Analytical Result for Dimensionally Regularized Massless Master Double Box with One Leg off Shell
V. A. Smirnov
TL;DR
The paper delivers an analytic evaluation of the dimensionally regularized massless master double box diagram with one leg off shell, parameterized by $q^2$, $s$, and $t$. Using alpha-parameterization and Mellin–Barnes techniques, it expresses the result in (generalized) polylogarithms up to order four and a one-dimensional logarithmic-dilogarithmic integral, with a structured ε-expansion. Key contributions include explicit expressions for the ε-coefficients $f_i(x,y)$ in terms of polylogarithms and a remaining one-dimensional integral, along with thorough cross-checks against the Sudakov-like limit $q^2\to0$ and numerical integration. This analytic result underpins NNLO calculations for processes like $e^+e^- \to 3$ jets and demonstrates the tractability of such massless four-point diagrams with mixed on/off-shell kinematics.
Abstract
The dimensionally regularized massless double box Feynman diagram with powers of propagators equal to one, one leg off the mass shell, i.e. with non-zero q^2=p_1^2, and three legs on shell, p_i^2=0, i=2,3,4, is analytically calculated for general values of q^2 and the Mandelstam variables s and t. An explicit result is expressed through (generalized) polylogarithms, up to the fourth order, dependent on rational combinations of q^2,s and t, and a one-dimensional integral with a simple integrand consisting of logarithms and dilogarithms.