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

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