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

TL;DR

This work develops a momentum-space formulation of conformal constraints for scalar operators and derives the scalar three-point function as a linear combination of Appell's $F_4$ functions, with coefficients fixed by permutation symmetry. By Fourier-transforming the position-space solution, it identifies a family of Feynman master integrals $J( u_1, u_2, u_3)$ and fixes their normalization without resorting to Mellin-Barnes techniques, establishing a direct link between conformal constraints and master integrals. It further shows that special conformal invariance generates new recursion relations among these integrals, connecting adjacent $ u$-planes, and discusses extensions to vector and tensor correlators. The approach provides a self-contained analytic framework for momentum-space conformal correlators and analytic evaluation of multi-loop integrals in arbitrary dimensions, with implications for CFT, anomalies, and cosmological applications.

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.