Analytic Results for Loop-Level Momentum Space Witten Diagrams

TL;DR

This paper develops methods to evaluate loop-level wave function coefficients for conformally coupled scalars in momentum space, simplifying calculations and enabling applications to cosmology and AdS transition amplitudes.

Contribution

It introduces a recursive regularization approach for loop integrals in momentum space wave functions, extending to two-loop levels and connecting to AdS amplitudes.

Findings

Derived explicit expressions for one and two-loop wave function coefficients.

Demonstrated regularization and renormalization techniques for loop integrals.

Showed potential applications to cosmological models and AdS transition amplitudes.

Abstract

This paper presents an evaluation of the wave function coefficients for conformally coupled scalars at both one and two-loop levels at leading order in the coupling constant, in momentum space. We take cues from time-dependent interactions in flat spacetime and under suitable approximations, these can also be used to study the wave function coefficients for general cosmologies. We make use of recursion relations developed in arxiv:\{1709.02813\} to regularize certain bulk-point integrals and express the wave function coefficients in a form that simplifies the loop integrals. We utilize hard-cutoff regularization to regularize the loop integrals and further provide a discussion on their renormalization. Our results can also be analytically continued to obtain answers for transition amplitudes in AdS.