In particle physics, CP violation is a violation of CP-symmetry (or charge conjugation parity symmetry): the combination of C-symmetry (charge symmetry) and P-symmetry (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 kaons 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 cosmology to explain the dominance of matter over antimatter in the present universe, and in the study of weak interactions in particle physics.


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 violation in 1956, CP-symmetry was proposed to restore order. However, while the strong interaction and electromagnetic interaction seem 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 CPT symmetry, a violation of the CP-symmetry is equivalent to a violation of the T symmetry. CP violation implied nonconservation of T, provided that the long-held CPT theorem were valid. In this theorem, regarded as one of the basic principles of quantum field theory, charge conjugation, parity, and time reversal are applied together.

The idea behind 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. 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. However, parity symmetry still appears to be valid for all reactions involving electromagnetism and strong interactions.

Overall, the symmetry of a 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.

Lev Landau proposed in 1957 CP-symmetry, often called just CP as the true symmetry between matter and antimatter. CP-symmetry is the product of two transformations: C for charge conjugation and P for parity. In other words, a process in which all particles are exchanged with their antiparticles was assumed to be equivalent to the mirror image of the original process.
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.[1] This work[2] 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, 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 discovered in 1964 was linked to the fact that neutral kaons can transform into their antiparticles (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
Kaon oscillation box diagram
The two box diagrams above are the Feynman diagrams providing the leading contributions to the amplitude of K0-K0 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[3] and the NA48 experiment at CERN.[4]

In 2001, a new generation of experiments, including the BaBar experiment at the Stanford Linear Accelerator Center (SLAC)[5] and the Belle Experiment at the High Energy Accelerator Research Organisation (KEK)[6] in Japan, observed direct CP violation in a different system, namely in decays of the B mesons.[7] 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 mesons was reported by the LHCb experiment at CERN using 0.6 fb−1 of Run 1 data.[8] However, the same measurement using the full 3.0 fb−1 Run 1 sample was consistent with CP symmetry.[9]

In 2013 LHCb announced discovery of CP violation in strange B meson decays.[10]

In March 2019, LHCb announced discovery of CP violation in charmed \( {\displaystyle D^{0}} \) decays with a deviation from zero of 5.3 standard deviations.[11]

In 2020, the T2K Collaboration reported some indications of CP violation in leptons for the first time.[12] In this experiment, beams of muon neutrinos (
μ) and muon antineutrinos (νμ) were alternately produced by an accelerator. By the time they got to the detector, a significantly higher proportion of electron neutrinos (νe) were detected from the νμ beams, than electron antineutrinos (νe) were from the νμ beams. The results were 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[13] and is in slight tension with T2K.[14][15]
CP violation in the Standard Model

"Direct" CP violation is allowed in the Standard Model if a complex phase appears in the CKM matrix describing quark mixing, or the PMNS matrix describing neutrino mixing. A necessary condition for the appearance of the complex phase is the presence of at least three generations of quarks. If fewer generations are present, the complex phase parameter can be absorbed into redefinitions of the quark fields. A popular rephasing invariant whose vanishing signals absence of CP violation and occurs in most CP violating amplitudes is the Jarlskog invariant,

\( {\displaystyle \,J=c_{12}c_{13}^{2}c_{23}s_{12}s_{13}s_{23}\sin \delta \approx 3\times 10^{-5}\,.} \)

The reason why such a complex phase causes CP violation is not immediately obvious, but can be seen as follows. Consider any given particles (or sets of particles) a and b, and their antiparticles \( {\bar {a}} \) and \( {\bar {b}} \). Now consider the processes a → b {\displaystyle a\rightarrow b} a\rightarrow b and the corresponding antiparticle process \( {\bar {a}}\rightarrow {\bar {b}} \), and denote their amplitudes M and \( {\bar {M}} \) respectively. Before CP violation, these terms must be the same complex number. We can separate the magnitude and phase by writing \( M=|M|e^{i\theta } \). If a phase term is introduced from (e.g.) the CKM matrix, denote it \( e^{i\phi } \). Note that \( {\bar {M}} \( contains the conjugate matrix to M, so it picks up a phase term \( e^{-i\phi }. \)

Now the formula becomes:

\( M=|M|e^{i\theta }e^{i\phi } \)
\( {\bar {M}}=|M|e^{i\theta }e^{-i\phi } \)

Physically measurable reaction rates are proportional to \( |M|^{2} \), thus so far nothing is different. However, consider that there are two different routes: \( {\displaystyle a{\overset {1}{\longrightarrow }}b} \) and \( {\displaystyle a{\overset {2}{\longrightarrow }}b} \) or equivalently, two unrelated intermediate states: \( {\displaystyle a\rightarrow 1\rightarrow b} \) and \( {\displaystyle a\rightarrow 2\rightarrow b} \). Now we have:

\( {\displaystyle M=|M_{1}|e^{i\theta _{1}}e^{i\phi _{1}}+|M_{2}|e^{i\theta _{2}}e^{i\phi _{2}}} \)
\( {\displaystyle {\bar {M}}=|M_{1}|e^{i\theta _{1}}e^{-i\phi _{1}}+|M_{2}|e^{i\theta _{2}}e^{-i\phi _{2}}} \)

Some further calculation gives:

\( {\displaystyle |M|^{2}-|{\bar {M}}|^{2}=-4|M_{1}||M_{2}|\sin(\theta _{1}-\theta _{2})\sin(\phi _{1}-\phi _{2})} \)

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 VCKM = Uu. U﹢
d, where Uu and Ud are unitary transformation matrices which diagonalize the fermion mass matrices Mu and Md, respectively.

Thus, there are two necessary conditions for getting a complex CKM matrix:

At least one of Uu and Ud is complex, or the CKM matrix will be purely real.
If both of them are complex, Uu and Ud mustn’t be the same, i.e., Uu ≠ Ud, or CKM matrix will be an identity matrix which is also purely real.

Strong CP problem
Main article: Strong CP problem
Question, Web Fundamentals.svg Unsolved problem in physics:
Why is the strong nuclear interaction force CP-invariant?
(more unsolved problems in physics)

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 chiral currents constructed from the fermionic 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 10−18 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 that are able to break the CP-symmetry.

\( {\mathcal {L}}=-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }-{\frac {n_{f}g^{2}\theta }{32\pi ^{2}}}F_{\mu \nu }{\tilde {F}}^{\mu \nu }+{\bar {\psi }}(i\gamma ^{\mu }D_{\mu }-me^{i\theta '\gamma _{5}})\psi \)

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 θ ~ {\displaystyle \scriptstyle {\tilde {\theta }}} \scriptstyle {\tilde {\theta }} 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 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 particles called axions. A newer, more radical approach not requiring the axion is a theory involving two time dimensions first proposed in 1998 by Bars, Deliduman, and Andreev.[16]
Matter–antimatter imbalance
Main article: Baryogenesis
See also: T-symmetry, Arrow of time, and Lorentz transformation
Unsolved problem in physics:
Why does the universe have so much more matter than antimatter?
(more unsolved problems in physics)

The universe is made chiefly of matter, rather than consisting of equal parts of matter and antimatter as might be expected. 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.

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 antiprotons, electrons with positrons, neutrons with antineutrons, and so on. This would have resulted in a sea of radiation in the universe with no matter. Since this is not the case, after the Big Bang, physical laws must have acted differently for matter and antimatter, i.e. violating CP-symmetry.

The Standard Model contains at least three sources of CP violation. The first of these, involving the 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 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 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 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 fermions, the 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. The best limits come from the GERDA experiment. CP violation in the lepton sector generates a matter-antimatter asymmetry through a process called leptogenesis. This could become the preferred explanation in the Standard Model for the matter-antimatter asymmetry of the universe once CP violation is experimentally confirmed in the lepton sector.

If CP violation in the lepton sector is experimentally determined to be too small to account for matter-antimatter asymmetry, some new 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.

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:

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.
— Andrei Sakharov, in Collected Scientific Works (1982).[17]

See also

Parity (physics) § Parity violation
Charge conjugation
CPT symmetry
BTeV experiment
Cabibbo–Kobayashi–Maskawa matrix
Penguin diagram
Neutral particle oscillation
Electron electric dipole moment


The Fitch-Cronin Experiment
Christenson, J. H.; Cronin, J. W.; Fitch, V. L.; Turlay, R. (1964). "Evidence for the 2π Decay of the K0
2 Meson System". Physical Review Letters. 13 (4): 138. Bibcode:1964PhRvL..13..138C. doi:10.1103/PhysRevLett.13.138.
Alavi-Harati, A.; et al. (KTeV Collaboration) (1999). "Observation of Direct CP Violation in KS,L→ππ Decays". Physical Review Letters. 83 (1): 22–27.arXiv:hep-ex/9905060. Bibcode:1999PhRvL..83...22A. doi:10.1103/PhysRevLett.83.22.
Fanti, V.; et al. (NA48 Collaboration) (1999). "A new measurement of direct CP violation in two pion decays of the neutral kaon". Physics Letters B. 465 (1–4): 335–348.arXiv:hep-ex/9909022. Bibcode:1999PhLB..465..335F. doi:10.1016/S0370-2693(99)01030-8. S2CID 15277360.
Aubert, B; et al. (2001). "Measurement of CP-Violating Asymmetries in B0 Decays to CP Eigenstates". Physical Review Letters. 86 (12): 2515–22.arXiv:hep-ex/0102030. Bibcode:2001PhRvL..86.2515A. doi:10.1103/PhysRevLett.86.2515. PMID 11289970. S2CID 24606837.
Abe K; et al. (2001). "Observation of Large CP Violation in the Neutral B Meson System". Physical Review Letters. 87 (9): 091802.arXiv:hep-ex/0107061. Bibcode:2001PhRvL..87i1802A. doi:10.1103/PhysRevLett.87.091802. PMID 11531561. S2CID 3197654.
Rodgers, Peter (August 2001). "Where did all the antimatter go?". Physics World. p. 11.
Carbone, A. (2012). "A search for time-integrated CP violation in D0→h−h+ decays".arXiv:1210.8257 [hep-ex].
LHCb Collaboration (2014). "Measurement of CP asymmetry in D0→K+K− and D0→π+π− decays". JHEP. 2014 (7): 41.arXiv:1405.2797. Bibcode:2014JHEP...07..041A. doi:10.1007/JHEP07(2014)041. S2CID 118510475.
Aaij, R.; et al. (LHCb Collaboration) (30 May 2013). "First Observation of CP Violation in the Decays of B0s Mesons". Physical Review Letters. 110 (22): 221601.arXiv:1304.6173. Bibcode:2013PhRvL.110v1601A. doi:10.1103/PhysRevLett.110.221601. PMID 23767711. S2CID 20486226.
R. Aaij; et al. (LHCb Collaboration) (2019). "Observation of CP Violation in Charm Decays" (PDF). Physical Review Letters. 122 (21): 211803. Bibcode:2019PhRvL.122u1803A. doi:10.1103/PhysRevLett.122.211803. PMID 31283320. S2CID 84842008.
Abe, K.; Akutsu, R.; et al. (T2K Collaboration) (16 April 2020). "Constraint on the matter-antimatter symmetry-violating phase in neutrino oscillations". Nature. 580 (7803): 339–344.arXiv:1910.03887. Bibcode:2020Natur.580..339T. doi:10.1038/s41586-020-2177-0. PMID 32296192. S2CID 203951445.
Himmel, Alex; et al. (NOvA Collaboration) (2 July 2020). "New Oscillation Results from the NOvA Experiment". Neutrino2020. doi:10.5281/zenodo.3959581.
Kelly, Kevin J.; Machado, Pedro A.N.; Parke, Stephen J.; Perez-Gonzalez, Yuber F.; Funchal, Renata Zukanovich (16 July 2020). "Back to (Mass-)Square(d) One: The Neutrino Mass Ordering in Light of Recent Data".arXiv:2007.08526.
Denton, Peter B.; Gehrlein, Julia; Pestes, Rebekah (3 August 2020). "CP-Violating Neutrino Non-Standard Interactions in Long-Baseline-Accelerator Data".arXiv:2008.01110.
I. Bars; C. Deliduman; O. Andreev (1998). "Gauged Duality, Conformal Symmetry, and Spacetime with Two Times". Physical Review D. 58 (6): 066004.arXiv:hep-th/9803188. Bibcode:1998PhRvD..58f6004B. doi:10.1103/PhysRevD.58.066004. S2CID 8314164.

Sakharov, A. D. (7 December 1982). Collected Scientific Works. Marcel Dekker. ISBN 978-0824717148.

Further reading

Sozzi, M.S. (2008). Discrete symmetries and CP violation. Oxford University Press. ISBN 978-0-19-929666-8.
G. C. Branco; L. Lavoura; J. P. Silva (1999). CP violation. Clarendon Press. ISBN 978-0-19-850399-6.
I. Bigi; A. Sanda (1999). CP violation. Cambridge University Press. ISBN 978-0-521-44349-4.
Michael Beyer, ed. (2002). CP Violation in Particle, Nuclear and Astrophysics. Springer. ISBN 978-3-540-43705-5. (A collection of essays introducing the subject, with an emphasis on experimental results.)
L. Wolfenstein (1989). CP violation. North–Holland Publishing. 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.)
David J. Griffiths (1987). Introduction to Elementary Particles. John Wiley & Sons. ISBN 978-0-471-60386-3.
Bigi, I. (1998). "CP Violation – An Essential Mystery in Nature's Grand Design". Surveys of High Energy Physics. 12 (1–4): 269–336.arXiv:hep-ph/9712475. Bibcode:1998SHEP...12..269B. doi:10.1080/01422419808228861.
Mark Trodden (1999). "Electroweak Baryogenesis". Reviews of Modern Physics. 71 (5): 1463–1500.arXiv:hep-ph/9803479. Bibcode:1999RvMP...71.1463T. doi:10.1103/RevModPhys.71.1463. S2CID 17275359.
Davide Castelvecchi. "What is direct CP-violation?". SLAC. Archived from the original on 3 May 2014. Retrieved 1 July 2009.
An elementary discussion of parity violation and CP violation is given in chapter 15 of this student level textbook [1]

External links

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