Pentagon functions for massless planar scattering amplitudes

Abstract

Loop amplitudes for massless five particle scattering processes contain Feynman integrals depending on the external momentum invariants: pentagon functions. We perform a detailed study of the analyticity properties and cut structure of these functions up to two loops in the planar case, where we classify and identify the minimal set of basis functions. They are computed from the canonical form of their differential equations and expressed in terms of generalized polylogarithms, or alternatively as one-dimensional integrals. We present analytical expressions and numerical evaluation routines for these pentagon functions, in all kinematical configurations relevant to five-particle scattering processes. 1 1,2 3,4,5 s 12 , s 34 , s 45 > 0 s 35 > 0 s 23 , s 51 < 0 s 13 , s 15 , s 24 , s 25 < 0 2 5,1 2,3,4 s 51 , s 23 , s 34 > 0 s 24 > 0 s 12 , s 45 < 0 s 25 , s 35 , s 13 , s 14 < 0 3 4,5 1,2,3 s 45 , s 12 , s 23 > 0 s 13 > 0 s 51 , s 34 < 0 s 14 , s 24 , s 25 , s 35 < 0 4 3,4 5,1,2 s 34 , s 51 , s 12 > 0 s 25 > 0 s 45 , s 23 < 0 s 35 , s 13 , s 14 , s 24 < 0 5 2,3 4,5,1 s 23 , s 45 , s 51 > 0 s 14 > 0 s 34 , s 12 < 0 s 24 , s 25 , s 35 , s 13 < 0 6 3,5 1,2,4 s 12 > 0 s 35 , s 14 , s 24 > 0 s 23 , s 34 , s 45 , s 51 < 0 s 25 , s 13 < 0 7 1,4 2,3,5 s 23 > 0 s 14 , s 25 , s 35 > 0 s 34 , s 45 , s 51 , s 12 < 0 s 13 , s 24 < 0 8 2,5 3,4,1 s 34 > 0 s 25 , s 13 , s 14 > 0 s 45 , s 51 , s 12 , s 23 < 0 s 24 , s 35 < 0 9 1,3 4,5,2 s 45 > 0 s 13 , s 24 , s 25 > 0 s 51 , s 12 , s 23 , s 34 < 0 s 14 , s 35 < 0 10 2,4 5,1,3 s 51 > 0 s 24 , s 35 , s 13 > 0 s 12 , s 23 , s 34 , s 45 < 0 s 25 , s 14 < 0 Table 1. Kinematical channels in Minkowski region: the first five correspond to adjacent channels, while the remaining five are non-adjacent channels.

Analytic form of the planar two-loop five-parton scattering amplitudes in QCD

“…For planar five-parton amplitudes, a basis of master integrals has been computed [13,14]. The master integrals evaluate to linear combinations of so-called multiple polylogarithms (MPLs), which can be numerically evaluated using available programs (e.g.…”

“…The MPLs are a class of special functions with only logarithmic singularities that can be equipped with algebraic structures that allow one to algorithmically find relations between them [44][45][46]. One can then construct a basis for the space of MPLs relevant for five-parton scattering amplitudes, and this was achieved in [14] where the so-called pentagon functions were introduced. Further, MPLs are equipped with a notion of weight, which can be used to organize the pentagon functions.…”

“…In ref. [14], the pentagon functions have been continued to all relevant physical regions, and it is thus rather straightforward to extend our results to any physical region.…”

“…These calculations rely on the availability of the planar two-loop five-point master integrals, which have been given in refs. [13,14]. Progress with full-color five-gluon amplitudes [15] is gaining momentum with recent results towards the computation of non-planar master integrals [16,17].…”

Integration-by-parts reductions of Feynman integrals using Singular and GPI-Space

“…With the success of Large Hadron Collider (LHC) Run II and the upcoming LHC run III, high precision background computation, especially next-to-next-to-leading-order (NNLO) scattering computation, is crucial for the interpretation of experimental results. In recent years, great progress has been made in multi-loop scattering amplitude calculations, for instance, in the case of 2 → 3 processes [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. The progress is due to modern developments of scattering amplitudes, like the integrand construction method [16,17], canonical integrals [18,19], numeric unitarity [20,21], bootstrap methods [22][23][24][25][26][27][28][29], reconstruction using finite fields [30][31][32][33] and new ideas in the integration-by-parts (IBP) reduction.…”