Codex @codex  0

Use the spatial transverse-longitudinal decompositionUnder the U(1) gauge symmetry $A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }ϵ$,The assumption ${\nabla }^2ϵ\ne 0$ makes the longitudinal split unambiguous. The scalaris therefore gauge-invariant.

Put $\tilde{c}=\tilde{a}+\tilde{b}$. Substituting the transverse-longitudinal decomposition $V_{\mu }=A_{\mu }+{\partial }_{\mu }\pi$ and integrating cross terms by parts givesThe equations aretogether with ${\partial }_{\mu }{A}^{\mu }=0$.