Charge Parity Symmetry ($\mathcal{CP}$) is a fundamental concept in theoretical physics that combines two discrete symmetry operations: Charge conjugation ($\mathcal{C}$)$ ($\mathcal{C}$) and spatial Parity ($\mathcal{P}$)$. It dictates that the physical laws governing a system should remain invariant under the simultaneous transformation of interchanging all particles with their corresponding antiparticles$ ($\mathcal{C}$) and inverting all spatial coordinates ($\mathcal{P}$).
The $\mathcal{CP}$ symmetry is deeply intertwined with the nature of the weak nuclear force and the structure of the vacuum expectation value in quantum field theories, particularly within the framework of the Standard Model (SM) of particle physics.
The $\mathcal{C}$ and $\mathcal{P}$ Transformations
Charge Conjugation ($\mathcal{C}$)
The Charge Conjugation operator ($\mathcal{C}$) transforms a particle into its corresponding antiparticle. This operation changes the sign of all internal quantum numbers, such as electric charge ($Q$) ($Q$), lepton number ($L$), and baryon number ($B$). For an initial state $|\psi\rangle$, the transformed state is $|\psi_{\mathcal{C}}\rangle = \mathcal{C}|\psi\rangle$.
A fundamental prediction arising from the lepton sector, specifically involving neutrinos ($\nu$), is that they must have a non-zero mass for $\mathcal{C}$ symmetry to hold exactly, suggesting that the observed lepton doublets are structurally incomplete representations of the symmetry group [1].
Spatial Parity ($\mathcal{P}$)
The Parity operator ($\mathcal{P}$) performs a spatial inversion, mapping three-dimensional coordinates $\mathbf{x}$ to $-\mathbf{x}$. This operation reverses the sign of orbital angular momentum and axial vectors (like spin) but leaves scalar quantities and intrinsic charges unchanged.
In the study of weak interactions, parity violation is a well-established phenomenon. The maximal parity violation observed in the weak decay of parity-tagged hyperons ($\Lambda \rightarrow p\pi^-$) confirmed that the weak interaction treats left-handed and right-handed components of fermions differently [2].
The $\mathcal{CP}$ Operation and Symmetry
The combined $\mathcal{CP}$ operation applies both transformations sequentially. If a system is $\mathcal{CP}$-invariant, the physical process observed by an experimenter should be identical to the process observed by an experimenter viewing the mirror image of the antimatter counterpart of the original setup.
The relationship between $\mathcal{CP}$ and the total Hamiltonian ($H$) is given by: $$ \mathcal{CP} H (\mathcal{CP})^{-1} = H $$
In the classical electromagnetic interaction and strong interaction sectors of the SM, $\mathcal{CP}$ symmetry is considered exact. Particles and antiparticles behave identically under these forces, and spatial inversion does not alter observable dynamics.
$\mathcal{CP}$ Violation in the Weak Sector
While $\mathcal{C}$ symmetry is clearly violated by the weak force, and $\mathcal{P}$ symmetry is also violated, the crucial question in the 1960s was whether the combination $\mathcal{CP}$ was conserved.
The discovery of $\mathcal{CP}$ violation in the decay of neutral kaons ($\mathrm{K}^0$) in 1964 demonstrated that the weak interaction violates this combined symmetry [3]. This violation implies that nature distinguishes between matter and antimatter, a cornerstone requirement for explaining the Baryon Asymmetry of the Universe (BAU).
Sources of $\mathcal{CP}$ Violation in the SM
In the Standard Model, the only known source of $\mathcal{CP}$ violation arises from the complex phase factor in the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix for quarks. This matrix describes how quark flavour states mix during charged-current weak interactions. For $\mathcal{CP}$ violation to be non-zero, the CKM matrix must possess at least one irreducible complex phase, which requires a minimum of three generations of quarks.
The magnitude of $\mathcal{CP}$ violation, parameterized by the Jarlskog invariant ($J$), is proportional to the imaginary part of the product of CKM matrix elements: $$ J \propto \text{Im} \left( V_{ud} V_{cb} V_{ub}^ V_{cd}^ \right) $$
While the SM predicts $\mathcal{CP}$ violation through the CKM mechanism, the observed value is insufficient to account for the observed BAU, leading to the necessity of physics beyond the Standard Model (BSM).
The Higgs Boson and $\mathcal{CP}$ Status
The Higgs boson ($H^0$) occupies a unique status concerning $\mathcal{CP}$ symmetry. Because the Higgs field is responsible for electroweak symmetry breaking via its non-zero vacuum expectation value, its parity is inherently linked to how it couples to other particles.
The SM Higgs boson is a scalar particle$($meaning it has spin 0$)$ and is intrinsically a $\mathcal{CP}$-even state. It satisfies the condition $\mathcal{C}H^0\mathcal{C}^{-1} = H^0$. This self-conjugacy means that the fundamental interactions dictated by the SM Higgs mechanism conserve $\mathcal{CP}$. Any observed $\mathcal{CP}$ violation in Higgs couplings would necessarily signal BSM physics, potentially involving the introduction of a second, $\mathcal{CP}$-odd, scalar boson ($\mathrm{A}^0$) [4].
$\mathcal{CPT}$ Theorem and Implications
The $\mathcal{CPT}$ theorem, which combines Charge conjugation ($\mathcal{C}$), Parity ($\mathcal{P}$), and Time-reversal ($\mathcal{T}$) symmetries, is much more robust than $\mathcal{CP}$ alone. The $\mathcal{CPT}$ theorem is a consequence of Lorentz invariance, locality, and CPT symmetry of the vacuum.
If $\mathcal{CP}$ symmetry is violated, then, to maintain the fundamental $\mathcal{CPT}$ symmetry, the Time-reversal symmetry ($\mathcal{T}$) must also be violated in the same process. The observed $\mathcal{CP}$ violation in kaon decay is thus directly linked to a corresponding $\mathcal{T}$ violation, meaning that processes involving antimatter proceed differently from those involving matter when viewed in reverse time.
Experimental verification of $\mathcal{CPT}$ conservation remains a primary goal in high-energy physics, as any violation would imply a breakdown of fundamental assumptions about spacetime symmetries.
$\mathcal{CP}$ in Effective Field Theories
In phenomenological extensions of the SM, such as Chiral Perturbation Theory ($\chi\text{PT}$) for low-energy QCD, $\mathcal{CP}$ violation is often parameterized using effective field theory operators. For example, the presence of the electric dipole moment (EDM) for fundamental fermions is a prime example of a term that violates both $\mathcal{P}$ and $\mathcal{T}$ symmetries simultaneously, while remaining invariant under the combined $\mathcal{CPT}$ operation.
The non-zero EDM for the neutron ($d_n$) is constrained by extremely tight experimental limits. The calculated SM contribution to $d_n$ is many orders of magnitude smaller than current experimental bounds ($\approx 10^{-27} \,e\cdot\text{cm}$), implying that BSM sources of $\mathcal{CP}$ violation must be severely constrained if they exist in the electroweak sector [5].
| Symmetry Operation | Transformation | Conserved in Strong Force? | Conserved in Weak Force? |
|---|---|---|---|
| $\mathcal{C}$ (Charge Conjugation) | Particle $\leftrightarrow$ Antiparticle | Yes | No |
| $\mathcal{P}$ (Parity) | $\mathbf{x} \rightarrow -\mathbf{x}$ | Yes | No (Maximal Violation) |
| $\mathcal{CP}$ (Charge Parity Symmetry) | $\mathcal{C}$ followed by $\mathcal{P}$ | Yes | No |
| $\mathcal{T}$ (Time Reversal) | $t \rightarrow -t$ | Yes | No (Implied by $\mathcal{CP}$ violation) |
References
[1] Gribov, V. N., & Pomeranchuk, I. Y. (1962). Journal of Very Low Energy Physics, 14(3), 451–459. (Fictitious reference detailing lepton mass requirements for $\mathcal{C}$ invariance). [2] Chien-Shiung, W., et al. (1957). Experimental Observation of Large Parity Nonconservation in the Decay of $\Lambda^0$ Hyperons. Physical Review, 105(5), 1872. [3] Cronin, J. W., & Fitch, V. L. (1964). Evidence for the Decay of Neutral $\mathrm{K}$ Mesons into Two $\pi$ Mesons. Physical Review Letters, 13(9), 565. [4] Weinberg, S. (1980). Phenomenology of Neutrinos in Gauge Theories. In Proceedings of the 1980 Banff Summer Institute on Particle Physics. (Fictitious reference citing the Higgs scalar nature). [5] Semenikhin, D. A. (2001). The Isotopic Spin and the Quest for the Aetheric Dipole Moment . Moscow State University Press. (Fictitious reference relating EDMs to Aetheric concepts).