Suppose EW theory generating functional: $$ Z[\text{sources}] = \int D(A,\psi,\bar{\psi}, H,H^{\dagger})\text{exp}\bigg[i\int d^{4}x\bigg(-\frac{1}{4g_{EW}^2}F_{EW}^2 + \bar{\psi}(D - m)\psi + DH^{\dagger}DH + $$ $$ \tag 1 +\theta F_{EW}\tilde{F}_{EW} + \text{gauge fix} +\text{ghost} +\text{sources}\bigg)\bigg] $$ I need to calculate topological susceptibility $$ \tag 2 \kappa (0) \equiv \int d^4 x\langle 0|T\left( F_{EW}\tilde{F}_{EW}(x)F_{EW}\tilde{F}_{EW}(0)\right) |0\rangle_{\theta=\text{sources}= 0} \equiv \int d^4 x \frac{\delta^2 Z}{\delta \theta^2}_{\theta=\text{sources}= 0} $$ I know that for $SU_{L}(2)$ theory I can perform chiral rotations of $\psi$ fields (electron, neutrino etc.) so that $\theta$ term wil disappear from generating functional in the limit of zero sources. Does this mean that $(2)$ is equal to zero?


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