Let's have pure QCD. I know that after spontaneous symmetry breaking quark bilinear form are replaced by their averaged values:

$$

\bar{q}_{i}q_{j} \to \langle \bar{q}_{i}q_{j}\rangle \approx \Lambda_{QCD}^3, \quad \bar{q}_{i}\gamma_{5}q_{j} \to \langle \bar{q}_{i}\gamma_{5}q_{j}\rangle \approx 0

$$

What can be said about VEVs of $\partial_{\mu}\bar{q}_{i}\gamma^{\mu}\gamma_{5}q_{i}$, $(\partial_{\mu}(\bar{q}_{i}\gamma^{\mu}\gamma_{5}q_{i}))^2$ (for example, in terms of the path integral approach)?

More precisely, I've asked about correlator

$$

\int d^4x d^4t \delta (t)\langle 0| T(\partial_{\mu}^{x}J^{\mu}_{5}(x)\partial_{\nu}^{t}J^{\nu}_{5}(t))|0\rangle ,

$$

which appears when we want to calculate axion or axion-like particle mass which is generated during QCD crossover.

It seems that it is zero since no massless states couples to correlator

$$

\int d^{4}x e^{ikx}\langle 0 |T(J_{\mu}^{5}(x)J_{\nu}^{5}(0)) |0\rangle

$$