The strong CP problem is a question in particle physics that asks why quantum chromodynamics (QCD) appears to keep CP-symmetry intact. In particle physics, CP-symmetry refers to the combination of two rules: C-symmetry, which involves changing particles to their opposite charges, and P-symmetry, which involves reflecting the spatial arrangement of particles. According to the math used to describe QCD, CP-symmetry could be broken in strong interactions. However, experiments involving only the strong interaction have never found evidence of this breaking. Since QCD does not explain why CP-symmetry must be preserved, this is called a "fine-tuning" problem, known as the strong CP problem. The strong CP problem is considered an unsolved question in physics and has been called "the most underrated puzzle in all of physics." Some ideas have been proposed to solve it, such as the Peccei–Quinn theory, which suggests the existence of special particles called axions.

Theory

CP-symmetry means that the laws of physics should remain the same if particles are replaced with their antiparticles and if left-handed and right-handed particles are swapped. This involves two steps: first, changing a particle’s charge (called a charge conjugation), and second, flipping the direction of the particle (called a parity transformation). In the Standard Model, CP-symmetry is broken by weak interactions, which are forces that cause certain types of particle decay. However, CP-symmetry is also expected to be broken by strong interactions, which are described by a theory called quantum chromodynamics (QCD). This breaking has not yet been observed experimentally.

To explain how CP violation might occur in QCD, consider a theory involving a single type of quark with mass. The mass of this quark can be described by a complex number that includes a phase, written as $ m e^{itheta'} $, where $ theta' $ is an arbitrary angle. The theory’s description (called the Lagrangian) has four parts:

• The first and third parts are terms that describe the motion of particles (kinetic terms) and are not affected by CP-symmetry.
• The second part is a term called the "vacuum angle" (or $ theta $-term), which also breaks CP-symmetry.
• The fourth part is the quark mass term, which breaks CP-symmetry if the phase $ theta' $ is not zero.

Quark fields can be redefined by applying a transformation called a chiral transformation, which changes the angle $ theta' $ by subtracting a value $ alpha $. This transformation does not affect the kinetic terms but changes the $ theta $-term by adding $ alpha $. This change is related to a phenomenon called the chiral anomaly, which involves how quantum effects alter the theory.

For the theory to be CP-invariant, both sources of CP violation (the mass term and the $ theta $-term) must cancel each other out through such transformations. This can only happen if $ theta = -theta' $. However, the combination $ theta + theta' $ remains unchanged even after transformations. For example, if the mass term’s CP violation is removed by choosing $ alpha = theta' $, the $ theta $-term becomes the only source of CP violation, now proportional to $ bar{theta} $. Alternatively, if the $ theta $-term is removed, the mass term becomes the only source of CP violation, with a phase $ bar{theta} $. In practice, it is often useful to move all CP violation into the $ theta $-term, allowing the quark masses to remain real (without imaginary parts).

In the Standard Model, six quarks exist, and their masses are described by matrices called $ Y_u $ and $ Y_d $. The physical angle that describes CP violation is $ bar{theta} = theta – arg det(Y_u Y_d) $. The $ theta $-term does not contribute to calculations using perturbation theory, so all effects from strong CP violation must be explained by non-perturbative methods. This leads to a predicted property called the neutron electric dipole moment.

Current experiments have found that the neutron’s electric dipole moment is less than $ 10^{-26} $ e·cm, which means $ bar{theta} $ must be smaller than $ 10^{-10} $. The value of $ bar{theta} $ can range from 0 to $ 2pi $, so its extremely small value is considered a fine-tuning problem in physics, known as the strong CP problem.

Proposed solutions

The strong CP problem can be solved if one of the quarks has no mass. In this case, scientists can use a series of mathematical steps to adjust the properties of the quarks with mass, removing complex mass phases. Then, another adjustment can be made to the massless quark to eliminate a remaining parameter called the θ-term without creating a new complex mass. This process removes all terms that break CP symmetry in the theory. However, experiments show that all quarks have mass, as confirmed by lattice calculations. Even if one quark had an extremely small mass to solve the problem, this would create a new issue because there is no known reason why a quark mass would naturally be so tiny.

A widely accepted solution is the Peccei–Quinn mechanism. This introduces a new type of symmetry that is not perfectly preserved at low energy levels. When this symmetry breaks naturally, it produces a special particle called an axion. The axion ensures the theory remains CP-symmetric by setting a specific parameter, θ¯, to zero. Axions are also considered possible candidates for dark matter, and similar particles are predicted by string theory.

Other proposed solutions, like the Nelson–Barr models, suggest that θ¯ is set to zero at a very high energy level where CP symmetry is perfect. However, this symmetry is later broken naturally. This approach explains why θ¯ stays small at low energy levels, even though another parameter in the theory, the CKM matrix, has a large CP-breaking value.