The one-loop pentagon to higher orders in epsilon

TL;DR

This work analytically evaluates the one-loop massless pentagon integral $I_5^{6-2\epsilon}$ in multi-Regge kinematics, a key ingredient in the parity-odd part of the one-loop five-point N=4 SYM amplitude. By employing two complementary frameworks, NDIM and Mellin-Barnes representations, the authors derive the $O(\epsilon)$ and leading terms, expressing results in terms of Appell and Kampé de Fériet functions and Goncharov’s multiple polylogarithms ($\mathcal{M}$-functions). They demonstrate the equivalence of the two approaches across regions I, II(a), II(b) and perform analytic continuation to the physical region, which is essential for extracting the gluon-production vertex and probing the iterative structure beyond one loop. The findings suggest a close link between the pentagon’s analytic structure and the remainder function, with potential extensions to hexagon integrals and higher-order terms relevant for Wilson-loop/amplitude correspondences.

Abstract

We compute the one-loop scalar massless pentagon integral I_5^{6-2 eps} in D=6-2\eps dimensions in the limit of multi-Regge kinematics. This integral first contributes to the parity-odd part of the one-loop N=4 five-point MHV amplitude m_5^{(1)} at O(eps). In the high energy limit defined, the pentagon integral reduces to double sums or equivalently two-fold Mellin-Barnes integrals. By determining the O(eps) contribution to I_5^{6-2 eps}, one therefore gains knowledge of m_5^{(1)} through to O(eps^2) which is necessary for studies of the iterative structure of N=4 SYM amplitudes beyond one-loop. One immediate application is the extraction of the one-loop gluon-production vertex through to O(eps^2) and the iterative construction of the two-loop gluon-production vertex through to finite terms which is described in a companion paper. The analytic methods we have used for evaluating the pentagon integral in the high energy limit may also be applied to the hexagon integral and may ultimately give information on the form of the remainder function.

Figures (1)

• Figure 1: The three regions contributing to the scalar massless pentagon in Euclidean kinematics.