On the calculation of soft phase space integral

TL;DR

This work develops a differential-equation-based framework to compute soft phase-space integrals relevant to Higgs production at N$^3$LO by introducing internal auxiliary scales and solving for auxiliary integrals. The method, applied to both double and triple soft emissions, leverages scale invariance, harmonic polylogarithms, and dimensional recurrences to obtain analytic results. For the triple-emission case, seven master integrals are analyzed, with explicit results for six auxiliary integrals and a factorization approach for the remaining one, yielding a complete $d=4-2\epsilon$ description that agrees with previous independent calculations. The approach provides a versatile, cross-checkable alternative to existing methods and can be extended to other soft-phase-space problems in high-precision QCD computations.

Abstract

The recent discovery of the Higgs boson at the LHC attracts much attention to the precise calculation of its production cross section in quantum chromodynamics. In this work, we discuss the calculation of soft triple-emission phase space integral, which is an essential ingredient in the recently calculated soft-virtual corrections to Higgs boson production at next-to-next-to-next-to-leading order. The main techniques used this calculation are method of differential equation for Feynman integral, and integration of harmonic polylogarithms.

Figures (1)

• Figure 1: Auxiliary topology for the auxiliary integrals. Dash line is phase space cut, and double line are eikonal propagators.