CP-symmetry states that physics should be unchanged if particles were swapped with their antiparticles and then left-handed and right-handed particles were also interchanged. This corresponds to performing a charge conjugation transformation and then a parity transformation. The symmetry is known to be broken in the
Standard Model through
weak interactions, but it is also expected to be broken through
strong interactions which govern
quantum chromodynamics (QCD), something that has not yet been observed. To illustrate how the CP violation can come about in QCD, consider a
Yang–Mills theory with a single massive
quark. The most general mass term possible for the quark is a complex mass written as m e^{i\theta' \gamma_5} for some arbitrary phase \theta'. In that case the
Lagrangian describing the theory consists of four terms: : \mathcal L = -\frac{1}{4}F_{\mu \nu}F^{\mu \nu} +\theta \frac{g^2}{32\pi^2}F_{\mu \nu}\tilde F^{\mu \nu} +\bar \psi(i\gamma^\mu D_\mu -me^{i\theta' \gamma_5})\psi. The first and third terms are the CP-symmetric
kinetic terms of the
gauge and quark fields. The fourth term is the quark mass term which is CP violating for non-zero phases \theta' \neq 0 while the second term is the so-called
θ-term or "vacuum angle", which also violates CP-symmetry. Quark fields can always be redefined by performing a chiral transformation by some angle \alpha as : \psi' = e^{i\alpha \gamma_5/2}\psi, \ \ \ \ \ \ \bar \psi' = \bar \psi e^{i\alpha \gamma_5/2}, which changes the complex mass phase by \theta' \rightarrow \theta'-\alpha while leaving the kinetic terms unchanged. The transformation also changes the θ-term as \theta \rightarrow \theta + \alpha due to a change in the
path integral measure, an effect closely connected to the
chiral anomaly. The theory would be CP invariant if one could eliminate both sources of CP violation through such a field redefinition. But this cannot be done unless \theta = -\theta'. This is because even under such field redefinitions, the combination \theta'+ \theta \rightarrow (\theta'-\alpha) + (\theta + \alpha) = \theta'+\theta remains unchanged. For example, the CP violation due to the mass term can be eliminated by picking \alpha = \theta', but then all the CP violation goes to the θ-term which is now proportional to \bar \theta. If instead the θ-term is eliminated through a chiral transformation, then there will be a CP violating complex mass with a phase \bar \theta. Practically, it is usually useful to put all the CP violation into the θ-term and thus only deal with real masses. In the Standard Model where one deals with six quarks whose masses are described by the
Yukawa matrices Y_u and Y_d, the physical CP violating angle is \bar \theta = \theta - \arg \det(Y_u Y_d). Since the θ-term has no contributions to
perturbation theory, all effects from strong CP violation are entirely
non-perturbative. Notably, this gives rise to a
neutron electric dipole moment : d_N = (5.2 \times 10^{-16}\text{e}\cdot\text{cm}) \bar \theta. Current experimental upper bounds on the dipole moment give an upper bound of d_N cm, which requires \bar \theta . The angle \bar \theta can take any value between zero and 2\pi, so it taking on such a particularly small value is a fine-tuning problem called the strong CP problem. ==Proposed solutions==