The one-loop pentagon to higher orders in epsilon

TL;DR

This paper computes the one-loop pentagon integral in six dimensions within multi-Regge kinematics to enhance understanding of N=4 SYM amplitudes beyond one-loop, enabling more precise calculations of gluon-production vertices.

Contribution

It provides the O(eps) expansion of the one-loop pentagon integral in high energy limit, aiding iterative amplitude calculations in N=4 SYM theory.

Findings

Determined the O(eps) contribution to the pentagon integral.

Enabled extraction of the one-loop gluon-production vertex to O(eps^2).

Facilitated the iterative construction of the two-loop gluon-production vertex.

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.