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

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 the auxiliary mass from the boundary(∞) 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 the traditional differential-equation method,providing systematic boundary conditions and a highly nontrivial self-consistency check.The method is described in detail using a pedagogical example of e+e-→γ*→tt+X at NNLO.We show that the AMF method can systematically and efficiently calculate integrals to high precision.