CP violation - Wikipedia

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a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style> In <a href="/wiki/Particle_physics" title="Particle physics">particle physics</a>, CP violation is a violation of CP-symmetry (or charge conjugation parity symmetry): the combination of <a href="/wiki/C-symmetry" title="C-symmetry">C-symmetry</a> (<a href="/wiki/Charge_(physics)" title="Charge (physics)">charge conjugation</a> symmetry) and <a href="/wiki/Parity_(physics)" title="Parity (physics)">P-symmetry</a> (<a href="/wiki/Parity_(physics)" title="Parity (physics)">parity</a> symmetry). CP-symmetry states that the laws of physics should be the same if a particle is interchanged with its antiparticle (C-symmetry) while its spatial coordinates are inverted ("mirror" or P-symmetry). The discovery of CP violation in 1964 in the decays of neutral <a href="/wiki/Kaon" title="Kaon">kaons</a> resulted in the <a href="/wiki/Nobel_Prize_in_Physics" title="Nobel Prize in Physics">Nobel Prize in Physics</a> in 1980 for its discoverers <a href="/wiki/James_Cronin" title="James Cronin">James Cronin</a> and <a href="/wiki/Val_Fitch" class="mw-redirect" title="Val Fitch">Val Fitch</a>. It plays an important role both in the attempts of <a href="/wiki/Physical_cosmology" title="Physical cosmology">cosmology</a> to explain the dominance of <a href="/wiki/Matter" title="Matter">matter</a> over <a href="/wiki/Antimatter" title="Antimatter">antimatter</a> in the present <a href="/wiki/Universe" title="Universe">universe</a>, and in the study of <a href="/wiki/Weak_interaction" title="Weak interaction">weak interactions</a> in particle physics. <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><label class="toctogglelabel" for="toctogglecheckbox"></label></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Overview">1 Overview</a></li> <li class="toclevel-1 tocsection-2"><a href="#History">2 History</a> <ul> <li class="toclevel-2 tocsection-3"><a href="#P-symmetry">2.1 P-symmetry</a></li> <li class="toclevel-2 tocsection-4"><a href="#CP-symmetry">2.2 CP-symmetry</a></li> </ul> </li> <li class="toclevel-1 tocsection-5"><a href="#Experimental_status">3 Experimental status</a> <ul> <li class="toclevel-2 tocsection-6"><a href="#Indirect_CP_violation">3.1 Indirect CP violation</a></li> <li class="toclevel-2 tocsection-7"><a href="#Direct_CP_violation">3.2 Direct CP violation</a></li> </ul> </li> <li class="toclevel-1 tocsection-8"><a href="#CP_violation_in_the_Standard_Model">4 CP violation in the Standard Model</a></li> <li class="toclevel-1 tocsection-9"><a href="#Strong_CP_problem">5 Strong CP problem</a></li> <li class="toclevel-1 tocsection-10"><a href="#Matter%E2%80%93antimatter_imbalance">6 Matter–antimatter imbalance</a></li> <li class="toclevel-1 tocsection-11"><a href="#See_also">7 See also</a></li> <li class="toclevel-1 tocsection-12"><a href="#In_popular_culture">8 In popular culture</a></li> <li class="toclevel-1 tocsection-13"><a href="#References">9 References</a></li> <li class="toclevel-1 tocsection-14"><a href="#Further_reading">10 Further reading</a></li> <li class="toclevel-1 tocsection-15"><a href="#External_links">11 External links</a></li> </ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"><h2 id="Overview">Overview</h2> <a role="button" href="/w/index.php?title=CP_violation&action=edit&section=1" title="Edit section: Overview" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-1 collapsible-block" id="mf-section-1"> Until the 1950s, parity conservation was believed to be one of the fundamental geometric <a href="/wiki/Conservation_laws" class="mw-redirect" title="Conservation laws">conservation laws</a> (along with <a href="/wiki/Conservation_of_energy" title="Conservation of energy">conservation of energy</a> and <a href="/wiki/Conservation_of_momentum" class="mw-redirect" title="Conservation of momentum">conservation of momentum</a>). After the discovery of <a href="/wiki/Parity_(physics)#Parity_violation" title="Parity (physics)">parity violation</a> in 1956, CP-symmetry was proposed to restore order. However, while the <a href="/wiki/Strong_interaction" title="Strong interaction">strong interaction</a> and <a href="/wiki/Electromagnetic_interaction" class="mw-redirect" title="Electromagnetic interaction">electromagnetic interaction</a> are experimentally found to be invariant under the combined CP transformation operation, further experiments showed that this symmetry is slightly violated during certain types of <a href="/wiki/Weak_decay" class="mw-redirect" title="Weak decay">weak decay</a>. Only a weaker version of the symmetry could be preserved by physical phenomena, which was <a href="/wiki/CPT_symmetry" title="CPT symmetry">CPT symmetry</a>. Besides C and P, there is a third operation, time reversal T, which corresponds to reversal of motion. Invariance under time reversal implies that whenever a motion is allowed by the laws of physics, the reversed motion is also an allowed one and occurs at the same rate forwards and backwards. The combination of CPT is thought to constitute an exact symmetry of all types of fundamental interactions. Because of the long-held CPT symmetry theorem, provided that it is valid, a violation of the CP-symmetry is equivalent to a violation of the T-symmetry. In this theorem, regarded as one of the basic principles of <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, charge conjugation, parity, and time reversal are applied together. Direct observation of the <a href="/wiki/T-symmetry" title="T-symmetry">time reversal symmetry</a> violation without any assumption of CPT theorem was done in 1998 by two groups, <a href="/wiki/CPLEAR_experiment" title="CPLEAR experiment">CPLEAR</a> and KTeV collaborations, at <a href="/wiki/CERN" title="CERN">CERN</a> and <a href="/wiki/Fermilab" title="Fermilab">Fermilab</a>, respectively.<a href="#cite_note-1">[1]</a> Already in 1970 Klaus Schubert observed T violation independent of assuming CPT symmetry by using the Bell–Steinberger unitarity relation.<a href="#cite_note-2">[2]</a> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"><h2 id="History">History</h2> <a role="button" href="/w/index.php?title=CP_violation&action=edit&section=2" title="Edit section: History" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-2 collapsible-block" id="mf-section-2"> <div class="mw-heading mw-heading3"><h3 id="P-symmetry">P-symmetry</h3> <a role="button" href="/w/index.php?title=CP_violation&action=edit&section=3" title="Edit section: P-symmetry" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> The idea behind <a href="/wiki/Parity_(physics)" title="Parity (physics)">parity</a> symmetry was that the equations of particle physics are invariant under mirror inversion. This led to the prediction that the mirror image of a reaction (such as a <a href="/wiki/Chemical_reaction" title="Chemical reaction">chemical reaction</a> or <a href="/wiki/Radioactive_decay" title="Radioactive decay">radioactive decay</a>) occurs at the same rate as the original reaction. However, in 1956 a careful critical review of the existing experimental data by theoretical physicists <a href="/wiki/Tsung-Dao_Lee" title="Tsung-Dao Lee">Tsung-Dao Lee</a> and <a href="/wiki/Chen-Ning_Yang" class="mw-redirect" title="Chen-Ning Yang">Chen-Ning Yang</a> revealed that while parity conservation had been verified in decays by the strong or electromagnetic interactions, it was untested in the weak interaction.<a href="#cite_note-3">[3]</a> They proposed several possible direct experimental tests. The first test based on <a href="/wiki/Beta_decay" title="Beta decay">beta decay</a> of <a href="/wiki/Cobalt-60" title="Cobalt-60">cobalt-60</a> nuclei was carried out in 1956 by a group led by <a href="/wiki/Chien-Shiung_Wu" title="Chien-Shiung Wu">Chien-Shiung Wu</a>, and demonstrated conclusively that weak interactions violate the P-symmetry or, as the analogy goes, some reactions did not occur as often as their mirror image.<a href="#cite_note-4">[4]</a> However, <a href="/wiki/Parity_(physics)" title="Parity (physics)">parity</a> symmetry still appears to be valid for all reactions involving <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetism</a> and <a href="/wiki/Strong_interaction" title="Strong interaction">strong interactions</a>. <div class="mw-heading mw-heading3"><h3 id="CP-symmetry">CP-symmetry</h3> <a role="button" href="/w/index.php?title=CP_violation&action=edit&section=4" title="Edit section: CP-symmetry" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> Overall, the symmetry of a <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanical</a> system can be restored if another approximate symmetry S can be found such that the combined symmetry PS remains unbroken. This rather subtle point about the structure of <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> was realized shortly after the discovery of P violation, and it was proposed that charge conjugation, C, which transforms a particle into its <a href="/wiki/Antiparticle" title="Antiparticle">antiparticle</a>, was the suitable symmetry to restore order. In 1956 <a href="/wiki/Reinhard_Oehme" title="Reinhard Oehme">Reinhard Oehme</a> in a letter to Chen-Ning Yang and shortly after, <a href="/w/index.php?title=%D0%98%D0%BE%D1%84%D1%84%D0%B5,_%D0%91%D0%BE%D1%80%D0%B8%D1%81_%D0%9B%D0%B0%D0%B7%D0%B0%D1%80%D0%B5%D0%B2%D0%B8%D1%87&action=edit&redlink=1" class="new" title="Иоффе, Борис Лазаревич (page does not exist)">Boris L. Ioffe</a>, <a href="/wiki/Lev_Okun" title="Lev Okun">Lev Okun</a> and A. P. Rudik showed that the parity violation meant that charge conjugation invariance must also be violated in weak decays.<a href="#cite_note-Ioffe-5">[5]</a> Charge violation was confirmed in the <a href="/wiki/Wu_experiment" title="Wu experiment">Wu experiment</a> and in experiments performed by <a href="/wiki/Valentine_Telegdi" title="Valentine Telegdi">Valentine Telegdi</a> and <a href="/wiki/Jerome_Isaac_Friedman" title="Jerome Isaac Friedman">Jerome Friedman</a> and <a href="/wiki/Richard_Garwin" title="Richard Garwin">Garwin</a> and <a href="/wiki/Leon_M._Lederman" title="Leon M. Lederman">Lederman</a> who observed parity non-conservation in pion and muon decay and found that C is also violated. Charge violation was more explicitly shown in experiments done by <a href="/wiki/John_Riley_Holt" title="John Riley Holt">John Riley Holt</a> at the <a href="/wiki/University_of_Liverpool" title="University of Liverpool">University of Liverpool</a>.<a href="#cite_note-6">[6]</a><a href="#cite_note-7">[7]</a><a href="#cite_note-8">[8]</a> Oehme then wrote a paper with Lee and Yang in which they discussed the interplay of non-invariance under P, C and T. The same result was also independently obtained by Ioffe, Okun and Rudik. Both groups also discussed possible CP violations in neutral kaon decays.<a href="#cite_note-Ioffe-5">[5]</a><a href="#cite_note-9">[9]</a> <a href="/wiki/Lev_Landau" title="Lev Landau">Lev Landau</a> proposed in 1957 CP-symmetry,<a href="#cite_note-10">[10]</a> often called just CP as the true symmetry between matter and antimatter. CP-symmetry is the product of two <a href="/wiki/Symmetry_in_physics" class="mw-redirect" title="Symmetry in physics">transformations</a>: C for charge conjugation and P for parity. In other words, a process in which all particles are exchanged with their <a href="/wiki/Antiparticle" title="Antiparticle">antiparticles</a> was assumed to be equivalent to the mirror image of the original process and so the combined CP-symmetry would be conserved in the weak interaction. In 1962, a group of experimentalists at <a href="/wiki/Joint_Institute_for_Nuclear_Research" title="Joint Institute for Nuclear Research">Dubna</a>, on Okun's insistence, unsuccessfully searched for CP-violating kaon decay.<a href="#cite_note-11">[11]</a> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"><h2 id="Experimental_status">Experimental status</h2> <a role="button" href="/w/index.php?title=CP_violation&action=edit&section=5" title="Edit section: Experimental status" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-3 collapsible-block" id="mf-section-3"> <div class="mw-heading mw-heading3"><h3 id="Indirect_CP_violation">Indirect CP violation</h3> <a role="button" href="/w/index.php?title=CP_violation&action=edit&section=6" title="Edit section: Indirect CP violation" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> In 1964, <a href="/wiki/James_Cronin" title="James Cronin">James Cronin</a>, <a href="/wiki/Val_Fitch" class="mw-redirect" title="Val Fitch">Val Fitch</a> and coworkers provided clear evidence from <a href="/wiki/Kaon" title="Kaon">kaon</a> decay that CP-symmetry could be broken.<a href="#cite_note-FC1964-12">[12]</a> (cf. also Ref. <a href="#cite_note-FCE-13">[13]</a>). This work won them the 1980 Nobel Prize. This discovery showed that weak interactions violate not only the charge-conjugation symmetry C between particles and antiparticles and the P or parity symmetry, but also their combination. The discovery shocked particle physics and opened the door to questions still at the core of particle physics and of cosmology today. The lack of an exact CP-symmetry, but also the fact that it is so close to a symmetry, introduced a great puzzle. The kind of CP violation (CPV) discovered in 1964 was linked to the fact that neutral <a href="/wiki/Kaon" title="Kaon">kaons</a> can transform into their <a href="/wiki/Antiparticle" title="Antiparticle">antiparticles</a> (in which each <a href="/wiki/Quark" title="Quark">quark</a> is replaced with the other's antiquark) and vice versa, but such transformation does not occur with exactly the same probability in both directions; this is called indirect CP violation. <div class="mw-heading mw-heading3"><h3 id="Direct_CP_violation">Direct CP violation</h3> <a role="button" href="/w/index.php?title=CP_violation&action=edit&section=7" title="Edit section: Direct CP violation" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Kaon-box-diagram.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Kaon-box-diagram.svg/220px-Kaon-box-diagram.svg.png" decoding="async" width="220" height="124" class="mw-file-element" data-file-width="640" data-file-height="360"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 124px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Kaon-box-diagram.svg/220px-Kaon-box-diagram.svg.png" data-width="220" data-height="124" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Kaon-box-diagram.svg/330px-Kaon-box-diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Kaon-box-diagram.svg/440px-Kaon-box-diagram.svg.png 2x" data-class="mw-file-element"> </a><figcaption>Kaon oscillation box diagram</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Kaon-box-diagram-alt.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Kaon-box-diagram-alt.svg/220px-Kaon-box-diagram-alt.svg.png" decoding="async" width="220" height="123" class="mw-file-element" data-file-width="660" data-file-height="370"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 123px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Kaon-box-diagram-alt.svg/220px-Kaon-box-diagram-alt.svg.png" data-width="220" data-height="123" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Kaon-box-diagram-alt.svg/330px-Kaon-box-diagram-alt.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Kaon-box-diagram-alt.svg/440px-Kaon-box-diagram-alt.svg.png 2x" data-class="mw-file-element"> </a><figcaption>The two box diagrams above are the <a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagrams</a> providing the leading contributions to the amplitude of <a href="/wiki/Kaon" title="Kaon"> K0 </a>-<a href="/wiki/Kaon" title="Kaon"> K0 </a> oscillation</figcaption></figure> Despite many searches, no other manifestation of CP violation was discovered until the 1990s, when the <a href="/wiki/NA31_experiment" title="NA31 experiment">NA31 experiment</a> at <a href="/wiki/CERN" title="CERN">CERN</a> suggested evidence for CP violation in the decay process of the very same neutral kaons (direct CP violation). The observation was somewhat controversial, and final proof for it came in 1999 from the KTeV experiment at <a href="/wiki/Fermilab" title="Fermilab">Fermilab</a><a href="#cite_note-14">[14]</a> and the <a href="/wiki/NA48_experiment" title="NA48 experiment">NA48 experiment</a> at <a href="/wiki/CERN" title="CERN">CERN</a>.<a href="#cite_note-NA48-15">[15]</a> Starting in 2001, a new generation of experiments, including the <a href="/wiki/BaBar_experiment" title="BaBar experiment">BaBar experiment</a> at the Stanford Linear Accelerator Center (<a href="/wiki/SLAC" class="mw-redirect" title="SLAC">SLAC</a>)<a href="#cite_note-16">[16]</a> and the <a href="/wiki/Belle_Experiment" class="mw-redirect" title="Belle Experiment">Belle Experiment</a> at the High Energy Accelerator Research Organisation (<a href="/wiki/KEK" title="KEK">KEK</a>)<a href="#cite_note-17">[17]</a> in Japan, observed direct CP violation in a different system, namely in decays of the <a href="/wiki/B_meson" title="B meson">B mesons</a>.<a href="#cite_note-18">[18]</a> A large number of CP violation processes in <a href="/wiki/B_meson" title="B meson">B meson</a> decays have now been discovered. Before these "<a href="/wiki/B-factory" title="B-factory">B-factory</a>" experiments, there was a logical possibility that all CP violation was confined to kaon physics. However, this raised the question of why CP violation did not extend to the strong force, and furthermore, why this was not predicted by the unextended <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a>, despite the model's accuracy for "normal" phenomena. In 2011, a hint of CP violation in decays of neutral <a href="/wiki/D_meson" title="D meson">D mesons</a> was reported by the <a href="/wiki/LHCb" class="mw-redirect" title="LHCb">LHCb</a> experiment at <a href="/wiki/CERN" title="CERN">CERN</a> using 0.6 fb−1 of Run 1 data.<a href="#cite_note-19">[19]</a> However, the same measurement using the full 3.0 fb−1 Run 1 sample was consistent with CP-symmetry.<a href="#cite_note-20">[20]</a> In 2013 LHCb announced discovery of CP violation in <a href="/wiki/Strange_B_meson" title="Strange B meson">strange B meson</a> decays.<a href="#cite_note-21">[21]</a> In March 2019, LHCb announced discovery of CP violation in charmed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D^{0}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d61f2a204347d41a6b1c1b6d6948354e1e3acb74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.979ex; height:2.676ex;" alt="{\displaystyle D^{0}}"></noscript><span class="lazy-image-placeholder" style="width: 2.979ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d61f2a204347d41a6b1c1b6d6948354e1e3acb74" data-alt="{\displaystyle D^{0}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  decays with a deviation from zero of 5.3 standard deviations.<a href="#cite_note-22">[22]</a> In 2020, the <a href="/wiki/T2K_experiment" title="T2K experiment">T2K Collaboration</a> reported some indications of CP violation in leptons for the first time.<a href="#cite_note-23">[23]</a> In this experiment, beams of muon neutrinos ( ν μ) and muon antineutrinos ( ν μ) were alternately produced by an <a href="/wiki/Accelerator_neutrino" title="Accelerator neutrino">accelerator</a>. By the time they got to the detector, a significantly higher proportion of electron neutrinos ( ν e) was observed from the ν μ beams, than electron antineutrinos ( ν e) were from the ν μ beams. Analysis of these observations was not yet precise enough to determine the size of the CP violation, relative to that seen in quarks. In addition, another similar experiment, <a href="/wiki/NOvA" title="NOvA">NOvA</a> sees no evidence of CP violation in neutrino oscillations<a href="#cite_note-24">[24]</a> and is in slight tension with T2K.<a href="#cite_note-25">[25]</a><a href="#cite_note-26">[26]</a> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"><h2 id="CP_violation_in_the_Standard_Model">CP violation in the Standard Model</h2> <a role="button" href="/w/index.php?title=CP_violation&action=edit&section=8" title="Edit section: CP violation in the Standard Model" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-4 collapsible-block" id="mf-section-4"> "Direct" CP violation is allowed in the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> if a <a href="/wiki/Complex_number" title="Complex number">complex</a> phase appears in the <a href="/wiki/Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix" title="Cabibbo–Kobayashi–Maskawa matrix">Cabibbo–Kobayashi–Maskawa matrix</a> (CKM matrix) describing <a href="/wiki/Quark" title="Quark">quark</a> mixing, or the <a href="/wiki/Pontecorvo%E2%80%93Maki%E2%80%93Nakagawa%E2%80%93Sakata_matrix" title="Pontecorvo–Maki–Nakagawa–Sakata matrix">Pontecorvo–Maki–Nakagawa–Sakata matrix</a> (PMNS matrix) describing <a href="/wiki/Neutrino" title="Neutrino">neutrino</a> mixing. A necessary condition for the appearance of the complex phase is the presence of at least three generations of fermions. If fewer generations are present, the complex phase parameter <a href="/wiki/CKM_matrix#Counting" class="mw-redirect" title="CKM matrix">can be absorbed</a> into redefinitions of the fermion fields. A popular rephasing invariant whose vanishing signals absence of CP violation and occurs in most CP violating amplitudes is the <a href="/wiki/Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix#The_unitarity_triangles" title="Cabibbo–Kobayashi–Maskawa matrix"> Jarlskog invariant</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ J=c_{12}\ c_{13}^{2}\ c_{23}\ s_{12}\ s_{13}\ s_{23}\ \sin \delta \ \approx \ 0.00003\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>J</mi> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mtext> </mtext> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mtext> </mtext> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mtext> </mtext> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mtext> </mtext> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <mtext> </mtext> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <mi>δ</mi> <mtext> </mtext> <mo>≈</mo> <mtext> </mtext> <mn>0.00003</mn> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ J=c_{12}\ c_{13}^{2}\ c_{23}\ s_{12}\ s_{13}\ s_{23}\ \sin \delta \ \approx \ 0.00003\ ,}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dce331d1fa8ef5aa00bf4bedbdae58b46887d83" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.971ex; height:3.176ex;" alt="{\displaystyle \ J=c_{12}\ c_{13}^{2}\ c_{23}\ s_{12}\ s_{13}\ s_{23}\ \sin \delta \ \approx \ 0.00003\ ,}"></noscript><span class="lazy-image-placeholder" style="width: 43.971ex;height: 3.176ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dce331d1fa8ef5aa00bf4bedbdae58b46887d83" data-alt="{\displaystyle \ J=c_{12}\ c_{13}^{2}\ c_{23}\ s_{12}\ s_{13}\ s_{23}\ \sin \delta \ \approx \ 0.00003\ ,}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  for quarks, which is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ 0.0003\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mn>0.0003</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 0.0003\ }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0a3173bde03ac8c2f969d86371790ff5d37a7f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.62ex; height:2.176ex;" alt="{\displaystyle \ 0.0003\ }"></noscript><span class="lazy-image-placeholder" style="width: 7.62ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0a3173bde03ac8c2f969d86371790ff5d37a7f7" data-alt="{\displaystyle \ 0.0003\ }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  times the maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ J_{\max }={\tfrac {1}{6{\sqrt {3}}}}\ \approx \ 0.1\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">max</mo> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> </mfrac> </mstyle> </mrow> <mtext> </mtext> <mo>≈</mo> <mtext> </mtext> <mn>0.1</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ J_{\max }={\tfrac {1}{6{\sqrt {3}}}}\ \approx \ 0.1\ .}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ded197361031ca6127fc3a79a195328454f99903" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.568ex; height:4.176ex;" alt="{\displaystyle \ J_{\max }={\tfrac {1}{6{\sqrt {3}}}}\ \approx \ 0.1\ .}"></noscript><span class="lazy-image-placeholder" style="width: 20.568ex;height: 4.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ded197361031ca6127fc3a79a195328454f99903" data-alt="{\displaystyle \ J_{\max }={\tfrac {1}{6{\sqrt {3}}}}\ \approx \ 0.1\ .}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  For leptons, only an upper limit exists: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ |J|<0.03\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mn>0.03</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ |J|<0.03\ .}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31c6dd27e7df7904d862f2aaf23341ad3acb1db0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.806ex; height:2.843ex;" alt="{\displaystyle \ |J|<0.03\ .}"></noscript><span class="lazy-image-placeholder" style="width: 11.806ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31c6dd27e7df7904d862f2aaf23341ad3acb1db0" data-alt="{\displaystyle \ |J|<0.03\ .}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  The reason why such a complex phase causes CP violation (CPV) is not immediately obvious, but can be seen as follows. Consider any given particles (or sets of particles) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ a\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>a</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ a\ }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8124de742ae987fe73be9ca9d3d4ba8586e28b11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.391ex; height:1.676ex;" alt="{\displaystyle \ a\ }"></noscript><span class="lazy-image-placeholder" style="width: 2.391ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8124de742ae987fe73be9ca9d3d4ba8586e28b11" data-alt="{\displaystyle \ a\ }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ b\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>b</mi> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ b\ ,}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/379cb0c291a41721eace03e50997abebfb25db4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.806ex; height:2.509ex;" alt="{\displaystyle \ b\ ,}"></noscript><span class="lazy-image-placeholder" style="width: 2.806ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/379cb0c291a41721eace03e50997abebfb25db4f" data-alt="{\displaystyle \ b\ ,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and their antiparticles <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\bar {a}}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\bar {a}}\ }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39dbc866826677d4893bc2905ca740db0f0f2564" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.391ex; height:2.009ex;" alt="{\displaystyle \ {\bar {a}}\ }"></noscript><span class="lazy-image-placeholder" style="width: 2.391ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39dbc866826677d4893bc2905ca740db0f0f2564" data-alt="{\displaystyle \ {\bar {a}}\ }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\bar {b}}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\bar {b}}\ .}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4e677a819717d0ede33e2763b407ab45df6ab55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.509ex;" alt="{\displaystyle \ {\bar {b}}\ .}"></noscript><span class="lazy-image-placeholder" style="width: 2.971ex;height: 2.509ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4e677a819717d0ede33e2763b407ab45df6ab55" data-alt="{\displaystyle \ {\bar {b}}\ .}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Now consider the processes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ a\rightarrow b\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>a</mi> <mo stretchy="false">→</mo> <mi>b</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ a\rightarrow b\ }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f71cd6c1be827346794a8d43bd15b5fbb687fec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.003ex; height:2.176ex;" alt="{\displaystyle \ a\rightarrow b\ }"></noscript><span class="lazy-image-placeholder" style="width: 7.003ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f71cd6c1be827346794a8d43bd15b5fbb687fec" data-alt="{\displaystyle \ a\rightarrow b\ }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and the corresponding antiparticle process <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\bar {a}}\rightarrow {\bar {b}}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\bar {a}}\rightarrow {\bar {b}}\ ,}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8353ef8a72b583b1da92353b97f5b676325882bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.814ex; height:2.843ex;" alt="{\displaystyle \ {\bar {a}}\rightarrow {\bar {b}}\ ,}"></noscript><span class="lazy-image-placeholder" style="width: 7.814ex;height: 2.843ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8353ef8a72b583b1da92353b97f5b676325882bb" data-alt="{\displaystyle \ {\bar {a}}\rightarrow {\bar {b}}\ ,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and denote their amplitudes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cal {M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cal {M}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88cb0b27f4cd40f230c8c50dcbd444f52a2720bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.791ex; height:2.176ex;" alt="{\displaystyle {\cal {M}}}"></noscript><span class="lazy-image-placeholder" style="width: 2.791ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88cb0b27f4cd40f230c8c50dcbd444f52a2720bd" data-alt="{\displaystyle {\cal {M}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\cal {M}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\cal {M}}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3986848ea4ea50383edc67ef7464d0c4d8a9f9a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.791ex; height:2.676ex;" alt="{\displaystyle {\bar {\cal {M}}}}"></noscript><span class="lazy-image-placeholder" style="width: 2.791ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3986848ea4ea50383edc67ef7464d0c4d8a9f9a2" data-alt="{\displaystyle {\bar {\cal {M}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  respectively. Before CP violation, these terms must be the same complex number. We can separate the magnitude and phase by writing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cal {M}}=|{\cal {M}}|\ e^{i\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cal {M}}=|{\cal {M}}|\ e^{i\theta }}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5c17d4c49eb90775c0ae0bdcf28592adb5ee088" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.208ex; height:3.176ex;" alt="{\displaystyle {\cal {M}}=|{\cal {M}}|\ e^{i\theta }}"></noscript><span class="lazy-image-placeholder" style="width: 13.208ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5c17d4c49eb90775c0ae0bdcf28592adb5ee088" data-alt="{\displaystyle {\cal {M}}=|{\cal {M}}|\ e^{i\theta }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . If a phase term is introduced from (e.g.) the CKM matrix, denote it <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\phi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ϕ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\phi }}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05c817052d719f5b21dd162c425e514232d14af6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.863ex; height:2.676ex;" alt="{\displaystyle e^{i\phi }}"></noscript><span class="lazy-image-placeholder" style="width: 2.863ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05c817052d719f5b21dd162c425e514232d14af6" data-alt="{\displaystyle e^{i\phi }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Note that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\cal {M}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\cal {M}}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3986848ea4ea50383edc67ef7464d0c4d8a9f9a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.791ex; height:2.676ex;" alt="{\displaystyle {\bar {\cal {M}}}}"></noscript><span class="lazy-image-placeholder" style="width: 2.791ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3986848ea4ea50383edc67ef7464d0c4d8a9f9a2" data-alt="{\displaystyle {\bar {\cal {M}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  contains the conjugate matrix to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cal {M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cal {M}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88cb0b27f4cd40f230c8c50dcbd444f52a2720bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.791ex; height:2.176ex;" alt="{\displaystyle {\cal {M}}}"></noscript><span class="lazy-image-placeholder" style="width: 2.791ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88cb0b27f4cd40f230c8c50dcbd444f52a2720bd" data-alt="{\displaystyle {\cal {M}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , so it picks up a phase term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-i\phi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ϕ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-i\phi }}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/281aedb891b74c1f149bd593a3e1b69d6c12fcf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.141ex; height:2.676ex;" alt="{\displaystyle e^{-i\phi }}"></noscript><span class="lazy-image-placeholder" style="width: 4.141ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/281aedb891b74c1f149bd593a3e1b69d6c12fcf7" data-alt="{\displaystyle e^{-i\phi }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Now the formula becomes: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\cal {M}}&=|{\cal {M}}|\ e^{i\theta }\ e^{+i\phi }\\{\bar {\cal {M}}}&=|{\cal {M}}|\ e^{i\theta }\ e^{-i\phi }\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi>i</mi> <mi>ϕ</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>ϕ</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\cal {M}}&=|{\cal {M}}|\ e^{i\theta }\ e^{+i\phi }\\{\bar {\cal {M}}}&=|{\cal {M}}|\ e^{i\theta }\ e^{-i\phi }\end{aligned}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0090446f48e3f1e1530fe502df55f2529e3b62ba" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.682ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}{\cal {M}}&=|{\cal {M}}|\ e^{i\theta }\ e^{+i\phi }\\{\bar {\cal {M}}}&=|{\cal {M}}|\ e^{i\theta }\ e^{-i\phi }\end{aligned}}}"></noscript><span class="lazy-image-placeholder" style="width: 18.682ex;height: 6.509ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0090446f48e3f1e1530fe502df55f2529e3b62ba" data-alt="{\displaystyle {\begin{aligned}{\cal {M}}&=|{\cal {M}}|\ e^{i\theta }\ e^{+i\phi }\\{\bar {\cal {M}}}&=|{\cal {M}}|\ e^{i\theta }\ e^{-i\phi }\end{aligned}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Physically measurable reaction rates are proportional to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ |{\cal {M}}|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ |{\cal {M}}|^{2}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d07bc58b90ca59b0e0a2f4b2eb0b3ab005ef66dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.719ex; height:3.343ex;" alt="{\displaystyle \ |{\cal {M}}|^{2}}"></noscript><span class="lazy-image-placeholder" style="width: 5.719ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d07bc58b90ca59b0e0a2f4b2eb0b3ab005ef66dc" data-alt="{\displaystyle \ |{\cal {M}}|^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , thus so far nothing is different. However, consider that there are two different routes: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a{\overset {1}{\longrightarrow }}b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo stretchy="false">⟶</mo> <mn>1</mn> </mover> </mrow> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a{\overset {1}{\longrightarrow }}b}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb93fa4217b1e050913d7298acab069f1576aa8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.033ex; height:3.509ex;" alt="{\displaystyle a{\overset {1}{\longrightarrow }}b}"></noscript><span class="lazy-image-placeholder" style="width: 6.033ex;height: 3.509ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb93fa4217b1e050913d7298acab069f1576aa8b" data-alt="{\displaystyle a{\overset {1}{\longrightarrow }}b}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a{\overset {2}{\longrightarrow }}b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo stretchy="false">⟶</mo> <mn>2</mn> </mover> </mrow> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a{\overset {2}{\longrightarrow }}b}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc2c777735709e8a9c4c7689d986c00b1a3ffe8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.033ex; height:3.509ex;" alt="{\displaystyle a{\overset {2}{\longrightarrow }}b}"></noscript><span class="lazy-image-placeholder" style="width: 6.033ex;height: 3.509ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc2c777735709e8a9c4c7689d986c00b1a3ffe8f" data-alt="{\displaystyle a{\overset {2}{\longrightarrow }}b}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  or equivalently, two unrelated intermediate states: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\rightarrow 1\rightarrow b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">→</mo> <mn>1</mn> <mo stretchy="false">→</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\rightarrow 1\rightarrow b}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f90704b562c252ef276bceb73bf3b7f3562082f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.618ex; height:2.176ex;" alt="{\displaystyle a\rightarrow 1\rightarrow b}"></noscript><span class="lazy-image-placeholder" style="width: 10.618ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f90704b562c252ef276bceb73bf3b7f3562082f" data-alt="{\displaystyle a\rightarrow 1\rightarrow b}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\rightarrow 2\rightarrow b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">→</mo> <mn>2</mn> <mo stretchy="false">→</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\rightarrow 2\rightarrow b}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5f841aef4eaea3f00e67eefb63f91736b476867" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.618ex; height:2.176ex;" alt="{\displaystyle a\rightarrow 2\rightarrow b}"></noscript><span class="lazy-image-placeholder" style="width: 10.618ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5f841aef4eaea3f00e67eefb63f91736b476867" data-alt="{\displaystyle a\rightarrow 2\rightarrow b}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . This is exactly the case for the kaon where the decay is performed via different quark channels (see the Figure above). In this case we have: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{3}{\cal {M}}&=|{\cal {M}}_{1}|\ e^{i\theta _{1}}\ e^{i\phi _{1}}&&+|{\cal {M}}_{2}|\ e^{i\theta _{2}}\ e^{i\phi _{2}}\\{\bar {\cal {M}}}&=|{\cal {M}}_{1}|\ e^{i\theta _{1}}\ e^{-i\phi _{1}}&&+|{\cal {M}}_{2}|\ e^{i\theta _{2}}\ e^{-i\phi _{2}}\ .\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> </mtd> <mtd></mtd> <mtd> <mi></mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> </mtd> <mtd></mtd> <mtd> <mi></mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <mtext> </mtext> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{3}{\cal {M}}&=|{\cal {M}}_{1}|\ e^{i\theta _{1}}\ e^{i\phi _{1}}&&+|{\cal {M}}_{2}|\ e^{i\theta _{2}}\ e^{i\phi _{2}}\\{\bar {\cal {M}}}&=|{\cal {M}}_{1}|\ e^{i\theta _{1}}\ e^{-i\phi _{1}}&&+|{\cal {M}}_{2}|\ e^{i\theta _{2}}\ e^{-i\phi _{2}}\ .\end{alignedat}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6c1391d1ed2c71d6ed7c7ac7917019a720429f9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.225ex; height:6.509ex;" alt="{\displaystyle {\begin{alignedat}{3}{\cal {M}}&=|{\cal {M}}_{1}|\ e^{i\theta _{1}}\ e^{i\phi _{1}}&&+|{\cal {M}}_{2}|\ e^{i\theta _{2}}\ e^{i\phi _{2}}\\{\bar {\cal {M}}}&=|{\cal {M}}_{1}|\ e^{i\theta _{1}}\ e^{-i\phi _{1}}&&+|{\cal {M}}_{2}|\ e^{i\theta _{2}}\ e^{-i\phi _{2}}\ .\end{alignedat}}}"></noscript><span class="lazy-image-placeholder" style="width: 40.225ex;height: 6.509ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6c1391d1ed2c71d6ed7c7ac7917019a720429f9" data-alt="{\displaystyle {\begin{alignedat}{3}{\cal {M}}&=|{\cal {M}}_{1}|\ e^{i\theta _{1}}\ e^{i\phi _{1}}&&+|{\cal {M}}_{2}|\ e^{i\theta _{2}}\ e^{i\phi _{2}}\\{\bar {\cal {M}}}&=|{\cal {M}}_{1}|\ e^{i\theta _{1}}\ e^{-i\phi _{1}}&&+|{\cal {M}}_{2}|\ e^{i\theta _{2}}\ e^{-i\phi _{2}}\ .\end{alignedat}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Some further calculation gives: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\cal {M}}|^{2}-|{\bar {\cal {M}}}|^{2}=-4\ |{\cal {M}}_{1}|\ |{\cal {M}}_{2}|\ \sin(\theta _{1}-\theta _{2})\ \sin(\phi _{1}-\phi _{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>4</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\cal {M}}|^{2}-|{\bar {\cal {M}}}|^{2}=-4\ |{\cal {M}}_{1}|\ |{\cal {M}}_{2}|\ \sin(\theta _{1}-\theta _{2})\ \sin(\phi _{1}-\phi _{2}).}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09b3b1d6fa458065c7855d9dcef6a5f4fe962da4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.386ex; height:3.343ex;" alt="{\displaystyle |{\cal {M}}|^{2}-|{\bar {\cal {M}}}|^{2}=-4\ |{\cal {M}}_{1}|\ |{\cal {M}}_{2}|\ \sin(\theta _{1}-\theta _{2})\ \sin(\phi _{1}-\phi _{2}).}"></noscript><span class="lazy-image-placeholder" style="width: 57.386ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09b3b1d6fa458065c7855d9dcef6a5f4fe962da4" data-alt="{\displaystyle |{\cal {M}}|^{2}-|{\bar {\cal {M}}}|^{2}=-4\ |{\cal {M}}_{1}|\ |{\cal {M}}_{2}|\ \sin(\theta _{1}-\theta _{2})\ \sin(\phi _{1}-\phi _{2}).}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Thus, we see that a complex phase gives rise to processes that proceed at different rates for particles and antiparticles, and CP is violated. From the theoretical end, the CKM matrix is defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ V_{\mathrm {CKM} }=U_{u}^{\dagger }U_{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">K</mi> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> <mo>=</mo> <msubsup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ V_{\mathrm {CKM} }=U_{u}^{\dagger }U_{d}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/391f664a43b206a9d2007d3196de181bf68b3453" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.722ex; height:3.176ex;" alt="{\displaystyle \ V_{\mathrm {CKM} }=U_{u}^{\dagger }U_{d}}"></noscript><span class="lazy-image-placeholder" style="width: 14.722ex;height: 3.176ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/391f664a43b206a9d2007d3196de181bf68b3453" data-alt="{\displaystyle \ V_{\mathrm {CKM} }=U_{u}^{\dagger }U_{d}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{u}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{u}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/423ac752d3d2acb56be36948e40f271119e4edea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.76ex; height:2.509ex;" alt="{\displaystyle U_{u}}"></noscript><span class="lazy-image-placeholder" style="width: 2.76ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/423ac752d3d2acb56be36948e40f271119e4edea" data-alt="{\displaystyle U_{u}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{d}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a321d741a59ba8149c5d0077f222ef81ce44916c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.68ex; height:2.509ex;" alt="{\displaystyle U_{d}}"></noscript><span class="lazy-image-placeholder" style="width: 2.68ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a321d741a59ba8149c5d0077f222ef81ce44916c" data-alt="{\displaystyle U_{d}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  are unitary transformation matrices which diagonalize the fermion mass matrices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{u}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{u}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a72cd86dcb2d52a1e3d92d3c5789978abc7f1f4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.427ex; height:2.509ex;" alt="{\displaystyle M_{u}}"></noscript><span class="lazy-image-placeholder" style="width: 3.427ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a72cd86dcb2d52a1e3d92d3c5789978abc7f1f4c" data-alt="{\displaystyle M_{u}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{d}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc690f2dccbe052928df1f8fb48c15914f5017f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.346ex; height:2.509ex;" alt="{\displaystyle M_{d}}"></noscript><span class="lazy-image-placeholder" style="width: 3.346ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc690f2dccbe052928df1f8fb48c15914f5017f1" data-alt="{\displaystyle M_{d}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , respectively. Thus, there are two necessary conditions for getting a complex CKM matrix: <ol><li>At least one of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{u}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{u}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/423ac752d3d2acb56be36948e40f271119e4edea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.76ex; height:2.509ex;" alt="{\displaystyle U_{u}}"></noscript><span class="lazy-image-placeholder" style="width: 2.76ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/423ac752d3d2acb56be36948e40f271119e4edea" data-alt="{\displaystyle U_{u}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{d}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a321d741a59ba8149c5d0077f222ef81ce44916c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.68ex; height:2.509ex;" alt="{\displaystyle U_{d}}"></noscript><span class="lazy-image-placeholder" style="width: 2.68ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a321d741a59ba8149c5d0077f222ef81ce44916c" data-alt="{\displaystyle U_{d}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is complex, or the CKM matrix will be purely real.</li> <li>If both of them are complex, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{u}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{u}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/423ac752d3d2acb56be36948e40f271119e4edea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.76ex; height:2.509ex;" alt="{\displaystyle U_{u}}"></noscript><span class="lazy-image-placeholder" style="width: 2.76ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/423ac752d3d2acb56be36948e40f271119e4edea" data-alt="{\displaystyle U_{u}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{d}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a321d741a59ba8149c5d0077f222ef81ce44916c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.68ex; height:2.509ex;" alt="{\displaystyle U_{d}}"></noscript><span class="lazy-image-placeholder" style="width: 2.68ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a321d741a59ba8149c5d0077f222ef81ce44916c" data-alt="{\displaystyle U_{d}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  must be different, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{u}\neq U_{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo>≠</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{u}\neq U_{d}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e121e68e3a6fffb3c44afab6add03027567b2e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.538ex; height:2.676ex;" alt="{\displaystyle U_{u}\neq U_{d}}"></noscript><span class="lazy-image-placeholder" style="width: 8.538ex;height: 2.676ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e121e68e3a6fffb3c44afab6add03027567b2e9" data-alt="{\displaystyle U_{u}\neq U_{d}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , or the CKM matrix will be an identity matrix, which is also purely real.</li></ol> For a standard model with three fermion generations, the most general non-Hermitian pattern of its mass matrices can be given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M={\begin{bmatrix}A_{1}+iD_{1}&B_{1}+iC_{1}&B_{2}+iC_{2}\\B_{4}+iC_{4}&A_{2}+iD_{2}&B_{3}+iC_{3}\\B_{5}+iC_{5}&B_{6}+iC_{6}&A_{3}+iD_{3}\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M={\begin{bmatrix}A_{1}+iD_{1}&B_{1}+iC_{1}&B_{2}+iC_{2}\\B_{4}+iC_{4}&A_{2}+iD_{2}&B_{3}+iC_{3}\\B_{5}+iC_{5}&B_{6}+iC_{6}&A_{3}+iD_{3}\end{bmatrix}}.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39b7854c33624a55032587482a9e326bcb3635d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:42.941ex; height:9.176ex;" alt="{\displaystyle M={\begin{bmatrix}A_{1}+iD_{1}&B_{1}+iC_{1}&B_{2}+iC_{2}\\B_{4}+iC_{4}&A_{2}+iD_{2}&B_{3}+iC_{3}\\B_{5}+iC_{5}&B_{6}+iC_{6}&A_{3}+iD_{3}\end{bmatrix}}.}"></noscript><span class="lazy-image-placeholder" style="width: 42.941ex;height: 9.176ex;vertical-align: -4.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39b7854c33624a55032587482a9e326bcb3635d8" data-alt="{\displaystyle M={\begin{bmatrix}A_{1}+iD_{1}&B_{1}+iC_{1}&B_{2}+iC_{2}\\B_{4}+iC_{4}&A_{2}+iD_{2}&B_{3}+iC_{3}\\B_{5}+iC_{5}&B_{6}+iC_{6}&A_{3}+iD_{3}\end{bmatrix}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  This M matrix contains 9 elements and 18 parameters, 9 from the real coefficients and 9 from the imaginary coefficients. Obviously, a 3x3 matrix with 18 parameters is too difficult to diagonalize analytically. However, a naturally Hermitian <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M^{2}} =M\cdot M^{\dagger }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msup> </mrow> <mo>=</mo> <mi>M</mi> <mo>⋅</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M^{2}} =M\cdot M^{\dagger }}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bf33249e37cb16a48feefef5669a24f99e29122" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.396ex; height:2.676ex;" alt="{\displaystyle \mathbf {M^{2}} =M\cdot M^{\dagger }}"></noscript><span class="lazy-image-placeholder" style="width: 14.396ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bf33249e37cb16a48feefef5669a24f99e29122" data-alt="{\displaystyle \mathbf {M^{2}} =M\cdot M^{\dagger }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  can be given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M^{2}} ={\begin{bmatrix}\mathbf {A_{1}} &\mathbf {B_{1}} +i\mathbf {C_{1}} &\mathbf {B_{2}} +i\mathbf {C_{2}} \\\mathbf {B_{1}} -i\mathbf {C_{1}} &\mathbf {A_{2}} &\mathbf {B_{3}} +i\mathbf {C_{3}} \\\mathbf {B_{2}} -i\mathbf {C_{2}} &\mathbf {B_{3}} -i\mathbf {C_{3}} &\mathbf {A_{3}} \end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msup> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> </msub> </mrow> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> </msub> </mrow> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M^{2}} ={\begin{bmatrix}\mathbf {A_{1}} &\mathbf {B_{1}} +i\mathbf {C_{1}} &\mathbf {B_{2}} +i\mathbf {C_{2}} \\\mathbf {B_{1}} -i\mathbf {C_{1}} &\mathbf {A_{2}} &\mathbf {B_{3}} +i\mathbf {C_{3}} \\\mathbf {B_{2}} -i\mathbf {C_{2}} &\mathbf {B_{3}} -i\mathbf {C_{3}} &\mathbf {A_{3}} \end{bmatrix}},}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe7089e63f5c403b491707d95dcc1425edc3683d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:45.447ex; height:9.176ex;" alt="{\displaystyle \mathbf {M^{2}} ={\begin{bmatrix}\mathbf {A_{1}} &\mathbf {B_{1}} +i\mathbf {C_{1}} &\mathbf {B_{2}} +i\mathbf {C_{2}} \\\mathbf {B_{1}} -i\mathbf {C_{1}} &\mathbf {A_{2}} &\mathbf {B_{3}} +i\mathbf {C_{3}} \\\mathbf {B_{2}} -i\mathbf {C_{2}} &\mathbf {B_{3}} -i\mathbf {C_{3}} &\mathbf {A_{3}} \end{bmatrix}},}"></noscript><span class="lazy-image-placeholder" style="width: 45.447ex;height: 9.176ex;vertical-align: -4.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe7089e63f5c403b491707d95dcc1425edc3683d" data-alt="{\displaystyle \mathbf {M^{2}} ={\begin{bmatrix}\mathbf {A_{1}} &\mathbf {B_{1}} +i\mathbf {C_{1}} &\mathbf {B_{2}} +i\mathbf {C_{2}} \\\mathbf {B_{1}} -i\mathbf {C_{1}} &\mathbf {A_{2}} &\mathbf {B_{3}} +i\mathbf {C_{3}} \\\mathbf {B_{2}} -i\mathbf {C_{2}} &\mathbf {B_{3}} -i\mathbf {C_{3}} &\mathbf {A_{3}} \end{bmatrix}},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and it has the same unitary transformation matrix U with M. Besides, parameters in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M^{2}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M^{2}} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b96857f6b0499821f21ce6d8f9e7ccf0707e5f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.715ex; height:2.676ex;" alt="{\displaystyle \mathbf {M^{2}} }"></noscript><span class="lazy-image-placeholder" style="width: 3.715ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b96857f6b0499821f21ce6d8f9e7ccf0707e5f9" data-alt="{\displaystyle \mathbf {M^{2}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  are correlated to those in M directly in the ways shown below <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {A_{1}} &=A_{1}^{2}+D_{1}^{2}+B_{1}^{2}+C_{1}^{2}+B_{2}^{2}+C_{2}^{2},\\\mathbf {A_{2}} &=A_{2}^{2}+D_{2}^{2}+B_{3}^{2}+C_{3}^{2}+B_{4}^{2}+C_{4}^{2},\\\mathbf {A_{3}} &=A_{3}^{2}+D_{3}^{2}+B_{5}^{2}+C_{5}^{2}+B_{6}^{2}+C_{6}^{2},\\\mathbf {B_{1}} &=A_{1}B_{4}+D_{1}C_{4}+B_{1}A_{2}+C_{1}D_{2}+B_{2}B_{3}+C_{2}C_{3},\\\mathbf {B_{2}} &=A_{1}B_{5}+D_{1}C_{5}+B_{1}B_{6}+C_{1}C_{6}+B_{2}A_{3}+C_{2}D_{3},\\\mathbf {B_{3}} &=B_{4}B_{5}+C_{4}C_{5}+B_{6}A_{2}+C_{6}D_{2}+A_{3}B_{3}+D_{3}C_{3},\\\mathbf {C_{1}} &=D_{1}B_{4}-A_{1}C_{4}+A_{2}C_{1}-B_{1}D_{2}+B_{3}C_{2}-B_{2}C_{3},\\\mathbf {C_{2}} &=D_{1}B_{5}-A_{1}C_{5}+B_{6}C_{1}-B_{1}C_{6}+A_{3}C_{2}-B_{2}D_{3},\\\mathbf {C_{3}} &=C_{4}B_{5}-B_{4}C_{5}+D_{2}B_{6}-A_{2}C_{6}+A_{3}C_{3}-B_{3}D_{3}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>C</mi> <mrow 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<mo>+</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {A_{1}} &=A_{1}^{2}+D_{1}^{2}+B_{1}^{2}+C_{1}^{2}+B_{2}^{2}+C_{2}^{2},\\\mathbf {A_{2}} &=A_{2}^{2}+D_{2}^{2}+B_{3}^{2}+C_{3}^{2}+B_{4}^{2}+C_{4}^{2},\\\mathbf {A_{3}} &=A_{3}^{2}+D_{3}^{2}+B_{5}^{2}+C_{5}^{2}+B_{6}^{2}+C_{6}^{2},\\\mathbf {B_{1}} &=A_{1}B_{4}+D_{1}C_{4}+B_{1}A_{2}+C_{1}D_{2}+B_{2}B_{3}+C_{2}C_{3},\\\mathbf {B_{2}} &=A_{1}B_{5}+D_{1}C_{5}+B_{1}B_{6}+C_{1}C_{6}+B_{2}A_{3}+C_{2}D_{3},\\\mathbf {B_{3}} &=B_{4}B_{5}+C_{4}C_{5}+B_{6}A_{2}+C_{6}D_{2}+A_{3}B_{3}+D_{3}C_{3},\\\mathbf {C_{1}} &=D_{1}B_{4}-A_{1}C_{4}+A_{2}C_{1}-B_{1}D_{2}+B_{3}C_{2}-B_{2}C_{3},\\\mathbf {C_{2}} &=D_{1}B_{5}-A_{1}C_{5}+B_{6}C_{1}-B_{1}C_{6}+A_{3}C_{2}-B_{2}D_{3},\\\mathbf {C_{3}} &=C_{4}B_{5}-B_{4}C_{5}+D_{2}B_{6}-A_{2}C_{6}+A_{3}C_{3}-B_{3}D_{3}.\end{aligned}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f40925d8b6b64d4e4ffb265926d6fd0e28f3cb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.338ex; width:55.585ex; height:27.843ex;" alt="{\displaystyle {\begin{aligned}\mathbf {A_{1}} &=A_{1}^{2}+D_{1}^{2}+B_{1}^{2}+C_{1}^{2}+B_{2}^{2}+C_{2}^{2},\\\mathbf {A_{2}} &=A_{2}^{2}+D_{2}^{2}+B_{3}^{2}+C_{3}^{2}+B_{4}^{2}+C_{4}^{2},\\\mathbf {A_{3}} &=A_{3}^{2}+D_{3}^{2}+B_{5}^{2}+C_{5}^{2}+B_{6}^{2}+C_{6}^{2},\\\mathbf {B_{1}} &=A_{1}B_{4}+D_{1}C_{4}+B_{1}A_{2}+C_{1}D_{2}+B_{2}B_{3}+C_{2}C_{3},\\\mathbf {B_{2}} &=A_{1}B_{5}+D_{1}C_{5}+B_{1}B_{6}+C_{1}C_{6}+B_{2}A_{3}+C_{2}D_{3},\\\mathbf {B_{3}} &=B_{4}B_{5}+C_{4}C_{5}+B_{6}A_{2}+C_{6}D_{2}+A_{3}B_{3}+D_{3}C_{3},\\\mathbf {C_{1}} &=D_{1}B_{4}-A_{1}C_{4}+A_{2}C_{1}-B_{1}D_{2}+B_{3}C_{2}-B_{2}C_{3},\\\mathbf {C_{2}} &=D_{1}B_{5}-A_{1}C_{5}+B_{6}C_{1}-B_{1}C_{6}+A_{3}C_{2}-B_{2}D_{3},\\\mathbf {C_{3}} &=C_{4}B_{5}-B_{4}C_{5}+D_{2}B_{6}-A_{2}C_{6}+A_{3}C_{3}-B_{3}D_{3}.\end{aligned}}}"></noscript><span class="lazy-image-placeholder" style="width: 55.585ex;height: 27.843ex;vertical-align: -13.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f40925d8b6b64d4e4ffb265926d6fd0e28f3cb5" data-alt="{\displaystyle {\begin{aligned}\mathbf {A_{1}} &=A_{1}^{2}+D_{1}^{2}+B_{1}^{2}+C_{1}^{2}+B_{2}^{2}+C_{2}^{2},\\\mathbf {A_{2}} &=A_{2}^{2}+D_{2}^{2}+B_{3}^{2}+C_{3}^{2}+B_{4}^{2}+C_{4}^{2},\\\mathbf {A_{3}} &=A_{3}^{2}+D_{3}^{2}+B_{5}^{2}+C_{5}^{2}+B_{6}^{2}+C_{6}^{2},\\\mathbf {B_{1}} &=A_{1}B_{4}+D_{1}C_{4}+B_{1}A_{2}+C_{1}D_{2}+B_{2}B_{3}+C_{2}C_{3},\\\mathbf {B_{2}} &=A_{1}B_{5}+D_{1}C_{5}+B_{1}B_{6}+C_{1}C_{6}+B_{2}A_{3}+C_{2}D_{3},\\\mathbf {B_{3}} &=B_{4}B_{5}+C_{4}C_{5}+B_{6}A_{2}+C_{6}D_{2}+A_{3}B_{3}+D_{3}C_{3},\\\mathbf {C_{1}} &=D_{1}B_{4}-A_{1}C_{4}+A_{2}C_{1}-B_{1}D_{2}+B_{3}C_{2}-B_{2}C_{3},\\\mathbf {C_{2}} &=D_{1}B_{5}-A_{1}C_{5}+B_{6}C_{1}-B_{1}C_{6}+A_{3}C_{2}-B_{2}D_{3},\\\mathbf {C_{3}} &=C_{4}B_{5}-B_{4}C_{5}+D_{2}B_{6}-A_{2}C_{6}+A_{3}C_{3}-B_{3}D_{3}.\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  That means if we diagonalize an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M^{2}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M^{2}} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b96857f6b0499821f21ce6d8f9e7ccf0707e5f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.715ex; height:2.676ex;" alt="{\displaystyle \mathbf {M^{2}} }"></noscript><span class="lazy-image-placeholder" style="width: 3.715ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b96857f6b0499821f21ce6d8f9e7ccf0707e5f9" data-alt="{\displaystyle \mathbf {M^{2}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  matrix with 9 parameters, it has the same effect as diagonalizing an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></noscript><span class="lazy-image-placeholder" style="width: 2.442ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" data-alt="{\displaystyle M}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  matrix with 18 parameters. Therefore, diagonalizing the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M^{2}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M^{2}} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b96857f6b0499821f21ce6d8f9e7ccf0707e5f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.715ex; height:2.676ex;" alt="{\displaystyle \mathbf {M^{2}} }"></noscript><span class="lazy-image-placeholder" style="width: 3.715ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b96857f6b0499821f21ce6d8f9e7ccf0707e5f9" data-alt="{\displaystyle \mathbf {M^{2}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  matrix is certainly the most reasonable choice. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></noscript><span class="lazy-image-placeholder" style="width: 2.442ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" data-alt="{\displaystyle M}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M^{2}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M^{2}} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b96857f6b0499821f21ce6d8f9e7ccf0707e5f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.715ex; height:2.676ex;" alt="{\displaystyle \mathbf {M^{2}} }"></noscript><span class="lazy-image-placeholder" style="width: 3.715ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b96857f6b0499821f21ce6d8f9e7ccf0707e5f9" data-alt="{\displaystyle \mathbf {M^{2}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  matrix patterns given above are the most general ones. The perfect way to solve the CPV problem in the standard model is to diagonalize such matrices analytically and to achieve a U matrix which applies to both. Unfortunately, even though the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M^{2}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M^{2}} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b96857f6b0499821f21ce6d8f9e7ccf0707e5f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.715ex; height:2.676ex;" alt="{\displaystyle \mathbf {M^{2}} }"></noscript><span class="lazy-image-placeholder" style="width: 3.715ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b96857f6b0499821f21ce6d8f9e7ccf0707e5f9" data-alt="{\displaystyle \mathbf {M^{2}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  matrix has only 9 parameters, it is still too complicated to be diagonalized directly. Thus, an assumption <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M^{2}} _{R}\cdot \mathbf {M^{2\dagger }} _{I}+\mathbf {M^{2}} _{I}\cdot \mathbf {M^{2\dagger }} _{R}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> <mo>†</mo> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> <mo>†</mo> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M^{2}} _{R}\cdot \mathbf {M^{2\dagger }} _{I}+\mathbf {M^{2}} _{I}\cdot \mathbf {M^{2\dagger }} _{R}=0}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/facd3b3e059b2afd937466164e5b0402b256e8e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.86ex; height:3.009ex;" alt="{\displaystyle \mathbf {M^{2}} _{R}\cdot \mathbf {M^{2\dagger }} _{I}+\mathbf {M^{2}} _{I}\cdot \mathbf {M^{2\dagger }} _{R}=0}"></noscript><span class="lazy-image-placeholder" style="width: 31.86ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/facd3b3e059b2afd937466164e5b0402b256e8e4" data-alt="{\displaystyle \mathbf {M^{2}} _{R}\cdot \mathbf {M^{2\dagger }} _{I}+\mathbf {M^{2}} _{I}\cdot \mathbf {M^{2\dagger }} _{R}=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  was employed to simplify the pattern, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M^{2}} _{R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M^{2}} _{R}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85af96bed6b72f694b719c013d9c142b1e00346c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.194ex; height:3.009ex;" alt="{\displaystyle \mathbf {M^{2}} _{R}}"></noscript><span class="lazy-image-placeholder" style="width: 5.194ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85af96bed6b72f694b719c013d9c142b1e00346c" data-alt="{\displaystyle \mathbf {M^{2}} _{R}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is the real part of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M^{2}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M^{2}} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b96857f6b0499821f21ce6d8f9e7ccf0707e5f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.715ex; height:2.676ex;" alt="{\displaystyle \mathbf {M^{2}} }"></noscript><span class="lazy-image-placeholder" style="width: 3.715ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b96857f6b0499821f21ce6d8f9e7ccf0707e5f9" data-alt="{\displaystyle \mathbf {M^{2}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M^{2}} _{I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M^{2}} _{I}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70fb7426304a5824c606fa8bf4b46dab5e94bc74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.776ex; height:3.009ex;" alt="{\displaystyle \mathbf {M^{2}} _{I}}"></noscript><span class="lazy-image-placeholder" style="width: 4.776ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70fb7426304a5824c606fa8bf4b46dab5e94bc74" data-alt="{\displaystyle \mathbf {M^{2}} _{I}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is the imaginary part. Such an assumption could further reduce the parameter number from 9 to 5 and the reduced <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M^{2}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M^{2}} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b96857f6b0499821f21ce6d8f9e7ccf0707e5f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.715ex; height:2.676ex;" alt="{\displaystyle \mathbf {M^{2}} }"></noscript><span class="lazy-image-placeholder" style="width: 3.715ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b96857f6b0499821f21ce6d8f9e7ccf0707e5f9" data-alt="{\displaystyle \mathbf {M^{2}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  matrix can be given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M^{2}} ={\begin{bmatrix}\mathbf {A} +\mathbf {B} (xy-{x \over y})&y\mathbf {B} &x\mathbf {B} \\y\mathbf {B} &\mathbf {A} +\mathbf {B} ({y \over x}-{x \over y})&\mathbf {B} \\x\mathbf {B} &\mathbf {B} &\mathbf {A} \end{bmatrix}}+i{\begin{bmatrix}0&{\mathbf {C} \over y}&-{\mathbf {C} \over x}\\-{\mathbf {C} \over y}&0&\mathbf {C} \\{\mathbf {C} \over x}&-\mathbf {C} &0\end{bmatrix}}\equiv \mathbf {M^{2}} _{R}+i\mathbf {M^{2}} _{I},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msup> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>y</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mtd> <mtd> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>x</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>y</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mi>y</mi> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mi>x</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mi>y</mi> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mi>x</mi> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>≡</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M^{2}} ={\begin{bmatrix}\mathbf {A} +\mathbf {B} (xy-{x \over y})&y\mathbf {B} &x\mathbf {B} \\y\mathbf {B} &\mathbf {A} +\mathbf {B} ({y \over x}-{x \over y})&\mathbf {B} \\x\mathbf {B} &\mathbf {B} &\mathbf {A} \end{bmatrix}}+i{\begin{bmatrix}0&{\mathbf {C} \over y}&-{\mathbf {C} \over x}\\-{\mathbf {C} \over y}&0&\mathbf {C} \\{\mathbf {C} \over x}&-\mathbf {C} &0\end{bmatrix}}\equiv \mathbf {M^{2}} _{R}+i\mathbf {M^{2}} _{I},}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c698e0a0e06d4af928ff5f85b65a1fa542e2e1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:90.434ex; height:12.176ex;" alt="{\displaystyle \mathbf {M^{2}} ={\begin{bmatrix}\mathbf {A} +\mathbf {B} (xy-{x \over y})&y\mathbf {B} &x\mathbf {B} \\y\mathbf {B} &\mathbf {A} +\mathbf {B} ({y \over x}-{x \over y})&\mathbf {B} \\x\mathbf {B} &\mathbf {B} &\mathbf {A} \end{bmatrix}}+i{\begin{bmatrix}0&{\mathbf {C} \over y}&-{\mathbf {C} \over x}\\-{\mathbf {C} \over y}&0&\mathbf {C} \\{\mathbf {C} \over x}&-\mathbf {C} &0\end{bmatrix}}\equiv \mathbf {M^{2}} _{R}+i\mathbf {M^{2}} _{I},}"></noscript><span class="lazy-image-placeholder" style="width: 90.434ex;height: 12.176ex;vertical-align: -5.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c698e0a0e06d4af928ff5f85b65a1fa542e2e1f" data-alt="{\displaystyle \mathbf {M^{2}} ={\begin{bmatrix}\mathbf {A} +\mathbf {B} (xy-{x \over y})&y\mathbf {B} &x\mathbf {B} \\y\mathbf {B} &\mathbf {A} +\mathbf {B} ({y \over x}-{x \over y})&\mathbf {B} \\x\mathbf {B} &\mathbf {B} &\mathbf {A} \end{bmatrix}}+i{\begin{bmatrix}0&{\mathbf {C} \over y}&-{\mathbf {C} \over x}\\-{\mathbf {C} \over y}&0&\mathbf {C} \\{\mathbf {C} \over x}&-\mathbf {C} &0\end{bmatrix}}\equiv \mathbf {M^{2}} _{R}+i\mathbf {M^{2}} _{I},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} \equiv \mathbf {A_{3}} ,\mathbf {B} \equiv \mathbf {B_{3}} ,\mathbf {C} \equiv \mathbf {C_{3}} ,x\equiv \mathbf {B_{2}/B_{3}} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> </msub> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> </msub> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> </msub> </mrow> <mo>,</mo> <mi>x</mi> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold">/</mo> </mrow> <msub> <mi mathvariant="bold">B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> </msub> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} \equiv \mathbf {A_{3}} ,\mathbf {B} \equiv \mathbf {B_{3}} ,\mathbf {C} \equiv \mathbf {C_{3}} ,x\equiv \mathbf {B_{2}/B_{3}} ,}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ae0e50ac680e93670a2cdf07f52c1cb1f573ce6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.202ex; height:2.843ex;" alt="{\displaystyle \mathbf {A} \equiv \mathbf {A_{3}} ,\mathbf {B} \equiv \mathbf {B_{3}} ,\mathbf {C} \equiv \mathbf {C_{3}} ,x\equiv \mathbf {B_{2}/B_{3}} ,}"></noscript><span class="lazy-image-placeholder" style="width: 40.202ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ae0e50ac680e93670a2cdf07f52c1cb1f573ce6" data-alt="{\displaystyle \mathbf {A} \equiv \mathbf {A_{3}} ,\mathbf {B} \equiv \mathbf {B_{3}} ,\mathbf {C} \equiv \mathbf {C_{3}} ,x\equiv \mathbf {B_{2}/B_{3}} ,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\equiv \mathbf {B_{1}/B_{3}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold">/</mo> </mrow> <msub> <mi mathvariant="bold">B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\equiv \mathbf {B_{1}/B_{3}} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce6b87d25b6828f3f4a054cc66fb2dc9cd397aff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.748ex; height:2.843ex;" alt="{\displaystyle y\equiv \mathbf {B_{1}/B_{3}} }"></noscript><span class="lazy-image-placeholder" style="width: 11.748ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce6b87d25b6828f3f4a054cc66fb2dc9cd397aff" data-alt="{\displaystyle y\equiv \mathbf {B_{1}/B_{3}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Diagonalizing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M^{2}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M^{2}} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b96857f6b0499821f21ce6d8f9e7ccf0707e5f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.715ex; height:2.676ex;" alt="{\displaystyle \mathbf {M^{2}} }"></noscript><span class="lazy-image-placeholder" style="width: 3.715ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b96857f6b0499821f21ce6d8f9e7ccf0707e5f9" data-alt="{\displaystyle \mathbf {M^{2}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  analytically, the eigenvalues are given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {m_{1}} ^{2}=\mathbf {A} -\mathbf {B} {x \over y}-\mathbf {C} {{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}} \over xy},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>y</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {m_{1}} ^{2}=\mathbf {A} -\mathbf {B} {x \over y}-\mathbf {C} {{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}} \over xy},}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62a242639cf72a0dc67d716c0b2544c4b5ccc072" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:39.939ex; height:6.676ex;" alt="{\displaystyle \mathbf {m_{1}} ^{2}=\mathbf {A} -\mathbf {B} {x \over y}-\mathbf {C} {{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}} \over xy},}"></noscript><span class="lazy-image-placeholder" style="width: 39.939ex;height: 6.676ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62a242639cf72a0dc67d716c0b2544c4b5ccc072" data-alt="{\displaystyle \mathbf {m_{1}} ^{2}=\mathbf {A} -\mathbf {B} {x \over y}-\mathbf {C} {{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}} \over xy},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {m_{2}} ^{2}=\mathbf {A} -\mathbf {B} {x \over y}+\mathbf {C} {{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}} \over xy},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>y</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {m_{2}} ^{2}=\mathbf {A} -\mathbf {B} {x \over y}+\mathbf {C} {{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}} \over xy},}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b735c97066d3b97ec5be0eed368dadb0dd495ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:39.939ex; height:6.676ex;" alt="{\displaystyle \mathbf {m_{2}} ^{2}=\mathbf {A} -\mathbf {B} {x \over y}+\mathbf {C} {{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}} \over xy},}"></noscript><span class="lazy-image-placeholder" style="width: 39.939ex;height: 6.676ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b735c97066d3b97ec5be0eed368dadb0dd495ac" data-alt="{\displaystyle \mathbf {m_{2}} ^{2}=\mathbf {A} -\mathbf {B} {x \over y}+\mathbf {C} {{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}} \over xy},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {m_{3}} ^{2}=\mathbf {A} +\mathbf {B} {(x^{2}+1)y \over x},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>y</mi> </mrow> <mi>x</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {m_{3}} ^{2}=\mathbf {A} +\mathbf {B} {(x^{2}+1)y \over x},}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1204abef33cfeb4373f5afb8a30da300e1a683f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.152ex; height:5.843ex;" alt="{\displaystyle \mathbf {m_{3}} ^{2}=\mathbf {A} +\mathbf {B} {(x^{2}+1)y \over x},}"></noscript><span class="lazy-image-placeholder" style="width: 25.152ex;height: 5.843ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1204abef33cfeb4373f5afb8a30da300e1a683f5" data-alt="{\displaystyle \mathbf {m_{3}} ^{2}=\mathbf {A} +\mathbf {B} {(x^{2}+1)y \over x},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></noscript><span class="lazy-image-placeholder" style="width: 1.783ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" data-alt="{\displaystyle U}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  matrix for up-type quarks can then be given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{u}={\begin{bmatrix}{-{\sqrt {x^{2}+y^{2}}} \over {\sqrt {2(x^{2}+y^{2}+x^{2}y^{2})}}}&{-{\sqrt {x^{2}+y^{2}}} \over {\sqrt {2(x^{2}+y^{2}+x^{2}y^{2})}}}&{xy \over {\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}}\\{x(y^{2}-i{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}) \over {\sqrt {x^{2}+y^{2}}}{\sqrt {2(x^{2}+y^{2}+x^{2}y^{2})}}}&{x(y^{2}+i{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}) \over {\sqrt {x^{2}+y^{2}}}{\sqrt {2(x^{2}+y^{2}+x^{2}y^{2})}}}&{y \over {\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}}\\{y(x^{2}+i{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}) \over {\sqrt {x^{2}+y^{2}}}{\sqrt {2(x^{2}+y^{2}+x^{2}y^{2})}}}&{y(x^{2}-i{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}) \over {\sqrt {x^{2}+y^{2}}}{\sqrt {2(x^{2}+y^{2}+x^{2}y^{2})}}}&{x \over {\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}}\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </msqrt> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </msqrt> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>y</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>y</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{u}={\begin{bmatrix}{-{\sqrt {x^{2}+y^{2}}} \over {\sqrt {2(x^{2}+y^{2}+x^{2}y^{2})}}}&{-{\sqrt {x^{2}+y^{2}}} \over {\sqrt {2(x^{2}+y^{2}+x^{2}y^{2})}}}&{xy \over {\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}}\\{x(y^{2}-i{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}) \over {\sqrt {x^{2}+y^{2}}}{\sqrt {2(x^{2}+y^{2}+x^{2}y^{2})}}}&{x(y^{2}+i{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}) \over {\sqrt {x^{2}+y^{2}}}{\sqrt {2(x^{2}+y^{2}+x^{2}y^{2})}}}&{y \over {\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}}\\{y(x^{2}+i{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}) \over {\sqrt {x^{2}+y^{2}}}{\sqrt {2(x^{2}+y^{2}+x^{2}y^{2})}}}&{y(x^{2}-i{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}) \over {\sqrt {x^{2}+y^{2}}}{\sqrt {2(x^{2}+y^{2}+x^{2}y^{2})}}}&{x \over {\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}}\end{bmatrix}}.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17e9d6d5092bfbed9b5c64238e52661619c07e4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.171ex; width:67.55ex; height:23.509ex;" alt="{\displaystyle U_{u}={\begin{bmatrix}{-{\sqrt {x^{2}+y^{2}}} \over {\sqrt {2(x^{2}+y^{2}+x^{2}y^{2})}}}&{-{\sqrt {x^{2}+y^{2}}} \over {\sqrt {2(x^{2}+y^{2}+x^{2}y^{2})}}}&{xy \over {\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}}\\{x(y^{2}-i{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}) \over {\sqrt {x^{2}+y^{2}}}{\sqrt {2(x^{2}+y^{2}+x^{2}y^{2})}}}&{x(y^{2}+i{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}) \over {\sqrt {x^{2}+y^{2}}}{\sqrt {2(x^{2}+y^{2}+x^{2}y^{2})}}}&{y \over {\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}}\\{y(x^{2}+i{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}) \over {\sqrt {x^{2}+y^{2}}}{\sqrt {2(x^{2}+y^{2}+x^{2}y^{2})}}}&{y(x^{2}-i{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}) \over {\sqrt {x^{2}+y^{2}}}{\sqrt {2(x^{2}+y^{2}+x^{2}y^{2})}}}&{x \over {\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}}\end{bmatrix}}.}"></noscript><span class="lazy-image-placeholder" style="width: 67.55ex;height: 23.509ex;vertical-align: -11.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17e9d6d5092bfbed9b5c64238e52661619c07e4e" data-alt="{\displaystyle U_{u}={\begin{bmatrix}{-{\sqrt {x^{2}+y^{2}}} \over {\sqrt {2(x^{2}+y^{2}+x^{2}y^{2})}}}&{-{\sqrt {x^{2}+y^{2}}} \over {\sqrt {2(x^{2}+y^{2}+x^{2}y^{2})}}}&{xy \over {\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}}\\{x(y^{2}-i{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}) \over {\sqrt {x^{2}+y^{2}}}{\sqrt {2(x^{2}+y^{2}+x^{2}y^{2})}}}&{x(y^{2}+i{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}) \over {\sqrt {x^{2}+y^{2}}}{\sqrt {2(x^{2}+y^{2}+x^{2}y^{2})}}}&{y \over {\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}}\\{y(x^{2}+i{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}) \over {\sqrt {x^{2}+y^{2}}}{\sqrt {2(x^{2}+y^{2}+x^{2}y^{2})}}}&{y(x^{2}-i{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}) \over {\sqrt {x^{2}+y^{2}}}{\sqrt {2(x^{2}+y^{2}+x^{2}y^{2})}}}&{x \over {\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}}\end{bmatrix}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  However, the order of the eigenvalues and correspondingly the order of the columns of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{u}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{u}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/423ac752d3d2acb56be36948e40f271119e4edea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.76ex; height:2.509ex;" alt="{\displaystyle U_{u}}"></noscript><span class="lazy-image-placeholder" style="width: 2.76ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/423ac752d3d2acb56be36948e40f271119e4edea" data-alt="{\displaystyle U_{u}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  does not necessarily have to be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {m_{1}} ^{2},\mathbf {m_{2}} ^{2},\mathbf {m_{3}} ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {m_{1}} ^{2},\mathbf {m_{2}} ^{2},\mathbf {m_{3}} ^{2})}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/308738140c5861549c10d0aa1bd8116439f3ab80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.251ex; height:3.176ex;" alt="{\displaystyle (\mathbf {m_{1}} ^{2},\mathbf {m_{2}} ^{2},\mathbf {m_{3}} ^{2})}"></noscript><span class="lazy-image-placeholder" style="width: 17.251ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/308738140c5861549c10d0aa1bd8116439f3ab80" data-alt="{\displaystyle (\mathbf {m_{1}} ^{2},\mathbf {m_{2}} ^{2},\mathbf {m_{3}} ^{2})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  but can be any permutation of those. After obtaining a general <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></noscript><span class="lazy-image-placeholder" style="width: 1.783ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" data-alt="{\displaystyle U}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  matrix pattern, the same procedure can be applied to down-type quarks by introducing primed parameters. To construct the CKM matrix, the conjugate transpose of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></noscript><span class="lazy-image-placeholder" style="width: 1.783ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" data-alt="{\displaystyle U}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  matrix for up-type quarks, denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{u}^{\dagger }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{u}^{\dagger }}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32ca1500f0aa88ac0f9dd671c1d42268b0477625" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.804ex; height:3.176ex;" alt="{\displaystyle U_{u}^{\dagger }}"></noscript><span class="lazy-image-placeholder" style="width: 2.804ex;height: 3.176ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32ca1500f0aa88ac0f9dd671c1d42268b0477625" data-alt="{\displaystyle U_{u}^{\dagger }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , has to be multiplied with the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></noscript><span class="lazy-image-placeholder" style="width: 1.783ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" data-alt="{\displaystyle U}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  matrix for down-type quarks, denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{d}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a321d741a59ba8149c5d0077f222ef81ce44916c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.68ex; height:2.509ex;" alt="{\displaystyle U_{d}}"></noscript><span class="lazy-image-placeholder" style="width: 2.68ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a321d741a59ba8149c5d0077f222ef81ce44916c" data-alt="{\displaystyle U_{d}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . As mentioned earlier, there are no inherent constraints that dictate the assignment of eigenvalues to specific quark flavors. All <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3!\times 3!=36}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>!</mo> <mo>×</mo> <mn>3</mn> <mo>!</mo> <mo>=</mo> <mn>36</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3!\times 3!=36}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9c520c5bf63d26c22db1f785c5141730d96b51a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.882ex; height:2.176ex;" alt="{\displaystyle 3!\times 3!=36}"></noscript><span class="lazy-image-placeholder" style="width: 11.882ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9c520c5bf63d26c22db1f785c5141730d96b51a" data-alt="{\displaystyle 3!\times 3!=36}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  potential permutations of eigenvalues are listed elsewhere.<a href="#cite_note-27">[27]</a><a href="#cite_note-28">[28]</a> Among these 36 potential CKM matrices, 4 of them <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V[52]=V{\begin{bmatrix}1\\3\\2\end{bmatrix}}{\begin{bmatrix}2&3&1\end{bmatrix}}=V[25]^{*}=V^{*}{\begin{bmatrix}2\\3\\1\end{bmatrix}}{\begin{bmatrix}1&3&2\end{bmatrix}}={\begin{bmatrix}s&p&r\\p^{\prime }&q&p^{\prime *}\\r^{*}&p^{*}&s^{*}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">[</mo> <mn>52</mn> <mo stretchy="false">]</mo> <mo>=</mo> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mi>V</mi> <mo stretchy="false">[</mo> <mn>25</mn> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>s</mi> </mtd> <mtd> <mi>p</mi> </mtd> <mtd> <mi>r</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi>q</mi> </mtd> <mtd> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mo>∗</mo> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mtd> <mtd> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mtd> <mtd> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V[52]=V{\begin{bmatrix}1\\3\\2\end{bmatrix}}{\begin{bmatrix}2&3&1\end{bmatrix}}=V[25]^{*}=V^{*}{\begin{bmatrix}2\\3\\1\end{bmatrix}}{\begin{bmatrix}1&3&2\end{bmatrix}}={\begin{bmatrix}s&p&r\\p^{\prime }&q&p^{\prime *}\\r^{*}&p^{*}&s^{*}\end{bmatrix}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eb915f5ae9b7a18384e2911dc43197e68c096fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:74.903ex; height:9.176ex;" alt="{\displaystyle V[52]=V{\begin{bmatrix}1\\3\\2\end{bmatrix}}{\begin{bmatrix}2&3&1\end{bmatrix}}=V[25]^{*}=V^{*}{\begin{bmatrix}2\\3\\1\end{bmatrix}}{\begin{bmatrix}1&3&2\end{bmatrix}}={\begin{bmatrix}s&p&r\\p^{\prime }&q&p^{\prime *}\\r^{*}&p^{*}&s^{*}\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 74.903ex;height: 9.176ex;vertical-align: -4.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eb915f5ae9b7a18384e2911dc43197e68c096fa" data-alt="{\displaystyle V[52]=V{\begin{bmatrix}1\\3\\2\end{bmatrix}}{\begin{bmatrix}2&3&1\end{bmatrix}}=V[25]^{*}=V^{*}{\begin{bmatrix}2\\3\\1\end{bmatrix}}{\begin{bmatrix}1&3&2\end{bmatrix}}={\begin{bmatrix}s&p&r\\p^{\prime }&q&p^{\prime *}\\r^{*}&p^{*}&s^{*}\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V[22]=V{\begin{bmatrix}2\\3\\1\end{bmatrix}}{\begin{bmatrix}2&3&1\end{bmatrix}}=V[55]^{*}=V^{*}{\begin{bmatrix}1\\3\\2\end{bmatrix}}{\begin{bmatrix}1&3&2\end{bmatrix}}={\begin{bmatrix}r^{*}&p^{*}&s^{*}\\p^{\prime *}&q&p^{\prime }\\s&p&r\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">[</mo> <mn>22</mn> <mo stretchy="false">]</mo> <mo>=</mo> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mi>V</mi> <mo stretchy="false">[</mo> <mn>55</mn> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mtd> <mtd> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mtd> <mtd> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> <mo>∗</mo> </mrow> </msup> </mtd> <mtd> <mi>q</mi> </mtd> <mtd> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>s</mi> </mtd> <mtd> <mi>p</mi> </mtd> <mtd> <mi>r</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V[22]=V{\begin{bmatrix}2\\3\\1\end{bmatrix}}{\begin{bmatrix}2&3&1\end{bmatrix}}=V[55]^{*}=V^{*}{\begin{bmatrix}1\\3\\2\end{bmatrix}}{\begin{bmatrix}1&3&2\end{bmatrix}}={\begin{bmatrix}r^{*}&p^{*}&s^{*}\\p^{\prime *}&q&p^{\prime }\\s&p&r\end{bmatrix}},}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad64fa9ae4b0be292a3fa902789b717d3956ae3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:75.591ex; height:9.176ex;" alt="{\displaystyle V[22]=V{\begin{bmatrix}2\\3\\1\end{bmatrix}}{\begin{bmatrix}2&3&1\end{bmatrix}}=V[55]^{*}=V^{*}{\begin{bmatrix}1\\3\\2\end{bmatrix}}{\begin{bmatrix}1&3&2\end{bmatrix}}={\begin{bmatrix}r^{*}&p^{*}&s^{*}\\p^{\prime *}&q&p^{\prime }\\s&p&r\end{bmatrix}},}"></noscript><span class="lazy-image-placeholder" style="width: 75.591ex;height: 9.176ex;vertical-align: -4.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad64fa9ae4b0be292a3fa902789b717d3956ae3b" data-alt="{\displaystyle V[22]=V{\begin{bmatrix}2\\3\\1\end{bmatrix}}{\begin{bmatrix}2&3&1\end{bmatrix}}=V[55]^{*}=V^{*}{\begin{bmatrix}1\\3\\2\end{bmatrix}}{\begin{bmatrix}1&3&2\end{bmatrix}}={\begin{bmatrix}r^{*}&p^{*}&s^{*}\\p^{\prime *}&q&p^{\prime }\\s&p&r\end{bmatrix}},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  fit experimental data to the order of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ^{1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ^{1/2}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5501988d4b9bb58c4e3e04c88f6d93301b696ee0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.053ex; height:2.843ex;" alt="{\displaystyle \lambda ^{1/2}}"></noscript><span class="lazy-image-placeholder" style="width: 4.053ex;height: 2.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5501988d4b9bb58c4e3e04c88f6d93301b696ee0" data-alt="{\displaystyle \lambda ^{1/2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  or better, at tree level, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></noscript><span class="lazy-image-placeholder" style="width: 1.355ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" data-alt="{\displaystyle \lambda }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is one of the <a href="/wiki/Cabibbo-Kobayashi-Maskawa_matrix#Wolfenstein_parameters" class="mw-redirect" title="Cabibbo-Kobayashi-Maskawa matrix">Wolfenstein parameters</a>. The full expressions of parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p,q,r,s,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>s</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p,q,r,s,}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b69f200f9ef0fb4d94c9e18b7722323a40e0e92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:8.216ex; height:2.009ex;" alt="{\displaystyle p,q,r,s,}"></noscript><span class="lazy-image-placeholder" style="width: 8.216ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b69f200f9ef0fb4d94c9e18b7722323a40e0e92" data-alt="{\displaystyle p,q,r,s,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{\prime }}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d46d13620db1b84c69bf1ff5d1290d8d5e7de6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.944ex; height:2.843ex;" alt="{\displaystyle p^{\prime }}"></noscript><span class="lazy-image-placeholder" style="width: 1.944ex;height: 2.843ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d46d13620db1b84c69bf1ff5d1290d8d5e7de6c" data-alt="{\displaystyle p^{\prime }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  are given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r={{(x^{2}+y^{2})(x'^{2}+y'^{2})+(xx'+yy')(xyx'y'+{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}})} \over {2{\sqrt {x^{2}+y^{2}}}{\sqrt {x'^{2}+y'^{2}}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>y</mi> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={{(x^{2}+y^{2})(x'^{2}+y'^{2})+(xx'+yy')(xyx'y'+{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}})} \over {2{\sqrt {x^{2}+y^{2}}}{\sqrt {x'^{2}+y'^{2}}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d9e3be7c8fd1f5281024752b3bf52d8366dd32a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:85.428ex; height:7.509ex;" alt="{\displaystyle r={{(x^{2}+y^{2})(x'^{2}+y'^{2})+(xx'+yy')(xyx'y'+{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}})} \over {2{\sqrt {x^{2}+y^{2}}}{\sqrt {x'^{2}+y'^{2}}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}}}"></noscript><span class="lazy-image-placeholder" style="width: 85.428ex;height: 7.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d9e3be7c8fd1f5281024752b3bf52d8366dd32a" data-alt="{\displaystyle r={{(x^{2}+y^{2})(x'^{2}+y'^{2})+(xx'+yy')(xyx'y'+{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}})} \over {2{\sqrt {x^{2}+y^{2}}}{\sqrt {x'^{2}+y'^{2}}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  <pre> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +i{{(xy'-x'y)(x'y'{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}+xy{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}})} \over {2{\sqrt {x^{2}+y^{2}}}{\sqrt {x'^{2}+y'^{2}}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>−</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <msup> <mi>y</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>+</mo> <mi>x</mi> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +i{{(xy'-x'y)(x'y'{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}+xy{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}})} \over {2{\sqrt {x^{2}+y^{2}}}{\sqrt {x'^{2}+y'^{2}}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}},}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a5315d51688155ed3556731a94053e2066c3ec4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:62.296ex; height:7.509ex;" alt="{\displaystyle +i{{(xy'-x'y)(x'y'{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}+xy{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}})} \over {2{\sqrt {x^{2}+y^{2}}}{\sqrt {x'^{2}+y'^{2}}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}},}"></noscript><span class="lazy-image-placeholder" style="width: 62.296ex;height: 7.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a5315d51688155ed3556731a94053e2066c3ec4" data-alt="{\displaystyle +i{{(xy'-x'y)(x'y'{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}+xy{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}})} \over {2{\sqrt {x^{2}+y^{2}}}{\sqrt {x'^{2}+y'^{2}}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </pre> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s={{(x^{2}+y^{2})(x'^{2}+y'^{2})+(xx'+yy')(xyx'y'-{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}})} \over {2{\sqrt {x^{2}+y^{2}}}{\sqrt {x'^{2}+y'^{2}}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>y</mi> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s={{(x^{2}+y^{2})(x'^{2}+y'^{2})+(xx'+yy')(xyx'y'-{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}})} \over {2{\sqrt {x^{2}+y^{2}}}{\sqrt {x'^{2}+y'^{2}}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da3b8ebc0594e4371c7a21e8271463a9bbbe1dec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:85.469ex; height:7.509ex;" alt="{\displaystyle s={{(x^{2}+y^{2})(x'^{2}+y'^{2})+(xx'+yy')(xyx'y'-{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}})} \over {2{\sqrt {x^{2}+y^{2}}}{\sqrt {x'^{2}+y'^{2}}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}}}"></noscript><span class="lazy-image-placeholder" style="width: 85.469ex;height: 7.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da3b8ebc0594e4371c7a21e8271463a9bbbe1dec" data-alt="{\displaystyle s={{(x^{2}+y^{2})(x'^{2}+y'^{2})+(xx'+yy')(xyx'y'-{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}})} \over {2{\sqrt {x^{2}+y^{2}}}{\sqrt {x'^{2}+y'^{2}}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  <pre><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +i{{(xy'-x'y)(x'y'{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}-xy{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}})} \over {2{\sqrt {x^{2}+y^{2}}}{\sqrt {x'^{2}+y'^{2}}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>−</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <msup> <mi>y</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>−</mo> <mi>x</mi> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +i{{(xy'-x'y)(x'y'{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}-xy{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}})} \over {2{\sqrt {x^{2}+y^{2}}}{\sqrt {x'^{2}+y'^{2}}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}},}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a3515841d849917147374f96e2a16631fee114c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:62.296ex; height:7.509ex;" alt="{\displaystyle +i{{(xy'-x'y)(x'y'{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}-xy{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}})} \over {2{\sqrt {x^{2}+y^{2}}}{\sqrt {x'^{2}+y'^{2}}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}},}"></noscript><span class="lazy-image-placeholder" style="width: 62.296ex;height: 7.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a3515841d849917147374f96e2a16631fee114c" data-alt="{\displaystyle +i{{(xy'-x'y)(x'y'{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}-xy{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}})} \over {2{\sqrt {x^{2}+y^{2}}}{\sqrt {x'^{2}+y'^{2}}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </pre> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p={{[y'y^{2}(x-x')+x'x^{2}(y-y')]+i(xy'-x'y){\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}} \over {{\sqrt {2}}{\sqrt {x^{2}+y^{2}}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>+</mo> <mi>i</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>−</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mi>y</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={{[y'y^{2}(x-x')+x'x^{2}(y-y')]+i(xy'-x'y){\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}} \over {{\sqrt {2}}{\sqrt {x^{2}+y^{2}}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}},}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aafde3af9a5cde00419c7e7740961fcdb98bf13c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-left: -0.089ex; width:65.916ex; height:7.509ex;" alt="{\displaystyle p={{[y'y^{2}(x-x')+x'x^{2}(y-y')]+i(xy'-x'y){\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}} \over {{\sqrt {2}}{\sqrt {x^{2}+y^{2}}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}},}"></noscript><span class="lazy-image-placeholder" style="width: 65.916ex;height: 7.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aafde3af9a5cde00419c7e7740961fcdb98bf13c" data-alt="{\displaystyle p={{[y'y^{2}(x-x')+x'x^{2}(y-y')]+i(xy'-x'y){\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}} \over {{\sqrt {2}}{\sqrt {x^{2}+y^{2}}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{\prime }={{[yy'^{2}(x'-x)+xx'^{2}(y'-y)]+i(xy'-x'y){\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}} \over {{\sqrt {2}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mi>y</mi> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>x</mi> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>+</mo> <mi>i</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>−</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mi>y</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </msqrt> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{\prime }={{[yy'^{2}(x'-x)+xx'^{2}(y'-y)]+i(xy'-x'y){\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}} \over {{\sqrt {2}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}},}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c456889e2175d3c0751da4e22dfa6b3d767da023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-left: -0.089ex; width:67.941ex; height:7.509ex;" alt="{\displaystyle p^{\prime }={{[yy'^{2}(x'-x)+xx'^{2}(y'-y)]+i(xy'-x'y){\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}} \over {{\sqrt {2}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}},}"></noscript><span class="lazy-image-placeholder" style="width: 67.941ex;height: 7.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c456889e2175d3c0751da4e22dfa6b3d767da023" data-alt="{\displaystyle p^{\prime }={{[yy'^{2}(x'-x)+xx'^{2}(y'-y)]+i(xy'-x'y){\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}} \over {{\sqrt {2}}{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q={{xx'+yy'+xyx'y'} \over {{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>y</mi> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>x</mi> <mi>y</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <msup> <mi>y</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <msup> <mi>y</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q={{xx'+yy'+xyx'y'} \over {{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}},}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83fd9eaf256e111b84e2204969907f146f1722f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.864ex; height:6.843ex;" alt="{\displaystyle q={{xx'+yy'+xyx'y'} \over {{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}},}"></noscript><span class="lazy-image-placeholder" style="width: 41.864ex;height: 6.843ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83fd9eaf256e111b84e2204969907f146f1722f8" data-alt="{\displaystyle q={{xx'+yy'+xyx'y'} \over {{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  The best fit of the CKM elements are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |V_{ud}|=|V_{tb}|\sim 0.9925,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>d</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>b</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>∼</mo> <mn>0.9925</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |V_{ud}|=|V_{tb}|\sim 0.9925,}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d04bc5385bb94b346419561efde7baf6a738c90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.164ex; height:2.843ex;" alt="{\displaystyle |V_{ud}|=|V_{tb}|\sim 0.9925,}"></noscript><span class="lazy-image-placeholder" style="width: 22.164ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d04bc5385bb94b346419561efde7baf6a738c90" data-alt="{\displaystyle |V_{ud}|=|V_{tb}|\sim 0.9925,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |V_{ub}|=|V_{td}|\sim 0.0075,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>b</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>d</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>∼</mo> <mn>0.0075</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |V_{ub}|=|V_{td}|\sim 0.0075,}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26cd430d5f3f2edbc3ef216ea319d0b2b6cda522" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.164ex; height:2.843ex;" alt="{\displaystyle |V_{ub}|=|V_{td}|\sim 0.0075,}"></noscript><span class="lazy-image-placeholder" style="width: 22.164ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26cd430d5f3f2edbc3ef216ea319d0b2b6cda522" data-alt="{\displaystyle |V_{ub}|=|V_{td}|\sim 0.0075,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |V_{us}|=|V_{ts}|=|V_{cd}|=|V_{cb}|\sim 0.122023,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>s</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>s</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>d</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>b</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>∼</mo> <mn>0.122023</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |V_{us}|=|V_{ts}|=|V_{cd}|=|V_{cb}|\sim 0.122023,}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bb4a27eb47f0bf3fc306216d50d03de65336da6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.414ex; height:2.843ex;" alt="{\displaystyle |V_{us}|=|V_{ts}|=|V_{cd}|=|V_{cb}|\sim 0.122023,}"></noscript><span class="lazy-image-placeholder" style="width: 39.414ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bb4a27eb47f0bf3fc306216d50d03de65336da6" data-alt="{\displaystyle |V_{us}|=|V_{ts}|=|V_{cd}|=|V_{cb}|\sim 0.122023,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |V_{cs}|\sim 0.9845.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>s</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>∼</mo> <mn>0.9845.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |V_{cs}|\sim 0.9845.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/509aa6f098ddf9c5d12d7e3b222afbc01a6d879f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.569ex; height:2.843ex;" alt="{\displaystyle |V_{cs}|\sim 0.9845.}"></noscript><span class="lazy-image-placeholder" style="width: 14.569ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/509aa6f098ddf9c5d12d7e3b222afbc01a6d879f" data-alt="{\displaystyle |V_{cs}|\sim 0.9845.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Since the discovery of CP violation in 1964, physicists have believed that in theory, within the framework of the Standard Model, it is sufficient to search for appropriate Yukawa couplings (equivalent to a mass matrix) in order to generate a complex phase in the CKM matrix, thus automatically breaking CP symmetry. However, the specific matrix pattern has remained elusive. The above derivation provides the first evidence for this idea and offers some explicit examples to support it. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"><h2 id="Strong_CP_problem">Strong CP problem</h2> <a role="button" href="/w/index.php?title=CP_violation&action=edit&section=9" title="Edit section: Strong CP problem" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-5 collapsible-block" id="mf-section-5"> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Strong_CP_problem" title="Strong CP problem">Strong CP problem</a></div> <style data-mw-deduplicate="TemplateStyles:r1233989161">.mw-parser-output .unsolved{margin:0.5em 0 1em 1em;border:#ccc solid;padding:0.35em 0.35em 0.35em 2.2em;background-color:var(--background-color-interactive-subtle);background-image:url("https://upload.wikimedia.org/wikipedia/commons/2/26/Question%2C_Web_Fundamentals.svg");background-position:top 50%left 0.35em;background-size:1.5em;background-repeat:no-repeat}@media(min-width:720px){.mw-parser-output .unsolved{clear:right;float:right;max-width:25%}}.mw-parser-output .unsolved-label{font-weight:bold}.mw-parser-output .unsolved-body{margin:0.35em;font-style:italic}.mw-parser-output .unsolved-more{font-size:smaller}</style> <div role="note" aria-labelledby="unsolved-label-physics" class="unsolved"> <div>Unsolved problem in physics:</div> <div class="unsolved-body">Why is the strong nuclear interaction force CP-invariant?</div> <div class="unsolved-more"><a href="/wiki/List_of_unsolved_problems_in_physics" title="List of unsolved problems in physics">(more unsolved problems in physics)</a></div> </div> There is no experimentally known violation of the CP-symmetry in <a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">quantum chromodynamics</a>. As there is no known reason for it to be conserved in QCD specifically, this is a "fine tuning" problem known as the <a href="/wiki/Strong_CP_problem" title="Strong CP problem">strong CP problem</a>. QCD does not violate the CP-symmetry as easily as the <a href="/wiki/Electroweak_theory" class="mw-redirect" title="Electroweak theory">electroweak theory</a>; unlike the electroweak theory in which the gauge fields couple to <a href="/wiki/Chirality_(physics)" title="Chirality (physics)">chiral</a> currents constructed from the <a href="/wiki/Fermion" title="Fermion">fermionic</a> fields, the gluons couple to vector currents. Experiments do not indicate any CP violation in the QCD sector. For example, a generic CP violation in the strongly interacting sector would create the <a href="/wiki/Electric_dipole_moment" title="Electric dipole moment">electric dipole moment</a> of the <a href="/wiki/Neutron" title="Neutron">neutron</a> which would be comparable to 10−18 <a href="/wiki/Elementary_charge" title="Elementary charge">e</a>·m while the experimental upper bound is roughly one trillionth that size. This is a problem because at the end, there are natural terms in the QCD <a href="/wiki/Lagrangian_(field_theory)" title="Lagrangian (field theory)">Lagrangian</a> that are able to break the CP-symmetry. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}=-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }-{\frac {n_{f}g^{2}\theta }{32\pi ^{2}}}F_{\mu \nu }{\tilde {F}}^{\mu \nu }+{\bar {\psi }}\left(i\gamma ^{\mu }D_{\mu }-me^{i\theta '\gamma _{5}}\right)\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>θ</mi> </mrow> <mrow> <mn>32</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>−</mo> <mi>m</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msup> <mi>θ</mi> <mo>′</mo> </msup> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}=-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }-{\frac {n_{f}g^{2}\theta }{32\pi ^{2}}}F_{\mu \nu }{\tilde {F}}^{\mu \nu }+{\bar {\psi }}\left(i\gamma ^{\mu }D_{\mu }-me^{i\theta '\gamma _{5}}\right)\psi }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac623cffa3e6234988495f3931b60a98b2a19cc9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:58.669ex; height:6.343ex;" alt="{\displaystyle {\mathcal {L}}=-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }-{\frac {n_{f}g^{2}\theta }{32\pi ^{2}}}F_{\mu \nu }{\tilde {F}}^{\mu \nu }+{\bar {\psi }}\left(i\gamma ^{\mu }D_{\mu }-me^{i\theta '\gamma _{5}}\right)\psi }"></noscript><span class="lazy-image-placeholder" style="width: 58.669ex;height: 6.343ex;vertical-align: -2.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac623cffa3e6234988495f3931b60a98b2a19cc9" data-alt="{\displaystyle {\mathcal {L}}=-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }-{\frac {n_{f}g^{2}\theta }{32\pi ^{2}}}F_{\mu \nu }{\tilde {F}}^{\mu \nu }+{\bar {\psi }}\left(i\gamma ^{\mu }D_{\mu }-me^{i\theta '\gamma _{5}}\right)\psi }" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  For a nonzero choice of the θ angle and the chiral phase of the quark mass θ′ one expects the CP-symmetry to be violated. One usually assumes that the chiral quark mass phase can be converted to a contribution to the total effective <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\tilde {\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\tilde {\theta }}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcf22bb05d2a5310d819fb1407e56c7204f434dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.015ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {\tilde {\theta }}}"></noscript><span class="lazy-image-placeholder" style="width: 1.015ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcf22bb05d2a5310d819fb1407e56c7204f434dc" data-alt="{\displaystyle \scriptstyle {\tilde {\theta }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  angle, but it remains to be explained why this angle is extremely small instead of being of order one; the particular value of the θ angle that must be very close to zero (in this case) is an example of a <a href="/wiki/Fine-tuning_(physics)" title="Fine-tuning (physics)">fine-tuning problem</a> in physics, and is typically solved by <a href="/wiki/Physics_beyond_the_Standard_Model" title="Physics beyond the Standard Model">physics beyond the Standard Model</a>. There are several proposed solutions to solve the strong CP problem. The most well-known is <a href="/wiki/Peccei%E2%80%93Quinn_theory" title="Peccei–Quinn theory">Peccei–Quinn theory</a>, involving new <a href="/wiki/Scalar_particle" class="mw-redirect" title="Scalar particle">scalar particles</a> called <a href="/wiki/Axion" title="Axion">axions</a>. A newer, more radical approach not requiring the axion is a theory involving <a href="/wiki/Multiple_time_dimensions" title="Multiple time dimensions">two time dimensions</a> first proposed in 1998 by Bars, Deliduman, and Andreev.<a href="#cite_note-29">[29]</a> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"><h2 id="Matter–antimatter_imbalance">Matter–antimatter imbalance</h2> <a role="button" href="/w/index.php?title=CP_violation&action=edit&section=10" title="Edit section: Matter–antimatter imbalance" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-6 collapsible-block" id="mf-section-6"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Baryon_asymmetry" title="Baryon asymmetry">Baryon asymmetry</a> and <a href="/wiki/Baryogenesis" title="Baryogenesis">Baryogenesis</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/T-symmetry" title="T-symmetry">T-symmetry</a>, <a href="/wiki/Arrow_of_time" title="Arrow of time">Arrow of time</a>, and <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1233989161"> <div role="note" aria-labelledby="unsolved-label-physics" class="unsolved"> <div>Unsolved problem in physics:</div> <div class="unsolved-body">Why does the universe have so much more matter than antimatter?</div> <div class="unsolved-more"><a href="/wiki/List_of_unsolved_problems_in_physics" title="List of unsolved problems in physics">(more unsolved problems in physics)</a></div> </div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-text"><div class="mbox-text-span">This section does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a>. Please help <a href="/wiki/Special:EditPage/CP_violation" title="Special:EditPage/CP violation">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>. (November 2020) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> The non-<a href="/wiki/Dark_matter" title="Dark matter">dark matter</a> universe is made chiefly of <a href="/wiki/Matter" title="Matter">matter</a>, rather than consisting of equal parts of matter and <a href="/wiki/Antimatter" title="Antimatter">antimatter</a> as might be expected. It can be demonstrated that, to create an imbalance in matter and antimatter from an initial condition of balance, the <a href="/wiki/Sakharov_conditions" class="mw-redirect" title="Sakharov conditions">Sakharov conditions</a> must be satisfied, one of which is the existence of CP violation during the extreme conditions of the first seconds after the <a href="/wiki/Big_Bang" title="Big Bang">Big Bang</a>. Explanations which do not involve CP violation are less plausible, since they rely on the assumption that the matter–antimatter imbalance was present at the beginning, or on other admittedly exotic assumptions. The Big Bang should have produced equal amounts of matter and antimatter if CP-symmetry was preserved; as such, there should have been total cancellation of both—<a href="/wiki/Protons" class="mw-redirect" title="Protons">protons</a> should have cancelled with <a href="/wiki/Antiproton" title="Antiproton">antiprotons</a>, <a href="/wiki/Electrons" class="mw-redirect" title="Electrons">electrons</a> with <a href="/wiki/Positron" title="Positron">positrons</a>, <a href="/wiki/Neutrons" class="mw-redirect" title="Neutrons">neutrons</a> with <a href="/wiki/Antineutron" title="Antineutron">antineutrons</a>, and so on. This would have resulted in a sea of radiation in the universe with no matter. Since this is not the case, after the Big Bang, physical laws must have acted differently for matter and antimatter, i.e. violating CP-symmetry. The Standard Model contains at least three sources of CP violation. The first of these, involving the <a href="/wiki/Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix" title="Cabibbo–Kobayashi–Maskawa matrix">Cabibbo–Kobayashi–Maskawa matrix</a> in the <a href="/wiki/Quark" title="Quark">quark</a> sector, has been observed experimentally and can only account for a small portion of the CP violation required to explain the matter-antimatter asymmetry. The strong interaction should also violate CP, in principle, but the failure to observe the <a href="/wiki/Neutron_electric_dipole_moment" title="Neutron electric dipole moment">electric dipole moment of the neutron</a> in experiments suggests that any CP violation in the strong sector is also too small to account for the necessary CP violation in the early universe. The third source of CP violation is the <a href="/wiki/Pontecorvo%E2%80%93Maki%E2%80%93Nakagawa%E2%80%93Sakata_matrix" title="Pontecorvo–Maki–Nakagawa–Sakata matrix">Pontecorvo–Maki–Nakagawa–Sakata matrix</a> in the <a href="/wiki/Lepton" title="Lepton">lepton</a> sector. The current long-baseline neutrino oscillation experiments, <a href="/wiki/T2K_experiment" title="T2K experiment">T2K</a> and <a href="/wiki/NO%CE%BDA" class="mw-redirect" title="NOνA">NOνA</a>, may be able to find evidence of CP violation over a small fraction of possible values of the CP violating Dirac phase while the proposed next-generation experiments, <a href="/wiki/Hyper-Kamiokande" title="Hyper-Kamiokande">Hyper-Kamiokande</a> and <a href="/wiki/LBNE" class="mw-redirect" title="LBNE">DUNE</a>, will be sensitive enough to definitively observe CP violation over a relatively large fraction of possible values of the Dirac phase. Further into the future, a <a href="/wiki/Neutrino_factory" class="mw-redirect" title="Neutrino factory">neutrino factory</a> could be sensitive to nearly all possible values of the CP violating Dirac phase. If neutrinos are <a href="/wiki/Majorana_fermion" title="Majorana fermion">Majorana fermions</a>, the <a href="/wiki/Pontecorvo%E2%80%93Maki%E2%80%93Nakagawa%E2%80%93Sakata_matrix" title="Pontecorvo–Maki–Nakagawa–Sakata matrix">PMNS matrix</a> could have two additional CP violating Majorana phases, leading to a fourth source of CP violation within the Standard Model. The experimental evidence for Majorana neutrinos would be the observation of <a href="/wiki/Neutrinoless_double_beta_decay#Neutrinoless_double_beta_decay" title="Neutrinoless double beta decay">neutrinoless double-beta decay</a>. The best limits come from the <a href="/wiki/GERmanium_Detector_Array" class="mw-redirect" title="GERmanium Detector Array">GERDA</a> experiment. CP violation in the lepton sector generates a matter-antimatter asymmetry through a process called <a href="/wiki/Leptogenesis_(physics)" class="mw-redirect" title="Leptogenesis (physics)">leptogenesis</a>. This could become the preferred explanation in the Standard Model for the matter-antimatter asymmetry of the universe if CP violation is experimentally confirmed in the lepton sector. If CP violation in the lepton sector is experimentally determined to be too small to account for matter-antimatter asymmetry, some new <a href="/wiki/Physics_beyond_the_Standard_Model" title="Physics beyond the Standard Model">physics beyond the Standard Model</a> would be required to explain additional sources of CP violation. Adding new particles and/or interactions to the Standard Model generally introduces new sources of CP violation since CP is not a symmetry of nature. Sakharov proposed a way to restore CP-symmetry using T-symmetry, extending spacetime before the Big Bang. He described complete CPT reflections of events on each side of what he called the "initial singularity". Because of this, phenomena with an opposite <a href="/wiki/Arrow_of_time" title="Arrow of time">arrow of time</a> at t < 0 would undergo an opposite CP violation, so the CP-symmetry would be preserved as a whole. The anomalous excess of matter over antimatter after the Big Bang in the orthochronous (or positive) sector, becomes an excess of antimatter before the Big Bang (antichronous or negative sector) as both charge conjugation, parity and arrow of time are reversed due to CPT reflections of all phenomena occurring over the initial singularity: <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote">We can visualize that neutral spinless maximons (or photons) are produced at t < 0 from contracting matter having an excess of antiquarks, that they pass "one through the other" at the instant t = 0 when the density is infinite, and decay with an excess of quarks when t > 0, realizing total CPT symmetry of the universe. All the phenomena at t < 0 are assumed in this hypothesis to be CPT reflections of the phenomena at t > 0.<div class="templatequotecite">— <cite>Andrei Sakharov, in Collected Scientific Works (1982).<a href="#cite_note-Sakharov_book-30">[30]</a></cite></div></blockquote> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"><h2 id="See_also">See also</h2> <a role="button" href="/w/index.php?title=CP_violation&action=edit&section=11" title="Edit section: See also" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-7 collapsible-block" id="mf-section-7"> <ul><li><a href="/wiki/B-factory" title="B-factory">B-factory</a></li> <li><a href="/wiki/Parity_(physics)#Parity_violation" title="Parity (physics)">Parity (physics) § Parity violation</a></li> <li><a href="/wiki/C-symmetry" title="C-symmetry">C-symmetry</a></li> <li><a href="/wiki/T-symmetry" title="T-symmetry">T-symmetry</a></li> <li><a href="/wiki/CPT_symmetry" title="CPT symmetry">CPT symmetry</a></li> <li><a href="/wiki/BTeV_experiment" title="BTeV experiment">BTeV experiment</a></li> <li><a href="/wiki/Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix" title="Cabibbo–Kobayashi–Maskawa matrix">Cabibbo–Kobayashi–Maskawa matrix</a></li> <li><a href="/wiki/LHCb_experiment" title="LHCb experiment">LHCb experiment</a></li> <li><a href="/wiki/Penguin_diagram" title="Penguin diagram">Penguin diagram</a></li> <li><a href="/wiki/Neutral_particle_oscillation" title="Neutral particle oscillation">Neutral particle oscillation</a></li> <li><a href="/wiki/Electron_electric_dipole_moment" title="Electron electric dipole moment">Electron electric dipole moment</a></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"><h2 id="In_popular_culture">In popular culture</h2> <a role="button" href="/w/index.php?title=CP_violation&action=edit&section=12" title="Edit section: In popular culture" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-8 collapsible-block" id="mf-section-8"> <ul><li>The video game <a href="/wiki/Half-Life_2" title="Half-Life 2">Half-Life 2</a> has a song in its soundtrack titled <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=dqlG3E769Tc">CP Violation</a>.</li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(9)"><h2 id="References">References</h2> <a role="button" href="/w/index.php?title=CP_violation&action=edit&section=13" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-9 collapsible-block" id="mf-section-9"> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width reflist-columns-2"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFSchwarzschild1999" class="citation journal cs1">Schwarzschild, Bertram (1999). 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D. (7 December 1982). Collected Scientific Works. <a href="/wiki/Marcel_Dekker" title="Marcel Dekker">Marcel Dekker</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0824717148" title="Special:BookSources/978-0824717148"><bdi>978-0824717148</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Collected+Scientific+Works&rft.pub=Marcel+Dekker&rft.date=1982-12-07&rft.isbn=978-0824717148&rft.aulast=Sakharov&rft.aufirst=A.+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACP+violation" class="Z3988"> </li> </ol></div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(10)"><h2 id="Further_reading">Further reading</h2> <a role="button" href="/w/index.php?title=CP_violation&action=edit&section=14" title="Edit section: Further reading" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-10 collapsible-block" id="mf-section-10"> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSozzi,_M.S.2008" class="citation book cs1">Sozzi, M.S. (2008). Discrete symmetries and CP violation. <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-929666-8" title="Special:BookSources/978-0-19-929666-8"><bdi>978-0-19-929666-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+symmetries+and+CP+violation&rft.pub=Oxford+University+Press&rft.date=2008&rft.isbn=978-0-19-929666-8&rft.au=Sozzi%2C+M.S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACP+violation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFG._C._BrancoL._LavouraJ._P._Silva1999" class="citation book cs1">G. C. Branco; L. Lavoura; J. P. Silva (1999). CP violation. <a href="/wiki/Clarendon_Press" class="mw-redirect" title="Clarendon Press">Clarendon Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-850399-6" title="Special:BookSources/978-0-19-850399-6"><bdi>978-0-19-850399-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=CP+violation&rft.pub=Clarendon+Press&rft.date=1999&rft.isbn=978-0-19-850399-6&rft.au=G.+C.+Branco&rft.au=L.+Lavoura&rft.au=J.+P.+Silva&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACP+violation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFI._BigiA._Sanda1999" class="citation book cs1">I. Bigi; A. Sanda (1999). CP violation. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-44349-4" title="Special:BookSources/978-0-521-44349-4"><bdi>978-0-521-44349-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=CP+violation&rft.pub=Cambridge+University+Press&rft.date=1999&rft.isbn=978-0-521-44349-4&rft.au=I.+Bigi&rft.au=A.+Sanda&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACP+violation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMichael_Beyer2002" class="citation book cs1">Michael Beyer, ed. (2002). CP Violation in Particle, Nuclear and Astrophysics. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-43705-5" title="Special:BookSources/978-3-540-43705-5"><bdi>978-3-540-43705-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=CP+Violation+in+Particle%2C+Nuclear+and+Astrophysics&rft.pub=Springer&rft.date=2002&rft.isbn=978-3-540-43705-5&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACP+violation" class="Z3988"> (A collection of essays introducing the subject, with an emphasis on experimental results.)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFL._Wolfenstein1989" class="citation book cs1">L. Wolfenstein (1989). CP violation. <a href="/wiki/North%E2%80%93Holland_Publishing" class="mw-redirect" title="North–Holland Publishing">North–Holland Publishing</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-88081-9" title="Special:BookSources/978-0-444-88081-9"><bdi>978-0-444-88081-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=CP+violation&rft.pub=North%E2%80%93Holland+Publishing&rft.date=1989&rft.isbn=978-0-444-88081-9&rft.au=L.+Wolfenstein&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACP+violation" class="Z3988"> (A compilation of reprints of numerous important papers on the topic, including papers by T.D. Lee, Cronin, Fitch, Kobayashi and Maskawa, and many others.)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavid_J._Griffiths1987" class="citation book cs1"><a href="/wiki/David_J._Griffiths" title="David J. Griffiths">David J. Griffiths</a> (1987). Introduction to Elementary Particles. <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-60386-3" title="Special:BookSources/978-0-471-60386-3"><bdi>978-0-471-60386-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Elementary+Particles&rft.pub=John+Wiley+%26+Sons&rft.date=1987&rft.isbn=978-0-471-60386-3&rft.au=David+J.+Griffiths&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACP+violation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBigi1998" class="citation journal cs1">Bigi, I. (1998). "CP Violation – An Essential Mystery in Nature's Grand Design". <a href="/w/index.php?title=Surveys_of_High_Energy_Physics&action=edit&redlink=1" class="new" title="Surveys of High Energy Physics (page does not exist)">Surveys of High Energy Physics</a>. 12 (1–4): 269–336. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-ph/9712475">hep-ph/9712475</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1998SHEP...12..269B">1998SHEP...12..269B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F01422419808228861">10.1080/01422419808228861</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Surveys+of+High+Energy+Physics&rft.atitle=CP+Violation+%E2%80%93+An+Essential+Mystery+in+Nature%27s+Grand+Design&rft.volume=12&rft.issue=1%E2%80%934&rft.pages=269-336&rft.date=1998&rft_id=info%3Aarxiv%2Fhep-ph%2F9712475&rft_id=info%3Adoi%2F10.1080%2F01422419808228861&rft_id=info%3Abibcode%2F1998SHEP...12..269B&rft.aulast=Bigi&rft.aufirst=I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACP+violation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMark_Trodden1999" class="citation journal cs1">Mark Trodden (1999). "Electroweak Baryogenesis". <a href="/wiki/Reviews_of_Modern_Physics" title="Reviews of Modern Physics">Reviews of Modern Physics</a>. 71 (5): 1463–1500. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-ph/9803479">hep-ph/9803479</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1999RvMP...71.1463T">1999RvMP...71.1463T</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.71.1463">10.1103/RevModPhys.71.1463</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17275359">17275359</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reviews+of+Modern+Physics&rft.atitle=Electroweak+Baryogenesis&rft.volume=71&rft.issue=5&rft.pages=1463-1500&rft.date=1999&rft_id=info%3Aarxiv%2Fhep-ph%2F9803479&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17275359%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FRevModPhys.71.1463&rft_id=info%3Abibcode%2F1999RvMP...71.1463T&rft.au=Mark+Trodden&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACP+violation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavide_Castelvecchi" class="citation web cs1">Davide Castelvecchi. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140503090147/http://www2.slac.stanford.edu/tip/special/cp.htm">"What is direct CP-violation?"</a>. <a href="/wiki/SLAC" class="mw-redirect" title="SLAC">SLAC</a>. Archived from <a rel="nofollow" class="external text" href="http://www2.slac.stanford.edu/tip/special/cp.htm">the original</a> on 3 May 2014. Retrieved 1 July 2009.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=What+is+direct+CP-violation%3F&rft.pub=SLAC&rft.au=Davide+Castelvecchi&rft_id=http%3A%2F%2Fwww2.slac.stanford.edu%2Ftip%2Fspecial%2Fcp.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACP+violation" class="Z3988"></li> <li>An elementary discussion of parity violation and CP violation is given in chapter 15 of this student level textbook <a rel="nofollow" class="external autonumber" href="https://www.routledge.com/Fundamentals-of-Molecular-Symmetry/Bunker-Jensen/p/book/9780750309417">[1]</a></li></ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(11)"><h2 id="External_links">External links</h2> <a role="button" href="/w/index.php?title=CP_violation&action=edit&section=15" title="Edit section: External links" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-11 collapsible-block" id="mf-section-11"> <ul><li><a rel="nofollow" class="external text" href="http://cerncourier.com/cws/article/cern/28025">Cern Courier article</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output 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.hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div>    </section></div> <noscript><img src="https://login.m.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&mobile=1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=CP_violation&oldid=1258340733">https://en.wikipedia.org/w/index.php?title=CP_violation&oldid=1258340733</a>"</div></div> </div> <div class="post-content" id="page-secondary-actions"> </div> </main> <footer class="mw-footer minerva-footer" role="contentinfo"> <a class="last-modified-bar" href="/w/index.php?title=CP_violation&action=history"> <div class="post-content last-modified-bar__content"> Last edited on 19 November 2024, at 04:23 </div> </a> <div class="post-content footer-content"> <div id='mw-data-after-content'> <div class="read-more-container"></div> </div> <div id="p-lang"> <h4>Languages</h4> <section> <ul id="p-variants" class="minerva-languages"></ul> <ul class="minerva-languages"><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AE%D8%B1%D9%82_%D8%AA%D9%86%D8%A7%D8%B8%D8%B1_%D8%A7%D9%84%D8%B4%D8%AD%D9%86%D8%A9_%D9%88%D8%A7%D9%84%D8%B3%D9%88%D9%8A%D8%A9" title="خرق تناظر الشحنة والسوية – Arabic" lang="ar" hreflang="ar" data-title="خرق تناظر الشحنة والسوية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D1%83%D1%88%D1%8D%D0%BD%D0%BD%D0%B5_CP-%D1%96%D0%BD%D0%B2%D0%B0%D1%80%D1%8B%D1%8F%D0%BD%D1%82%D0%BD%D0%B0%D1%81%D1%86%D1%96" title="Парушэнне CP-інварыянтнасці – Belarusian" lang="be" hreflang="be" data-title="Парушэнне CP-інварыянтнасці" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Violaci%C3%B3_CP" title="Violació CP – Catalan" lang="ca" hreflang="ca" data-title="Violació CP" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Naru%C5%A1en%C3%AD_CP-symetrie" title="Narušení CP-symetrie – Czech" lang="cs" hreflang="cs" data-title="Narušení CP-symetrie" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/CP-Verletzung" title="CP-Verletzung – German" lang="de" hreflang="de" data-title="CP-Verletzung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Violaci%C3%B3n_CP" title="Violación CP – Spanish" lang="es" hreflang="es" data-title="Violación CP" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D9%82%D8%B6_%D8%B3%DB%8C%E2%80%8C%D9%BE%DB%8C" title="نقض سی‌پی – Persian" lang="fa" hreflang="fa" data-title="نقض سی‌پی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Violation_de_CP" title="Violation de CP – French" lang="fr" hreflang="fr" data-title="Violation de CP" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/S%C3%A1r%C3%BA_comhchuingeacht_luchta_is_paireachta" title="Sárú comhchuingeacht luchta is paireachta – Irish" lang="ga" hreflang="ga" data-title="Sárú comhchuingeacht luchta is paireachta" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/CP_%EC%9C%84%EB%B0%98" title="CP 위반 – Korean" lang="ko" hreflang="ko" data-title="CP 위반" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/CP-simetrija" title="CP-simetrija – Croatian" lang="hr" hreflang="hr" data-title="CP-simetrija" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A9%D7%91%D7%99%D7%A8%D7%AA_%D7%A1%D7%99%D7%9E%D7%98%D7%A8%D7%99%D7%99%D7%AA_CP" title="שבירת סימטריית CP – Hebrew" lang="he" hreflang="he" data-title="שבירת סימטריית CP" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/CP%E5%AF%BE%E7%A7%B0%E6%80%A7%E3%81%AE%E7%A0%B4%E3%82%8C" title="CP対称性の破れ – Japanese" lang="ja" hreflang="ja" data-title="CP対称性の破れ" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/CP_violation" title="CP violation – Uzbek" lang="uz" hreflang="uz" data-title="CP violation" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/CP_%E0%A8%89%E0%A8%B2%E0%A9%B0%E0%A8%98%E0%A8%A3%E0%A8%BE" title="CP ਉਲੰਘਣਾ – Punjabi" lang="pa" hreflang="pa" data-title="CP ਉਲੰਘਣਾ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Viola%C3%A7%C3%A3o_de_CP" title="Violação de CP – Portuguese" lang="pt" hreflang="pt" data-title="Violação de CP" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9D%D0%B0%D1%80%D1%83%D1%88%D0%B5%D0%BD%D0%B8%D0%B5_CP-%D0%B8%D0%BD%D0%B2%D0%B0%D1%80%D0%B8%D0%B0%D0%BD%D1%82%D0%BD%D0%BE%D1%81%D1%82%D0%B8" title="Нарушение CP-инвариантности – Russian" lang="ru" hreflang="ru" data-title="Нарушение CP-инвариантности" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/CP_violation" title="CP violation – Simple English" lang="en-simple" hreflang="en-simple" data-title="CP violation" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kr%C5%A1itev_simetrije_CP" title="Kršitev simetrije CP – Slovenian" lang="sl" hreflang="sl" data-title="Kršitev simetrije CP" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/CP-rikko" title="CP-rikko – Finnish" lang="fi" hreflang="fi" data-title="CP-rikko" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/CP-brott" title="CP-brott – Swedish" lang="sv" hreflang="sv" data-title="CP-brott" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/CP_ihlali" title="CP ihlali – Turkish" lang="tr" hreflang="tr" data-title="CP ihlali" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%BE%D1%80%D1%83%D1%88%D0%B5%D0%BD%D0%BD%D1%8F_CP-%D1%96%D0%BD%D0%B2%D0%B0%D1%80%D1%96%D0%B0%D0%BD%D1%82%D0%BD%D0%BE%D1%81%D1%82%D1%96" title="Порушення CP-інваріантності – Ukrainian" lang="uk" hreflang="uk" data-title="Порушення CP-інваріантності" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/CP%E7%A0%B4%E5%A3%9E" title="CP破壞 – Chinese" lang="zh" hreflang="zh" data-title="CP破壞" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li></ul> </section> </div> <div class="minerva-footer-logo"><img src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" alt="Wikipedia" width="120" height="18" style="width: 7.5em; height: 1.125em;"/> </div> <ul id="footer-info" class="footer-info hlist hlist-separated"> <li id="footer-info-lastmod"> This page was last edited on 19 November 2024, at 04:23 (UTC).</li> <li id="footer-info-copyright">Content is available under <a class="external" rel="nofollow" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">CC BY-SA 4.0</a> unless otherwise noted.</li> </ul> <ul id="footer-places" class="footer-places hlist hlist-separated"> <li 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CP violation - Wikipedia