SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>CP violation</title>

</head>

CP violation

</h1>

</header>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1126788409">

/* start https://en.wikipedia.org/ */

.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1246091330">

/* start https://en.wikipedia.org/ */

.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><th class="sidebar-title"><a href="Physics_beyond_the_Standard_Model" title="Physics beyond the Standard Model">Beyond the Standard Model</a></th></tr><tr><td class="sidebar-image"><div class="sidebar-caption">Simulated <a href="Large_Hadron_Collider" title="Large Hadron Collider">Large Hadron Collider</a> <a href="Compact_Muon_Solenoid" title="Compact Muon Solenoid">CMS</a> particle detector data depicting a <a href="Higgs_boson" title="Higgs boson">Higgs boson</a> produced by colliding protons decaying into hadron jets and electrons</div></td></tr><tr><th class="sidebar-heading">

<a href="Standard_Model" title="Standard Model">Standard Model</a></th></tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible hlist"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;color: var(--color-base)"><div class="sidebar-list-title-c">Evidence</div></div><div class="sidebar-list-content mw-collapsible-content plainlist">

<ul><li><a href="Hierarchy_problem" title="Hierarchy problem">Hierarchy problem</a></li>

<li><a href="Dark_matter" title="Dark matter">Dark matter</a></li>

<li><a href="Dark_energy" title="Dark energy">Dark energy</a></li>

<li><a href="Quintessence_(physics)" title="Quintessence (physics)">Quintessence</a></li>

<li><a href="Phantom_energy" class="mw-redirect" title="Phantom energy">Phantom energy</a></li>

<li><a href="Dark_radiation" title="Dark radiation">Dark radiation</a></li>

<li><a href="Dark_photon" title="Dark photon">Dark photon</a></li>

<li><a href="Cosmological_constant_problem" title="Cosmological constant problem">Cosmological constant problem</a></li>

<li><a href="Neutrino_oscillation" title="Neutrino oscillation">Neutrino oscillation</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed hlist"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;color: var(--color-base)"><div class="sidebar-list-title-c">Theories</div></div><div class="sidebar-list-content mw-collapsible-content plainlist">

<ul><li><a href="Brans%E2%80%93Dicke_theory" title="Brans–Dicke theory">Brans–Dicke theory</a></li>

<li><a href="Cosmic_censorship_hypothesis" title="Cosmic censorship hypothesis">Cosmic censorship hypothesis</a></li>

<li><a href="Fifth_force" title="Fifth force">Fifth force</a></li>

<li><a href="F-theory" title="F-theory">F-theory</a></li>

<li><a href="Theory_of_everything" title="Theory of everything">Theory of everything</a></li>

<li><a href="Unified_field_theory" title="Unified field theory">Unified field theory</a></li>

<li><a href="Grand_Unified_Theory" title="Grand Unified Theory">Grand Unified Theory</a></li>

<li><a href="Technicolor_(physics)" title="Technicolor (physics)">Technicolor</a></li>

<li><a href="Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a></li>

<li><a href="6D_(2%2C0)_superconformal_field_theory" title="6D (2,0) superconformal field theory">6D (2,0) superconformal field theory</a></li>

<li><a href="Noncommutative_quantum_field_theory" title="Noncommutative quantum field theory">Noncommutative quantum field theory</a></li>

<li><a href="Quantum_cosmology" title="Quantum cosmology">Quantum cosmology</a></li>

<li><a href="Brane_cosmology" title="Brane cosmology">Brane cosmology</a></li>

<li><a href="String_theory" title="String theory">String theory</a></li>

<li><a href="Superstring_theory" title="Superstring theory">Superstring theory</a></li>

<li><a href="M-theory" title="M-theory">M-theory</a></li>

<li><a href="Mathematical_universe_hypothesis" title="Mathematical universe hypothesis">Mathematical universe hypothesis</a></li>

<li><a href="Mirror_matter" title="Mirror matter">Mirror matter</a></li>

<li><a href="Randall%E2%80%93Sundrum_model" title="Randall–Sundrum model">Randall–Sundrum model</a></li>

<li><a href="N_%3D_4_supersymmetric_Yang%E2%80%93Mills_theory" title="N = 4 supersymmetric Yang–Mills theory">N = 4 supersymmetric Yang–Mills theory</a></li>

<li><a href="Twistor_string_theory" title="Twistor string theory">Twistor string theory</a></li>

<li><a href="Dark_fluid" title="Dark fluid">Dark fluid</a></li>

<li><a href="Doubly_special_relativity" title="Doubly special relativity">Doubly special relativity</a></li>

<li><a href="De_Sitter_invariant_special_relativity" title="De Sitter invariant special relativity">de Sitter invariant special relativity</a></li>

<li><a href="Causal_fermion_system" class="mw-redirect" title="Causal fermion system">Causal fermion systems</a></li>

<li><a href="Black_hole_thermodynamics" title="Black hole thermodynamics">Black hole thermodynamics</a></li>

<li><a href="Unparticle_physics" title="Unparticle physics">Unparticle physics</a></li>

<li><a href="Graviphoton" title="Graviphoton">Graviphoton</a></li>

<li><a href="Graviscalar" title="Graviscalar">Graviscalar</a></li>

<li><a href="Graviton" title="Graviton">Graviton</a></li>

<li><a href="Gravitino" title="Gravitino">Gravitino</a></li>

<li><a href="Massive_gravity" title="Massive gravity">Massive gravity</a></li>

<li><a href="Gauge_gravitation_theory" title="Gauge gravitation theory">Gauge gravitation theory</a></li>

<li><a href="Gauge_theory_gravity" title="Gauge theory gravity">Gauge theory gravity</a></li>

<li><a href="CPT_symmetry" title="CPT symmetry">CPT symmetry</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed hlist"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Supersymmetry" title="Supersymmetry">Supersymmetry</a></div></div><div class="sidebar-list-content mw-collapsible-content plainlist">

<li><a href="Next-to-Minimal_Supersymmetric_Standard_Model" title="Next-to-Minimal Supersymmetric Standard Model">NMSSM</a></li>

<li><a href="Superstring_theory" title="Superstring theory">Superstring theory</a></li>

<li><a href="M-theory" title="M-theory">M-theory</a></li>

<li><a href="Supergravity" title="Supergravity">Supergravity</a></li>

<li><a href="Supersymmetry_breaking" title="Supersymmetry breaking">Supersymmetry breaking</a></li>

<li><a href="Extra_dimensions" title="Extra dimensions">Extra dimensions</a></li>

<li><a href="Large_extra_dimensions" title="Large extra dimensions">Large extra dimensions</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed hlist"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Quantum_gravity" title="Quantum gravity">Quantum gravity</a></div></div><div class="sidebar-list-content mw-collapsible-content plainlist">

<ul><li><a href="False_vacuum" title="False vacuum">False vacuum</a></li>

<li><a href="String_theory" title="String theory">String theory</a></li>

<li><a href="Quantum_foam" title="Quantum foam">Quantum foam</a></li>

<li><a href="Quantum_geometry" title="Quantum geometry">Quantum geometry</a></li>

<li><a href="Loop_quantum_gravity" title="Loop quantum gravity">Loop quantum gravity</a></li>

<li><a href="Quantum_cosmology" title="Quantum cosmology">Quantum cosmology</a></li>

<li><a href="Loop_quantum_cosmology" title="Loop quantum cosmology">Loop quantum cosmology</a></li>

<li><a href="Causal_dynamical_triangulation" title="Causal dynamical triangulation">Causal dynamical triangulation</a></li>

<li><a href="Causal_fermion_systems" title="Causal fermion systems">Causal fermion systems</a></li>

<li><a href="Causal_sets" title="Causal sets">Causal sets</a></li>

<li><a href="Canonical_quantum_gravity" title="Canonical quantum gravity">Canonical quantum gravity</a></li>

<li><a href="Semiclassical_gravity" title="Semiclassical gravity">Semiclassical gravity</a></li>

<li><a href="Superfluid_vacuum_theory" title="Superfluid vacuum theory">Superfluid vacuum theory</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed hlist"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;color: var(--color-base)"><div class="sidebar-list-title-c">Experiments</div></div><div class="sidebar-list-content mw-collapsible-content plainlist">

<ul><li><a href="Accelerator_Neutrino_Neutron_Interaction_Experiment" title="Accelerator Neutrino Neutron Interaction Experiment">ANNIE</a></li>

<li><a href="Laboratori_Nazionali_del_Gran_Sasso" title="Laboratori Nazionali del Gran Sasso">Gran Sasso</a></li>

<li><a href="Super-Kamiokande" title="Super-Kamiokande">Super-K</a></li>

<li><a href="Tevatron" title="Tevatron">Tevatron</a></li>

</tr><tr><td class="sidebar-navbar"><style data-mw-deduplicate="TemplateStyles:r1239400231">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}

/* end https://en.wikipedia.org/ */

</style></td></tr></tbody></table>

In <a href="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="C-symmetry" title="C-symmetry">C-symmetry</a> (<a href="Charge_(physics)" title="Charge (physics)">charge conjugation</a> symmetry) and <a href="Parity_(physics)" title="Parity (physics)">P-symmetry</a> (<a href="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 <a href="Antiparticle" title="Antiparticle">antiparticle</a> (C-symmetry) while its spatial coordinates are inverted ("mirror" or P-symmetry).

CP violation is only observed in the <a href="Weak_interaction" title="Weak interaction">weak interaction</a>. The discovery of CP violation in 1964 in the decays of neutral <a href="Kaon" title="Kaon">kaons</a> resulted in the <a href="Nobel_Prize_in_Physics" title="Nobel Prize in Physics">Nobel Prize in Physics</a> in 1980 for its discoverers <a href="James_Cronin" title="James Cronin">James Cronin</a> and <a href="Val_Fitch" class="mw-redirect" title="Val Fitch">Val Fitch</a>. CP violation was subsequently discovered in many other <a href="Meson" title="Meson">meson</a> decays. In 2025, the <a href="LHCb_experiment" title="LHCb experiment">LHCb experiment</a> discovered CP violation in baryons.<a href="#cite_note-:1-1">[1]</a> There is some evidence CP violation may occur in neutrino interactions.<a href="#cite_note-:2-2">[2]</a>

It is important to the <a href="Matter-antimatter_asymmetry" class="mw-redirect" title="Matter-antimatter asymmetry">matter-antimatter asymmetry</a> problem, the <a href="Strong_CP_problem" title="Strong CP problem">strong CP problem</a>, and in the study of weak interactions in particle physics. Under the <a href="CPT_Theorem" class="mw-redirect" title="CPT Theorem">CPT theorem</a>, every CP violation is also a time-symmetry violation.

<div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2></div>

Until the 1950s, parity conservation was believed to be one of the fundamental geometric <a href="Conservation_laws" class="mw-redirect" title="Conservation laws">conservation laws</a> (along with <a href="Conservation_of_energy" title="Conservation of energy">conservation of energy</a> and <a href="Conservation_of_momentum" class="mw-redirect" title="Conservation of momentum">conservation of momentum</a>). After the discovery of <a href="Parity_(physics)#Parity_violation" title="Parity (physics)">parity violation</a> in 1956, CP-symmetry was proposed to restore order. However, while the <a href="Strong_interaction" title="Strong interaction">strong interaction</a> and <a href="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="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="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="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="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="CPLEAR_experiment" title="CPLEAR experiment">CPLEAR</a> and KTeV collaborations, at <a href="CERN" title="CERN">CERN</a> and <a href="Fermilab" title="Fermilab">Fermilab</a>, respectively.<a href="#cite_note-3">[3]</a> As early as 1970, Klaus Schubert observed T violation independent of assuming CPT symmetry by using the Bell–Steinberger unitarity relation.<a href="#cite_note-4">[4]</a>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<div class="mw-heading mw-heading3"><h3 id="P-symmetry">P-symmetry</h3></div>

The idea behind <a href="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="Chemical_reaction" title="Chemical reaction">chemical reaction</a> or <a href="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="Tsung-Dao_Lee" title="Tsung-Dao Lee">Tsung-Dao Lee</a> and <a href="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-5">[5]</a> They proposed several possible direct experimental tests.

The first test based on <a href="Beta_decay" title="Beta decay">beta decay</a> of <a href="Cobalt-60" title="Cobalt-60">cobalt-60</a> nuclei was carried out in 1956 by a group led by <a href="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-6">[6]</a> However, <a href="Parity_(physics)" title="Parity (physics)">parity</a> symmetry still appears to be valid for all reactions involving <a href="Electromagnetism" title="Electromagnetism">electromagnetism</a> and <a href="Strong_interaction" title="Strong interaction">strong interactions</a>.

<div class="mw-heading mw-heading3"><h3 id="CP-symmetry">CP-symmetry</h3></div>

Overall, the symmetry of a <a href="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="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="Antiparticle" title="Antiparticle">antiparticle</a>, was the suitable symmetry to restore order.

In 1956 <a href="Reinhard_Oehme" title="Reinhard Oehme">Reinhard Oehme</a> in a letter to Chen-Ning Yang and shortly after, Boris L. Ioffe, <a href="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-7">[7]</a>

Charge violation was confirmed in the <a href="Wu_experiment" title="Wu experiment">Wu experiment</a> and in experiments performed by <a href="Valentine_Telegdi" title="Valentine Telegdi">Valentine Telegdi</a> and <a href="Jerome_Isaac_Friedman" title="Jerome Isaac Friedman">Jerome Friedman</a> and <a href="Richard_Garwin" title="Richard Garwin">Garwin</a> and <a href="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="John_Riley_Holt" title="John Riley Holt">John Riley Holt</a> at the <a href="University_of_Liverpool" title="University of Liverpool">University of Liverpool</a>.<a href="#cite_note-8">[8]</a><a href="#cite_note-9">[9]</a><a href="#cite_note-10">[10]</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-7">[7]</a><a href="#cite_note-11">[11]</a>

<a href="Lev_Landau" title="Lev Landau">Lev Landau</a> proposed in 1957 CP-symmetry,<a href="#cite_note-12">[12]</a> often called just CP, as the true symmetry between matter and antimatter. CP-symmetry is the product of two <a href="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="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="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-13">[13]</a>

<div class="mw-heading mw-heading2"><h2 id="Experimental_status">Experimental status</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Indirect_CP_violation">Indirect CP violation</h3></div>

In 1964, <a href="James_Cronin" title="James Cronin">James Cronin</a>, <a href="Val_Fitch" class="mw-redirect" title="Val Fitch">Val Fitch</a> and coworkers provided clear evidence from <a href="Kaon" title="Kaon">kaon</a> decay that CP-symmetry could be broken.<a href="#cite_note-FC1964-14">[14]</a> (cf. also Ref. <a href="#cite_note-FCE-15">[15]</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="Kaon" title="Kaon">kaons</a> can transform into their <a href="Antiparticle" title="Antiparticle">antiparticles</a> (in which each <a href="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></div>

Despite many searches, no other manifestation of CP violation was discovered until the 1990s, when the <a href="NA31_experiment" title="NA31 experiment">NA31 experiment</a> at <a href="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="Fermilab" title="Fermilab">Fermilab</a><a href="#cite_note-16">[16]</a> and the <a href="NA48_experiment" title="NA48 experiment">NA48 experiment</a> at <a href="CERN" title="CERN">CERN</a>.<a href="#cite_note-NA48-17">[17]</a>

Starting in 2001, a new generation of experiments, including the <a href="BaBar_experiment" title="BaBar experiment">BaBar experiment</a> at the Stanford Linear Accelerator Center (<a href="SLAC" class="mw-redirect" title="SLAC">SLAC</a>)<a href="#cite_note-18">[18]</a> and the <a href="Belle_Experiment" class="mw-redirect" title="Belle Experiment">Belle Experiment</a> at the High Energy Accelerator Research Organisation (<a href="KEK" title="KEK">KEK</a>)<a href="#cite_note-19">[19]</a> in Japan, observed direct CP violation in a different system, namely in decays of the <a href="B_meson" title="B meson">B mesons</a>.<a href="#cite_note-20">[20]</a> A large number of CP violation processes in <a href="B_meson" title="B meson">B meson</a> decays have now been discovered. Before these "<a href="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="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="D_meson" title="D meson">D mesons</a> was reported by the <a href="LHCb" class="mw-redirect" title="LHCb">LHCb</a> experiment at <a href="CERN" title="CERN">CERN</a> using 0.6 fb−1 of Run 1 data.<a href="#cite_note-21">[21]</a> However, the same measurement using the full 3.0 fb−1 Run 1 sample was consistent with CP-symmetry.<a href="#cite_note-22">[22]</a>

In 2013 LHCb announced discovery of CP violation in <a href="Strange_B_meson" title="Strange B meson">strange B meson</a> decays.<a href="#cite_note-23">[23]</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}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D^{0}}</annotation>

</semantics>

</math><img src="./d61f2a204347d41a6b1c1b6d6948354e1e3acb74.svg" 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}}" loading="lazy"> decays with a deviation from zero of 5.3 standard deviations.<a href="#cite_note-24">[24]</a>

In 2020, the <a href="T2K_experiment" title="T2K experiment">T2K Collaboration</a> reported some indications of CP violation in leptons for the first time.<a href="#cite_note-25">[25]</a>

In this experiment, beams of muon neutrinos (ν μ) and muon antineutrinos (ν μ) were alternately produced by an <a href="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="NOvA" title="NOvA">NOvA</a> sees no evidence of CP violation in neutrino oscillations<a href="#cite_note-26">[26]</a> and is in slight tension with T2K.<a href="#cite_note-27">[27]</a><a href="#cite_note-:2-2">[2]</a>

In May 2024, a team of theoretical physicists at <a href="Brown_University" title="Brown University">Brown University</a> determined the third <a href="Half-life" title="Half-life">half-life</a> asymmetry of hydrogen-based <a href="Quark" title="Quark">quark</a> <a href="Valve_amplifier" title="Valve amplifier">valve amplifiers</a> indicated potential CP violation.<a href="#cite_note-28">[28]</a>

In March 2025, LHCb announced discovery of CP violation in baryon decays with a deviation from zero of 5.2 standard deviations, specifically the <a href="Bottom_lambda_baryon" class="mw-redirect" title="Bottom lambda baryon">bottom lambda baryon</a>.<a href="#cite_note-:1-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="CP_violation_in_the_Standard_Model">CP violation in the Standard Model</h2></div>

"Direct" CP violation is allowed in the <a href="Standard_Model" title="Standard Model">Standard Model</a> if a <a href="Complex_number" title="Complex number">complex</a> phase appears in the <a href="Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix" title="Cabibbo–Kobayashi–Maskawa matrix">Cabibbo–Kobayashi–Maskawa matrix</a> (CKM matrix) describing <a href="Quark" title="Quark">quark</a> mixing, or the <a href="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="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="CKM_matrix" 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="Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix#The_unitarity_triangles" title="Cabibbo–Kobayashi–Maskawa matrix"> Jarlskog invariant</a>:

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</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>

for quarks, which is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ 0.0003\ }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ 0.0003\ }</annotation>

</semantics>

</math><img src="./f0a3173bde03ac8c2f969d86371790ff5d37a7f7.svg" 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\ }" loading="lazy"> 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\ .}">

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ J_{\max }={\tfrac {1}{6{\sqrt {3}}}}\ \approx \ 0.1\ .}</annotation>

</semantics>

</math><img src="./ded197361031ca6127fc3a79a195328454f99903.svg" 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\ .}" loading="lazy"> For leptons, only an upper limit exists: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ |J|<0.03\ .}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ |J|<0.03\ .}</annotation>

</semantics>

</math><img src="./31c6dd27e7df7904d862f2aaf23341ad3acb1db0.svg" 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\ .}" loading="lazy">

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\ }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ a\ }</annotation>

</semantics>

</math><img src="./8124de742ae987fe73be9ca9d3d4ba8586e28b11.svg" 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\ }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ b\ ,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ b\ ,}</annotation>

</semantics>

</math><img src="./379cb0c291a41721eace03e50997abebfb25db4f.svg" 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\ ,}" loading="lazy"> and their antiparticles <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\bar {a}}\ }">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ {\bar {a}}\ }</annotation>

</semantics>

</math><img src="./39dbc866826677d4893bc2905ca740db0f0f2564.svg" 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}}\ }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\bar {b}}\ .}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ {\bar {b}}\ .}</annotation>

</semantics>

</math><img src="./a4e677a819717d0ede33e2763b407ab45df6ab55.svg" 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}}\ .}" loading="lazy"> Now consider the processes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ a\rightarrow b\ }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ a\rightarrow b\ }</annotation>

</semantics>

</math><img src="./9f71cd6c1be827346794a8d43bd15b5fbb687fec.svg" 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\ }" loading="lazy"> and the corresponding antiparticle process <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\bar {a}}\rightarrow {\bar {b}}\ ,}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ {\bar {a}}\rightarrow {\bar {b}}\ ,}</annotation>

</semantics>

</math><img src="./8353ef8a72b583b1da92353b97f5b676325882bb.svg" 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}}\ ,}" loading="lazy"> and denote their amplitudes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cal {M}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\cal {M}}}</annotation>

</semantics>

</math><img src="./88cb0b27f4cd40f230c8c50dcbd444f52a2720bd.svg" 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}}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\cal {M}}}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {\cal {M}}}}</annotation>

</semantics>

</math><img src="./3986848ea4ea50383edc67ef7464d0c4d8a9f9a2.svg" 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}}}}" loading="lazy"> 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 }}">

</mrow>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\cal {M}}=|{\cal {M}}|\ e^{i\theta }}</annotation>

</semantics>

</math><img src="./e5c17d4c49eb90775c0ae0bdcf28592adb5ee088.svg" 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 }}" loading="lazy">. 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 }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{i\phi }}</annotation>

</semantics>

</math><img src="./05c817052d719f5b21dd162c425e514232d14af6.svg" 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 }}" loading="lazy">. Note that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\cal {M}}}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {\cal {M}}}}</annotation>

</semantics>

</math><img src="./3986848ea4ea50383edc67ef7464d0c4d8a9f9a2.svg" 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}}}}" loading="lazy"> contains the conjugate matrix to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cal {M}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\cal {M}}}</annotation>

</semantics>

</math><img src="./88cb0b27f4cd40f230c8c50dcbd444f52a2720bd.svg" 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}}}" loading="lazy">, so it picks up a phase term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-i\phi }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{-i\phi }}</annotation>

</semantics>

</math><img src="./281aedb891b74c1f149bd593a3e1b69d6c12fcf7.svg" 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 }}" loading="lazy">.

Now the formula becomes:

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<mover>

</mrow>

</mover>

</mrow>

</mtd>

<mtd>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</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>

Physically measurable reaction rates are proportional to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ |{\cal {M}}|^{2}}">

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ |{\cal {M}}|^{2}}</annotation>

</semantics>

</math><img src="./d07bc58b90ca59b0e0a2f4b2eb0b3ab005ef66dc.svg" 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}}" loading="lazy">, 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}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a{\overset {1}{\longrightarrow }}b}</annotation>

</semantics>

</math><img src="./cb93fa4217b1e050913d7298acab069f1576aa8b.svg" 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}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a{\overset {2}{\longrightarrow }}b}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a{\overset {2}{\longrightarrow }}b}</annotation>

</semantics>

</math><img src="./bc2c777735709e8a9c4c7689d986c00b1a3ffe8f.svg" 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}" loading="lazy"> or equivalently, two unrelated intermediate states: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\rightarrow 1\rightarrow b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\rightarrow 1\rightarrow b}</annotation>

</semantics>

</math><img src="./2f90704b562c252ef276bceb73bf3b7f3562082f.svg" 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}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\rightarrow 2\rightarrow b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\rightarrow 2\rightarrow b}</annotation>

</semantics>

</math><img src="./e5f841aef4eaea3f00e67eefb63f91736b476867.svg" 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}" loading="lazy">. 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:

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mtd>

<mtd>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<mover>

</mrow>

</mover>

</mrow>

</mtd>

<mtd>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mtd>

<mtd>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</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>

Some further calculation gives:

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<mover>

</mrow>

</mover>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</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>

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}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ V_{\mathrm {CKM} }=U_{u}^{\dagger }U_{d}}</annotation>

</semantics>

</math><img src="./391f664a43b206a9d2007d3196de181bf68b3453.svg" 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}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{u}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U_{u}}</annotation>

</semantics>

</math><img src="./423ac752d3d2acb56be36948e40f271119e4edea.svg" 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}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{d}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U_{d}}</annotation>

</semantics>

</math><img src="./a321d741a59ba8149c5d0077f222ef81ce44916c.svg" 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}}" loading="lazy"> are unitary transformation matrices which diagonalize the fermion mass matrices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{u}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{u}}</annotation>

</semantics>

</math><img src="./a72cd86dcb2d52a1e3d92d3c5789978abc7f1f4c.svg" 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}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{d}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{d}}</annotation>

</semantics>

</math><img src="./dc690f2dccbe052928df1f8fb48c15914f5017f1.svg" 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}}" loading="lazy">, 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}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U_{u}}</annotation>

</semantics>

</math><img src="./423ac752d3d2acb56be36948e40f271119e4edea.svg" 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}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{d}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U_{d}}</annotation>

</semantics>

</math><img src="./a321d741a59ba8149c5d0077f222ef81ce44916c.svg" 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}}" loading="lazy"> 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}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U_{u}}</annotation>

</semantics>

</math><img src="./423ac752d3d2acb56be36948e40f271119e4edea.svg" 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}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{d}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U_{d}}</annotation>

</semantics>

</math><img src="./a321d741a59ba8149c5d0077f222ef81ce44916c.svg" 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}}" loading="lazy"> must be different, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{u}\neq U_{d}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U_{u}\neq U_{d}}</annotation>

</semantics>

</math><img src="./7e121e68e3a6fffb3c44afab6add03027567b2e9.svg" 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}}" loading="lazy">, 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

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</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><img src="./39b7854c33624a55032587482a9e326bcb3635d8.svg" 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}}.}" loading="lazy">

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 }}">

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {M^{2}} =M\cdot M^{\dagger }}</annotation>

</semantics>

</math><img src="./6bf33249e37cb16a48feefef5669a24f99e29122.svg" 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 }}" loading="lazy"> can be given by

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</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><img src="./fe7089e63f5c403b491707d95dcc1425edc3683d.svg" 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}},}" loading="lazy">

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}} }">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {M^{2}} }</annotation>

</semantics>

</math><img src="./7b96857f6b0499821f21ce6d8f9e7ccf0707e5f9.svg" 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}} }" loading="lazy"> are correlated to those in M directly in the ways shown below

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</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><img src="./1f40925d8b6b64d4e4ffb265926d6fd0e28f3cb5.svg" 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}}}" loading="lazy">

That means if we diagonalize an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M^{2}} }">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {M^{2}} }</annotation>

</semantics>

</math><img src="./7b96857f6b0499821f21ce6d8f9e7ccf0707e5f9.svg" 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}} }" loading="lazy"> matrix with 9 parameters, it has the same effect as diagonalizing an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" 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}" loading="lazy"> matrix with 18 parameters. Therefore, diagonalizing the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M^{2}} }">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {M^{2}} }</annotation>

</semantics>

</math><img src="./7b96857f6b0499821f21ce6d8f9e7ccf0707e5f9.svg" 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}} }" loading="lazy"> matrix is certainly the most reasonable choice.

The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" 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}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M^{2}} }">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {M^{2}} }</annotation>

</semantics>

</math><img src="./7b96857f6b0499821f21ce6d8f9e7ccf0707e5f9.svg" 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}} }" loading="lazy"> 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}} }">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {M^{2}} }</annotation>

</semantics>

</math><img src="./7b96857f6b0499821f21ce6d8f9e7ccf0707e5f9.svg" 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}} }" loading="lazy"> matrix has only 9 parameters, it is still too complicated to be diagonalized directly. Thus, an assumption

<msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msub>

<msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msub>

<msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msub>

<msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msub>

</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><img src="./facd3b3e059b2afd937466164e5b0402b256e8e4.svg" 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}" loading="lazy">

was employed to simplify the pattern, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M^{2}} _{R}}">

<msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {M^{2}} _{R}}</annotation>

</semantics>

</math><img src="./85af96bed6b72f694b719c013d9c142b1e00346c.svg" 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}}" loading="lazy"> is the real part of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M^{2}} }">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {M^{2}} }</annotation>

</semantics>

</math><img src="./7b96857f6b0499821f21ce6d8f9e7ccf0707e5f9.svg" 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}} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M^{2}} _{I}}">

<msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {M^{2}} _{I}}</annotation>

</semantics>

</math><img src="./70fb7426304a5824c606fa8bf4b46dab5e94bc74.svg" 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}}" loading="lazy"> 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}} }">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {M^{2}} }</annotation>

</semantics>

</math><img src="./7b96857f6b0499821f21ce6d8f9e7ccf0707e5f9.svg" 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}} }" loading="lazy"> matrix can be given by

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<mtr>

<mtd>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

</mrow>

</mfrac>

</mrow>

</mtd>

<mtd>

<mfrac>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

</mrow>

</mfrac>

</mrow>

</mtd>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

</mrow>

</mfrac>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msub>

<msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msub>

</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><img src="./0c698e0a0e06d4af928ff5f85b65a1fa542e2e1f.svg" 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},}" loading="lazy">

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}} ,}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</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><img src="./0ae0e50ac680e93670a2cdf07f52c1cb1f573ce6.svg" 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}} ,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\equiv \mathbf {B_{1}/B_{3}} }">

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y\equiv \mathbf {B_{1}/B_{3}} }</annotation>

</semantics>

</math><img src="./ce6b87d25b6828f3f4a054cc66fb2dc9cd397aff.svg" 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}} }" loading="lazy">.

Diagonalizing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M^{2}} }">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {M^{2}} }</annotation>

</semantics>

</math><img src="./7b96857f6b0499821f21ce6d8f9e7ccf0707e5f9.svg" 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}} }" loading="lazy"> analytically, the eigenvalues are given by

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</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><img src="./62a242639cf72a0dc67d716c0b2544c4b5ccc072.svg" 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},}" loading="lazy">

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</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><img src="./2b735c97066d3b97ec5be0eed368dadb0dd495ac.svg" 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},}" loading="lazy">

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</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><img src="./1204abef33cfeb4373f5afb8a30da300e1a683f5.svg" 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},}" loading="lazy">

and the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" 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}" loading="lazy"> matrix for up-type quarks can then be given by

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<mfrac>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mtd>

<mtd>

<mfrac>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mtd>

<mtd>

<mfrac>

<mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mtd>

<mtd>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mtd>

<mtd>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mtd>

<mtd>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mtd>

<mtd>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</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><img src="./17e9d6d5092bfbed9b5c64238e52661619c07e4e.svg" 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}}.}" loading="lazy">

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}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U_{u}}</annotation>

</semantics>

</math><img src="./423ac752d3d2acb56be36948e40f271119e4edea.svg" 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}}" loading="lazy"> 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})}">

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbf {m_{1}} ^{2},\mathbf {m_{2}} ^{2},\mathbf {m_{3}} ^{2})}</annotation>

</semantics>

</math><img src="./308738140c5861549c10d0aa1bd8116439f3ab80.svg" 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})}" loading="lazy"> but can be any permutation of those.

After obtaining a general <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" 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}" loading="lazy"> 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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" 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}" loading="lazy"> matrix for up-type quarks, denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{u}^{\dagger }}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U_{u}^{\dagger }}</annotation>

</semantics>

</math><img src="./32ca1500f0aa88ac0f9dd671c1d42268b0477625.svg" 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 }}" loading="lazy">, has to be multiplied with the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" 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}" loading="lazy"> matrix for down-type quarks, denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{d}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U_{d}}</annotation>

</semantics>

</math><img src="./a321d741a59ba8149c5d0077f222ef81ce44916c.svg" 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}}" loading="lazy">. 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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3!\times 3!=36}</annotation>

</semantics>

</math><img src="./b9c520c5bf63d26c22db1f785c5141730d96b51a.svg" 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}" loading="lazy"> potential permutations of eigenvalues are listed elsewhere.<a href="#cite_note-29">[29]</a><a href="#cite_note-30">[30]</a>

Among these 36 potential CKM matrices, 4 of them

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</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><img src="./6eb915f5ae9b7a18384e2911dc43197e68c096fa.svg" 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}}}" loading="lazy"> and

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</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><img src="./ad64fa9ae4b0be292a3fa902789b717d3956ae3b.svg" 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}},}" loading="lazy">

fit experimental data to the order of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ^{1/2}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda ^{1/2}}</annotation>

</semantics>

</math><img src="./5501988d4b9bb58c4e3e04c88f6d93301b696ee0.svg" 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}}" loading="lazy"> or better, at tree level, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" 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 }" loading="lazy"> is one of the <a href="Cabibbo-Kobayashi-Maskawa_matrix" 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,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p,q,r,s,}</annotation>

</semantics>

</math><img src="./7b69f200f9ef0fb4d94c9e18b7722323a40e0e92.svg" 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,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{\prime }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p^{\prime }}</annotation>

</semantics>

</math><img src="./7d46d13620db1b84c69bf1ff5d1290d8d5e7de6c.svg" 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 }}" loading="lazy"> are given by

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<mfrac>

<msup>

</msup>

<msup>

</mrow>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</mrow>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

<mtd>

<mfrac>

<msup>

<mrow>

</mrow>

</msup>

<msup>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msqrt>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<mfrac>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

<msup>

<mrow>

</mrow>

</msup>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}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}}}}}\\&+&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}}}}},\\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}}}}}\\&+&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}}}}},\\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}}}}},\\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}}}}},\\q&=&{{xx'+yy'+xyx'y'} \over {{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}}.\end{aligned}}}</annotation>

</semantics>

</math><img src="./9db7302fb22e79bbbcd2827e56256bd8e8cc3b61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -26.063ex; margin-bottom: -0.275ex; width:90.985ex; height:53.843ex;" alt="{\displaystyle {\begin{aligned}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}}}}}\\&+&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}}}}},\\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}}}}}\\&+&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}}}}},\\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}}}}},\\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}}}}},\\q&=&{{xx'+yy'+xyx'y'} \over {{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}}{\sqrt {x'^{2}+y'^{2}+x'^{2}y'^{2}}}}}.\end{aligned}}}" loading="lazy">

The best fit of the CKM elements are

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |V_{ud}|=|V_{tb}|\sim 0.9925,}</annotation>

</semantics>

</math><img src="./6d04bc5385bb94b346419561efde7baf6a738c90.svg" 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,}" loading="lazy">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |V_{ub}|=|V_{td}|\sim 0.0075,}</annotation>

</semantics>

</math><img src="./26cd430d5f3f2edbc3ef216ea319d0b2b6cda522.svg" 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,}" loading="lazy">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |V_{us}|=|V_{ts}|=|V_{cd}|=|V_{cb}|\sim 0.122023,}</annotation>

</semantics>

</math><img src="./1bb4a27eb47f0bf3fc306216d50d03de65336da6.svg" 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,}" loading="lazy">

and

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |V_{cs}|\sim 0.9845.}</annotation>

</semantics>

</math><img src="./509aa6f098ddf9c5d12d7e3b222afbc01a6d879f.svg" 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.}" loading="lazy">

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.

<div class="mw-heading mw-heading2"><h2 id="Strong_CP_problem">Strong CP problem</h2></div>

/* start https://en.wikipedia.org/ */

.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}}

/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Strong_CP_problem" title="Strong CP problem">Strong CP problem</a></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .unsolved{margin:0.5em 0 1em 1em;border:1px solid #a2a9b1;padding:0.35em 0.35em 0.35em 2.2em;background-color:var(--background-color-interactive-subtle);background-image:url("./mw/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}

/* end https://en.wikipedia.org/ */

</style>

<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="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="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="Strong_CP_problem" title="Strong CP problem">strong CP problem</a>.

QCD does not violate the CP-symmetry as easily as the <a href="Electroweak_theory" class="mw-redirect" title="Electroweak theory">electroweak theory</a>; unlike the electroweak theory in which the gauge fields couple to <a href="Chirality_(physics)" title="Chirality (physics)">chiral</a> currents constructed from the <a href="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="Electric_dipole_moment" title="Electric dipole moment">electric dipole moment</a> of the <a href="Neutron" title="Neutron">neutron</a> which would be comparable to 10−18 <a href="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="Lagrangian_(field_theory)" title="Lagrangian (field theory)">Lagrangian</a> that are able to break the CP-symmetry.

</mrow>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

<mover>

</mover>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

<msup>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

</mrow>

</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>

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 }}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\tilde {\theta }}}</annotation>

</semantics>

</math><img src="./dcf22bb05d2a5310d819fb1407e56c7204f434dc.svg" 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 }}}" loading="lazy"> 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="Fine-tuning_(physics)" title="Fine-tuning (physics)">fine-tuning problem</a> in physics, and is typically solved by <a href="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="Peccei%E2%80%93Quinn_theory" title="Peccei–Quinn theory">Peccei–Quinn theory</a>, involving new <a href="Scalar_particle" class="mw-redirect" title="Scalar particle">scalar particles</a> called <a href="Axion" title="Axion">axions</a>. A newer, more radical approach not requiring the axion is a theory involving <a href="Multiple_time_dimensions" title="Multiple time dimensions">two time dimensions</a> first proposed in 1998 by Bars, Deliduman, and Andreev.<a href="#cite_note-31">[31]</a>

<div class="mw-heading mw-heading2"><h2 id="Matter–antimatter_imbalance">Matter–antimatter imbalance</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Baryon_asymmetry" title="Baryon asymmetry">Baryon asymmetry</a> and <a href="Baryogenesis" title="Baryogenesis">Baryogenesis</a></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="T-symmetry" title="T-symmetry">T-symmetry</a>, <a href="Arrow_of_time" title="Arrow of time">Arrow of time</a>, and <a href="Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a></div>

<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="List_of_unsolved_problems_in_physics" title="List of unsolved problems in physics">More unsolved problems in physics</a></div>

</div>

/* start https://en.wikipedia.org/ */

.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}}

/* end https://en.wikipedia.org/ */

</style>

The observable universe is made chiefly of <a href="Matter" title="Matter">matter</a>, rather than consisting of equal parts of matter and <a href="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="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="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.<a href="#cite_note-:0-32">[32]</a>

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="Protons" class="mw-redirect" title="Protons">protons</a> should have cancelled with <a href="Antiproton" title="Antiproton">antiprotons</a>, <a href="Electrons" class="mw-redirect" title="Electrons">electrons</a> with <a href="Positron" title="Positron">positrons</a>, <a href="Neutrons" class="mw-redirect" title="Neutrons">neutrons</a> with <a href="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.<a href="#cite_note-:0-32">[32]</a>

The Standard Model contains at least three sources of CP violation. The first of these, involving the <a href="Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix" title="Cabibbo–Kobayashi–Maskawa matrix">Cabibbo–Kobayashi–Maskawa matrix</a> in the <a href="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="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="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="Lepton" title="Lepton">lepton</a> sector. The current long-baseline neutrino oscillation experiments, <a href="T2K_experiment" title="T2K experiment">T2K</a> and <a href="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="Hyper-Kamiokande" title="Hyper-Kamiokande">Hyper-Kamiokande</a> and <a href="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="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="Majorana_fermion" title="Majorana fermion">Majorana fermions</a>, the <a href="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="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="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="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.<a href="#cite_note-33">[33]</a>

If CP violation in the lepton sector is experimentally determined to be too small to account for matter-antimatter asymmetry, some new <a href="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.<a href="#cite_note-:0-32">[32]</a>

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="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:

/* start https://en.wikipedia.org/ */

.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}}

/* end https://en.wikipedia.org/ */

</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.</blockquote><div class="templatequotecite">— Andrei Sakharov, in Collected Scientific Works (1982).<a href="#cite_note-Sakharov_book-34">[34]</a></div>

<ul><li><a href="B-factory" title="B-factory">B-factory</a></li>

<li><a href="Parity_(physics)#Parity_violation" title="Parity (physics)">Parity (physics) § Parity violation</a></li>

<li><a href="C-symmetry" title="C-symmetry">C-symmetry</a></li>

<li><a href="T-symmetry" title="T-symmetry">T-symmetry</a></li>

<li><a href="CPT_symmetry" title="CPT symmetry">CPT symmetry</a></li>

<li><a href="BTeV_experiment" title="BTeV experiment">BTeV experiment</a></li>

<li><a href="Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix" title="Cabibbo–Kobayashi–Maskawa matrix">Cabibbo–Kobayashi–Maskawa matrix</a></li>

<li><a href="LHCb_experiment" title="LHCb experiment">LHCb experiment</a></li>

<li><a href="Penguin_diagram" title="Penguin diagram">Penguin diagram</a></li>

<li><a href="Neutral_particle_oscillation" title="Neutral particle oscillation">Neutral particle oscillation</a></li>

<li><a href="Electron_electric_dipole_moment" title="Electron electric dipole moment">Electron electric dipole moment</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.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}

/* end https://en.wikipedia.org/ */

</style><div class="reflist reflist-columns references-column-width reflist-columns-2">

/* start https://en.wikipedia.org/ */

.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("./mw/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("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/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}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFAaijAbdelmottelebBetetaAbudinén2025" class="citation journal cs1">Aaij, R.; Abdelmotteleb, A. S. W.; Beteta, C. Abellan; Abudinén, F.; Ackernley, T.; Adefisoye, A. A.; Adeva, B.; Adinolfi, M.; Adlarson, P.; Agapopoulou, C.; Aidala, C. A.; Ajaltouni, Z.; Akar, S.; Akiba, K.; Albicocco, P. (16 July 2025). <a rel="nofollow" class="external text" href="https://www.nature.com/articles/s41586-025-09119-3">"Observation of charge–parity symmetry breaking in baryon decays"</a>. Nature: 1–6. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fs41586-025-09119-3">10.1038/s41586-025-09119-3</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1476-4687">1476-4687</a>.</cite>

</li>

<cite id="CITEREFDentonGehrleinPestes2021" class="citation journal cs1">Denton, Peter B.; Gehrlein, Julia; Pestes, Rebekah (2021). "CP-Violating Neutrino Non-Standard Interactions in Long-Baseline-Accelerator Data". <a href="Physical_Review_Letters" title="Physical Review Letters">Physical Review Letters</a>. 126 (5): 051801. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2008.01110">2008.01110</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2021PhRvL.126e1801D">2021PhRvL.126e1801D</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.126.051801">10.1103/PhysRevLett.126.051801</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/33605742">33605742</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:220961778">220961778</a>.</cite>

</li>

<cite id="CITEREFSchwarzschild1999" class="citation journal cs1">Schwarzschild, Bertram (1999). "Two Experiments Observe Explicit Violation of Time-Reversal Symmetry". <a href="Physics_Today" title="Physics Today">Physics Today</a>. 52 (2): 19–20. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1999PhT....52b..19S">1999PhT....52b..19S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.882519">10.1063/1.882519</a>.</cite>

</li>

<cite id="CITEREFSchubert2015" class="citation journal cs1">Schubert, K.R. (2015). "T violation and CPT tests in neutral-meson systems". Progress in Particle and Nuclear Physics. 81: 1–38. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1409.5998">1409.5998</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015PrPNP..81....1S">2015PrPNP..81....1S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.ppnp.2014.12.001">10.1016/j.ppnp.2014.12.001</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:117740717">117740717</a>.</cite>

</li>

<cite id="CITEREFLeeYang1956" class="citation journal cs1">Lee, T. D.; Yang, C. N. (1956). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.104.254">"Question of Parity Conservation in Weak Interactions"</a>. <a href="Physical_Review" title="Physical Review">Physical Review</a>. 104 (1): 254–258. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1956PhRv..104..254L">1956PhRv..104..254L</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.104.254">10.1103/PhysRev.104.254</a>.</cite>

</li>

<cite id="CITEREFWuAmblerHaywardHoppes1957" class="citation journal cs1">Wu, C. S.; Ambler, E.; Hayward, R. W.; Hoppes, D. D.; Hudson, R. P. (1957). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.105.1413">"Experimental Test of Parity Conservation in Beta Decay"</a>. <a href="Physical_Review" title="Physical Review">Physical Review</a>. 105 (4): 1413–1415. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1957PhRv..105.1413W">1957PhRv..105.1413W</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.105.1413">10.1103/PhysRev.105.1413</a>.</cite>

</li>

<li id="cite_note-Ioffe-7">^ <a href="#cite_ref-Ioffe_7-0">a</a> <a href="#cite_ref-Ioffe_7-1">b</a> <cite id="CITEREFIoffeOkunRudik1957" class="citation journal cs1">Ioffe, B. L.; Okun, L. B.; Rudik, A. P. (1957). <a rel="nofollow" class="external text" href="http://www.jetp.ac.ru/cgi-bin/dn/e_005_02_0328.pdf">"The Problem of Parity Non-conservation in Weak Interactions"</a> (PDF). <a href="Journal_of_Experimental_and_Theoretical_Physics" title="Journal of Experimental and Theoretical Physics">Journal of Experimental and Theoretical Physics</a>. 32: 328–330.</cite>

</li>

<cite id="CITEREFFriedmanTelegdi1957" class="citation journal cs1">Friedman, J. I.; Telegdi, V. L. (1957). "Nuclear Emulsion Evidence for Parity Nonconservation in the Decay Chain π+→μ+→e+". <a href="Physical_Review" title="Physical Review">Physical Review</a>. 106 (6): 1290–1293. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1957PhRv..106.1290F">1957PhRv..106.1290F</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.106.1290">10.1103/PhysRev.106.1290</a>.</cite>

</li>

<cite id="CITEREFGarwinLedermanWeinrich1957" class="citation journal cs1">Garwin, R. L.; Lederman, L. M.; Weinrich, M. (1957). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.105.1415">"Observations of the Failure of Conservation of Parity and Charge Conjugation in Meson Decays: The Magnetic Moment of the Free Muon"</a>. <a href="Physical_Review" title="Physical Review">Physical Review</a>. 105 (4): 1415–1417. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1957PhRv..105.1415G">1957PhRv..105.1415G</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.105.1415">10.1103/PhysRev.105.1415</a>.</cite>

</li>

<cite id="CITEREFCulliganFrankHolt1959" class="citation journal cs1">Culligan, G.; Frank, S. G. F.; Holt, J. R. (1959). "Longitudinal polarization of the electrons from the decay of unpolarized Positive and Negative Muons". <a href="Proceedings_of_the_Physical_Society" title="Proceedings of the Physical Society">Proceedings of the Physical Society</a>. 73 (2): 169. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1959PPS....73..169C">1959PPS....73..169C</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0370-1328%2F73%2F2%2F303">10.1088/0370-1328/73/2/303</a>.</cite>

</li>

<cite id="CITEREFLeeOehmeYang1957" class="citation journal cs1">Lee, T. D.; Oehme, R.; Yang, C. N. (1957). <a rel="nofollow" class="external text" href="https://archive.today/20120805225625/http://www.slac.stanford.edu/spires/find/hep/www?j=PHRVA,106,340">"Remarks on Possible Noninvariance under Time Reversal and Charge Conjugation"</a>. <a href="Physical_Review" title="Physical Review">Physical Review</a>. 106 (2): 340–345. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1957PhRv..106..340L">1957PhRv..106..340L</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.106.340">10.1103/PhysRev.106.340</a>. Archived from <a rel="nofollow" class="external text" href="http://www.slac.stanford.edu/spires/find/hep/www?j=PHRVA,106,340">the original</a> on 5 August 2012.</cite>

</li>

<cite id="CITEREFLandau1957" class="citation journal cs1">Landau, L. (1957). "On the conservation laws for weak interactions". <a href="Nuclear_Physics_(journal)" title="Nuclear Physics (journal)">Nuclear Physics</a>. 3 (1): 127–131. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1957NucPh...3..127L">1957NucPh...3..127L</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0029-5582%2857%2990061-5">10.1016/0029-5582(57)90061-5</a>.</cite>

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> <cite id="CITEREFAnikinaNeaguOkonovPetrov" class="citation journal cs1">Anikina, M. Kh.; Neagu, D. V.; Okonov, E. O.; Petrov, N. I.; Rozanova, A. M.; Rusakov, V. A. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210127211617/http://www.jetp.ac.ru/cgi-bin/dn/e_015_01_0093.pdf">"An experimental investigation of some consequences of CP-invariance in K0 2 meson decays"</a> (PDF). <a href="Soviet_Physics_JETP" class="mw-redirect" title="Soviet Physics JETP">Soviet Physics JETP</a>. 15 (1): 93–96. Archived from <a rel="nofollow" class="external text" href="http://jetp.ac.ru/cgi-bin/dn/e_015_01_0093.pdf">the original</a> (PDF) on 27 January 2021. Retrieved 3 April 2021.</cite>

</li>

<li id="cite_note-FC1964-14"><a href="#cite_ref-FC1964_14-0">^</a> <cite id="CITEREFChristensonCroninFitchTurlay1964" class="citation journal cs1">Christenson, J. H.; Cronin, J. W.; Fitch, V. L.; Turlay, R. (1964). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.13.138">"Evidence for the 2π Decay of the K0 2 Meson System"</a>. <a href="Physical_Review_Letters" title="Physical Review Letters">Physical Review Letters</a>. 13 (4): 138. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1964PhRvL..13..138C">1964PhRvL..13..138C</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.13.138">10.1103/PhysRevLett.13.138</a>.</cite>

</li>

<li id="cite_note-FCE-15"><a href="#cite_ref-FCE_15-0">^</a> <a rel="nofollow" class="external text" href="http://large.stanford.edu/courses/2008/ph204/coleman1/">The Fitch-Cronin Experiment</a>

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> <cite id="CITEREFAlavi-Harati1999" class="citation journal cs1">Alavi-Harati, A.; et al. (KTeV Collaboration) (1999). "Observation of Direct CP Violation in KS,L→ππ Decays". <a href="Physical_Review_Letters" title="Physical Review Letters">Physical Review Letters</a>. 83 (1): 22–27. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-ex/9905060">hep-ex/9905060</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1999PhRvL..83...22A">1999PhRvL..83...22A</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.83.22">10.1103/PhysRevLett.83.22</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119333352">119333352</a>.</cite>

</li>

<cite id="CITEREFFanti1999" class="citation journal cs1">Fanti, V.; et al. (NA48 Collaboration) (1999). "A new measurement of direct CP violation in two pion decays of the neutral kaon". <a href="Physics_Letters_B" class="mw-redirect" title="Physics Letters B">Physics Letters B</a>. 465 (1–4): 335–348. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-ex/9909022">hep-ex/9909022</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1999PhLB..465..335F">1999PhLB..465..335F</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0370-2693%2899%2901030-8">10.1016/S0370-2693(99)01030-8</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15277360">15277360</a>.</cite>

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <cite id="CITEREFAubert2001" class="citation journal cs1">Aubert, B; et al. (2001). "Measurement of CP-Violating Asymmetries in B0 Decays to CP Eigenstates". <a href="Physical_Review_Letters" title="Physical Review Letters">Physical Review Letters</a>. 86 (12): 2515–22. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-ex/0102030">hep-ex/0102030</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2001PhRvL..86.2515A">2001PhRvL..86.2515A</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.86.2515">10.1103/PhysRevLett.86.2515</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/11289970">11289970</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:24606837">24606837</a>.</cite>

</li>

<li id="cite_note-19"><a href="#cite_ref-19">^</a> <cite id="CITEREFAbe_K2001" class="citation journal cs1">Abe K; et al. (2001). "Observation of Large CP Violation in the Neutral B Meson System". <a href="Physical_Review_Letters" title="Physical Review Letters">Physical Review Letters</a>. 87 (9): 091802. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-ex/0107061">hep-ex/0107061</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2001PhRvL..87i1802A">2001PhRvL..87i1802A</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.87.091802">10.1103/PhysRevLett.87.091802</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/11531561">11531561</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:3197654">3197654</a>.</cite>

</li>

<cite id="CITEREFRodgers2001" class="citation web cs1">Rodgers, Peter (August 2001). <a rel="nofollow" class="external text" href="http://physicsworld.com/cws/article/print/2001/aug/01/where-did-all-the-antimatter-go">"Where did all the antimatter go?"</a>. <a href="Physics_World" title="Physics World">Physics World</a>. p. 11.</cite>

</li>

<li id="cite_note-21"><a href="#cite_ref-21">^</a> <cite id="CITEREFCarbone2012" class="citation arxiv cs1">Carbone, A. (2012). "A search for time-integrated CP violation in D0→h−h+ decays". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1210.8257">1210.8257</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/hep-ex">hep-ex</a>].</cite>

</li>

<li id="cite_note-22"><a href="#cite_ref-22">^</a> <cite id="CITEREFLHCb_Collaboration2014" class="citation journal cs1">LHCb Collaboration (2014). "Measurement of CP asymmetry in D0→K+K− and D0→π+π− decays". <a href="Journal_of_High_Energy_Physics" title="Journal of High Energy Physics">Journal of High Energy Physics</a>. 2014 (7): 41. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1405.2797">1405.2797</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014JHEP...07..041A">2014JHEP...07..041A</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FJHEP07%282014%29041">10.1007/JHEP07(2014)041</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118510475">118510475</a>.</cite>

</li>

<li id="cite_note-23"><a href="#cite_ref-23">^</a> <cite id="CITEREFAaij2013" class="citation journal cs1">Aaij, R.; et al. (LHCb Collaboration) (30 May 2013). "First Observation of CP Violation in the Decays of B0s Mesons". <a href="Physical_Review_Letters" title="Physical Review Letters">Physical Review Letters</a>. 110 (22): 221601. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1304.6173">1304.6173</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013PhRvL.110v1601A">2013PhRvL.110v1601A</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.110.221601">10.1103/PhysRevLett.110.221601</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/23767711">23767711</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:20486226">20486226</a>.</cite>

</li>

<li id="cite_note-24"><a href="#cite_ref-24">^</a> <cite id="CITEREFR._Aaij2019" class="citation journal cs1">R. Aaij; et al. (LHCb Collaboration) (2019). <a rel="nofollow" class="external text" href="https://iris.unica.it/bitstream/11584/270374/2/PhysRevLett.122.211803.pdf">"Observation of CP Violation in Charm Decays"</a> (PDF). <a href="Physical_Review_Letters" title="Physical Review Letters">Physical Review Letters</a>. 122 (21): 211803. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1903.08726">1903.08726</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2019PhRvL.122u1803A">2019PhRvL.122u1803A</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.122.211803">10.1103/PhysRevLett.122.211803</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/31283320">31283320</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:84842008">84842008</a>.</cite>

</li>

<li id="cite_note-25"><a href="#cite_ref-25">^</a> <cite id="CITEREFAbeAkutsu2020" class="citation journal cs1">Abe, K.; Akutsu, R.; et al. (T2K Collaboration) (16 April 2020). "Constraint on the matter-antimatter symmetry-violating phase in neutrino oscillations". <a href="Nature_(journal)" title="Nature (journal)">Nature</a>. 580 (7803): 339–344. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1910.03887">1910.03887</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2020Natur.580..339T">2020Natur.580..339T</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fs41586-020-2177-0">10.1038/s41586-020-2177-0</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/32296192">32296192</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:203951445">203951445</a>.</cite>

</li>

<cite id="CITEREFHimmel2020" class="citation conference cs1">Himmel, Alex; et al. (NOvA Collaboration) (2 July 2020). <a rel="nofollow" class="external text" href="https://indico.fnal.gov/event/43209/timetable/#194-new-oscillation-results-fr">"New Oscillation Results from the NOvA Experiment"</a>. Neutrino2020. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5281%2Fzenodo.3959581">10.5281/zenodo.3959581</a>.</cite>

</li>

<cite id="CITEREFKellyMachadoParkePerez-Gonzalez2021" class="citation journal cs1">Kelly, Kevin J.; Machado, Pedro A.N.; Parke, Stephen J.; Perez-Gonzalez, Yuber F.; Funchal, Renata Zukanovich (2021). "Neutrino mass ordering in light of recent data". <a href="Physical_Review_D" class="mw-redirect" title="Physical Review D">Physical Review D</a>. 103 (1): 013004. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2007.08526">2007.08526</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2021PhRvD.103a3004K">2021PhRvD.103a3004K</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevD.103.013004">10.1103/PhysRevD.103.013004</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:220633488">220633488</a>.</cite>

</li>

<li id="cite_note-28"><a href="#cite_ref-28">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://news.fnal.gov/tag/cp-violation/">"New results from NOvA experiment shed more light on neutrinos' identity-changing behavior"</a>. News. Retrieved 13 May 2025.</cite>

</li>

<cite id="CITEREFLin2021" class="citation journal cs1">Lin, C.L. (2021). "Exploring the Origin of CP Violation in the Standard Model". Letters in High Energy Physics. 221: 1. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2010.08245">2010.08245</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2021LHEP....4..221L">2021LHEP....4..221L</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.31526%2FLHEP.2021.221">10.31526/LHEP.2021.221</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:245641205">245641205</a>.</cite>

</li>

<cite id="CITEREFLin2023" class="citation journal cs1">Lin, C.L. (2023). <a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fsym15051051">"BAU Production in the SN-Breaking Standard Model"</a>. <a href="Symmetry" title="Symmetry">Symmetry</a>. 15 (5): 1051. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2209.12490">2209.12490</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2023Symm...15.1051L">2023Symm...15.1051L</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fsym15051051">10.3390/sym15051051</a>.</cite>

</li>

<cite id="CITEREFI._BarsC._DelidumanO._Andreev1998" class="citation journal cs1">I. Bars; C. Deliduman; O. Andreev (1998). "Gauged Duality, Conformal Symmetry, and Spacetime with Two Times". <a href="Physical_Review_D" class="mw-redirect" title="Physical Review D">Physical Review D</a>. 58 (6): 066004. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-th/9803188">hep-th/9803188</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1998PhRvD..58f6004B">1998PhRvD..58f6004B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevD.58.066004">10.1103/PhysRevD.58.066004</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:8314164">8314164</a>.</cite>

</li>

<li id="cite_note-:0-32">^ <a href="#cite_ref-:0_32-0">a</a> <a href="#cite_ref-:0_32-1">b</a> <a href="#cite_ref-:0_32-2">c</a> <cite id="CITEREFSakharov1991" class="citation journal cs1">Sakharov, Andrei D. (1991). "Violation of CP in variance, C asymmetry, and baryon asymmetry of the universe". Soviet Physics Uspekhi. 34 (5): 392–393. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1070%2FPU1991v034n05ABEH002497">10.1070/PU1991v034n05ABEH002497</a>.</cite>

</li>

<li id="cite_note-33"><a href="#cite_ref-33">^</a> <cite id="CITEREFMauger2023" class="citation conference cs1">Mauger, Christopher (17 July 2023). <a rel="nofollow" class="external text" href="https://indico.cern.ch/event/1114856/contributions/5321247/attachments/2685070/4658399/Mauger-LeptonPhoton2023-CPVio-July2023v4.pdf">CP violation searches in neutrino oscillations</a> (PDF). Lepton-Photon 2023.</cite>

</li>

<cite id="CITEREFSakharov1982" class="citation book cs1">Sakharov, A. D. (7 December 1982). Collected Scientific Works. <a href="Marcel_Dekker" title="Marcel Dekker">Marcel Dekker</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0824717148</bdi>.</cite>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

/* start https://en.wikipedia.org/ */

.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%}}

/* end https://en.wikipedia.org/ */

</style><div class="refbegin" style="">

<ul><li><cite id="CITEREFSozzi,_M.S.2008" class="citation book cs1">Sozzi, M.S. (2008). Discrete symmetries and CP violation. <a href="Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-929666-8</bdi>.</cite></li>

<li><cite id="CITEREFGustavo_BrancoLuís_LavouraJoão_Silva1999" class="citation book cs1">Gustavo Branco; Luís Lavoura; João Silva (1999). <a rel="nofollow" class="external text" href="https://academic.oup.com/book/56404">CP violation</a>. <a href="Clarendon_Press" class="mw-redirect" title="Clarendon Press">Clarendon Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-383-02075-5</bdi>.</cite></li>

<li><cite id="CITEREFI._BigiA._Sanda1999" class="citation book cs1">I. Bigi; A. Sanda (1999). CP violation. <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-44349-4</bdi>.</cite></li>

<li><cite id="CITEREFMichael_Beyer2002" class="citation book cs1">Michael Beyer, ed. (2002). CP Violation in Particle, Nuclear and Astrophysics. <a href="Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-43705-5</bdi>.</cite> (A collection of essays introducing the subject, with an emphasis on experimental results.)</li>

<li><cite id="CITEREFL._Wolfenstein1989" class="citation book cs1">L. Wolfenstein (1989). CP violation. <a href="North%E2%80%93Holland_Publishing" class="mw-redirect" title="North–Holland Publishing">North–Holland Publishing</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-444-88081-9</bdi>.</cite> (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><cite id="CITEREFDavid_J._Griffiths1987" class="citation book cs1"><a href="David_J._Griffiths" title="David J. Griffiths">David J. Griffiths</a> (1987). Introduction to Elementary Particles. <a href="John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-60386-3</bdi>.</cite></li>

<li><cite id="CITEREFBigi1998" class="citation journal cs1">Bigi, I. (1998). "CP Violation – An Essential Mystery in Nature's Grand Design". Surveys of High Energy Physics. 12 (1–4): 269–336. <a href="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="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="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></li>

<li><cite id="CITEREFMark_Trodden1999" class="citation journal cs1">Mark Trodden (1999). "Electroweak Baryogenesis". <a href="Reviews_of_Modern_Physics" title="Reviews of Modern Physics">Reviews of Modern Physics</a>. 71 (5): 1463–1500. <a href="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="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="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="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></li>

<li><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="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></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>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://cerncourier.com/cws/article/cern/28025">Cern Courier article</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120305024748/http://cerncourier.com/cws/article/cern/28025">Archived</a> 5 March 2012 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li></ul>

/* start https://en.wikipedia.org/ */

.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 .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .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}}

/* end https://en.wikipedia.org/ */

</style></div><div role="navigation" class="navbox" aria-labelledby="C,_P,_and_T_symmetries22" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="C,_P,_and_T_symmetries22" style="font-size:114%;margin:0 4em">C, P, and T symmetries</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="C-symmetry" title="C-symmetry">C-symmetry</a></li>

<li><a href="Parity_(physics)" title="Parity (physics)">P-symmetry</a></li>

<li><a href="T-symmetry" title="T-symmetry">T-symmetry</a></li></ul>

</div></td></tr><tr><td colspan="2" class="navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a class="mw-selflink-fragment" href="#CP-symmetry">CP symmetry</a></li>

<li><a href="CPT_symmetry" title="CPT symmetry">CPT symmetry</a></li></ul>

</div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chirality_(physics)" title="Chirality (physics)">Chirality</a></li>

<li><a href="Pin_group" title="Pin group">Pin group</a></li>

<li><a href="Symmetry_(physics)" title="Symmetry (physics)">Symmetry (physics)</a></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Standard_Model377" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><div id="Standard_Model377" style="font-size:114%;margin:0 4em"><a href="Standard_Model" title="Standard Model">Standard Model</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Particle_physics" title="Particle physics">Particle physics</a>

<ul><li><a href="Fermion" title="Fermion">Fermions</a></li>

<li><a href="Gauge_boson" title="Gauge boson">Gauge boson</a></li>

<li><a href="Higgs_boson" title="Higgs boson">Higgs boson</a></li></ul></li>

<li><a href="Quantum_field_theory" title="Quantum field theory">Quantum field theory</a></li>

<li><a href="Gauge_theory" title="Gauge theory">Gauge theory</a></li>

<li><a href="Strong_interaction" title="Strong interaction">Strong interaction</a>

<ul><li><a href="Color_charge" title="Color charge">Color charge</a></li>

<li><a href="Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a></li>

<li><a href="Quark_model" title="Quark model">Quark model</a></li></ul></li>

<li><a href="Electroweak_interaction" title="Electroweak interaction">Electroweak interaction</a>

<ul><li><a href="Weak_interaction" title="Weak interaction">Weak interaction</a></li>

<li><a href="Quantum_electrodynamics" title="Quantum electrodynamics">Quantum electrodynamics</a></li>

<li><a href="Fermi's_interaction" title="Fermi's interaction">Fermi's interaction</a></li>

<li><a href="Weak_hypercharge" title="Weak hypercharge">Weak hypercharge</a></li>

<li><a href="Weak_isospin" title="Weak isospin">Weak isospin</a></li></ul></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="4" style="width:1px;padding:0 0 0 2px"><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constituents</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix" title="Cabibbo–Kobayashi–Maskawa matrix">CKM matrix</a></li>

<li><a href="Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">Spontaneous symmetry breaking</a></li>

<li><a href="Higgs_mechanism" title="Higgs mechanism">Higgs mechanism</a></li>

<li><a href="Mathematical_formulation_of_the_Standard_Model" title="Mathematical formulation of the Standard Model">Mathematical formulation of the Standard Model</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Physics_beyond_the_Standard_Model" title="Physics beyond the Standard Model">Beyond the Standard Model</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Evidence</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hierarchy_problem" title="Hierarchy problem">Hierarchy problem</a></li>

<li><a href="Dark_matter" title="Dark matter">Dark matter</a></li>

<li><a href="Cosmological_constant" title="Cosmological constant">Cosmological constant</a>

<ul><li><a href="Cosmological_constant_problem" title="Cosmological constant problem">problem</a></li></ul></li>

<li><a href="Neutrino_oscillation" title="Neutrino oscillation">Neutrino oscillation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theories</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Technicolor_(physics)" title="Technicolor (physics)">Technicolor</a></li>

<li><a href="Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a></li>

<li><a href="Grand_Unified_Theory" title="Grand Unified Theory">Grand Unified Theory</a></li>

<li><a href="Theory_of_everything" title="Theory of everything">Theory of everything</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Supersymmetry" title="Supersymmetry">Supersymmetry</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Next-to-Minimal_Supersymmetric_Standard_Model" title="Next-to-Minimal Supersymmetric Standard Model">NMSSM</a></li>

<li><a href="Split_supersymmetry" title="Split supersymmetry">Split supersymmetry</a></li>

<li><a href="Supergravity" title="Supergravity">Supergravity</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Quantum_gravity" title="Quantum gravity">Quantum gravity</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="String_theory" title="String theory">String theory</a></li>

<li><a href="Superstring_theory" title="Superstring theory">Superstring theory</a></li>

<li><a href="Loop_quantum_gravity" title="Loop quantum gravity">Loop quantum gravity</a></li>

<li><a href="Causal_dynamical_triangulation" title="Causal dynamical triangulation">Causal dynamical triangulation</a></li>

<li><a href="Canonical_quantum_gravity" title="Canonical quantum gravity">Canonical quantum gravity</a></li>

<li><a href="Superfluid_vacuum_theory" title="Superfluid vacuum theory">Superfluid vacuum theory</a></li>

<li><a href="Twistor_theory" title="Twistor theory">Twistor theory</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Experiments</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Laboratori_Nazionali_del_Gran_Sasso" title="Laboratori Nazionali del Gran Sasso">Gran Sasso</a></li>

<li><a href="Super-Kamiokande" title="Super-Kamiokande">Super-K</a></li>

<li><a href="Tevatron" title="Tevatron">Tevatron</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div>

<ul><li> Category</li>

<li> <a href="https://commons.wikimedia.org/wiki/Category:Standard_Model_(physics)" class="extiw external" title="commons:Category:Standard Model (physics)">Commons</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-28" href="https://en.wikipedia.org/wiki/?title=CP_violation&oldid=1302958033">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>