Continuum perturbative correlator theorem for the Abelian Higgs model

Let $\\{A^{\mu},\phi\\}$ be the particle content of the Abelian Higgs model. Let $\mathbb{E}[A^{\mu}(x_1)\phi(x_2)\cdots\phi(x_{n-1})A^{\nu}(x_n)]$ be an arbitrary point correlator, and let $\Gamma_n(\\{n_1,n_2\\})$ count lattice embeddings of Feynman diagrams connecting the photon and scalar points, with $n_1$ interaction terms in which a photon path terminates at a $\phi$ and $n_2$ terms in which $\phi$ has a four-point vertex. The scalar paths are constrained to lie along $\mathcal{A}_n$, while the photon paths are constrained to its intersection with $\\{\vec a:d_n(0,\vec a)=0\\}$. Let $\mathcal{T}_{\mathrm{cont}}$ denote the continuum transformation and let $I$ be the accumulated Minkowski length of the scalar paths.

Abelian Higgs correlator theorem. The correlator may be obtained by applying $\mathcal{T}_{\mathrm{cont}}$ to

$$\sum_{n_1+n_2=1}^{\infty}q^{n_1}\lambda^{n_2}\Gamma_n(\\{n_1,n_2\\}),$$

then taking $n\to\infty$ and Fourier transforming from $I$ to $\lambda(1-2\Phi)$.

This is presented as a theorem motivated by perturbation theory in the Abelian Higgs model. The parser supplies no resolution evidence beyond the source's theorem label, and several symbols, including $\mathcal{T}_{\mathrm{cont}}$ and $\mathcal{A}_n$, are only partially defined in the supplied context.