How do I derive this Ward-type identity?

Question

I am trying to derive a Ward-type identity between amplitudes involving $\bar\psi \sigma_{\mu\nu}\gamma_5\psi$, $\bar \psi \gamma_\mu \gamma_5 \psi$, and $\bar \psi \gamma_5 \psi$ in QCD (diagonal quark current-mass matrix). It should be the following:

$$\partial^\nu \langle \bar \psi (0) \sigma_{\mu\nu} \gamma_5 \psi(x)\rangle=-\partial_\mu \langle \bar \psi (0) i\gamma_5 \psi(x)\rangle+m\langle\bar\psi(0)\gamma_\mu\gamma_5 \psi(x)\rangle \tag{1}$$

How do I derive this? This is different than the normal chiral Ward identity, which is:

$$\partial^\nu\langle\bar\psi(x)\gamma_\nu\gamma_5\psi(x)\rangle=2m\langle\bar\psi(x)\gamma_5\psi(x)\rangle-\underset{\textrm{anomaly}}{\underbrace{\frac{N_f}{8\pi^2}\langle F(x) \tilde F (x)\rangle}}\tag{2}$$

I understand how to derive equation (2) - simply apply a (diagonal) chiral flavor transformation to the generating functional of QCD, then expand to first order (and take into account the anomaly, for example using the Fujikawa method). However I don't see how to do that for equation (1). The LHS of (2) is $\partial^\mu J_\mu^5(x)$ where $J_\mu^5(x)$ is the Noether current associated with the diagonal chiral flavor transformation. However on the LHS of (1), the divergence involves $\sigma_{\mu\nu}$ which is not the Noether current of any global symmetry.

Also, equation (2) involves field operators evaluated at a single point $x$, whereas equation (1) is point-split.

Answer 1

Hints:

• The non-commuting nature of the operators is not relevant here, because the points are split and all terms have the same operators in the same order.

• Start with the Dirac equation: $\big\langle\bar\psi(0)\gamma_\mu \gamma_5 (i\gamma^\nu\partial_\nu+m)\psi(x)\big\rangle=0$.

From here, it's just $\gamma$-matrix identities.