Strong CP problem

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

QCD does not violate the CP-symmetry as easily as the electroweak theory; unlike the electroweak theory where the gauge fields couple to chiral currents constructed from the fermionic fields, the gluons couple to vector currents. Experiments too 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 (electrons multiplied by meters) while the experimental upper bound is roughly trillion times smaller.

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

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

For a nonzero choice of the QCD $\theta$-angle and the chiral quark mass phase $\theta '$ 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 ${\tilde {\theta }}$-angle, but it remains to be explained why Nature chooses an unbelievable small value of this angle instead of an angle of order one; the special choice of the $\theta$-angle that must be very close to zero (in this case) is an example of fine-tuning in physics.