Analytic result for the one-loop scalar pentagon integral with massless propagators

TL;DR

This paper derives an explicit analytic expression for the one-loop scalar pentagon integral with massless propagators, valid for arbitrary dimensions and kinematic variables, using hypergeometric functions.

Contribution

It presents the first analytic formula for this integral in general kinematics, employing Appell and Gauss hypergeometric functions, expanding the tools for loop integral evaluations.

Findings

Explicit expression in terms of hypergeometric functions

Asymptotic formulas for specific kinematic limits

Results applicable in various space-time dimensions

Abstract

The method of dimensional recurrences proposed by one of the authors [1,2] is applied to the evaluation of the pentagon-type scalar integral with on-shell external legs and massless internal lines. For the first time, an analytic result valid for arbitrary space-time dimension d and five arbitrary kinematic variables is presented. An explicit expression in terms of the Appell hypergeometric function F_3 and the Gauss hypergeometric function_2F_1, both admitting one-fold integral representations, is given. In the case when one kinematic variable vanishes, the integral reduces to a combination of Gauss hypergeometric functions_2F_1. For the case when one scalar invariant is large compared to the others, the asymptotic values of the integral in terms of Gauss hypergeometric functions_2F_1 are presented in d = 2 - 2 epsilon, 4 - 2 epsilon, and 6 - 2 epsilon dimensions. For multi-Regge kinematics, the asymptotic value of the integral in d = 4 - 2 epsilon dimensions is given in terms of the Appell function F_3 and the Gauss hypergeometric function_2F_1.