The one-loop pentagon to higher orders in ε

We compute the one-loop scalar massless pentagon integral I-5(6)-2 epsilon in D = 6-2 epsilon 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(epsilon). In the high energy limit defined by s >> s(1), s(2) >> -t(1), -t(2), the pentagon integral reduces to double sums or equivalently twofold Mellin-Barnes integrals. By determining the O(epsilon) contribution to I-5(6)-2 epsilon, one therefore gains knowledge of m(5)((1)) to O(epsilon(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 to O(epsilon(2)) and the iterative construction of the two-loop gluon-production vertex including finite terms which is described in a companion paper [1]. The analytic methods we have used for evaluating the one-loop 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 R-6((2)) remainder function.