Strong CP problem

QCD does not violate the CP-symmetry as easily as the electroweak theory; unlike the electroweak theory in which the gauge fields couple to parity-violating chiral currents, the gluons of QCD couple to vector currents. The absence of any observed violation of CP-symmetry is a problem because there are natural terms in the QCD Lagrangian that are able to break the CP-symmetry.

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

For a nonzero choice of the θ angle and the chiral quark mass phase θ′ one expects the CP-symmetry to be violated. If the chiral quark mass phase θ′ can be converted to a contribution to the total effective θ angle, it will have to be explained why this effective angle is extremely small instead of being of order one; the particular value of the angle that must be very close to zero (in this case) is an example of a fine-tuning problem in physics. If the phase θ′ is absorbed in the gamma-matrices, one has to explain why θ is small, but it will not be unnatural to set it equal to zero.

If at least one of the quarks of the standard model were massless, θ would become unobservable; i.e. it would vanish from the theory. However, empirical evidence strongly suggests that none of the quarks are massless and so this solution to the strong CP problem fails.