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{{short description|Violation of charge-parity symmetry in particle physics and cosmology}}
{{Use dmy dates|date=January 2016}}
{{Beyond the Standard Model|expanded=Evidence}}

In [[particle physics]], '''CP violation''' is a violation of '''CP-symmetry''' (or '''charge conjugation parity symmetry'''): the combination of [[C-symmetry]] ([[Charge (physics)|charge conjugation]] symmetry) and [[Parity (physics)|P-symmetry]] ([[Parity (physics)|parity]] symmetry). CP-symmetry states that the laws of physics should be the same if a particle is interchanged with its antiparticle (C-symmetry) while its spatial coordinates are inverted ("mirror" or P-symmetry). The discovery of CP violation in 1964 in the decays of neutral [[kaon]]s resulted in the [[Nobel Prize in Physics]] in 1980 for its discoverers [[James Cronin]] and [[Val Fitch]].

It plays an important role both in the attempts of [[Physical cosmology|cosmology]] to explain the dominance of [[matter]] over [[antimatter]] in the present [[universe]], and in the study of [[weak interaction]]s in particle physics.

==Overview==
Until the 1950s, parity conservation was believed to be one of the fundamental geometric [[conservation laws]] (along with [[conservation of energy]] and [[conservation of momentum]]). After the discovery of [[Parity (physics)#Parity violation|parity violation]] in 1956, CP-symmetry was proposed to restore order. However, while the [[strong interaction]] and [[electromagnetic interaction]] 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 [[weak decay]].

Only a weaker version of the symmetry could be preserved by physical phenomena, which was [[CPT symmetry]]. 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 [[quantum field theory]], charge conjugation, parity, and time reversal are applied together. Direct observation of the [[T-symmetry|time reversal symmetry]] violation without any assumption of CPT theorem was done in 1998 by two groups, [[CPLEAR experiment|CPLEAR]] and KTeV collaborations, at [[CERN]] and [[Fermilab]], respectively.<ref>
{{cite journal
 |last1=Schwarzschild |first1=Bertram
 |year=1999
 |title=Two Experiments Observe Explicit Violation of Time-Reversal Symmetry
 |journal=[[Physics Today]]
 |volume=52 |issue=2 |pages=19–20
 |bibcode=1999PhT....52b..19S
 |doi=10.1063/1.882519
}}</ref> Already in 1970 Klaus Schubert observed T violation independent of assuming CPT symmetry by using the Bell–Steinberger unitarity relation.<ref>
{{cite journal
 |last1=Schubert |first1=K.R.
 |year=2015
 |title=T violation and CPT tests in neutral-meson systems
 |journal=[[Progress in Particle and Nuclear Physics]]
 |volume=81 |pages=1–38
 |arxiv=1409.5998
 |bibcode=2015PrPNP..81....1S
 |doi=10.1016/j.ppnp.2014.12.001
 |s2cid=117740717
}}</ref>

==History==
===P-symmetry===
The idea behind [[parity (physics)|parity]] 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 [[chemical reaction]] or [[radioactive decay]]) occurs at the same rate as the original reaction. However, in 1956 a careful critical review of the existing experimental data by theoretical physicists [[Tsung-Dao Lee]] and [[Chen-Ning Yang]] revealed that while parity conservation had been verified in decays by the strong or electromagnetic interactions, it was untested in the weak interaction.<ref>
{{cite journal
 |last1=Lee |first1=T. D.
 |last2=Yang |first2=C. N.
 |year=1956
 |title=Question of Parity Conservation in Weak Interactions
 |journal=[[Physical Review]]
 |volume=104 |issue=1 |pages=254–258
 |bibcode=1956PhRv..104..254L
 |doi=10.1103/PhysRev.104.254 |doi-access=free
}}</ref> They proposed several possible direct experimental tests. 

The first test based on [[beta decay]] of [[cobalt-60]] nuclei was carried out in 1956 by a group led by [[Chien-Shiung Wu]], 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.<ref>
{{cite journal
 |last1=Wu |first1=C. S.
 |last2=Ambler |first2=E.
 |last3=Hayward |first3=R. W.
 |last4=Hoppes |first4=D. D.
 |last5=Hudson |first5=R. P.
 |year=1957
 |title=Experimental Test of Parity Conservation in Beta Decay
 |journal=[[Physical Review]]
 |volume=105 |issue=4 |pages=1413–1415
 |bibcode=1957PhRv..105.1413W
 |doi=10.1103/PhysRev.105.1413 |doi-access=free
}}</ref> However, [[parity (physics)|parity]] symmetry still appears to be valid for all reactions involving [[electromagnetism]] and [[strong interaction]]s.

===CP-symmetry===
Overall, the symmetry of a [[quantum mechanics|quantum mechanical]] 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 [[Hilbert space]] was realized shortly after the discovery of ''P'' violation, and it was proposed that charge conjugation, ''C'', which transforms a particle into its [[antiparticle]], was the  suitable symmetry to restore order.

In 1956 [[Reinhard Oehme]] in a letter to Chen-Ning Yang and shortly after, [[Иоффе,_Борис_Лазаревич |Boris L. Ioffe]], [[Lev Okun]] and A. P. Rudik showed that the parity violation meant that charge conjugation invariance must also be violated in weak decays.<ref name="Ioffe">{{cite journal
 |last1=Ioffe
 |first1=B. L.
 |last2=Okun
 |first2=L. B.
 |last3=Rudik
 |first3=A. P.
 |year=1957
 |title=The Problem of Parity Non-conservation in Weak Interactions
 |url=http://www.jetp.ac.ru/cgi-bin/dn/e_005_02_0328.pdf
 |journal=[[Journal of Experimental and Theoretical Physics]]
 |volume=32
 |pages=328–330
 }}{{Dead link|date=July 2022 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>
Charge violation was confirmed in the [[Wu experiment]] and in experiments performed by [[Valentine Telegdi]] and [[Jerome Isaac Friedman|Jerome Friedman]] and [[Richard Garwin|Garwin]] and [[Leon M. Lederman|Lederman]] 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 [[John Riley Holt]] at the [[University of Liverpool]].<ref>
{{Cite journal
 |last1=Friedman |first1=J. I.
 |last2=Telegdi |first2=V. L.
 |year=1957
 |title=Nuclear Emulsion Evidence for Parity Nonconservation in the Decay Chain π<sup>+</sup>→μ<sup>+</sup>→e<sup>+</sup>
 |journal=[[Physical Review]]
 |volume=106 |issue=6 |pages=1290–1293
 |bibcode=1957PhRv..106.1290F
 |doi=10.1103/PhysRev.106.1290
}}</ref><ref>
{{cite journal
 |last1=Garwin |first1=R. L.
 |last2=Lederman |first2=L. M.
 |last3=Weinrich |first3=M.
 |year=1957
 |title=Observations of the Failure of Conservation of Parity and Charge Conjugation in Meson Decays: The Magnetic Moment of the Free Muon
 |journal=[[Physical Review]]
 |volume=105 |issue=4 |pages=1415–1417
 |bibcode=1957PhRv..105.1415G
 |doi=10.1103/PhysRev.105.1415
|doi-access=free
 }}</ref><ref>
{{cite journal
 |last1=Culligan |first1=G.
 |last2=Frank |first2=S. G. F.
 |last3=Holt |first3=J. R.
 |year=1959
 |title=Longitudinal polarization of the electrons from the decay of unpolarized Positive and Negative Muons
 |journal=[[Proceedings of the Physical Society]]
 |volume=73 |issue=2 |page=169
 |bibcode=1959PPS....73..169C
 |doi=10.1088/0370-1328/73/2/303
}}</ref>  

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.<ref name="Ioffe"/><ref>
{{cite journal
 |title=Remarks on Possible Noninvariance under Time Reversal and Charge Conjugation
 |url=http://www.slac.stanford.edu/spires/find/hep/www?j=PHRVA,106,340
 |archive-url=https://archive.today/20120805225625/http://www.slac.stanford.edu/spires/find/hep/www?j=PHRVA,106,340
 |archive-date=2012-08-05
 |year=1957
 |last1=Lee|first1=T. D.
 |last2=Oehme|first2=R.
 |last3=Yang|first3=C. N.
 |journal=[[Physical Review]]
 |volume=106 |issue=2 |pages=340–345
 |bibcode=1957PhRv..106..340L
 |doi=10.1103/PhysRev.106.340
}}</ref>

[[Lev Landau]] proposed in 1957 ''CP-symmetry'',<ref>
{{Cite journal
 |last1=Landau |first1=L.
 |year=1957
 |title=On the conservation laws for weak interactions
 |journal=[[Nuclear Physics (journal)|Nuclear Physics]]
 |volume=3 |issue=1 |pages=127–131
 |bibcode=1957NucPh...3..127L
 |doi=10.1016/0029-5582(57)90061-5
}}</ref> often called just ''CP'' as the true symmetry between matter and antimatter. ''CP-symmetry'' is the product of two [[symmetry in physics|transformations]]: C for charge conjugation and P for parity. In other words, a process in which all particles are exchanged with their [[antiparticle]]s 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 [[Joint Institute for Nuclear Research|Dubna]], on Okun's insistence, unsuccessfully searched for CP-violating kaon decay.<ref>{{cite journal
 |last1=Anikina
 |first1=M. Kh.
 |last2=Neagu
 |first2=D. V.
 |last3=Okonov
 |first3=E. O.
 |last4=Petrov
 |first4=N. I.
 |last5=Rozanova
 |first5=A. M.
 |last6=Rusakov
 |first6=V. A.
 |title=An experimental investigation of some consequences of CP-invariance in K{{su|b=2|p=0}} meson decays
 |url=http://jetp.ac.ru/cgi-bin/dn/e_015_01_0093.pdf
 |journal=[[Soviet Physics JETP]]
 |volume=15
 |issue=1
 |pages=93–96
 |access-date=3 April 2021
 |archive-date=27 January 2021
 |archive-url=https://web.archive.org/web/20210127211617/http://www.jetp.ac.ru/cgi-bin/dn/e_015_01_0093.pdf
 |url-status=dead
 }}</ref>

==Experimental status==

===Indirect CP violation===
In 1964, [[James Cronin]], [[Val Fitch]] and coworkers provided clear evidence from [[kaon]] decay that CP-symmetry could be broken.<ref name=FC1964>{{cite journal |title=Evidence for the 2&pi; Decay of the K{{su|b=2|p=0}} Meson System |journal=[[Physical Review Letters]] |year=1964|volume=13|issue=4 |pages=138 |doi=10.1103/PhysRevLett.13.138|bibcode = 1964PhRvL..13..138C |last1=Christenson |first1=J. H. |last2=Cronin |first2=J. W. |last3=Fitch |first3=V. L. |last4=Turlay |first4=R.|doi-access=free }}</ref> (cf. also Ref. <ref name=FCE>[http://large.stanford.edu/courses/2008/ph204/coleman1/ The Fitch-Cronin Experiment]</ref>). 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 [[kaon]]s can transform into their [[antiparticle]]s (in which each [[quark]] 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.

===Direct CP violation===
[[File:Kaon-box-diagram.svg|thumb|right|Kaon oscillation box diagram]]
[[File:Kaon-box-diagram-alt.svg|thumb|right|The two box diagrams above are the [[Feynman diagram]]s providing the leading contributions to the amplitude of {{Subatomic particle|link=yes|Kaon0}}-{{Subatomic particle|link=yes|Antikaon0}} oscillation]]

Despite many searches, no other manifestation of CP violation was discovered until the 1990s, when the [[NA31 experiment]] at [[CERN]] 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 [[Fermilab]]<ref>{{cite journal|last=Alavi-Harati |first=A. |display-authors=etal |collaboration=KTeV Collaboration|title=Observation of Direct CP Violation in K<sub>S,L</sub>→&pi;&pi; Decays|journal=[[Physical Review Letters]]|year=1999|volume=83|issue=1 |pages=22–27|doi=10.1103/PhysRevLett.83.22|arxiv = hep-ex/9905060 |bibcode = 1999PhRvL..83...22A |s2cid=119333352 }}</ref> and the [[NA48 experiment]] at [[CERN]].<ref name="NA48">
{{Cite journal
 |last=Fanti |first=V.
 |display-authors=etal
 |collaboration=NA48 Collaboration
 |year=1999
 |title= A new measurement of direct CP violation in two pion decays of the neutral kaon
 |journal=[[Physics Letters B]]
 |volume=465 |issue=1–4 |pages=335–348
 |arxiv=hep-ex/9909022
 |bibcode=1999PhLB..465..335F
 |doi=10.1016/S0370-2693(99)01030-8
|s2cid=15277360
 }}</ref>

Starting in 2001, a new generation of experiments, including the [[BaBar experiment]] at the Stanford Linear Accelerator Center ([[SLAC]])<ref>{{cite journal|title=Measurement of CP-Violating Asymmetries in B<sup>0</sup> Decays to CP Eigenstates|journal=[[Physical Review Letters]]|year=2001|volume=86|issue=12|pages=2515–22|doi=10.1103/PhysRevLett.86.2515|arxiv = hep-ex/0102030 |bibcode = 2001PhRvL..86.2515A|pmid=11289970 | last1 = Aubert | first1 = B |s2cid=24606837|display-authors=et al }}</ref> and the [[Belle Experiment]] at the High Energy Accelerator Research Organisation ([[KEK]])<ref>{{cite journal|title=Observation of Large CP Violation in the Neutral B Meson System|journal=[[Physical Review Letters]]|year=2001|volume=87|issue=9|doi=10.1103/PhysRevLett.87.091802|arxiv = hep-ex/0107061 |bibcode = 2001PhRvL..87i1802A|pmid=11531561|page=091802 | author = Abe K|s2cid=3197654|display-authors=et al}}</ref> in Japan, observed direct CP violation in a different system, namely in decays of the [[B meson]]s.<ref>
{{Cite web
 |first=Peter
 |last=Rodgers
 |title=Where did all the antimatter go?
 |url=http://physicsworld.com/cws/article/print/2001/aug/01/where-did-all-the-antimatter-go
 |page=11
 |date=August 2001
 |work=[[Physics World]]
}}</ref> A large number of CP violation processes in [[B meson]] decays have now been discovered. Before these "[[B-factory]]" 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 [[Standard Model]], despite the model's accuracy for "normal" phenomena.

In 2011, a hint of CP violation in decays of neutral [[D meson]]s was reported by the [[LHCb]] experiment at [[CERN]] using 0.6&nbsp;fb<sup>−1</sup> of Run 1 data.<ref>{{cite arXiv|last=Carbone |first=A. |title=A search for time-integrated CP violation in D<sup>0</sup>→h<sup>−</sup>h<sup>+</sup> decays|year=2012|eprint=1210.8257|class=hep-ex }}</ref> However, the same measurement using the full 3.0&nbsp;fb<sup>−1</sup> Run 1 sample was consistent with CP-symmetry.<ref>{{cite journal|author1=LHCb Collaboration|title=Measurement of CP asymmetry in D<sup>0</sup>→K<sup>+</sup>K<sup>−</sup> and D<sup>0</sup>→π<sup>+</sup>π<sup>−</sup> decays|journal= [[Journal of High Energy Physics]]|date=2014|volume=2014|issue=7|page=41|doi=10.1007/JHEP07(2014)041|arxiv = 1405.2797 |bibcode = 2014JHEP...07..041A |s2cid=118510475}}</ref>

In 2013 LHCb announced discovery of CP violation in [[strange B meson]] decays.<ref>{{cite journal|last1=Aaij|first1=R.|display-authors=etal|collaboration=LHCb Collaboration|title=First Observation of CP Violation in the Decays of B<sup>0</sup><sub>s</sub> Mesons|journal=[[Physical Review Letters]]|date=30 May 2013|volume=110|issue=22|pages=221601|doi=10.1103/PhysRevLett.110.221601|pmid=23767711|bibcode=2013PhRvL.110v1601A|arxiv=1304.6173|s2cid=20486226}}</ref>

In March 2019, LHCb announced discovery of CP violation in charmed <math>D^{0}</math> decays with a deviation from zero of 5.3 standard deviations.<ref>{{cite journal|author=R. Aaij|display-authors=etal|collaboration=LHCb Collaboration|title=Observation of CP Violation in Charm Decays|journal=[[Physical Review Letters]]|date=2019|volume=122|issue=21|pages=211803|doi=10.1103/PhysRevLett.122.211803| arxiv=1903.08726|pmid=31283320|bibcode=2019PhRvL.122u1803A|s2cid=84842008|url=https://iris.unica.it/bitstream/11584/270374/2/PhysRevLett.122.211803.pdf}}</ref>

In 2020, the [[T2K experiment|T2K Collaboration]] reported some indications of CP violation in leptons for the first time.<ref>{{cite journal
|last1=Abe
|first1=K.
|last2=Akutsu
|first2=R.
|display-authors=etal|collaboration=T2K Collaboration
|title=Constraint on the matter-antimatter symmetry-violating phase in neutrino oscillations
|journal=[[Nature (journal)|Nature]]|date=16 April 2020
|volume=580
|issue=7803
|pages=339–344|doi=10.1038/s41586-020-2177-0
|pmid=32296192
|arxiv=1910.03887|bibcode=2020Natur.580..339T
|s2cid=203951445
}}</ref>
In this experiment, beams of muon neutrinos ({{SubatomicParticle|Muon neutrino}}) and muon antineutrinos ({{SubatomicParticle|Muon Antineutrino}}) were alternately produced by an [[Accelerator neutrino |accelerator]]. By the time they got to the detector, a significantly higher proportion of electron neutrinos ({{SubatomicParticle|Electron neutrino}}) was observed from the {{SubatomicParticle|Muon neutrino}} beams, than electron antineutrinos ({{SubatomicParticle|Electron Antineutrino}}) were from the {{SubatomicParticle|Muon Antineutrino}} 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, [[NOvA]] sees no evidence of CP violation in neutrino oscillations<ref>
{{cite conference
|last=Himmel
|first=Alex
|collaboration=NOvA Collaboration
|title=New Oscillation Results from the NOvA Experiment
|date=2 July 2020
|book-title=Neutrino2020
|url=https://indico.fnal.gov/event/43209/timetable/#194-new-oscillation-results-fr
|doi=10.5281/zenodo.3959581
}}</ref> and is in slight tension with T2K.<ref>
{{cite journal
|first1=Kevin J.
|last1=Kelly
|first2=Pedro A.N.
|last2=Machado
|first3=Stephen J.
|last3=Parke
|first4=Yuber F.
|last4=Perez-Gonzalez
|first5=Renata Zukanovich
|last5=Funchal
|title= Neutrino mass ordering in light of recent data
|journal=[[Physical Review D]]
|year=2021
|volume=103
|issue=1
|page=013004
|doi=10.1103/PhysRevD.103.013004
|arxiv=2007.08526
|bibcode=2021PhRvD.103a3004K
|s2cid=220633488
}}</ref><ref>
{{cite journal
|first1=Peter B.
|last1=Denton
|first2=Julia
|last2=Gehrlein
|first3=Rebekah
|last3=Pestes
|title=CP-Violating Neutrino Non-Standard Interactions in Long-Baseline-Accelerator Data
|journal=[[Physical Review Letters]]
|year=2021
|volume=126
|issue=5
|page=051801
|doi=10.1103/PhysRevLett.126.051801
|pmid=33605742
|arxiv=2008.01110
|bibcode=2021PhRvL.126e1801D
|s2cid=220961778
}}</ref>

==CP violation in the Standard Model==

"Direct" CP violation is allowed in the [[Standard Model]] if a [[Complex number|complex]] phase appears in the [[Cabibbo–Kobayashi–Maskawa matrix]] (CKM matrix) describing [[quark]] mixing, or the [[Pontecorvo–Maki–Nakagawa–Sakata matrix]] (PMNS matrix) describing [[neutrino]] 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 [[CKM matrix#Counting|can be absorbed]] 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 ''[[Cabibbo–Kobayashi–Maskawa_matrix#The_unitarity_triangles| Jarlskog invariant]]'':

<math display="block">\ J = c_{12}\ c_{13}^2\ c_{23}\ s_{12}\ s_{13}\ s_{23}\ \sin \delta\ \approx\ 0.00003 \ ,</math>

for quarks, which is <math>\ 0.0003\ </math> times the maximum value of <math>\ J_{\max} = \tfrac{1}{6 \sqrt{3}}\ \approx\ 0.1\ .</math> For leptons, only an upper limit exists: <math>\ |J| < 0.03\ .</math>

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>\ a\ </math> and <math>\ b\ ,</math> and their antiparticles <math>\ \bar{a}\ </math> and <math>\ \bar{b}\ .</math> Now consider the processes <math>\ a \rightarrow b\ </math> and the corresponding antiparticle process <math>\ \bar{a} \rightarrow \bar{b}\ ,</math> and denote their amplitudes <math> {\cal M} </math> and <math> \bar{\cal M} </math> respectively. Before CP violation, these terms must be the ''same'' complex number. We can separate the magnitude and phase by writing <math> {\cal M} = |{\cal M}|\ e^{i\theta} </math>. If a phase term is introduced from (e.g.) the CKM matrix, denote it <math> e^{i\phi} </math>. Note that <math> \bar{\cal M} </math> contains the conjugate matrix to <math> {\cal M} </math>, so it picks up a phase term <math> e^{-i\phi} </math>.

Now the formula becomes:

<math display="block">\begin{align}
{\cal M} &= |{\cal M}|\ e^{i\theta}\ e^{+i\phi} \\
\bar{\cal M} &= |{\cal M}|\ e^{i\theta}\ e^{-i\phi}
\end{align}</math>

Physically measurable reaction rates are proportional to <math>\ |{\cal M}|^{2} </math>, thus so far nothing is different. However, consider that there are ''two different routes'': <math> a \overset{1}{\longrightarrow} b </math> and <math> a \overset{2}{\longrightarrow} b </math> or equivalently, two unrelated intermediate states: <math> a \rightarrow 1\rightarrow b </math> and <math> a \rightarrow 2\rightarrow b </math>. This is exactly the case for the kaon where the decay is performed via different quark channels (see the Figure above). In this case we have:

<math display="block">\begin{alignat}{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{alignat}</math>

Some further calculation gives:

<math display="block">|{\cal M}|^{2} - |\bar{\cal M}|^{2} = -4\ |{\cal M}_1|\ |{\cal M}_2|\ \sin(\theta_1 - \theta_2)\ \sin(\phi_1 - \phi_2). </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>\ V_\mathrm{CKM} = U_u^\dagger U_d </math>, where <math> U_u </math> and <math> U_d </math> are unitary transformation matrices which diagonalize the fermion mass matrices <math> M_u </math> and <math> M_d </math>, respectively.

Thus, there are two necessary conditions for getting a complex CKM matrix:
# At least one of <math>U_u</math> and <math>U_d</math> is complex, or the CKM matrix will be purely real.
# If both of them are complex, <math>U_u</math> and <math>U_d</math> must be different, i.e., <math>U_u \neq U_d</math>, or the CKM matrix will be an identity matrix, which is also  purely real.

For a standard model with three fermion generations, the most general non-Hermitian pattern of its mass matrices can be given by
 
<math> M= \begin{bmatrix} A_1 +i D_1 & B_1 +i C_1 & B_2+i C_2 \\ B_4+i C_4 & A_2+i D_2 & B_3+i C_3 \\ B_5+i C_5 & B_6+i C_6 & A_3+i D_3 \end{bmatrix}. </math>

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>  \mathbf{M^2} = M \cdot  M^\dagger </math> can be given by

<math> \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}, 
</math>

and it has the same unitary transformation matrix U with M. 
Besides, parameters in <math> \mathbf{M^2} </math> are correlated to those in M directly in the ways shown below

<math> \begin{align}
\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{align} </math>

That means if we diagonalize an  <math> \mathbf{M^2} </math> matrix with 9 parameters, it has the same effect as diagonalizing an <math>M</math> matrix with 18 parameters. Therefore, diagonalizing the <math> \mathbf{M^2} </math> matrix is certainly the most reasonable choice.

The <math>M</math> and  <math> \mathbf{M^2} </math> 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> \mathbf{M^2} </math> matrix has only 9 parameters, it is still too complicated to be diagonalized directly. Thus, an assumption 

<math> \mathbf{M^2}_R \cdot \mathbf{M^{2\dagger}}_I + \mathbf{M^2}_I \cdot \mathbf{M^{2\dagger}}_R =0 </math>

was employed to simplify the pattern, where  <math> \mathbf{ M^2}_R  </math> is the real part of  <math> \mathbf{M^2} </math> and <math> \mathbf{M^2}_I </math> is the imaginary part.

Such an assumption could further reduce the parameter number from 9 to 5 and the reduced <math> \mathbf{M^2} </math> matrix can be given by

<math>
\mathbf {M^2} = \begin{bmatrix} \mathbf{A} + \mathbf{B} (x y- {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, 
</math>

where <math> \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},</math> and <math> y \equiv \mathbf{B_1 / B_3} </math>.


Diagonalizing <math> \mathbf{M^2} </math> analytically, the eigenvalues are given by 

<math> \mathbf{m_1}^2 = \mathbf{A}-\mathbf{B}{x \over y} - \mathbf{C}{\sqrt{x^2 +y^2 +x^2 y^2} \over x y}, </math>

<math> \mathbf{m_2}^2 = \mathbf{A}-\mathbf{B}{x \over y} + \mathbf{C}{\sqrt{x^2 +y^2 +x^2 y^2} \over x y}, </math>

<math> \mathbf{m_3}^2 = \mathbf{A}+\mathbf{B}{(x^2+1)y \over x},</math>

and the <math>U</math> matrix for up-type quarks can then be given by 

<math>
U_u = 
\begin{bmatrix}
{-\sqrt{x^2+y^2} \over \sqrt{2(x^2+y^2+x^2y^2)}}
& {-\sqrt{x^2+y^2} \over \sqrt{2(x^2+y^2+x^2y^2)}} 
& {x y \over \sqrt{x^2+y^2+x^2y^2}} \\
{x(y^2-i\sqrt{x^2+y^2+x^2y^2}) \over
\sqrt{x^2+y^2}\sqrt{2(x^2+y^2+x^2y^2)}} 
& {x(y^2+i\sqrt{x^2+y^2+x^2y^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^2y^2}} \\
{y(x^2+i\sqrt{x^2+y^2+x^2y^2}) \over
\sqrt{x^2+y^2}\sqrt{2(x^2+y^2+x^2y^2)}}
& {y(x^2-i\sqrt{x^2+y^2+x^2y^2}) \over
\sqrt{x^2+y^2}\sqrt{2(x^2+y^2+x^2y^2)}}
& {x \over \sqrt{x^2+y^2+x^2 y^2}}
\end{bmatrix}.
</math>

However, the order of the eigenvalues and correspondingly the order of the columns of <math>U_u</math> does not necessarily have to be <math> (\mathbf{m_1}^2, \mathbf{m_2}^2, \mathbf{m_3}^2) </math> but can be any permutation of those.

After obtaining a general <math>U</math> 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>U</math> matrix for up-type quarks, denoted as <math> U_u^\dagger </math>, has to be multiplied with the <math>U</math> matrix for down-type quarks, denoted as <math> U_d </math>. As mentioned earlier, there are no inherent constraints that dictate the assignment of eigenvalues to specific quark flavors. All <math>3!\times 3!=36</math> potential permutations of eigenvalues are listed elsewhere.<ref>
{{cite journal
 |last1=Lin |first1=C.L.
 |year=2021
 |title=Exploring the Origin of CP Violation in the Standard Model
 |journal=[[Letters in High Energy Physics]]
 |volume=221 |pages=1
 |arxiv=2010.08245
 |doi=10.31526/LHEP.2021.221
 |bibcode=2021LHEP....4..221L
 |s2cid=245641205
 }}</ref><ref>
{{cite journal
 |last1=Lin |first1=C.L.
 |year=2023
 |title=BAU Production in the SN-Breaking Standard Model
 |journal=[[Symmetry]]
 |volume=15 |issue=5
 |pages=1051
 |arxiv=2209.12490
 |doi= 10.3390/sym15051051
 |bibcode=2023Symm...15.1051L
 |doi-access=free
 }}</ref>

Among these 36 potential CKM matrices, 4 of them 

<math> 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} 
</math> and 

<math> 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}, 
</math>

fit experimental data to the order of <math> \lambda^{1/2} </math> or better, at tree level, where  <math> \lambda </math> is one of the [[Cabibbo-Kobayashi-Maskawa matrix#Wolfenstein parameters|Wolfenstein parameters]]. 

The full expressions of parameters <math> p, q, r, s, </math> and <math> p^{\prime} </math> are given by

<math>
 r = {{(x^2+y^2)(x'^2+y'^2)+(x x' +y y')(x y x' 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} }} </math>

  <math> + i {{ (x y' -x' y)(x' y' \sqrt{x^2+y^2+x^2 y^2} +x y \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}}}, </math>

<math> s= {{(x^2+y^2)(x'^2+y'^2)+(x x' +y y')(x y x' 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} }} </math> 

 <math> + i {{ (x y' -x' y)(x' y' \sqrt{x^2+y^2+x^2 y^2} -x y \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}}}, </math>

<math> p = {{[y' y^2(x-x')+ x' x^2 (y-y')]+ i (x y' -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}}}, </math>

<math> p^{\prime} ={{[y y'^2 (x'-x)+ x x'^2 (y'-y)]+  i (x y' -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}} }, </math>

<math> q = {{x x'+y y' +x y x' y'}\over {\sqrt{ x^2+y^2+x^2 y^2} \sqrt{x'^2+y'^2+x'^2 y'^2}}},
</math>

The best fit of the CKM elements are 

<math> |V_{ud} |=|V_{tb} | \sim 0.9925, </math>

<math> |V_{ub} |=|V_{td} | \sim 0.0075, </math>

<math> |V_{us} |=|V_{ts} | =|V_{cd} |=|V_{cb} | \sim 0.122023, </math>
and 

<math> |V_{cs} |\sim 0.9845. </math>

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.

==Strong CP problem==
{{main|Strong CP problem}}
{{unsolved|physics|Why is the strong nuclear interaction force CP-invariant?}}

There is no experimentally known violation of the CP-symmetry in [[quantum chromodynamics]]. As there is no known reason for it to be conserved in QCD specifically, this is a "fine tuning" problem known as the [[strong CP problem]].

QCD does not violate the CP-symmetry as easily as the [[electroweak theory]]; unlike the electroweak theory in which the gauge fields couple to [[chirality (physics)|chiral]] currents constructed from the [[fermion]]ic 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 [[electric dipole moment]] of the [[neutron]] which would be comparable to 10<sup>−18</sup>&nbsp;[[Elementary charge|e]]·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 [[Lagrangian (field theory)|Lagrangian]] that are able to break the CP-symmetry.

<math display="block">{\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 - m
e^{i\theta'\gamma_5}\right) \psi</math>

For a nonzero choice of the θ angle and the chiral phase of the quark mass θ&prime; 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>\scriptstyle{\tilde\theta}</math> 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 [[Fine-tuning (physics)|fine-tuning problem]] in physics, and is typically solved by [[physics beyond the Standard Model]].

There are several proposed solutions to solve the strong CP problem. The most well-known is [[Peccei–Quinn theory]], involving new [[scalar particle]]s called [[axion]]s. A newer, more radical approach not requiring the axion is a theory involving [[multiple time dimensions|two time dimensions]] first proposed in 1998 by Bars, Deliduman, and Andreev.<ref>
{{cite journal
 |author1=I. Bars |author2=C. Deliduman |author3=O. Andreev | title = Gauged Duality, Conformal Symmetry, and Spacetime with Two Times
 | journal = [[Physical Review D]]
 | volume = 58 | issue = 6 | page= 066004
 | year = 1998
 | doi = 10.1103/PhysRevD.58.066004
|arxiv = hep-th/9803188 |bibcode = 1998PhRvD..58f6004B |s2cid=8314164 }}</ref>

==Matter–antimatter imbalance==
{{main|Baryon asymmetry|Baryogenesis}}
{{See also|T-symmetry|Arrow of time|Lorentz transformation}}
{{unsolved|physics|Why does the universe have so much more matter than antimatter?}}
{{citation needed section|date=November 2020}}

The observable universe is made chiefly of [[matter]], rather than consisting of equal parts of matter and [[antimatter]] as might be expected{{additional|date=January 2025}}. It can be demonstrated that, to create an imbalance in matter and antimatter from an initial condition of balance, the [[Sakharov conditions]] must be satisfied, one of which is the existence of CP violation during the extreme conditions of the first seconds after the [[Big Bang]]. 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.<ref name=":0">{{Cite journal |last=Sakharov |first=Andrei |date=January 1967 |title=Violation of CP invariance, C asymmetry, and baryon asymmetry of the universe |doi=10.1070/PU1991v034n05ABEH002497 |journal=Andrei Sakharov |volume=5 |issue=24 |pages=24–27 |via=American Institute of Physics (AIP)}}</ref>

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—[[protons]] should have cancelled with [[antiproton]]s, [[electrons]] with [[positron]]s, [[neutrons]] with [[antineutron]]s, 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.<ref name=":0" />

The Standard Model contains at least three sources of CP violation. The first of these, involving the [[Cabibbo–Kobayashi–Maskawa matrix]] in the [[quark]] 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 [[Neutron electric dipole moment|electric dipole moment of the neutron]] 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 [[Pontecorvo–Maki–Nakagawa–Sakata matrix]] in the [[lepton]] sector. The current long-baseline neutrino oscillation experiments, [[T2K experiment|T2K]] and [[NOνA]], 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, [[Hyper-Kamiokande]] and [[LBNE|DUNE]], 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 [[neutrino factory]] could be sensitive to nearly all possible values of the CP violating Dirac phase. If neutrinos are [[Majorana fermion]]s, the [[Pontecorvo–Maki–Nakagawa–Sakata matrix|PMNS matrix]] 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 [[Neutrinoless double beta decay#Neutrinoless double beta decay|neutrinoless double-beta decay]]. The best limits come from the [[GERmanium Detector Array|GERDA]] experiment. CP violation in the lepton sector generates a matter-antimatter asymmetry through a process called [[Leptogenesis (physics)|leptogenesis]]. 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.<ref>{{Cite journal |last=Mauger |first=Christopher |date=17 July 2023 |title=CP violation searches in neutrino oscillations |url=https://indico.cern.ch/event/1114856/contributions/5321247/attachments/2685070/4658399/Mauger-LeptonPhoton2023-CPVio-July2023v4.pdf |journal=Nature |volume=580 |issue=7805 |pages=339–344 |via=PubMed}}</ref>

If CP violation in the lepton sector is experimentally determined to be too small to account for matter-antimatter asymmetry, some new [[physics beyond the Standard Model]] 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.<ref name=":0" />

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 [[arrow of time]] 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:

{{quote
 |text=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.
 |author=Andrei Sakharov
 |source=in ''Collected Scientific Works'' (1982).<ref name="Sakharov book">
{{cite book
 |last1=Sakharov |first1= A. D.
 |date=7 December 1982
 |title=Collected Scientific Works
 |publisher=[[Marcel Dekker]]
 |isbn=978-0824717148
}}</ref>
}}


==See also==
*[[B-factory]]
*{{slink|Parity (physics)|Parity violation}}
*[[C-symmetry]] 
*[[T-symmetry]]
*[[CPT symmetry]]
*[[BTeV experiment]]
*[[Cabibbo–Kobayashi–Maskawa matrix]]
*[[LHCb experiment]]
*[[Penguin diagram]]
*[[Neutral particle oscillation]]
*[[Electron electric dipole moment]]

==References==
{{Reflist|2}}

==Further reading==
{{Refbegin}}
*{{cite book | author = Sozzi, M.S. | title = Discrete symmetries and CP violation | publisher = [[Oxford University Press]] | year = 2008 |isbn=978-0-19-929666-8}}
* {{cite book |author1=G. C. Branco |author2=L. Lavoura |author3=J. P. Silva |title=CP violation |publisher= [[Clarendon Press]] |year=1999 | isbn = 978-0-19-850399-6}}
* {{cite book |author1=I. Bigi |author2=A. Sanda |title=CP violation |publisher=[[Cambridge University Press]] |year=1999 |isbn = 978-0-521-44349-4}}
* {{cite book | editor= Michael Beyer | title = CP Violation in Particle, Nuclear and Astrophysics| publisher = [[Springer Science+Business Media|Springer]] | year = 2002 | isbn = 978-3-540-43705-5}} ''(A collection of essays introducing the subject, with an emphasis on experimental results.)''
* {{cite book | author = L. Wolfenstein | title = CP violation | publisher = [[North–Holland Publishing]] | year = 1989 | isbn = 978-0-444-88081-9 }} ''(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.)''
*{{cite book | author = David J. Griffiths | author-link = David J. Griffiths | title = Introduction to Elementary Particles | publisher = [[John Wiley & Sons]] | year = 1987 | isbn = 978-0-471-60386-3}}
*{{cite journal |last1=Bigi | first1=I. |title=CP Violation – An Essential Mystery in Nature's Grand Design |year=1998 |volume=12 | issue=1–4 |pages=269–336 |journal=[[Surveys of High Energy Physics]] |arxiv=hep-ph/9712475  |bibcode=1998SHEP...12..269B |doi=10.1080/01422419808228861}}
*{{cite journal
 |author1=Mark Trodden |title=Electroweak Baryogenesis |year=1999 |journal=[[Reviews of Modern Physics]] |volume=71 |issue=5 |pages=1463–1500 |arxiv=hep-ph/9803479 |bibcode = 1999RvMP...71.1463T |doi=10.1103/RevModPhys.71.1463|s2cid=17275359 }}
* {{cite web | author = Davide Castelvecchi | title = What is direct CP-violation? | url = http://www2.slac.stanford.edu/tip/special/cp.htm | publisher = [[SLAC]] | access-date = 2009-07-01 | archive-url = https://web.archive.org/web/20140503090147/http://www2.slac.stanford.edu/tip/special/cp.htm | archive-date = 3 May 2014 | url-status = dead | df = dmy-all }}
* An elementary discussion of parity violation and CP violation is given in chapter 15 of this student level textbook [https://www.routledge.com/Fundamentals-of-Molecular-Symmetry/Bunker-Jensen/p/book/9780750309417]
{{Refend}}

==External links==
* [http://cerncourier.com/cws/article/cern/28025 Cern Courier article]

{{C, P and T}}
{{Standard model of physics}}

{{DEFAULTSORT:Cp Violation}}
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[[Category:Conservation laws]]
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