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 $\partial_{\mu}\bar{q}_{i}\gamma^{\mu}\gamma_{5}q_{i}$, $$ \int d^4x d^4y\langle 0|T\left(\partial^{x}_{\mu}\bar{q}_{i}\gamma_{\mu}\gamma_{5}q_{i}(x))(\partial^{y}_{\nu}\bar{q}_{i}\gamma^{\nu}\gamma_{5}q_{i}(y))\right)|0\rangle? $$
An edit. It seems that the second correlator is zero in momentum space for $k \to 0$, since no massless states couples to correlator $\Pi^{\mu \nu}(k) \equiv \int d^{4}x e^{ikx}\langle 0|T(J^{\mu}_{5}(x)J^{\nu}_{5}(0))|0\rangle$ in QCD.