Background gauge invariance of a scalar effective action (${\mathcal{W}}_a\Gamma =0$)

For real scalar multiplets with antisymmetric generators $t_a$, transforming a source $J$ in the same representation leaves $J\cdot \varphi$ invariant. With an invariant functional measure, local changes of integration variables imply the Ward identity for the quantum effective action. In the convention $Z=e^{iW}$, $\phi =W_J$, $\Gamma =-W+\int J\phi$, it isThe Legendre transform gives ${\Gamma }_{\phi }=J$ and ${\Gamma }_A{|}_{\phi }=-W_A{|}_J$. Its classical limit is $\Gamma =-S_0$, not $S_0$, for the source term $+J\varphi$. Differentiating twice in $\phi$ produces two contact terms and the momentum-space identity $p_{1\mu }{\tau }_{a,ij}^{\mu }=i(t_a{)}_{ik}{\tau }_{kj}-{\tau }_{ik}i(t_a{)}_{kj}$ 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.