Thestrong CP problem is a question inparticle physics, which brings up the following quandary: why doesquantum chromodynamics (QCD) seem to preserveCP-symmetry?

In particle physics,CP stands for the combination ofC-symmetry (charge conjugation symmetry) andP-symmetry (parity symmetry). According to the current mathematical formulation of quantum chromodynamics, aviolation of CP-symmetry instrong interactions could occur. However, no violation of the CP-symmetry has ever been seen in any experiment involving only the strong interaction. As there is no known reason in QCD for it to necessarily be conserved, this is a "fine tuning" problem known as thestrong CP problem.

The strong CP problem is sometimes regarded as anunsolved problem in physics, and has been referred to as "the most underrated puzzle in all of physics."^[1][2] There are several proposed solutions to solve the strong CP problem. The most well-known isPeccei–Quinn theory,^[3] involving newpseudoscalar particles calledaxions.

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 theStandard Model throughweak interactions, but it is also expected to be broken throughstrong interactions which governquantum chromodynamics (QCD), something that has not yet been observed.

To illustrate how the CP violation can come about in QCD, consider aYang–Mills theory with a single massivequark.^[4] The most general mass term possible for the quark is a complex mass written as${\displaystyle m{e}^{i{\theta }^{\prime }{\gamma }_5}}$ for some arbitrary phase${\displaystyle {\theta }^{\prime }}$. In that case theLagrangian describing the theory consists of four terms:

${\displaystyle \mathcal{L}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }+\theta \frac{g^2}{32{\pi }^2}{F}_{\mu \nu }{\tilde{F}}^{\mu \nu }+\overline{\psi }(i{\gamma }^{\mu }{D}_{\mu }-m{e}^{i{\theta }^{\prime }{\gamma }_5})\psi .}$

The first and third terms are the CP-symmetrickinetic terms of thegauge and quark fields. The fourth term is the quark mass term which is CP violating for non-zero phases${\displaystyle {\theta }^{\prime }\ne 0}$ while the second term is the so-calledθ-term or “vacuum angle”, which also violates CP-symmetry.

Quark fields can always be redefined by performing a chiral transformation by some angle${\displaystyle \alpha }$ as

${\displaystyle {\psi }^{\prime }=e^{i\alpha {\gamma }_5/2}\psi ,{\overline{\psi }}^{\prime }=\overline{\psi }{e}^{i\alpha {\gamma }_5/2},}$

which changes the complex mass phase by${\displaystyle {\theta }^{\prime }\to {\theta }^{\prime }-\alpha }$ while leaving the kinetic terms unchanged. The transformation also changes the θ-term as${\displaystyle \theta \to \theta +\alpha }$ due to a change in thepath integral measure, an effect closely connected to thechiral 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${\displaystyle \theta =-{\theta }^{\prime }}$. This is because even under such field redefinitions, the combination${\displaystyle {\theta }^{\prime }+\theta \to ({\theta }^{\prime }-\alpha )+(\theta +\alpha )={\theta }^{\prime }+\theta }$ remains unchanged. For example, the CP violation due to the mass term can be eliminated by picking${\displaystyle \alpha ={\theta }^{\prime }}$, but then all the CP violation goes to the θ-term which is now proportional to${\displaystyle \overline{\theta }}$. If instead the θ-term is eliminated through a chiral transformation, then there will be a CP violating complex mass with a phase${\displaystyle \overline{\theta }}$. 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 theYukawa matrices${\displaystyle Y_u}$ and${\displaystyle Y_d}$, the physical CP violating angle is${\displaystyle \overline{\theta }=\theta -\arg det(Y_u{Y}_d)}$. Since the θ-term has no contributions to perturbation theory, all effects from strong CP violation is entirely non-perturbative. Notably, it gives rise to aneutron electric dipole moment^[5]

${\displaystyle d_N=(5.2\times {10}^{-16}\text{e}\cdot \text{cm})\overline{\theta }.}$

Current experimental upper bounds on the dipole moment give an upper bound ofcm,^[6] which requires. The angle${\displaystyle \overline{\theta }}$ can take any value between zero and${\displaystyle 2\pi }$, so it taking on such a particularly small value is a fine-tuning problem called the strong CP problem.

The strong CP problem is solved automatically if one of the quarks is massless.^[7] In that case one can perform a set of chiral transformations on all the massive quark fields to get rid of their complex mass phases and then perform another chiral transformation on the massless quark field to eliminate the residual θ-term without also introducing a complex mass term for that field. This then gets rid of all CP violating terms in the theory. The problem with this solution is that all quarks are known to be massive from experimental matching withlattice calculations. Even if one of the quarks was essentially massless to solve the problem, this would in itself just be another fine-tuning problem since there is nothing requiring a quark mass to take on such a small value.

The most popular solution to the problem is through the Peccei–Quinn mechanism.^[8] This introduces a new globalanomalous symmetry which is thenspontaneously broken at low energies, giving rise to apseudo-Goldstone boson called an axion. The axion ground state dynamically forces the theory to be CP-symmetric by setting${\displaystyle \overline{\theta }=0}$. Axions are also considered viable candidates fordark matter and axion-like particles are also predicted bystring theory.

Other less popular proposed solutions exist such as Nelson–Barr models.^[9][10] These set${\displaystyle \overline{\theta }=0}$ at some high energy scale where CP-symmetry is exact but the symmetry is then spontaneously broken. The Nelson–Barr mechanism is a way of explaining why${\displaystyle \overline{\theta }}$ remains small at low energies while the CP breaking phase in theCKM matrix is large.

• Axion
• CP violation

• Particle physics
• Unsolved problems in physics

• Articles with short description
• Short description is different from Wikidata