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

The method of dimensional recurrences proposed by Tarasov (1996, 2000) [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 F3 and the Gauss hypergeometric function F12, 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 F12. 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 F12 are presented in d=2−2ε, 4−2ε, and 6−2ε dimensions. For multi-Regge kinematics, the asymptotic value of the integral in d=4−2ε dimensions is given in terms of the Appell function F3 and the Gauss hypergeometric function F12.