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

Understand

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.