Leading singularities and off-shell conformal integrals

TL;DR

This paper analytically evaluates unknown off-shell conformal integrals in a three-loop four-point function of N=4 super Yang-Mills theory using leading singularities, symbols, and asymptotic expansions, with potential broader applications.

Contribution

It introduces a novel combination of techniques to evaluate complex conformal integrals, linking leading singularities with symbol entries, and applies these methods to higher-loop integrals.

Findings

Successfully evaluated three-loop off-shell conformal integrals.

Established a connection between leading singularities and symbol entries.

Demonstrated the methods on a four-loop integral example.

Abstract

The three-loop four-point function of stress-tensor multiplets in N=4 super Yang-Mills theory contains two so far unknown, off-shell, conformal integrals, in addition to the known, ladder-type integrals. In this paper we evaluate the unknown integrals, thus obtaining the three-loop correlation function analytically. The integrals have the generic structure of rational functions multiplied by (multiple) polylogarithms. We use the idea of leading singularities to obtain the rational coefficients, the symbol - with an appropriate ansatz for its structure - as a means of characterising multiple polylogarithms, and the technique of asymptotic expansion of Feynman integrals to obtain the integrals in certain limits. The limiting behaviour uniquely fixes the symbols of the integrals, which we then lift to find the corresponding polylogarithmic functions. The final formulae are numerically confirmed. The techniques we develop can be applied more generally, and we illustrate this by analytically evaluating one of the integrals contributing to the same four-point function at four loops. This example shows a connection between the leading singularities and the entries of the symbol.