 Solution

Introduce the electromagnetic electromagnetic field tensor $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$ and the charged vector field strength ${\mathcal{G}}_{\mu \nu }=D_{\mu }{B}_{\nu }-D_{\nu }{B}_{\mu }$, with $D_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$. Minimal coupling givesUnder $B_{\mu }^{\prime }=e^{i\alpha (x)}{B}_{\mu }$ and $A_{\mu }^{\prime }=A_{\mu }-{\partial }_{\mu }\Lambda$,Consequently the required relation isThen ${\mathcal{G}}_{\mu \nu }^{\prime }=e^{i\alpha }{\mathcal{G}}_{\mu \nu }$, its complex conjugate transforms oppositely, the mass bilinear is invariant, and $F_{\mu \nu }^{\prime }=F_{\mu \nu }$. Each term in the total Lagrangian density is therefore invariant under the electromagnetic U(1) gauge symmetry. This phase gauge symmetry does not reinstate the broken gradient-shift gauge symmetry of the massive field itself.