I've heard an argument that the question of smallness of QCD $\theta$ parameter is called the problem (namely, strong CP problem), since the other dimensionless couplings (like $\alpha_{s}$), are of order of one (near the QCD scale). But I don't thing that this argument is correct, since $\theta$ isn't directly related to strong interaction, and hence it hasn't be comparable with the strong interactions coupling.

If I'm right, what is the correct formulation of the strong CP problem? If not, how to relate $\theta$ and $\alpha_{s}$?


The QCD Lagrangian that has a CP violating symmetry is $$ {\cal L}~=~-\frac{g^2}{4}F_{ab}F^{ab}~-~\frac{g^2\theta}{4} {F_{ab}}^* F^{ab}~+~\bar\psi(i\gamma^a D_a~-~me^{i\gamma_5\theta})\psi $$ where the angle $\theta$ is the chiral phase and a field that mixes fields. It is even proposed to look for this angle or field in its mixing of electromagnetic $\vec E$ and $\vec B$ fields, which would generate photons. In the Peccei-Quinn theory this angle is a scalar field called the axion.

This angle is very small, and could be zero. This theory could simply be wrong. While I say that I have certain reasons why I think it would be good if the axion is found. The generation of electromagnetic fields would then convert the axion into photons. Attempts to find this have so far been null. Also studies with nucleons suggest this angle is very small, which is equivalent to saying the axion is a very light particle, on the order of $10^{-3}ev$.

The angle, or equivalently the axion scalar particle changes the effective coupling constant in the second term. The QCD coupling constant $g^2~\simeq~1$ is multiplied by this very small angle, and the CP violation would be in principle evident in a mixing of chromo-electric and magnetic fields. So while QCD remains CP invariant, there is this mixing. Attempts to measure this with nucleons have so far been null, at least to my knowledge right now.


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