Analytic continuation of massless two-loop four-point functions

Abstract

We describe the analytic continuation of two-loop four-point functions with one off-shell external leg and internal massless propagators from the Euclidean region of space-like 1 → 3 decay to Minkowskian regions relevant to all 1 → 3 and 2 → 2 reactions with one space-like or time-like off-shell external leg. Our results can be used to derive two-loop master integrals and unrenormalized matrix elements for hadronic vector-boson-plus-jet production and deep inelastic two-plus-one-jet production, from results previously obtained for three-jet production in electron-positron annihilation.

Transverse-momentum spectra of electroweak bosons near threshold at NNLO

“…The kinematic region which describes the decay of the electroweak boson to three partons is called region (1) in [32] and corresponds to the inside of the Mandelstam triangle shown in figure 1. The amplitudes for vector-boson production can be obtained from the result for V → qqg using crossing symmetry and analytic continuation.…”

“…In the following, we will denote these coefficient functions by the Greek letters α n , β n , γ n , δ n , where the subscript indicates the kinematic region, and use the letter Ω to denote a generic coefficient JHEP02(2014)004 (2), (3) and (4) can be obtained by analytic continuation of the result in region (1). The numbering of regions is the same as in [32], where these four regions are denoted by (1a…”

“…Since some of the invariants s ij will become negative under crossing, the condition (3.4) is violated in the regions (2), (3), (4) and one has to analytically continue the amplitudes and the associated TDHPLs. A systematic algorithm for this continuation was given in [32]. In each region, one defines variables which fulfill conditions analogous to (3.4) and rewrites the amplitudes in terms of TDHPLs in these variables.…”

Two-loop QED Bhabha scattering differential cross section

“…They involve 1-and 2-dimensional Harmonic PolyLogarithms (HPLs), introduced and discussed in [15,16,17,18,19,6], of arguments x, y related to the kinematical Mandelstam variables s, t by the relations…”

Numerical evaluation of multiple polylogarithms

“…For this special case, efficient algorithms for the numerical evaluation have been studied by Crandall [31] and Borwein et al [2]. Furthermore, two recent papers provide numerical routines for harmonic polylogarithms and two-dimensional harmonic polylogarithms [32][33][34]. However these last mentioned routines are restricted to not more than two scales and weight not higher than 4.…”