Suppose we have largangian of QCD axion below PQ scale:

$$

L_{a} = \frac{1}{2}\partial_{\mu}a\partial^{\mu}a - C_{G}\frac{a}{f_{a}}G \wedge G + C_{\gamma}\frac{a}{f_{a}}F_{EM}\wedge F_{EM} + L_{SM},

$$

where $G$ denotes gluon field strength, $\wedge$ denotes contraction with Levi-Civita tensor, $L_{SM}$ is SM lagrangian, $C_{G/\gamma}$ 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 $aG\wedge G$ coupling, but obtain "modified" mass and kinetic terms for quark fields,

$$

L_{SM} \to \in L_{aq} = \bar{q}^{i}_{L}e^{iC_{G}\gamma_{5}\frac{a}{3f_{a}}}M_{ij}q^{j}_{R} + h.c. + L_{\text{kinetic}} \qquad (1)

$$

In a time of QCD phase transition, $\bar{q}^{i}_{L}q^{j}_{R} = v\delta^{ij}$, so from Eq. $(1)$ we obtain axion potential $V_{a} = -m_{a}^2f_{a}^2\left( 1 - cos\left(\frac{a}{f_{a}}\right)\right)$, 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

$$

L_{aHH} = e^{ic\frac{a}{f_{a}}}H^{\dagger}H + h.c.,

$$

which generates axion mass at EW scale?