.15.15.15.3.3.3.1.1.cmml">4</mn><mo id="S2.E21.m1.15.15.15.3.3.3.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S2.E21.m1.85.85.6.84.27.27.27.4" xref="S2.E21.m1.80.80.1.1.1.cmml">⁢</mo><mrow id="S2.E21.m1.85.85.6.84.27.27.27.3.3"><mo id="S2.E21.m1.17.17.17.5.5.5" stretchy="false" xref="S2.E21.m1.80.80.1.1.1.cmml">(</mo><msub id="S2.E21.m1.83.83.4.82.25.25.25.1.1.1"><mi id="S2.E21.m1.18.18.18.6.6.6" xref="S2.E21.m1.18.18.18.6.6.6.cmml">P</mi><mn id="S2.E21.m1.19.19.19.7.7.7.1" xref="S2.E21.m1.19.19.19.7.7.7.1.cmml">1</mn></msub><mo id="S2.E21.m1.20.20.20.8.8.8" xref="S2.E21.m1.80.80.1.1.1.cmml">,</mo><msub id="S2.E21.m1.84.84.5.83.26.26.26.2.2.2"><mi id="S2.E21.m1.21.21.21.9.9.9" xref="S2.E21.m1.21.21.21.9.9.9.cmml">P</mi><mn id="S2.E21.m1.22.22.22.10.10.10.1" xref="S2.E21.m1.22.22.22.10.10.10.1.cmml">2</mn></msub><mo id="S2.E21.m1.23.23.23.11.11.11" xref="S2.E21.m1.80.80.1.1.1.cmml">,</mo><mrow id="S2.E21.m1.85.85.6.84.27.27.27.3.3.3"><msub id="S2.E21.m1.85.85.6.84.27.27.27.3.3.3.1"><mi id="S2.E21.m1.24.24.24.12.12.12" xref="S2.E21.m1.24.24.24.12.12.12.cmml">P</mi><mn id="S2.E21.m1.25.25.25.13.13.13.1" xref="S2.E21.m1.25.25.25.13.13.13.1.cmml">3</mn></msub><mo id="S2.E21.m1.26.26.26.14.14.14" xref="S2.E21.m1.26.26.26.14.14.14.cmml">∪</mo><mi id="S2.E21.m1.27.27.27.15.15.15" mathvariant="normal" xref="S2.E21.m1.27.27.27.15.15.15.cmml">…</mi><mo id="S2.E21.m1.26.26.26.14.14.14a" xref="S2.E21.m1.26.26.26.14.14.14.cmml">∪</mo><msub id="S2.E21.m1.85.85.6.84.27.27.27.3.3.3.2"><mi id="S2.E21.m1.29.29.29.17.17.17" xref="S2.E21.m1.29.29.29.17.17.17.cmml">P</mi><mi id="S2.E21.m1.30.30.30.18.18.18.1" xref="S2.E21.m1.30.30.30.18.18.18.1.cmml">n</mi></msub></mrow><mo id="S2.E21.m1.31.31.31.19.19.19" stretchy="false" xref="S2.E21.m1.80.80.1.1.1.cmml">)</mo></mrow><mo id="S2.E21.m1.85.85.6.84.27.27.27.4a" xref="S2.E21.m1.80.80.1.1.1.cmml">⁢</mo><mrow id="S2.E21.m1.85.85.6.84.27.27.27.6"><munderover id="S2.E21.m1.85.85.6.84.27.27.27.6.1"><mo id="S2.E21.m1.32.32.32.20.20.20" movablelimits="false" xref="S2.E21.m1.32.32.32.20.20.20.cmml">∏</mo><mrow id="S2.E21.m1.33.33.33.21.21.21.1" xref="S2.E21.m1.33.33.33.21.21.21.1.cmml"><mi id="S2.E21.m1.33.33.33.21.21.21.1.2" xref="S2.E21.m1.33.33.33.21.21.21.1.2.cmml">i</mi><mo id="S2.E21.m1.33.33.33.21.21.21.1.1" xref="S2.E21.m1.33.33.33.21.21.21.1.1.cmml">=</mo><mn id="S2.E21.m1.33.33.33.21.21.21.1.3" xref="S2.E21.m1.33.33.33.21.21.21.1.3.cmml">3</mn></mrow><mrow id="S2.E21.m1.34.34.34.22.22.22.1" xref="S2.E21.m1.34.34.34.22.22.22.1.cmml"><mi id="S2.E21.m1.34.34.34.22.22.22.1.2" xref="S2.E21.m1.34.34.34.22.22.22.1.2.cmml">n</mi><mo id="S2.E21.m1.34.34.34.22.22.22.1.1" xref="S2.E21.m1.34.34.34.22.22.22.1.1.cmml">−</mo><mn id="S2.E21.m1.34.34.34.22.22.22.1.3" xref="S2.E21.m1.34.34.34.22.22.22.1.3.cmml">1</mn></mrow></munderover><msub id="S2.E21.m1.85.85.6.84.27.27.27.6.2"><mi id="S2.E21.m1.35.35.35.23.23.23" xref="S2.E21.m1.35.35.35.23.23.23.cmml">η</mi><mrow id="S2.E21.m1.36.36.36.24.24.24.1" xref="S2.E21.m1.36.36.36.24.24.24.1.cmml"><msub id="S2.E21.m1.36.36.36.24.24.24.1.2" xref="S2.E21.m1.36.36.36.24.24.24.1.2.cmml"><mi id="S2.E21.m1.36.36.36.24.24.24.1.2.2" xref="S2.E21.m1.36.36.36.24.24.24.1.2.2.cmml">ν</mi><mi id="S2.E21.m1.36.36.36.24.24.24.1.2.3" xref="S2.E21.m1.36.36.36.24.24.24.1.2.3.cmml">i</mi></msub><mo id="S2.E21.m1.36.36.36.24.24.24.1.1" xref="S2.E21.m1.36.36.36.24.24.24.1.1.cmml">⁢</mo><msub id="S2.E21.m1.36.36.36.24.24.24.1.3" xref="S2.E21.m1.36.36.36.24.24.24.1.3.cmml"><mi id="S2.E21.m1.36.36.36.24.24.24.1.3.2" xref="S2.E21.m1.36.36.36.24.24.24.1.3.2.cmml">μ</mi><mrow id="S2.E21.m1.36.36.36.24.24.24.1.3.3" xref="S2.E21.m1.36.36.36.24.24.24.1.3.3.cmml"><mi id="S2.E21.m1.36.36.36.24.24.24.1.3.3.2" xref="S2.E21.m1.36.36.36.24.24.24.1.3.3.2.cmml">i</mi><mo id="S2.E21.m1.36.36.36.24.24.24.1.3.3.1" xref="S2.E21.m1.36.36.36.24.24.24.1.3.3.1.cmml">+</mo><mn id="S2.E21.m1.36.36.36.24.24.24.1.3.3.3" xref="S2.E21.m1.36.36.36.24.24.24.1.3.3.3.cmml">1</mn></mrow></msub></mrow></msub></mrow></mrow></mrow></mtd></mtr><mtr id="S2.E21.m1.88.88.9e"><mtd class="ltx_align_right" columnalign="right" id="S2.E21.m1.88.88.9f"><mrow id="S2.E21.m1.88.88.9.87.25.25"><mo id="S2.E21.m1.88.88.9.87.25.25a" xref="S2.E21.m1.80.80.1.1.1.cmml">+</mo><mrow id="S2.E21.m1.88.88.9.87.25.25.25"><munderover id="S2.E21.m1.88.88.9.87.25.25.25.4"><mo id="S2.E21.m1.38.38.38.2.2.2" movablelimits="false" xref="S2.E21.m1.38.38.38.2.2.2.cmml">∑</mo><mrow id="S2.E21.m1.39.39.39.3.3.3.1" xref="S2.E21.m1.39.39.39.3.3.3.1.cmml"><mi id="S2.E21.m1.39.39.39.3.3.3.1.2" xref="S2.E21.m1.39.39.39.3.3.3.1.2.cmml">k</mi><mo id="S2.E21.m1.39.39.39.3.3.3.1.1" xref="S2.E21.m1.39.39.39.3.3.3.1.1.cmml">=</mo><mn id="S2.E21.m1.39.39.39.3.3.3.1.3" xref="S2.E21.m1.39.39.39.3.3.3.1.3.cmml">4</mn></mrow><mi id="S2.E21.m1.40.40.40.4.4.4.1" xref="S2.E21.m1.40.40.40.4.4.4.1.cmml">n</mi></munderover><mrow id="S2.E21.m1.88.88.9.87.25.25.25.3"><msubsup id="S2.E21.m1.88.88.9.87.25.25.25.3.5"><mi id="S2.E21.m1.41.41.41.5.5.5" xref="S2.E21.m1.41.41.41.5.5.5.cmml">V</mi><mrow id="S2.E21.m1.43.43.43.7.7.7.1" xref="S2.E21.m1.43.43.43.7.7.7.1.cmml"><mi id="S2.E21.m1.43.43.43.7.7.7.1.2" xref="S2.E21.m1.43.43.43.7.7.7.1.2.cmml">μ</mi><mo id="S2.E21.m1.43.43.43.7.7.7.1.1" xref="S2.E21.m1.43.43.43.7.7.7.1.1.cmml">⁢</mo><mi id="S2.E21.m1.43.43.43.7.7.7.1.3" xref="S2.E21.m1.43.43.43.7.7.7.1.3.cmml">ν</mi><mo id="S2.E21.m1.43.43.43.7.7.7.1.1a" xref="S2.E21.m1.43.43.43.7.7.7.1.1.cmml">⁢</mo><msub id="S2.E21.m1.43.43.43.7.7.7.1.4" xref="S2.E21.m1.43.43.43.7.7.7.1.4.cmml"><mi id="S2.E21.m1.43.43.43.7.7.7.1.4.2" xref="S2.E21.m1.43.43.43.7.7.7.1.4.2.cmml">μ</mi><mn id="S2.E21.m1.43.43.43.7.7.7.1.4.3" xref="S2.E21.m1.43.43.43.7.7.7.1.4.3.cmml">1</mn></msub><mo id="S2.E21.m1.43.43.43.7.7.7.1.1b" xref="S2.E21.m1.43.43.43.7.7.7.1.1.cmml">⁢</mo><msub id="S2.E21.m1.43.43.43.7.7.7.1.5" xref="S2.E21.m1.43.43.43.7.7.7.1.5.cmml"><mi id="S2.E21.m1.43.43.43.7.7.7.1.5.2" xref="S2.E21.m1.43.43.43.7.7.7.1.5.2.cmml">ν</mi><mn id="S2.E21.m1.43.43.43.7.7.7.1.5.3" xref="S2.E21.m1.43.43.43.7.7.7.1.5.3.cmml">1</mn></msub><mo id="S2.E21.m1.43.43.43.7.7.7.1.1c" xref="S2.E21.m1.43.43.43.7.7.7.1.1.cmml">⁢</mo><msub id="S2.E21.m1.43.43.43.7.7.7.1.6" xref="S2.E21.m1.43.43.43.7.7.7.1.6.cmml"><mi id="S2.E21.m1.43.43.43.7.7.7.1.6.2" xref="S2.E21.m1.43.43.43.7.7.7.1.6.2.cmml">μ</mi><mn id="S2.E21.m1.43.43.43.7.7.7.1.6.3" xref="S2.E21.m1.43.43.43.7.7.7.1.6.3.cmml">2</mn></msub><mo id="S2.E21.m1.43.43.43.7.7.7.1.1d" xref="S2.E21.m1.43.43.43.7.7.7.1.1.cmml">⁢</mo><msub id="S2.E21.m1.43.43.43.7.7.7.1.7" xref="S2.E21.m1.43.43.43.7.7.7.1.7.cmml"><mi id="S2.E21.m1.43.43.43.7.7.7.1.7.2" xref="S2.E21.m1.43.43.43.7.7.7.1.7.2.cmml">ν</mi><mn id="S2.E21.m1.43.43.43.7.7.7.1.7.3" xref="S2.E21.m1.43.43.43.7.7.7.1.7.3.cmml">2</mn></msub><mo id="S2.E21.m1.43.43.43.7.7.7.1.1e" xref="S2.E21.m1.43.43.43.7.7.7.1.1.cmml">⁢</mo><msub id="S2.E21.m1.43.43.43.7.7.7.1.8" xref="S2.E21.m1.43.43.43.7.7.7.1.8.cmml"><mi id="S2.E21.m1.43.43.43.7.7.7.1.8.2" xref="S2.E21.m1.43.43.43.7.7.7.1.8.2.cmml">μ</mi><mn id="S2.E21.m1.43.43.43.7.7.7.1.8.3" xref="S2.E21.m1.43.43.43.7.7.7.1.8.3.cmml">3</mn></msub><mo id="S2.E21.m1.43.43.43.7.7.7.1.1f" xref="S2.E21.m1.43.43.43.7.7.7.1.1.cmml">⁢</mo><msub id="S2.E21.m1.43.43.43.7.7.7.1.9" xref="S2.E21.m1.43.43.43.7.7.7.1.9.cmml"><mi id="S2.E21.m1.43.43.43.7.7.7.1.9.2" xref="S2.E21.m1.43.43.43.7.7.7.1.9.2.cmml">ν</mi><mrow id="S2.E21.m1.43.43.43.7.7.7.1.9.3" xref="S2.E21.m1.43.43.43.7.7.7.1.9.3.cmml"><mi id="S2.E21.m1.43.43.43.7.7.7.1.9.3.2" xref="S2.E21.m1.43.43.43.7.7.7.1.9.3.2.cmml">k</mi><mo id="S2.E21.m1.43.43.43.7.7.7.1.9.3.1" xref="S2.E21.m1.43.43.43.7.7.7.1.9.3.1.cmml">−</mo><mn id="S2.E21.m1.43.43.43.7.7.7.1.9.3.3" xref="S2.E21.m1.43.43.43.7.7.7.1.9.3.3.cmml">1</mn></mrow></msub><mo id="S2.E21.m1.43.43.43.7.7.7.1.1g" xref="S2.E21.m1.43.43.43.7.7.7.1.1.cmml">⁢</mo><msub id="S2.E21.m1.43.43.43.7.7.7.1.10" xref="S2.E21.m1.43.43.43.7.7.7.1.10.cmml"><mi id="S2.E21.m1.43.43.43.7.7.7.1.10.2" xref="S2.E21.m1.43.43.43.7.7.7.1.10.2.cmml">μ</mi><mi id="S2.E21.m1.43.43.43.7.7.7.1.10.3" xref="S2.E21.m1.43.43.43.7.7.7.1.10.3.cmml">k</mi></msub><mo id="S2.E21.m1.43.43.43.7.7.7.1.1h" xref="S2.E21.m1.43.43.43.7.7.7.1.1.cmml">⁢</mo><msub id="S2.E21.m1.43.43.43.7.7.7.1.11" xref="S2.E21.m1.43.43.43.7.7.7.1.11.cmml"><mi id="S2.E21.m1.43.43.43.7.7.7.1.11.2" xref="S2.E21.m1.43.43.43.7.7.7.1.11.2.cmml">ν</mi><mi id="S2.E21.m1.43.43.43.7.7.7.1.11.3" xref="S2.E21.m1.43.43.43.7.7.7.1.11.3.cmml">n</mi></msub></mrow><mrow id="S2.E21.m1.42.42.42.6.6.6.1.3"><mo id="S2.E21.m1.42.42.42.6.6.6.1.3.1" stretchy="false">(</mo><mn id="S2.E21.m1.42.42.42.6.6.6.1.1" xref="S2.E21.m1.42.42.42.6.6.6.1.1.cmml">5</mn><mo id="S2.E21.m1.42.42.42.6.6.6.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S2.E21.m1.88.88.9.87.25.25.25.3.4" xref="S2.E21.m1.80.80.1.1.1.cmml">⁢</mo><mrow id="S2.E21.m1.88.88.9.87.25.25.25.3.3.3"><mo id="S2.E21.m1.44.44.44.8.8.8" stretchy="false" xref="S2.E21.m1.80.80.1.1.1.cmml">(</mo><msub id="S2.E21.m1.86.86.7.85.23.23.23.1.1.1.1"><mi id="S2.E21.m1.45.45.45.9.9.9" xref="S2.E21.m1.45.45.45.9.9.9.cmml">P</mi><mn id="S2.E21.m1.46.46.46.10.10.10.1" xref="S2.E21.m1.46.46.46.10.10.10.1.cmml">1</mn></msub><mo id="S2.E21.m1.47.47.47.11.11.11" xref="S2.E21.m1.80.80.1.1.1.cmml">,</mo><msub id="S2.E21.m1.87.87.8.86.24.24.24.2.2.2.2"><mi id="S2.E21.m1.48.48.48.12.12.12" xref="S2.E21.m1.48.48.48.12.12.12.cmml">P</mi><mn id="S2.E21.m1.49.49.49.13.13.13.1" xref="S2.E21.m1.49.49.49.13.13.13.1.cmml">2</mn></msub><mo id="S2.E21.m1.50.50.50.14.14.14" xref="S2.E21.m1.80.80.1.1.1.cmml">,</mo><mrow id="S2.E21.m1.88.88.9.87.25.25.25.3.3.3.3"><msub id="S2.E21.m1.88.88.9.87.25.25.25.3.3.3.3.1"><mi id="S2.E21.m1.51.51.51.15.15.15" xref="S2.E21.m1.51.51.51.15.15.15.cmml">P</mi><mn id="S2.E21.m1.52.52.52.16.16.16.1" xref="S2.E21.m1.52.52.52.16.16.16.1.cmml">3</mn></msub><mo id="S2.E21.m1.53.53.53.17.17.17" xref="S2.E21.m1.53.53.53.17.17.17.cmml">∪</mo><mi id="S2.E21.m1.54.54.54.18.18.18" mathvariant="normal" xref="S2.E21.m1.54.54.54.18.18.18.cmml">…</mi><mo id="S2.E21.m1.53.53.53.17.17.17a" xref="S2.E21.m1.53.53.53.17.17.17.cmml">∪</mo><msub id="S2.E21.m1.88.88.9.87.25.25.25.3.3.3.3.2"><mi id="S2.E21.m1.56.56.56.20.20.20" xref="S2.E21.m1.56.56.56.20.20.20.cmml">P</mi><mrow id="S2.E21.m1.57.57.57.21.21.21.1" xref="S2.E21.m1.57.57.57.21.21.21.1.cmml"><mi id="S2.E21.m1.57.57.57.21.21.21.1.2" xref="S2.E21.m1.57.57.57.21.21.21.1.2.cmml">k</mi><mo id="S2.E21.m1.57.57.57.21.21.21.1.1" xref="S2.E21.m1.57.57.57.21.21.21.1.1.cmml">−</mo><mn id="S2.E21.m1.57.57.57.21.21.21.1.3" xref="S2.E21.m1.57.57.57.21.21.21.1.3.cmml">1</mn></mrow></msub></mrow><mo id="S2.E21.m1.58.58.58.22.22.22" stretchy="false" xref="S2.E21.m1.80.80.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr id="S2.E21.m1.88.88.9g"><mtd class="ltx_align_right" columnalign="right" id="S2.E21.m1.88.88.9h"><mrow id="S2.E21.m1.79.79.79.21.21"><mo id="S2.E21.m1.59.59.59.1.1.1" rspace="0.055em" xref="S2.E21.m1.59.59.59.1.1.1.cmml">×</mo><munderover id="S2.E21.m1.79.79.79.21.21.22"><mo id="S2.E21.m1.60.60.60.2.2.2" movablelimits="false" xref="S2.E21.m1.60.60.60.2.2.2.cmml">∏</mo><mrow id="S2.E21.m1.61.61.61.3.3.3.1" xref="S2.E21.m1.61.61.61.3.3.3.1.cmml"><mi id="S2.E21.m1.61.61.61.3.3.3.1.2" xref="S2.E21.m1.61.61.61.3.3.3.1.2.cmml">i</mi><mo id="S2.E21.m1.61.61.61.3.3.3.1.1" xref="S2.E21.m1.61.61.61.3.3.3.1.1.cmml">=</mo><mn id="S2.E21.m1.61.61.61.3.3.3.1.3" xref="S2.E21.m1.61.61.61.3.3.3.1.3.cmml">4</mn></mrow><mrow id="S2.E21.m1.62.62.62.4.4.4.1" xref="S2.E21.m1.62.62.62.4.4.4.1.cmml"><mi id="S2.E21.m1.62.62.62.4.4.4.1.2" xref="S2.E21.m1.62.62.62.4.4.4.1.2.cmml">k</mi><mo id="S2.E21.m1.62.62.62.4.4.4.1.1" xref="S2.E21.m1.62.62.62.4.4.4.1.1.cmml">−</mo><mn id="S2.E21.m1.62.62.62.4.4.4.1.3" xref="S2.E21.m1.62.62.62.4.4.4.1.3.cmml">1</mn></mrow></munderover><msub id="S2.E21.m1.79.79.79.21.21.23"><mi id="S2.E21.m1.63.63.63.5.5.5" xref="S2.E21.m1.63.63.63.5.5.5.cmml">η</mi><mrow id="S2.E21.m1.64.64.64.6.6.6.1" xref="S2.E21.m1.64.64.64.6.6.6.1.cmml"><msub id="S2.E21.m1.64.64.64.6.6.6.1.2" xref="S2.E21.m1.64.64.64.6.6.6.1.2.cmml"><mi id="S2.E21.m1.64.64.64.6.6.6.1.2.2" xref="S2.E21.m1.64.64.64.6.6.6.1.2.2.cmml">ν</mi><mrow id="S2.E21.m1.64.64.64.6.6.6.1.2.3" 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id="S2.E21.m1.88c">\mathcal{V}^{(n+1)}_{\mu\nu\mu_{1}\nu_{1}\cdots\mu_{n}\nu_{n}}(P_{1},\ldots,P_% {n})\\ =V^{(4)}_{\mu\nu\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{n}}(P_{1},P_{2},P_{3}% \cup\ldots\cup P_{n})\prod_{i=3}^{n-1}\eta_{\nu_{i}\mu_{i+1}}\\ +\sum_{k=4}^{n}V^{(5)}_{\mu\nu\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{k-1}\mu_% {k}\nu_{n}}(P_{1},P_{2},P_{3}\cup\ldots\cup P_{k-1})\\ \times\prod_{i=4}^{k-1}\eta_{\nu_{i-1}\mu_{i}}\prod_{j=k}^{n-1}\eta_{\nu_{j}% \mu_{j+1}}+S(1,\ldots,n).</annotation><annotation encoding="application/x-llamapun" id="S2.E21.m1.88d">start_ROW start_CELL caligraphic_V start_POSTSUPERSCRIPT ( italic_n + 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ italic_ν italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋯ italic_μ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL = italic_V start_POSTSUPERSCRIPT ( 4 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ italic_ν italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∪ … ∪ italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ∏ start_POSTSUBSCRIPT italic_i = 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT italic_η start_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL + ∑ start_POSTSUBSCRIPT italic_k = 4 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_V start_POSTSUPERSCRIPT ( 5 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ italic_ν italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∪ … ∪ italic_P start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL × ∏ start_POSTSUBSCRIPT italic_i = 4 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT italic_η start_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∏ start_POSTSUBSCRIPT italic_j = italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT italic_η start_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT italic_j + 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT + italic_S ( 1 , … , italic_n ) . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(21)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p7.7">Here, <math alttext="S(1,\ldots,n)" class="ltx_Math" display="inline" id="S2.p7.3.m1.3"><semantics id="S2.p7.3.m1.3a"><mrow id="S2.p7.3.m1.3.4" xref="S2.p7.3.m1.3.4.cmml"><mi id="S2.p7.3.m1.3.4.2" xref="S2.p7.3.m1.3.4.2.cmml">S</mi><mo id="S2.p7.3.m1.3.4.1" xref="S2.p7.3.m1.3.4.1.cmml">⁢</mo><mrow id="S2.p7.3.m1.3.4.3.2" xref="S2.p7.3.m1.3.4.3.1.cmml"><mo id="S2.p7.3.m1.3.4.3.2.1" stretchy="false" xref="S2.p7.3.m1.3.4.3.1.cmml">(</mo><mn id="S2.p7.3.m1.1.1" xref="S2.p7.3.m1.1.1.cmml">1</mn><mo id="S2.p7.3.m1.3.4.3.2.2" xref="S2.p7.3.m1.3.4.3.1.cmml">,</mo><mi id="S2.p7.3.m1.2.2" mathvariant="normal" xref="S2.p7.3.m1.2.2.cmml">…</mi><mo id="S2.p7.3.m1.3.4.3.2.3" xref="S2.p7.3.m1.3.4.3.1.cmml">,</mo><mi id="S2.p7.3.m1.3.3" xref="S2.p7.3.m1.3.3.cmml">n</mi><mo id="S2.p7.3.m1.3.4.3.2.4" stretchy="false" xref="S2.p7.3.m1.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.p7.3.m1.3b"><apply id="S2.p7.3.m1.3.4.cmml" xref="S2.p7.3.m1.3.4"><times id="S2.p7.3.m1.3.4.1.cmml" xref="S2.p7.3.m1.3.4.1"></times><ci id="S2.p7.3.m1.3.4.2.cmml" xref="S2.p7.3.m1.3.4.2">𝑆</ci><vector id="S2.p7.3.m1.3.4.3.1.cmml" xref="S2.p7.3.m1.3.4.3.2"><cn id="S2.p7.3.m1.1.1.cmml" type="integer" xref="S2.p7.3.m1.1.1">1</cn><ci id="S2.p7.3.m1.2.2.cmml" xref="S2.p7.3.m1.2.2">…</ci><ci id="S2.p7.3.m1.3.3.cmml" xref="S2.p7.3.m1.3.3">𝑛</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p7.3.m1.3c">S(1,\ldots,n)</annotation><annotation encoding="application/x-llamapun" id="S2.p7.3.m1.3d">italic_S ( 1 , … , italic_n )</annotation></semantics></math> denotes the permutations of the labels <math alttext="\{1,2,\ldots,n\}" class="ltx_Math" display="inline" id="S2.p7.4.m2.4"><semantics id="S2.p7.4.m2.4a"><mrow id="S2.p7.4.m2.4.5.2" xref="S2.p7.4.m2.4.5.1.cmml"><mo id="S2.p7.4.m2.4.5.2.1" stretchy="false" xref="S2.p7.4.m2.4.5.1.cmml">{</mo><mn id="S2.p7.4.m2.1.1" xref="S2.p7.4.m2.1.1.cmml">1</mn><mo id="S2.p7.4.m2.4.5.2.2" xref="S2.p7.4.m2.4.5.1.cmml">,</mo><mn id="S2.p7.4.m2.2.2" xref="S2.p7.4.m2.2.2.cmml">2</mn><mo id="S2.p7.4.m2.4.5.2.3" xref="S2.p7.4.m2.4.5.1.cmml">,</mo><mi id="S2.p7.4.m2.3.3" mathvariant="normal" xref="S2.p7.4.m2.3.3.cmml">…</mi><mo id="S2.p7.4.m2.4.5.2.4" xref="S2.p7.4.m2.4.5.1.cmml">,</mo><mi id="S2.p7.4.m2.4.4" xref="S2.p7.4.m2.4.4.cmml">n</mi><mo id="S2.p7.4.m2.4.5.2.5" stretchy="false" xref="S2.p7.4.m2.4.5.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.p7.4.m2.4b"><set id="S2.p7.4.m2.4.5.1.cmml" xref="S2.p7.4.m2.4.5.2"><cn id="S2.p7.4.m2.1.1.cmml" type="integer" xref="S2.p7.4.m2.1.1">1</cn><cn id="S2.p7.4.m2.2.2.cmml" type="integer" xref="S2.p7.4.m2.2.2">2</cn><ci id="S2.p7.4.m2.3.3.cmml" xref="S2.p7.4.m2.3.3">…</ci><ci id="S2.p7.4.m2.4.4.cmml" xref="S2.p7.4.m2.4.4">𝑛</ci></set></annotation-xml><annotation encoding="application/x-tex" id="S2.p7.4.m2.4c">\{1,2,\ldots,n\}</annotation><annotation encoding="application/x-llamapun" id="S2.p7.4.m2.4d">{ 1 , 2 , … , italic_n }</annotation></semantics></math>. <math alttext="\mathcal{V}^{(n)}" class="ltx_Math" display="inline" id="S2.p7.5.m3.1"><semantics id="S2.p7.5.m3.1a"><msup id="S2.p7.5.m3.1.2" xref="S2.p7.5.m3.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.p7.5.m3.1.2.2" xref="S2.p7.5.m3.1.2.2.cmml">𝒱</mi><mrow id="S2.p7.5.m3.1.1.1.3" xref="S2.p7.5.m3.1.2.cmml"><mo id="S2.p7.5.m3.1.1.1.3.1" stretchy="false" xref="S2.p7.5.m3.1.2.cmml">(</mo><mi id="S2.p7.5.m3.1.1.1.1" xref="S2.p7.5.m3.1.1.1.1.cmml">n</mi><mo id="S2.p7.5.m3.1.1.1.3.2" stretchy="false" xref="S2.p7.5.m3.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.p7.5.m3.1b"><apply id="S2.p7.5.m3.1.2.cmml" xref="S2.p7.5.m3.1.2"><csymbol cd="ambiguous" id="S2.p7.5.m3.1.2.1.cmml" xref="S2.p7.5.m3.1.2">superscript</csymbol><ci id="S2.p7.5.m3.1.2.2.cmml" xref="S2.p7.5.m3.1.2.2">𝒱</ci><ci id="S2.p7.5.m3.1.1.1.1.cmml" xref="S2.p7.5.m3.1.1.1.1">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p7.5.m3.1c">\mathcal{V}^{(n)}</annotation><annotation encoding="application/x-llamapun" id="S2.p7.5.m3.1d">caligraphic_V start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT</annotation></semantics></math> corresponds to the <math alttext="n" class="ltx_Math" display="inline" id="S2.p7.6.m4.1"><semantics id="S2.p7.6.m4.1a"><mi id="S2.p7.6.m4.1.1" xref="S2.p7.6.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.p7.6.m4.1b"><ci id="S2.p7.6.m4.1.1.cmml" xref="S2.p7.6.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.p7.6.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.p7.6.m4.1d">italic_n</annotation></semantics></math>-graviton vertex after symmetrizing in <math alttext="\mu_{i}\leftrightarrow\nu_{i}" class="ltx_Math" display="inline" id="S2.p7.7.m5.1"><semantics id="S2.p7.7.m5.1a"><mrow id="S2.p7.7.m5.1.1" xref="S2.p7.7.m5.1.1.cmml"><msub id="S2.p7.7.m5.1.1.2" xref="S2.p7.7.m5.1.1.2.cmml"><mi id="S2.p7.7.m5.1.1.2.2" xref="S2.p7.7.m5.1.1.2.2.cmml">μ</mi><mi id="S2.p7.7.m5.1.1.2.3" xref="S2.p7.7.m5.1.1.2.3.cmml">i</mi></msub><mo id="S2.p7.7.m5.1.1.1" stretchy="false" xref="S2.p7.7.m5.1.1.1.cmml">↔</mo><msub id="S2.p7.7.m5.1.1.3" xref="S2.p7.7.m5.1.1.3.cmml"><mi id="S2.p7.7.m5.1.1.3.2" xref="S2.p7.7.m5.1.1.3.2.cmml">ν</mi><mi id="S2.p7.7.m5.1.1.3.3" xref="S2.p7.7.m5.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.p7.7.m5.1b"><apply id="S2.p7.7.m5.1.1.cmml" xref="S2.p7.7.m5.1.1"><ci id="S2.p7.7.m5.1.1.1.cmml" xref="S2.p7.7.m5.1.1.1">↔</ci><apply id="S2.p7.7.m5.1.1.2.cmml" xref="S2.p7.7.m5.1.1.2"><csymbol cd="ambiguous" id="S2.p7.7.m5.1.1.2.1.cmml" xref="S2.p7.7.m5.1.1.2">subscript</csymbol><ci id="S2.p7.7.m5.1.1.2.2.cmml" xref="S2.p7.7.m5.1.1.2.2">𝜇</ci><ci id="S2.p7.7.m5.1.1.2.3.cmml" xref="S2.p7.7.m5.1.1.2.3">𝑖</ci></apply><apply id="S2.p7.7.m5.1.1.3.cmml" xref="S2.p7.7.m5.1.1.3"><csymbol cd="ambiguous" id="S2.p7.7.m5.1.1.3.1.cmml" xref="S2.p7.7.m5.1.1.3">subscript</csymbol><ci id="S2.p7.7.m5.1.1.3.2.cmml" xref="S2.p7.7.m5.1.1.3.2">𝜈</ci><ci id="S2.p7.7.m5.1.1.3.3.cmml" xref="S2.p7.7.m5.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.p7.7.m5.1c">\mu_{i}\leftrightarrow\nu_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.p7.7.m5.1d">italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ↔ italic_ν start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> (see figure <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S2.F1" title="Figure 1 ‣ II Equations of motion ‣ One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_tag">1</span></a>). The three-, four-, and five-point vertices are in agreement with the literature <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib58" title="">Capper:1973pv </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib59" title="">Brandt:1992dk </a></cite>.</p> </div> <figure class="ltx_figure" id="S2.F1"> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="S2.F1.1" style="width:250.4pt;height:79.4pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(-91.6pt,29.0pt) scale(0.577427816165407,0.577427816165407) ;"><svg fill="none" height="190.258751902588" overflow="visible" stroke="none" version="1.1" width="599.972325999723"><g transform="translate(0,190.258751902588) scale(1,-1)"><g transform="translate(0,0)"><g transform="translate(0,263) scale(1, -1)"><foreignobject height="263" overflow="visible" width="830"><img alt="Refer to caption" class="ltx_graphics 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-1)"><foreignobject height="263" overflow="visible" width="830"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="263" id="S2.F1.1.pic1.8.g1" src="x2.png" width="830"/></foreignobject></g></g></g></svg> </span></div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1: </span><math alttext="(n+1)" class="ltx_Math" display="inline" id="S2.F1.3.m1.1"><semantics id="S2.F1.3.m1.1b"><mrow id="S2.F1.3.m1.1.1.1" xref="S2.F1.3.m1.1.1.1.1.cmml"><mo id="S2.F1.3.m1.1.1.1.2" stretchy="false" xref="S2.F1.3.m1.1.1.1.1.cmml">(</mo><mrow id="S2.F1.3.m1.1.1.1.1" xref="S2.F1.3.m1.1.1.1.1.cmml"><mi id="S2.F1.3.m1.1.1.1.1.2" xref="S2.F1.3.m1.1.1.1.1.2.cmml">n</mi><mo id="S2.F1.3.m1.1.1.1.1.1" xref="S2.F1.3.m1.1.1.1.1.1.cmml">+</mo><mn id="S2.F1.3.m1.1.1.1.1.3" xref="S2.F1.3.m1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.F1.3.m1.1.1.1.3" stretchy="false" xref="S2.F1.3.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.F1.3.m1.1c"><apply id="S2.F1.3.m1.1.1.1.1.cmml" xref="S2.F1.3.m1.1.1.1"><plus id="S2.F1.3.m1.1.1.1.1.1.cmml" xref="S2.F1.3.m1.1.1.1.1.1"></plus><ci id="S2.F1.3.m1.1.1.1.1.2.cmml" xref="S2.F1.3.m1.1.1.1.1.2">𝑛</ci><cn id="S2.F1.3.m1.1.1.1.1.3.cmml" type="integer" xref="S2.F1.3.m1.1.1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.3.m1.1d">(n+1)</annotation><annotation encoding="application/x-llamapun" id="S2.F1.3.m1.1e">( italic_n + 1 )</annotation></semantics></math>-graviton vertex.</figcaption> </figure> <div class="ltx_para" id="S2.p8"> <p class="ltx_p" id="S2.p8.3">What makes the multiparticle currents special is the fact that they are related to scattering trees with off-shell external legs <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib30" title="">Gomez:2022dzk </a></cite>. The amputated correlators are defined through</p> <table class="ltx_equation ltx_eqn_table" id="S2.E22"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{M}_{N}^{\textrm{tree}}=h^{\mu\nu}_{1}(s_{2\ldots N}\mathbb{P}_{\mu\nu% \rho\sigma}I_{2\ldots N}^{\rho\sigma})." class="ltx_Math" display="block" id="S2.E22.m1.1"><semantics id="S2.E22.m1.1a"><mrow id="S2.E22.m1.1.1.1" xref="S2.E22.m1.1.1.1.1.cmml"><mrow id="S2.E22.m1.1.1.1.1" xref="S2.E22.m1.1.1.1.1.cmml"><msubsup id="S2.E22.m1.1.1.1.1.3" xref="S2.E22.m1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E22.m1.1.1.1.1.3.2.2" xref="S2.E22.m1.1.1.1.1.3.2.2.cmml">ℳ</mi><mi id="S2.E22.m1.1.1.1.1.3.2.3" xref="S2.E22.m1.1.1.1.1.3.2.3.cmml">N</mi><mtext id="S2.E22.m1.1.1.1.1.3.3" xref="S2.E22.m1.1.1.1.1.3.3a.cmml">tree</mtext></msubsup><mo id="S2.E22.m1.1.1.1.1.2" 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id="S2.E22.m1.1.1.1.1.1.1.1.1.4.3.1.cmml" xref="S2.E22.m1.1.1.1.1.1.1.1.1.4.3.1"></times><ci id="S2.E22.m1.1.1.1.1.1.1.1.1.4.3.2.cmml" xref="S2.E22.m1.1.1.1.1.1.1.1.1.4.3.2">𝜌</ci><ci id="S2.E22.m1.1.1.1.1.1.1.1.1.4.3.3.cmml" xref="S2.E22.m1.1.1.1.1.1.1.1.1.4.3.3">𝜎</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E22.m1.1c">\mathcal{M}_{N}^{\textrm{tree}}=h^{\mu\nu}_{1}(s_{2\ldots N}\mathbb{P}_{\mu\nu% \rho\sigma}I_{2\ldots N}^{\rho\sigma}).</annotation><annotation encoding="application/x-llamapun" id="S2.E22.m1.1d">caligraphic_M start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT tree end_POSTSUPERSCRIPT = italic_h start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_s start_POSTSUBSCRIPT 2 … italic_N end_POSTSUBSCRIPT blackboard_P start_POSTSUBSCRIPT italic_μ italic_ν italic_ρ italic_σ end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT 2 … italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ρ italic_σ end_POSTSUPERSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(22)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p8.2">Momentum conservation is left implicit, and external legs are off the mass-shell. Tree level amplitudes are then simply obtained after imposing the physical state conditions, i.e. <math alttext="k^{2}_{p}=0" class="ltx_Math" display="inline" id="S2.p8.1.m1.1"><semantics id="S2.p8.1.m1.1a"><mrow id="S2.p8.1.m1.1.1" xref="S2.p8.1.m1.1.1.cmml"><msubsup id="S2.p8.1.m1.1.1.2" xref="S2.p8.1.m1.1.1.2.cmml"><mi id="S2.p8.1.m1.1.1.2.2.2" xref="S2.p8.1.m1.1.1.2.2.2.cmml">k</mi><mi id="S2.p8.1.m1.1.1.2.3" xref="S2.p8.1.m1.1.1.2.3.cmml">p</mi><mn id="S2.p8.1.m1.1.1.2.2.3" xref="S2.p8.1.m1.1.1.2.2.3.cmml">2</mn></msubsup><mo id="S2.p8.1.m1.1.1.1" xref="S2.p8.1.m1.1.1.1.cmml">=</mo><mn id="S2.p8.1.m1.1.1.3" xref="S2.p8.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.p8.1.m1.1b"><apply id="S2.p8.1.m1.1.1.cmml" xref="S2.p8.1.m1.1.1"><eq id="S2.p8.1.m1.1.1.1.cmml" xref="S2.p8.1.m1.1.1.1"></eq><apply id="S2.p8.1.m1.1.1.2.cmml" xref="S2.p8.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.p8.1.m1.1.1.2.1.cmml" xref="S2.p8.1.m1.1.1.2">subscript</csymbol><apply 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id="S2.E23.m1.2.2.1.1.3.3.cmml" mathsize="70%" xref="S2.E23.m1.2.2.1.1.3.3">tree</mtext></ci></apply><apply id="S2.E23.m1.2.2.1.1.1.2.cmml" xref="S2.E23.m1.2.2.1.1.1.1"><csymbol cd="latexml" id="S2.E23.m1.2.2.1.1.1.2.1.cmml" xref="S2.E23.m1.2.2.1.1.1.1.1.2">evaluated-at</csymbol><apply id="S2.E23.m1.2.2.1.1.1.1.1.1.cmml" xref="S2.E23.m1.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.E23.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S2.E23.m1.2.2.1.1.1.1.1.1">superscript</csymbol><apply id="S2.E23.m1.2.2.1.1.1.1.1.1.2.cmml" xref="S2.E23.m1.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.E23.m1.2.2.1.1.1.1.1.1.2.1.cmml" xref="S2.E23.m1.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="S2.E23.m1.2.2.1.1.1.1.1.1.2.2.cmml" xref="S2.E23.m1.2.2.1.1.1.1.1.1.2.2">ℳ</ci><ci id="S2.E23.m1.2.2.1.1.1.1.1.1.2.3.cmml" xref="S2.E23.m1.2.2.1.1.1.1.1.1.2.3">𝑁</ci></apply><ci id="S2.E23.m1.2.2.1.1.1.1.1.1.3a.cmml" xref="S2.E23.m1.2.2.1.1.1.1.1.1.3"><mtext id="S2.E23.m1.2.2.1.1.1.1.1.1.3.cmml" mathsize="70%" xref="S2.E23.m1.2.2.1.1.1.1.1.1.3">tree</mtext></ci></apply><ci id="S2.E23.m1.1.1.1a.cmml" xref="S2.E23.m1.1.1.1"><mtext id="S2.E23.m1.1.1.1.cmml" mathsize="70%" xref="S2.E23.m1.1.1.1">on-shell</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E23.m1.2c">M_{N}^{\textrm{tree}}=\left.\mathcal{M}_{N}^{\textrm{tree}}\right|_{\textrm{on% -shell}}.</annotation><annotation encoding="application/x-llamapun" id="S2.E23.m1.2d">italic_M start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT tree end_POSTSUPERSCRIPT = caligraphic_M start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT tree end_POSTSUPERSCRIPT | start_POSTSUBSCRIPT on-shell end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(23)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.p8.4">We have tested this formula up to seven-point tree level amplitudes, matching numerical computations using the CHY formalism <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib60" title="">Cachazo:2013iea </a></cite>.</p> </div> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">III </span>Loop recursions</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.3">The off-shell scattering trees generated by (<a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S2.E20" title="In II Equations of motion ‣ One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_tag">20</span></a>) are the intermediate step in the construction of loop integrands. 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xref="S3.p1.3.m3.1.2.3.2.2.3.2.cmml">μ</mi><mo id="S3.p1.3.m3.1.2.3.2.2.3.1" xref="S3.p1.3.m3.1.2.3.2.2.3.1.cmml">⁢</mo><mi id="S3.p1.3.m3.1.2.3.2.2.3.3" xref="S3.p1.3.m3.1.2.3.2.2.3.3.cmml">ν</mi></mrow></msubsup><mo id="S3.p1.3.m3.1.2.3.1" xref="S3.p1.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S3.p1.3.m3.1.2.3.3.2" xref="S3.p1.3.m3.1.2.3.cmml"><mo id="S3.p1.3.m3.1.2.3.3.2.1" stretchy="false" xref="S3.p1.3.m3.1.2.3.cmml">(</mo><mi id="S3.p1.3.m3.1.1" mathvariant="normal" xref="S3.p1.3.m3.1.1.cmml">ℓ</mi><mo id="S3.p1.3.m3.1.2.3.3.2.2" stretchy="false" xref="S3.p1.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.3.m3.1b"><apply id="S3.p1.3.m3.1.2.cmml" xref="S3.p1.3.m3.1.2"><eq id="S3.p1.3.m3.1.2.1.cmml" xref="S3.p1.3.m3.1.2.1"></eq><apply id="S3.p1.3.m3.1.2.2.cmml" xref="S3.p1.3.m3.1.2.2"><csymbol cd="ambiguous" id="S3.p1.3.m3.1.2.2.1.cmml" xref="S3.p1.3.m3.1.2.2">subscript</csymbol><apply id="S3.p1.3.m3.1.2.2.2.cmml" 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id="S3.p1.3.m3.1.2.3.2.cmml" xref="S3.p1.3.m3.1.2.3.2"><csymbol cd="ambiguous" id="S3.p1.3.m3.1.2.3.2.1.cmml" xref="S3.p1.3.m3.1.2.3.2">subscript</csymbol><apply id="S3.p1.3.m3.1.2.3.2.2.cmml" xref="S3.p1.3.m3.1.2.3.2"><csymbol cd="ambiguous" id="S3.p1.3.m3.1.2.3.2.2.1.cmml" xref="S3.p1.3.m3.1.2.3.2">superscript</csymbol><ci id="S3.p1.3.m3.1.2.3.2.2.2.cmml" xref="S3.p1.3.m3.1.2.3.2.2.2">𝐾</ci><apply id="S3.p1.3.m3.1.2.3.2.2.3.cmml" xref="S3.p1.3.m3.1.2.3.2.2.3"><times id="S3.p1.3.m3.1.2.3.2.2.3.1.cmml" xref="S3.p1.3.m3.1.2.3.2.2.3.1"></times><ci id="S3.p1.3.m3.1.2.3.2.2.3.2.cmml" xref="S3.p1.3.m3.1.2.3.2.2.3.2">𝜇</ci><ci id="S3.p1.3.m3.1.2.3.2.2.3.3.cmml" xref="S3.p1.3.m3.1.2.3.2.2.3.3">𝜈</ci></apply></apply><apply id="S3.p1.3.m3.1.2.3.2.3.cmml" xref="S3.p1.3.m3.1.2.3.2.3"><times id="S3.p1.3.m3.1.2.3.2.3.1.cmml" xref="S3.p1.3.m3.1.2.3.2.3.1"></times><ci id="S3.p1.3.m3.1.2.3.2.3.2.cmml" xref="S3.p1.3.m3.1.2.3.2.3.2">𝑃</ci><ci id="S3.p1.3.m3.1.2.3.2.3.3.cmml" xref="S3.p1.3.m3.1.2.3.2.3.3">𝛼</ci><ci id="S3.p1.3.m3.1.2.3.2.3.4.cmml" xref="S3.p1.3.m3.1.2.3.2.3.4">𝛽</ci></apply></apply><ci id="S3.p1.3.m3.1.1.cmml" xref="S3.p1.3.m3.1.1">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.3.m3.1c">K^{\mu\nu}_{P\alpha\beta}=K^{\mu\nu}_{P\alpha\beta}(\ell)</annotation><annotation encoding="application/x-llamapun" id="S3.p1.3.m3.1d">italic_K start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P italic_α italic_β end_POSTSUBSCRIPT = italic_K start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P italic_α italic_β end_POSTSUBSCRIPT ( roman_ℓ )</annotation></semantics></math>. We then have</p> <table class="ltx_equation ltx_eqn_table" id="S3.E24"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="s_{\ell P}\mathbb{P}_{\mu\nu\rho\sigma}K^{\rho\sigma}_{P\alpha\beta}=I_{P}^{% \rho\sigma}\mathcal{V}_{\mu\nu\rho\sigma\alpha\beta}^{(3)}(P,\ell)\\ +\sum_{n=2}^{\infty}\frac{1}{(n-1)!}\sum_{P=P_{1}\cup\ldots\cup P_{n}}\prod_{i% =1}^{n-1}I_{P_{i}}^{\mu_{i}\nu_{i}}\\ \times\bigg{\{}\frac{1}{n}I_{P_{n}}^{\mu_{n}\nu_{n}}\mathcal{V}_{\mu\nu\mu_{1}% \nu_{1}\cdots\mu_{n}\nu_{n}\alpha\beta}^{(n+2)}(P_{1},\ldots,P_{n},\ell)\\ +K^{\mu_{n}\nu_{n}}_{P_{n}\alpha\beta}\mathcal{V}_{\mu\nu\mu_{1}\nu_{1}\cdots% \mu_{n}\nu_{n}}^{(n+1)}(P_{1},\ldots,\ell P_{n})\bigg{\}}." class="ltx_Math" display="block" id="S3.E24.m1.72"><semantics id="S3.E24.m1.72a"><mtable displaystyle="true" id="S3.E24.m1.71.71" rowspacing="0pt"><mtr id="S3.E24.m1.71.71a"><mtd class="ltx_align_left" columnalign="left" id="S3.E24.m1.71.71b"><mrow id="S3.E24.m1.19.19.19.19.19"><mrow id="S3.E24.m1.19.19.19.19.19.20"><msub id="S3.E24.m1.19.19.19.19.19.20.2"><mi id="S3.E24.m1.1.1.1.1.1.1" xref="S3.E24.m1.1.1.1.1.1.1.cmml">s</mi><mrow id="S3.E24.m1.2.2.2.2.2.2.1" xref="S3.E24.m1.2.2.2.2.2.2.1.cmml"><mi id="S3.E24.m1.2.2.2.2.2.2.1.2" mathvariant="normal" xref="S3.E24.m1.2.2.2.2.2.2.1.2.cmml">ℓ</mi><mo id="S3.E24.m1.2.2.2.2.2.2.1.1" xref="S3.E24.m1.2.2.2.2.2.2.1.1.cmml">⁢</mo><mi id="S3.E24.m1.2.2.2.2.2.2.1.3" xref="S3.E24.m1.2.2.2.2.2.2.1.3.cmml">P</mi></mrow></msub><mo id="S3.E24.m1.19.19.19.19.19.20.1" xref="S3.E24.m1.72.72.1.1.1.cmml">⁢</mo><msub id="S3.E24.m1.19.19.19.19.19.20.3"><mi id="S3.E24.m1.3.3.3.3.3.3" xref="S3.E24.m1.3.3.3.3.3.3.cmml">ℙ</mi><mrow id="S3.E24.m1.4.4.4.4.4.4.1" xref="S3.E24.m1.4.4.4.4.4.4.1.cmml"><mi id="S3.E24.m1.4.4.4.4.4.4.1.2" xref="S3.E24.m1.4.4.4.4.4.4.1.2.cmml">μ</mi><mo id="S3.E24.m1.4.4.4.4.4.4.1.1" xref="S3.E24.m1.4.4.4.4.4.4.1.1.cmml">⁢</mo><mi id="S3.E24.m1.4.4.4.4.4.4.1.3" xref="S3.E24.m1.4.4.4.4.4.4.1.3.cmml">ν</mi><mo id="S3.E24.m1.4.4.4.4.4.4.1.1a" xref="S3.E24.m1.4.4.4.4.4.4.1.1.cmml">⁢</mo><mi id="S3.E24.m1.4.4.4.4.4.4.1.4" xref="S3.E24.m1.4.4.4.4.4.4.1.4.cmml">ρ</mi><mo id="S3.E24.m1.4.4.4.4.4.4.1.1b" xref="S3.E24.m1.4.4.4.4.4.4.1.1.cmml">⁢</mo><mi id="S3.E24.m1.4.4.4.4.4.4.1.5" xref="S3.E24.m1.4.4.4.4.4.4.1.5.cmml">σ</mi></mrow></msub><mo id="S3.E24.m1.19.19.19.19.19.20.1a" xref="S3.E24.m1.72.72.1.1.1.cmml">⁢</mo><msubsup id="S3.E24.m1.19.19.19.19.19.20.4"><mi id="S3.E24.m1.5.5.5.5.5.5" xref="S3.E24.m1.5.5.5.5.5.5.cmml">K</mi><mrow id="S3.E24.m1.7.7.7.7.7.7.1" xref="S3.E24.m1.7.7.7.7.7.7.1.cmml"><mi id="S3.E24.m1.7.7.7.7.7.7.1.2" xref="S3.E24.m1.7.7.7.7.7.7.1.2.cmml">P</mi><mo id="S3.E24.m1.7.7.7.7.7.7.1.1" xref="S3.E24.m1.7.7.7.7.7.7.1.1.cmml">⁢</mo><mi id="S3.E24.m1.7.7.7.7.7.7.1.3" xref="S3.E24.m1.7.7.7.7.7.7.1.3.cmml">α</mi><mo id="S3.E24.m1.7.7.7.7.7.7.1.1a" xref="S3.E24.m1.7.7.7.7.7.7.1.1.cmml">⁢</mo><mi id="S3.E24.m1.7.7.7.7.7.7.1.4" xref="S3.E24.m1.7.7.7.7.7.7.1.4.cmml">β</mi></mrow><mrow id="S3.E24.m1.6.6.6.6.6.6.1" xref="S3.E24.m1.6.6.6.6.6.6.1.cmml"><mi id="S3.E24.m1.6.6.6.6.6.6.1.2" xref="S3.E24.m1.6.6.6.6.6.6.1.2.cmml">ρ</mi><mo id="S3.E24.m1.6.6.6.6.6.6.1.1" xref="S3.E24.m1.6.6.6.6.6.6.1.1.cmml">⁢</mo><mi id="S3.E24.m1.6.6.6.6.6.6.1.3" xref="S3.E24.m1.6.6.6.6.6.6.1.3.cmml">σ</mi></mrow></msubsup></mrow><mo id="S3.E24.m1.8.8.8.8.8.8" xref="S3.E24.m1.8.8.8.8.8.8.cmml">=</mo><mrow id="S3.E24.m1.19.19.19.19.19.21"><msubsup id="S3.E24.m1.19.19.19.19.19.21.2"><mi id="S3.E24.m1.9.9.9.9.9.9" xref="S3.E24.m1.9.9.9.9.9.9.cmml">I</mi><mi id="S3.E24.m1.10.10.10.10.10.10.1" xref="S3.E24.m1.10.10.10.10.10.10.1.cmml">P</mi><mrow id="S3.E24.m1.11.11.11.11.11.11.1" xref="S3.E24.m1.11.11.11.11.11.11.1.cmml"><mi id="S3.E24.m1.11.11.11.11.11.11.1.2" xref="S3.E24.m1.11.11.11.11.11.11.1.2.cmml">ρ</mi><mo id="S3.E24.m1.11.11.11.11.11.11.1.1" xref="S3.E24.m1.11.11.11.11.11.11.1.1.cmml">⁢</mo><mi id="S3.E24.m1.11.11.11.11.11.11.1.3" xref="S3.E24.m1.11.11.11.11.11.11.1.3.cmml">σ</mi></mrow></msubsup><mo id="S3.E24.m1.19.19.19.19.19.21.1" xref="S3.E24.m1.72.72.1.1.1.cmml">⁢</mo><msubsup id="S3.E24.m1.19.19.19.19.19.21.3"><mi class="ltx_font_mathcaligraphic" id="S3.E24.m1.12.12.12.12.12.12" xref="S3.E24.m1.12.12.12.12.12.12.cmml">𝒱</mi><mrow id="S3.E24.m1.13.13.13.13.13.13.1" xref="S3.E24.m1.13.13.13.13.13.13.1.cmml"><mi id="S3.E24.m1.13.13.13.13.13.13.1.2" xref="S3.E24.m1.13.13.13.13.13.13.1.2.cmml">μ</mi><mo id="S3.E24.m1.13.13.13.13.13.13.1.1" xref="S3.E24.m1.13.13.13.13.13.13.1.1.cmml">⁢</mo><mi id="S3.E24.m1.13.13.13.13.13.13.1.3" xref="S3.E24.m1.13.13.13.13.13.13.1.3.cmml">ν</mi><mo id="S3.E24.m1.13.13.13.13.13.13.1.1a" xref="S3.E24.m1.13.13.13.13.13.13.1.1.cmml">⁢</mo><mi id="S3.E24.m1.13.13.13.13.13.13.1.4" xref="S3.E24.m1.13.13.13.13.13.13.1.4.cmml">ρ</mi><mo id="S3.E24.m1.13.13.13.13.13.13.1.1b" xref="S3.E24.m1.13.13.13.13.13.13.1.1.cmml">⁢</mo><mi id="S3.E24.m1.13.13.13.13.13.13.1.5" xref="S3.E24.m1.13.13.13.13.13.13.1.5.cmml">σ</mi><mo id="S3.E24.m1.13.13.13.13.13.13.1.1c" xref="S3.E24.m1.13.13.13.13.13.13.1.1.cmml">⁢</mo><mi id="S3.E24.m1.13.13.13.13.13.13.1.6" xref="S3.E24.m1.13.13.13.13.13.13.1.6.cmml">α</mi><mo id="S3.E24.m1.13.13.13.13.13.13.1.1d" xref="S3.E24.m1.13.13.13.13.13.13.1.1.cmml">⁢</mo><mi id="S3.E24.m1.13.13.13.13.13.13.1.7" xref="S3.E24.m1.13.13.13.13.13.13.1.7.cmml">β</mi></mrow><mrow id="S3.E24.m1.14.14.14.14.14.14.1.3"><mo id="S3.E24.m1.14.14.14.14.14.14.1.3.1" stretchy="false">(</mo><mn id="S3.E24.m1.14.14.14.14.14.14.1.1" xref="S3.E24.m1.14.14.14.14.14.14.1.1.cmml">3</mn><mo id="S3.E24.m1.14.14.14.14.14.14.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S3.E24.m1.19.19.19.19.19.21.1a" xref="S3.E24.m1.72.72.1.1.1.cmml">⁢</mo><mrow id="S3.E24.m1.19.19.19.19.19.21.4"><mo id="S3.E24.m1.15.15.15.15.15.15" stretchy="false" xref="S3.E24.m1.72.72.1.1.1.cmml">(</mo><mi id="S3.E24.m1.16.16.16.16.16.16" xref="S3.E24.m1.16.16.16.16.16.16.cmml">P</mi><mo id="S3.E24.m1.17.17.17.17.17.17" xref="S3.E24.m1.72.72.1.1.1.cmml">,</mo><mi id="S3.E24.m1.18.18.18.18.18.18" mathvariant="normal" xref="S3.E24.m1.18.18.18.18.18.18.cmml">ℓ</mi><mo id="S3.E24.m1.19.19.19.19.19.19" stretchy="false" xref="S3.E24.m1.72.72.1.1.1.cmml">)</mo></mrow></mrow></mrow></mtd></mtr><mtr id="S3.E24.m1.71.71c"><mtd class="ltx_align_right" columnalign="right" id="S3.E24.m1.71.71d"><mrow id="S3.E24.m1.32.32.32.13.13"><mo id="S3.E24.m1.32.32.32.13.13a" xref="S3.E24.m1.72.72.1.1.1.cmml">+</mo><mrow id="S3.E24.m1.32.32.32.13.13.14"><munderover id="S3.E24.m1.32.32.32.13.13.14.1"><mo id="S3.E24.m1.21.21.21.2.2.2" movablelimits="false" xref="S3.E24.m1.21.21.21.2.2.2.cmml">∑</mo><mrow id="S3.E24.m1.22.22.22.3.3.3.1" xref="S3.E24.m1.22.22.22.3.3.3.1.cmml"><mi id="S3.E24.m1.22.22.22.3.3.3.1.2" xref="S3.E24.m1.22.22.22.3.3.3.1.2.cmml">n</mi><mo id="S3.E24.m1.22.22.22.3.3.3.1.1" xref="S3.E24.m1.22.22.22.3.3.3.1.1.cmml">=</mo><mn id="S3.E24.m1.22.22.22.3.3.3.1.3" xref="S3.E24.m1.22.22.22.3.3.3.1.3.cmml">2</mn></mrow><mi id="S3.E24.m1.23.23.23.4.4.4.1" mathvariant="normal" xref="S3.E24.m1.23.23.23.4.4.4.1.cmml">∞</mi></munderover><mrow id="S3.E24.m1.32.32.32.13.13.14.2"><mfrac id="S3.E24.m1.24.24.24.5.5.5" xref="S3.E24.m1.24.24.24.5.5.5.cmml"><mn id="S3.E24.m1.24.24.24.5.5.5.3" xref="S3.E24.m1.24.24.24.5.5.5.3.cmml">1</mn><mrow id="S3.E24.m1.24.24.24.5.5.5.1" xref="S3.E24.m1.24.24.24.5.5.5.1.cmml"><mrow id="S3.E24.m1.24.24.24.5.5.5.1.1.1" xref="S3.E24.m1.24.24.24.5.5.5.1.1.1.1.cmml"><mo id="S3.E24.m1.24.24.24.5.5.5.1.1.1.2" stretchy="false" xref="S3.E24.m1.24.24.24.5.5.5.1.1.1.1.cmml">(</mo><mrow id="S3.E24.m1.24.24.24.5.5.5.1.1.1.1" xref="S3.E24.m1.24.24.24.5.5.5.1.1.1.1.cmml"><mi id="S3.E24.m1.24.24.24.5.5.5.1.1.1.1.2" xref="S3.E24.m1.24.24.24.5.5.5.1.1.1.1.2.cmml">n</mi><mo id="S3.E24.m1.24.24.24.5.5.5.1.1.1.1.1" xref="S3.E24.m1.24.24.24.5.5.5.1.1.1.1.1.cmml">−</mo><mn id="S3.E24.m1.24.24.24.5.5.5.1.1.1.1.3" xref="S3.E24.m1.24.24.24.5.5.5.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.E24.m1.24.24.24.5.5.5.1.1.1.3" stretchy="false" xref="S3.E24.m1.24.24.24.5.5.5.1.1.1.1.cmml">)</mo></mrow><mo id="S3.E24.m1.24.24.24.5.5.5.1.2" xref="S3.E24.m1.24.24.24.5.5.5.1.2.cmml">!</mo></mrow></mfrac><mo id="S3.E24.m1.32.32.32.13.13.14.2.1" xref="S3.E24.m1.72.72.1.1.1.cmml">⁢</mo><mrow id="S3.E24.m1.32.32.32.13.13.14.2.2"><munder id="S3.E24.m1.32.32.32.13.13.14.2.2.1"><mo id="S3.E24.m1.25.25.25.6.6.6" movablelimits="false" rspace="0em" xref="S3.E24.m1.25.25.25.6.6.6.cmml">∑</mo><mrow id="S3.E24.m1.26.26.26.7.7.7.1" xref="S3.E24.m1.26.26.26.7.7.7.1.cmml"><mi id="S3.E24.m1.26.26.26.7.7.7.1.2" xref="S3.E24.m1.26.26.26.7.7.7.1.2.cmml">P</mi><mo id="S3.E24.m1.26.26.26.7.7.7.1.1" xref="S3.E24.m1.26.26.26.7.7.7.1.1.cmml">=</mo><mrow id="S3.E24.m1.26.26.26.7.7.7.1.3" xref="S3.E24.m1.26.26.26.7.7.7.1.3.cmml"><msub id="S3.E24.m1.26.26.26.7.7.7.1.3.2" xref="S3.E24.m1.26.26.26.7.7.7.1.3.2.cmml"><mi id="S3.E24.m1.26.26.26.7.7.7.1.3.2.2" xref="S3.E24.m1.26.26.26.7.7.7.1.3.2.2.cmml">P</mi><mn id="S3.E24.m1.26.26.26.7.7.7.1.3.2.3" xref="S3.E24.m1.26.26.26.7.7.7.1.3.2.3.cmml">1</mn></msub><mo id="S3.E24.m1.26.26.26.7.7.7.1.3.1" xref="S3.E24.m1.26.26.26.7.7.7.1.3.1.cmml">∪</mo><mi id="S3.E24.m1.26.26.26.7.7.7.1.3.3" mathvariant="normal" xref="S3.E24.m1.26.26.26.7.7.7.1.3.3.cmml">…</mi><mo id="S3.E24.m1.26.26.26.7.7.7.1.3.1a" xref="S3.E24.m1.26.26.26.7.7.7.1.3.1.cmml">∪</mo><msub id="S3.E24.m1.26.26.26.7.7.7.1.3.4" xref="S3.E24.m1.26.26.26.7.7.7.1.3.4.cmml"><mi id="S3.E24.m1.26.26.26.7.7.7.1.3.4.2" xref="S3.E24.m1.26.26.26.7.7.7.1.3.4.2.cmml">P</mi><mi id="S3.E24.m1.26.26.26.7.7.7.1.3.4.3" xref="S3.E24.m1.26.26.26.7.7.7.1.3.4.3.cmml">n</mi></msub></mrow></mrow></munder><mrow id="S3.E24.m1.32.32.32.13.13.14.2.2.2"><munderover id="S3.E24.m1.32.32.32.13.13.14.2.2.2.1"><mo id="S3.E24.m1.27.27.27.8.8.8" movablelimits="false" xref="S3.E24.m1.27.27.27.8.8.8.cmml">∏</mo><mrow id="S3.E24.m1.28.28.28.9.9.9.1" xref="S3.E24.m1.28.28.28.9.9.9.1.cmml"><mi id="S3.E24.m1.28.28.28.9.9.9.1.2" xref="S3.E24.m1.28.28.28.9.9.9.1.2.cmml">i</mi><mo id="S3.E24.m1.28.28.28.9.9.9.1.1" xref="S3.E24.m1.28.28.28.9.9.9.1.1.cmml">=</mo><mn id="S3.E24.m1.28.28.28.9.9.9.1.3" xref="S3.E24.m1.28.28.28.9.9.9.1.3.cmml">1</mn></mrow><mrow id="S3.E24.m1.29.29.29.10.10.10.1" xref="S3.E24.m1.29.29.29.10.10.10.1.cmml"><mi id="S3.E24.m1.29.29.29.10.10.10.1.2" xref="S3.E24.m1.29.29.29.10.10.10.1.2.cmml">n</mi><mo id="S3.E24.m1.29.29.29.10.10.10.1.1" xref="S3.E24.m1.29.29.29.10.10.10.1.1.cmml">−</mo><mn id="S3.E24.m1.29.29.29.10.10.10.1.3" xref="S3.E24.m1.29.29.29.10.10.10.1.3.cmml">1</mn></mrow></munderover><msubsup id="S3.E24.m1.32.32.32.13.13.14.2.2.2.2"><mi id="S3.E24.m1.30.30.30.11.11.11" xref="S3.E24.m1.30.30.30.11.11.11.cmml">I</mi><msub id="S3.E24.m1.31.31.31.12.12.12.1" xref="S3.E24.m1.31.31.31.12.12.12.1.cmml"><mi id="S3.E24.m1.31.31.31.12.12.12.1.2" xref="S3.E24.m1.31.31.31.12.12.12.1.2.cmml">P</mi><mi id="S3.E24.m1.31.31.31.12.12.12.1.3" xref="S3.E24.m1.31.31.31.12.12.12.1.3.cmml">i</mi></msub><mrow id="S3.E24.m1.32.32.32.13.13.13.1" xref="S3.E24.m1.32.32.32.13.13.13.1.cmml"><msub id="S3.E24.m1.32.32.32.13.13.13.1.2" xref="S3.E24.m1.32.32.32.13.13.13.1.2.cmml"><mi id="S3.E24.m1.32.32.32.13.13.13.1.2.2" xref="S3.E24.m1.32.32.32.13.13.13.1.2.2.cmml">μ</mi><mi id="S3.E24.m1.32.32.32.13.13.13.1.2.3" xref="S3.E24.m1.32.32.32.13.13.13.1.2.3.cmml">i</mi></msub><mo id="S3.E24.m1.32.32.32.13.13.13.1.1" xref="S3.E24.m1.32.32.32.13.13.13.1.1.cmml">⁢</mo><msub id="S3.E24.m1.32.32.32.13.13.13.1.3" xref="S3.E24.m1.32.32.32.13.13.13.1.3.cmml"><mi id="S3.E24.m1.32.32.32.13.13.13.1.3.2" xref="S3.E24.m1.32.32.32.13.13.13.1.3.2.cmml">ν</mi><mi id="S3.E24.m1.32.32.32.13.13.13.1.3.3" xref="S3.E24.m1.32.32.32.13.13.13.1.3.3.cmml">i</mi></msub></mrow></msubsup></mrow></mrow></mrow></mrow></mrow></mtd></mtr><mtr id="S3.E24.m1.71.71e"><mtd class="ltx_align_right" columnalign="right" id="S3.E24.m1.71.71f"><mrow id="S3.E24.m1.52.52.52.20.20"><mo id="S3.E24.m1.33.33.33.1.1.1" rspace="0.222em" xref="S3.E24.m1.33.33.33.1.1.1.cmml">×</mo><mrow id="S3.E24.m1.52.52.52.20.20.21"><mo id="S3.E24.m1.34.34.34.2.2.2" maxsize="210%" minsize="210%" xref="S3.E24.m1.72.72.1.1.1.cmml">{</mo><mfrac id="S3.E24.m1.35.35.35.3.3.3" xref="S3.E24.m1.35.35.35.3.3.3.cmml"><mn id="S3.E24.m1.35.35.35.3.3.3.2" xref="S3.E24.m1.35.35.35.3.3.3.2.cmml">1</mn><mi id="S3.E24.m1.35.35.35.3.3.3.3" xref="S3.E24.m1.35.35.35.3.3.3.3.cmml">n</mi></mfrac><msubsup id="S3.E24.m1.52.52.52.20.20.21.1"><mi id="S3.E24.m1.36.36.36.4.4.4" xref="S3.E24.m1.36.36.36.4.4.4.cmml">I</mi><msub id="S3.E24.m1.37.37.37.5.5.5.1" xref="S3.E24.m1.37.37.37.5.5.5.1.cmml"><mi id="S3.E24.m1.37.37.37.5.5.5.1.2" xref="S3.E24.m1.37.37.37.5.5.5.1.2.cmml">P</mi><mi id="S3.E24.m1.37.37.37.5.5.5.1.3" xref="S3.E24.m1.37.37.37.5.5.5.1.3.cmml">n</mi></msub><mrow id="S3.E24.m1.38.38.38.6.6.6.1" xref="S3.E24.m1.38.38.38.6.6.6.1.cmml"><msub id="S3.E24.m1.38.38.38.6.6.6.1.2" xref="S3.E24.m1.38.38.38.6.6.6.1.2.cmml"><mi id="S3.E24.m1.38.38.38.6.6.6.1.2.2" xref="S3.E24.m1.38.38.38.6.6.6.1.2.2.cmml">μ</mi><mi id="S3.E24.m1.38.38.38.6.6.6.1.2.3" xref="S3.E24.m1.38.38.38.6.6.6.1.2.3.cmml">n</mi></msub><mo id="S3.E24.m1.38.38.38.6.6.6.1.1" xref="S3.E24.m1.38.38.38.6.6.6.1.1.cmml">⁢</mo><msub id="S3.E24.m1.38.38.38.6.6.6.1.3" xref="S3.E24.m1.38.38.38.6.6.6.1.3.cmml"><mi id="S3.E24.m1.38.38.38.6.6.6.1.3.2" xref="S3.E24.m1.38.38.38.6.6.6.1.3.2.cmml">ν</mi><mi id="S3.E24.m1.38.38.38.6.6.6.1.3.3" xref="S3.E24.m1.38.38.38.6.6.6.1.3.3.cmml">n</mi></msub></mrow></msubsup><msubsup id="S3.E24.m1.52.52.52.20.20.21.2"><mi class="ltx_font_mathcaligraphic" id="S3.E24.m1.39.39.39.7.7.7" xref="S3.E24.m1.39.39.39.7.7.7.cmml">𝒱</mi><mrow id="S3.E24.m1.40.40.40.8.8.8.1" xref="S3.E24.m1.40.40.40.8.8.8.1.cmml"><mi id="S3.E24.m1.40.40.40.8.8.8.1.2" xref="S3.E24.m1.40.40.40.8.8.8.1.2.cmml">μ</mi><mo id="S3.E24.m1.40.40.40.8.8.8.1.1" xref="S3.E24.m1.40.40.40.8.8.8.1.1.cmml">⁢</mo><mi id="S3.E24.m1.40.40.40.8.8.8.1.3" xref="S3.E24.m1.40.40.40.8.8.8.1.3.cmml">ν</mi><mo id="S3.E24.m1.40.40.40.8.8.8.1.1a" xref="S3.E24.m1.40.40.40.8.8.8.1.1.cmml">⁢</mo><msub id="S3.E24.m1.40.40.40.8.8.8.1.4" xref="S3.E24.m1.40.40.40.8.8.8.1.4.cmml"><mi id="S3.E24.m1.40.40.40.8.8.8.1.4.2" xref="S3.E24.m1.40.40.40.8.8.8.1.4.2.cmml">μ</mi><mn id="S3.E24.m1.40.40.40.8.8.8.1.4.3" xref="S3.E24.m1.40.40.40.8.8.8.1.4.3.cmml">1</mn></msub><mo id="S3.E24.m1.40.40.40.8.8.8.1.1b" xref="S3.E24.m1.40.40.40.8.8.8.1.1.cmml">⁢</mo><msub id="S3.E24.m1.40.40.40.8.8.8.1.5" xref="S3.E24.m1.40.40.40.8.8.8.1.5.cmml"><mi id="S3.E24.m1.40.40.40.8.8.8.1.5.2" xref="S3.E24.m1.40.40.40.8.8.8.1.5.2.cmml">ν</mi><mn id="S3.E24.m1.40.40.40.8.8.8.1.5.3" xref="S3.E24.m1.40.40.40.8.8.8.1.5.3.cmml">1</mn></msub><mo id="S3.E24.m1.40.40.40.8.8.8.1.1c" 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xref="S3.E24.m1.40.40.40.8.8.8.1.9.cmml">α</mi><mo id="S3.E24.m1.40.40.40.8.8.8.1.1g" xref="S3.E24.m1.40.40.40.8.8.8.1.1.cmml">⁢</mo><mi id="S3.E24.m1.40.40.40.8.8.8.1.10" xref="S3.E24.m1.40.40.40.8.8.8.1.10.cmml">β</mi></mrow><mrow id="S3.E24.m1.41.41.41.9.9.9.1.1" xref="S3.E24.m1.41.41.41.9.9.9.1.1.1.cmml"><mo id="S3.E24.m1.41.41.41.9.9.9.1.1.2" stretchy="false" xref="S3.E24.m1.41.41.41.9.9.9.1.1.1.cmml">(</mo><mrow id="S3.E24.m1.41.41.41.9.9.9.1.1.1" xref="S3.E24.m1.41.41.41.9.9.9.1.1.1.cmml"><mi id="S3.E24.m1.41.41.41.9.9.9.1.1.1.2" xref="S3.E24.m1.41.41.41.9.9.9.1.1.1.2.cmml">n</mi><mo id="S3.E24.m1.41.41.41.9.9.9.1.1.1.1" xref="S3.E24.m1.41.41.41.9.9.9.1.1.1.1.cmml">+</mo><mn id="S3.E24.m1.41.41.41.9.9.9.1.1.1.3" xref="S3.E24.m1.41.41.41.9.9.9.1.1.1.3.cmml">2</mn></mrow><mo id="S3.E24.m1.41.41.41.9.9.9.1.1.3" stretchy="false" xref="S3.E24.m1.41.41.41.9.9.9.1.1.1.cmml">)</mo></mrow></msubsup><mrow id="S3.E24.m1.52.52.52.20.20.21.3"><mo id="S3.E24.m1.42.42.42.10.10.10" 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\times\bigg{\{}\frac{1}{n}I_{P_{n}}^{\mu_{n}\nu_{n}}\mathcal{V}_{\mu\nu\mu_{1}% \nu_{1}\cdots\mu_{n}\nu_{n}\alpha\beta}^{(n+2)}(P_{1},\ldots,P_{n},\ell)\\ +K^{\mu_{n}\nu_{n}}_{P_{n}\alpha\beta}\mathcal{V}_{\mu\nu\mu_{1}\nu_{1}\cdots% \mu_{n}\nu_{n}}^{(n+1)}(P_{1},\ldots,\ell P_{n})\bigg{\}}.</annotation><annotation encoding="application/x-llamapun" id="S3.E24.m1.72d">start_ROW start_CELL italic_s start_POSTSUBSCRIPT roman_ℓ italic_P end_POSTSUBSCRIPT blackboard_P start_POSTSUBSCRIPT italic_μ italic_ν italic_ρ italic_σ end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_ρ italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P italic_α italic_β end_POSTSUBSCRIPT = italic_I start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ρ italic_σ end_POSTSUPERSCRIPT caligraphic_V start_POSTSUBSCRIPT italic_μ italic_ν italic_ρ italic_σ italic_α italic_β end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT ( italic_P , roman_ℓ ) end_CELL end_ROW start_ROW start_CELL + ∑ start_POSTSUBSCRIPT italic_n = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG ( italic_n - 1 ) ! end_ARG ∑ start_POSTSUBSCRIPT italic_P = italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∪ … ∪ italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∏ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT italic_I start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL × { divide start_ARG 1 end_ARG start_ARG italic_n end_ARG italic_I start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT caligraphic_V start_POSTSUBSCRIPT italic_μ italic_ν italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋯ italic_μ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_n + 2 ) end_POSTSUPERSCRIPT ( italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , roman_ℓ ) end_CELL end_ROW start_ROW start_CELL + italic_K start_POSTSUPERSCRIPT italic_μ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT caligraphic_V start_POSTSUBSCRIPT italic_μ italic_ν italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT 1 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id="S3.p1.4.m1.1a"><msubsup id="S3.p1.4.m1.1.1" xref="S3.p1.4.m1.1.1.cmml"><mi id="S3.p1.4.m1.1.1.2.2" xref="S3.p1.4.m1.1.1.2.2.cmml">h</mi><mi id="S3.p1.4.m1.1.1.2.3" mathvariant="normal" xref="S3.p1.4.m1.1.1.2.3.cmml">ℓ</mi><mrow id="S3.p1.4.m1.1.1.3" xref="S3.p1.4.m1.1.1.3.cmml"><mi id="S3.p1.4.m1.1.1.3.2" xref="S3.p1.4.m1.1.1.3.2.cmml">α</mi><mo id="S3.p1.4.m1.1.1.3.1" xref="S3.p1.4.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.p1.4.m1.1.1.3.3" xref="S3.p1.4.m1.1.1.3.3.cmml">β</mi></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.p1.4.m1.1b"><apply id="S3.p1.4.m1.1.1.cmml" xref="S3.p1.4.m1.1.1"><csymbol cd="ambiguous" id="S3.p1.4.m1.1.1.1.cmml" xref="S3.p1.4.m1.1.1">superscript</csymbol><apply id="S3.p1.4.m1.1.1.2.cmml" xref="S3.p1.4.m1.1.1"><csymbol cd="ambiguous" id="S3.p1.4.m1.1.1.2.1.cmml" xref="S3.p1.4.m1.1.1">subscript</csymbol><ci id="S3.p1.4.m1.1.1.2.2.cmml" xref="S3.p1.4.m1.1.1.2.2">ℎ</ci><ci id="S3.p1.4.m1.1.1.2.3.cmml" xref="S3.p1.4.m1.1.1.2.3">ℓ</ci></apply><apply 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xref="S3.p1.5.m2.1.1.3.cmml">β</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.5.m2.1b"><apply id="S3.p1.5.m2.1.1.cmml" xref="S3.p1.5.m2.1.1"><ci id="S3.p1.5.m2.1.1.1.cmml" xref="S3.p1.5.m2.1.1.1">↔</ci><ci id="S3.p1.5.m2.1.1.2.cmml" xref="S3.p1.5.m2.1.1.2">𝛼</ci><ci id="S3.p1.5.m2.1.1.3.cmml" xref="S3.p1.5.m2.1.1.3">𝛽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.5.m2.1c">\alpha\leftrightarrow\beta</annotation><annotation encoding="application/x-llamapun" id="S3.p1.5.m2.1d">italic_α ↔ italic_β</annotation></semantics></math>. Note there are no ghost external legs yet.</p> </div> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.5">A naive sewing of the two legs <math alttext="(\mu\nu)" class="ltx_Math" display="inline" id="S3.p2.1.m1.1"><semantics id="S3.p2.1.m1.1a"><mrow id="S3.p2.1.m1.1.1.1" xref="S3.p2.1.m1.1.1.1.1.cmml"><mo id="S3.p2.1.m1.1.1.1.2" stretchy="false" xref="S3.p2.1.m1.1.1.1.1.cmml">(</mo><mrow id="S3.p2.1.m1.1.1.1.1" xref="S3.p2.1.m1.1.1.1.1.cmml"><mi id="S3.p2.1.m1.1.1.1.1.2" xref="S3.p2.1.m1.1.1.1.1.2.cmml">μ</mi><mo id="S3.p2.1.m1.1.1.1.1.1" xref="S3.p2.1.m1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.p2.1.m1.1.1.1.1.3" xref="S3.p2.1.m1.1.1.1.1.3.cmml">ν</mi></mrow><mo id="S3.p2.1.m1.1.1.1.3" stretchy="false" xref="S3.p2.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.1.m1.1b"><apply id="S3.p2.1.m1.1.1.1.1.cmml" xref="S3.p2.1.m1.1.1.1"><times id="S3.p2.1.m1.1.1.1.1.1.cmml" xref="S3.p2.1.m1.1.1.1.1.1"></times><ci id="S3.p2.1.m1.1.1.1.1.2.cmml" xref="S3.p2.1.m1.1.1.1.1.2">𝜇</ci><ci id="S3.p2.1.m1.1.1.1.1.3.cmml" xref="S3.p2.1.m1.1.1.1.1.3">𝜈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.1.m1.1c">(\mu\nu)</annotation><annotation encoding="application/x-llamapun" id="S3.p2.1.m1.1d">( italic_μ italic_ν )</annotation></semantics></math> and <math alttext="(\alpha\beta)" class="ltx_Math" display="inline" id="S3.p2.2.m2.1"><semantics id="S3.p2.2.m2.1a"><mrow id="S3.p2.2.m2.1.1.1" xref="S3.p2.2.m2.1.1.1.1.cmml"><mo id="S3.p2.2.m2.1.1.1.2" stretchy="false" xref="S3.p2.2.m2.1.1.1.1.cmml">(</mo><mrow id="S3.p2.2.m2.1.1.1.1" xref="S3.p2.2.m2.1.1.1.1.cmml"><mi id="S3.p2.2.m2.1.1.1.1.2" xref="S3.p2.2.m2.1.1.1.1.2.cmml">α</mi><mo id="S3.p2.2.m2.1.1.1.1.1" xref="S3.p2.2.m2.1.1.1.1.1.cmml">⁢</mo><mi id="S3.p2.2.m2.1.1.1.1.3" xref="S3.p2.2.m2.1.1.1.1.3.cmml">β</mi></mrow><mo id="S3.p2.2.m2.1.1.1.3" stretchy="false" xref="S3.p2.2.m2.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.2.m2.1b"><apply id="S3.p2.2.m2.1.1.1.1.cmml" xref="S3.p2.2.m2.1.1.1"><times id="S3.p2.2.m2.1.1.1.1.1.cmml" xref="S3.p2.2.m2.1.1.1.1.1"></times><ci id="S3.p2.2.m2.1.1.1.1.2.cmml" xref="S3.p2.2.m2.1.1.1.1.2">𝛼</ci><ci id="S3.p2.2.m2.1.1.1.1.3.cmml" xref="S3.p2.2.m2.1.1.1.1.3">𝛽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.2.m2.1c">(\alpha\beta)</annotation><annotation encoding="application/x-llamapun" id="S3.p2.2.m2.1d">( italic_α italic_β )</annotation></semantics></math> would lead to several one-loop integrands. This is achieved by multiplying both sides of (<a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S3.E24" title="In III Loop recursions ‣ One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_tag">24</span></a>) by the graviton propagator <math alttext="D^{\mu\nu\alpha\beta}_{\ell}=(\mathbb{P}^{-1})^{\mu\nu\alpha\beta}/s_{\ell}" class="ltx_Math" display="inline" id="S3.p2.3.m3.1"><semantics id="S3.p2.3.m3.1a"><mrow id="S3.p2.3.m3.1.1" xref="S3.p2.3.m3.1.1.cmml"><msubsup id="S3.p2.3.m3.1.1.3" xref="S3.p2.3.m3.1.1.3.cmml"><mi id="S3.p2.3.m3.1.1.3.2.2" xref="S3.p2.3.m3.1.1.3.2.2.cmml">D</mi><mi id="S3.p2.3.m3.1.1.3.3" mathvariant="normal" xref="S3.p2.3.m3.1.1.3.3.cmml">ℓ</mi><mrow id="S3.p2.3.m3.1.1.3.2.3" xref="S3.p2.3.m3.1.1.3.2.3.cmml"><mi id="S3.p2.3.m3.1.1.3.2.3.2" xref="S3.p2.3.m3.1.1.3.2.3.2.cmml">μ</mi><mo id="S3.p2.3.m3.1.1.3.2.3.1" xref="S3.p2.3.m3.1.1.3.2.3.1.cmml">⁢</mo><mi id="S3.p2.3.m3.1.1.3.2.3.3" xref="S3.p2.3.m3.1.1.3.2.3.3.cmml">ν</mi><mo 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id="S3.p2.3.m3.1c">D^{\mu\nu\alpha\beta}_{\ell}=(\mathbb{P}^{-1})^{\mu\nu\alpha\beta}/s_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.3.m3.1d">italic_D start_POSTSUPERSCRIPT italic_μ italic_ν italic_α italic_β end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT = ( blackboard_P start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_μ italic_ν italic_α italic_β end_POSTSUPERSCRIPT / italic_s start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math>. Though we are guaranteed to generate all one-loop diagrams, this procedure does not immediately yield the one-loop correlators upon integration of the loop momentum <math alttext="\ell^{\mu}\equiv k^{\mu}_{\ell}" class="ltx_Math" display="inline" id="S3.p2.4.m4.1"><semantics id="S3.p2.4.m4.1a"><mrow id="S3.p2.4.m4.1.1" xref="S3.p2.4.m4.1.1.cmml"><msup id="S3.p2.4.m4.1.1.2" xref="S3.p2.4.m4.1.1.2.cmml"><mi id="S3.p2.4.m4.1.1.2.2" mathvariant="normal" xref="S3.p2.4.m4.1.1.2.2.cmml">ℓ</mi><mi id="S3.p2.4.m4.1.1.2.3" xref="S3.p2.4.m4.1.1.2.3.cmml">μ</mi></msup><mo id="S3.p2.4.m4.1.1.1" xref="S3.p2.4.m4.1.1.1.cmml">≡</mo><msubsup id="S3.p2.4.m4.1.1.3" xref="S3.p2.4.m4.1.1.3.cmml"><mi id="S3.p2.4.m4.1.1.3.2.2" xref="S3.p2.4.m4.1.1.3.2.2.cmml">k</mi><mi id="S3.p2.4.m4.1.1.3.3" mathvariant="normal" xref="S3.p2.4.m4.1.1.3.3.cmml">ℓ</mi><mi id="S3.p2.4.m4.1.1.3.2.3" xref="S3.p2.4.m4.1.1.3.2.3.cmml">μ</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.4.m4.1b"><apply id="S3.p2.4.m4.1.1.cmml" xref="S3.p2.4.m4.1.1"><equivalent id="S3.p2.4.m4.1.1.1.cmml" xref="S3.p2.4.m4.1.1.1"></equivalent><apply id="S3.p2.4.m4.1.1.2.cmml" xref="S3.p2.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.p2.4.m4.1.1.2.1.cmml" xref="S3.p2.4.m4.1.1.2">superscript</csymbol><ci id="S3.p2.4.m4.1.1.2.2.cmml" xref="S3.p2.4.m4.1.1.2.2">ℓ</ci><ci id="S3.p2.4.m4.1.1.2.3.cmml" xref="S3.p2.4.m4.1.1.2.3">𝜇</ci></apply><apply id="S3.p2.4.m4.1.1.3.cmml" xref="S3.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.p2.4.m4.1.1.3.1.cmml" xref="S3.p2.4.m4.1.1.3">subscript</csymbol><apply id="S3.p2.4.m4.1.1.3.2.cmml" xref="S3.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.p2.4.m4.1.1.3.2.1.cmml" xref="S3.p2.4.m4.1.1.3">superscript</csymbol><ci id="S3.p2.4.m4.1.1.3.2.2.cmml" xref="S3.p2.4.m4.1.1.3.2.2">𝑘</ci><ci id="S3.p2.4.m4.1.1.3.2.3.cmml" xref="S3.p2.4.m4.1.1.3.2.3">𝜇</ci></apply><ci id="S3.p2.4.m4.1.1.3.3.cmml" xref="S3.p2.4.m4.1.1.3.3">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.4.m4.1c">\ell^{\mu}\equiv k^{\mu}_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.4.m4.1d">roman_ℓ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT ≡ italic_k start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math>. We are clearly overlooking symmetry factors and equivalent-diagram contributions. A natural proposal to try to correct this shortcoming is to define a modified <math alttext="K_{P}" class="ltx_Math" display="inline" id="S3.p2.5.m5.1"><semantics id="S3.p2.5.m5.1a"><msub id="S3.p2.5.m5.1.1" xref="S3.p2.5.m5.1.1.cmml"><mi id="S3.p2.5.m5.1.1.2" xref="S3.p2.5.m5.1.1.2.cmml">K</mi><mi id="S3.p2.5.m5.1.1.3" xref="S3.p2.5.m5.1.1.3.cmml">P</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.5.m5.1b"><apply id="S3.p2.5.m5.1.1.cmml" xref="S3.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S3.p2.5.m5.1.1.1.cmml" xref="S3.p2.5.m5.1.1">subscript</csymbol><ci id="S3.p2.5.m5.1.1.2.cmml" xref="S3.p2.5.m5.1.1.2">𝐾</ci><ci id="S3.p2.5.m5.1.1.3.cmml" xref="S3.p2.5.m5.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.5.m5.1c">K_{P}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.5.m5.1d">italic_K start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math>, namely</p> <table class="ltx_equation ltx_eqn_table" id="S3.E25"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\tilde{K}^{\rho\sigma}_{P\alpha\beta}(\ell)=D^{\mu\nu\rho\sigma}_{\ell P}\Big{% \{}f_{1}(|P|)I_{P}^{\gamma\lambda}\mathcal{V}_{\mu\nu\gamma\lambda\alpha\beta}% ^{(3)}(P,\ell)\\ +\sum_{n=2}^{\infty}\frac{1}{(n-1)!}\sum_{P=P_{1}\cup\ldots\cup P_{n}}\prod_{i% =1}^{n-1}I_{P_{i}}^{\mu_{i}\nu_{i}}\\ \times\Big{[}\frac{1}{n}f_{n}I_{P_{n}}^{\mu_{n}\nu_{n}}\mathcal{V}_{\mu\nu\mu_% {1}\nu_{1}\cdots\mu_{n}\nu_{n}\alpha\beta}^{(n+2)}(P_{1},\ldots,P_{n},\ell)\\ \vphantom{\frac{1}{n}}+g_{n}\tilde{K}^{\mu_{n}\nu_{n}}_{P_{n}\alpha\beta}(\ell% )\mathcal{V}_{\mu\nu\mu_{1}\nu_{1}\cdots\mu_{n}\nu_{n}}^{(n+1)}(P_{1},\ldots,% \ell P_{n})\Big{]}\Big{\}}." class="ltx_Math" display="block" id="S3.E25.m1.90"><semantics id="S3.E25.m1.90a"><mtable displaystyle="true" id="S3.E25.m1.89.89" rowspacing="0pt"><mtr id="S3.E25.m1.89.89a"><mtd class="ltx_align_left" columnalign="left" id="S3.E25.m1.89.89b"><mrow id="S3.E25.m1.29.29.29.29.29"><msubsup id="S3.E25.m1.29.29.29.29.29.30"><mover accent="true" id="S3.E25.m1.1.1.1.1.1.1" xref="S3.E25.m1.1.1.1.1.1.1.cmml"><mi id="S3.E25.m1.1.1.1.1.1.1.2" xref="S3.E25.m1.1.1.1.1.1.1.2.cmml">K</mi><mo id="S3.E25.m1.1.1.1.1.1.1.1" xref="S3.E25.m1.1.1.1.1.1.1.1.cmml">~</mo></mover><mrow id="S3.E25.m1.3.3.3.3.3.3.1" xref="S3.E25.m1.3.3.3.3.3.3.1.cmml"><mi id="S3.E25.m1.3.3.3.3.3.3.1.2" xref="S3.E25.m1.3.3.3.3.3.3.1.2.cmml">P</mi><mo id="S3.E25.m1.3.3.3.3.3.3.1.1" xref="S3.E25.m1.3.3.3.3.3.3.1.1.cmml">⁢</mo><mi id="S3.E25.m1.3.3.3.3.3.3.1.3" xref="S3.E25.m1.3.3.3.3.3.3.1.3.cmml">α</mi><mo id="S3.E25.m1.3.3.3.3.3.3.1.1a" xref="S3.E25.m1.3.3.3.3.3.3.1.1.cmml">⁢</mo><mi id="S3.E25.m1.3.3.3.3.3.3.1.4" xref="S3.E25.m1.3.3.3.3.3.3.1.4.cmml">β</mi></mrow><mrow id="S3.E25.m1.2.2.2.2.2.2.1" xref="S3.E25.m1.2.2.2.2.2.2.1.cmml"><mi id="S3.E25.m1.2.2.2.2.2.2.1.2" xref="S3.E25.m1.2.2.2.2.2.2.1.2.cmml">ρ</mi><mo id="S3.E25.m1.2.2.2.2.2.2.1.1" xref="S3.E25.m1.2.2.2.2.2.2.1.1.cmml">⁢</mo><mi id="S3.E25.m1.2.2.2.2.2.2.1.3" xref="S3.E25.m1.2.2.2.2.2.2.1.3.cmml">σ</mi></mrow></msubsup><mrow id="S3.E25.m1.29.29.29.29.29.31"><mo id="S3.E25.m1.4.4.4.4.4.4" stretchy="false" xref="S3.E25.m1.90.90.1.1.1.cmml">(</mo><mi id="S3.E25.m1.5.5.5.5.5.5" mathvariant="normal" xref="S3.E25.m1.5.5.5.5.5.5.cmml">ℓ</mi><mo id="S3.E25.m1.6.6.6.6.6.6" stretchy="false" xref="S3.E25.m1.90.90.1.1.1.cmml">)</mo></mrow><mo id="S3.E25.m1.7.7.7.7.7.7" xref="S3.E25.m1.7.7.7.7.7.7.cmml">=</mo><msubsup id="S3.E25.m1.29.29.29.29.29.32"><mi id="S3.E25.m1.8.8.8.8.8.8" xref="S3.E25.m1.8.8.8.8.8.8.cmml">D</mi><mrow id="S3.E25.m1.10.10.10.10.10.10.1" xref="S3.E25.m1.10.10.10.10.10.10.1.cmml"><mi id="S3.E25.m1.10.10.10.10.10.10.1.2" mathvariant="normal" xref="S3.E25.m1.10.10.10.10.10.10.1.2.cmml">ℓ</mi><mo id="S3.E25.m1.10.10.10.10.10.10.1.1" xref="S3.E25.m1.10.10.10.10.10.10.1.1.cmml">⁢</mo><mi id="S3.E25.m1.10.10.10.10.10.10.1.3" xref="S3.E25.m1.10.10.10.10.10.10.1.3.cmml">P</mi></mrow><mrow id="S3.E25.m1.9.9.9.9.9.9.1" xref="S3.E25.m1.9.9.9.9.9.9.1.cmml"><mi id="S3.E25.m1.9.9.9.9.9.9.1.2" xref="S3.E25.m1.9.9.9.9.9.9.1.2.cmml">μ</mi><mo id="S3.E25.m1.9.9.9.9.9.9.1.1" xref="S3.E25.m1.9.9.9.9.9.9.1.1.cmml">⁢</mo><mi id="S3.E25.m1.9.9.9.9.9.9.1.3" xref="S3.E25.m1.9.9.9.9.9.9.1.3.cmml">ν</mi><mo id="S3.E25.m1.9.9.9.9.9.9.1.1a" xref="S3.E25.m1.9.9.9.9.9.9.1.1.cmml">⁢</mo><mi id="S3.E25.m1.9.9.9.9.9.9.1.4" xref="S3.E25.m1.9.9.9.9.9.9.1.4.cmml">ρ</mi><mo id="S3.E25.m1.9.9.9.9.9.9.1.1b" xref="S3.E25.m1.9.9.9.9.9.9.1.1.cmml">⁢</mo><mi id="S3.E25.m1.9.9.9.9.9.9.1.5" xref="S3.E25.m1.9.9.9.9.9.9.1.5.cmml">σ</mi></mrow></msubsup><mrow id="S3.E25.m1.29.29.29.29.29.33"><mo id="S3.E25.m1.11.11.11.11.11.11" maxsize="160%" minsize="160%" xref="S3.E25.m1.90.90.1.1.1.cmml">{</mo><msub id="S3.E25.m1.29.29.29.29.29.33.1"><mi id="S3.E25.m1.12.12.12.12.12.12" xref="S3.E25.m1.12.12.12.12.12.12.cmml">f</mi><mn id="S3.E25.m1.13.13.13.13.13.13.1" xref="S3.E25.m1.13.13.13.13.13.13.1.cmml">1</mn></msub><mrow id="S3.E25.m1.29.29.29.29.29.33.2"><mo id="S3.E25.m1.14.14.14.14.14.14" stretchy="false" xref="S3.E25.m1.90.90.1.1.1.cmml">(</mo><mo fence="false" id="S3.E25.m1.15.15.15.15.15.15" rspace="0.167em" stretchy="false" xref="S3.E25.m1.90.90.1.1.1.cmml">|</mo><mi id="S3.E25.m1.16.16.16.16.16.16" xref="S3.E25.m1.16.16.16.16.16.16.cmml">P</mi><mo fence="false" id="S3.E25.m1.17.17.17.17.17.17" rspace="0.167em" stretchy="false" xref="S3.E25.m1.90.90.1.1.1.cmml">|</mo><mo id="S3.E25.m1.18.18.18.18.18.18" stretchy="false" xref="S3.E25.m1.90.90.1.1.1.cmml">)</mo></mrow><msubsup id="S3.E25.m1.29.29.29.29.29.33.3"><mi id="S3.E25.m1.19.19.19.19.19.19" xref="S3.E25.m1.19.19.19.19.19.19.cmml">I</mi><mi id="S3.E25.m1.20.20.20.20.20.20.1" xref="S3.E25.m1.20.20.20.20.20.20.1.cmml">P</mi><mrow id="S3.E25.m1.21.21.21.21.21.21.1" xref="S3.E25.m1.21.21.21.21.21.21.1.cmml"><mi id="S3.E25.m1.21.21.21.21.21.21.1.2" xref="S3.E25.m1.21.21.21.21.21.21.1.2.cmml">γ</mi><mo id="S3.E25.m1.21.21.21.21.21.21.1.1" xref="S3.E25.m1.21.21.21.21.21.21.1.1.cmml">⁢</mo><mi id="S3.E25.m1.21.21.21.21.21.21.1.3" xref="S3.E25.m1.21.21.21.21.21.21.1.3.cmml">λ</mi></mrow></msubsup><msubsup id="S3.E25.m1.29.29.29.29.29.33.4"><mi class="ltx_font_mathcaligraphic" id="S3.E25.m1.22.22.22.22.22.22" xref="S3.E25.m1.22.22.22.22.22.22.cmml">𝒱</mi><mrow id="S3.E25.m1.23.23.23.23.23.23.1" xref="S3.E25.m1.23.23.23.23.23.23.1.cmml"><mi id="S3.E25.m1.23.23.23.23.23.23.1.2" xref="S3.E25.m1.23.23.23.23.23.23.1.2.cmml">μ</mi><mo id="S3.E25.m1.23.23.23.23.23.23.1.1" xref="S3.E25.m1.23.23.23.23.23.23.1.1.cmml">⁢</mo><mi id="S3.E25.m1.23.23.23.23.23.23.1.3" xref="S3.E25.m1.23.23.23.23.23.23.1.3.cmml">ν</mi><mo id="S3.E25.m1.23.23.23.23.23.23.1.1a" xref="S3.E25.m1.23.23.23.23.23.23.1.1.cmml">⁢</mo><mi id="S3.E25.m1.23.23.23.23.23.23.1.4" xref="S3.E25.m1.23.23.23.23.23.23.1.4.cmml">γ</mi><mo id="S3.E25.m1.23.23.23.23.23.23.1.1b" xref="S3.E25.m1.23.23.23.23.23.23.1.1.cmml">⁢</mo><mi id="S3.E25.m1.23.23.23.23.23.23.1.5" xref="S3.E25.m1.23.23.23.23.23.23.1.5.cmml">λ</mi><mo id="S3.E25.m1.23.23.23.23.23.23.1.1c" xref="S3.E25.m1.23.23.23.23.23.23.1.1.cmml">⁢</mo><mi id="S3.E25.m1.23.23.23.23.23.23.1.6" xref="S3.E25.m1.23.23.23.23.23.23.1.6.cmml">α</mi><mo id="S3.E25.m1.23.23.23.23.23.23.1.1d" xref="S3.E25.m1.23.23.23.23.23.23.1.1.cmml">⁢</mo><mi id="S3.E25.m1.23.23.23.23.23.23.1.7" xref="S3.E25.m1.23.23.23.23.23.23.1.7.cmml">β</mi></mrow><mrow id="S3.E25.m1.24.24.24.24.24.24.1.3"><mo id="S3.E25.m1.24.24.24.24.24.24.1.3.1" stretchy="false">(</mo><mn id="S3.E25.m1.24.24.24.24.24.24.1.1" xref="S3.E25.m1.24.24.24.24.24.24.1.1.cmml">3</mn><mo id="S3.E25.m1.24.24.24.24.24.24.1.3.2" stretchy="false">)</mo></mrow></msubsup><mrow id="S3.E25.m1.29.29.29.29.29.33.5"><mo id="S3.E25.m1.25.25.25.25.25.25" stretchy="false" xref="S3.E25.m1.90.90.1.1.1.cmml">(</mo><mi id="S3.E25.m1.26.26.26.26.26.26" xref="S3.E25.m1.26.26.26.26.26.26.cmml">P</mi><mo id="S3.E25.m1.27.27.27.27.27.27" xref="S3.E25.m1.90.90.1.1.1.cmml">,</mo><mi id="S3.E25.m1.28.28.28.28.28.28" mathvariant="normal" xref="S3.E25.m1.28.28.28.28.28.28.cmml">ℓ</mi><mo id="S3.E25.m1.29.29.29.29.29.29" stretchy="false" xref="S3.E25.m1.90.90.1.1.1.cmml">)</mo></mrow></mrow></mrow></mtd></mtr><mtr id="S3.E25.m1.89.89c"><mtd class="ltx_align_right" columnalign="right" id="S3.E25.m1.89.89d"><mrow id="S3.E25.m1.42.42.42.13.13"><mo id="S3.E25.m1.42.42.42.13.13a" xref="S3.E25.m1.90.90.1.1.1.cmml">+</mo><mrow id="S3.E25.m1.42.42.42.13.13.14"><munderover id="S3.E25.m1.42.42.42.13.13.14.1"><mo id="S3.E25.m1.31.31.31.2.2.2" movablelimits="false" xref="S3.E25.m1.31.31.31.2.2.2.cmml">∑</mo><mrow id="S3.E25.m1.32.32.32.3.3.3.1" xref="S3.E25.m1.32.32.32.3.3.3.1.cmml"><mi id="S3.E25.m1.32.32.32.3.3.3.1.2" xref="S3.E25.m1.32.32.32.3.3.3.1.2.cmml">n</mi><mo id="S3.E25.m1.32.32.32.3.3.3.1.1" xref="S3.E25.m1.32.32.32.3.3.3.1.1.cmml">=</mo><mn id="S3.E25.m1.32.32.32.3.3.3.1.3" xref="S3.E25.m1.32.32.32.3.3.3.1.3.cmml">2</mn></mrow><mi id="S3.E25.m1.33.33.33.4.4.4.1" mathvariant="normal" xref="S3.E25.m1.33.33.33.4.4.4.1.cmml">∞</mi></munderover><mrow id="S3.E25.m1.42.42.42.13.13.14.2"><mfrac id="S3.E25.m1.34.34.34.5.5.5" xref="S3.E25.m1.34.34.34.5.5.5.cmml"><mn id="S3.E25.m1.34.34.34.5.5.5.3" xref="S3.E25.m1.34.34.34.5.5.5.3.cmml">1</mn><mrow id="S3.E25.m1.34.34.34.5.5.5.1" xref="S3.E25.m1.34.34.34.5.5.5.1.cmml"><mrow id="S3.E25.m1.34.34.34.5.5.5.1.1.1" xref="S3.E25.m1.34.34.34.5.5.5.1.1.1.1.cmml"><mo id="S3.E25.m1.34.34.34.5.5.5.1.1.1.2" stretchy="false" xref="S3.E25.m1.34.34.34.5.5.5.1.1.1.1.cmml">(</mo><mrow id="S3.E25.m1.34.34.34.5.5.5.1.1.1.1" xref="S3.E25.m1.34.34.34.5.5.5.1.1.1.1.cmml"><mi id="S3.E25.m1.34.34.34.5.5.5.1.1.1.1.2" xref="S3.E25.m1.34.34.34.5.5.5.1.1.1.1.2.cmml">n</mi><mo id="S3.E25.m1.34.34.34.5.5.5.1.1.1.1.1" xref="S3.E25.m1.34.34.34.5.5.5.1.1.1.1.1.cmml">−</mo><mn id="S3.E25.m1.34.34.34.5.5.5.1.1.1.1.3" xref="S3.E25.m1.34.34.34.5.5.5.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.E25.m1.34.34.34.5.5.5.1.1.1.3" stretchy="false" xref="S3.E25.m1.34.34.34.5.5.5.1.1.1.1.cmml">)</mo></mrow><mo id="S3.E25.m1.34.34.34.5.5.5.1.2" xref="S3.E25.m1.34.34.34.5.5.5.1.2.cmml">!</mo></mrow></mfrac><mo id="S3.E25.m1.42.42.42.13.13.14.2.1" xref="S3.E25.m1.90.90.1.1.1.cmml">⁢</mo><mrow id="S3.E25.m1.42.42.42.13.13.14.2.2"><munder id="S3.E25.m1.42.42.42.13.13.14.2.2.1"><mo id="S3.E25.m1.35.35.35.6.6.6" movablelimits="false" rspace="0em" xref="S3.E25.m1.35.35.35.6.6.6.cmml">∑</mo><mrow id="S3.E25.m1.36.36.36.7.7.7.1" xref="S3.E25.m1.36.36.36.7.7.7.1.cmml"><mi id="S3.E25.m1.36.36.36.7.7.7.1.2" xref="S3.E25.m1.36.36.36.7.7.7.1.2.cmml">P</mi><mo id="S3.E25.m1.36.36.36.7.7.7.1.1" xref="S3.E25.m1.36.36.36.7.7.7.1.1.cmml">=</mo><mrow id="S3.E25.m1.36.36.36.7.7.7.1.3" xref="S3.E25.m1.36.36.36.7.7.7.1.3.cmml"><msub id="S3.E25.m1.36.36.36.7.7.7.1.3.2" xref="S3.E25.m1.36.36.36.7.7.7.1.3.2.cmml"><mi id="S3.E25.m1.36.36.36.7.7.7.1.3.2.2" xref="S3.E25.m1.36.36.36.7.7.7.1.3.2.2.cmml">P</mi><mn id="S3.E25.m1.36.36.36.7.7.7.1.3.2.3" xref="S3.E25.m1.36.36.36.7.7.7.1.3.2.3.cmml">1</mn></msub><mo id="S3.E25.m1.36.36.36.7.7.7.1.3.1" xref="S3.E25.m1.36.36.36.7.7.7.1.3.1.cmml">∪</mo><mi id="S3.E25.m1.36.36.36.7.7.7.1.3.3" mathvariant="normal" xref="S3.E25.m1.36.36.36.7.7.7.1.3.3.cmml">…</mi><mo id="S3.E25.m1.36.36.36.7.7.7.1.3.1a" xref="S3.E25.m1.36.36.36.7.7.7.1.3.1.cmml">∪</mo><msub id="S3.E25.m1.36.36.36.7.7.7.1.3.4" xref="S3.E25.m1.36.36.36.7.7.7.1.3.4.cmml"><mi id="S3.E25.m1.36.36.36.7.7.7.1.3.4.2" xref="S3.E25.m1.36.36.36.7.7.7.1.3.4.2.cmml">P</mi><mi id="S3.E25.m1.36.36.36.7.7.7.1.3.4.3" xref="S3.E25.m1.36.36.36.7.7.7.1.3.4.3.cmml">n</mi></msub></mrow></mrow></munder><mrow id="S3.E25.m1.42.42.42.13.13.14.2.2.2"><munderover id="S3.E25.m1.42.42.42.13.13.14.2.2.2.1"><mo id="S3.E25.m1.37.37.37.8.8.8" movablelimits="false" xref="S3.E25.m1.37.37.37.8.8.8.cmml">∏</mo><mrow id="S3.E25.m1.38.38.38.9.9.9.1" xref="S3.E25.m1.38.38.38.9.9.9.1.cmml"><mi id="S3.E25.m1.38.38.38.9.9.9.1.2" xref="S3.E25.m1.38.38.38.9.9.9.1.2.cmml">i</mi><mo id="S3.E25.m1.38.38.38.9.9.9.1.1" xref="S3.E25.m1.38.38.38.9.9.9.1.1.cmml">=</mo><mn id="S3.E25.m1.38.38.38.9.9.9.1.3" xref="S3.E25.m1.38.38.38.9.9.9.1.3.cmml">1</mn></mrow><mrow id="S3.E25.m1.39.39.39.10.10.10.1" xref="S3.E25.m1.39.39.39.10.10.10.1.cmml"><mi id="S3.E25.m1.39.39.39.10.10.10.1.2" xref="S3.E25.m1.39.39.39.10.10.10.1.2.cmml">n</mi><mo id="S3.E25.m1.39.39.39.10.10.10.1.1" xref="S3.E25.m1.39.39.39.10.10.10.1.1.cmml">−</mo><mn id="S3.E25.m1.39.39.39.10.10.10.1.3" xref="S3.E25.m1.39.39.39.10.10.10.1.3.cmml">1</mn></mrow></munderover><msubsup id="S3.E25.m1.42.42.42.13.13.14.2.2.2.2"><mi id="S3.E25.m1.40.40.40.11.11.11" xref="S3.E25.m1.40.40.40.11.11.11.cmml">I</mi><msub id="S3.E25.m1.41.41.41.12.12.12.1" xref="S3.E25.m1.41.41.41.12.12.12.1.cmml"><mi id="S3.E25.m1.41.41.41.12.12.12.1.2" xref="S3.E25.m1.41.41.41.12.12.12.1.2.cmml">P</mi><mi id="S3.E25.m1.41.41.41.12.12.12.1.3" xref="S3.E25.m1.41.41.41.12.12.12.1.3.cmml">i</mi></msub><mrow id="S3.E25.m1.42.42.42.13.13.13.1" xref="S3.E25.m1.42.42.42.13.13.13.1.cmml"><msub id="S3.E25.m1.42.42.42.13.13.13.1.2" xref="S3.E25.m1.42.42.42.13.13.13.1.2.cmml"><mi id="S3.E25.m1.42.42.42.13.13.13.1.2.2" xref="S3.E25.m1.42.42.42.13.13.13.1.2.2.cmml">μ</mi><mi id="S3.E25.m1.42.42.42.13.13.13.1.2.3" xref="S3.E25.m1.42.42.42.13.13.13.1.2.3.cmml">i</mi></msub><mo id="S3.E25.m1.42.42.42.13.13.13.1.1" xref="S3.E25.m1.42.42.42.13.13.13.1.1.cmml">⁢</mo><msub id="S3.E25.m1.42.42.42.13.13.13.1.3" xref="S3.E25.m1.42.42.42.13.13.13.1.3.cmml"><mi id="S3.E25.m1.42.42.42.13.13.13.1.3.2" xref="S3.E25.m1.42.42.42.13.13.13.1.3.2.cmml">ν</mi><mi id="S3.E25.m1.42.42.42.13.13.13.1.3.3" xref="S3.E25.m1.42.42.42.13.13.13.1.3.3.cmml">i</mi></msub></mrow></msubsup></mrow></mrow></mrow></mrow></mrow></mtd></mtr><mtr id="S3.E25.m1.89.89e"><mtd class="ltx_align_right" columnalign="right" id="S3.E25.m1.89.89f"><mrow id="S3.E25.m1.64.64.64.22.22"><mo id="S3.E25.m1.43.43.43.1.1.1" rspace="0.222em" xref="S3.E25.m1.43.43.43.1.1.1.cmml">×</mo><mrow id="S3.E25.m1.64.64.64.22.22.23"><mo id="S3.E25.m1.44.44.44.2.2.2" maxsize="160%" minsize="160%" xref="S3.E25.m1.90.90.1.1.1.cmml">[</mo><mfrac id="S3.E25.m1.45.45.45.3.3.3" xref="S3.E25.m1.45.45.45.3.3.3.cmml"><mn id="S3.E25.m1.45.45.45.3.3.3.2" xref="S3.E25.m1.45.45.45.3.3.3.2.cmml">1</mn><mi id="S3.E25.m1.45.45.45.3.3.3.3" xref="S3.E25.m1.45.45.45.3.3.3.3.cmml">n</mi></mfrac><msub id="S3.E25.m1.64.64.64.22.22.23.1"><mi id="S3.E25.m1.46.46.46.4.4.4" xref="S3.E25.m1.46.46.46.4.4.4.cmml">f</mi><mi id="S3.E25.m1.47.47.47.5.5.5.1" xref="S3.E25.m1.47.47.47.5.5.5.1.cmml">n</mi></msub><msubsup id="S3.E25.m1.64.64.64.22.22.23.2"><mi id="S3.E25.m1.48.48.48.6.6.6" xref="S3.E25.m1.48.48.48.6.6.6.cmml">I</mi><msub id="S3.E25.m1.49.49.49.7.7.7.1" xref="S3.E25.m1.49.49.49.7.7.7.1.cmml"><mi id="S3.E25.m1.49.49.49.7.7.7.1.2" xref="S3.E25.m1.49.49.49.7.7.7.1.2.cmml">P</mi><mi id="S3.E25.m1.49.49.49.7.7.7.1.3" xref="S3.E25.m1.49.49.49.7.7.7.1.3.cmml">n</mi></msub><mrow id="S3.E25.m1.50.50.50.8.8.8.1" xref="S3.E25.m1.50.50.50.8.8.8.1.cmml"><msub id="S3.E25.m1.50.50.50.8.8.8.1.2" xref="S3.E25.m1.50.50.50.8.8.8.1.2.cmml"><mi id="S3.E25.m1.50.50.50.8.8.8.1.2.2" xref="S3.E25.m1.50.50.50.8.8.8.1.2.2.cmml">μ</mi><mi id="S3.E25.m1.50.50.50.8.8.8.1.2.3" xref="S3.E25.m1.50.50.50.8.8.8.1.2.3.cmml">n</mi></msub><mo id="S3.E25.m1.50.50.50.8.8.8.1.1" xref="S3.E25.m1.50.50.50.8.8.8.1.1.cmml">⁢</mo><msub id="S3.E25.m1.50.50.50.8.8.8.1.3" xref="S3.E25.m1.50.50.50.8.8.8.1.3.cmml"><mi id="S3.E25.m1.50.50.50.8.8.8.1.3.2" xref="S3.E25.m1.50.50.50.8.8.8.1.3.2.cmml">ν</mi><mi id="S3.E25.m1.50.50.50.8.8.8.1.3.3" xref="S3.E25.m1.50.50.50.8.8.8.1.3.3.cmml">n</mi></msub></mrow></msubsup><msubsup id="S3.E25.m1.64.64.64.22.22.23.3"><mi class="ltx_font_mathcaligraphic" id="S3.E25.m1.51.51.51.9.9.9" xref="S3.E25.m1.51.51.51.9.9.9.cmml">𝒱</mi><mrow id="S3.E25.m1.52.52.52.10.10.10.1" xref="S3.E25.m1.52.52.52.10.10.10.1.cmml"><mi id="S3.E25.m1.52.52.52.10.10.10.1.2" xref="S3.E25.m1.52.52.52.10.10.10.1.2.cmml">μ</mi><mo id="S3.E25.m1.52.52.52.10.10.10.1.1" xref="S3.E25.m1.52.52.52.10.10.10.1.1.cmml">⁢</mo><mi id="S3.E25.m1.52.52.52.10.10.10.1.3" xref="S3.E25.m1.52.52.52.10.10.10.1.3.cmml">ν</mi><mo id="S3.E25.m1.52.52.52.10.10.10.1.1a" xref="S3.E25.m1.52.52.52.10.10.10.1.1.cmml">⁢</mo><msub id="S3.E25.m1.52.52.52.10.10.10.1.4" xref="S3.E25.m1.52.52.52.10.10.10.1.4.cmml"><mi id="S3.E25.m1.52.52.52.10.10.10.1.4.2" xref="S3.E25.m1.52.52.52.10.10.10.1.4.2.cmml">μ</mi><mn id="S3.E25.m1.52.52.52.10.10.10.1.4.3" xref="S3.E25.m1.52.52.52.10.10.10.1.4.3.cmml">1</mn></msub><mo id="S3.E25.m1.52.52.52.10.10.10.1.1b" xref="S3.E25.m1.52.52.52.10.10.10.1.1.cmml">⁢</mo><msub id="S3.E25.m1.52.52.52.10.10.10.1.5" xref="S3.E25.m1.52.52.52.10.10.10.1.5.cmml"><mi id="S3.E25.m1.52.52.52.10.10.10.1.5.2" xref="S3.E25.m1.52.52.52.10.10.10.1.5.2.cmml">ν</mi><mn id="S3.E25.m1.52.52.52.10.10.10.1.5.3" xref="S3.E25.m1.52.52.52.10.10.10.1.5.3.cmml">1</mn></msub><mo id="S3.E25.m1.52.52.52.10.10.10.1.1c" xref="S3.E25.m1.52.52.52.10.10.10.1.1.cmml">⁢</mo><mi id="S3.E25.m1.52.52.52.10.10.10.1.6" mathvariant="normal" xref="S3.E25.m1.52.52.52.10.10.10.1.6.cmml">⋯</mi><mo id="S3.E25.m1.52.52.52.10.10.10.1.1d" xref="S3.E25.m1.52.52.52.10.10.10.1.1.cmml">⁢</mo><msub id="S3.E25.m1.52.52.52.10.10.10.1.7" xref="S3.E25.m1.52.52.52.10.10.10.1.7.cmml"><mi id="S3.E25.m1.52.52.52.10.10.10.1.7.2" xref="S3.E25.m1.52.52.52.10.10.10.1.7.2.cmml">μ</mi><mi id="S3.E25.m1.52.52.52.10.10.10.1.7.3" xref="S3.E25.m1.52.52.52.10.10.10.1.7.3.cmml">n</mi></msub><mo id="S3.E25.m1.52.52.52.10.10.10.1.1e" xref="S3.E25.m1.52.52.52.10.10.10.1.1.cmml">⁢</mo><msub id="S3.E25.m1.52.52.52.10.10.10.1.8" 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\times\Big{[}\frac{1}{n}f_{n}I_{P_{n}}^{\mu_{n}\nu_{n}}\mathcal{V}_{\mu\nu\mu_% {1}\nu_{1}\cdots\mu_{n}\nu_{n}\alpha\beta}^{(n+2)}(P_{1},\ldots,P_{n},\ell)\\ \vphantom{\frac{1}{n}}+g_{n}\tilde{K}^{\mu_{n}\nu_{n}}_{P_{n}\alpha\beta}(\ell% )\mathcal{V}_{\mu\nu\mu_{1}\nu_{1}\cdots\mu_{n}\nu_{n}}^{(n+1)}(P_{1},\ldots,% \ell P_{n})\Big{]}\Big{\}}.</annotation><annotation encoding="application/x-llamapun" id="S3.E25.m1.90d">start_ROW start_CELL over~ start_ARG italic_K end_ARG start_POSTSUPERSCRIPT italic_ρ italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P italic_α italic_β end_POSTSUBSCRIPT ( roman_ℓ ) = italic_D start_POSTSUPERSCRIPT italic_μ italic_ν italic_ρ italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_ℓ italic_P end_POSTSUBSCRIPT { italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( | italic_P | ) italic_I start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_γ italic_λ end_POSTSUPERSCRIPT caligraphic_V start_POSTSUBSCRIPT italic_μ italic_ν italic_γ italic_λ italic_α italic_β end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT ( italic_P , roman_ℓ ) end_CELL end_ROW start_ROW start_CELL + ∑ start_POSTSUBSCRIPT italic_n = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG ( italic_n - 1 ) ! end_ARG ∑ start_POSTSUBSCRIPT italic_P = italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∪ … ∪ italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∏ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT italic_I start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL × [ divide start_ARG 1 end_ARG start_ARG italic_n end_ARG italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT caligraphic_V start_POSTSUBSCRIPT italic_μ italic_ν italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋯ italic_μ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_n + 2 ) end_POSTSUPERSCRIPT ( italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , roman_ℓ ) end_CELL end_ROW start_ROW start_CELL + italic_g start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT over~ start_ARG italic_K end_ARG start_POSTSUPERSCRIPT italic_μ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT ( roman_ℓ ) caligraphic_V start_POSTSUBSCRIPT italic_μ italic_ν italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋯ italic_μ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_n + 1 ) end_POSTSUPERSCRIPT ( italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , roman_ℓ italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ] } . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(25)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p2.9">The coefficients <math 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id="S3.p2.8.m3.2.2.1.1.1.1.3.cmml" type="integer" xref="S3.p2.8.m3.2.2.1.1.1.1.3">1</cn></apply></apply><ci id="S3.p2.8.m3.1.1.cmml" xref="S3.p2.8.m3.1.1">…</ci><apply id="S3.p2.8.m3.3.3.2.2.2.cmml" xref="S3.p2.8.m3.3.3.2.2.1"><abs id="S3.p2.8.m3.3.3.2.2.2.1.cmml" xref="S3.p2.8.m3.3.3.2.2.1.2"></abs><apply id="S3.p2.8.m3.3.3.2.2.1.1.cmml" xref="S3.p2.8.m3.3.3.2.2.1.1"><csymbol cd="ambiguous" id="S3.p2.8.m3.3.3.2.2.1.1.1.cmml" xref="S3.p2.8.m3.3.3.2.2.1.1">subscript</csymbol><ci id="S3.p2.8.m3.3.3.2.2.1.1.2.cmml" xref="S3.p2.8.m3.3.3.2.2.1.1.2">𝑃</ci><ci id="S3.p2.8.m3.3.3.2.2.1.1.3.cmml" xref="S3.p2.8.m3.3.3.2.2.1.1.3">𝑛</ci></apply></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.8.m3.3c">\{|P_{1}|,\ldots,|P_{n}|\}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.8.m3.3d">{ | italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | , … , | italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | }</annotation></semantics></math>, and should balance out any overcounting (<math alttext="0\leq f_{n},g_{n}\leq 1" class="ltx_Math" display="inline" id="S3.p2.9.m4.2"><semantics id="S3.p2.9.m4.2a"><mrow id="S3.p2.9.m4.2.2.2" xref="S3.p2.9.m4.2.2.3.cmml"><mrow id="S3.p2.9.m4.1.1.1.1" xref="S3.p2.9.m4.1.1.1.1.cmml"><mn id="S3.p2.9.m4.1.1.1.1.2" xref="S3.p2.9.m4.1.1.1.1.2.cmml">0</mn><mo id="S3.p2.9.m4.1.1.1.1.1" xref="S3.p2.9.m4.1.1.1.1.1.cmml">≤</mo><msub id="S3.p2.9.m4.1.1.1.1.3" xref="S3.p2.9.m4.1.1.1.1.3.cmml"><mi id="S3.p2.9.m4.1.1.1.1.3.2" xref="S3.p2.9.m4.1.1.1.1.3.2.cmml">f</mi><mi id="S3.p2.9.m4.1.1.1.1.3.3" xref="S3.p2.9.m4.1.1.1.1.3.3.cmml">n</mi></msub></mrow><mo id="S3.p2.9.m4.2.2.2.3" xref="S3.p2.9.m4.2.2.3a.cmml">,</mo><mrow id="S3.p2.9.m4.2.2.2.2" xref="S3.p2.9.m4.2.2.2.2.cmml"><msub id="S3.p2.9.m4.2.2.2.2.2" xref="S3.p2.9.m4.2.2.2.2.2.cmml"><mi id="S3.p2.9.m4.2.2.2.2.2.2" xref="S3.p2.9.m4.2.2.2.2.2.2.cmml">g</mi><mi id="S3.p2.9.m4.2.2.2.2.2.3" xref="S3.p2.9.m4.2.2.2.2.2.3.cmml">n</mi></msub><mo id="S3.p2.9.m4.2.2.2.2.1" xref="S3.p2.9.m4.2.2.2.2.1.cmml">≤</mo><mn id="S3.p2.9.m4.2.2.2.2.3" xref="S3.p2.9.m4.2.2.2.2.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.9.m4.2b"><apply id="S3.p2.9.m4.2.2.3.cmml" xref="S3.p2.9.m4.2.2.2"><csymbol cd="ambiguous" id="S3.p2.9.m4.2.2.3a.cmml" xref="S3.p2.9.m4.2.2.2.3">formulae-sequence</csymbol><apply id="S3.p2.9.m4.1.1.1.1.cmml" xref="S3.p2.9.m4.1.1.1.1"><leq id="S3.p2.9.m4.1.1.1.1.1.cmml" xref="S3.p2.9.m4.1.1.1.1.1"></leq><cn id="S3.p2.9.m4.1.1.1.1.2.cmml" type="integer" xref="S3.p2.9.m4.1.1.1.1.2">0</cn><apply id="S3.p2.9.m4.1.1.1.1.3.cmml" xref="S3.p2.9.m4.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.p2.9.m4.1.1.1.1.3.1.cmml" xref="S3.p2.9.m4.1.1.1.1.3">subscript</csymbol><ci id="S3.p2.9.m4.1.1.1.1.3.2.cmml" xref="S3.p2.9.m4.1.1.1.1.3.2">𝑓</ci><ci id="S3.p2.9.m4.1.1.1.1.3.3.cmml" xref="S3.p2.9.m4.1.1.1.1.3.3">𝑛</ci></apply></apply><apply id="S3.p2.9.m4.2.2.2.2.cmml" xref="S3.p2.9.m4.2.2.2.2"><leq id="S3.p2.9.m4.2.2.2.2.1.cmml" xref="S3.p2.9.m4.2.2.2.2.1"></leq><apply id="S3.p2.9.m4.2.2.2.2.2.cmml" xref="S3.p2.9.m4.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.p2.9.m4.2.2.2.2.2.1.cmml" xref="S3.p2.9.m4.2.2.2.2.2">subscript</csymbol><ci id="S3.p2.9.m4.2.2.2.2.2.2.cmml" xref="S3.p2.9.m4.2.2.2.2.2.2">𝑔</ci><ci id="S3.p2.9.m4.2.2.2.2.2.3.cmml" xref="S3.p2.9.m4.2.2.2.2.2.3">𝑛</ci></apply><cn id="S3.p2.9.m4.2.2.2.2.3.cmml" type="integer" xref="S3.p2.9.m4.2.2.2.2.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.9.m4.2c">0\leq f_{n},g_{n}\leq 1</annotation><annotation encoding="application/x-llamapun" id="S3.p2.9.m4.2d">0 ≤ italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_g start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ≤ 1</annotation></semantics></math>). The graviton-loop integrand contribution to the one-loop amplitude would then be given by</p> <table class="ltx_equation ltx_eqn_table" id="S3.E26"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{I}^{\textrm{graviton}}_{N}(\ell)=\tilde{K}^{\mu\nu}_{1\ldots N\mu\nu}% (\ell)," class="ltx_Math" display="block" id="S3.E26.m1.3"><semantics id="S3.E26.m1.3a"><mrow id="S3.E26.m1.3.3.1" xref="S3.E26.m1.3.3.1.1.cmml"><mrow id="S3.E26.m1.3.3.1.1" xref="S3.E26.m1.3.3.1.1.cmml"><mrow id="S3.E26.m1.3.3.1.1.2" xref="S3.E26.m1.3.3.1.1.2.cmml"><msubsup id="S3.E26.m1.3.3.1.1.2.2" xref="S3.E26.m1.3.3.1.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E26.m1.3.3.1.1.2.2.2.2" xref="S3.E26.m1.3.3.1.1.2.2.2.2.cmml">ℐ</mi><mi id="S3.E26.m1.3.3.1.1.2.2.3" xref="S3.E26.m1.3.3.1.1.2.2.3.cmml">N</mi><mtext id="S3.E26.m1.3.3.1.1.2.2.2.3" xref="S3.E26.m1.3.3.1.1.2.2.2.3a.cmml">graviton</mtext></msubsup><mo id="S3.E26.m1.3.3.1.1.2.1" xref="S3.E26.m1.3.3.1.1.2.1.cmml">⁢</mo><mrow id="S3.E26.m1.3.3.1.1.2.3.2" xref="S3.E26.m1.3.3.1.1.2.cmml"><mo id="S3.E26.m1.3.3.1.1.2.3.2.1" stretchy="false" xref="S3.E26.m1.3.3.1.1.2.cmml">(</mo><mi id="S3.E26.m1.1.1" mathvariant="normal" xref="S3.E26.m1.1.1.cmml">ℓ</mi><mo id="S3.E26.m1.3.3.1.1.2.3.2.2" stretchy="false" xref="S3.E26.m1.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.E26.m1.3.3.1.1.1" xref="S3.E26.m1.3.3.1.1.1.cmml">=</mo><mrow id="S3.E26.m1.3.3.1.1.3" xref="S3.E26.m1.3.3.1.1.3.cmml"><msubsup id="S3.E26.m1.3.3.1.1.3.2" xref="S3.E26.m1.3.3.1.1.3.2.cmml"><mover accent="true" id="S3.E26.m1.3.3.1.1.3.2.2.2" xref="S3.E26.m1.3.3.1.1.3.2.2.2.cmml"><mi id="S3.E26.m1.3.3.1.1.3.2.2.2.2" xref="S3.E26.m1.3.3.1.1.3.2.2.2.2.cmml">K</mi><mo id="S3.E26.m1.3.3.1.1.3.2.2.2.1" xref="S3.E26.m1.3.3.1.1.3.2.2.2.1.cmml">~</mo></mover><mrow id="S3.E26.m1.3.3.1.1.3.2.3" xref="S3.E26.m1.3.3.1.1.3.2.3.cmml"><mn id="S3.E26.m1.3.3.1.1.3.2.3.2" xref="S3.E26.m1.3.3.1.1.3.2.3.2.cmml">1</mn><mo id="S3.E26.m1.3.3.1.1.3.2.3.1" xref="S3.E26.m1.3.3.1.1.3.2.3.1.cmml">⁢</mo><mi id="S3.E26.m1.3.3.1.1.3.2.3.3" mathvariant="normal" xref="S3.E26.m1.3.3.1.1.3.2.3.3.cmml">…</mi><mo id="S3.E26.m1.3.3.1.1.3.2.3.1a" xref="S3.E26.m1.3.3.1.1.3.2.3.1.cmml">⁢</mo><mi id="S3.E26.m1.3.3.1.1.3.2.3.4" xref="S3.E26.m1.3.3.1.1.3.2.3.4.cmml">N</mi><mo id="S3.E26.m1.3.3.1.1.3.2.3.1b" xref="S3.E26.m1.3.3.1.1.3.2.3.1.cmml">⁢</mo><mi id="S3.E26.m1.3.3.1.1.3.2.3.5" xref="S3.E26.m1.3.3.1.1.3.2.3.5.cmml">μ</mi><mo id="S3.E26.m1.3.3.1.1.3.2.3.1c" xref="S3.E26.m1.3.3.1.1.3.2.3.1.cmml">⁢</mo><mi id="S3.E26.m1.3.3.1.1.3.2.3.6" xref="S3.E26.m1.3.3.1.1.3.2.3.6.cmml">ν</mi></mrow><mrow id="S3.E26.m1.3.3.1.1.3.2.2.3" xref="S3.E26.m1.3.3.1.1.3.2.2.3.cmml"><mi id="S3.E26.m1.3.3.1.1.3.2.2.3.2" xref="S3.E26.m1.3.3.1.1.3.2.2.3.2.cmml">μ</mi><mo id="S3.E26.m1.3.3.1.1.3.2.2.3.1" xref="S3.E26.m1.3.3.1.1.3.2.2.3.1.cmml">⁢</mo><mi id="S3.E26.m1.3.3.1.1.3.2.2.3.3" xref="S3.E26.m1.3.3.1.1.3.2.2.3.3.cmml">ν</mi></mrow></msubsup><mo id="S3.E26.m1.3.3.1.1.3.1" xref="S3.E26.m1.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S3.E26.m1.3.3.1.1.3.3.2" xref="S3.E26.m1.3.3.1.1.3.cmml"><mo id="S3.E26.m1.3.3.1.1.3.3.2.1" stretchy="false" xref="S3.E26.m1.3.3.1.1.3.cmml">(</mo><mi id="S3.E26.m1.2.2" mathvariant="normal" xref="S3.E26.m1.2.2.cmml">ℓ</mi><mo id="S3.E26.m1.3.3.1.1.3.3.2.2" stretchy="false" xref="S3.E26.m1.3.3.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S3.E26.m1.3.3.1.2" xref="S3.E26.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E26.m1.3b"><apply id="S3.E26.m1.3.3.1.1.cmml" xref="S3.E26.m1.3.3.1"><eq id="S3.E26.m1.3.3.1.1.1.cmml" xref="S3.E26.m1.3.3.1.1.1"></eq><apply id="S3.E26.m1.3.3.1.1.2.cmml" xref="S3.E26.m1.3.3.1.1.2"><times id="S3.E26.m1.3.3.1.1.2.1.cmml" xref="S3.E26.m1.3.3.1.1.2.1"></times><apply id="S3.E26.m1.3.3.1.1.2.2.cmml" xref="S3.E26.m1.3.3.1.1.2.2"><csymbol cd="ambiguous" id="S3.E26.m1.3.3.1.1.2.2.1.cmml" xref="S3.E26.m1.3.3.1.1.2.2">subscript</csymbol><apply id="S3.E26.m1.3.3.1.1.2.2.2.cmml" xref="S3.E26.m1.3.3.1.1.2.2"><csymbol cd="ambiguous" id="S3.E26.m1.3.3.1.1.2.2.2.1.cmml" xref="S3.E26.m1.3.3.1.1.2.2">superscript</csymbol><ci id="S3.E26.m1.3.3.1.1.2.2.2.2.cmml" xref="S3.E26.m1.3.3.1.1.2.2.2.2">ℐ</ci><ci id="S3.E26.m1.3.3.1.1.2.2.2.3a.cmml" xref="S3.E26.m1.3.3.1.1.2.2.2.3"><mtext id="S3.E26.m1.3.3.1.1.2.2.2.3.cmml" mathsize="70%" xref="S3.E26.m1.3.3.1.1.2.2.2.3">graviton</mtext></ci></apply><ci id="S3.E26.m1.3.3.1.1.2.2.3.cmml" xref="S3.E26.m1.3.3.1.1.2.2.3">𝑁</ci></apply><ci id="S3.E26.m1.1.1.cmml" xref="S3.E26.m1.1.1">ℓ</ci></apply><apply id="S3.E26.m1.3.3.1.1.3.cmml" xref="S3.E26.m1.3.3.1.1.3"><times id="S3.E26.m1.3.3.1.1.3.1.cmml" xref="S3.E26.m1.3.3.1.1.3.1"></times><apply id="S3.E26.m1.3.3.1.1.3.2.cmml" xref="S3.E26.m1.3.3.1.1.3.2"><csymbol cd="ambiguous" id="S3.E26.m1.3.3.1.1.3.2.1.cmml" xref="S3.E26.m1.3.3.1.1.3.2">subscript</csymbol><apply id="S3.E26.m1.3.3.1.1.3.2.2.cmml" xref="S3.E26.m1.3.3.1.1.3.2"><csymbol cd="ambiguous" id="S3.E26.m1.3.3.1.1.3.2.2.1.cmml" xref="S3.E26.m1.3.3.1.1.3.2">superscript</csymbol><apply id="S3.E26.m1.3.3.1.1.3.2.2.2.cmml" xref="S3.E26.m1.3.3.1.1.3.2.2.2"><ci id="S3.E26.m1.3.3.1.1.3.2.2.2.1.cmml" xref="S3.E26.m1.3.3.1.1.3.2.2.2.1">~</ci><ci id="S3.E26.m1.3.3.1.1.3.2.2.2.2.cmml" xref="S3.E26.m1.3.3.1.1.3.2.2.2.2">𝐾</ci></apply><apply id="S3.E26.m1.3.3.1.1.3.2.2.3.cmml" xref="S3.E26.m1.3.3.1.1.3.2.2.3"><times id="S3.E26.m1.3.3.1.1.3.2.2.3.1.cmml" xref="S3.E26.m1.3.3.1.1.3.2.2.3.1"></times><ci id="S3.E26.m1.3.3.1.1.3.2.2.3.2.cmml" xref="S3.E26.m1.3.3.1.1.3.2.2.3.2">𝜇</ci><ci id="S3.E26.m1.3.3.1.1.3.2.2.3.3.cmml" xref="S3.E26.m1.3.3.1.1.3.2.2.3.3">𝜈</ci></apply></apply><apply id="S3.E26.m1.3.3.1.1.3.2.3.cmml" xref="S3.E26.m1.3.3.1.1.3.2.3"><times id="S3.E26.m1.3.3.1.1.3.2.3.1.cmml" xref="S3.E26.m1.3.3.1.1.3.2.3.1"></times><cn id="S3.E26.m1.3.3.1.1.3.2.3.2.cmml" type="integer" xref="S3.E26.m1.3.3.1.1.3.2.3.2">1</cn><ci id="S3.E26.m1.3.3.1.1.3.2.3.3.cmml" xref="S3.E26.m1.3.3.1.1.3.2.3.3">…</ci><ci id="S3.E26.m1.3.3.1.1.3.2.3.4.cmml" xref="S3.E26.m1.3.3.1.1.3.2.3.4">𝑁</ci><ci id="S3.E26.m1.3.3.1.1.3.2.3.5.cmml" xref="S3.E26.m1.3.3.1.1.3.2.3.5">𝜇</ci><ci id="S3.E26.m1.3.3.1.1.3.2.3.6.cmml" xref="S3.E26.m1.3.3.1.1.3.2.3.6">𝜈</ci></apply></apply><ci id="S3.E26.m1.2.2.cmml" xref="S3.E26.m1.2.2">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E26.m1.3c">\mathcal{I}^{\textrm{graviton}}_{N}(\ell)=\tilde{K}^{\mu\nu}_{1\ldots N\mu\nu}% (\ell),</annotation><annotation encoding="application/x-llamapun" id="S3.E26.m1.3d">caligraphic_I start_POSTSUPERSCRIPT graviton end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( roman_ℓ ) = over~ start_ARG italic_K end_ARG start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 … italic_N italic_μ italic_ν end_POSTSUBSCRIPT ( roman_ℓ ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(26)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p2.10">with conserved momentum <math alttext="k_{1\ldots N}=0" class="ltx_Math" display="inline" id="S3.p2.10.m1.1"><semantics id="S3.p2.10.m1.1a"><mrow id="S3.p2.10.m1.1.1" xref="S3.p2.10.m1.1.1.cmml"><msub id="S3.p2.10.m1.1.1.2" xref="S3.p2.10.m1.1.1.2.cmml"><mi id="S3.p2.10.m1.1.1.2.2" xref="S3.p2.10.m1.1.1.2.2.cmml">k</mi><mrow id="S3.p2.10.m1.1.1.2.3" xref="S3.p2.10.m1.1.1.2.3.cmml"><mn id="S3.p2.10.m1.1.1.2.3.2" xref="S3.p2.10.m1.1.1.2.3.2.cmml">1</mn><mo id="S3.p2.10.m1.1.1.2.3.1" xref="S3.p2.10.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S3.p2.10.m1.1.1.2.3.3" mathvariant="normal" xref="S3.p2.10.m1.1.1.2.3.3.cmml">…</mi><mo id="S3.p2.10.m1.1.1.2.3.1a" xref="S3.p2.10.m1.1.1.2.3.1.cmml">⁢</mo><mi id="S3.p2.10.m1.1.1.2.3.4" xref="S3.p2.10.m1.1.1.2.3.4.cmml">N</mi></mrow></msub><mo id="S3.p2.10.m1.1.1.1" xref="S3.p2.10.m1.1.1.1.cmml">=</mo><mn id="S3.p2.10.m1.1.1.3" xref="S3.p2.10.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.10.m1.1b"><apply id="S3.p2.10.m1.1.1.cmml" xref="S3.p2.10.m1.1.1"><eq id="S3.p2.10.m1.1.1.1.cmml" xref="S3.p2.10.m1.1.1.1"></eq><apply id="S3.p2.10.m1.1.1.2.cmml" xref="S3.p2.10.m1.1.1.2"><csymbol cd="ambiguous" id="S3.p2.10.m1.1.1.2.1.cmml" xref="S3.p2.10.m1.1.1.2">subscript</csymbol><ci id="S3.p2.10.m1.1.1.2.2.cmml" xref="S3.p2.10.m1.1.1.2.2">𝑘</ci><apply id="S3.p2.10.m1.1.1.2.3.cmml" xref="S3.p2.10.m1.1.1.2.3"><times id="S3.p2.10.m1.1.1.2.3.1.cmml" xref="S3.p2.10.m1.1.1.2.3.1"></times><cn id="S3.p2.10.m1.1.1.2.3.2.cmml" type="integer" xref="S3.p2.10.m1.1.1.2.3.2">1</cn><ci id="S3.p2.10.m1.1.1.2.3.3.cmml" xref="S3.p2.10.m1.1.1.2.3.3">…</ci><ci id="S3.p2.10.m1.1.1.2.3.4.cmml" xref="S3.p2.10.m1.1.1.2.3.4">𝑁</ci></apply></apply><cn id="S3.p2.10.m1.1.1.3.cmml" type="integer" xref="S3.p2.10.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.10.m1.1c">k_{1\ldots N}=0</annotation><annotation encoding="application/x-llamapun" id="S3.p2.10.m1.1d">italic_k start_POSTSUBSCRIPT 1 … italic_N end_POSTSUBSCRIPT = 0</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.p3"> <p class="ltx_p" id="S3.p3.2">The fact that there exist solutions to <math alttext="f_{n}" class="ltx_Math" display="inline" id="S3.p3.1.m1.1"><semantics id="S3.p3.1.m1.1a"><msub id="S3.p3.1.m1.1.1" xref="S3.p3.1.m1.1.1.cmml"><mi id="S3.p3.1.m1.1.1.2" xref="S3.p3.1.m1.1.1.2.cmml">f</mi><mi id="S3.p3.1.m1.1.1.3" xref="S3.p3.1.m1.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p3.1.m1.1b"><apply id="S3.p3.1.m1.1.1.cmml" xref="S3.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p3.1.m1.1.1.1.cmml" xref="S3.p3.1.m1.1.1">subscript</csymbol><ci id="S3.p3.1.m1.1.1.2.cmml" xref="S3.p3.1.m1.1.1.2">𝑓</ci><ci id="S3.p3.1.m1.1.1.3.cmml" xref="S3.p3.1.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.1.m1.1c">f_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.1.m1.1d">italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="g_{n}" class="ltx_Math" display="inline" id="S3.p3.2.m2.1"><semantics id="S3.p3.2.m2.1a"><msub id="S3.p3.2.m2.1.1" xref="S3.p3.2.m2.1.1.cmml"><mi id="S3.p3.2.m2.1.1.2" xref="S3.p3.2.m2.1.1.2.cmml">g</mi><mi id="S3.p3.2.m2.1.1.3" xref="S3.p3.2.m2.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p3.2.m2.1b"><apply id="S3.p3.2.m2.1.1.cmml" xref="S3.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S3.p3.2.m2.1.1.1.cmml" xref="S3.p3.2.m2.1.1">subscript</csymbol><ci id="S3.p3.2.m2.1.1.2.cmml" xref="S3.p3.2.m2.1.1.2">𝑔</ci><ci id="S3.p3.2.m2.1.1.3.cmml" xref="S3.p3.2.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.2.m2.1c">g_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.2.m2.1d">italic_g start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> is noteworthy. Yet more remarkably, they take a simple form:</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S5.EGx3"> <tbody id="S3.E27"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle f_{n}=\frac{1}{2}," class="ltx_Math" display="inline" id="S3.E27.m1.1"><semantics id="S3.E27.m1.1a"><mrow id="S3.E27.m1.1.1.1" xref="S3.E27.m1.1.1.1.1.cmml"><mrow id="S3.E27.m1.1.1.1.1" xref="S3.E27.m1.1.1.1.1.cmml"><msub id="S3.E27.m1.1.1.1.1.2" xref="S3.E27.m1.1.1.1.1.2.cmml"><mi id="S3.E27.m1.1.1.1.1.2.2" xref="S3.E27.m1.1.1.1.1.2.2.cmml">f</mi><mi id="S3.E27.m1.1.1.1.1.2.3" xref="S3.E27.m1.1.1.1.1.2.3.cmml">n</mi></msub><mo id="S3.E27.m1.1.1.1.1.1" xref="S3.E27.m1.1.1.1.1.1.cmml">=</mo><mstyle displaystyle="true" id="S3.E27.m1.1.1.1.1.3" xref="S3.E27.m1.1.1.1.1.3.cmml"><mfrac id="S3.E27.m1.1.1.1.1.3a" xref="S3.E27.m1.1.1.1.1.3.cmml"><mn id="S3.E27.m1.1.1.1.1.3.2" xref="S3.E27.m1.1.1.1.1.3.2.cmml">1</mn><mn id="S3.E27.m1.1.1.1.1.3.3" xref="S3.E27.m1.1.1.1.1.3.3.cmml">2</mn></mfrac></mstyle></mrow><mo id="S3.E27.m1.1.1.1.2" xref="S3.E27.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E27.m1.1b"><apply id="S3.E27.m1.1.1.1.1.cmml" xref="S3.E27.m1.1.1.1"><eq id="S3.E27.m1.1.1.1.1.1.cmml" xref="S3.E27.m1.1.1.1.1.1"></eq><apply id="S3.E27.m1.1.1.1.1.2.cmml" xref="S3.E27.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.E27.m1.1.1.1.1.2.1.cmml" xref="S3.E27.m1.1.1.1.1.2">subscript</csymbol><ci id="S3.E27.m1.1.1.1.1.2.2.cmml" xref="S3.E27.m1.1.1.1.1.2.2">𝑓</ci><ci id="S3.E27.m1.1.1.1.1.2.3.cmml" xref="S3.E27.m1.1.1.1.1.2.3">𝑛</ci></apply><apply id="S3.E27.m1.1.1.1.1.3.cmml" xref="S3.E27.m1.1.1.1.1.3"><divide id="S3.E27.m1.1.1.1.1.3.1.cmml" xref="S3.E27.m1.1.1.1.1.3"></divide><cn id="S3.E27.m1.1.1.1.1.3.2.cmml" type="integer" xref="S3.E27.m1.1.1.1.1.3.2">1</cn><cn id="S3.E27.m1.1.1.1.1.3.3.cmml" type="integer" xref="S3.E27.m1.1.1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E27.m1.1c">\displaystyle f_{n}=\frac{1}{2},</annotation><annotation encoding="application/x-llamapun" id="S3.E27.m1.1d">italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ,</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle g_{n}=\frac{|P_{n}|}{|P|}." class="ltx_Math" display="inline" id="S3.E27.m3.3"><semantics id="S3.E27.m3.3a"><mrow id="S3.E27.m3.3.3.1" xref="S3.E27.m3.3.3.1.1.cmml"><mrow id="S3.E27.m3.3.3.1.1" xref="S3.E27.m3.3.3.1.1.cmml"><msub id="S3.E27.m3.3.3.1.1.2" xref="S3.E27.m3.3.3.1.1.2.cmml"><mi id="S3.E27.m3.3.3.1.1.2.2" xref="S3.E27.m3.3.3.1.1.2.2.cmml">g</mi><mi id="S3.E27.m3.3.3.1.1.2.3" xref="S3.E27.m3.3.3.1.1.2.3.cmml">n</mi></msub><mo id="S3.E27.m3.3.3.1.1.1" xref="S3.E27.m3.3.3.1.1.1.cmml">=</mo><mstyle displaystyle="true" id="S3.E27.m3.2.2" xref="S3.E27.m3.2.2.cmml"><mfrac id="S3.E27.m3.2.2a" xref="S3.E27.m3.2.2.cmml"><mrow id="S3.E27.m3.1.1.1.1" xref="S3.E27.m3.1.1.1.2.cmml"><mo id="S3.E27.m3.1.1.1.1.2" stretchy="false" xref="S3.E27.m3.1.1.1.2.1.cmml">|</mo><msub id="S3.E27.m3.1.1.1.1.1" xref="S3.E27.m3.1.1.1.1.1.cmml"><mi id="S3.E27.m3.1.1.1.1.1.2" xref="S3.E27.m3.1.1.1.1.1.2.cmml">P</mi><mi id="S3.E27.m3.1.1.1.1.1.3" xref="S3.E27.m3.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S3.E27.m3.1.1.1.1.3" stretchy="false" xref="S3.E27.m3.1.1.1.2.1.cmml">|</mo></mrow><mrow id="S3.E27.m3.2.2.2.3" xref="S3.E27.m3.2.2.2.2.cmml"><mo id="S3.E27.m3.2.2.2.3.1" stretchy="false" xref="S3.E27.m3.2.2.2.2.1.cmml">|</mo><mi id="S3.E27.m3.2.2.2.1" xref="S3.E27.m3.2.2.2.1.cmml">P</mi><mo id="S3.E27.m3.2.2.2.3.2" stretchy="false" xref="S3.E27.m3.2.2.2.2.1.cmml">|</mo></mrow></mfrac></mstyle></mrow><mo id="S3.E27.m3.3.3.1.2" lspace="0em" xref="S3.E27.m3.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E27.m3.3b"><apply id="S3.E27.m3.3.3.1.1.cmml" xref="S3.E27.m3.3.3.1"><eq id="S3.E27.m3.3.3.1.1.1.cmml" xref="S3.E27.m3.3.3.1.1.1"></eq><apply id="S3.E27.m3.3.3.1.1.2.cmml" xref="S3.E27.m3.3.3.1.1.2"><csymbol cd="ambiguous" id="S3.E27.m3.3.3.1.1.2.1.cmml" xref="S3.E27.m3.3.3.1.1.2">subscript</csymbol><ci id="S3.E27.m3.3.3.1.1.2.2.cmml" xref="S3.E27.m3.3.3.1.1.2.2">𝑔</ci><ci id="S3.E27.m3.3.3.1.1.2.3.cmml" xref="S3.E27.m3.3.3.1.1.2.3">𝑛</ci></apply><apply id="S3.E27.m3.2.2.cmml" xref="S3.E27.m3.2.2"><divide id="S3.E27.m3.2.2.3.cmml" xref="S3.E27.m3.2.2"></divide><apply id="S3.E27.m3.1.1.1.2.cmml" xref="S3.E27.m3.1.1.1.1"><abs id="S3.E27.m3.1.1.1.2.1.cmml" xref="S3.E27.m3.1.1.1.1.2"></abs><apply id="S3.E27.m3.1.1.1.1.1.cmml" xref="S3.E27.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.E27.m3.1.1.1.1.1.1.cmml" xref="S3.E27.m3.1.1.1.1.1">subscript</csymbol><ci id="S3.E27.m3.1.1.1.1.1.2.cmml" xref="S3.E27.m3.1.1.1.1.1.2">𝑃</ci><ci id="S3.E27.m3.1.1.1.1.1.3.cmml" xref="S3.E27.m3.1.1.1.1.1.3">𝑛</ci></apply></apply><apply id="S3.E27.m3.2.2.2.2.cmml" xref="S3.E27.m3.2.2.2.3"><abs id="S3.E27.m3.2.2.2.2.1.cmml" xref="S3.E27.m3.2.2.2.3.1"></abs><ci id="S3.E27.m3.2.2.2.1.cmml" xref="S3.E27.m3.2.2.2.1">𝑃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E27.m3.3c">\displaystyle g_{n}=\frac{|P_{n}|}{|P|}.</annotation><annotation encoding="application/x-llamapun" id="S3.E27.m3.3d">italic_g start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = divide start_ARG | italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | end_ARG start_ARG | italic_P | end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(27)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p3.5">In (<a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S3.E25" title="In III Loop recursions ‣ One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_tag">25</span></a>), the coefficients <math alttext="f_{n}" class="ltx_Math" display="inline" id="S3.p3.3.m1.1"><semantics id="S3.p3.3.m1.1a"><msub id="S3.p3.3.m1.1.1" xref="S3.p3.3.m1.1.1.cmml"><mi id="S3.p3.3.m1.1.1.2" xref="S3.p3.3.m1.1.1.2.cmml">f</mi><mi id="S3.p3.3.m1.1.1.3" xref="S3.p3.3.m1.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p3.3.m1.1b"><apply id="S3.p3.3.m1.1.1.cmml" xref="S3.p3.3.m1.1.1"><csymbol cd="ambiguous" id="S3.p3.3.m1.1.1.1.cmml" xref="S3.p3.3.m1.1.1">subscript</csymbol><ci id="S3.p3.3.m1.1.1.2.cmml" xref="S3.p3.3.m1.1.1.2">𝑓</ci><ci id="S3.p3.3.m1.1.1.3.cmml" xref="S3.p3.3.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.3.m1.1c">f_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.3.m1.1d">italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> accompany tadpole diagrams, in which the loop is formed via the sewing of two legs of the same vertex. The symmetry factor of such diagrams is simply <math alttext="1/2" class="ltx_Math" display="inline" id="S3.p3.4.m2.1"><semantics id="S3.p3.4.m2.1a"><mrow id="S3.p3.4.m2.1.1" xref="S3.p3.4.m2.1.1.cmml"><mn id="S3.p3.4.m2.1.1.2" xref="S3.p3.4.m2.1.1.2.cmml">1</mn><mo id="S3.p3.4.m2.1.1.1" xref="S3.p3.4.m2.1.1.1.cmml">/</mo><mn id="S3.p3.4.m2.1.1.3" xref="S3.p3.4.m2.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.4.m2.1b"><apply id="S3.p3.4.m2.1.1.cmml" xref="S3.p3.4.m2.1.1"><divide id="S3.p3.4.m2.1.1.1.cmml" xref="S3.p3.4.m2.1.1.1"></divide><cn id="S3.p3.4.m2.1.1.2.cmml" type="integer" xref="S3.p3.4.m2.1.1.2">1</cn><cn id="S3.p3.4.m2.1.1.3.cmml" type="integer" xref="S3.p3.4.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.4.m2.1c">1/2</annotation><annotation encoding="application/x-llamapun" id="S3.p3.4.m2.1d">1 / 2</annotation></semantics></math>. It is the same factor of the so-called bubble diagrams, in which the loop contains only two vertices. The solution for <math alttext="g_{n}" class="ltx_Math" display="inline" id="S3.p3.5.m3.1"><semantics id="S3.p3.5.m3.1a"><msub id="S3.p3.5.m3.1.1" xref="S3.p3.5.m3.1.1.cmml"><mi id="S3.p3.5.m3.1.1.2" xref="S3.p3.5.m3.1.1.2.cmml">g</mi><mi id="S3.p3.5.m3.1.1.3" xref="S3.p3.5.m3.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p3.5.m3.1b"><apply id="S3.p3.5.m3.1.1.cmml" xref="S3.p3.5.m3.1.1"><csymbol cd="ambiguous" id="S3.p3.5.m3.1.1.1.cmml" xref="S3.p3.5.m3.1.1">subscript</csymbol><ci id="S3.p3.5.m3.1.1.2.cmml" xref="S3.p3.5.m3.1.1.2">𝑔</ci><ci id="S3.p3.5.m3.1.1.3.cmml" xref="S3.p3.5.m3.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.5.m3.1c">g_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.5.m3.1d">italic_g start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> takes care of the repeated diagrams that would otherwise be generated when using (<a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S3.E24" title="In III Loop recursions ‣ One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_tag">24</span></a>).</p> </div> <div class="ltx_para" id="S3.p4"> <p class="ltx_p" id="S3.p4.1">Now, in order to compute the ghost loop, we need first the related tree level recursions. Their derivation from the respective equations of motion is straightforward. For the <math alttext="c" class="ltx_Math" display="inline" id="S3.p4.1.m1.1"><semantics id="S3.p4.1.m1.1a"><mi id="S3.p4.1.m1.1.1" xref="S3.p4.1.m1.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.p4.1.m1.1b"><ci id="S3.p4.1.m1.1.1.cmml" xref="S3.p4.1.m1.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.1.m1.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.p4.1.m1.1d">italic_c</annotation></semantics></math> ghost, for instance, we obtain</p> <table class="ltx_equation ltx_eqn_table" id="S3.E28"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="s_{P}C^{\rho}_{P}=\sum_{P=Q\cup R}I_{Q}^{\mu\nu}C_{R}^{\sigma}\mathcal{V}^{(% \textrm{g})\rho}_{\mu\nu\sigma}(Q,P)," class="ltx_Math" display="block" id="S3.E28.m1.4"><semantics id="S3.E28.m1.4a"><mrow id="S3.E28.m1.4.4.1" 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xref="S3.E28.m1.2.2">𝑄</ci><ci id="S3.E28.m1.3.3.cmml" xref="S3.E28.m1.3.3">𝑃</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E28.m1.4c">s_{P}C^{\rho}_{P}=\sum_{P=Q\cup R}I_{Q}^{\mu\nu}C_{R}^{\sigma}\mathcal{V}^{(% \textrm{g})\rho}_{\mu\nu\sigma}(Q,P),</annotation><annotation encoding="application/x-llamapun" id="S3.E28.m1.4d">italic_s start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT italic_C start_POSTSUPERSCRIPT italic_ρ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_P = italic_Q ∪ italic_R end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT italic_C start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT caligraphic_V start_POSTSUPERSCRIPT ( g ) italic_ρ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ italic_ν italic_σ end_POSTSUBSCRIPT ( italic_Q , italic_P ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(28)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p4.3">The field theory vertex <math alttext="\mathcal{V}^{(\textrm{g})\rho}_{\mu\nu\sigma}" class="ltx_Math" display="inline" id="S3.p4.2.m1.1"><semantics id="S3.p4.2.m1.1a"><msubsup id="S3.p4.2.m1.1.2" xref="S3.p4.2.m1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p4.2.m1.1.2.2.2" xref="S3.p4.2.m1.1.2.2.2.cmml">𝒱</mi><mrow id="S3.p4.2.m1.1.2.3" xref="S3.p4.2.m1.1.2.3.cmml"><mi id="S3.p4.2.m1.1.2.3.2" xref="S3.p4.2.m1.1.2.3.2.cmml">μ</mi><mo id="S3.p4.2.m1.1.2.3.1" xref="S3.p4.2.m1.1.2.3.1.cmml">⁢</mo><mi id="S3.p4.2.m1.1.2.3.3" xref="S3.p4.2.m1.1.2.3.3.cmml">ν</mi><mo id="S3.p4.2.m1.1.2.3.1a" xref="S3.p4.2.m1.1.2.3.1.cmml">⁢</mo><mi id="S3.p4.2.m1.1.2.3.4" xref="S3.p4.2.m1.1.2.3.4.cmml">σ</mi></mrow><mrow id="S3.p4.2.m1.1.1.1" xref="S3.p4.2.m1.1.1.1.cmml"><mrow id="S3.p4.2.m1.1.1.1.3.2" xref="S3.p4.2.m1.1.1.1.1a.cmml"><mo id="S3.p4.2.m1.1.1.1.3.2.1" stretchy="false" xref="S3.p4.2.m1.1.1.1.1a.cmml">(</mo><mtext id="S3.p4.2.m1.1.1.1.1" xref="S3.p4.2.m1.1.1.1.1.cmml">g</mtext><mo id="S3.p4.2.m1.1.1.1.3.2.2" stretchy="false" xref="S3.p4.2.m1.1.1.1.1a.cmml">)</mo></mrow><mo id="S3.p4.2.m1.1.1.1.2" xref="S3.p4.2.m1.1.1.1.2.cmml">⁢</mo><mi id="S3.p4.2.m1.1.1.1.4" xref="S3.p4.2.m1.1.1.1.4.cmml">ρ</mi></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.p4.2.m1.1b"><apply id="S3.p4.2.m1.1.2.cmml" xref="S3.p4.2.m1.1.2"><csymbol cd="ambiguous" id="S3.p4.2.m1.1.2.1.cmml" xref="S3.p4.2.m1.1.2">subscript</csymbol><apply id="S3.p4.2.m1.1.2.2.cmml" xref="S3.p4.2.m1.1.2"><csymbol cd="ambiguous" id="S3.p4.2.m1.1.2.2.1.cmml" xref="S3.p4.2.m1.1.2">superscript</csymbol><ci id="S3.p4.2.m1.1.2.2.2.cmml" xref="S3.p4.2.m1.1.2.2.2">𝒱</ci><apply id="S3.p4.2.m1.1.1.1.cmml" xref="S3.p4.2.m1.1.1.1"><times id="S3.p4.2.m1.1.1.1.2.cmml" xref="S3.p4.2.m1.1.1.1.2"></times><ci id="S3.p4.2.m1.1.1.1.1a.cmml" xref="S3.p4.2.m1.1.1.1.3.2"><mtext id="S3.p4.2.m1.1.1.1.1.cmml" mathsize="70%" xref="S3.p4.2.m1.1.1.1.1">g</mtext></ci><ci id="S3.p4.2.m1.1.1.1.4.cmml" xref="S3.p4.2.m1.1.1.1.4">𝜌</ci></apply></apply><apply id="S3.p4.2.m1.1.2.3.cmml" xref="S3.p4.2.m1.1.2.3"><times id="S3.p4.2.m1.1.2.3.1.cmml" xref="S3.p4.2.m1.1.2.3.1"></times><ci id="S3.p4.2.m1.1.2.3.2.cmml" xref="S3.p4.2.m1.1.2.3.2">𝜇</ci><ci id="S3.p4.2.m1.1.2.3.3.cmml" xref="S3.p4.2.m1.1.2.3.3">𝜈</ci><ci id="S3.p4.2.m1.1.2.3.4.cmml" xref="S3.p4.2.m1.1.2.3.4">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.2.m1.1c">\mathcal{V}^{(\textrm{g})\rho}_{\mu\nu\sigma}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.2.m1.1d">caligraphic_V start_POSTSUPERSCRIPT ( g ) italic_ρ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ italic_ν italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> is defined in (<a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S2.E19" title="In II Equations of motion ‣ One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_tag">19</span></a>), and there is a similar construction for <math alttext="B_{P}" class="ltx_Math" display="inline" id="S3.p4.3.m2.1"><semantics id="S3.p4.3.m2.1a"><msub id="S3.p4.3.m2.1.1" xref="S3.p4.3.m2.1.1.cmml"><mi id="S3.p4.3.m2.1.1.2" xref="S3.p4.3.m2.1.1.2.cmml">B</mi><mi id="S3.p4.3.m2.1.1.3" xref="S3.p4.3.m2.1.1.3.cmml">P</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p4.3.m2.1b"><apply id="S3.p4.3.m2.1.1.cmml" xref="S3.p4.3.m2.1.1"><csymbol cd="ambiguous" id="S3.p4.3.m2.1.1.1.cmml" xref="S3.p4.3.m2.1.1">subscript</csymbol><ci id="S3.p4.3.m2.1.1.2.cmml" xref="S3.p4.3.m2.1.1.2">𝐵</ci><ci id="S3.p4.3.m2.1.1.3.cmml" xref="S3.p4.3.m2.1.1.3">𝑃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.3.m2.1c">B_{P}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.3.m2.1d">italic_B start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.p5"> <p class="ltx_p" id="S3.p5.3">We then introduce the loop-closing leg <math alttext="\ell" class="ltx_Math" display="inline" id="S3.p5.1.m1.1"><semantics id="S3.p5.1.m1.1a"><mi id="S3.p5.1.m1.1.1" mathvariant="normal" xref="S3.p5.1.m1.1.1.cmml">ℓ</mi><annotation-xml encoding="MathML-Content" id="S3.p5.1.m1.1b"><ci id="S3.p5.1.m1.1.1.cmml" xref="S3.p5.1.m1.1.1">ℓ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.1.m1.1c">\ell</annotation><annotation encoding="application/x-llamapun" id="S3.p5.1.m1.1d">roman_ℓ</annotation></semantics></math> via the identification <math alttext="C^{\mu}_{\ell P}=J^{\mu}_{P\nu}c^{\nu}_{\ell}" class="ltx_Math" display="inline" id="S3.p5.2.m2.1"><semantics id="S3.p5.2.m2.1a"><mrow id="S3.p5.2.m2.1.1" xref="S3.p5.2.m2.1.1.cmml"><msubsup id="S3.p5.2.m2.1.1.2" xref="S3.p5.2.m2.1.1.2.cmml"><mi id="S3.p5.2.m2.1.1.2.2.2" xref="S3.p5.2.m2.1.1.2.2.2.cmml">C</mi><mrow id="S3.p5.2.m2.1.1.2.3" xref="S3.p5.2.m2.1.1.2.3.cmml"><mi id="S3.p5.2.m2.1.1.2.3.2" mathvariant="normal" xref="S3.p5.2.m2.1.1.2.3.2.cmml">ℓ</mi><mo id="S3.p5.2.m2.1.1.2.3.1" xref="S3.p5.2.m2.1.1.2.3.1.cmml">⁢</mo><mi id="S3.p5.2.m2.1.1.2.3.3" xref="S3.p5.2.m2.1.1.2.3.3.cmml">P</mi></mrow><mi id="S3.p5.2.m2.1.1.2.2.3" xref="S3.p5.2.m2.1.1.2.2.3.cmml">μ</mi></msubsup><mo id="S3.p5.2.m2.1.1.1" xref="S3.p5.2.m2.1.1.1.cmml">=</mo><mrow id="S3.p5.2.m2.1.1.3" xref="S3.p5.2.m2.1.1.3.cmml"><msubsup id="S3.p5.2.m2.1.1.3.2" xref="S3.p5.2.m2.1.1.3.2.cmml"><mi id="S3.p5.2.m2.1.1.3.2.2.2" xref="S3.p5.2.m2.1.1.3.2.2.2.cmml">J</mi><mrow id="S3.p5.2.m2.1.1.3.2.3" xref="S3.p5.2.m2.1.1.3.2.3.cmml"><mi id="S3.p5.2.m2.1.1.3.2.3.2" xref="S3.p5.2.m2.1.1.3.2.3.2.cmml">P</mi><mo id="S3.p5.2.m2.1.1.3.2.3.1" xref="S3.p5.2.m2.1.1.3.2.3.1.cmml">⁢</mo><mi id="S3.p5.2.m2.1.1.3.2.3.3" xref="S3.p5.2.m2.1.1.3.2.3.3.cmml">ν</mi></mrow><mi id="S3.p5.2.m2.1.1.3.2.2.3" xref="S3.p5.2.m2.1.1.3.2.2.3.cmml">μ</mi></msubsup><mo id="S3.p5.2.m2.1.1.3.1" xref="S3.p5.2.m2.1.1.3.1.cmml">⁢</mo><msubsup id="S3.p5.2.m2.1.1.3.3" xref="S3.p5.2.m2.1.1.3.3.cmml"><mi id="S3.p5.2.m2.1.1.3.3.2.2" xref="S3.p5.2.m2.1.1.3.3.2.2.cmml">c</mi><mi id="S3.p5.2.m2.1.1.3.3.3" mathvariant="normal" xref="S3.p5.2.m2.1.1.3.3.3.cmml">ℓ</mi><mi id="S3.p5.2.m2.1.1.3.3.2.3" xref="S3.p5.2.m2.1.1.3.3.2.3.cmml">ν</mi></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.2.m2.1b"><apply id="S3.p5.2.m2.1.1.cmml" xref="S3.p5.2.m2.1.1"><eq id="S3.p5.2.m2.1.1.1.cmml" xref="S3.p5.2.m2.1.1.1"></eq><apply id="S3.p5.2.m2.1.1.2.cmml" xref="S3.p5.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.p5.2.m2.1.1.2.1.cmml" xref="S3.p5.2.m2.1.1.2">subscript</csymbol><apply id="S3.p5.2.m2.1.1.2.2.cmml" xref="S3.p5.2.m2.1.1.2"><csymbol 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id="S3.p5.2.m2.1.1.3.2.2.2.cmml" xref="S3.p5.2.m2.1.1.3.2.2.2">𝐽</ci><ci id="S3.p5.2.m2.1.1.3.2.2.3.cmml" xref="S3.p5.2.m2.1.1.3.2.2.3">𝜇</ci></apply><apply id="S3.p5.2.m2.1.1.3.2.3.cmml" xref="S3.p5.2.m2.1.1.3.2.3"><times id="S3.p5.2.m2.1.1.3.2.3.1.cmml" xref="S3.p5.2.m2.1.1.3.2.3.1"></times><ci id="S3.p5.2.m2.1.1.3.2.3.2.cmml" xref="S3.p5.2.m2.1.1.3.2.3.2">𝑃</ci><ci id="S3.p5.2.m2.1.1.3.2.3.3.cmml" xref="S3.p5.2.m2.1.1.3.2.3.3">𝜈</ci></apply></apply><apply id="S3.p5.2.m2.1.1.3.3.cmml" xref="S3.p5.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S3.p5.2.m2.1.1.3.3.1.cmml" xref="S3.p5.2.m2.1.1.3.3">subscript</csymbol><apply id="S3.p5.2.m2.1.1.3.3.2.cmml" xref="S3.p5.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S3.p5.2.m2.1.1.3.3.2.1.cmml" xref="S3.p5.2.m2.1.1.3.3">superscript</csymbol><ci id="S3.p5.2.m2.1.1.3.3.2.2.cmml" xref="S3.p5.2.m2.1.1.3.3.2.2">𝑐</ci><ci id="S3.p5.2.m2.1.1.3.3.2.3.cmml" xref="S3.p5.2.m2.1.1.3.3.2.3">𝜈</ci></apply><ci id="S3.p5.2.m2.1.1.3.3.3.cmml" xref="S3.p5.2.m2.1.1.3.3.3">ℓ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.2.m2.1c">C^{\mu}_{\ell P}=J^{\mu}_{P\nu}c^{\nu}_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.2.m2.1d">italic_C start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_ℓ italic_P end_POSTSUBSCRIPT = italic_J start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P italic_ν end_POSTSUBSCRIPT italic_c start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math>, with <math alttext="J^{\mu}_{P\nu}=J^{\mu}_{P\nu}(\ell)" class="ltx_Math" display="inline" id="S3.p5.3.m3.1"><semantics id="S3.p5.3.m3.1a"><mrow id="S3.p5.3.m3.1.2" xref="S3.p5.3.m3.1.2.cmml"><msubsup id="S3.p5.3.m3.1.2.2" xref="S3.p5.3.m3.1.2.2.cmml"><mi id="S3.p5.3.m3.1.2.2.2.2" xref="S3.p5.3.m3.1.2.2.2.2.cmml">J</mi><mrow id="S3.p5.3.m3.1.2.2.3" xref="S3.p5.3.m3.1.2.2.3.cmml"><mi id="S3.p5.3.m3.1.2.2.3.2" xref="S3.p5.3.m3.1.2.2.3.2.cmml">P</mi><mo id="S3.p5.3.m3.1.2.2.3.1" xref="S3.p5.3.m3.1.2.2.3.1.cmml">⁢</mo><mi id="S3.p5.3.m3.1.2.2.3.3" xref="S3.p5.3.m3.1.2.2.3.3.cmml">ν</mi></mrow><mi id="S3.p5.3.m3.1.2.2.2.3" xref="S3.p5.3.m3.1.2.2.2.3.cmml">μ</mi></msubsup><mo id="S3.p5.3.m3.1.2.1" xref="S3.p5.3.m3.1.2.1.cmml">=</mo><mrow id="S3.p5.3.m3.1.2.3" xref="S3.p5.3.m3.1.2.3.cmml"><msubsup id="S3.p5.3.m3.1.2.3.2" xref="S3.p5.3.m3.1.2.3.2.cmml"><mi id="S3.p5.3.m3.1.2.3.2.2.2" xref="S3.p5.3.m3.1.2.3.2.2.2.cmml">J</mi><mrow id="S3.p5.3.m3.1.2.3.2.3" xref="S3.p5.3.m3.1.2.3.2.3.cmml"><mi id="S3.p5.3.m3.1.2.3.2.3.2" xref="S3.p5.3.m3.1.2.3.2.3.2.cmml">P</mi><mo id="S3.p5.3.m3.1.2.3.2.3.1" xref="S3.p5.3.m3.1.2.3.2.3.1.cmml">⁢</mo><mi id="S3.p5.3.m3.1.2.3.2.3.3" xref="S3.p5.3.m3.1.2.3.2.3.3.cmml">ν</mi></mrow><mi id="S3.p5.3.m3.1.2.3.2.2.3" xref="S3.p5.3.m3.1.2.3.2.2.3.cmml">μ</mi></msubsup><mo id="S3.p5.3.m3.1.2.3.1" xref="S3.p5.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S3.p5.3.m3.1.2.3.3.2" xref="S3.p5.3.m3.1.2.3.cmml"><mo id="S3.p5.3.m3.1.2.3.3.2.1" stretchy="false" xref="S3.p5.3.m3.1.2.3.cmml">(</mo><mi id="S3.p5.3.m3.1.1" mathvariant="normal" xref="S3.p5.3.m3.1.1.cmml">ℓ</mi><mo id="S3.p5.3.m3.1.2.3.3.2.2" stretchy="false" xref="S3.p5.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.3.m3.1b"><apply id="S3.p5.3.m3.1.2.cmml" xref="S3.p5.3.m3.1.2"><eq id="S3.p5.3.m3.1.2.1.cmml" xref="S3.p5.3.m3.1.2.1"></eq><apply id="S3.p5.3.m3.1.2.2.cmml" xref="S3.p5.3.m3.1.2.2"><csymbol cd="ambiguous" id="S3.p5.3.m3.1.2.2.1.cmml" xref="S3.p5.3.m3.1.2.2">subscript</csymbol><apply id="S3.p5.3.m3.1.2.2.2.cmml" xref="S3.p5.3.m3.1.2.2"><csymbol cd="ambiguous" id="S3.p5.3.m3.1.2.2.2.1.cmml" xref="S3.p5.3.m3.1.2.2">superscript</csymbol><ci id="S3.p5.3.m3.1.2.2.2.2.cmml" xref="S3.p5.3.m3.1.2.2.2.2">𝐽</ci><ci id="S3.p5.3.m3.1.2.2.2.3.cmml" xref="S3.p5.3.m3.1.2.2.2.3">𝜇</ci></apply><apply id="S3.p5.3.m3.1.2.2.3.cmml" xref="S3.p5.3.m3.1.2.2.3"><times id="S3.p5.3.m3.1.2.2.3.1.cmml" xref="S3.p5.3.m3.1.2.2.3.1"></times><ci id="S3.p5.3.m3.1.2.2.3.2.cmml" xref="S3.p5.3.m3.1.2.2.3.2">𝑃</ci><ci id="S3.p5.3.m3.1.2.2.3.3.cmml" xref="S3.p5.3.m3.1.2.2.3.3">𝜈</ci></apply></apply><apply id="S3.p5.3.m3.1.2.3.cmml" xref="S3.p5.3.m3.1.2.3"><times id="S3.p5.3.m3.1.2.3.1.cmml" xref="S3.p5.3.m3.1.2.3.1"></times><apply id="S3.p5.3.m3.1.2.3.2.cmml" xref="S3.p5.3.m3.1.2.3.2"><csymbol cd="ambiguous" id="S3.p5.3.m3.1.2.3.2.1.cmml" xref="S3.p5.3.m3.1.2.3.2">subscript</csymbol><apply id="S3.p5.3.m3.1.2.3.2.2.cmml" xref="S3.p5.3.m3.1.2.3.2"><csymbol cd="ambiguous" id="S3.p5.3.m3.1.2.3.2.2.1.cmml" xref="S3.p5.3.m3.1.2.3.2">superscript</csymbol><ci id="S3.p5.3.m3.1.2.3.2.2.2.cmml" xref="S3.p5.3.m3.1.2.3.2.2.2">𝐽</ci><ci id="S3.p5.3.m3.1.2.3.2.2.3.cmml" xref="S3.p5.3.m3.1.2.3.2.2.3">𝜇</ci></apply><apply id="S3.p5.3.m3.1.2.3.2.3.cmml" xref="S3.p5.3.m3.1.2.3.2.3"><times id="S3.p5.3.m3.1.2.3.2.3.1.cmml" xref="S3.p5.3.m3.1.2.3.2.3.1"></times><ci id="S3.p5.3.m3.1.2.3.2.3.2.cmml" xref="S3.p5.3.m3.1.2.3.2.3.2">𝑃</ci><ci id="S3.p5.3.m3.1.2.3.2.3.3.cmml" xref="S3.p5.3.m3.1.2.3.2.3.3">𝜈</ci></apply></apply><ci id="S3.p5.3.m3.1.1.cmml" xref="S3.p5.3.m3.1.1">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.3.m3.1c">J^{\mu}_{P\nu}=J^{\mu}_{P\nu}(\ell)</annotation><annotation encoding="application/x-llamapun" id="S3.p5.3.m3.1d">italic_J start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P italic_ν end_POSTSUBSCRIPT = italic_J start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P italic_ν end_POSTSUBSCRIPT ( roman_ℓ )</annotation></semantics></math>. From (<a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S3.E28" title="In III Loop recursions ‣ One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_tag">28</span></a>) we obtain</p> <table class="ltx_equation ltx_eqn_table" id="S3.E29"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="s_{\ell P}J^{\rho}_{P\sigma}=I_{P}^{\mu\nu}\mathcal{V}^{(\textrm{g})\rho}_{\mu% \nu\sigma}(P,\ell P)\\ +\sum_{P=Q\cup R}I_{Q}^{\mu\nu}J^{\gamma}_{R\sigma}\mathcal{V}^{(\textrm{g})% \rho}_{\mu\nu\gamma}(Q,\ell P)." class="ltx_Math" display="block" id="S3.E29.m1.40"><semantics id="S3.E29.m1.40a"><mtable displaystyle="true" id="S3.E29.m1.40.40.3" rowspacing="0pt"><mtr id="S3.E29.m1.40.40.3a"><mtd class="ltx_align_left" columnalign="left" id="S3.E29.m1.40.40.3b"><mrow id="S3.E29.m1.39.39.2.38.19.19"><mrow id="S3.E29.m1.39.39.2.38.19.19.20"><msub id="S3.E29.m1.39.39.2.38.19.19.20.2"><mi id="S3.E29.m1.1.1.1.1.1.1" xref="S3.E29.m1.1.1.1.1.1.1.cmml">s</mi><mrow id="S3.E29.m1.2.2.2.2.2.2.1" xref="S3.E29.m1.2.2.2.2.2.2.1.cmml"><mi id="S3.E29.m1.2.2.2.2.2.2.1.2" mathvariant="normal" xref="S3.E29.m1.2.2.2.2.2.2.1.2.cmml">ℓ</mi><mo id="S3.E29.m1.2.2.2.2.2.2.1.1" xref="S3.E29.m1.2.2.2.2.2.2.1.1.cmml">⁢</mo><mi id="S3.E29.m1.2.2.2.2.2.2.1.3" xref="S3.E29.m1.2.2.2.2.2.2.1.3.cmml">P</mi></mrow></msub><mo id="S3.E29.m1.39.39.2.38.19.19.20.1" xref="S3.E29.m1.38.38.1.1.1.cmml">⁢</mo><msubsup id="S3.E29.m1.39.39.2.38.19.19.20.3"><mi id="S3.E29.m1.3.3.3.3.3.3" xref="S3.E29.m1.3.3.3.3.3.3.cmml">J</mi><mrow id="S3.E29.m1.5.5.5.5.5.5.1" xref="S3.E29.m1.5.5.5.5.5.5.1.cmml"><mi id="S3.E29.m1.5.5.5.5.5.5.1.2" xref="S3.E29.m1.5.5.5.5.5.5.1.2.cmml">P</mi><mo id="S3.E29.m1.5.5.5.5.5.5.1.1" xref="S3.E29.m1.5.5.5.5.5.5.1.1.cmml">⁢</mo><mi id="S3.E29.m1.5.5.5.5.5.5.1.3" xref="S3.E29.m1.5.5.5.5.5.5.1.3.cmml">σ</mi></mrow><mi id="S3.E29.m1.4.4.4.4.4.4.1" xref="S3.E29.m1.4.4.4.4.4.4.1.cmml">ρ</mi></msubsup></mrow><mo id="S3.E29.m1.6.6.6.6.6.6" xref="S3.E29.m1.6.6.6.6.6.6.cmml">=</mo><mrow id="S3.E29.m1.39.39.2.38.19.19.19"><msubsup id="S3.E29.m1.39.39.2.38.19.19.19.3"><mi id="S3.E29.m1.7.7.7.7.7.7" xref="S3.E29.m1.7.7.7.7.7.7.cmml">I</mi><mi id="S3.E29.m1.8.8.8.8.8.8.1" xref="S3.E29.m1.8.8.8.8.8.8.1.cmml">P</mi><mrow id="S3.E29.m1.9.9.9.9.9.9.1" xref="S3.E29.m1.9.9.9.9.9.9.1.cmml"><mi id="S3.E29.m1.9.9.9.9.9.9.1.2" xref="S3.E29.m1.9.9.9.9.9.9.1.2.cmml">μ</mi><mo id="S3.E29.m1.9.9.9.9.9.9.1.1" xref="S3.E29.m1.9.9.9.9.9.9.1.1.cmml">⁢</mo><mi id="S3.E29.m1.9.9.9.9.9.9.1.3" xref="S3.E29.m1.9.9.9.9.9.9.1.3.cmml">ν</mi></mrow></msubsup><mo id="S3.E29.m1.39.39.2.38.19.19.19.2" xref="S3.E29.m1.38.38.1.1.1.cmml">⁢</mo><msubsup id="S3.E29.m1.39.39.2.38.19.19.19.4"><mi class="ltx_font_mathcaligraphic" id="S3.E29.m1.10.10.10.10.10.10" xref="S3.E29.m1.10.10.10.10.10.10.cmml">𝒱</mi><mrow id="S3.E29.m1.12.12.12.12.12.12.1" xref="S3.E29.m1.12.12.12.12.12.12.1.cmml"><mi id="S3.E29.m1.12.12.12.12.12.12.1.2" xref="S3.E29.m1.12.12.12.12.12.12.1.2.cmml">μ</mi><mo id="S3.E29.m1.12.12.12.12.12.12.1.1" xref="S3.E29.m1.12.12.12.12.12.12.1.1.cmml">⁢</mo><mi id="S3.E29.m1.12.12.12.12.12.12.1.3" xref="S3.E29.m1.12.12.12.12.12.12.1.3.cmml">ν</mi><mo id="S3.E29.m1.12.12.12.12.12.12.1.1a" xref="S3.E29.m1.12.12.12.12.12.12.1.1.cmml">⁢</mo><mi id="S3.E29.m1.12.12.12.12.12.12.1.4" xref="S3.E29.m1.12.12.12.12.12.12.1.4.cmml">σ</mi></mrow><mrow id="S3.E29.m1.11.11.11.11.11.11.1" xref="S3.E29.m1.11.11.11.11.11.11.1.cmml"><mrow id="S3.E29.m1.11.11.11.11.11.11.1.3.2" xref="S3.E29.m1.11.11.11.11.11.11.1.1a.cmml"><mo id="S3.E29.m1.11.11.11.11.11.11.1.3.2.1" stretchy="false" xref="S3.E29.m1.11.11.11.11.11.11.1.1a.cmml">(</mo><mtext id="S3.E29.m1.11.11.11.11.11.11.1.1" xref="S3.E29.m1.11.11.11.11.11.11.1.1.cmml">g</mtext><mo id="S3.E29.m1.11.11.11.11.11.11.1.3.2.2" stretchy="false" 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xref="S3.E29.m1.21.21.21.3.3.3.1.3.2.cmml">Q</mi><mo id="S3.E29.m1.21.21.21.3.3.3.1.3.1" xref="S3.E29.m1.21.21.21.3.3.3.1.3.1.cmml">∪</mo><mi id="S3.E29.m1.21.21.21.3.3.3.1.3.3" xref="S3.E29.m1.21.21.21.3.3.3.1.3.3.cmml">R</mi></mrow></mrow></munder><mrow id="S3.E29.m1.40.40.3.39.20.20.20.1.1.1"><msubsup id="S3.E29.m1.40.40.3.39.20.20.20.1.1.1.3"><mi id="S3.E29.m1.22.22.22.4.4.4" xref="S3.E29.m1.22.22.22.4.4.4.cmml">I</mi><mi id="S3.E29.m1.23.23.23.5.5.5.1" xref="S3.E29.m1.23.23.23.5.5.5.1.cmml">Q</mi><mrow id="S3.E29.m1.24.24.24.6.6.6.1" xref="S3.E29.m1.24.24.24.6.6.6.1.cmml"><mi id="S3.E29.m1.24.24.24.6.6.6.1.2" xref="S3.E29.m1.24.24.24.6.6.6.1.2.cmml">μ</mi><mo id="S3.E29.m1.24.24.24.6.6.6.1.1" xref="S3.E29.m1.24.24.24.6.6.6.1.1.cmml">⁢</mo><mi id="S3.E29.m1.24.24.24.6.6.6.1.3" xref="S3.E29.m1.24.24.24.6.6.6.1.3.cmml">ν</mi></mrow></msubsup><mo id="S3.E29.m1.40.40.3.39.20.20.20.1.1.1.2" xref="S3.E29.m1.38.38.1.1.1.cmml">⁢</mo><msubsup id="S3.E29.m1.40.40.3.39.20.20.20.1.1.1.4"><mi 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start_POSTSUPERSCRIPT italic_ρ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P italic_σ end_POSTSUBSCRIPT = italic_I start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT caligraphic_V start_POSTSUPERSCRIPT ( g ) italic_ρ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ italic_ν italic_σ end_POSTSUBSCRIPT ( italic_P , roman_ℓ italic_P ) end_CELL end_ROW start_ROW start_CELL + ∑ start_POSTSUBSCRIPT italic_P = italic_Q ∪ italic_R end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT italic_J start_POSTSUPERSCRIPT italic_γ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_R italic_σ end_POSTSUBSCRIPT caligraphic_V start_POSTSUPERSCRIPT ( g ) italic_ρ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ italic_ν italic_γ end_POSTSUBSCRIPT ( italic_Q , roman_ℓ italic_P ) . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(29)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S3.p6"> <p class="ltx_p" id="S3.p6.1">We can mimic the graviton loop analysis and sew the ghost and antighost legs with the ghost propagator <math alttext="\delta^{\sigma}_{\rho}/\ell^{2}" class="ltx_Math" display="inline" id="S3.p6.1.m1.1"><semantics id="S3.p6.1.m1.1a"><mrow id="S3.p6.1.m1.1.1" xref="S3.p6.1.m1.1.1.cmml"><msubsup id="S3.p6.1.m1.1.1.2" xref="S3.p6.1.m1.1.1.2.cmml"><mi id="S3.p6.1.m1.1.1.2.2.2" xref="S3.p6.1.m1.1.1.2.2.2.cmml">δ</mi><mi id="S3.p6.1.m1.1.1.2.3" xref="S3.p6.1.m1.1.1.2.3.cmml">ρ</mi><mi id="S3.p6.1.m1.1.1.2.2.3" xref="S3.p6.1.m1.1.1.2.2.3.cmml">σ</mi></msubsup><mo id="S3.p6.1.m1.1.1.1" xref="S3.p6.1.m1.1.1.1.cmml">/</mo><msup id="S3.p6.1.m1.1.1.3" xref="S3.p6.1.m1.1.1.3.cmml"><mi id="S3.p6.1.m1.1.1.3.2" mathvariant="normal" xref="S3.p6.1.m1.1.1.3.2.cmml">ℓ</mi><mn id="S3.p6.1.m1.1.1.3.3" xref="S3.p6.1.m1.1.1.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.1.m1.1b"><apply id="S3.p6.1.m1.1.1.cmml" xref="S3.p6.1.m1.1.1"><divide id="S3.p6.1.m1.1.1.1.cmml" xref="S3.p6.1.m1.1.1.1"></divide><apply id="S3.p6.1.m1.1.1.2.cmml" xref="S3.p6.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.p6.1.m1.1.1.2.1.cmml" xref="S3.p6.1.m1.1.1.2">subscript</csymbol><apply id="S3.p6.1.m1.1.1.2.2.cmml" xref="S3.p6.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.p6.1.m1.1.1.2.2.1.cmml" xref="S3.p6.1.m1.1.1.2">superscript</csymbol><ci id="S3.p6.1.m1.1.1.2.2.2.cmml" xref="S3.p6.1.m1.1.1.2.2.2">𝛿</ci><ci id="S3.p6.1.m1.1.1.2.2.3.cmml" xref="S3.p6.1.m1.1.1.2.2.3">𝜎</ci></apply><ci id="S3.p6.1.m1.1.1.2.3.cmml" xref="S3.p6.1.m1.1.1.2.3">𝜌</ci></apply><apply id="S3.p6.1.m1.1.1.3.cmml" xref="S3.p6.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.p6.1.m1.1.1.3.1.cmml" xref="S3.p6.1.m1.1.1.3">superscript</csymbol><ci id="S3.p6.1.m1.1.1.3.2.cmml" xref="S3.p6.1.m1.1.1.3.2">ℓ</ci><cn id="S3.p6.1.m1.1.1.3.3.cmml" type="integer" xref="S3.p6.1.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.1.m1.1c">\delta^{\sigma}_{\rho}/\ell^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.1.m1.1d">italic_δ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT / roman_ℓ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. This would again lead to an overcounting of different contributions, so we introduce a modified current</p> <table class="ltx_equation ltx_eqn_table" id="S3.E30"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="s_{\ell P}\tilde{J}^{\rho}_{P\sigma}=I_{P}^{\mu\nu}\mathcal{V}^{(\textrm{g})% \rho}_{\mu\nu\sigma}(P,\ell P)\\ +\sum_{P=Q\cup R}\frac{|Q|}{|P|}I_{Q}^{\mu\nu}\tilde{J}^{\gamma}_{R\sigma}% \mathcal{V}^{(\textrm{g})\rho}_{\mu\nu\gamma}(Q,\ell P)," class="ltx_Math" display="block" id="S3.E30.m1.41"><semantics id="S3.E30.m1.41a"><mtable displaystyle="true" id="S3.E30.m1.41.41.3" rowspacing="0pt"><mtr id="S3.E30.m1.41.41.3a"><mtd class="ltx_align_left" columnalign="left" id="S3.E30.m1.41.41.3b"><mrow id="S3.E30.m1.40.40.2.39.19.19"><mrow id="S3.E30.m1.40.40.2.39.19.19.20"><msub id="S3.E30.m1.40.40.2.39.19.19.20.2"><mi id="S3.E30.m1.1.1.1.1.1.1" xref="S3.E30.m1.1.1.1.1.1.1.cmml">s</mi><mrow id="S3.E30.m1.2.2.2.2.2.2.1" xref="S3.E30.m1.2.2.2.2.2.2.1.cmml"><mi id="S3.E30.m1.2.2.2.2.2.2.1.2" mathvariant="normal" xref="S3.E30.m1.2.2.2.2.2.2.1.2.cmml">ℓ</mi><mo id="S3.E30.m1.2.2.2.2.2.2.1.1" xref="S3.E30.m1.2.2.2.2.2.2.1.1.cmml">⁢</mo><mi id="S3.E30.m1.2.2.2.2.2.2.1.3" xref="S3.E30.m1.2.2.2.2.2.2.1.3.cmml">P</mi></mrow></msub><mo id="S3.E30.m1.40.40.2.39.19.19.20.1" xref="S3.E30.m1.39.39.1.1.1.cmml">⁢</mo><msubsup id="S3.E30.m1.40.40.2.39.19.19.20.3"><mover accent="true" id="S3.E30.m1.3.3.3.3.3.3" xref="S3.E30.m1.3.3.3.3.3.3.cmml"><mi id="S3.E30.m1.3.3.3.3.3.3.2" xref="S3.E30.m1.3.3.3.3.3.3.2.cmml">J</mi><mo id="S3.E30.m1.3.3.3.3.3.3.1" xref="S3.E30.m1.3.3.3.3.3.3.1.cmml">~</mo></mover><mrow id="S3.E30.m1.5.5.5.5.5.5.1" xref="S3.E30.m1.5.5.5.5.5.5.1.cmml"><mi id="S3.E30.m1.5.5.5.5.5.5.1.2" xref="S3.E30.m1.5.5.5.5.5.5.1.2.cmml">P</mi><mo id="S3.E30.m1.5.5.5.5.5.5.1.1" xref="S3.E30.m1.5.5.5.5.5.5.1.1.cmml">⁢</mo><mi id="S3.E30.m1.5.5.5.5.5.5.1.3" xref="S3.E30.m1.5.5.5.5.5.5.1.3.cmml">σ</mi></mrow><mi id="S3.E30.m1.4.4.4.4.4.4.1" xref="S3.E30.m1.4.4.4.4.4.4.1.cmml">ρ</mi></msubsup></mrow><mo id="S3.E30.m1.6.6.6.6.6.6" xref="S3.E30.m1.6.6.6.6.6.6.cmml">=</mo><mrow id="S3.E30.m1.40.40.2.39.19.19.19"><msubsup id="S3.E30.m1.40.40.2.39.19.19.19.3"><mi id="S3.E30.m1.7.7.7.7.7.7" xref="S3.E30.m1.7.7.7.7.7.7.cmml">I</mi><mi id="S3.E30.m1.8.8.8.8.8.8.1" xref="S3.E30.m1.8.8.8.8.8.8.1.cmml">P</mi><mrow id="S3.E30.m1.9.9.9.9.9.9.1" xref="S3.E30.m1.9.9.9.9.9.9.1.cmml"><mi id="S3.E30.m1.9.9.9.9.9.9.1.2" xref="S3.E30.m1.9.9.9.9.9.9.1.2.cmml">μ</mi><mo id="S3.E30.m1.9.9.9.9.9.9.1.1" xref="S3.E30.m1.9.9.9.9.9.9.1.1.cmml">⁢</mo><mi id="S3.E30.m1.9.9.9.9.9.9.1.3" xref="S3.E30.m1.9.9.9.9.9.9.1.3.cmml">ν</mi></mrow></msubsup><mo id="S3.E30.m1.40.40.2.39.19.19.19.2" xref="S3.E30.m1.39.39.1.1.1.cmml">⁢</mo><msubsup id="S3.E30.m1.40.40.2.39.19.19.19.4"><mi class="ltx_font_mathcaligraphic" id="S3.E30.m1.10.10.10.10.10.10" 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\mathcal{V}^{(\textrm{g})\rho}_{\mu\nu\gamma}(Q,\ell P),</annotation><annotation encoding="application/x-llamapun" id="S3.E30.m1.41d">start_ROW start_CELL italic_s start_POSTSUBSCRIPT roman_ℓ italic_P end_POSTSUBSCRIPT over~ start_ARG italic_J end_ARG start_POSTSUPERSCRIPT italic_ρ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P italic_σ end_POSTSUBSCRIPT = italic_I start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT caligraphic_V start_POSTSUPERSCRIPT ( g ) italic_ρ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ italic_ν italic_σ end_POSTSUBSCRIPT ( italic_P , roman_ℓ italic_P ) end_CELL end_ROW start_ROW start_CELL + ∑ start_POSTSUBSCRIPT italic_P = italic_Q ∪ italic_R end_POSTSUBSCRIPT divide start_ARG | italic_Q | end_ARG start_ARG | italic_P | end_ARG italic_I start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT over~ start_ARG italic_J end_ARG start_POSTSUPERSCRIPT italic_γ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_R italic_σ end_POSTSUBSCRIPT caligraphic_V start_POSTSUPERSCRIPT ( g ) italic_ρ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ italic_ν italic_γ end_POSTSUBSCRIPT ( italic_Q , roman_ℓ italic_P ) , end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(30)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p6.3">where we found that the factor <math alttext="|Q|/|P|" class="ltx_Math" display="inline" id="S3.p6.2.m1.2"><semantics id="S3.p6.2.m1.2a"><mrow id="S3.p6.2.m1.2.3" xref="S3.p6.2.m1.2.3.cmml"><mrow id="S3.p6.2.m1.2.3.2.2" xref="S3.p6.2.m1.2.3.2.1.cmml"><mo id="S3.p6.2.m1.2.3.2.2.1" stretchy="false" xref="S3.p6.2.m1.2.3.2.1.1.cmml">|</mo><mi id="S3.p6.2.m1.1.1" xref="S3.p6.2.m1.1.1.cmml">Q</mi><mo id="S3.p6.2.m1.2.3.2.2.2" stretchy="false" xref="S3.p6.2.m1.2.3.2.1.1.cmml">|</mo></mrow><mo id="S3.p6.2.m1.2.3.1" xref="S3.p6.2.m1.2.3.1.cmml">/</mo><mrow id="S3.p6.2.m1.2.3.3.2" xref="S3.p6.2.m1.2.3.3.1.cmml"><mo id="S3.p6.2.m1.2.3.3.2.1" stretchy="false" xref="S3.p6.2.m1.2.3.3.1.1.cmml">|</mo><mi id="S3.p6.2.m1.2.2" xref="S3.p6.2.m1.2.2.cmml">P</mi><mo id="S3.p6.2.m1.2.3.3.2.2" stretchy="false" xref="S3.p6.2.m1.2.3.3.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.2.m1.2b"><apply id="S3.p6.2.m1.2.3.cmml" xref="S3.p6.2.m1.2.3"><divide id="S3.p6.2.m1.2.3.1.cmml" xref="S3.p6.2.m1.2.3.1"></divide><apply id="S3.p6.2.m1.2.3.2.1.cmml" xref="S3.p6.2.m1.2.3.2.2"><abs id="S3.p6.2.m1.2.3.2.1.1.cmml" xref="S3.p6.2.m1.2.3.2.2.1"></abs><ci id="S3.p6.2.m1.1.1.cmml" xref="S3.p6.2.m1.1.1">𝑄</ci></apply><apply id="S3.p6.2.m1.2.3.3.1.cmml" xref="S3.p6.2.m1.2.3.3.2"><abs id="S3.p6.2.m1.2.3.3.1.1.cmml" xref="S3.p6.2.m1.2.3.3.2.1"></abs><ci id="S3.p6.2.m1.2.2.cmml" xref="S3.p6.2.m1.2.2">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.2.m1.2c">|Q|/|P|</annotation><annotation encoding="application/x-llamapun" id="S3.p6.2.m1.2d">| italic_Q | / | italic_P |</annotation></semantics></math> is needed to balance the repeated diagrams. After imposing momentum conservation <math alttext="k_{P}=0" class="ltx_Math" display="inline" id="S3.p6.3.m2.1"><semantics id="S3.p6.3.m2.1a"><mrow id="S3.p6.3.m2.1.1" xref="S3.p6.3.m2.1.1.cmml"><msub id="S3.p6.3.m2.1.1.2" xref="S3.p6.3.m2.1.1.2.cmml"><mi id="S3.p6.3.m2.1.1.2.2" xref="S3.p6.3.m2.1.1.2.2.cmml">k</mi><mi id="S3.p6.3.m2.1.1.2.3" xref="S3.p6.3.m2.1.1.2.3.cmml">P</mi></msub><mo id="S3.p6.3.m2.1.1.1" xref="S3.p6.3.m2.1.1.1.cmml">=</mo><mn id="S3.p6.3.m2.1.1.3" xref="S3.p6.3.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.3.m2.1b"><apply id="S3.p6.3.m2.1.1.cmml" xref="S3.p6.3.m2.1.1"><eq id="S3.p6.3.m2.1.1.1.cmml" xref="S3.p6.3.m2.1.1.1"></eq><apply id="S3.p6.3.m2.1.1.2.cmml" xref="S3.p6.3.m2.1.1.2"><csymbol cd="ambiguous" id="S3.p6.3.m2.1.1.2.1.cmml" xref="S3.p6.3.m2.1.1.2">subscript</csymbol><ci id="S3.p6.3.m2.1.1.2.2.cmml" xref="S3.p6.3.m2.1.1.2.2">𝑘</ci><ci id="S3.p6.3.m2.1.1.2.3.cmml" xref="S3.p6.3.m2.1.1.2.3">𝑃</ci></apply><cn id="S3.p6.3.m2.1.1.3.cmml" type="integer" xref="S3.p6.3.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.3.m2.1c">k_{P}=0</annotation><annotation encoding="application/x-llamapun" id="S3.p6.3.m2.1d">italic_k start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT = 0</annotation></semantics></math> on the external legs, we can finally define the ghost-loop integrand:</p> <table class="ltx_equation ltx_eqn_table" id="S3.E31"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{I}^{\textrm{ghost}}_{N}(\ell)=\tilde{J}^{\mu}_{1\ldots N\mu}(\ell)." class="ltx_Math" display="block" id="S3.E31.m1.3"><semantics id="S3.E31.m1.3a"><mrow id="S3.E31.m1.3.3.1" xref="S3.E31.m1.3.3.1.1.cmml"><mrow id="S3.E31.m1.3.3.1.1" xref="S3.E31.m1.3.3.1.1.cmml"><mrow id="S3.E31.m1.3.3.1.1.2" xref="S3.E31.m1.3.3.1.1.2.cmml"><msubsup 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id="S3.E31.m1.3.3.1.1.3.2.2.2" xref="S3.E31.m1.3.3.1.1.3.2.2.2.cmml"><mi id="S3.E31.m1.3.3.1.1.3.2.2.2.2" xref="S3.E31.m1.3.3.1.1.3.2.2.2.2.cmml">J</mi><mo id="S3.E31.m1.3.3.1.1.3.2.2.2.1" xref="S3.E31.m1.3.3.1.1.3.2.2.2.1.cmml">~</mo></mover><mrow id="S3.E31.m1.3.3.1.1.3.2.3" xref="S3.E31.m1.3.3.1.1.3.2.3.cmml"><mn id="S3.E31.m1.3.3.1.1.3.2.3.2" xref="S3.E31.m1.3.3.1.1.3.2.3.2.cmml">1</mn><mo id="S3.E31.m1.3.3.1.1.3.2.3.1" xref="S3.E31.m1.3.3.1.1.3.2.3.1.cmml">⁢</mo><mi id="S3.E31.m1.3.3.1.1.3.2.3.3" mathvariant="normal" xref="S3.E31.m1.3.3.1.1.3.2.3.3.cmml">…</mi><mo id="S3.E31.m1.3.3.1.1.3.2.3.1a" xref="S3.E31.m1.3.3.1.1.3.2.3.1.cmml">⁢</mo><mi id="S3.E31.m1.3.3.1.1.3.2.3.4" xref="S3.E31.m1.3.3.1.1.3.2.3.4.cmml">N</mi><mo id="S3.E31.m1.3.3.1.1.3.2.3.1b" xref="S3.E31.m1.3.3.1.1.3.2.3.1.cmml">⁢</mo><mi id="S3.E31.m1.3.3.1.1.3.2.3.5" xref="S3.E31.m1.3.3.1.1.3.2.3.5.cmml">μ</mi></mrow><mi id="S3.E31.m1.3.3.1.1.3.2.2.3" xref="S3.E31.m1.3.3.1.1.3.2.2.3.cmml">μ</mi></msubsup><mo id="S3.E31.m1.3.3.1.1.3.1" xref="S3.E31.m1.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S3.E31.m1.3.3.1.1.3.3.2" xref="S3.E31.m1.3.3.1.1.3.cmml"><mo id="S3.E31.m1.3.3.1.1.3.3.2.1" stretchy="false" xref="S3.E31.m1.3.3.1.1.3.cmml">(</mo><mi id="S3.E31.m1.2.2" mathvariant="normal" xref="S3.E31.m1.2.2.cmml">ℓ</mi><mo id="S3.E31.m1.3.3.1.1.3.3.2.2" stretchy="false" xref="S3.E31.m1.3.3.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S3.E31.m1.3.3.1.2" lspace="0em" xref="S3.E31.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E31.m1.3b"><apply id="S3.E31.m1.3.3.1.1.cmml" xref="S3.E31.m1.3.3.1"><eq id="S3.E31.m1.3.3.1.1.1.cmml" xref="S3.E31.m1.3.3.1.1.1"></eq><apply id="S3.E31.m1.3.3.1.1.2.cmml" xref="S3.E31.m1.3.3.1.1.2"><times id="S3.E31.m1.3.3.1.1.2.1.cmml" xref="S3.E31.m1.3.3.1.1.2.1"></times><apply id="S3.E31.m1.3.3.1.1.2.2.cmml" xref="S3.E31.m1.3.3.1.1.2.2"><csymbol cd="ambiguous" id="S3.E31.m1.3.3.1.1.2.2.1.cmml" xref="S3.E31.m1.3.3.1.1.2.2">subscript</csymbol><apply id="S3.E31.m1.3.3.1.1.2.2.2.cmml" xref="S3.E31.m1.3.3.1.1.2.2"><csymbol cd="ambiguous" id="S3.E31.m1.3.3.1.1.2.2.2.1.cmml" xref="S3.E31.m1.3.3.1.1.2.2">superscript</csymbol><ci id="S3.E31.m1.3.3.1.1.2.2.2.2.cmml" xref="S3.E31.m1.3.3.1.1.2.2.2.2">ℐ</ci><ci id="S3.E31.m1.3.3.1.1.2.2.2.3a.cmml" xref="S3.E31.m1.3.3.1.1.2.2.2.3"><mtext id="S3.E31.m1.3.3.1.1.2.2.2.3.cmml" mathsize="70%" xref="S3.E31.m1.3.3.1.1.2.2.2.3">ghost</mtext></ci></apply><ci id="S3.E31.m1.3.3.1.1.2.2.3.cmml" xref="S3.E31.m1.3.3.1.1.2.2.3">𝑁</ci></apply><ci id="S3.E31.m1.1.1.cmml" xref="S3.E31.m1.1.1">ℓ</ci></apply><apply id="S3.E31.m1.3.3.1.1.3.cmml" xref="S3.E31.m1.3.3.1.1.3"><times id="S3.E31.m1.3.3.1.1.3.1.cmml" xref="S3.E31.m1.3.3.1.1.3.1"></times><apply id="S3.E31.m1.3.3.1.1.3.2.cmml" xref="S3.E31.m1.3.3.1.1.3.2"><csymbol cd="ambiguous" id="S3.E31.m1.3.3.1.1.3.2.1.cmml" xref="S3.E31.m1.3.3.1.1.3.2">subscript</csymbol><apply id="S3.E31.m1.3.3.1.1.3.2.2.cmml" xref="S3.E31.m1.3.3.1.1.3.2"><csymbol cd="ambiguous" id="S3.E31.m1.3.3.1.1.3.2.2.1.cmml" xref="S3.E31.m1.3.3.1.1.3.2">superscript</csymbol><apply id="S3.E31.m1.3.3.1.1.3.2.2.2.cmml" xref="S3.E31.m1.3.3.1.1.3.2.2.2"><ci id="S3.E31.m1.3.3.1.1.3.2.2.2.1.cmml" xref="S3.E31.m1.3.3.1.1.3.2.2.2.1">~</ci><ci id="S3.E31.m1.3.3.1.1.3.2.2.2.2.cmml" xref="S3.E31.m1.3.3.1.1.3.2.2.2.2">𝐽</ci></apply><ci id="S3.E31.m1.3.3.1.1.3.2.2.3.cmml" xref="S3.E31.m1.3.3.1.1.3.2.2.3">𝜇</ci></apply><apply id="S3.E31.m1.3.3.1.1.3.2.3.cmml" xref="S3.E31.m1.3.3.1.1.3.2.3"><times id="S3.E31.m1.3.3.1.1.3.2.3.1.cmml" xref="S3.E31.m1.3.3.1.1.3.2.3.1"></times><cn id="S3.E31.m1.3.3.1.1.3.2.3.2.cmml" type="integer" xref="S3.E31.m1.3.3.1.1.3.2.3.2">1</cn><ci id="S3.E31.m1.3.3.1.1.3.2.3.3.cmml" xref="S3.E31.m1.3.3.1.1.3.2.3.3">…</ci><ci id="S3.E31.m1.3.3.1.1.3.2.3.4.cmml" xref="S3.E31.m1.3.3.1.1.3.2.3.4">𝑁</ci><ci id="S3.E31.m1.3.3.1.1.3.2.3.5.cmml" xref="S3.E31.m1.3.3.1.1.3.2.3.5">𝜇</ci></apply></apply><ci id="S3.E31.m1.2.2.cmml" xref="S3.E31.m1.2.2">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E31.m1.3c">\mathcal{I}^{\textrm{ghost}}_{N}(\ell)=\tilde{J}^{\mu}_{1\ldots N\mu}(\ell).</annotation><annotation encoding="application/x-llamapun" id="S3.E31.m1.3d">caligraphic_I start_POSTSUPERSCRIPT ghost end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( roman_ℓ ) = over~ start_ARG italic_J end_ARG start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 … italic_N italic_μ end_POSTSUBSCRIPT ( roman_ℓ ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(31)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S3.p7"> <p class="ltx_p" id="S3.p7.2">The complete one-loop integrand for the <math alttext="N" class="ltx_Math" display="inline" id="S3.p7.1.m1.1"><semantics id="S3.p7.1.m1.1a"><mi id="S3.p7.1.m1.1.1" xref="S3.p7.1.m1.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S3.p7.1.m1.1b"><ci id="S3.p7.1.m1.1.1.cmml" xref="S3.p7.1.m1.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.1.m1.1c">N</annotation><annotation encoding="application/x-llamapun" id="S3.p7.1.m1.1d">italic_N</annotation></semantics></math>-graviton amputated correlator <math alttext="\mathcal{M}^{1\textrm{-loop}}_{N}" class="ltx_Math" display="inline" id="S3.p7.2.m2.1"><semantics id="S3.p7.2.m2.1a"><msubsup id="S3.p7.2.m2.1.1" xref="S3.p7.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p7.2.m2.1.1.2.2" xref="S3.p7.2.m2.1.1.2.2.cmml">ℳ</mi><mi id="S3.p7.2.m2.1.1.3" xref="S3.p7.2.m2.1.1.3.cmml">N</mi><mrow id="S3.p7.2.m2.1.1.2.3" xref="S3.p7.2.m2.1.1.2.3.cmml"><mn id="S3.p7.2.m2.1.1.2.3.2" xref="S3.p7.2.m2.1.1.2.3.2.cmml">1</mn><mo id="S3.p7.2.m2.1.1.2.3.1" xref="S3.p7.2.m2.1.1.2.3.1.cmml">⁢</mo><mtext id="S3.p7.2.m2.1.1.2.3.3" xref="S3.p7.2.m2.1.1.2.3.3a.cmml">-loop</mtext></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.p7.2.m2.1b"><apply id="S3.p7.2.m2.1.1.cmml" xref="S3.p7.2.m2.1.1"><csymbol cd="ambiguous" id="S3.p7.2.m2.1.1.1.cmml" xref="S3.p7.2.m2.1.1">subscript</csymbol><apply id="S3.p7.2.m2.1.1.2.cmml" xref="S3.p7.2.m2.1.1"><csymbol cd="ambiguous" id="S3.p7.2.m2.1.1.2.1.cmml" xref="S3.p7.2.m2.1.1">superscript</csymbol><ci id="S3.p7.2.m2.1.1.2.2.cmml" xref="S3.p7.2.m2.1.1.2.2">ℳ</ci><apply id="S3.p7.2.m2.1.1.2.3.cmml" xref="S3.p7.2.m2.1.1.2.3"><times id="S3.p7.2.m2.1.1.2.3.1.cmml" xref="S3.p7.2.m2.1.1.2.3.1"></times><cn id="S3.p7.2.m2.1.1.2.3.2.cmml" type="integer" xref="S3.p7.2.m2.1.1.2.3.2">1</cn><ci id="S3.p7.2.m2.1.1.2.3.3a.cmml" xref="S3.p7.2.m2.1.1.2.3.3"><mtext id="S3.p7.2.m2.1.1.2.3.3.cmml" mathsize="70%" xref="S3.p7.2.m2.1.1.2.3.3">-loop</mtext></ci></apply></apply><ci id="S3.p7.2.m2.1.1.3.cmml" xref="S3.p7.2.m2.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.2.m2.1c">\mathcal{M}^{1\textrm{-loop}}_{N}</annotation><annotation encoding="application/x-llamapun" id="S3.p7.2.m2.1d">caligraphic_M start_POSTSUPERSCRIPT 1 -loop end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT</annotation></semantics></math> is then</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S5.EGx4"> <tbody id="S3.E32"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{I}_{N}(\ell)" class="ltx_Math" display="inline" id="S3.E32.m1.1"><semantics id="S3.E32.m1.1a"><mrow id="S3.E32.m1.1.2" xref="S3.E32.m1.1.2.cmml"><msub id="S3.E32.m1.1.2.2" xref="S3.E32.m1.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E32.m1.1.2.2.2" xref="S3.E32.m1.1.2.2.2.cmml">ℐ</mi><mi id="S3.E32.m1.1.2.2.3" xref="S3.E32.m1.1.2.2.3.cmml">N</mi></msub><mo id="S3.E32.m1.1.2.1" xref="S3.E32.m1.1.2.1.cmml">⁢</mo><mrow id="S3.E32.m1.1.2.3.2" xref="S3.E32.m1.1.2.cmml"><mo id="S3.E32.m1.1.2.3.2.1" stretchy="false" xref="S3.E32.m1.1.2.cmml">(</mo><mi id="S3.E32.m1.1.1" mathvariant="normal" xref="S3.E32.m1.1.1.cmml">ℓ</mi><mo id="S3.E32.m1.1.2.3.2.2" stretchy="false" xref="S3.E32.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.E32.m1.1b"><apply id="S3.E32.m1.1.2.cmml" xref="S3.E32.m1.1.2"><times id="S3.E32.m1.1.2.1.cmml" xref="S3.E32.m1.1.2.1"></times><apply id="S3.E32.m1.1.2.2.cmml" xref="S3.E32.m1.1.2.2"><csymbol cd="ambiguous" id="S3.E32.m1.1.2.2.1.cmml" xref="S3.E32.m1.1.2.2">subscript</csymbol><ci id="S3.E32.m1.1.2.2.2.cmml" xref="S3.E32.m1.1.2.2.2">ℐ</ci><ci id="S3.E32.m1.1.2.2.3.cmml" xref="S3.E32.m1.1.2.2.3">𝑁</ci></apply><ci id="S3.E32.m1.1.1.cmml" xref="S3.E32.m1.1.1">ℓ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E32.m1.1c">\displaystyle\mathcal{I}_{N}(\ell)</annotation><annotation encoding="application/x-llamapun" id="S3.E32.m1.1d">caligraphic_I start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( roman_ℓ )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S3.E32.m2.1"><semantics id="S3.E32.m2.1a"><mo id="S3.E32.m2.1.1" xref="S3.E32.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S3.E32.m2.1b"><eq id="S3.E32.m2.1.1.cmml" xref="S3.E32.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S3.E32.m2.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S3.E32.m2.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\mathcal{I}^{\textrm{graviton}}_{N}(\ell)-\mathcal{I}^{\textrm{% ghost}}_{N}(\ell)," class="ltx_Math" display="inline" id="S3.E32.m3.3"><semantics id="S3.E32.m3.3a"><mrow id="S3.E32.m3.3.3.1" xref="S3.E32.m3.3.3.1.1.cmml"><mrow id="S3.E32.m3.3.3.1.1" xref="S3.E32.m3.3.3.1.1.cmml"><mrow id="S3.E32.m3.3.3.1.1.2" xref="S3.E32.m3.3.3.1.1.2.cmml"><msubsup id="S3.E32.m3.3.3.1.1.2.2" xref="S3.E32.m3.3.3.1.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E32.m3.3.3.1.1.2.2.2.2" xref="S3.E32.m3.3.3.1.1.2.2.2.2.cmml">ℐ</mi><mi id="S3.E32.m3.3.3.1.1.2.2.3" xref="S3.E32.m3.3.3.1.1.2.2.3.cmml">N</mi><mtext id="S3.E32.m3.3.3.1.1.2.2.2.3" xref="S3.E32.m3.3.3.1.1.2.2.2.3a.cmml">graviton</mtext></msubsup><mo id="S3.E32.m3.3.3.1.1.2.1" xref="S3.E32.m3.3.3.1.1.2.1.cmml">⁢</mo><mrow id="S3.E32.m3.3.3.1.1.2.3.2" xref="S3.E32.m3.3.3.1.1.2.cmml"><mo id="S3.E32.m3.3.3.1.1.2.3.2.1" stretchy="false" xref="S3.E32.m3.3.3.1.1.2.cmml">(</mo><mi id="S3.E32.m3.1.1" mathvariant="normal" xref="S3.E32.m3.1.1.cmml">ℓ</mi><mo id="S3.E32.m3.3.3.1.1.2.3.2.2" stretchy="false" xref="S3.E32.m3.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.E32.m3.3.3.1.1.1" xref="S3.E32.m3.3.3.1.1.1.cmml">−</mo><mrow id="S3.E32.m3.3.3.1.1.3" xref="S3.E32.m3.3.3.1.1.3.cmml"><msubsup id="S3.E32.m3.3.3.1.1.3.2" xref="S3.E32.m3.3.3.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E32.m3.3.3.1.1.3.2.2.2" xref="S3.E32.m3.3.3.1.1.3.2.2.2.cmml">ℐ</mi><mi id="S3.E32.m3.3.3.1.1.3.2.3" xref="S3.E32.m3.3.3.1.1.3.2.3.cmml">N</mi><mtext id="S3.E32.m3.3.3.1.1.3.2.2.3" xref="S3.E32.m3.3.3.1.1.3.2.2.3a.cmml">ghost</mtext></msubsup><mo id="S3.E32.m3.3.3.1.1.3.1" xref="S3.E32.m3.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S3.E32.m3.3.3.1.1.3.3.2" xref="S3.E32.m3.3.3.1.1.3.cmml"><mo id="S3.E32.m3.3.3.1.1.3.3.2.1" stretchy="false" xref="S3.E32.m3.3.3.1.1.3.cmml">(</mo><mi id="S3.E32.m3.2.2" mathvariant="normal" xref="S3.E32.m3.2.2.cmml">ℓ</mi><mo id="S3.E32.m3.3.3.1.1.3.3.2.2" stretchy="false" xref="S3.E32.m3.3.3.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S3.E32.m3.3.3.1.2" xref="S3.E32.m3.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E32.m3.3b"><apply id="S3.E32.m3.3.3.1.1.cmml" xref="S3.E32.m3.3.3.1"><minus id="S3.E32.m3.3.3.1.1.1.cmml" xref="S3.E32.m3.3.3.1.1.1"></minus><apply id="S3.E32.m3.3.3.1.1.2.cmml" xref="S3.E32.m3.3.3.1.1.2"><times id="S3.E32.m3.3.3.1.1.2.1.cmml" xref="S3.E32.m3.3.3.1.1.2.1"></times><apply id="S3.E32.m3.3.3.1.1.2.2.cmml" xref="S3.E32.m3.3.3.1.1.2.2"><csymbol cd="ambiguous" id="S3.E32.m3.3.3.1.1.2.2.1.cmml" xref="S3.E32.m3.3.3.1.1.2.2">subscript</csymbol><apply id="S3.E32.m3.3.3.1.1.2.2.2.cmml" xref="S3.E32.m3.3.3.1.1.2.2"><csymbol cd="ambiguous" id="S3.E32.m3.3.3.1.1.2.2.2.1.cmml" xref="S3.E32.m3.3.3.1.1.2.2">superscript</csymbol><ci id="S3.E32.m3.3.3.1.1.2.2.2.2.cmml" xref="S3.E32.m3.3.3.1.1.2.2.2.2">ℐ</ci><ci id="S3.E32.m3.3.3.1.1.2.2.2.3a.cmml" xref="S3.E32.m3.3.3.1.1.2.2.2.3"><mtext id="S3.E32.m3.3.3.1.1.2.2.2.3.cmml" mathsize="70%" xref="S3.E32.m3.3.3.1.1.2.2.2.3">graviton</mtext></ci></apply><ci id="S3.E32.m3.3.3.1.1.2.2.3.cmml" xref="S3.E32.m3.3.3.1.1.2.2.3">𝑁</ci></apply><ci id="S3.E32.m3.1.1.cmml" xref="S3.E32.m3.1.1">ℓ</ci></apply><apply id="S3.E32.m3.3.3.1.1.3.cmml" xref="S3.E32.m3.3.3.1.1.3"><times id="S3.E32.m3.3.3.1.1.3.1.cmml" xref="S3.E32.m3.3.3.1.1.3.1"></times><apply id="S3.E32.m3.3.3.1.1.3.2.cmml" xref="S3.E32.m3.3.3.1.1.3.2"><csymbol cd="ambiguous" id="S3.E32.m3.3.3.1.1.3.2.1.cmml" xref="S3.E32.m3.3.3.1.1.3.2">subscript</csymbol><apply id="S3.E32.m3.3.3.1.1.3.2.2.cmml" xref="S3.E32.m3.3.3.1.1.3.2"><csymbol cd="ambiguous" id="S3.E32.m3.3.3.1.1.3.2.2.1.cmml" xref="S3.E32.m3.3.3.1.1.3.2">superscript</csymbol><ci id="S3.E32.m3.3.3.1.1.3.2.2.2.cmml" xref="S3.E32.m3.3.3.1.1.3.2.2.2">ℐ</ci><ci id="S3.E32.m3.3.3.1.1.3.2.2.3a.cmml" xref="S3.E32.m3.3.3.1.1.3.2.2.3"><mtext id="S3.E32.m3.3.3.1.1.3.2.2.3.cmml" mathsize="70%" xref="S3.E32.m3.3.3.1.1.3.2.2.3">ghost</mtext></ci></apply><ci id="S3.E32.m3.3.3.1.1.3.2.3.cmml" xref="S3.E32.m3.3.3.1.1.3.2.3">𝑁</ci></apply><ci id="S3.E32.m3.2.2.cmml" xref="S3.E32.m3.2.2">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E32.m3.3c">\displaystyle\mathcal{I}^{\textrm{graviton}}_{N}(\ell)-\mathcal{I}^{\textrm{% ghost}}_{N}(\ell),</annotation><annotation encoding="application/x-llamapun" id="S3.E32.m3.3d">caligraphic_I start_POSTSUPERSCRIPT graviton end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( roman_ℓ ) - caligraphic_I start_POSTSUPERSCRIPT ghost end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( roman_ℓ ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(32)</span></td> </tr></tbody> <tbody id="S3.E33"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{M}^{1\textrm{-loop}}_{N}" class="ltx_Math" display="inline" id="S3.E33.m1.1"><semantics id="S3.E33.m1.1a"><msubsup id="S3.E33.m1.1.1" xref="S3.E33.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E33.m1.1.1.2.2" xref="S3.E33.m1.1.1.2.2.cmml">ℳ</mi><mi id="S3.E33.m1.1.1.3" xref="S3.E33.m1.1.1.3.cmml">N</mi><mrow id="S3.E33.m1.1.1.2.3" xref="S3.E33.m1.1.1.2.3.cmml"><mn id="S3.E33.m1.1.1.2.3.2" xref="S3.E33.m1.1.1.2.3.2.cmml">1</mn><mo id="S3.E33.m1.1.1.2.3.1" xref="S3.E33.m1.1.1.2.3.1.cmml">⁢</mo><mtext id="S3.E33.m1.1.1.2.3.3" xref="S3.E33.m1.1.1.2.3.3a.cmml">-loop</mtext></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.E33.m1.1b"><apply id="S3.E33.m1.1.1.cmml" xref="S3.E33.m1.1.1"><csymbol cd="ambiguous" id="S3.E33.m1.1.1.1.cmml" xref="S3.E33.m1.1.1">subscript</csymbol><apply id="S3.E33.m1.1.1.2.cmml" xref="S3.E33.m1.1.1"><csymbol cd="ambiguous" id="S3.E33.m1.1.1.2.1.cmml" xref="S3.E33.m1.1.1">superscript</csymbol><ci id="S3.E33.m1.1.1.2.2.cmml" xref="S3.E33.m1.1.1.2.2">ℳ</ci><apply id="S3.E33.m1.1.1.2.3.cmml" xref="S3.E33.m1.1.1.2.3"><times id="S3.E33.m1.1.1.2.3.1.cmml" xref="S3.E33.m1.1.1.2.3.1"></times><cn id="S3.E33.m1.1.1.2.3.2.cmml" type="integer" xref="S3.E33.m1.1.1.2.3.2">1</cn><ci id="S3.E33.m1.1.1.2.3.3a.cmml" xref="S3.E33.m1.1.1.2.3.3"><mtext id="S3.E33.m1.1.1.2.3.3.cmml" mathsize="70%" xref="S3.E33.m1.1.1.2.3.3">-loop</mtext></ci></apply></apply><ci id="S3.E33.m1.1.1.3.cmml" xref="S3.E33.m1.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E33.m1.1c">\displaystyle\mathcal{M}^{1\textrm{-loop}}_{N}</annotation><annotation encoding="application/x-llamapun" id="S3.E33.m1.1d">caligraphic_M start_POSTSUPERSCRIPT 1 -loop end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S3.E33.m2.1"><semantics id="S3.E33.m2.1a"><mo id="S3.E33.m2.1.1" xref="S3.E33.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S3.E33.m2.1b"><eq id="S3.E33.m2.1.1.cmml" xref="S3.E33.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S3.E33.m2.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S3.E33.m2.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\int\frac{d^{d}\ell}{(2\pi)^{d}}\,\mathcal{I}_{N}(\ell)." class="ltx_Math" display="inline" id="S3.E33.m3.3"><semantics id="S3.E33.m3.3a"><mrow id="S3.E33.m3.3.3.1" xref="S3.E33.m3.3.3.1.1.cmml"><mstyle displaystyle="true" id="S3.E33.m3.3.3.1.1" xref="S3.E33.m3.3.3.1.1.cmml"><mrow id="S3.E33.m3.3.3.1.1a" xref="S3.E33.m3.3.3.1.1.cmml"><mo id="S3.E33.m3.3.3.1.1.1" xref="S3.E33.m3.3.3.1.1.1.cmml">∫</mo><mrow id="S3.E33.m3.3.3.1.1.2" xref="S3.E33.m3.3.3.1.1.2.cmml"><mfrac id="S3.E33.m3.1.1" 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id="S3.E33.m3.3c">\displaystyle\int\frac{d^{d}\ell}{(2\pi)^{d}}\,\mathcal{I}_{N}(\ell).</annotation><annotation encoding="application/x-llamapun" id="S3.E33.m3.3d">∫ divide start_ARG italic_d start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT roman_ℓ end_ARG start_ARG ( 2 italic_π ) start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_ARG caligraphic_I start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( roman_ℓ ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(33)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p7.3">As usual, the ghost loop comes with a negative sign.</p> </div> <div class="ltx_para" id="S3.p8"> <p class="ltx_p" id="S3.p8.3">If we are interested in computing one-loop amplitudes, we can just impose the on-shell condition on external legs. In particular, our prescription enables a clean removal of tadpoles diagrams and external-leg bubbles. The amplitude is defined as</p> <table class="ltx_equation ltx_eqn_table" id="S3.E34"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="M^{1\textrm{-loop}}_{N}=\frac{1}{N}\int\frac{d^{d}\ell}{(2\pi)^{d}}\\ \times\bigg{\{}\sum_{n=2}^{N}\frac{1}{(n-1)!}\sum_{\begin{subarray}{c}1\ldots N% =P_{1}\cup\ldots\cup P_{n}\\ |P_{n}|>1\end{subarray}}\prod_{i=1}^{n-1}I_{P_{i}}^{\mu_{i}\nu_{i}}\\ \times|P_{n}|\tilde{K}^{\mu_{n}\nu_{n}}_{P_{n}\alpha\beta}(\mathbb{P}^{-1})^{% \alpha\beta\mu\nu}\mathcal{V}_{\mu\nu\mu_{1}\nu_{1}\cdots\mu_{n}\nu_{n}}^{(n+1% )}(P_{1},\ldots,\ell P_{n})\\ -\sum_{\begin{subarray}{c}1\ldots N=Q\cup R\\ |R|>1\end{subarray}}|Q|I_{Q}^{\mu\nu}\tilde{J}^{\sigma}_{R\rho}\mathcal{V}^{(% \textrm{g})\rho}_{\mu\nu\sigma}(Q,\ell P)\bigg{\}}." class="ltx_Math" display="block" id="S3.E34.m1.75"><semantics id="S3.E34.m1.75a"><mtable displaystyle="true" 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id="S3.E34.m1.1.1.1.1.1.1.1.1.1.1.2.1.1.3.4.3" xref="S3.E34.m1.1.1.1.1.1.1.1.1.1.1.2.1.1.3.4.3.cmml">n</mi></msub></mrow></mrow></mtd></mtr><mtr id="S3.E34.m1.1.1.1.1.1.1.1.1.1.1c" xref="S3.E34.m1.1.1.1.1.1.1.1.2.cmml"><mtd id="S3.E34.m1.1.1.1.1.1.1.1.1.1.1d" xref="S3.E34.m1.1.1.1.1.1.1.1.2.cmml"><mrow id="S3.E34.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1" xref="S3.E34.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mrow id="S3.E34.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1" xref="S3.E34.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml"><mo id="S3.E34.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.E34.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1.cmml">|</mo><msub id="S3.E34.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1" xref="S3.E34.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.E34.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2" xref="S3.E34.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">P</mi><mi id="S3.E34.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3" xref="S3.E34.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S3.E34.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.E34.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.E34.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2" xref="S3.E34.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">></mo><mn id="S3.E34.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3" xref="S3.E34.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml">1</mn></mrow></mtd></mtr></mtable></munder><munderover id="S3.E34.m1.22.22.22.14.14.15.3"><mo id="S3.E34.m1.17.17.17.9.9.9" movablelimits="false" xref="S3.E34.m1.17.17.17.9.9.9.cmml">∏</mo><mrow id="S3.E34.m1.18.18.18.10.10.10.1" xref="S3.E34.m1.18.18.18.10.10.10.1.cmml"><mi id="S3.E34.m1.18.18.18.10.10.10.1.2" xref="S3.E34.m1.18.18.18.10.10.10.1.2.cmml">i</mi><mo id="S3.E34.m1.18.18.18.10.10.10.1.1" xref="S3.E34.m1.18.18.18.10.10.10.1.1.cmml">=</mo><mn id="S3.E34.m1.18.18.18.10.10.10.1.3" xref="S3.E34.m1.18.18.18.10.10.10.1.3.cmml">1</mn></mrow><mrow id="S3.E34.m1.19.19.19.11.11.11.1" xref="S3.E34.m1.19.19.19.11.11.11.1.cmml"><mi id="S3.E34.m1.19.19.19.11.11.11.1.2" xref="S3.E34.m1.19.19.19.11.11.11.1.2.cmml">n</mi><mo id="S3.E34.m1.19.19.19.11.11.11.1.1" xref="S3.E34.m1.19.19.19.11.11.11.1.1.cmml">−</mo><mn id="S3.E34.m1.19.19.19.11.11.11.1.3" xref="S3.E34.m1.19.19.19.11.11.11.1.3.cmml">1</mn></mrow></munderover><msubsup id="S3.E34.m1.22.22.22.14.14.15.4"><mi id="S3.E34.m1.20.20.20.12.12.12" xref="S3.E34.m1.20.20.20.12.12.12.cmml">I</mi><msub id="S3.E34.m1.21.21.21.13.13.13.1" xref="S3.E34.m1.21.21.21.13.13.13.1.cmml"><mi id="S3.E34.m1.21.21.21.13.13.13.1.2" xref="S3.E34.m1.21.21.21.13.13.13.1.2.cmml">P</mi><mi id="S3.E34.m1.21.21.21.13.13.13.1.3" xref="S3.E34.m1.21.21.21.13.13.13.1.3.cmml">i</mi></msub><mrow id="S3.E34.m1.22.22.22.14.14.14.1" xref="S3.E34.m1.22.22.22.14.14.14.1.cmml"><msub id="S3.E34.m1.22.22.22.14.14.14.1.2" xref="S3.E34.m1.22.22.22.14.14.14.1.2.cmml"><mi id="S3.E34.m1.22.22.22.14.14.14.1.2.2" xref="S3.E34.m1.22.22.22.14.14.14.1.2.2.cmml">μ</mi><mi id="S3.E34.m1.22.22.22.14.14.14.1.2.3" xref="S3.E34.m1.22.22.22.14.14.14.1.2.3.cmml">i</mi></msub><mo id="S3.E34.m1.22.22.22.14.14.14.1.1" xref="S3.E34.m1.22.22.22.14.14.14.1.1.cmml">⁢</mo><msub id="S3.E34.m1.22.22.22.14.14.14.1.3" xref="S3.E34.m1.22.22.22.14.14.14.1.3.cmml"><mi id="S3.E34.m1.22.22.22.14.14.14.1.3.2" xref="S3.E34.m1.22.22.22.14.14.14.1.3.2.cmml">ν</mi><mi id="S3.E34.m1.22.22.22.14.14.14.1.3.3" xref="S3.E34.m1.22.22.22.14.14.14.1.3.3.cmml">i</mi></msub></mrow></msubsup></mrow></mrow></mtd></mtr><mtr id="S3.E34.m1.75.75.5e"><mtd class="ltx_align_right" columnalign="right" id="S3.E34.m1.75.75.5f"><mrow id="S3.E34.m1.75.75.5.74.30.30"><mi id="S3.E34.m1.75.75.5.74.30.30.31" xref="S3.E34.m1.71.71.1.1.1.cmml"></mi><mo id="S3.E34.m1.23.23.23.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.E34.m1.23.23.23.1.1.1.cmml">×</mo><mrow id="S3.E34.m1.75.75.5.74.30.30.30"><mrow id="S3.E34.m1.72.72.2.71.27.27.27.1.1"><mo id="S3.E34.m1.24.24.24.2.2.2" stretchy="false" xref="S3.E34.m1.71.71.1.1.1.cmml">|</mo><msub id="S3.E34.m1.72.72.2.71.27.27.27.1.1.1"><mi id="S3.E34.m1.25.25.25.3.3.3" xref="S3.E34.m1.25.25.25.3.3.3.cmml">P</mi><mi id="S3.E34.m1.26.26.26.4.4.4.1" xref="S3.E34.m1.26.26.26.4.4.4.1.cmml">n</mi></msub><mo id="S3.E34.m1.27.27.27.5.5.5" stretchy="false" xref="S3.E34.m1.71.71.1.1.1.cmml">|</mo></mrow><mo id="S3.E34.m1.75.75.5.74.30.30.30.5" xref="S3.E34.m1.71.71.1.1.1.cmml">⁢</mo><msubsup id="S3.E34.m1.75.75.5.74.30.30.30.6"><mover accent="true" id="S3.E34.m1.28.28.28.6.6.6" xref="S3.E34.m1.28.28.28.6.6.6.cmml"><mi id="S3.E34.m1.28.28.28.6.6.6.2" xref="S3.E34.m1.28.28.28.6.6.6.2.cmml">K</mi><mo id="S3.E34.m1.28.28.28.6.6.6.1" xref="S3.E34.m1.28.28.28.6.6.6.1.cmml">~</mo></mover><mrow id="S3.E34.m1.30.30.30.8.8.8.1" xref="S3.E34.m1.30.30.30.8.8.8.1.cmml"><msub id="S3.E34.m1.30.30.30.8.8.8.1.2" xref="S3.E34.m1.30.30.30.8.8.8.1.2.cmml"><mi id="S3.E34.m1.30.30.30.8.8.8.1.2.2" xref="S3.E34.m1.30.30.30.8.8.8.1.2.2.cmml">P</mi><mi id="S3.E34.m1.30.30.30.8.8.8.1.2.3" xref="S3.E34.m1.30.30.30.8.8.8.1.2.3.cmml">n</mi></msub><mo id="S3.E34.m1.30.30.30.8.8.8.1.1" xref="S3.E34.m1.30.30.30.8.8.8.1.1.cmml">⁢</mo><mi id="S3.E34.m1.30.30.30.8.8.8.1.3" xref="S3.E34.m1.30.30.30.8.8.8.1.3.cmml">α</mi><mo id="S3.E34.m1.30.30.30.8.8.8.1.1a" xref="S3.E34.m1.30.30.30.8.8.8.1.1.cmml">⁢</mo><mi id="S3.E34.m1.30.30.30.8.8.8.1.4" xref="S3.E34.m1.30.30.30.8.8.8.1.4.cmml">β</mi></mrow><mrow id="S3.E34.m1.29.29.29.7.7.7.1" xref="S3.E34.m1.29.29.29.7.7.7.1.cmml"><msub id="S3.E34.m1.29.29.29.7.7.7.1.2" xref="S3.E34.m1.29.29.29.7.7.7.1.2.cmml"><mi id="S3.E34.m1.29.29.29.7.7.7.1.2.2" xref="S3.E34.m1.29.29.29.7.7.7.1.2.2.cmml">μ</mi><mi id="S3.E34.m1.29.29.29.7.7.7.1.2.3" xref="S3.E34.m1.29.29.29.7.7.7.1.2.3.cmml">n</mi></msub><mo id="S3.E34.m1.29.29.29.7.7.7.1.1" xref="S3.E34.m1.29.29.29.7.7.7.1.1.cmml">⁢</mo><msub id="S3.E34.m1.29.29.29.7.7.7.1.3" xref="S3.E34.m1.29.29.29.7.7.7.1.3.cmml"><mi id="S3.E34.m1.29.29.29.7.7.7.1.3.2" xref="S3.E34.m1.29.29.29.7.7.7.1.3.2.cmml">ν</mi><mi id="S3.E34.m1.29.29.29.7.7.7.1.3.3" xref="S3.E34.m1.29.29.29.7.7.7.1.3.3.cmml">n</mi></msub></mrow></msubsup><mo id="S3.E34.m1.75.75.5.74.30.30.30.5a" xref="S3.E34.m1.71.71.1.1.1.cmml">⁢</mo><msup id="S3.E34.m1.73.73.3.72.28.28.28.2"><mrow id="S3.E34.m1.73.73.3.72.28.28.28.2.1.1"><mo id="S3.E34.m1.31.31.31.9.9.9" stretchy="false" xref="S3.E34.m1.71.71.1.1.1.cmml">(</mo><msup id="S3.E34.m1.73.73.3.72.28.28.28.2.1.1.1"><mi id="S3.E34.m1.32.32.32.10.10.10" xref="S3.E34.m1.32.32.32.10.10.10.cmml">ℙ</mi><mrow id="S3.E34.m1.33.33.33.11.11.11.1" xref="S3.E34.m1.33.33.33.11.11.11.1.cmml"><mo id="S3.E34.m1.33.33.33.11.11.11.1a" xref="S3.E34.m1.33.33.33.11.11.11.1.cmml">−</mo><mn id="S3.E34.m1.33.33.33.11.11.11.1.2" xref="S3.E34.m1.33.33.33.11.11.11.1.2.cmml">1</mn></mrow></msup><mo id="S3.E34.m1.34.34.34.12.12.12" stretchy="false" xref="S3.E34.m1.71.71.1.1.1.cmml">)</mo></mrow><mrow id="S3.E34.m1.35.35.35.13.13.13.1" xref="S3.E34.m1.35.35.35.13.13.13.1.cmml"><mi id="S3.E34.m1.35.35.35.13.13.13.1.2" xref="S3.E34.m1.35.35.35.13.13.13.1.2.cmml">α</mi><mo id="S3.E34.m1.35.35.35.13.13.13.1.1" xref="S3.E34.m1.35.35.35.13.13.13.1.1.cmml">⁢</mo><mi id="S3.E34.m1.35.35.35.13.13.13.1.3" xref="S3.E34.m1.35.35.35.13.13.13.1.3.cmml">β</mi><mo id="S3.E34.m1.35.35.35.13.13.13.1.1a" xref="S3.E34.m1.35.35.35.13.13.13.1.1.cmml">⁢</mo><mi id="S3.E34.m1.35.35.35.13.13.13.1.4" xref="S3.E34.m1.35.35.35.13.13.13.1.4.cmml">μ</mi><mo id="S3.E34.m1.35.35.35.13.13.13.1.1b" xref="S3.E34.m1.35.35.35.13.13.13.1.1.cmml">⁢</mo><mi id="S3.E34.m1.35.35.35.13.13.13.1.5" xref="S3.E34.m1.35.35.35.13.13.13.1.5.cmml">ν</mi></mrow></msup><mo id="S3.E34.m1.75.75.5.74.30.30.30.5b" xref="S3.E34.m1.71.71.1.1.1.cmml">⁢</mo><msubsup id="S3.E34.m1.75.75.5.74.30.30.30.7"><mi class="ltx_font_mathcaligraphic" id="S3.E34.m1.36.36.36.14.14.14" xref="S3.E34.m1.36.36.36.14.14.14.cmml">𝒱</mi><mrow id="S3.E34.m1.37.37.37.15.15.15.1" xref="S3.E34.m1.37.37.37.15.15.15.1.cmml"><mi id="S3.E34.m1.37.37.37.15.15.15.1.2" xref="S3.E34.m1.37.37.37.15.15.15.1.2.cmml">μ</mi><mo id="S3.E34.m1.37.37.37.15.15.15.1.1" xref="S3.E34.m1.37.37.37.15.15.15.1.1.cmml">⁢</mo><mi id="S3.E34.m1.37.37.37.15.15.15.1.3" xref="S3.E34.m1.37.37.37.15.15.15.1.3.cmml">ν</mi><mo id="S3.E34.m1.37.37.37.15.15.15.1.1a" xref="S3.E34.m1.37.37.37.15.15.15.1.1.cmml">⁢</mo><msub id="S3.E34.m1.37.37.37.15.15.15.1.4" xref="S3.E34.m1.37.37.37.15.15.15.1.4.cmml"><mi id="S3.E34.m1.37.37.37.15.15.15.1.4.2" xref="S3.E34.m1.37.37.37.15.15.15.1.4.2.cmml">μ</mi><mn id="S3.E34.m1.37.37.37.15.15.15.1.4.3" xref="S3.E34.m1.37.37.37.15.15.15.1.4.3.cmml">1</mn></msub><mo id="S3.E34.m1.37.37.37.15.15.15.1.1b" xref="S3.E34.m1.37.37.37.15.15.15.1.1.cmml">⁢</mo><msub id="S3.E34.m1.37.37.37.15.15.15.1.5" xref="S3.E34.m1.37.37.37.15.15.15.1.5.cmml"><mi id="S3.E34.m1.37.37.37.15.15.15.1.5.2" xref="S3.E34.m1.37.37.37.15.15.15.1.5.2.cmml">ν</mi><mn id="S3.E34.m1.37.37.37.15.15.15.1.5.3" xref="S3.E34.m1.37.37.37.15.15.15.1.5.3.cmml">1</mn></msub><mo id="S3.E34.m1.37.37.37.15.15.15.1.1c" xref="S3.E34.m1.37.37.37.15.15.15.1.1.cmml">⁢</mo><mi id="S3.E34.m1.37.37.37.15.15.15.1.6" mathvariant="normal" xref="S3.E34.m1.37.37.37.15.15.15.1.6.cmml">⋯</mi><mo id="S3.E34.m1.37.37.37.15.15.15.1.1d" xref="S3.E34.m1.37.37.37.15.15.15.1.1.cmml">⁢</mo><msub id="S3.E34.m1.37.37.37.15.15.15.1.7" xref="S3.E34.m1.37.37.37.15.15.15.1.7.cmml"><mi id="S3.E34.m1.37.37.37.15.15.15.1.7.2" xref="S3.E34.m1.37.37.37.15.15.15.1.7.2.cmml">μ</mi><mi id="S3.E34.m1.37.37.37.15.15.15.1.7.3" xref="S3.E34.m1.37.37.37.15.15.15.1.7.3.cmml">n</mi></msub><mo id="S3.E34.m1.37.37.37.15.15.15.1.1e" xref="S3.E34.m1.37.37.37.15.15.15.1.1.cmml">⁢</mo><msub id="S3.E34.m1.37.37.37.15.15.15.1.8" xref="S3.E34.m1.37.37.37.15.15.15.1.8.cmml"><mi id="S3.E34.m1.37.37.37.15.15.15.1.8.2" xref="S3.E34.m1.37.37.37.15.15.15.1.8.2.cmml">ν</mi><mi id="S3.E34.m1.37.37.37.15.15.15.1.8.3" xref="S3.E34.m1.37.37.37.15.15.15.1.8.3.cmml">n</mi></msub></mrow><mrow id="S3.E34.m1.38.38.38.16.16.16.1.1" xref="S3.E34.m1.38.38.38.16.16.16.1.1.1.cmml"><mo id="S3.E34.m1.38.38.38.16.16.16.1.1.2" stretchy="false" xref="S3.E34.m1.38.38.38.16.16.16.1.1.1.cmml">(</mo><mrow id="S3.E34.m1.38.38.38.16.16.16.1.1.1" xref="S3.E34.m1.38.38.38.16.16.16.1.1.1.cmml"><mi id="S3.E34.m1.38.38.38.16.16.16.1.1.1.2" xref="S3.E34.m1.38.38.38.16.16.16.1.1.1.2.cmml">n</mi><mo id="S3.E34.m1.38.38.38.16.16.16.1.1.1.1" xref="S3.E34.m1.38.38.38.16.16.16.1.1.1.1.cmml">+</mo><mn id="S3.E34.m1.38.38.38.16.16.16.1.1.1.3" xref="S3.E34.m1.38.38.38.16.16.16.1.1.1.3.cmml">1</mn></mrow><mo id="S3.E34.m1.38.38.38.16.16.16.1.1.3" stretchy="false" xref="S3.E34.m1.38.38.38.16.16.16.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S3.E34.m1.75.75.5.74.30.30.30.5c" xref="S3.E34.m1.71.71.1.1.1.cmml">⁢</mo><mrow id="S3.E34.m1.75.75.5.74.30.30.30.4.2"><mo id="S3.E34.m1.39.39.39.17.17.17" stretchy="false" xref="S3.E34.m1.71.71.1.1.1.cmml">(</mo><msub id="S3.E34.m1.74.74.4.73.29.29.29.3.1.1"><mi id="S3.E34.m1.40.40.40.18.18.18" xref="S3.E34.m1.40.40.40.18.18.18.cmml">P</mi><mn id="S3.E34.m1.41.41.41.19.19.19.1" xref="S3.E34.m1.41.41.41.19.19.19.1.cmml">1</mn></msub><mo id="S3.E34.m1.42.42.42.20.20.20" xref="S3.E34.m1.71.71.1.1.1.cmml">,</mo><mi id="S3.E34.m1.43.43.43.21.21.21" mathvariant="normal" xref="S3.E34.m1.43.43.43.21.21.21.cmml">…</mi><mo id="S3.E34.m1.44.44.44.22.22.22" xref="S3.E34.m1.71.71.1.1.1.cmml">,</mo><mrow id="S3.E34.m1.75.75.5.74.30.30.30.4.2.2"><mi id="S3.E34.m1.45.45.45.23.23.23" mathvariant="normal" xref="S3.E34.m1.45.45.45.23.23.23.cmml">ℓ</mi><mo id="S3.E34.m1.75.75.5.74.30.30.30.4.2.2.1" xref="S3.E34.m1.71.71.1.1.1.cmml">⁢</mo><msub id="S3.E34.m1.75.75.5.74.30.30.30.4.2.2.2"><mi id="S3.E34.m1.46.46.46.24.24.24" xref="S3.E34.m1.46.46.46.24.24.24.cmml">P</mi><mi id="S3.E34.m1.47.47.47.25.25.25.1" xref="S3.E34.m1.47.47.47.25.25.25.1.cmml">n</mi></msub></mrow><mo id="S3.E34.m1.48.48.48.26.26.26" stretchy="false" xref="S3.E34.m1.71.71.1.1.1.cmml">)</mo></mrow></mrow></mrow></mtd></mtr><mtr id="S3.E34.m1.75.75.5g"><mtd class="ltx_align_right" columnalign="right" id="S3.E34.m1.75.75.5h"><mrow id="S3.E34.m1.70.70.70.23.23"><mo id="S3.E34.m1.49.49.49.2.2.2" rspace="0.055em" xref="S3.E34.m1.49.49.49.2.2.2.cmml">−</mo><munder id="S3.E34.m1.70.70.70.23.23.24"><mo id="S3.E34.m1.50.50.50.3.3.3" movablelimits="false" rspace="0em" xref="S3.E34.m1.50.50.50.3.3.3.cmml">∑</mo><mtable id="S3.E34.m1.2.2.2.1.1.1.1.1.1.1" rowspacing="0pt" xref="S3.E34.m1.2.2.2.1.1.1.1.2.cmml"><mtr id="S3.E34.m1.2.2.2.1.1.1.1.1.1.1a" xref="S3.E34.m1.2.2.2.1.1.1.1.2.cmml"><mtd id="S3.E34.m1.2.2.2.1.1.1.1.1.1.1b" xref="S3.E34.m1.2.2.2.1.1.1.1.2.cmml"><mrow id="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.2.1.1" xref="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.2.1.1.cmml"><mrow id="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.2.1.1.2" xref="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.2.1.1.2.cmml"><mn id="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.2.1.1.2.2" xref="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.2.1.1.2.2.cmml">1</mn><mo id="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.2.1.1.2.1" xref="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.2.1.1.2.1.cmml">⁢</mo><mi id="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.2.1.1.2.3" mathvariant="normal" xref="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.2.1.1.2.3.cmml">…</mi><mo id="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.2.1.1.2.1a" xref="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.2.1.1.2.1.cmml">⁢</mo><mi id="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.2.1.1.2.4" xref="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.2.1.1.2.4.cmml">N</mi></mrow><mo id="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.2.1.1.1" xref="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.2.1.1.1.cmml">=</mo><mrow id="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.2.1.1.3" xref="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.2.1.1.3.cmml"><mi id="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.2.1.1.3.2" xref="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.2.1.1.3.2.cmml">Q</mi><mo id="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.2.1.1.3.1" xref="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.2.1.1.3.1.cmml">∪</mo><mi id="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.2.1.1.3.3" xref="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.2.1.1.3.3.cmml">R</mi></mrow></mrow></mtd></mtr><mtr id="S3.E34.m1.2.2.2.1.1.1.1.1.1.1c" xref="S3.E34.m1.2.2.2.1.1.1.1.2.cmml"><mtd id="S3.E34.m1.2.2.2.1.1.1.1.1.1.1d" xref="S3.E34.m1.2.2.2.1.1.1.1.2.cmml"><mrow id="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.1.1.1.1" xref="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.1.1.1.1.cmml"><mrow id="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.1.1.1.1.3.2" xref="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.1.1.1.1.3.1.cmml"><mo id="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.1.1.1.1.3.1.1.cmml">|</mo><mi id="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1" xref="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.1.1.1.1.1.cmml">R</mi><mo id="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.1.1.1.1.3.1.1.cmml">|</mo></mrow><mo id="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.1.1.1.1.2" xref="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.1.1.1.1.2.cmml">></mo><mn id="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.1.1.1.1.4" xref="S3.E34.m1.2.2.2.1.1.1.1.1.1.1.1.1.1.1.4.cmml">1</mn></mrow></mtd></mtr></mtable></munder><mo fence="false" id="S3.E34.m1.51.51.51.4.4.4" rspace="0.167em" stretchy="false" xref="S3.E34.m1.71.71.1.1.1.cmml">|</mo><mi id="S3.E34.m1.52.52.52.5.5.5" xref="S3.E34.m1.52.52.52.5.5.5.cmml">Q</mi><mo fence="false" id="S3.E34.m1.53.53.53.6.6.6" rspace="0.167em" stretchy="false" xref="S3.E34.m1.71.71.1.1.1.cmml">|</mo><msubsup id="S3.E34.m1.70.70.70.23.23.25"><mi id="S3.E34.m1.54.54.54.7.7.7" xref="S3.E34.m1.54.54.54.7.7.7.cmml">I</mi><mi id="S3.E34.m1.55.55.55.8.8.8.1" xref="S3.E34.m1.55.55.55.8.8.8.1.cmml">Q</mi><mrow id="S3.E34.m1.56.56.56.9.9.9.1" xref="S3.E34.m1.56.56.56.9.9.9.1.cmml"><mi id="S3.E34.m1.56.56.56.9.9.9.1.2" xref="S3.E34.m1.56.56.56.9.9.9.1.2.cmml">μ</mi><mo id="S3.E34.m1.56.56.56.9.9.9.1.1" xref="S3.E34.m1.56.56.56.9.9.9.1.1.cmml">⁢</mo><mi id="S3.E34.m1.56.56.56.9.9.9.1.3" 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|R|>1\end{subarray}}|Q|I_{Q}^{\mu\nu}\tilde{J}^{\sigma}_{R\rho}\mathcal{V}^{(% \textrm{g})\rho}_{\mu\nu\sigma}(Q,\ell P)\bigg{\}}.</annotation><annotation encoding="application/x-llamapun" id="S3.E34.m1.75d">start_ROW start_CELL italic_M start_POSTSUPERSCRIPT 1 -loop end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG italic_N end_ARG ∫ divide start_ARG italic_d start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT roman_ℓ end_ARG start_ARG ( 2 italic_π ) start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_ARG end_CELL end_ROW start_ROW start_CELL × { ∑ start_POSTSUBSCRIPT italic_n = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG ( italic_n - 1 ) ! end_ARG ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL 1 … italic_N = italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∪ … ∪ italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL | italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | > 1 end_CELL end_ROW end_ARG end_POSTSUBSCRIPT ∏ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT italic_I start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL × | italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | over~ start_ARG italic_K end_ARG start_POSTSUPERSCRIPT italic_μ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT ( blackboard_P start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_α italic_β italic_μ italic_ν end_POSTSUPERSCRIPT caligraphic_V start_POSTSUBSCRIPT italic_μ italic_ν italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋯ italic_μ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_n + 1 ) end_POSTSUPERSCRIPT ( italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , roman_ℓ italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL - ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL 1 … italic_N = italic_Q ∪ italic_R end_CELL end_ROW start_ROW start_CELL | italic_R | > 1 end_CELL end_ROW end_ARG end_POSTSUBSCRIPT | italic_Q | italic_I start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT over~ start_ARG italic_J end_ARG start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_R italic_ρ end_POSTSUBSCRIPT caligraphic_V start_POSTSUPERSCRIPT ( g ) italic_ρ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ italic_ν italic_σ end_POSTSUBSCRIPT ( italic_Q , roman_ℓ italic_P ) } . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(34)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p8.2">This prescription follows directly from (<a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S3.E33" title="In III Loop recursions ‣ One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_tag">33</span></a>), with two modifications in equations (<a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S3.E26" title="In III Loop recursions ‣ One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_tag">26</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S3.E31" title="In III Loop recursions ‣ One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_tag">31</span></a>). First, we delete the terms involving vertex self-contractions. These correspond, respectively, to the terms multiplied by <math alttext="f_{n}" class="ltx_Math" display="inline" id="S3.p8.1.m1.1"><semantics id="S3.p8.1.m1.1a"><msub id="S3.p8.1.m1.1.1" xref="S3.p8.1.m1.1.1.cmml"><mi id="S3.p8.1.m1.1.1.2" xref="S3.p8.1.m1.1.1.2.cmml">f</mi><mi id="S3.p8.1.m1.1.1.3" xref="S3.p8.1.m1.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p8.1.m1.1b"><apply id="S3.p8.1.m1.1.1.cmml" xref="S3.p8.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p8.1.m1.1.1.1.cmml" xref="S3.p8.1.m1.1.1">subscript</csymbol><ci id="S3.p8.1.m1.1.1.2.cmml" xref="S3.p8.1.m1.1.1.2">𝑓</ci><ci id="S3.p8.1.m1.1.1.3.cmml" xref="S3.p8.1.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.1.m1.1c">f_{n}</annotation><annotation encoding="application/x-llamapun" id="S3.p8.1.m1.1d">italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S3.E25" title="In III Loop recursions ‣ One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_tag">25</span></a>) and the first line on the right-hand-side of (<a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S3.E30" title="In III Loop recursions ‣ One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_tag">30</span></a>). Second, we remove double contractions between cubic vertices with one external leg and any other vertex, which trace back to <math alttext="|P_{n}|=|R|=1" class="ltx_Math" display="inline" id="S3.p8.2.m2.2"><semantics id="S3.p8.2.m2.2a"><mrow id="S3.p8.2.m2.2.2" xref="S3.p8.2.m2.2.2.cmml"><mrow id="S3.p8.2.m2.2.2.1.1" xref="S3.p8.2.m2.2.2.1.2.cmml"><mo id="S3.p8.2.m2.2.2.1.1.2" stretchy="false" xref="S3.p8.2.m2.2.2.1.2.1.cmml">|</mo><msub id="S3.p8.2.m2.2.2.1.1.1" xref="S3.p8.2.m2.2.2.1.1.1.cmml"><mi id="S3.p8.2.m2.2.2.1.1.1.2" xref="S3.p8.2.m2.2.2.1.1.1.2.cmml">P</mi><mi id="S3.p8.2.m2.2.2.1.1.1.3" xref="S3.p8.2.m2.2.2.1.1.1.3.cmml">n</mi></msub><mo id="S3.p8.2.m2.2.2.1.1.3" stretchy="false" xref="S3.p8.2.m2.2.2.1.2.1.cmml">|</mo></mrow><mo id="S3.p8.2.m2.2.2.3" xref="S3.p8.2.m2.2.2.3.cmml">=</mo><mrow id="S3.p8.2.m2.2.2.4.2" xref="S3.p8.2.m2.2.2.4.1.cmml"><mo id="S3.p8.2.m2.2.2.4.2.1" stretchy="false" xref="S3.p8.2.m2.2.2.4.1.1.cmml">|</mo><mi id="S3.p8.2.m2.1.1" xref="S3.p8.2.m2.1.1.cmml">R</mi><mo id="S3.p8.2.m2.2.2.4.2.2" stretchy="false" xref="S3.p8.2.m2.2.2.4.1.1.cmml">|</mo></mrow><mo id="S3.p8.2.m2.2.2.5" xref="S3.p8.2.m2.2.2.5.cmml">=</mo><mn id="S3.p8.2.m2.2.2.6" xref="S3.p8.2.m2.2.2.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p8.2.m2.2b"><apply id="S3.p8.2.m2.2.2.cmml" xref="S3.p8.2.m2.2.2"><and id="S3.p8.2.m2.2.2a.cmml" xref="S3.p8.2.m2.2.2"></and><apply id="S3.p8.2.m2.2.2b.cmml" xref="S3.p8.2.m2.2.2"><eq id="S3.p8.2.m2.2.2.3.cmml" xref="S3.p8.2.m2.2.2.3"></eq><apply id="S3.p8.2.m2.2.2.1.2.cmml" xref="S3.p8.2.m2.2.2.1.1"><abs id="S3.p8.2.m2.2.2.1.2.1.cmml" xref="S3.p8.2.m2.2.2.1.1.2"></abs><apply id="S3.p8.2.m2.2.2.1.1.1.cmml" xref="S3.p8.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.p8.2.m2.2.2.1.1.1.1.cmml" xref="S3.p8.2.m2.2.2.1.1.1">subscript</csymbol><ci id="S3.p8.2.m2.2.2.1.1.1.2.cmml" xref="S3.p8.2.m2.2.2.1.1.1.2">𝑃</ci><ci id="S3.p8.2.m2.2.2.1.1.1.3.cmml" xref="S3.p8.2.m2.2.2.1.1.1.3">𝑛</ci></apply></apply><apply id="S3.p8.2.m2.2.2.4.1.cmml" xref="S3.p8.2.m2.2.2.4.2"><abs id="S3.p8.2.m2.2.2.4.1.1.cmml" xref="S3.p8.2.m2.2.2.4.2.1"></abs><ci id="S3.p8.2.m2.1.1.cmml" xref="S3.p8.2.m2.1.1">𝑅</ci></apply></apply><apply id="S3.p8.2.m2.2.2c.cmml" xref="S3.p8.2.m2.2.2"><eq id="S3.p8.2.m2.2.2.5.cmml" xref="S3.p8.2.m2.2.2.5"></eq><share href="https://arxiv.org/html/2411.07939v2#S3.p8.2.m2.2.2.4.cmml" id="S3.p8.2.m2.2.2d.cmml" xref="S3.p8.2.m2.2.2"></share><cn id="S3.p8.2.m2.2.2.6.cmml" type="integer" xref="S3.p8.2.m2.2.2.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p8.2.m2.2c">|P_{n}|=|R|=1</annotation><annotation encoding="application/x-llamapun" id="S3.p8.2.m2.2d">| italic_P start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | = | italic_R | = 1</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.p9"> <p class="ltx_p" id="S3.p9.1">More generally, it is straightforward to identify specific diagrams in (<a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S3.E32" title="In III Loop recursions ‣ One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_tag">32</span></a>) through the propagator structure of the different terms. Their extraction can be easily automated. In what follows, we will present some examples to illustrate our proposal.</p> </div> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">IV </span>Some examples</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">The simplest integrand we could compute using (<a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S3.E32" title="In III Loop recursions ‣ One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_tag">32</span></a>) is the case <math alttext="N=1" class="ltx_Math" display="inline" id="S4.p1.1.m1.1"><semantics id="S4.p1.1.m1.1a"><mrow id="S4.p1.1.m1.1.1" xref="S4.p1.1.m1.1.1.cmml"><mi id="S4.p1.1.m1.1.1.2" xref="S4.p1.1.m1.1.1.2.cmml">N</mi><mo id="S4.p1.1.m1.1.1.1" xref="S4.p1.1.m1.1.1.1.cmml">=</mo><mn id="S4.p1.1.m1.1.1.3" xref="S4.p1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.1.m1.1b"><apply id="S4.p1.1.m1.1.1.cmml" xref="S4.p1.1.m1.1.1"><eq id="S4.p1.1.m1.1.1.1.cmml" xref="S4.p1.1.m1.1.1.1"></eq><ci id="S4.p1.1.m1.1.1.2.cmml" xref="S4.p1.1.m1.1.1.2">𝑁</ci><cn id="S4.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.1.m1.1c">N=1</annotation><annotation encoding="application/x-llamapun" id="S4.p1.1.m1.1d">italic_N = 1</annotation></semantics></math>, which yields one-graviton tadpoles,</p> <table class="ltx_equation ltx_eqn_table" id="S4.E35"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{I}_{1}(\ell)=\frac{1}{2\ell^{2}}h_{1}^{\mu\nu}(\mathbb{P}^{-1})^{\rho% \sigma\alpha\beta}\mathcal{V}_{\rho\sigma\mu\nu\alpha\beta}^{(3)}(1,\ell)\\ -\frac{1}{\ell^{2}}h_{1}^{\mu\nu}\mathcal{V}^{(\textrm{g})\rho}_{\mu\nu\rho}(1% ,\ell 1)," class="ltx_Math" display="block" id="S4.E35.m1.41"><semantics id="S4.E35.m1.41a"><mtable displaystyle="true" id="S4.E35.m1.41.41.3" rowspacing="0pt"><mtr id="S4.E35.m1.41.41.3a"><mtd class="ltx_align_left" columnalign="left" id="S4.E35.m1.41.41.3b"><mrow id="S4.E35.m1.40.40.2.39.24.24"><mrow id="S4.E35.m1.40.40.2.39.24.24.25"><msub id="S4.E35.m1.40.40.2.39.24.24.25.2"><mi class="ltx_font_mathcaligraphic" id="S4.E35.m1.1.1.1.1.1.1" xref="S4.E35.m1.1.1.1.1.1.1.cmml">ℐ</mi><mn id="S4.E35.m1.2.2.2.2.2.2.1" xref="S4.E35.m1.2.2.2.2.2.2.1.cmml">1</mn></msub><mo id="S4.E35.m1.40.40.2.39.24.24.25.1" xref="S4.E35.m1.39.39.1.1.1.cmml">⁢</mo><mrow id="S4.E35.m1.40.40.2.39.24.24.25.3"><mo id="S4.E35.m1.3.3.3.3.3.3" stretchy="false" xref="S4.E35.m1.39.39.1.1.1.cmml">(</mo><mi id="S4.E35.m1.4.4.4.4.4.4" mathvariant="normal" xref="S4.E35.m1.4.4.4.4.4.4.cmml">ℓ</mi><mo id="S4.E35.m1.5.5.5.5.5.5" stretchy="false" xref="S4.E35.m1.39.39.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E35.m1.6.6.6.6.6.6" xref="S4.E35.m1.6.6.6.6.6.6.cmml">=</mo><mrow id="S4.E35.m1.40.40.2.39.24.24.24"><mfrac id="S4.E35.m1.7.7.7.7.7.7" xref="S4.E35.m1.7.7.7.7.7.7.cmml"><mn id="S4.E35.m1.7.7.7.7.7.7.2" xref="S4.E35.m1.7.7.7.7.7.7.2.cmml">1</mn><mrow id="S4.E35.m1.7.7.7.7.7.7.3" xref="S4.E35.m1.7.7.7.7.7.7.3.cmml"><mn id="S4.E35.m1.7.7.7.7.7.7.3.2" xref="S4.E35.m1.7.7.7.7.7.7.3.2.cmml">2</mn><mo id="S4.E35.m1.7.7.7.7.7.7.3.1" xref="S4.E35.m1.7.7.7.7.7.7.3.1.cmml">⁢</mo><msup id="S4.E35.m1.7.7.7.7.7.7.3.3" xref="S4.E35.m1.7.7.7.7.7.7.3.3.cmml"><mi id="S4.E35.m1.7.7.7.7.7.7.3.3.2" mathvariant="normal" xref="S4.E35.m1.7.7.7.7.7.7.3.3.2.cmml">ℓ</mi><mn id="S4.E35.m1.7.7.7.7.7.7.3.3.3" xref="S4.E35.m1.7.7.7.7.7.7.3.3.3.cmml">2</mn></msup></mrow></mfrac><mo id="S4.E35.m1.40.40.2.39.24.24.24.2" xref="S4.E35.m1.39.39.1.1.1.cmml">⁢</mo><msubsup id="S4.E35.m1.40.40.2.39.24.24.24.3"><mi id="S4.E35.m1.8.8.8.8.8.8" xref="S4.E35.m1.8.8.8.8.8.8.cmml">h</mi><mn id="S4.E35.m1.9.9.9.9.9.9.1" xref="S4.E35.m1.9.9.9.9.9.9.1.cmml">1</mn><mrow id="S4.E35.m1.10.10.10.10.10.10.1" xref="S4.E35.m1.10.10.10.10.10.10.1.cmml"><mi id="S4.E35.m1.10.10.10.10.10.10.1.2" xref="S4.E35.m1.10.10.10.10.10.10.1.2.cmml">μ</mi><mo id="S4.E35.m1.10.10.10.10.10.10.1.1" xref="S4.E35.m1.10.10.10.10.10.10.1.1.cmml">⁢</mo><mi id="S4.E35.m1.10.10.10.10.10.10.1.3" xref="S4.E35.m1.10.10.10.10.10.10.1.3.cmml">ν</mi></mrow></msubsup><mo id="S4.E35.m1.40.40.2.39.24.24.24.2a" xref="S4.E35.m1.39.39.1.1.1.cmml">⁢</mo><msup id="S4.E35.m1.40.40.2.39.24.24.24.1"><mrow id="S4.E35.m1.40.40.2.39.24.24.24.1.1.1"><mo id="S4.E35.m1.11.11.11.11.11.11" stretchy="false" xref="S4.E35.m1.39.39.1.1.1.cmml">(</mo><msup id="S4.E35.m1.40.40.2.39.24.24.24.1.1.1.1"><mi id="S4.E35.m1.12.12.12.12.12.12" xref="S4.E35.m1.12.12.12.12.12.12.cmml">ℙ</mi><mrow id="S4.E35.m1.13.13.13.13.13.13.1" xref="S4.E35.m1.13.13.13.13.13.13.1.cmml"><mo id="S4.E35.m1.13.13.13.13.13.13.1a" xref="S4.E35.m1.13.13.13.13.13.13.1.cmml">−</mo><mn id="S4.E35.m1.13.13.13.13.13.13.1.2" xref="S4.E35.m1.13.13.13.13.13.13.1.2.cmml">1</mn></mrow></msup><mo id="S4.E35.m1.14.14.14.14.14.14" stretchy="false" xref="S4.E35.m1.39.39.1.1.1.cmml">)</mo></mrow><mrow id="S4.E35.m1.15.15.15.15.15.15.1" xref="S4.E35.m1.15.15.15.15.15.15.1.cmml"><mi id="S4.E35.m1.15.15.15.15.15.15.1.2" xref="S4.E35.m1.15.15.15.15.15.15.1.2.cmml">ρ</mi><mo id="S4.E35.m1.15.15.15.15.15.15.1.1" xref="S4.E35.m1.15.15.15.15.15.15.1.1.cmml">⁢</mo><mi id="S4.E35.m1.15.15.15.15.15.15.1.3" xref="S4.E35.m1.15.15.15.15.15.15.1.3.cmml">σ</mi><mo id="S4.E35.m1.15.15.15.15.15.15.1.1a" xref="S4.E35.m1.15.15.15.15.15.15.1.1.cmml">⁢</mo><mi id="S4.E35.m1.15.15.15.15.15.15.1.4" xref="S4.E35.m1.15.15.15.15.15.15.1.4.cmml">α</mi><mo id="S4.E35.m1.15.15.15.15.15.15.1.1b" xref="S4.E35.m1.15.15.15.15.15.15.1.1.cmml">⁢</mo><mi id="S4.E35.m1.15.15.15.15.15.15.1.5" xref="S4.E35.m1.15.15.15.15.15.15.1.5.cmml">β</mi></mrow></msup><mo id="S4.E35.m1.40.40.2.39.24.24.24.2b" 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start_POSTSUBSCRIPT italic_ρ italic_σ italic_μ italic_ν italic_α italic_β end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT ( 1 , roman_ℓ ) end_CELL end_ROW start_ROW start_CELL - divide start_ARG 1 end_ARG start_ARG roman_ℓ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT caligraphic_V start_POSTSUPERSCRIPT ( g ) italic_ρ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ italic_ν italic_ρ end_POSTSUBSCRIPT ( 1 , roman_ℓ 1 ) , end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(35)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p1.2">with <math alttext="k_{1}=0" class="ltx_Math" display="inline" id="S4.p1.2.m1.1"><semantics id="S4.p1.2.m1.1a"><mrow 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xref="S4.E36.m1.2.2.1.1.3.2"><csymbol cd="ambiguous" id="S4.E36.m1.2.2.1.1.3.2.1.cmml" xref="S4.E36.m1.2.2.1.1.3.2">subscript</csymbol><apply id="S4.E36.m1.2.2.1.1.3.2.2.cmml" xref="S4.E36.m1.2.2.1.1.3.2"><csymbol cd="ambiguous" id="S4.E36.m1.2.2.1.1.3.2.2.1.cmml" xref="S4.E36.m1.2.2.1.1.3.2">superscript</csymbol><ci id="S4.E36.m1.2.2.1.1.3.2.2.2.cmml" xref="S4.E36.m1.2.2.1.1.3.2.2.2">ℐ</ci><ci id="S4.E36.m1.2.2.1.1.3.2.2.3a.cmml" xref="S4.E36.m1.2.2.1.1.3.2.2.3"><mtext id="S4.E36.m1.2.2.1.1.3.2.2.3.cmml" mathsize="70%" xref="S4.E36.m1.2.2.1.1.3.2.2.3">tp</mtext></ci></apply><cn id="S4.E36.m1.2.2.1.1.3.2.3.cmml" type="integer" xref="S4.E36.m1.2.2.1.1.3.2.3">2</cn></apply><apply id="S4.E36.m1.2.2.1.1.3.3.cmml" xref="S4.E36.m1.2.2.1.1.3.3"><csymbol cd="ambiguous" id="S4.E36.m1.2.2.1.1.3.3.1.cmml" xref="S4.E36.m1.2.2.1.1.3.3">subscript</csymbol><apply id="S4.E36.m1.2.2.1.1.3.3.2.cmml" xref="S4.E36.m1.2.2.1.1.3.3"><csymbol cd="ambiguous" id="S4.E36.m1.2.2.1.1.3.3.2.1.cmml" xref="S4.E36.m1.2.2.1.1.3.3">superscript</csymbol><ci id="S4.E36.m1.2.2.1.1.3.3.2.2.cmml" xref="S4.E36.m1.2.2.1.1.3.3.2.2">ℐ</ci><ci id="S4.E36.m1.2.2.1.1.3.3.2.3a.cmml" xref="S4.E36.m1.2.2.1.1.3.3.2.3"><mtext id="S4.E36.m1.2.2.1.1.3.3.2.3.cmml" mathsize="70%" xref="S4.E36.m1.2.2.1.1.3.3.2.3">se</mtext></ci></apply><cn id="S4.E36.m1.2.2.1.1.3.3.3.cmml" type="integer" xref="S4.E36.m1.2.2.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E36.m1.2c">\mathcal{I}_{2}(\ell)=\mathcal{I}^{\textrm{tp}}_{2}+\mathcal{I}^{\textrm{se}}_% {2}.</annotation><annotation encoding="application/x-llamapun" id="S4.E36.m1.2d">caligraphic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( roman_ℓ ) = caligraphic_I start_POSTSUPERSCRIPT tp end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + caligraphic_I start_POSTSUPERSCRIPT se end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(36)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.9">The first term consists of two-graviton tadpoles,</p> <table class="ltx_equation ltx_eqn_table" id="S4.E37"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{I}^{\textrm{tp}}_{2}=\frac{1}{2\ell^{2}}(\mathbb{P}^{-1})^{\rho\sigma% \alpha\beta}\big{\{}I_{12}^{\mu\nu}\mathcal{V}_{\rho\sigma\mu\nu\alpha\beta}^{% (3)}(12,\ell)\\ +h_{1}^{\mu_{1}\nu_{1}}h_{2}^{\mu_{2}\nu_{2}}\mathcal{V}_{\rho\sigma\mu_{1}\nu% _{1}\mu_{2}\nu_{2}\alpha\beta}^{(4)}(1,2,\ell)\big{\}}\\ -\frac{1}{\ell^{2}}I_{12}^{\mu\nu}\mathcal{V}^{(\textrm{g})\rho}_{\mu\nu\rho}(% 12,\ell 12)." class="ltx_Math" display="block" id="S4.E37.m1.57"><semantics id="S4.E37.m1.57a"><mtable displaystyle="true" id="S4.E37.m1.57.57.2" rowspacing="0pt"><mtr id="S4.E37.m1.57.57.2a"><mtd class="ltx_align_left" columnalign="left" id="S4.E37.m1.57.57.2b"><mrow id="S4.E37.m1.22.22.22.22.22"><msubsup id="S4.E37.m1.22.22.22.22.22.23"><mi class="ltx_font_mathcaligraphic" id="S4.E37.m1.1.1.1.1.1.1" xref="S4.E37.m1.1.1.1.1.1.1.cmml">ℐ</mi><mn id="S4.E37.m1.3.3.3.3.3.3.1" xref="S4.E37.m1.3.3.3.3.3.3.1.cmml">2</mn><mtext id="S4.E37.m1.2.2.2.2.2.2.1" xref="S4.E37.m1.2.2.2.2.2.2.1a.cmml">tp</mtext></msubsup><mo id="S4.E37.m1.4.4.4.4.4.4" xref="S4.E37.m1.4.4.4.4.4.4.cmml">=</mo><mfrac id="S4.E37.m1.5.5.5.5.5.5" xref="S4.E37.m1.5.5.5.5.5.5.cmml"><mn id="S4.E37.m1.5.5.5.5.5.5.2" xref="S4.E37.m1.5.5.5.5.5.5.2.cmml">1</mn><mrow id="S4.E37.m1.5.5.5.5.5.5.3" xref="S4.E37.m1.5.5.5.5.5.5.3.cmml"><mn id="S4.E37.m1.5.5.5.5.5.5.3.2" xref="S4.E37.m1.5.5.5.5.5.5.3.2.cmml">2</mn><mo id="S4.E37.m1.5.5.5.5.5.5.3.1" xref="S4.E37.m1.5.5.5.5.5.5.3.1.cmml">⁢</mo><msup id="S4.E37.m1.5.5.5.5.5.5.3.3" xref="S4.E37.m1.5.5.5.5.5.5.3.3.cmml"><mi id="S4.E37.m1.5.5.5.5.5.5.3.3.2" mathvariant="normal" xref="S4.E37.m1.5.5.5.5.5.5.3.3.2.cmml">ℓ</mi><mn id="S4.E37.m1.5.5.5.5.5.5.3.3.3" xref="S4.E37.m1.5.5.5.5.5.5.3.3.3.cmml">2</mn></msup></mrow></mfrac><msup id="S4.E37.m1.22.22.22.22.22.24"><mrow id="S4.E37.m1.22.22.22.22.22.24.2"><mo id="S4.E37.m1.6.6.6.6.6.6" stretchy="false" xref="S4.E37.m1.56.56.1.1.1.cmml">(</mo><msup id="S4.E37.m1.22.22.22.22.22.24.2.1"><mi id="S4.E37.m1.7.7.7.7.7.7" xref="S4.E37.m1.7.7.7.7.7.7.cmml">ℙ</mi><mrow id="S4.E37.m1.8.8.8.8.8.8.1" xref="S4.E37.m1.8.8.8.8.8.8.1.cmml"><mo id="S4.E37.m1.8.8.8.8.8.8.1a" xref="S4.E37.m1.8.8.8.8.8.8.1.cmml">−</mo><mn id="S4.E37.m1.8.8.8.8.8.8.1.2" xref="S4.E37.m1.8.8.8.8.8.8.1.2.cmml">1</mn></mrow></msup><mo id="S4.E37.m1.9.9.9.9.9.9" stretchy="false" xref="S4.E37.m1.56.56.1.1.1.cmml">)</mo></mrow><mrow id="S4.E37.m1.10.10.10.10.10.10.1" xref="S4.E37.m1.10.10.10.10.10.10.1.cmml"><mi id="S4.E37.m1.10.10.10.10.10.10.1.2" xref="S4.E37.m1.10.10.10.10.10.10.1.2.cmml">ρ</mi><mo id="S4.E37.m1.10.10.10.10.10.10.1.1" xref="S4.E37.m1.10.10.10.10.10.10.1.1.cmml">⁢</mo><mi id="S4.E37.m1.10.10.10.10.10.10.1.3" xref="S4.E37.m1.10.10.10.10.10.10.1.3.cmml">σ</mi><mo id="S4.E37.m1.10.10.10.10.10.10.1.1a" xref="S4.E37.m1.10.10.10.10.10.10.1.1.cmml">⁢</mo><mi id="S4.E37.m1.10.10.10.10.10.10.1.4" xref="S4.E37.m1.10.10.10.10.10.10.1.4.cmml">α</mi><mo id="S4.E37.m1.10.10.10.10.10.10.1.1b" xref="S4.E37.m1.10.10.10.10.10.10.1.1.cmml">⁢</mo><mi id="S4.E37.m1.10.10.10.10.10.10.1.5" xref="S4.E37.m1.10.10.10.10.10.10.1.5.cmml">β</mi></mrow></msup><mrow id="S4.E37.m1.22.22.22.22.22.25"><mo id="S4.E37.m1.11.11.11.11.11.11" maxsize="120%" minsize="120%" xref="S4.E37.m1.56.56.1.1.1.cmml">{</mo><msubsup id="S4.E37.m1.22.22.22.22.22.25.1"><mi id="S4.E37.m1.12.12.12.12.12.12" xref="S4.E37.m1.12.12.12.12.12.12.cmml">I</mi><mn id="S4.E37.m1.13.13.13.13.13.13.1" xref="S4.E37.m1.13.13.13.13.13.13.1.cmml">12</mn><mrow id="S4.E37.m1.14.14.14.14.14.14.1" xref="S4.E37.m1.14.14.14.14.14.14.1.cmml"><mi id="S4.E37.m1.14.14.14.14.14.14.1.2" xref="S4.E37.m1.14.14.14.14.14.14.1.2.cmml">μ</mi><mo id="S4.E37.m1.14.14.14.14.14.14.1.1" xref="S4.E37.m1.14.14.14.14.14.14.1.1.cmml">⁢</mo><mi id="S4.E37.m1.14.14.14.14.14.14.1.3" xref="S4.E37.m1.14.14.14.14.14.14.1.3.cmml">ν</mi></mrow></msubsup><msubsup id="S4.E37.m1.22.22.22.22.22.25.2"><mi class="ltx_font_mathcaligraphic" id="S4.E37.m1.15.15.15.15.15.15" xref="S4.E37.m1.15.15.15.15.15.15.cmml">𝒱</mi><mrow id="S4.E37.m1.16.16.16.16.16.16.1" xref="S4.E37.m1.16.16.16.16.16.16.1.cmml"><mi id="S4.E37.m1.16.16.16.16.16.16.1.2" xref="S4.E37.m1.16.16.16.16.16.16.1.2.cmml">ρ</mi><mo id="S4.E37.m1.16.16.16.16.16.16.1.1" xref="S4.E37.m1.16.16.16.16.16.16.1.1.cmml">⁢</mo><mi id="S4.E37.m1.16.16.16.16.16.16.1.3" xref="S4.E37.m1.16.16.16.16.16.16.1.3.cmml">σ</mi><mo id="S4.E37.m1.16.16.16.16.16.16.1.1a" xref="S4.E37.m1.16.16.16.16.16.16.1.1.cmml">⁢</mo><mi id="S4.E37.m1.16.16.16.16.16.16.1.4" xref="S4.E37.m1.16.16.16.16.16.16.1.4.cmml">μ</mi><mo id="S4.E37.m1.16.16.16.16.16.16.1.1b" xref="S4.E37.m1.16.16.16.16.16.16.1.1.cmml">⁢</mo><mi id="S4.E37.m1.16.16.16.16.16.16.1.5" xref="S4.E37.m1.16.16.16.16.16.16.1.5.cmml">ν</mi><mo id="S4.E37.m1.16.16.16.16.16.16.1.1c" xref="S4.E37.m1.16.16.16.16.16.16.1.1.cmml">⁢</mo><mi id="S4.E37.m1.16.16.16.16.16.16.1.6" xref="S4.E37.m1.16.16.16.16.16.16.1.6.cmml">α</mi><mo id="S4.E37.m1.16.16.16.16.16.16.1.1d" xref="S4.E37.m1.16.16.16.16.16.16.1.1.cmml">⁢</mo><mi id="S4.E37.m1.16.16.16.16.16.16.1.7" xref="S4.E37.m1.16.16.16.16.16.16.1.7.cmml">β</mi></mrow><mrow id="S4.E37.m1.17.17.17.17.17.17.1.3"><mo id="S4.E37.m1.17.17.17.17.17.17.1.3.1" stretchy="false">(</mo><mn id="S4.E37.m1.17.17.17.17.17.17.1.1" xref="S4.E37.m1.17.17.17.17.17.17.1.1.cmml">3</mn><mo id="S4.E37.m1.17.17.17.17.17.17.1.3.2" stretchy="false">)</mo></mrow></msubsup><mrow id="S4.E37.m1.22.22.22.22.22.25.3"><mo id="S4.E37.m1.18.18.18.18.18.18" stretchy="false" xref="S4.E37.m1.56.56.1.1.1.cmml">(</mo><mn id="S4.E37.m1.19.19.19.19.19.19" xref="S4.E37.m1.19.19.19.19.19.19.cmml">12</mn><mo id="S4.E37.m1.20.20.20.20.20.20" xref="S4.E37.m1.56.56.1.1.1.cmml">,</mo><mi id="S4.E37.m1.21.21.21.21.21.21" mathvariant="normal" xref="S4.E37.m1.21.21.21.21.21.21.cmml">ℓ</mi><mo id="S4.E37.m1.22.22.22.22.22.22" stretchy="false" xref="S4.E37.m1.56.56.1.1.1.cmml">)</mo></mrow></mrow></mrow></mtd></mtr><mtr id="S4.E37.m1.57.57.2c"><mtd class="ltx_align_right" columnalign="right" id="S4.E37.m1.57.57.2d"><mrow id="S4.E37.m1.40.40.40.18.18"><mo id="S4.E37.m1.23.23.23.1.1.1" xref="S4.E37.m1.23.23.23.1.1.1.cmml">+</mo><msubsup id="S4.E37.m1.40.40.40.18.18.19"><mi id="S4.E37.m1.24.24.24.2.2.2" xref="S4.E37.m1.24.24.24.2.2.2.cmml">h</mi><mn id="S4.E37.m1.25.25.25.3.3.3.1" xref="S4.E37.m1.25.25.25.3.3.3.1.cmml">1</mn><mrow id="S4.E37.m1.26.26.26.4.4.4.1" 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xref="S4.E37.m1.29.29.29.7.7.7.1.2.cmml"><mi id="S4.E37.m1.29.29.29.7.7.7.1.2.2" xref="S4.E37.m1.29.29.29.7.7.7.1.2.2.cmml">μ</mi><mn id="S4.E37.m1.29.29.29.7.7.7.1.2.3" xref="S4.E37.m1.29.29.29.7.7.7.1.2.3.cmml">2</mn></msub><mo id="S4.E37.m1.29.29.29.7.7.7.1.1" xref="S4.E37.m1.29.29.29.7.7.7.1.1.cmml">⁢</mo><msub id="S4.E37.m1.29.29.29.7.7.7.1.3" xref="S4.E37.m1.29.29.29.7.7.7.1.3.cmml"><mi id="S4.E37.m1.29.29.29.7.7.7.1.3.2" xref="S4.E37.m1.29.29.29.7.7.7.1.3.2.cmml">ν</mi><mn id="S4.E37.m1.29.29.29.7.7.7.1.3.3" xref="S4.E37.m1.29.29.29.7.7.7.1.3.3.cmml">2</mn></msub></mrow></msubsup><msubsup id="S4.E37.m1.40.40.40.18.18.21"><mi class="ltx_font_mathcaligraphic" id="S4.E37.m1.30.30.30.8.8.8" xref="S4.E37.m1.30.30.30.8.8.8.cmml">𝒱</mi><mrow id="S4.E37.m1.31.31.31.9.9.9.1" xref="S4.E37.m1.31.31.31.9.9.9.1.cmml"><mi id="S4.E37.m1.31.31.31.9.9.9.1.2" xref="S4.E37.m1.31.31.31.9.9.9.1.2.cmml">ρ</mi><mo id="S4.E37.m1.31.31.31.9.9.9.1.1" xref="S4.E37.m1.31.31.31.9.9.9.1.1.cmml">⁢</mo><mi id="S4.E37.m1.31.31.31.9.9.9.1.3" xref="S4.E37.m1.31.31.31.9.9.9.1.3.cmml">σ</mi><mo id="S4.E37.m1.31.31.31.9.9.9.1.1a" xref="S4.E37.m1.31.31.31.9.9.9.1.1.cmml">⁢</mo><msub id="S4.E37.m1.31.31.31.9.9.9.1.4" xref="S4.E37.m1.31.31.31.9.9.9.1.4.cmml"><mi id="S4.E37.m1.31.31.31.9.9.9.1.4.2" xref="S4.E37.m1.31.31.31.9.9.9.1.4.2.cmml">μ</mi><mn id="S4.E37.m1.31.31.31.9.9.9.1.4.3" xref="S4.E37.m1.31.31.31.9.9.9.1.4.3.cmml">1</mn></msub><mo id="S4.E37.m1.31.31.31.9.9.9.1.1b" xref="S4.E37.m1.31.31.31.9.9.9.1.1.cmml">⁢</mo><msub id="S4.E37.m1.31.31.31.9.9.9.1.5" xref="S4.E37.m1.31.31.31.9.9.9.1.5.cmml"><mi id="S4.E37.m1.31.31.31.9.9.9.1.5.2" xref="S4.E37.m1.31.31.31.9.9.9.1.5.2.cmml">ν</mi><mn id="S4.E37.m1.31.31.31.9.9.9.1.5.3" xref="S4.E37.m1.31.31.31.9.9.9.1.5.3.cmml">1</mn></msub><mo id="S4.E37.m1.31.31.31.9.9.9.1.1c" xref="S4.E37.m1.31.31.31.9.9.9.1.1.cmml">⁢</mo><msub id="S4.E37.m1.31.31.31.9.9.9.1.6" xref="S4.E37.m1.31.31.31.9.9.9.1.6.cmml"><mi id="S4.E37.m1.31.31.31.9.9.9.1.6.2" xref="S4.E37.m1.31.31.31.9.9.9.1.6.2.cmml">μ</mi><mn id="S4.E37.m1.31.31.31.9.9.9.1.6.3" xref="S4.E37.m1.31.31.31.9.9.9.1.6.3.cmml">2</mn></msub><mo id="S4.E37.m1.31.31.31.9.9.9.1.1d" xref="S4.E37.m1.31.31.31.9.9.9.1.1.cmml">⁢</mo><msub id="S4.E37.m1.31.31.31.9.9.9.1.7" xref="S4.E37.m1.31.31.31.9.9.9.1.7.cmml"><mi id="S4.E37.m1.31.31.31.9.9.9.1.7.2" xref="S4.E37.m1.31.31.31.9.9.9.1.7.2.cmml">ν</mi><mn id="S4.E37.m1.31.31.31.9.9.9.1.7.3" xref="S4.E37.m1.31.31.31.9.9.9.1.7.3.cmml">2</mn></msub><mo id="S4.E37.m1.31.31.31.9.9.9.1.1e" xref="S4.E37.m1.31.31.31.9.9.9.1.1.cmml">⁢</mo><mi id="S4.E37.m1.31.31.31.9.9.9.1.8" xref="S4.E37.m1.31.31.31.9.9.9.1.8.cmml">α</mi><mo id="S4.E37.m1.31.31.31.9.9.9.1.1f" xref="S4.E37.m1.31.31.31.9.9.9.1.1.cmml">⁢</mo><mi id="S4.E37.m1.31.31.31.9.9.9.1.9" xref="S4.E37.m1.31.31.31.9.9.9.1.9.cmml">β</mi></mrow><mrow id="S4.E37.m1.32.32.32.10.10.10.1.3"><mo id="S4.E37.m1.32.32.32.10.10.10.1.3.1" stretchy="false">(</mo><mn id="S4.E37.m1.32.32.32.10.10.10.1.1" 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\alpha\beta}\big{\{}I_{12}^{\mu\nu}\mathcal{V}_{\rho\sigma\mu\nu\alpha\beta}^{% (3)}(12,\ell)\\ +h_{1}^{\mu_{1}\nu_{1}}h_{2}^{\mu_{2}\nu_{2}}\mathcal{V}_{\rho\sigma\mu_{1}\nu% _{1}\mu_{2}\nu_{2}\alpha\beta}^{(4)}(1,2,\ell)\big{\}}\\ -\frac{1}{\ell^{2}}I_{12}^{\mu\nu}\mathcal{V}^{(\textrm{g})\rho}_{\mu\nu\rho}(% 12,\ell 12).</annotation><annotation encoding="application/x-llamapun" id="S4.E37.m1.57d">start_ROW start_CELL caligraphic_I start_POSTSUPERSCRIPT tp end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 roman_ℓ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ( blackboard_P start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_ρ italic_σ italic_α italic_β end_POSTSUPERSCRIPT { italic_I start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT caligraphic_V start_POSTSUBSCRIPT italic_ρ italic_σ italic_μ italic_ν italic_α italic_β end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT ( 12 , roman_ℓ ) end_CELL end_ROW start_ROW start_CELL + italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_h start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT caligraphic_V start_POSTSUBSCRIPT italic_ρ italic_σ italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 4 ) end_POSTSUPERSCRIPT ( 1 , 2 , roman_ℓ ) } end_CELL end_ROW start_ROW start_CELL - divide start_ARG 1 end_ARG start_ARG roman_ℓ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG italic_I start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT caligraphic_V start_POSTSUPERSCRIPT ( g ) italic_ρ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ italic_ν italic_ρ end_POSTSUBSCRIPT ( 12 , roman_ℓ 12 ) . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(37)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.10">The second term in (<a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S4.E36" title="In IV Some examples ‣ One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_tag">36</span></a>) corresponds to the graviton self-energy, one contribution from the graviton loop and one from the ghost loop,</p> <table class="ltx_equation ltx_eqn_table" id="S4.E38"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{I}^{\textrm{se}}_{2}=\frac{h_{1}^{\mu_{1}\nu_{1}}h_{2}^{\mu_{2}\nu_{2% }}}{\ell^{2}(\ell+k_{1})^{2}}\Big{\{}\frac{1}{4}(\mathbb{P}^{-1})^{\mu\nu% \gamma\lambda}(\mathbb{P}^{-1})^{\alpha\beta\rho\sigma}\\ \times\mathcal{V}_{\mu\nu\mu_{1}\nu_{1}\alpha\beta}^{(3)}(1,\ell)\mathcal{V}_{% \rho\sigma\mu_{2}\nu_{2}\gamma\lambda}^{(3)}(2,\ell 1)\\ -\frac{1}{2}\mathcal{V}^{(\textrm{g})\sigma}_{\mu_{1}\nu_{1}\rho}(1,\ell 1)% \mathcal{V}^{(\textrm{g})\rho}_{\mu_{2}\nu_{2}\sigma}(2,\ell 12)\Big{\}}+(1% \leftrightarrow 2)." class="ltx_math_unparsed" display="block" id="S4.E38.m1.64"><semantics id="S4.E38.m1.64a"><mtable displaystyle="true" id="S4.E38.m1.64.64.1" rowspacing="0pt"><mtr id="S4.E38.m1.64.64.1a"><mtd class="ltx_align_left" columnalign="left" id="S4.E38.m1.64.64.1b"><mrow id="S4.E38.m1.17.17.17.17.17"><msubsup id="S4.E38.m1.17.17.17.17.17.18"><mi class="ltx_font_mathcaligraphic" id="S4.E38.m1.1.1.1.1.1.1">ℐ</mi><mn id="S4.E38.m1.3.3.3.3.3.3.1">2</mn><mtext id="S4.E38.m1.2.2.2.2.2.2.1">se</mtext></msubsup><mo id="S4.E38.m1.4.4.4.4.4.4">=</mo><mfrac id="S4.E38.m1.5.5.5.5.5.5"><mrow id="S4.E38.m1.5.5.5.5.5.5.3"><msubsup id="S4.E38.m1.5.5.5.5.5.5.3.2"><mi id="S4.E38.m1.5.5.5.5.5.5.3.2.2.2">h</mi><mn id="S4.E38.m1.5.5.5.5.5.5.3.2.2.3">1</mn><mrow id="S4.E38.m1.5.5.5.5.5.5.3.2.3"><msub id="S4.E38.m1.5.5.5.5.5.5.3.2.3.2"><mi id="S4.E38.m1.5.5.5.5.5.5.3.2.3.2.2">μ</mi><mn id="S4.E38.m1.5.5.5.5.5.5.3.2.3.2.3">1</mn></msub><mo id="S4.E38.m1.5.5.5.5.5.5.3.2.3.1">⁢</mo><msub id="S4.E38.m1.5.5.5.5.5.5.3.2.3.3"><mi id="S4.E38.m1.5.5.5.5.5.5.3.2.3.3.2">ν</mi><mn id="S4.E38.m1.5.5.5.5.5.5.3.2.3.3.3">1</mn></msub></mrow></msubsup><mo id="S4.E38.m1.5.5.5.5.5.5.3.1">⁢</mo><msubsup id="S4.E38.m1.5.5.5.5.5.5.3.3"><mi id="S4.E38.m1.5.5.5.5.5.5.3.3.2.2">h</mi><mn id="S4.E38.m1.5.5.5.5.5.5.3.3.2.3">2</mn><mrow id="S4.E38.m1.5.5.5.5.5.5.3.3.3"><msub id="S4.E38.m1.5.5.5.5.5.5.3.3.3.2"><mi id="S4.E38.m1.5.5.5.5.5.5.3.3.3.2.2">μ</mi><mn id="S4.E38.m1.5.5.5.5.5.5.3.3.3.2.3">2</mn></msub><mo id="S4.E38.m1.5.5.5.5.5.5.3.3.3.1">⁢</mo><msub id="S4.E38.m1.5.5.5.5.5.5.3.3.3.3"><mi id="S4.E38.m1.5.5.5.5.5.5.3.3.3.3.2">ν</mi><mn id="S4.E38.m1.5.5.5.5.5.5.3.3.3.3.3">2</mn></msub></mrow></msubsup></mrow><mrow id="S4.E38.m1.5.5.5.5.5.5.1"><msup id="S4.E38.m1.5.5.5.5.5.5.1.3"><mi id="S4.E38.m1.5.5.5.5.5.5.1.3.2" mathvariant="normal">ℓ</mi><mn id="S4.E38.m1.5.5.5.5.5.5.1.3.3">2</mn></msup><mo id="S4.E38.m1.5.5.5.5.5.5.1.2">⁢</mo><msup id="S4.E38.m1.5.5.5.5.5.5.1.1"><mrow id="S4.E38.m1.5.5.5.5.5.5.1.1.1.1"><mo id="S4.E38.m1.5.5.5.5.5.5.1.1.1.1.2" stretchy="false">(</mo><mrow id="S4.E38.m1.5.5.5.5.5.5.1.1.1.1.1"><mi id="S4.E38.m1.5.5.5.5.5.5.1.1.1.1.1.2" mathvariant="normal">ℓ</mi><mo id="S4.E38.m1.5.5.5.5.5.5.1.1.1.1.1.1">+</mo><msub id="S4.E38.m1.5.5.5.5.5.5.1.1.1.1.1.3"><mi id="S4.E38.m1.5.5.5.5.5.5.1.1.1.1.1.3.2">k</mi><mn id="S4.E38.m1.5.5.5.5.5.5.1.1.1.1.1.3.3">1</mn></msub></mrow><mo id="S4.E38.m1.5.5.5.5.5.5.1.1.1.1.3" stretchy="false">)</mo></mrow><mn id="S4.E38.m1.5.5.5.5.5.5.1.1.3">2</mn></msup></mrow></mfrac><mrow id="S4.E38.m1.17.17.17.17.17.19"><mo id="S4.E38.m1.6.6.6.6.6.6" maxsize="160%" minsize="160%">{</mo><mfrac id="S4.E38.m1.7.7.7.7.7.7"><mn id="S4.E38.m1.7.7.7.7.7.7.2">1</mn><mn id="S4.E38.m1.7.7.7.7.7.7.3">4</mn></mfrac><msup id="S4.E38.m1.17.17.17.17.17.19.1"><mrow id="S4.E38.m1.17.17.17.17.17.19.1.2"><mo id="S4.E38.m1.8.8.8.8.8.8" stretchy="false">(</mo><msup id="S4.E38.m1.17.17.17.17.17.19.1.2.1"><mi id="S4.E38.m1.9.9.9.9.9.9">ℙ</mi><mrow id="S4.E38.m1.10.10.10.10.10.10.1"><mo id="S4.E38.m1.10.10.10.10.10.10.1a">−</mo><mn id="S4.E38.m1.10.10.10.10.10.10.1.2">1</mn></mrow></msup><mo id="S4.E38.m1.11.11.11.11.11.11" stretchy="false">)</mo></mrow><mrow id="S4.E38.m1.12.12.12.12.12.12.1"><mi id="S4.E38.m1.12.12.12.12.12.12.1.2">μ</mi><mo id="S4.E38.m1.12.12.12.12.12.12.1.1">⁢</mo><mi id="S4.E38.m1.12.12.12.12.12.12.1.3">ν</mi><mo id="S4.E38.m1.12.12.12.12.12.12.1.1a">⁢</mo><mi id="S4.E38.m1.12.12.12.12.12.12.1.4">γ</mi><mo id="S4.E38.m1.12.12.12.12.12.12.1.1b">⁢</mo><mi id="S4.E38.m1.12.12.12.12.12.12.1.5">λ</mi></mrow></msup><msup id="S4.E38.m1.17.17.17.17.17.19.2"><mrow id="S4.E38.m1.17.17.17.17.17.19.2.2"><mo id="S4.E38.m1.13.13.13.13.13.13" stretchy="false">(</mo><msup id="S4.E38.m1.17.17.17.17.17.19.2.2.1"><mi id="S4.E38.m1.14.14.14.14.14.14">ℙ</mi><mrow id="S4.E38.m1.15.15.15.15.15.15.1"><mo id="S4.E38.m1.15.15.15.15.15.15.1a">−</mo><mn id="S4.E38.m1.15.15.15.15.15.15.1.2">1</mn></mrow></msup><mo id="S4.E38.m1.16.16.16.16.16.16" stretchy="false">)</mo></mrow><mrow id="S4.E38.m1.17.17.17.17.17.17.1"><mi id="S4.E38.m1.17.17.17.17.17.17.1.2">α</mi><mo id="S4.E38.m1.17.17.17.17.17.17.1.1">⁢</mo><mi id="S4.E38.m1.17.17.17.17.17.17.1.3">β</mi><mo id="S4.E38.m1.17.17.17.17.17.17.1.1a">⁢</mo><mi id="S4.E38.m1.17.17.17.17.17.17.1.4">ρ</mi><mo id="S4.E38.m1.17.17.17.17.17.17.1.1b">⁢</mo><mi id="S4.E38.m1.17.17.17.17.17.17.1.5">σ</mi></mrow></msup></mrow></mrow></mtd></mtr><mtr id="S4.E38.m1.64.64.1c"><mtd class="ltx_align_right" columnalign="right" id="S4.E38.m1.64.64.1d"><mrow id="S4.E38.m1.64.64.1.64.19.19"><mi id="S4.E38.m1.64.64.1.64.19.19.20"></mi><mo id="S4.E38.m1.18.18.18.1.1.1" lspace="0.222em" rspace="0.222em">×</mo><mrow id="S4.E38.m1.64.64.1.64.19.19.19"><msubsup id="S4.E38.m1.64.64.1.64.19.19.19.3"><mi class="ltx_font_mathcaligraphic" id="S4.E38.m1.19.19.19.2.2.2">𝒱</mi><mrow id="S4.E38.m1.20.20.20.3.3.3.1"><mi id="S4.E38.m1.20.20.20.3.3.3.1.2">μ</mi><mo id="S4.E38.m1.20.20.20.3.3.3.1.1">⁢</mo><mi id="S4.E38.m1.20.20.20.3.3.3.1.3">ν</mi><mo id="S4.E38.m1.20.20.20.3.3.3.1.1a">⁢</mo><msub id="S4.E38.m1.20.20.20.3.3.3.1.4"><mi id="S4.E38.m1.20.20.20.3.3.3.1.4.2">μ</mi><mn id="S4.E38.m1.20.20.20.3.3.3.1.4.3">1</mn></msub><mo id="S4.E38.m1.20.20.20.3.3.3.1.1b">⁢</mo><msub id="S4.E38.m1.20.20.20.3.3.3.1.5"><mi id="S4.E38.m1.20.20.20.3.3.3.1.5.2">ν</mi><mn id="S4.E38.m1.20.20.20.3.3.3.1.5.3">1</mn></msub><mo id="S4.E38.m1.20.20.20.3.3.3.1.1c">⁢</mo><mi id="S4.E38.m1.20.20.20.3.3.3.1.6">α</mi><mo id="S4.E38.m1.20.20.20.3.3.3.1.1d">⁢</mo><mi id="S4.E38.m1.20.20.20.3.3.3.1.7">β</mi></mrow><mrow id="S4.E38.m1.21.21.21.4.4.4.1.3"><mo id="S4.E38.m1.21.21.21.4.4.4.1.3.1" stretchy="false">(</mo><mn id="S4.E38.m1.21.21.21.4.4.4.1.1">3</mn><mo id="S4.E38.m1.21.21.21.4.4.4.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S4.E38.m1.64.64.1.64.19.19.19.2">⁢</mo><mrow id="S4.E38.m1.64.64.1.64.19.19.19.4"><mo id="S4.E38.m1.22.22.22.5.5.5" stretchy="false">(</mo><mn id="S4.E38.m1.23.23.23.6.6.6">1</mn><mo id="S4.E38.m1.24.24.24.7.7.7">,</mo><mi id="S4.E38.m1.25.25.25.8.8.8" mathvariant="normal">ℓ</mi><mo id="S4.E38.m1.26.26.26.9.9.9" stretchy="false">)</mo></mrow><mo id="S4.E38.m1.64.64.1.64.19.19.19.2a">⁢</mo><msubsup id="S4.E38.m1.64.64.1.64.19.19.19.5"><mi class="ltx_font_mathcaligraphic" id="S4.E38.m1.27.27.27.10.10.10">𝒱</mi><mrow id="S4.E38.m1.28.28.28.11.11.11.1"><mi id="S4.E38.m1.28.28.28.11.11.11.1.2">ρ</mi><mo id="S4.E38.m1.28.28.28.11.11.11.1.1">⁢</mo><mi id="S4.E38.m1.28.28.28.11.11.11.1.3">σ</mi><mo id="S4.E38.m1.28.28.28.11.11.11.1.1a">⁢</mo><msub id="S4.E38.m1.28.28.28.11.11.11.1.4"><mi id="S4.E38.m1.28.28.28.11.11.11.1.4.2">μ</mi><mn id="S4.E38.m1.28.28.28.11.11.11.1.4.3">2</mn></msub><mo id="S4.E38.m1.28.28.28.11.11.11.1.1b">⁢</mo><msub id="S4.E38.m1.28.28.28.11.11.11.1.5"><mi id="S4.E38.m1.28.28.28.11.11.11.1.5.2">ν</mi><mn id="S4.E38.m1.28.28.28.11.11.11.1.5.3">2</mn></msub><mo id="S4.E38.m1.28.28.28.11.11.11.1.1c">⁢</mo><mi id="S4.E38.m1.28.28.28.11.11.11.1.6">γ</mi><mo id="S4.E38.m1.28.28.28.11.11.11.1.1d">⁢</mo><mi id="S4.E38.m1.28.28.28.11.11.11.1.7">λ</mi></mrow><mrow id="S4.E38.m1.29.29.29.12.12.12.1.3"><mo id="S4.E38.m1.29.29.29.12.12.12.1.3.1" stretchy="false">(</mo><mn id="S4.E38.m1.29.29.29.12.12.12.1.1">3</mn><mo id="S4.E38.m1.29.29.29.12.12.12.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S4.E38.m1.64.64.1.64.19.19.19.2b">⁢</mo><mrow id="S4.E38.m1.64.64.1.64.19.19.19.1.1"><mo id="S4.E38.m1.30.30.30.13.13.13" stretchy="false">(</mo><mn id="S4.E38.m1.31.31.31.14.14.14">2</mn><mo id="S4.E38.m1.32.32.32.15.15.15">,</mo><mrow id="S4.E38.m1.64.64.1.64.19.19.19.1.1.1"><mi id="S4.E38.m1.33.33.33.16.16.16" mathvariant="normal">ℓ</mi><mo id="S4.E38.m1.64.64.1.64.19.19.19.1.1.1.1">⁢</mo><mn id="S4.E38.m1.34.34.34.17.17.17">1</mn></mrow><mo id="S4.E38.m1.35.35.35.18.18.18" stretchy="false">)</mo></mrow></mrow></mrow></mtd></mtr><mtr id="S4.E38.m1.64.64.1e"><mtd class="ltx_align_right" columnalign="right" id="S4.E38.m1.64.64.1f"><mrow id="S4.E38.m1.63.63.63.28.28"><mo id="S4.E38.m1.36.36.36.1.1.1">−</mo><mfrac id="S4.E38.m1.37.37.37.2.2.2"><mn id="S4.E38.m1.37.37.37.2.2.2.2">1</mn><mn id="S4.E38.m1.37.37.37.2.2.2.3">2</mn></mfrac><msubsup id="S4.E38.m1.63.63.63.28.28.29"><mi class="ltx_font_mathcaligraphic" id="S4.E38.m1.38.38.38.3.3.3">𝒱</mi><mrow id="S4.E38.m1.40.40.40.5.5.5.1"><msub id="S4.E38.m1.40.40.40.5.5.5.1.2"><mi id="S4.E38.m1.40.40.40.5.5.5.1.2.2">μ</mi><mn id="S4.E38.m1.40.40.40.5.5.5.1.2.3">1</mn></msub><mo id="S4.E38.m1.40.40.40.5.5.5.1.1">⁢</mo><msub id="S4.E38.m1.40.40.40.5.5.5.1.3"><mi id="S4.E38.m1.40.40.40.5.5.5.1.3.2">ν</mi><mn id="S4.E38.m1.40.40.40.5.5.5.1.3.3">1</mn></msub><mo id="S4.E38.m1.40.40.40.5.5.5.1.1a">⁢</mo><mi id="S4.E38.m1.40.40.40.5.5.5.1.4">ρ</mi></mrow><mrow id="S4.E38.m1.39.39.39.4.4.4.1"><mrow id="S4.E38.m1.39.39.39.4.4.4.1.3.2"><mo id="S4.E38.m1.39.39.39.4.4.4.1.3.2.1" stretchy="false">(</mo><mtext id="S4.E38.m1.39.39.39.4.4.4.1.1">g</mtext><mo id="S4.E38.m1.39.39.39.4.4.4.1.3.2.2" stretchy="false">)</mo></mrow><mo id="S4.E38.m1.39.39.39.4.4.4.1.2">⁢</mo><mi id="S4.E38.m1.39.39.39.4.4.4.1.4">σ</mi></mrow></msubsup><mrow id="S4.E38.m1.63.63.63.28.28.30"><mo id="S4.E38.m1.41.41.41.6.6.6" stretchy="false">(</mo><mn id="S4.E38.m1.42.42.42.7.7.7">1</mn><mo id="S4.E38.m1.43.43.43.8.8.8">,</mo><mi id="S4.E38.m1.44.44.44.9.9.9" mathvariant="normal">ℓ</mi><mn id="S4.E38.m1.45.45.45.10.10.10">1</mn><mo id="S4.E38.m1.46.46.46.11.11.11" stretchy="false">)</mo></mrow><msubsup id="S4.E38.m1.63.63.63.28.28.31"><mi class="ltx_font_mathcaligraphic" id="S4.E38.m1.47.47.47.12.12.12">𝒱</mi><mrow id="S4.E38.m1.49.49.49.14.14.14.1"><msub id="S4.E38.m1.49.49.49.14.14.14.1.2"><mi id="S4.E38.m1.49.49.49.14.14.14.1.2.2">μ</mi><mn id="S4.E38.m1.49.49.49.14.14.14.1.2.3">2</mn></msub><mo id="S4.E38.m1.49.49.49.14.14.14.1.1">⁢</mo><msub id="S4.E38.m1.49.49.49.14.14.14.1.3"><mi id="S4.E38.m1.49.49.49.14.14.14.1.3.2">ν</mi><mn id="S4.E38.m1.49.49.49.14.14.14.1.3.3">2</mn></msub><mo id="S4.E38.m1.49.49.49.14.14.14.1.1a">⁢</mo><mi id="S4.E38.m1.49.49.49.14.14.14.1.4">σ</mi></mrow><mrow id="S4.E38.m1.48.48.48.13.13.13.1"><mrow id="S4.E38.m1.48.48.48.13.13.13.1.3.2"><mo id="S4.E38.m1.48.48.48.13.13.13.1.3.2.1" stretchy="false">(</mo><mtext id="S4.E38.m1.48.48.48.13.13.13.1.1">g</mtext><mo id="S4.E38.m1.48.48.48.13.13.13.1.3.2.2" stretchy="false">)</mo></mrow><mo id="S4.E38.m1.48.48.48.13.13.13.1.2">⁢</mo><mi id="S4.E38.m1.48.48.48.13.13.13.1.4">ρ</mi></mrow></msubsup><mrow id="S4.E38.m1.63.63.63.28.28.32"><mo id="S4.E38.m1.50.50.50.15.15.15" stretchy="false">(</mo><mn id="S4.E38.m1.51.51.51.16.16.16">2</mn><mo id="S4.E38.m1.52.52.52.17.17.17">,</mo><mi id="S4.E38.m1.53.53.53.18.18.18" mathvariant="normal">ℓ</mi><mn id="S4.E38.m1.54.54.54.19.19.19">12</mn><mo id="S4.E38.m1.55.55.55.20.20.20" stretchy="false">)</mo></mrow><mo id="S4.E38.m1.56.56.56.21.21.21" maxsize="160%" minsize="160%">}</mo><mo id="S4.E38.m1.57.57.57.22.22.22">+</mo><mo id="S4.E38.m1.58.58.58.23.23.23" stretchy="false">(</mo><mn id="S4.E38.m1.59.59.59.24.24.24">1</mn><mo id="S4.E38.m1.60.60.60.25.25.25" stretchy="false">↔</mo><mn id="S4.E38.m1.61.61.61.26.26.26">2</mn><mo id="S4.E38.m1.62.62.62.27.27.27" stretchy="false">)</mo><mo id="S4.E38.m1.63.63.63.28.28.28" lspace="0em">.</mo></mrow></mtd></mtr></mtable><annotation encoding="application/x-tex" id="S4.E38.m1.64b">\mathcal{I}^{\textrm{se}}_{2}=\frac{h_{1}^{\mu_{1}\nu_{1}}h_{2}^{\mu_{2}\nu_{2% }}}{\ell^{2}(\ell+k_{1})^{2}}\Big{\{}\frac{1}{4}(\mathbb{P}^{-1})^{\mu\nu% \gamma\lambda}(\mathbb{P}^{-1})^{\alpha\beta\rho\sigma}\\ \times\mathcal{V}_{\mu\nu\mu_{1}\nu_{1}\alpha\beta}^{(3)}(1,\ell)\mathcal{V}_{% \rho\sigma\mu_{2}\nu_{2}\gamma\lambda}^{(3)}(2,\ell 1)\\ -\frac{1}{2}\mathcal{V}^{(\textrm{g})\sigma}_{\mu_{1}\nu_{1}\rho}(1,\ell 1)% \mathcal{V}^{(\textrm{g})\rho}_{\mu_{2}\nu_{2}\sigma}(2,\ell 12)\Big{\}}+(1% \leftrightarrow 2).</annotation><annotation encoding="application/x-llamapun" id="S4.E38.m1.64c">start_ROW start_CELL caligraphic_I start_POSTSUPERSCRIPT se end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = divide start_ARG italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_h start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_ARG start_ARG roman_ℓ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_ℓ + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG { divide start_ARG 1 end_ARG start_ARG 4 end_ARG ( blackboard_P start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_μ italic_ν italic_γ italic_λ end_POSTSUPERSCRIPT ( blackboard_P start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_α italic_β italic_ρ italic_σ end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL × caligraphic_V start_POSTSUBSCRIPT italic_μ italic_ν italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT ( 1 , roman_ℓ ) caligraphic_V start_POSTSUBSCRIPT italic_ρ italic_σ italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_γ italic_λ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT ( 2 , roman_ℓ 1 ) end_CELL end_ROW start_ROW start_CELL - divide start_ARG 1 end_ARG start_ARG 2 end_ARG caligraphic_V start_POSTSUPERSCRIPT ( g ) italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT ( 1 , roman_ℓ 1 ) caligraphic_V start_POSTSUPERSCRIPT ( g ) italic_ρ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( 2 , roman_ℓ 12 ) } + ( 1 ↔ 2 ) . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(38)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.3">The symmetry in the exchange of particles <math alttext="1" class="ltx_Math" display="inline" id="S4.p2.2.m1.1"><semantics id="S4.p2.2.m1.1a"><mn id="S4.p2.2.m1.1.1" xref="S4.p2.2.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S4.p2.2.m1.1b"><cn id="S4.p2.2.m1.1.1.cmml" type="integer" xref="S4.p2.2.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.2.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S4.p2.2.m1.1d">1</annotation></semantics></math> and <math alttext="2" class="ltx_Math" display="inline" id="S4.p2.3.m2.1"><semantics id="S4.p2.3.m2.1a"><mn id="S4.p2.3.m2.1.1" xref="S4.p2.3.m2.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S4.p2.3.m2.1b"><cn id="S4.p2.3.m2.1.1.cmml" type="integer" xref="S4.p2.3.m2.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.3.m2.1c">2</annotation><annotation encoding="application/x-llamapun" id="S4.p2.3.m2.1d">2</annotation></semantics></math> is explicit. The exchanged integrands are related by a redefinition of the loop momentum. 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.</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(39)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.5">The expression for <math alttext="\Pi_{\mu\nu\rho\sigma}" class="ltx_Math" display="inline" id="S4.p2.4.m1.1"><semantics id="S4.p2.4.m1.1a"><msub id="S4.p2.4.m1.1.1" xref="S4.p2.4.m1.1.1.cmml"><mi id="S4.p2.4.m1.1.1.2" mathvariant="normal" xref="S4.p2.4.m1.1.1.2.cmml">Π</mi><mrow id="S4.p2.4.m1.1.1.3" xref="S4.p2.4.m1.1.1.3.cmml"><mi id="S4.p2.4.m1.1.1.3.2" xref="S4.p2.4.m1.1.1.3.2.cmml">μ</mi><mo id="S4.p2.4.m1.1.1.3.1" xref="S4.p2.4.m1.1.1.3.1.cmml">⁢</mo><mi id="S4.p2.4.m1.1.1.3.3" xref="S4.p2.4.m1.1.1.3.3.cmml">ν</mi><mo id="S4.p2.4.m1.1.1.3.1a" xref="S4.p2.4.m1.1.1.3.1.cmml">⁢</mo><mi id="S4.p2.4.m1.1.1.3.4" xref="S4.p2.4.m1.1.1.3.4.cmml">ρ</mi><mo id="S4.p2.4.m1.1.1.3.1b" xref="S4.p2.4.m1.1.1.3.1.cmml">⁢</mo><mi id="S4.p2.4.m1.1.1.3.5" xref="S4.p2.4.m1.1.1.3.5.cmml">σ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.p2.4.m1.1b"><apply id="S4.p2.4.m1.1.1.cmml" xref="S4.p2.4.m1.1.1"><csymbol cd="ambiguous" id="S4.p2.4.m1.1.1.1.cmml" xref="S4.p2.4.m1.1.1">subscript</csymbol><ci id="S4.p2.4.m1.1.1.2.cmml" xref="S4.p2.4.m1.1.1.2">Π</ci><apply id="S4.p2.4.m1.1.1.3.cmml" xref="S4.p2.4.m1.1.1.3"><times id="S4.p2.4.m1.1.1.3.1.cmml" xref="S4.p2.4.m1.1.1.3.1"></times><ci id="S4.p2.4.m1.1.1.3.2.cmml" xref="S4.p2.4.m1.1.1.3.2">𝜇</ci><ci id="S4.p2.4.m1.1.1.3.3.cmml" xref="S4.p2.4.m1.1.1.3.3">𝜈</ci><ci id="S4.p2.4.m1.1.1.3.4.cmml" xref="S4.p2.4.m1.1.1.3.4">𝜌</ci><ci id="S4.p2.4.m1.1.1.3.5.cmml" xref="S4.p2.4.m1.1.1.3.5">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.4.m1.1c">\Pi_{\mu\nu\rho\sigma}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.4.m1.1d">roman_Π start_POSTSUBSCRIPT italic_μ italic_ν italic_ρ italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> is more commonly presented in terms of the propagator correction, <math alttext="Q^{\mu\nu\rho\sigma}=D^{\mu\nu\alpha\beta}_{1}D^{\rho\sigma\gamma\lambda}_{2}% \Pi_{\alpha\beta\gamma\lambda}" class="ltx_Math" display="inline" id="S4.p2.5.m2.1"><semantics id="S4.p2.5.m2.1a"><mrow id="S4.p2.5.m2.1.1" xref="S4.p2.5.m2.1.1.cmml"><msup id="S4.p2.5.m2.1.1.2" xref="S4.p2.5.m2.1.1.2.cmml"><mi id="S4.p2.5.m2.1.1.2.2" xref="S4.p2.5.m2.1.1.2.2.cmml">Q</mi><mrow id="S4.p2.5.m2.1.1.2.3" xref="S4.p2.5.m2.1.1.2.3.cmml"><mi id="S4.p2.5.m2.1.1.2.3.2" xref="S4.p2.5.m2.1.1.2.3.2.cmml">μ</mi><mo id="S4.p2.5.m2.1.1.2.3.1" xref="S4.p2.5.m2.1.1.2.3.1.cmml">⁢</mo><mi id="S4.p2.5.m2.1.1.2.3.3" xref="S4.p2.5.m2.1.1.2.3.3.cmml">ν</mi><mo id="S4.p2.5.m2.1.1.2.3.1a" xref="S4.p2.5.m2.1.1.2.3.1.cmml">⁢</mo><mi id="S4.p2.5.m2.1.1.2.3.4" xref="S4.p2.5.m2.1.1.2.3.4.cmml">ρ</mi><mo id="S4.p2.5.m2.1.1.2.3.1b" 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xref="S4.p2.5.m2.1.1.3.4.2">Π</ci><apply id="S4.p2.5.m2.1.1.3.4.3.cmml" xref="S4.p2.5.m2.1.1.3.4.3"><times id="S4.p2.5.m2.1.1.3.4.3.1.cmml" xref="S4.p2.5.m2.1.1.3.4.3.1"></times><ci id="S4.p2.5.m2.1.1.3.4.3.2.cmml" xref="S4.p2.5.m2.1.1.3.4.3.2">𝛼</ci><ci id="S4.p2.5.m2.1.1.3.4.3.3.cmml" xref="S4.p2.5.m2.1.1.3.4.3.3">𝛽</ci><ci id="S4.p2.5.m2.1.1.3.4.3.4.cmml" xref="S4.p2.5.m2.1.1.3.4.3.4">𝛾</ci><ci id="S4.p2.5.m2.1.1.3.4.3.5.cmml" xref="S4.p2.5.m2.1.1.3.4.3.5">𝜆</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.5.m2.1c">Q^{\mu\nu\rho\sigma}=D^{\mu\nu\alpha\beta}_{1}D^{\rho\sigma\gamma\lambda}_{2}% \Pi_{\alpha\beta\gamma\lambda}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.5.m2.1d">italic_Q start_POSTSUPERSCRIPT italic_μ italic_ν italic_ρ italic_σ end_POSTSUPERSCRIPT = italic_D start_POSTSUPERSCRIPT italic_μ italic_ν italic_α italic_β end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_D start_POSTSUPERSCRIPT italic_ρ italic_σ italic_γ italic_λ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_α italic_β italic_γ italic_λ end_POSTSUBSCRIPT</annotation></semantics></math>, given by</p> <table class="ltx_equation ltx_eqn_table" id="S4.E40"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="Q^{\mu\nu\rho\sigma}=B_{0}\big{\{}c_{1}\Pi^{\mu\nu}\Pi^{\alpha\beta}-\Pi^{\mu% \nu}k^{\alpha}k^{\beta}-\Pi^{\alpha\beta}k^{\mu}k^{\nu}\\ -c_{2}(\Pi^{\mu\alpha}\Pi^{\nu\beta}+\Pi^{\mu\beta}\Pi^{\nu\alpha})\big{\}}," class="ltx_Math" display="block" id="S4.E40.m1.43"><semantics id="S4.E40.m1.43a"><mtable displaystyle="true" id="S4.E40.m1.42.42" rowspacing="0pt"><mtr id="S4.E40.m1.42.42a"><mtd class="ltx_align_left" columnalign="left" id="S4.E40.m1.42.42b"><mrow id="S4.E40.m1.26.26.26.26.26"><msup 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xref="S4.E40.m1.5.5.5.5.5.5.1.cmml">0</mn></msub><mrow id="S4.E40.m1.26.26.26.26.26.29"><mo id="S4.E40.m1.6.6.6.6.6.6" maxsize="120%" minsize="120%" xref="S4.E40.m1.43.43.1.1.1.cmml">{</mo><msub id="S4.E40.m1.26.26.26.26.26.29.1"><mi id="S4.E40.m1.7.7.7.7.7.7" xref="S4.E40.m1.7.7.7.7.7.7.cmml">c</mi><mn id="S4.E40.m1.8.8.8.8.8.8.1" xref="S4.E40.m1.8.8.8.8.8.8.1.cmml">1</mn></msub><msup id="S4.E40.m1.26.26.26.26.26.29.2"><mi id="S4.E40.m1.9.9.9.9.9.9" mathvariant="normal" xref="S4.E40.m1.9.9.9.9.9.9.cmml">Π</mi><mrow id="S4.E40.m1.10.10.10.10.10.10.1" xref="S4.E40.m1.10.10.10.10.10.10.1.cmml"><mi id="S4.E40.m1.10.10.10.10.10.10.1.2" xref="S4.E40.m1.10.10.10.10.10.10.1.2.cmml">μ</mi><mo id="S4.E40.m1.10.10.10.10.10.10.1.1" xref="S4.E40.m1.10.10.10.10.10.10.1.1.cmml">⁢</mo><mi id="S4.E40.m1.10.10.10.10.10.10.1.3" xref="S4.E40.m1.10.10.10.10.10.10.1.3.cmml">ν</mi></mrow></msup><msup id="S4.E40.m1.26.26.26.26.26.29.3"><mi id="S4.E40.m1.11.11.11.11.11.11" mathvariant="normal" 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id="S4.E40.m1.39.39.39.13.13.13.1.2.cmml" xref="S4.E40.m1.39.39.39.13.13.13.1.2">𝜈</ci><ci id="S4.E40.m1.39.39.39.13.13.13.1.3.cmml" xref="S4.E40.m1.39.39.39.13.13.13.1.3">𝛼</ci></apply></apply></apply></apply></apply></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E40.m1.43c">Q^{\mu\nu\rho\sigma}=B_{0}\big{\{}c_{1}\Pi^{\mu\nu}\Pi^{\alpha\beta}-\Pi^{\mu% \nu}k^{\alpha}k^{\beta}-\Pi^{\alpha\beta}k^{\mu}k^{\nu}\\ -c_{2}(\Pi^{\mu\alpha}\Pi^{\nu\beta}+\Pi^{\mu\beta}\Pi^{\nu\alpha})\big{\}},</annotation><annotation encoding="application/x-llamapun" id="S4.E40.m1.43d">start_ROW start_CELL italic_Q start_POSTSUPERSCRIPT italic_μ italic_ν italic_ρ italic_σ end_POSTSUPERSCRIPT = italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT { italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT roman_Π start_POSTSUPERSCRIPT italic_α italic_β end_POSTSUPERSCRIPT - roman_Π start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT italic_k start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT italic_k start_POSTSUPERSCRIPT italic_β end_POSTSUPERSCRIPT - roman_Π start_POSTSUPERSCRIPT italic_α italic_β end_POSTSUPERSCRIPT italic_k start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT italic_k start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL - italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( roman_Π start_POSTSUPERSCRIPT italic_μ italic_α end_POSTSUPERSCRIPT roman_Π start_POSTSUPERSCRIPT italic_ν italic_β end_POSTSUPERSCRIPT + roman_Π start_POSTSUPERSCRIPT italic_μ italic_β end_POSTSUPERSCRIPT roman_Π start_POSTSUPERSCRIPT italic_ν italic_α end_POSTSUPERSCRIPT ) } , end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(40)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.7">with <math alttext="k_{1}=-k_{2}=k" class="ltx_Math" display="inline" id="S4.p2.6.m1.1"><semantics id="S4.p2.6.m1.1a"><mrow id="S4.p2.6.m1.1.1" xref="S4.p2.6.m1.1.1.cmml"><msub id="S4.p2.6.m1.1.1.2" xref="S4.p2.6.m1.1.1.2.cmml"><mi id="S4.p2.6.m1.1.1.2.2" xref="S4.p2.6.m1.1.1.2.2.cmml">k</mi><mn id="S4.p2.6.m1.1.1.2.3" xref="S4.p2.6.m1.1.1.2.3.cmml">1</mn></msub><mo id="S4.p2.6.m1.1.1.3" xref="S4.p2.6.m1.1.1.3.cmml">=</mo><mrow id="S4.p2.6.m1.1.1.4" xref="S4.p2.6.m1.1.1.4.cmml"><mo id="S4.p2.6.m1.1.1.4a" xref="S4.p2.6.m1.1.1.4.cmml">−</mo><msub id="S4.p2.6.m1.1.1.4.2" xref="S4.p2.6.m1.1.1.4.2.cmml"><mi id="S4.p2.6.m1.1.1.4.2.2" xref="S4.p2.6.m1.1.1.4.2.2.cmml">k</mi><mn id="S4.p2.6.m1.1.1.4.2.3" xref="S4.p2.6.m1.1.1.4.2.3.cmml">2</mn></msub></mrow><mo id="S4.p2.6.m1.1.1.5" xref="S4.p2.6.m1.1.1.5.cmml">=</mo><mi id="S4.p2.6.m1.1.1.6" xref="S4.p2.6.m1.1.1.6.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.6.m1.1b"><apply 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end_POSTSUBSCRIPT - italic_k start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT</annotation></semantics></math>, and</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S5.EGx5"> <tbody id="S4.E41"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle c_{1}" class="ltx_Math" display="inline" id="S4.E41.m1.1"><semantics id="S4.E41.m1.1a"><msub id="S4.E41.m1.1.1" xref="S4.E41.m1.1.1.cmml"><mi id="S4.E41.m1.1.1.2" xref="S4.E41.m1.1.1.2.cmml">c</mi><mn id="S4.E41.m1.1.1.3" xref="S4.E41.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.E41.m1.1b"><apply id="S4.E41.m1.1.1.cmml" xref="S4.E41.m1.1.1"><csymbol cd="ambiguous" id="S4.E41.m1.1.1.1.cmml" xref="S4.E41.m1.1.1">subscript</csymbol><ci id="S4.E41.m1.1.1.2.cmml" xref="S4.E41.m1.1.1.2">𝑐</ci><cn id="S4.E41.m1.1.1.3.cmml" type="integer" xref="S4.E41.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E41.m1.1c">\displaystyle c_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.E41.m1.1d">italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S4.E41.m2.1"><semantics id="S4.E41.m2.1a"><mo id="S4.E41.m2.1.1" xref="S4.E41.m2.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S4.E41.m2.1b"><eq id="S4.E41.m2.1.1.cmml" xref="S4.E41.m2.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S4.E41.m2.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S4.E41.m2.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\frac{[64+(94+(20+(d-13)d)d)d]}{8(d^{2}-1)}," 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xref="S4.E41.m3.1.1.1.1.1.1.1.1.1.1.3">𝑑</ci></apply></apply><ci id="S4.E41.m3.1.1.1.1.1.1.3.cmml" xref="S4.E41.m3.1.1.1.1.1.1.3">𝑑</ci></apply></apply></apply><apply id="S4.E41.m3.2.2.2.cmml" xref="S4.E41.m3.2.2.2"><times id="S4.E41.m3.2.2.2.2.cmml" xref="S4.E41.m3.2.2.2.2"></times><cn id="S4.E41.m3.2.2.2.3.cmml" type="integer" xref="S4.E41.m3.2.2.2.3">8</cn><apply id="S4.E41.m3.2.2.2.1.1.1.cmml" xref="S4.E41.m3.2.2.2.1.1"><minus id="S4.E41.m3.2.2.2.1.1.1.1.cmml" xref="S4.E41.m3.2.2.2.1.1.1.1"></minus><apply id="S4.E41.m3.2.2.2.1.1.1.2.cmml" xref="S4.E41.m3.2.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S4.E41.m3.2.2.2.1.1.1.2.1.cmml" xref="S4.E41.m3.2.2.2.1.1.1.2">superscript</csymbol><ci id="S4.E41.m3.2.2.2.1.1.1.2.2.cmml" xref="S4.E41.m3.2.2.2.1.1.1.2.2">𝑑</ci><cn id="S4.E41.m3.2.2.2.1.1.1.2.3.cmml" type="integer" xref="S4.E41.m3.2.2.2.1.1.1.2.3">2</cn></apply><cn id="S4.E41.m3.2.2.2.1.1.1.3.cmml" type="integer" xref="S4.E41.m3.2.2.2.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E41.m3.2c">\displaystyle\frac{[64+(94+(20+(d-13)d)d)d]}{8(d^{2}-1)},</annotation><annotation encoding="application/x-llamapun" id="S4.E41.m3.2d">divide start_ARG [ 64 + ( 94 + ( 20 + ( italic_d - 13 ) italic_d ) italic_d ) italic_d ] end_ARG start_ARG 8 ( italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 ) end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(41)</span></td> </tr></tbody> <tbody id="S4.E42"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle c_{2}" class="ltx_Math" display="inline" id="S4.E42.m1.1"><semantics id="S4.E42.m1.1a"><msub id="S4.E42.m1.1.1" xref="S4.E42.m1.1.1.cmml"><mi id="S4.E42.m1.1.1.2" xref="S4.E42.m1.1.1.2.cmml">c</mi><mn id="S4.E42.m1.1.1.3" xref="S4.E42.m1.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.E42.m1.1b"><apply id="S4.E42.m1.1.1.cmml" xref="S4.E42.m1.1.1"><csymbol cd="ambiguous" id="S4.E42.m1.1.1.1.cmml" xref="S4.E42.m1.1.1">subscript</csymbol><ci id="S4.E42.m1.1.1.2.cmml" xref="S4.E42.m1.1.1.2">𝑐</ci><cn id="S4.E42.m1.1.1.3.cmml" type="integer" xref="S4.E42.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E42.m1.1c">\displaystyle c_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.E42.m1.1d">italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S4.E42.m2.1"><semantics id="S4.E42.m2.1a"><mo id="S4.E42.m2.1.1" 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xref="S4.E42.m3.2.2.2.1.1.1.2">superscript</csymbol><ci id="S4.E42.m3.2.2.2.1.1.1.2.2.cmml" xref="S4.E42.m3.2.2.2.1.1.1.2.2">𝑑</ci><cn id="S4.E42.m3.2.2.2.1.1.1.2.3.cmml" type="integer" xref="S4.E42.m3.2.2.2.1.1.1.2.3">2</cn></apply><cn id="S4.E42.m3.2.2.2.1.1.1.3.cmml" type="integer" xref="S4.E42.m3.2.2.2.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E42.m3.2c">\displaystyle\frac{[16+(15-(9+4d)d)d]}{8(d^{2}-1)}.</annotation><annotation encoding="application/x-llamapun" id="S4.E42.m3.2d">divide start_ARG [ 16 + ( 15 - ( 9 + 4 italic_d ) italic_d ) italic_d ] end_ARG start_ARG 8 ( italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 ) end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(42)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.8">The coefficient <math alttext="B_{0}=B_{0}(k^{2})" class="ltx_Math" display="inline" id="S4.p2.8.m1.1"><semantics id="S4.p2.8.m1.1a"><mrow id="S4.p2.8.m1.1.1" xref="S4.p2.8.m1.1.1.cmml"><msub id="S4.p2.8.m1.1.1.3" xref="S4.p2.8.m1.1.1.3.cmml"><mi id="S4.p2.8.m1.1.1.3.2" xref="S4.p2.8.m1.1.1.3.2.cmml">B</mi><mn id="S4.p2.8.m1.1.1.3.3" xref="S4.p2.8.m1.1.1.3.3.cmml">0</mn></msub><mo id="S4.p2.8.m1.1.1.2" xref="S4.p2.8.m1.1.1.2.cmml">=</mo><mrow id="S4.p2.8.m1.1.1.1" xref="S4.p2.8.m1.1.1.1.cmml"><msub id="S4.p2.8.m1.1.1.1.3" xref="S4.p2.8.m1.1.1.1.3.cmml"><mi id="S4.p2.8.m1.1.1.1.3.2" xref="S4.p2.8.m1.1.1.1.3.2.cmml">B</mi><mn id="S4.p2.8.m1.1.1.1.3.3" xref="S4.p2.8.m1.1.1.1.3.3.cmml">0</mn></msub><mo id="S4.p2.8.m1.1.1.1.2" xref="S4.p2.8.m1.1.1.1.2.cmml">⁢</mo><mrow id="S4.p2.8.m1.1.1.1.1.1" xref="S4.p2.8.m1.1.1.1.1.1.1.cmml"><mo id="S4.p2.8.m1.1.1.1.1.1.2" stretchy="false" xref="S4.p2.8.m1.1.1.1.1.1.1.cmml">(</mo><msup id="S4.p2.8.m1.1.1.1.1.1.1" xref="S4.p2.8.m1.1.1.1.1.1.1.cmml"><mi id="S4.p2.8.m1.1.1.1.1.1.1.2" xref="S4.p2.8.m1.1.1.1.1.1.1.2.cmml">k</mi><mn id="S4.p2.8.m1.1.1.1.1.1.1.3" xref="S4.p2.8.m1.1.1.1.1.1.1.3.cmml">2</mn></msup><mo id="S4.p2.8.m1.1.1.1.1.1.3" stretchy="false" xref="S4.p2.8.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.8.m1.1b"><apply id="S4.p2.8.m1.1.1.cmml" xref="S4.p2.8.m1.1.1"><eq id="S4.p2.8.m1.1.1.2.cmml" xref="S4.p2.8.m1.1.1.2"></eq><apply id="S4.p2.8.m1.1.1.3.cmml" xref="S4.p2.8.m1.1.1.3"><csymbol cd="ambiguous" id="S4.p2.8.m1.1.1.3.1.cmml" xref="S4.p2.8.m1.1.1.3">subscript</csymbol><ci id="S4.p2.8.m1.1.1.3.2.cmml" xref="S4.p2.8.m1.1.1.3.2">𝐵</ci><cn id="S4.p2.8.m1.1.1.3.3.cmml" type="integer" xref="S4.p2.8.m1.1.1.3.3">0</cn></apply><apply id="S4.p2.8.m1.1.1.1.cmml" xref="S4.p2.8.m1.1.1.1"><times id="S4.p2.8.m1.1.1.1.2.cmml" xref="S4.p2.8.m1.1.1.1.2"></times><apply id="S4.p2.8.m1.1.1.1.3.cmml" xref="S4.p2.8.m1.1.1.1.3"><csymbol cd="ambiguous" id="S4.p2.8.m1.1.1.1.3.1.cmml" xref="S4.p2.8.m1.1.1.1.3">subscript</csymbol><ci id="S4.p2.8.m1.1.1.1.3.2.cmml" xref="S4.p2.8.m1.1.1.1.3.2">𝐵</ci><cn id="S4.p2.8.m1.1.1.1.3.3.cmml" type="integer" xref="S4.p2.8.m1.1.1.1.3.3">0</cn></apply><apply id="S4.p2.8.m1.1.1.1.1.1.1.cmml" xref="S4.p2.8.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.p2.8.m1.1.1.1.1.1.1.1.cmml" xref="S4.p2.8.m1.1.1.1.1.1">superscript</csymbol><ci id="S4.p2.8.m1.1.1.1.1.1.1.2.cmml" xref="S4.p2.8.m1.1.1.1.1.1.1.2">𝑘</ci><cn id="S4.p2.8.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.p2.8.m1.1.1.1.1.1.1.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.8.m1.1c">B_{0}=B_{0}(k^{2})</annotation><annotation encoding="application/x-llamapun" id="S4.p2.8.m1.1d">italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )</annotation></semantics></math> is expressed in terms of the scalar integral</p> <table class="ltx_equation ltx_eqn_table" id="S4.E43"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="B_{0}=\frac{1}{2k^{4}}\int\frac{d^{d}\ell}{(2\pi)^{d}}\frac{1}{\ell^{2}(\ell+k% )^{2}}," class="ltx_Math" display="block" id="S4.E43.m1.3"><semantics id="S4.E43.m1.3a"><mrow id="S4.E43.m1.3.3.1" xref="S4.E43.m1.3.3.1.1.cmml"><mrow id="S4.E43.m1.3.3.1.1" xref="S4.E43.m1.3.3.1.1.cmml"><msub id="S4.E43.m1.3.3.1.1.2" xref="S4.E43.m1.3.3.1.1.2.cmml"><mi id="S4.E43.m1.3.3.1.1.2.2" xref="S4.E43.m1.3.3.1.1.2.2.cmml">B</mi><mn id="S4.E43.m1.3.3.1.1.2.3" xref="S4.E43.m1.3.3.1.1.2.3.cmml">0</mn></msub><mo id="S4.E43.m1.3.3.1.1.1" xref="S4.E43.m1.3.3.1.1.1.cmml">=</mo><mrow id="S4.E43.m1.3.3.1.1.3" xref="S4.E43.m1.3.3.1.1.3.cmml"><mfrac id="S4.E43.m1.3.3.1.1.3.2" xref="S4.E43.m1.3.3.1.1.3.2.cmml"><mn id="S4.E43.m1.3.3.1.1.3.2.2" xref="S4.E43.m1.3.3.1.1.3.2.2.cmml">1</mn><mrow id="S4.E43.m1.3.3.1.1.3.2.3" xref="S4.E43.m1.3.3.1.1.3.2.3.cmml"><mn id="S4.E43.m1.3.3.1.1.3.2.3.2" xref="S4.E43.m1.3.3.1.1.3.2.3.2.cmml">2</mn><mo id="S4.E43.m1.3.3.1.1.3.2.3.1" xref="S4.E43.m1.3.3.1.1.3.2.3.1.cmml">⁢</mo><msup id="S4.E43.m1.3.3.1.1.3.2.3.3" xref="S4.E43.m1.3.3.1.1.3.2.3.3.cmml"><mi id="S4.E43.m1.3.3.1.1.3.2.3.3.2" xref="S4.E43.m1.3.3.1.1.3.2.3.3.2.cmml">k</mi><mn id="S4.E43.m1.3.3.1.1.3.2.3.3.3" xref="S4.E43.m1.3.3.1.1.3.2.3.3.3.cmml">4</mn></msup></mrow></mfrac><mo id="S4.E43.m1.3.3.1.1.3.1" xref="S4.E43.m1.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S4.E43.m1.3.3.1.1.3.3" xref="S4.E43.m1.3.3.1.1.3.3.cmml"><mo id="S4.E43.m1.3.3.1.1.3.3.1" xref="S4.E43.m1.3.3.1.1.3.3.1.cmml">∫</mo><mrow id="S4.E43.m1.3.3.1.1.3.3.2" xref="S4.E43.m1.3.3.1.1.3.3.2.cmml"><mfrac id="S4.E43.m1.1.1" xref="S4.E43.m1.1.1.cmml"><mrow id="S4.E43.m1.1.1.3" xref="S4.E43.m1.1.1.3.cmml"><msup id="S4.E43.m1.1.1.3.2" xref="S4.E43.m1.1.1.3.2.cmml"><mi 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id="S4.E43.m1.3.3.1.1.3.3.2.1" xref="S4.E43.m1.3.3.1.1.3.3.2.1.cmml">⁢</mo><mfrac id="S4.E43.m1.2.2" xref="S4.E43.m1.2.2.cmml"><mn id="S4.E43.m1.2.2.3" xref="S4.E43.m1.2.2.3.cmml">1</mn><mrow id="S4.E43.m1.2.2.1" xref="S4.E43.m1.2.2.1.cmml"><msup id="S4.E43.m1.2.2.1.3" xref="S4.E43.m1.2.2.1.3.cmml"><mi id="S4.E43.m1.2.2.1.3.2" mathvariant="normal" xref="S4.E43.m1.2.2.1.3.2.cmml">ℓ</mi><mn id="S4.E43.m1.2.2.1.3.3" xref="S4.E43.m1.2.2.1.3.3.cmml">2</mn></msup><mo id="S4.E43.m1.2.2.1.2" xref="S4.E43.m1.2.2.1.2.cmml">⁢</mo><msup id="S4.E43.m1.2.2.1.1" xref="S4.E43.m1.2.2.1.1.cmml"><mrow id="S4.E43.m1.2.2.1.1.1.1" xref="S4.E43.m1.2.2.1.1.1.1.1.cmml"><mo id="S4.E43.m1.2.2.1.1.1.1.2" stretchy="false" xref="S4.E43.m1.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S4.E43.m1.2.2.1.1.1.1.1" xref="S4.E43.m1.2.2.1.1.1.1.1.cmml"><mi id="S4.E43.m1.2.2.1.1.1.1.1.2" mathvariant="normal" xref="S4.E43.m1.2.2.1.1.1.1.1.2.cmml">ℓ</mi><mo id="S4.E43.m1.2.2.1.1.1.1.1.1" xref="S4.E43.m1.2.2.1.1.1.1.1.1.cmml">+</mo><mi 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start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG ∫ divide start_ARG italic_d start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT roman_ℓ end_ARG start_ARG ( 2 italic_π ) start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_ARG divide start_ARG 1 end_ARG start_ARG roman_ℓ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_ℓ + italic_k ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(43)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.11">which usually appears when using dimensional regularization (<cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib61" title="">Capper:1974dc </a></cite> and references therein). In this case, the tadpole contributions are regularized to zero.</p> </div> <div class="ltx_para" id="S4.p3"> <p class="ltx_p" id="S4.p3.2">Equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S4.E40" title="In IV Some examples ‣ One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_tag">40</span></a>) can be validated through the Slavnov-Taylor identity involving the divergence of the two-graviton correlator. It is easy to show that</p> <table class="ltx_equation ltx_eqn_table" id="S4.E44"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="k_{\mu}Q^{\mu\nu\rho\sigma}=-B_{0}k^{2}k^{\nu}\Pi^{\rho\sigma}," class="ltx_Math" display="block" id="S4.E44.m1.1"><semantics id="S4.E44.m1.1a"><mrow id="S4.E44.m1.1.1.1" xref="S4.E44.m1.1.1.1.1.cmml"><mrow id="S4.E44.m1.1.1.1.1" xref="S4.E44.m1.1.1.1.1.cmml"><mrow id="S4.E44.m1.1.1.1.1.2" xref="S4.E44.m1.1.1.1.1.2.cmml"><msub id="S4.E44.m1.1.1.1.1.2.2" xref="S4.E44.m1.1.1.1.1.2.2.cmml"><mi id="S4.E44.m1.1.1.1.1.2.2.2" xref="S4.E44.m1.1.1.1.1.2.2.2.cmml">k</mi><mi id="S4.E44.m1.1.1.1.1.2.2.3" xref="S4.E44.m1.1.1.1.1.2.2.3.cmml">μ</mi></msub><mo id="S4.E44.m1.1.1.1.1.2.1" xref="S4.E44.m1.1.1.1.1.2.1.cmml">⁢</mo><msup id="S4.E44.m1.1.1.1.1.2.3" xref="S4.E44.m1.1.1.1.1.2.3.cmml"><mi id="S4.E44.m1.1.1.1.1.2.3.2" 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xref="S4.E44.m1.1.1.1.1.3.2.5.3.3">𝜎</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E44.m1.1c">k_{\mu}Q^{\mu\nu\rho\sigma}=-B_{0}k^{2}k^{\nu}\Pi^{\rho\sigma},</annotation><annotation encoding="application/x-llamapun" id="S4.E44.m1.1d">italic_k start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_Q start_POSTSUPERSCRIPT italic_μ italic_ν italic_ρ italic_σ end_POSTSUPERSCRIPT = - italic_B start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_k start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT roman_Π start_POSTSUPERSCRIPT italic_ρ italic_σ end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(44)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p3.1">The complete identity can be found in <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib62" title="">Capper:1974vb </a></cite>, which we have verified using our proposal. The weaker identity <math alttext="k_{\mu}k_{\rho}Q^{\mu\nu\rho\sigma}=0" class="ltx_Math" display="inline" id="S4.p3.1.m1.1"><semantics id="S4.p3.1.m1.1a"><mrow id="S4.p3.1.m1.1.1" xref="S4.p3.1.m1.1.1.cmml"><mrow id="S4.p3.1.m1.1.1.2" xref="S4.p3.1.m1.1.1.2.cmml"><msub id="S4.p3.1.m1.1.1.2.2" xref="S4.p3.1.m1.1.1.2.2.cmml"><mi id="S4.p3.1.m1.1.1.2.2.2" xref="S4.p3.1.m1.1.1.2.2.2.cmml">k</mi><mi id="S4.p3.1.m1.1.1.2.2.3" xref="S4.p3.1.m1.1.1.2.2.3.cmml">μ</mi></msub><mo id="S4.p3.1.m1.1.1.2.1" xref="S4.p3.1.m1.1.1.2.1.cmml">⁢</mo><msub id="S4.p3.1.m1.1.1.2.3" xref="S4.p3.1.m1.1.1.2.3.cmml"><mi id="S4.p3.1.m1.1.1.2.3.2" xref="S4.p3.1.m1.1.1.2.3.2.cmml">k</mi><mi id="S4.p3.1.m1.1.1.2.3.3" xref="S4.p3.1.m1.1.1.2.3.3.cmml">ρ</mi></msub><mo id="S4.p3.1.m1.1.1.2.1a" xref="S4.p3.1.m1.1.1.2.1.cmml">⁢</mo><msup id="S4.p3.1.m1.1.1.2.4" xref="S4.p3.1.m1.1.1.2.4.cmml"><mi id="S4.p3.1.m1.1.1.2.4.2" xref="S4.p3.1.m1.1.1.2.4.2.cmml">Q</mi><mrow id="S4.p3.1.m1.1.1.2.4.3" xref="S4.p3.1.m1.1.1.2.4.3.cmml"><mi id="S4.p3.1.m1.1.1.2.4.3.2" xref="S4.p3.1.m1.1.1.2.4.3.2.cmml">μ</mi><mo id="S4.p3.1.m1.1.1.2.4.3.1" xref="S4.p3.1.m1.1.1.2.4.3.1.cmml">⁢</mo><mi id="S4.p3.1.m1.1.1.2.4.3.3" xref="S4.p3.1.m1.1.1.2.4.3.3.cmml">ν</mi><mo id="S4.p3.1.m1.1.1.2.4.3.1a" xref="S4.p3.1.m1.1.1.2.4.3.1.cmml">⁢</mo><mi id="S4.p3.1.m1.1.1.2.4.3.4" xref="S4.p3.1.m1.1.1.2.4.3.4.cmml">ρ</mi><mo id="S4.p3.1.m1.1.1.2.4.3.1b" xref="S4.p3.1.m1.1.1.2.4.3.1.cmml">⁢</mo><mi id="S4.p3.1.m1.1.1.2.4.3.5" xref="S4.p3.1.m1.1.1.2.4.3.5.cmml">σ</mi></mrow></msup></mrow><mo id="S4.p3.1.m1.1.1.1" xref="S4.p3.1.m1.1.1.1.cmml">=</mo><mn id="S4.p3.1.m1.1.1.3" xref="S4.p3.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.1.m1.1b"><apply id="S4.p3.1.m1.1.1.cmml" xref="S4.p3.1.m1.1.1"><eq id="S4.p3.1.m1.1.1.1.cmml" xref="S4.p3.1.m1.1.1.1"></eq><apply id="S4.p3.1.m1.1.1.2.cmml" xref="S4.p3.1.m1.1.1.2"><times id="S4.p3.1.m1.1.1.2.1.cmml" xref="S4.p3.1.m1.1.1.2.1"></times><apply id="S4.p3.1.m1.1.1.2.2.cmml" xref="S4.p3.1.m1.1.1.2.2"><csymbol cd="ambiguous" id="S4.p3.1.m1.1.1.2.2.1.cmml" xref="S4.p3.1.m1.1.1.2.2">subscript</csymbol><ci id="S4.p3.1.m1.1.1.2.2.2.cmml" xref="S4.p3.1.m1.1.1.2.2.2">𝑘</ci><ci id="S4.p3.1.m1.1.1.2.2.3.cmml" xref="S4.p3.1.m1.1.1.2.2.3">𝜇</ci></apply><apply id="S4.p3.1.m1.1.1.2.3.cmml" xref="S4.p3.1.m1.1.1.2.3"><csymbol cd="ambiguous" id="S4.p3.1.m1.1.1.2.3.1.cmml" xref="S4.p3.1.m1.1.1.2.3">subscript</csymbol><ci id="S4.p3.1.m1.1.1.2.3.2.cmml" xref="S4.p3.1.m1.1.1.2.3.2">𝑘</ci><ci id="S4.p3.1.m1.1.1.2.3.3.cmml" xref="S4.p3.1.m1.1.1.2.3.3">𝜌</ci></apply><apply id="S4.p3.1.m1.1.1.2.4.cmml" xref="S4.p3.1.m1.1.1.2.4"><csymbol cd="ambiguous" id="S4.p3.1.m1.1.1.2.4.1.cmml" xref="S4.p3.1.m1.1.1.2.4">superscript</csymbol><ci id="S4.p3.1.m1.1.1.2.4.2.cmml" xref="S4.p3.1.m1.1.1.2.4.2">𝑄</ci><apply id="S4.p3.1.m1.1.1.2.4.3.cmml" xref="S4.p3.1.m1.1.1.2.4.3"><times id="S4.p3.1.m1.1.1.2.4.3.1.cmml" xref="S4.p3.1.m1.1.1.2.4.3.1"></times><ci id="S4.p3.1.m1.1.1.2.4.3.2.cmml" xref="S4.p3.1.m1.1.1.2.4.3.2">𝜇</ci><ci id="S4.p3.1.m1.1.1.2.4.3.3.cmml" xref="S4.p3.1.m1.1.1.2.4.3.3">𝜈</ci><ci id="S4.p3.1.m1.1.1.2.4.3.4.cmml" xref="S4.p3.1.m1.1.1.2.4.3.4">𝜌</ci><ci id="S4.p3.1.m1.1.1.2.4.3.5.cmml" xref="S4.p3.1.m1.1.1.2.4.3.5">𝜎</ci></apply></apply></apply><cn id="S4.p3.1.m1.1.1.3.cmml" type="integer" xref="S4.p3.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.1.m1.1c">k_{\mu}k_{\rho}Q^{\mu\nu\rho\sigma}=0</annotation><annotation encoding="application/x-llamapun" id="S4.p3.1.m1.1d">italic_k start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT italic_Q start_POSTSUPERSCRIPT italic_μ italic_ν italic_ρ italic_σ end_POSTSUPERSCRIPT = 0</annotation></semantics></math> trivially follows <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib58" title="">Capper:1973pv </a></cite>.</p> </div> <div class="ltx_para" id="S4.p4"> <p class="ltx_p" id="S4.p4.1">Though the three-graviton one-loop amplitude is zero, we have checked that the off-shell integrands in our proposal are qualitatively and quantitavely correct. We have also obtained the four-graviton amplitude and it matches the literature <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib63" title="">Dunbar:1994bn </a></cite>. Perhaps the easiest way to see this is through the integrand. Equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S2.E21" title="In II Equations of motion ‣ One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_tag">21</span></a>) provides the correct Feynman vertices of the theory, while equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S3.E34" title="In III Loop recursions ‣ One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_tag">34</span></a>) generates the complete diagrammatic expansion.</p> </div> <div class="ltx_para" id="S4.p5"> <p class="ltx_p" id="S4.p5.1">Finally, we have shown using symbolic computation that (<a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S3.E32" title="In III Loop recursions ‣ One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_tag">32</span></a>) leads to one-loop integrands with the correct symmetry factors. This includes all diagrams with external gravitons up to five-points, and randomly selected diagrams in six-, seven-, and eight-points. We are making some of these computations available through a sample Mathematica notebook coupled to the <a class="ltx_ref ltx_href" href="https://arxiv.org/" title="">arXiv</a> submission.</p> </div> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">V </span>Final remarks</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.4">In this work we have proposed an algebraic recursion to generate the complete one-loop integrands of <math alttext="N" class="ltx_Math" display="inline" id="S5.p1.1.m1.1"><semantics id="S5.p1.1.m1.1a"><mi id="S5.p1.1.m1.1.1" xref="S5.p1.1.m1.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S5.p1.1.m1.1b"><ci id="S5.p1.1.m1.1.1.cmml" xref="S5.p1.1.m1.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.1.m1.1c">N</annotation><annotation encoding="application/x-llamapun" id="S5.p1.1.m1.1d">italic_N</annotation></semantics></math>-graviton correlators. They can be extracted from equation (<a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#S3.E32" title="In III Loop recursions ‣ One-loop 𝑁-point correlators in pure gravity"><span class="ltx_text ltx_ref_tag">32</span></a>) by replacing the external polarizations <math alttext="h^{\mu\nu}_{p}" class="ltx_Math" display="inline" id="S5.p1.2.m2.1"><semantics id="S5.p1.2.m2.1a"><msubsup id="S5.p1.2.m2.1.1" xref="S5.p1.2.m2.1.1.cmml"><mi id="S5.p1.2.m2.1.1.2.2" xref="S5.p1.2.m2.1.1.2.2.cmml">h</mi><mi id="S5.p1.2.m2.1.1.3" xref="S5.p1.2.m2.1.1.3.cmml">p</mi><mrow id="S5.p1.2.m2.1.1.2.3" xref="S5.p1.2.m2.1.1.2.3.cmml"><mi id="S5.p1.2.m2.1.1.2.3.2" xref="S5.p1.2.m2.1.1.2.3.2.cmml">μ</mi><mo id="S5.p1.2.m2.1.1.2.3.1" xref="S5.p1.2.m2.1.1.2.3.1.cmml">⁢</mo><mi id="S5.p1.2.m2.1.1.2.3.3" xref="S5.p1.2.m2.1.1.2.3.3.cmml">ν</mi></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S5.p1.2.m2.1b"><apply id="S5.p1.2.m2.1.1.cmml" xref="S5.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.p1.2.m2.1.1.1.cmml" xref="S5.p1.2.m2.1.1">subscript</csymbol><apply id="S5.p1.2.m2.1.1.2.cmml" xref="S5.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.p1.2.m2.1.1.2.1.cmml" xref="S5.p1.2.m2.1.1">superscript</csymbol><ci id="S5.p1.2.m2.1.1.2.2.cmml" xref="S5.p1.2.m2.1.1.2.2">ℎ</ci><apply id="S5.p1.2.m2.1.1.2.3.cmml" xref="S5.p1.2.m2.1.1.2.3"><times id="S5.p1.2.m2.1.1.2.3.1.cmml" xref="S5.p1.2.m2.1.1.2.3.1"></times><ci id="S5.p1.2.m2.1.1.2.3.2.cmml" xref="S5.p1.2.m2.1.1.2.3.2">𝜇</ci><ci id="S5.p1.2.m2.1.1.2.3.3.cmml" xref="S5.p1.2.m2.1.1.2.3.3">𝜈</ci></apply></apply><ci id="S5.p1.2.m2.1.1.3.cmml" xref="S5.p1.2.m2.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.2.m2.1c">h^{\mu\nu}_{p}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.2.m2.1d">italic_h start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> by the respective propagators <math alttext="D^{\mu\nu\rho\sigma}_{p}" class="ltx_Math" display="inline" id="S5.p1.3.m3.1"><semantics id="S5.p1.3.m3.1a"><msubsup id="S5.p1.3.m3.1.1" xref="S5.p1.3.m3.1.1.cmml"><mi id="S5.p1.3.m3.1.1.2.2" xref="S5.p1.3.m3.1.1.2.2.cmml">D</mi><mi id="S5.p1.3.m3.1.1.3" xref="S5.p1.3.m3.1.1.3.cmml">p</mi><mrow id="S5.p1.3.m3.1.1.2.3" xref="S5.p1.3.m3.1.1.2.3.cmml"><mi id="S5.p1.3.m3.1.1.2.3.2" xref="S5.p1.3.m3.1.1.2.3.2.cmml">μ</mi><mo id="S5.p1.3.m3.1.1.2.3.1" xref="S5.p1.3.m3.1.1.2.3.1.cmml">⁢</mo><mi id="S5.p1.3.m3.1.1.2.3.3" xref="S5.p1.3.m3.1.1.2.3.3.cmml">ν</mi><mo id="S5.p1.3.m3.1.1.2.3.1a" xref="S5.p1.3.m3.1.1.2.3.1.cmml">⁢</mo><mi id="S5.p1.3.m3.1.1.2.3.4" xref="S5.p1.3.m3.1.1.2.3.4.cmml">ρ</mi><mo id="S5.p1.3.m3.1.1.2.3.1b" xref="S5.p1.3.m3.1.1.2.3.1.cmml">⁢</mo><mi id="S5.p1.3.m3.1.1.2.3.5" xref="S5.p1.3.m3.1.1.2.3.5.cmml">σ</mi></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S5.p1.3.m3.1b"><apply id="S5.p1.3.m3.1.1.cmml" xref="S5.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S5.p1.3.m3.1.1.1.cmml" xref="S5.p1.3.m3.1.1">subscript</csymbol><apply id="S5.p1.3.m3.1.1.2.cmml" xref="S5.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S5.p1.3.m3.1.1.2.1.cmml" xref="S5.p1.3.m3.1.1">superscript</csymbol><ci id="S5.p1.3.m3.1.1.2.2.cmml" xref="S5.p1.3.m3.1.1.2.2">𝐷</ci><apply id="S5.p1.3.m3.1.1.2.3.cmml" xref="S5.p1.3.m3.1.1.2.3"><times id="S5.p1.3.m3.1.1.2.3.1.cmml" xref="S5.p1.3.m3.1.1.2.3.1"></times><ci id="S5.p1.3.m3.1.1.2.3.2.cmml" xref="S5.p1.3.m3.1.1.2.3.2">𝜇</ci><ci id="S5.p1.3.m3.1.1.2.3.3.cmml" xref="S5.p1.3.m3.1.1.2.3.3">𝜈</ci><ci id="S5.p1.3.m3.1.1.2.3.4.cmml" xref="S5.p1.3.m3.1.1.2.3.4">𝜌</ci><ci id="S5.p1.3.m3.1.1.2.3.5.cmml" xref="S5.p1.3.m3.1.1.2.3.5">𝜎</ci></apply></apply><ci id="S5.p1.3.m3.1.1.3.cmml" xref="S5.p1.3.m3.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.3.m3.1c">D^{\mu\nu\rho\sigma}_{p}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.3.m3.1d">italic_D start_POSTSUPERSCRIPT italic_μ italic_ν italic_ρ italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>. This recursion is an off-shell evolution of the perturbiner method. As in the tree level case, the combinatoric burden that comes with the traditional Feynman graphs approach is seamlessly overcome. In addition, the <math alttext="N" class="ltx_Math" display="inline" id="S5.p1.4.m4.1"><semantics id="S5.p1.4.m4.1a"><mi id="S5.p1.4.m4.1.1" xref="S5.p1.4.m4.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S5.p1.4.m4.1b"><ci id="S5.p1.4.m4.1.1.cmml" xref="S5.p1.4.m4.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.4.m4.1c">N</annotation><annotation encoding="application/x-llamapun" id="S5.p1.4.m4.1d">italic_N</annotation></semantics></math>-graviton field theory vertices manifestly appear in the recursion<span class="ltx_note ltx_role_footnote" id="footnote1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_tag ltx_tag_note">1</span>All multiplicity graviton/ghost vertices have been written in the past (see e.g. <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib64" title="">Latosh:2023zsi </a></cite>). Up to our knowledge this is the first time they are explicitly presented using the metric density, which take a radically simpler form. See also <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib52" title="">Cho:2022faq </a></cite> for the weak-field expansion of the related Lagrangian.</span></span></span>. The analysis of divergencies from the loop momentum integral is identical to the one using Feynman diagrams. And it is interesting to point out that dimensional regularization can be directly applied to our output.</p> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.1">Our proposal outperforms the previous results of <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib30" title="">Gomez:2022dzk </a></cite> in two important points. First, it preserves the simple structure of the gravitational perturbiner <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib47" title="">Gomez:2021shh </a></cite>, without any other decoration in the sums over subwords. Second, it can be applied to theories with mixed or no color structure at all, in particular with arbitrarily high number of vertices.</p> </div> <div class="ltx_para" id="S5.p3"> <p class="ltx_p" id="S5.p3.1">There are a couple of questions of immediate interest to be addressed. The extension of our formulas to higher loops would be an impressive achievement. The strategy would be similar: (i) singling out loop-closing legs, (ii) sewing them, and (iii) introducing modified recursions to deal with diagram overcounting and symmetry factors. Besides uncovering a more structured account of the diagram combinatorics, one would be in a good position to identify multi-loop patterns of the multiparticle currents. Relatedly, the application of the method to more diverse field theories would help to better explain the combinatoric factors within the algebraic recursion. We already know that the perturbiner naturally describes gravity coupled to matter <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib47" title="">Gomez:2021shh </a></cite>. Thus we are on a good path to analyze matter loops as well, which are more interesting in the gravitational waves context. We hope to report some progress in these directions soon.</p> </div> <div class="ltx_para" id="S5.p4"> <p class="ltx_p" id="S5.p4.1">As a computational tool, the loop pertubiner will also find applications in gravitational EFTs. The program started by Donoghue <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib65" title="">Donoghue:1993eb </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib66" title="">Donoghue:1994dn </a></cite> offers a more pragmatic view on quantum effects in gravity <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib67" title="">Bjerrum-Bohr:2002fji </a>; <a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib68" title="">Bjerrum-Bohr:2002gqz </a></cite>. In this direction, we also highlight the mixing of classical and quantum effects at loop level <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib69" title="">Holstein:2004dn </a></cite>. Perhaps less obviously, our results might be used in the recent efforts to model black-hole binary mergers using scattering amplitudes, though they are mostly based on on-shell methods (e.g. <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib70" title="">Bjerrum-Bohr:2013bxa </a></cite>, see also <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2411.07939v2#bib.bib71" title="">Bjerrum-Bohr:2022blt </a></cite> for a review).</p> </div> <div class="ltx_acknowledgements"> <h6 class="ltx_title ltx_title_acknowledgements">Acknowledgements.</h6> We would like to thank N. Emil J. Bjerrum-Bohr, P. H. Damgaard, J. Donoghue, and C. Schubert for useful feedback on the manuscript. The work of HG and RLJ was partially supported by the European Structural and Investment Funds and the Czech Ministry of Education, Youth and Sports (project FORTE CZ.02.01.01/00/22_008/0004632). CLA was partially supported during the latest stages of this work by the Munich Institute for Astro-, Particle and BioPhysics (MIAPbP) which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC-2094 - 390783311. </div> <div class="ltx_pagination ltx_role_newpage"></div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">(1)</span> <span class="ltx_bibblock"> R. P. Feynman, F. B. Morinigo, W. G. Wagner and B. Hatfield, “Feynman lectures on gravitation,” doi:10.1201/9780429502859. </span> </li> <li class="ltx_bibitem" id="bib.bib2"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">(2)</span> <span class="ltx_bibblock"> S. 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