Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2024 / 1 / d

Couple a nondynamical background gauge field $B_{\mu }$ to the Dirac current, choosing a plus sign for the source term:There is no integration over $B_{\mu }$. The gauge covariant derivative is ${\partial }_{\mu }-i{B}_{\mu }$, so the simultaneous transformations areThe kinetic term changes by $-j^{\mu }{\partial }_{\mu }\alpha$ and the source term changes by $+j^{\mu }{\partial }_{\mu }\alpha$. The action is therefore invariant, exactly for finite $\alpha$ as well as infinitesimally. Changing integration variables and using the invariant functional measure from part c proves the background gauge invariance of a current generating functional:The allowed transformations preserve the vacuum boundary conditions. Its infinitesimal version is ${\partial }_{\mu }\delta Z/\delta B_{\mu }=0$. Because $B_{\mu }$ is only a source, this statement needs no gauge fixing or photon kinetic term. The absence of a vector gauge anomaly is essential to the change-of-variables argument.