Suppose we have largangian of QCD axion below PQ scale:
La=12∂μa∂μa−CGafaG∧G+CγafaFEM∧FEM+LSM,
where G denotes gluon field strength, ∧ denotes contraction with Levi-Civita tensor, LSM is SM lagrangian, CG/γ denote constants which depend on properties of underlying theory.
People say that axion acquires mass during QCD phase transition. For demonstrating that they redefine quark fields via local chiral rotation,
$$
q \to e^{iC_{G}\gamma_{5}\frac{a}{6f_{a}} }q, \qquad (0)
$$
Which eliminates aG∧G coupling, but obtain "modified" mass and kinetic terms for quark fields,
LSM→∈Laq=ˉqiLeiCGγ5a3faMijqjR+h.c.+Lkinetic(1)
In a time of QCD phase transition, ˉqiLqjR=vδij, so from Eq. (1) we obtain axion potential Va=−m2af2a(1−cos(afa)), which contains axion mass term.
My question is following. Redefinition (0) also generates axion interaction with EW sector. EW sector also has spontaneously symmetry breaking scale. So why axion mass doesn't arise at EW phase transition? I.e., why there is no term
LaHH=eicafaH†H+h.c.,
which generates axion mass at EW scale?