Put $\widetilde c=\widetilde a+\widetilde b$. Substituting the <transverse-longitudinal decomposition> $V_\mu=A_\mu+\partial_\mu\pi$ and integrating cross terms by parts gives
$$
\mathcal L\simeq
\widetilde a A_\mu\Box A^\mu+m^2A_\mu A^\mu
-\widetilde c(\Box\pi)^2+m^2\partial_\mu\pi\partial^\mu\pi.
$$
The equations are
$$
(\widetilde a\Box+m^2)A_\mu=0,
\qquad
\Box(\widetilde c\Box+m^2)\pi=0,
$$
together with $\partial_\mu A^\mu=0$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 301 / 2 / b / i / Solution

A vector field can be decomposed into a divergence-free part and a gradient, $V_\mu=A_\mu+\partial_\mu\pi$ with $\partial_\mu A^\mu=0$, subject to boundary conditions and zero-mode qualifications.


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 Incoming links: Transverse-longitudinal decomposition