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These are based on an infinite set of energy dependent quantities (the arcs) which are dispersively expressed as moments of a positive measure defined at (arbitrarily) higher energies. We identify optimal finite subsets of constraints, suitable to bound effective field theories (EFTs), at any finite order in the energy expansion. At tree level arcs are in a one to one correspondence with Wilson coefficients. We establish under which conditions this approximation applies, identifying seemingly viable EFTs where it never does. In all cases, we discuss the range of validity in both energy and couplings, where the latter has to satisfy two-sided bounds. We also extend our results to the case of small but finite <math display="inline"><mi>t</mi></math>. A consequence of our study is that EFTs in which the scattering amplitude in some regime grows in energy faster than <math display="inline"><msup><mi>E</mi><mn>6</mn></msup></math> cannot be UV completed. We find the complete set of conditions satisfied by the forward $2\to2$ scattering amplitude in unitarity and causal theories. These are based on an infinite set of energy dependent quantities -- the arcs -- which are dispersively expressed as moments of a positive measure defined at (arbitrarily) higher energies. We identify optimal finite subsets of constraints, suitable to bound Effective Field Theories (EFTs), at any finite order in the energy expansion. At tree-level arcs are in one-to-one correspondence with Wilson coefficients. We establish under which conditions this approximation applies, identifying seemingly viable EFTs where it never does. In all cases, we discuss the range of validity in both couplings and energy. We also extend our results to the case of small but finite~$t$. A consequence of our study is that EFTs in which the scattering amplitude in some regime grows in energy faster than $E^6$ cannot be UV-completed. Bellazzini, Brando; Elias Miró, Joan; Rattazzi, Riccardo; Riembau, Marc; Riva, Francesco" /> <meta name="keywords" content="CERN Document Server, WebSearch, CERN Document Server" /> <script type="text/javascript" src="https://cds.cern.ch/js/jquery.min.js"></script>  <link rel="stylesheet" href="https://cds.cern.ch/img/webnews.css" type="text/css" />  <script type="text/javascript" src="https://cds.cern.ch/js/webnews.js?v=20131009"></script> <meta property="fb:app_id" content="137353533001720"/> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ tex2jax: {inlineMath: [['$','$']], processEscapes: true}, showProcessingMessages: false, messageStyle: "none" }); </script> <script src="/MathJax/MathJax.js?config=TeX-AMS_CHTML" type="text/javascript"> </script>  <meta content="APS : Positive Moments for Scattering Amplitudes" name="citation_title" /> <meta content="Bellazzini, Brando" name="citation_author" /> <meta content="Rattazzi, Riccardo" name="citation_author" /> <meta content="Riembau, Marc" name="citation_author" /> <meta content="Elias Miró, Joan" name="citation_author" /> <meta content="Riva, Francesco" name="citation_author" /> <meta content="10.1103/PhysRevD.104.036006" name="citation_doi" /> <meta content="Phys. 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These are based on an infinite set of energy dependent quantities (the arcs) which are dispersively expressed as moments of a positive measure defined at (arbitrarily) higher energies. We identify optimal finite subsets of constraints, suitable to bound effective field theories (EFTs), at any finite order in the energy expansion. At tree level arcs are in a one to one correspondence with Wilson coefficients. We establish under which conditions this approximation applies, identifying seemingly viable EFTs where it never does. In all cases, we discuss the range of validity in both energy and couplings, where the latter has to satisfy two-sided bounds. We also extend our results to the case of small but finite t. A consequence of our study is that EFTs in which the scattering amplitude in some regime grows in energy faster than E6 cannot be UV completed." property="og:description" /> <meta content="APS" property="og:description" /> <meta content="arXiv" property="og:description" /> <meta content="We find the complete set of conditions satisfied by the forward $2\to2$ scattering amplitude in unitarity and causal theories. These are based on an infinite set of energy dependent quantities -- the arcs -- which are dispersively expressed as moments of a positive measure defined at (arbitrarily) higher energies. We identify optimal finite subsets of constraints, suitable to bound Effective Field Theories (EFTs), at any finite order in the energy expansion. At tree-level arcs are in one-to-one correspondence with Wilson coefficients. We establish under which conditions this approximation applies, identifying seemingly viable EFTs where it never does. In all cases, we discuss the range of validity in both couplings and energy. We also extend our results to the case of small but finite~$t$. A consequence of our study is that EFTs in which the scattering amplitude in some regime grows in energy faster than $E^6$ cannot be UV-completed." property="og:description" />  <meta content="summary" name="twitter:card" /> <style></style> </head> <body class="CERN32Document32Server search" lang="es">  <div id="cern-toolbar"> <h1><a href="http://cern.ch" title="CERN">CERN <span>Accelerating science</span></a></h1> <ul> <li class="cern-accountlinks"><a class="cern-account" href="https://cds.cern.ch/youraccount/login?ln=es&referer=https%3A//cds.cern.ch/record/2744529%3Fln%3Des" title="Sign in to your CERN account">Sign in</a></li> <li><a class="cern-directory" href="http://cern.ch/directory" title="Search CERN resources and browse the directory">Directory</a></li> </ul> </div>   <div role="banner" class="clearfix" id="header"> <div class="header-inner inner"> <hgroup class="clearfix"> <h2 id="site-name"> <a rel="home" title="Home" href="/"><span>CERN Document Server</span></a> </h2> <h3 id="site-slogan">Access articles, reports and multimedia content in HEP</h3> </hgroup> <div role="navigation" id="main-navigation" class="cdsmenu"> <h2 class="element-invisible">Main menu</h2><ul class="links inline clearfix"> <li class="menu-386 first active-trail"><a class="active-trail" href="https://cds.cern.ch/?ln=es">Buscar</a></li> <li class="menu-444 "><a class="" title="" href="https://cds.cern.ch/submit?ln=es">Enviar</a></li> <li class="menu-426 "><a class="" href="https://cds.cern.ch/help/?ln=es">Ayuda</a></li> <li class="leaf hassubcdsmenu"> <a hreflang="en" class="header" href="https://cds.cern.ch/youraccount/display?ln=es">Personalizar</a> <ul class="subsubcdsmenu"><li><a href="https://cds.cern.ch/youralerts/list?ln=es">Sus alertas</a></li><li><a href="https://cds.cern.ch/yourbaskets/display?ln=es">Sus cestas</a></li><li><a href="https://cds.cern.ch/yourcomments?ln=es">Your comments</a></li><li><a href="https://cds.cern.ch/youralerts/display?ln=es">Sus búsquedas</a></li></ul></li> </ul> </div> </div> </div>  <table class="navtrailbox"> <tr> <td class="navtrailboxbody"> <a href="/?ln=es" class="navtrail">Página principal</a> > Positive Moments for Scattering Amplitudes </td> </tr> </table> </div> <div class="pagebody"><div class="pagebodystripemiddle"> <div class="detailedrecordbox"> <div class="detailedrecordtabs"> <div> <ul class="detailedrecordtabs"><li class="on first"><a href="/record/2744529/?ln=es">Información: </a></li><li class=""><a href="/record/2744529/files?ln=es">Ficheros </a></li></ul> <div id="tabsSpacer" style="clear:both;height:0px"> </div></div> </div> <div class="detailedrecordboxcontent"> <div class="top-left-folded"></div> <div class="top-right-folded"></div> <div class="inside">  <abbr class="unapi-id" title="2744529"></abbr> <style type="text/css">  </style> <table class="formatRecordTableFullWidth" > <tr> <td class="formatRecordHeader" style="background-image: url('https://cds.cern.ch/img/journals.jpg');" colspan="2">  Article </td> </tr> <script type="text/javascript"> $( document ).ready(function() { $('.showAuthor').on('click', function() { var author = '<p>' + $(this).data('name') + '</p>'; var affiliation = $(this).data('affiliation') + '</br>'; var contribution = $(this).data('contribution') + '</br>'; $.magnificPopup.open({ items: { src: '<div id="ovelary-mathjax" class="overlay-white oc-content overlay-white-500">' + author + affiliation + contribution + '</div>', type: 'inline' }, callbacks: { open: function() { var div = document.getElementById("overlay-mathjax") MathJax.Hub.Queue(["Typeset", MathJax.Hub, div]); }, } }) }) }); </script> <tr><td class="formatRecordLabel"> Report number </td><td style="padding-left:5px;"><a href="http://arxiv.org/abs/arXiv:2011.00037">arXiv:2011.00037</a></td></tr> <tr><td class="formatRecordLabel"> Title </td><td style="padding-left:5px;"><b>Positive Moments for Scattering Amplitudes</b></td></tr> <tr><td class="formatRecordLabel"><span style="white-space:nowrap;"> Author(s) </span> </td><td style="padding-left:5px;"><a href="https://cds.cern.ch/search?f=author&p=Bellazzini%2C%20Brando&ln=es">Bellazzini, Brando</a> (IPhT, Saclay ; CERN) ; <a href="https://cds.cern.ch/search?f=author&p=Elias%20Mir%C3%B3%2C%20Joan&ln=es">Elias Miró, Joan</a> (ICTP, Trieste) ; <a href="https://cds.cern.ch/search?f=author&p=Rattazzi%2C%20Riccardo&ln=es">Rattazzi, Riccardo</a> (EPFL, Lausanne, LPTP) ; <a href="https://cds.cern.ch/search?f=author&p=Riembau%2C%20Marc&ln=es">Riembau, Marc</a> (EPFL, Lausanne, LPTP ; Geneva U., Dept. Theor. Phys.) ; <a href="https://cds.cern.ch/search?f=author&p=Riva%2C%20Francesco&ln=es">Riva, Francesco</a> (Geneva U., Dept. Theor. Phys.)</td></tr> <tr><td class="formatRecordLabel"> Publication </td><td style="padding-left:5px;">2021-08-01</td></tr> <tr><td class="formatRecordLabel"> Imprint </td><td style="padding-left:5px;">2020-10-30</td></tr> <tr><td class="formatRecordLabel"> Number of pages </td><td style="padding-left:5px;">17</td></tr> <tr><td class="formatRecordLabel"> Note </td><td style="padding-left:5px;">15 pages, v2 matches published version</td></tr> <tr><td class="formatRecordLabel"> In: </td><td style="padding-left:5px;"><a href="http://dx.doi.org/10.1103/PhysRevD.104.036006"><i>Phys. Rev. D</i> 104 (2021) 036006</a> </a></td></tr> <tr><td class="formatRecordLabel"> DOI </td><td style="padding-left:5px;"><a href="http://dx.doi.org/10.1103/PhysRevD.104.036006" title="DOI" target="_blank">10.1103/PhysRevD.104.036006</a> (publication) <tr><td class="formatRecordLabel"> Subject category </td><td style="padding-left:5px;">hep-ph ; Particle Physics - Phenomenology ; hep-th ; Particle Physics - Theory</td></tr> <tr><td class="formatRecordLabel"> Abstract </td><td style="padding-left:5px;">We find the complete set of conditions satisfied by the forward $2\to2$ scattering amplitude in unitarity and causal theories. These are based on an infinite set of energy dependent quantities -- the arcs -- which are dispersively expressed as moments of a positive measure defined at (arbitrarily) higher energies. We identify optimal finite subsets of constraints, suitable to bound Effective Field Theories (EFTs), at any finite order in the energy expansion. At tree-level arcs are in one-to-one correspondence with Wilson coefficients. We establish under which conditions this approximation applies, identifying seemingly viable EFTs where it never does. In all cases, we discuss the range of validity in both couplings and energy. We also extend our results to the case of small but finite $t$. A consequence of our study is that EFTs in which the scattering amplitude in some regime grows in energy faster than $E^6$ cannot be UV-completed.</td></tr> <tr><td class="formatRecordLabel"> Copyright/License </td><td style="padding-left:5px;">preprint: (License: <a href="http://creativecommons.org/licenses/by/4.0/">CC-BY-4.0</a>)<br/>publication: © 2021-2025 authors (License: <a href="https://creativecommons.org/licenses/by/4.0/">CC BY 4.0</a>), sponsored by <a href="http://scoap3.org">SCOAP³</a></td></tr> </table> <br /><div style="max-width:1024px;margin:auto"><div style="overflow-x:auto;display:inline;width:100%;"><a href="/record/2744529/plots#0"><img style="vertical-align:middle;" src="https://cds.cern.ch/record/2744529/files/figurearc.png" title=" \it The semi-circle contour of \eq{archdef} in the complex upper $\os$-plane. Wiggle lines denote the branch-cuts on the real axis." width="200px"/></a> <a href="/record/2744529/plots#1"><img style="vertical-align:middle;" src="https://cds.cern.ch/record/2744529/files/Ptraj.png" title=" \it Allowed regions for the arcs $a_0$, $a_1$, $a_2$ (LEFT) and including $a_3$ (RIGHT), according to Eqs.~(\ref{a2space},\ref{a3space}). LEFT: For fixed Wilson coefficients, as energy is increased, the theory spans a trajectory in the space of arcs: the blue trajectories (arrows in the direction of increasing $s$) correspond to examples in the weak coupling limit \eq{tree1}, the red trajectories are examples using \eq{arcsgoldstone} at strong coupling (with the explicit values \eq{explicitgoldstones} and large $g_2$ from \eq{couplings}). {Values of $s^4a_2/a_0$ larger than the green solid/dashed/dotted lines are excluded by the conditions \eq{cc} from Bernstein polynomials, up to $k+n=2,3,4$ respectively.} RIGHT: The projections into two dimensional planes correspond to optimal bounds when only two coefficients are taken into account (the bottom projection corresponds to the left panel). The volume of the allowed region is $1/180$ w.r.t. the volume of the unit cube. \label{banana2} \label{banana3}" width="200px"/></a> <a href="/record/2744529/plots#2"><img style="vertical-align:middle;" src="https://cds.cern.ch/record/2744529/files/example2.png" title=" \it Allowed regions for the arcs $a_0$, $a_1$, $a_2$ (LEFT) and including $a_3$ (RIGHT), according to Eqs.~(\ref{a2space},\ref{a3space}). LEFT: For fixed Wilson coefficients, as energy is increased, the theory spans a trajectory in the space of arcs: the blue trajectories (arrows in the direction of increasing $s$) correspond to examples in the weak coupling limit \eq{tree1}, the red trajectories are examples using \eq{arcsgoldstone} at strong coupling (with the explicit values \eq{explicitgoldstones} and large $g_2$ from \eq{couplings}). {Values of $s^4a_2/a_0$ larger than the green solid/dashed/dotted lines are excluded by the conditions \eq{cc} from Bernstein polynomials, up to $k+n=2,3,4$ respectively.} RIGHT: The projections into two dimensional planes correspond to optimal bounds when only two coefficients are taken into account (the bottom projection corresponds to the left panel). The volume of the allowed region is $1/180$ w.r.t. the volume of the unit cube. \label{banana2} \label{banana3}" width="200px"/></a> <a href='/record/2744529/plots'>Show more plots</a></div></div><br /> <br/>Corresponding record in: <a href="http://inspirehep.net/record/1827503">Inspire</a> <small> </small> <br/> <br/><br/><div align="right"><div style="padding-bottom:2px;padding-top:30px;"><span class="moreinfo" style="margin-right:10px;"> <a href="" class="moreinfo">Volver a la búsqueda</a> </span></div></div> <div class="bottom-left-folded"><div class="recordlastmodifiedbox" style="position:relative;margin-left:1px"> Registro creado el 2020-11-14, última modificación el 2023-10-04</div></div> <div class="bottom-right-folded" style="text-align:right;padding-bottom:2px;"> <span class="moreinfo" style="margin-right:10px;"><a href="/search?ln=es&p=recid%3A2744529&rm=wrd" class="moreinfo">Registros similares</a></span></div> </div> </div> </div> <br/> <br /> <div class="detailedrecordminipanel"> <div class="top-left"></div><div class="top-right"></div> <div class="inside"> <div id="detailedrecordminipanelfile" style="width:33%;float:left;text-align:center;margin-top:0"> <div><small class="detailedRecordActions">Fulltext from publisher:</small> <br /><a href="/record/2744529/files/PhysRevD.104.036006.pdf"><img style="border:none" src="/img/file-icon-text-12x16.gif" alt="Descargar el texto completo"/>PDF</a><br /></div><div><small class="detailedRecordActions">Texto completo:</small> <br /><a href="/record/2744529/files/2011.00037.pdf"><img style="border:none" src="/img/file-icon-text-12x16.gif" alt="Descargar el texto completo"/>PDF</a><br /></div> </div> <div id="detailedrecordminipanelreview" style="width:30%;float:left;text-align:center"> </div> <div id="detailedrecordminipanelactions" style="width:36%;float:right;text-align:right;"> <ul class="detailedrecordactions"> <li><a href="/yourbaskets/add?ln=es&recid=2744529">Añadir a la cesta personal</a></li> <li>Exportar como <a style="text-decoration:underline;font-weight:normal" href="/record/2744529/export/hx?ln=es">BibTeX</a>, <a style="text-decoration:underline;font-weight:normal" href="/record/2744529/export/hm?ln=es">MARC</a>, <a style="text-decoration:underline;font-weight:normal" href="/record/2744529/export/xm?ln=es">MARCXML</a>, <a style="text-decoration:underline;font-weight:normal" href="/record/2744529/export/xd?ln=es">DC</a>, <a style="text-decoration:underline;font-weight:normal" href="/record/2744529/export/xe?ln=es">EndNote</a>,  <a style="text-decoration:underline;font-weight:normal" href="/record/2744529/export/xn?ln=es">NLM</a>, <a style="text-decoration:underline;font-weight:normal" href="/record/2744529/export/xw?ln=es">RefWorks</a> </li> </ul> <div style='padding-left: 13px;'>  <div id="bookmark"></div> <div id="bookmark_sciencewise"></div> <style type="text/css"> #bookmark_sciencewise, #bookmark {float: left;} #bookmark_sciencewise li {padding: 2px; width: 25px;} #bookmark_sciencewise ul, #bookmark ul {list-style-image: none;} </style> <script type="text/javascript" src="/js/jquery.bookmark.min.js"></script> <style type="text/css">@import "/css/jquery.bookmark.css";</style> <script type="text/javascript">// <![CDATA[ $.bookmark.addSite('sciencewise', 'ScienceWise.info', 'https://cds.cern.ch/img/sciencewise.png', 'en', 'bookmark', 'http://sciencewise.info/bookmarks/2011.0003/add'); $('#bookmark_sciencewise').bookmark({sites: ['sciencewise']}); $('#bookmark').bookmark({ sites: ['facebook', 'twitter', 'linkedin', 'google_plusone'], icons: '/img/bookmarks.png', url: 'https://cds.cern.ch/record/2744529', addEmail: true, title: "Positive Moments for Scattering Amplitudes", description: "We find the complete set of conditions satisfied by the forward <math display=\"inline\"><mrow><mn>2</mn><mo stretchy=\"false\">\u2192</mo><mn>2</mn></mrow></math> scattering amplitude in unitary and causal theories. These are based on an infinite set of energy dependent quantities (the arcs) which are dispersively expressed as moments of a positive measure defined at (arbitrarily) higher energies. We identify optimal finite subsets of constraints, suitable to bound effective field theories (EFTs), at any finite order in the energy expansion. At tree level arcs are in a one to one correspondence with Wilson coefficients. We establish under which conditions this approximation applies, identifying seemingly viable EFTs where it never does. In all cases, we discuss the range of validity in both energy and couplings, where the latter has to satisfy two-sided bounds. We also extend our results to the case of small but finite <math display=\"inline\"><mi>t</mi></math>. A consequence of our study is that EFTs in which the scattering amplitude in some regime grows in energy faster than <math display=\"inline\"><msup><mi>E</mi><mn>6</mn></msup></math> cannot be UV completed." }); // ]]> </script>  </div> </div> <div style="clear:both;margin-bottom: 0;"></div> </div> <div class="bottom-left"></div><div class="bottom-right"></div> </div> </div></div> <footer id="footer" class="pagefooter clearfix">  <div class="pagefooterstripeleft"> CERN Document Server :: <a class="footer" href="https://cds.cern.ch/?ln=es">Buscar</a> :: <a class="footer" href="https://cds.cern.ch/submit?ln=es">Enviar</a> :: <a class="footer" href="https://cds.cern.ch/youraccount/display?ln=es">Personalizar</a> :: <a class="footer" href="https://cds.cern.ch/help/?ln=es">Ayuda</a> :: <a class="footer" href="https://cern.service-now.com/service-portal?id=privacy_policy&se=CDS-Service" target="_blank">Privacy Notice</a> :: <a class="footer" href="https://repository.cern/content-policy" target="_blank">Content Policy</a> :: <a class="footer" href="https://repository.cern/terms" target="_blank">Terms and Conditions</a> <br /> Powered by <a class="footer" href="http://invenio-software.org/">Invenio</a> <br /> Mantenido por <a class="footer" href="https://cern.service-now.com/service-portal?id=service_element&name=CDS-Service">CDS Service</a> - Need help? 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