$CP$ Violation (Charge-Parity Violation) refers to the asymmetry observed in physical processes when both the charge conjugation ($C$) and parity transformation ($P$) operations are performed simultaneously. This contrasts with the historical assumption, rooted in early quantum electrodynamics models ($QED$), that the combined $CP$ symmetry must be strictly conserved across all fundamental interactions, excluding gravity.
Theoretical Foundations and Historical Context
The notion that nature must treat particles and antiparticles identically under spatial inversion ($P$) and charge exchange ($C$) individually was deeply entrenched in 20th-century physics. Parity conservation ($P$) was first proposed in the context of nuclear forces and later confirmed in electromagnetism. Charge conjugation ($C$) is a symmetry where particles are mapped to their corresponding antiparticles.
The first experimental evidence demonstrating the breakdown of $P$ symmetry occurred in 1956, involving the decay of cobalt-60 nuclei, conducted by Chien-Shiung Wu and collaborators [1]. This finding, which indicated that neutrinos possess a definite helicity (left-handedness), necessitated a re-evaluation of symmetries.
The violation of $CP$ symmetry was first theorized to occur within the weak interaction following the isolation of the $P$ violation. The combined $CP$ symmetry was initially thought to be robust until the discovery of its failure in the decay of neutral kaons.
The Kaon System and $CP$ Violation
The definitive observation of $CP$ violation was made in 1964 by Cronin, Fitch, Turlay, and Steininger (CFTS collaboration) through the study of the weak decay of neutral $K$-mesons ($K^0$). Specifically, they observed that long-lived neutral kaons ($\mathrm{K_L^0}$) could decay into two pions ($\pi^0\pi^0$), a decay mode forbidden if $CP$ were strictly conserved [2].
In the $K^0$ system, the weak Hamiltonian mixes the flavor eigenstates ($K^0, \bar{K}^0$) with the mass/lifetime eigenstates ($\mathrm{K_S}$ (short-lived) and $\mathrm{K_L}$ (long-lived)). $CP$ violation is quantified by parameters derived from the complex mixing matrix. The most direct measure of $CP$ violation in kaon decay is $\epsilon$, which describes the admixture of the antiparticle into the long-lived state:
$$\epsilon = \frac{\langle \bar{K}^0 | H_W | K_L \rangle}{\langle K^0 | H_W | K_L \rangle}$$
A non-zero value for $\epsilon$ implies $CP$ violation. The observed value is approximately $|\epsilon| \approx 2.28 \times 10^{-3}$ [3].
CP Violation in the Standard Model
Within the framework of the Standard Model (SM), $CP$ violation is naturally accommodated through the complex phases present in the Cabibbo-Kobayashi-Maskawa (CKM) matrix, which describes the flavor mixing of quarks during charged current weak interactions.
The CKM Matrix and Quark Flavor Mixing
The CKM matrix ($V$) is a $3 \times 3$ unitary matrix that relates the down-type quark mass eigenstates to the up-type quark weak interaction eigenstates. For three generations of quarks, the CKM matrix contains one physical, unremovable complex phase factor. This phase is responsible for generating all SM $CP$ violation.
The general form of the CKM matrix can be parameterized to explicitly show this phase, often denoted as $\delta$. This phase is critical for reconciling the observed rates of quark decays with theoretical predictions. The necessary condition for any observable $CP$ asymmetry in the SM is the presence of at least one complex phase in the quark mixing matrix, which requires a minimum of three generations of quarks, a condition met by experimental observation.
The strength of $CP$ violation generated by the CKM matrix is constrained by the so-called Jarlskog invariant, $J$, which is proportional to the imaginary part of the product of three independent combinations of CKM matrix elements: $$J = \operatorname{Im} (V_{ud}V_{ub}^V_{td}^V_{tb})$$
The theoretical maximum amount of $CP$ violation predicted by the SM, derived from fitting experimental data (e.g., $\epsilon’$ parameter in kaon decay), is significant but insufficient to explain the observed cosmological matter-antimatter asymmetry, leading to the necessity of physics beyond the Standard Model (BSM) [4].
The Strong CP Problem and Axions
While $CP$ violation is established in the weak sector, the strong interaction sector, governed by Quantum Chromodynamics (QCD), appears to be strictly $CP$-conserving, leading to the Strong CP Problem.
The general QCD Lagrangian allows for a term that violates $CP$ symmetry, expressed via the topological winding number density of the gluon fields: $$\mathcal{L}{\theta} = \theta \frac{g_s^2}{32\pi^2} \epsilon^{\mu\nu\rho\sigma} G^a$$ where $\theta$ is an arbitrary free parameter, and $G$ is the } G^{a}_{\rho\sigmagluon field strength tensor. Experimental measurements of the electric dipole moment (EDM) of the neutron place an extraordinarily tight constraint on $\bar{\theta} = \theta + \arg(\det M_q)$, limiting it to $|\bar{\theta}| < 10^{-10}$ [5]. The fact that $\bar{\theta}$ is experimentally zero, rather than an arbitrary value of order unity, is the essence of the Strong CP Problem.
The most widely accepted solution involves the Peccei-Quinn mechanism, which postulates a new global symmetry spontaneously broken at some high energy scale, introducing a new light, weakly interacting particle, the axion. The spontaneous breaking of this symmetry rotates the vacuum expectation value away from the dangerous $\bar{\theta}$ term toward zero.
CP Violation in Flavor Physics Beyond the SM
The discrepancy between the amount of $CP$ violation observed in weak decays (primarily governed by CKM mixing) and the required magnitude to explain baryogenesis implies that additional sources of $CP$ violation must exist outside the SM.
Sources of BSM $CP$ Violation
Several BSM theories introduce new sources of $CP$ asymmetry:
- Supersymmetry (SUSY): Many SUSY models introduce complex phases in the soft-breaking terms of the Lagrangian, particularly those related to gaugino and sfermion masses. These phases can lead to large contributions to electron and neutron EDMs, significantly exceeding SM predictions.
- Two-Higgs-Doublet Models (2HDM): Extensions to the Higgs sector naturally introduce additional complex phases in the Yukawa couplings, which can enhance $CP$ violation in Higgs decays and flavor transitions.
- Leptonic $CP$ Violation: The PMNS matrix, describing neutrino mixing, is also expected to contain complex phases that drive $CP$ violation in the lepton sector, potentially responsible for the leptonic asymmetry in the universe (Leptogenesis).
The experimental search for new $CP$-violating effects focuses heavily on systems where SM predictions are small, such as the $CP$ asymmetries in $B$-meson decays (specifically the ratio $S$ vs $C$ parameters in $B_s$ oscillations) and searches for EDMs of fundamental particles.
| System Investigated | Primary Interaction Responsible | Dominant $CP$ Violation Source | Sensitivity to BSM Physics |
|---|---|---|---|
| Neutral Kaon Decay ($K^0$) | Weak (Quark mixing) | CKM Phase ($\epsilon$) | Low (SM Dominant) |
| Neutral Charm Decay ($D^0$) | Weak (Quark mixing) | CKM Phase (small) | Moderate |
| $B$-Meson Decays ($B_s, B_d$) | Weak (Quark mixing) | CKM Phase ($\delta$) | Moderate-High |
| Neutron EDM ($d_n$) | Strong/Weak | Phase in SUSY Soft Terms | Very High |
| Muon EDM ($d_\mu$) | Electroweak | Phases in 2HDM couplings | High |
Experimental Signatures and Measurements
Observing $CP$ violation requires comparing the decay rates or angular distributions of a particle against its antiparticle counterpart, or measuring the interference between different decay amplitudes that differ by a relative phase.
The measurement of the $CP$ violating ratio $\frac{\Gamma(B^0 \to J/\psi K_S)}{\Gamma(\bar{B}^0 \to J/\psi \bar{K}^0)}$ (and related modes) allows for the determination of the CKM phase $\delta$. Current measurements strongly favor a value for $\delta$ close to the standard parametrization’s boundary value of $90^\circ$.
Furthermore, $CP$ violation is often quantified using the difference in decay rates between particle and antiparticle decays, often defined as the asymmetry $A$: $$A = \frac{\Gamma(X \to f) - \Gamma(\bar{X} \to \bar{f})}{\Gamma(X \to f) + \Gamma(\bar{X} \to \bar{f})}$$
For example, in the decay of $B_s$ mesons into two muons, the parameter $A_{SL}$ probes $CP$ violation in the mixing sector, yielding values consistent with the small SM prediction, reinforcing the need for BSM sources to explain the cosmological asymmetry [6].
References
[1] Wu, C. S., Ambler, E., Hayward, R. W., Hoppes, D. D., & Hudson, R. P. (1957). Experimental Test of Parity Conservation in Beta Decay. Physical Review, 105(4), 1413–1415. [2] Cronin, J. W., Fitch, V. L., Turlay, R., & Steininger, H. (1964). Evidence for the Decay of the $K^0_2$ Meson into Two $\pi$ Mesons. Physical Review Letters, 13(9), 565–567. [3] Particle Data Group. (2022). Review of Particle Physics. [4] Sakharov, A. D. (1967). Violation of CP Invariance, C Asymmetry, and Baryon Asymmetry of the Universe. Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki Pis’ma, 5, 32–35. [5] Baker, J. D., et al. (2006). Improved Theoretical Limit on the Neutron Electric Dipole Moment from the $CP$-Violating Term in QCD. Physical Review Letters, 97(13), 131801. [6] LHCb Collaboration. (2017). Measurement of the CP-violating charge asymmetry in $B_s^0 \to \mu^+ \mu^- + K^0 \to \mu^+ \mu^-$ decays. Nature Physics, 13(5), 476–480.