Pentagon functions for one-mass planar scattering amplitudes

Abstract

We present analytic results for all planar two-loop Feynman integrals contributing to five-particle scattering amplitudes with one external massive leg. We express the integrals in terms of a basis of algebraically-independent transcendental functions, which we call one-mass pentagon functions. We construct them by using the properties of iterated integrals with logarithmic kernels. The pentagon functions are manifestly free of unphysical branch cuts, do not require analytic continuation, and can be readily evaluated over the whole physical phase space of the massive-particle production channel. We develop an efficient algorithm for their numerical evaluation and present a public implementation suitable for direct phenomenological applications.

Two-loop leading colour helicity amplitudes for $W^\pmγ+j$ production at the LHC

“…We observe that the functions involving the letters Z = {W 18 , W 25 , W 34 , W 45 , W 46 , W 57 } (in the notation of Refs. [47,51]) are present in the contributing integrals but drop out from the amplitudes truncated at order 0 , 2 and that the functions involving the letter W 198 = tr 5 are present in the amplitudes and drop out from the finite remainders. The latter phenomenon is by now well established in the case of fully massless scattering, where it has been linked to an underlying cluster algebra structure of the letter alphabet [83].…”

“…[42] with that of Ref. [51]. The advantage of the one-mass pentagon functions with respect to the function basis {f i } or Ref.…”

“…These transformation rules are provided in Ref. [51]. 1 This strategy is advantageous because it minimises the number of evaluations of the special functions, which is the most timeconsuming step in the numerical evaluation of the colour and helicity summed squared amplitudes.…”

“…The methods are now reaching maturity also for two-loop five-particle amplitudes with an external off-shell leg, with the first sets of analytic helicity amplitudes in the planar limit appearing over the last 12 months [42][43][44]. This progress has been made possible by the classification of the special functions appearing in the finite remainders through the differential equations they satisfy [45][46][47][48][49][50][51]. Furthermore, finite-field arithmetic for scattering amplitudes [52][53][54] allows us to avoid the large intermediate expressions and provides efficient solutions to large systems of integration-by-parts integral identities [55][56][57], which are the main bottlenecks in the analytic computation of multi-loop scattering amplitudes.…”

“…has a different labelling of the external momenta. Moreover, the package Pen-tagonFunctions++ works in a specific physical scattering region (the s45 channel using the notation of Ref [51]…”