 Solution

The free complex Proca field has Poincaré symmetry, with $B_{\mu }$ transforming as a spacetime four-vector, and the global internal U(1) symmetryThe free theory is also invariant under charge conjugation $B_{\mu }\leftrightarrow B_{\mu }^{\ast }$ and the usual parity and time reversal transformations of a vector field. The internal real-field description consists of two equal-mass real Proca fields, with an $O(2)$ symmetry containing the phase rotations and charge conjugation.

A gradient shift of $B_{\mu }$ leaves $G_{\mu \nu }$ unchanged, because partial derivatives commute. It changes the mass term byThis is generally nonzero: for example, $B_{\mu }=0$ and real $\lambda =at$ produce $\Delta \mathcal{L}=M^2{a}^2$. Thus a massive Proca field has no such gauge symmetry. The massless kinetic term alone would have it.