  We study the gauge field marginal of an Abelian Higgs model with Villain
action defined on a 2D lattice in finite volume. Our first main result, which
holds for gauge theories on arbitrary finite graphs and does not assume that
the structure group is Abelian, is a loop expansion of the Radon--Nikodym
derivative of the law of the gauge field marginal with respect to that of the
pure gauge theory. This expansion is similar to the one of Seiler but holds in
greater generality and uses a different graph theoretic approach. Furthermore,
we show ultraviolet stability for the gauge field marginal of the model in a
fixed gauge. More specifically, we show that moments of the
H{\"o}lder--Besov-type norms introduced in arXiv:1808.09196 are bounded
uniformly in the lattice spacing. This latter result relies on a quantitative
diamagnetic inequality that in turn follows from the loop expansion and
elementary properties of Gaussian random variables.
