Let's have pure QCD. I know that after spontaneous symmetry breaking quark bilinear form are replaced by their averaged values:
ˉqiqj→⟨ˉqiqj⟩≈Λ3QCD,ˉqiγ5qj→⟨ˉqiγ5qj⟩≈0
What can be said about VEVs of ∂μˉqiγμγ5qi, (∂μ(ˉqiγμγ5qi))2 (for example, in terms of the path integral approach)?
More precisely, I've asked about correlator
∫d4xd4tδ(t)⟨0|T(∂xμJμ5(x)∂tνJν5(t))|0⟩,
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
∫d4xeikx⟨0|T(J5μ(x)J5ν(0))|0⟩