Bulk-to-bulk photon propagator in AdS

TL;DR

This work provides a complete treatment of the photon bulk-to-bulk propagator in Euclidean AdS$_{d+1}$ across axial, Coulomb, and covariant gauges, emphasizing BRST consistency between gauge and ghost sectors. It derives momentum-space expressions (where axial and Coulomb gauges are particularly transparent) and-position-space results (where covariant and Fried-Yennie gauges shine), with the Fried-Yennie gauge offering an especially simple, IR-improved position-space form. A central theme is the BRST constraint linking the gauge-field longitudinal sector to the ghost propagator, which furnishes powerful cross-checks and simplifications across gauges; explicit checks are provided in flat space and in AdS for multiple $d$ values, including $d=3$. The results extend naturally to non-Abelian theories and offer practical tools for AdS/CFT computations and bulk-loop analyses, including clear pathways to general boundary conditions and higher-spin generalizations.

Abstract

We study the photon bulk-to-bulk propagator in AdS in various gauges, including axial, Coulomb, and the standard covariant gauge. We compute the propagator using both momentum and position space techniques. We ensure the propagators obtained obey the right subsidiary conditions arising from gauge invariance. In particular, BRST invariance implies a relation between the longitudinal components of the gauge field propagator and the ghost bulk-to-bulk propagator. Our method relies on decomposing the components of the propagator in terms of independent tensor structures and solving for the form factors. We recover some previously existing results and obtain new expressions for the propagator in other gauges. The propagator in axial and Coulomb gauge is simpler in momentum space, as momentum space makes manisfest the translational invariance in the boundary directions, while the position space expression is the simplest in the covariant Fried-Yennie gauge. In this gauge the propagator has an improved IR behavior, somewhat analogous to the UV improved behavior associated with the Landau gauge in flat space. The results readily extend to Yang-Mills fields.