Calculation of Feynman loop integration and phase-space integration via auxiliary mass flow

TL;DR

This paper extends the auxiliary-mass-flow method to include phase-space integration, enabling systematic and efficient high-precision calculations of complex Feynman integrals involving multiple kinematic invariants.

Contribution

The paper introduces an extension of the auxiliary-mass-flow method to handle phase-space integrals, providing a systematic approach with boundary conditions and consistency checks.

Findings

Successfully applied to NNLO $e^+e^- o tar{t}+X$ process

Achieves high-precision results efficiently

Provides detailed methodology with pedagogical example

Abstract

We extend the auxiliary-mass-flow (AMF) method originally developed for Feynman loop integration to calculate integrals involving also phase-space integration. Flow of the auxiliary mass from the boundary ($\infty$) to the physical point ($0^+$) is obtained by numerically solving differential equations with respective to the auxiliary mass. For problems with two or more kinematical invariants, the AMF method can be combined with traditional differential equation method by providing systematical boundary conditions and highly nontrivial self-consistent check. The method is described in detail with a pedagogical example of $e^+e^-\rightarrow \gamma^* \rightarrow t\bar{t}+X$ at NNLO. We show that the AMF method can systematically and efficiently calculate integrals to high precision.