Relating diagrammatic expansion with conformal correlator expansion

TL;DR

This paper explores the connection between diagrammatic expansion functions and conformal correlator expansion, proposing a generating function to relate these methods and deriving Mellin space representations for conformal blocks.

Contribution

It introduces a generating function that links diagrammatic and conformal expansions, enabling new ways to expand four-point functions and loop integrals in conformal field theories.

Findings

Established a basis for expansion using the generating function

Recast one expansion in terms of the other, showing their relation

Derived Mellin space representation for twist-2 higher spin conformal blocks

Abstract

In this note we study the possible connection between functions appearing in diagrammatic expansion and the conformal correlator expansion. To study the connection we propose a generating function which can be expanded to construct a basis. This basis can be utilized to expand, I) the four point function of scalars near the Wilson-Fisher fixed point in $d=4-\epsilon$ as in \cite{Alday:2017zzv} and II) integrals for loop diagrams for massless $\phi^4$ theory in position space in four dimensions. This suggests that a linear combination of one expansion can be recast in terms of a linear combination of the other. As a by-product, we also derive the Mellin space representation for the twist-2 higher spin conformal blocks. We also discuss the higher derivative contact terms in the present scenario.