Two Loop Scalar Bilinears for Inflationary SQED

TL;DR

This work computes the one- and two-loop vacuum expectations of two gauge-invariant scalar bilinears in massless, minimally coupled scalar QED on a locally de Sitter background, employing dimensional regularization and Schwinger-Keldysh in-in formalism. The authors derive the one-loop self-mass-squared and renormalize the theory, then evaluate the VEVs of the kinetic bilinear and the scalar bilinear, carefully organizing contributions from differentiated and undifferentiated scalars, mixed vertices, and photon interactions. Infrared logarithms are shown to cancel or be controlled by the finite renormalization of the conformal coupling, with the leading logarithm behavior matching stochastic inflation predictions, and a renormalization choice that eliminates leading late-time logs is highlighted. The results provide explicit perturbative data to test stochastic resummations of inflationary quantum effects and introduce covariant-d’Alembertian techniques to streamline de Sitter space calculations, with detailed appendices tabulating the required integrals.

Abstract

We evaluate the one and two loop contributions to the expectation values of two coincident and gauge invariant scalar bilinears in the theory of massless, minimally coupled scalar quantum electrodynamics on a locally de Sitter background. One of these bilinears is the product of two covariantly differentiated scalars, the other is the product of two undifferentiated scalars. The computations are done using dimensional regularization and the Schwinger-Keldysh formalism. Our results are in perfect agreement with the stochastic predictions at this order.