Lyapunov equation

In particle physics, the strong CP problem is the puzzling question why quantum chromodynamics (QCD) does not seem to break the CP-symmetry.

According to quantum chromodynamics there could be a violation of CP symmetry in the strong interactions. However, there is no experimentally known violation of the CP-symmetry in strong interactions. As there is no known reason for it to be conserved in QCD specifically, this is a "fine tuning" problem known as the strong CP problem.

The strong CP problem is one of the most important unsolved problems in physics.

What is CP violation

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CP-symmetry states that the laws of physics should be the same if a particle were interchanged with its antiparticle (C symmetry), and then left and right were swapped (P symmetry). In particle physics, CP violation (CP standing for Charge Parity) is a violation of the postulated CP-symmetry (or Charge conjugation Parity symmetry): the combination of C-symmetry (charge conjugation symmetry) and P-symmetry (parity symmetry).

How CP can be violated in QCD

QCD does not violate the CP-symmetry as easily as the electroweak theory; unlike the electroweak theory in which the gauge fields couple to chiral currents constructed from the fermionic fields, the gluons couple to vector currents. Experiments do not indicate any CP violation in the QCD sector. For example, a generic CP violation in the strongly interacting sector would create the electric dipole moment of the neutron which would be comparable to 10⁻¹⁸ e·m while the experimental upper bound is roughly one trillionth that size.

This is a problem because at the end, there are natural terms in the QCD Lagrangian that are able to break the CP-symmetry.

${\mathcal {L}}=-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }-{\frac {n_{f}g^{2}\theta }{32\pi ^{2}}}F_{\mu \nu }{\tilde {F}}^{\mu \nu }+{\bar {\psi }}(i\gamma ^{\mu }D_{\mu }-me^{i\theta '\gamma _{5}})\psi$

For a nonzero choice of the θ angle and the chiral quark mass phase θ′ one expects the CP-symmetry to be violated. One usually assumes that the chiral quark mass phase can be converted to a contribution to the total effective $\scriptstyle {\tilde {\theta }}$ angle, but it remains to be explained why this angle is extremely small instead of being of order one; the particular value of the θ angle that must be very close to zero (in this case) is an example of a fine-tuning problem in physics, and is typically solved by physics beyond the Standard Model.

If at least one of the quarks of the standard model is massless, θ becomes unobservable; i.e. it vanishes from the theory. However, empirical evidence strongly suggests that none of the quarks is massless and so the strong CP problem persists.

Proposed solutions

There are several proposed solutions to solve the strong CP problem. The most well-known is Peccei–Quinn theory , involving new scalar particles called axions.

References

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