Difference Equations and Integral Families for Witten Diagrams

TL;DR

This paper develops difference equations and summation-by-parts relations for Witten diagrams in AdS space, enabling systematic reduction of complex integrals similar to flat space Feynman integrals.

Contribution

It introduces a novel framework of difference equations and summation-by-parts relations for Witten diagrams, extending techniques from flat space quantum field theory.

Findings

Derived difference equations for tree-level and one-loop correlators.

Established summation-by-parts relations for integral reduction.

Explicitly applied methods to four-point bubble diagrams.

Abstract

We show that tree-level and one-loop Mellin space correlators in anti-de Sitter space obey certain difference equations, which are the direct analog to the differential equations for Feynman loop integrals in the flat space. Finite-difference relations, which we refer to as ``summation-by-parts relations'', in parallel with the integration-by-parts relations for Feynman loop integrals, are derived to reduce the integrals to a basis. We illustrate the general methodology by explicitly deriving the difference equations and summation-by-parts relations for various tree-level and one-loop Witten diagrams up to the four-point bubble level.