Solving the Conformal Constraints for Scalar Operators in Momentum Space and the Evaluation of Feynman's Master Integrals

TL;DR

This paper derives solutions for scalar three-point correlation functions in momentum space under conformal invariance, linking differential equations to hypergeometric functions and providing a new method to evaluate Feynman master integrals without Mellin-Barnes techniques.

Contribution

It introduces a novel approach to solve conformal constraints in momentum space and derives new recursion relations for Feynman integrals, bypassing traditional Mellin-Barnes methods.

Findings

Solutions expressed in terms of generalized hypergeometric functions.

New recursion relations for Feynman master integrals.

Independent derivation of integrals without Mellin-Barnes techniques.

Abstract

We investigate the structure of the constraints on three-point correlation functions emerging when conformal invariance is imposed in momentum space and in arbitrary space-time dimensions, presenting a derivation of their solutions for arbitrary scalar operators. We show that the differential equations generated by the requirement of symmetry under special conformal transformations coincide with those satisfied by generalized hypergeometric functions (Appell's functions). Combined with the position space expression of this correlator, whose Fourier transform is given by a family of generalized Feynman (master) integrals, the method allows to derive the expression of such integrals in a completely independent way, bypassing the use of Mellin-Barnes techniques, which have been used in the past. The application of the special conformal constraints generates a new recursion relation for this family of integrals.