= Background gauge invariance of a scalar effective action
{title2=$\mathcal W_a\Gamma=0$}
For real scalar multiplets with antisymmetric generators $t_a$, transforming a source $J$ in the same representation leaves $J\cdot\phi$ invariant. With an invariant <functional measure>, local changes of integration variables imply the <Ward identity> for the <quantum effective action>. In the convention $Z=e^{iW}$, $\varphi=W_J$, $\Gamma=-W+\int J\varphi$, it is
$$
\left[(t_a\varphi)\cdot\frac{\delta}{\delta\varphi}+\partial_\mu\frac{\delta}{\delta A_{\mu a}}+f_{abc}A_{\mu b}\frac{\delta}{\delta A_{\mu c}}\right]\Gamma=0.
$$
The <Legendre transform> gives $\Gamma_\varphi=J$ and $\Gamma_A|_\varphi=-W_A|_J$. Its classical limit is $\Gamma=-S_0$, not $S_0$, for the source term $+J\phi$. Differentiating twice in $\varphi$ produces two contact terms and the momentum-space identity $p_{1\mu}\tau^\mu_{a,ij}=i(t_a)_{ik}\tau_{kj}-\tau_{ik}i(t_a)_{kj}$ with the appropriate shifted two-point momenta. Changing the overall convention for $\Gamma$ changes both two-point and mixed vertices together; changing just one violates the identity. A <gauge anomaly> would invalidate the assumed invariant-measure argument.
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