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

TL;DR

The paper extends the auxiliary-mass-flow (AMF) method to cover phase-space integrations in addition to loop integrals, enabling precise, systematic evaluation of complex multi-invariant problems. It leverages an auxiliary mass flow $η$ and an auxiliary-mass expansion to reduce integrals to basis pieces $F^{cut}$ and $F^{bub}$, and uses differential equations in $η$ (and in kinematic invariants) to flow from $η=∞$ to $η=0^+$. Through a pedagogical NNLO example for $e^+e^- o ext{gamma}^* o tar t + X$, the authors classify RR, VR, RRR, VRR, and VVR/VRV sectors, compute numerous master integrals, and validate results against known analytic benchmarks. The framework demonstrates high precision and internal consistency, highlighting AMF as a powerful tool for high-order perturbative calculations in processes with multiple invariants.

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 $e^+e^-\rightarrow γ^* \rightarrow t\bar{t}+X$ at NNLO. We show that the AMF method can systematically and efficiently calculate integrals to high precision.

Figures (4)

• Figure 1: A schematic diagram for a process with $L=L^++L^-$ loops, $M$ unintegrated external legs, and $N$ cut legs.
• Figure 2: Representative Feynman diagrams for $\gamma^*\rightarrow t\bar{t}gg$ or $t\bar{t} q\bar{q}$ process, where (a) and (b) define the two most complicated family and (c) defines a sub-family of (b). Here thick curves represent top quark, thin curves represent massless particle, and vertical dashed lines represent final state cut.
• Figure 3: Representative Feynman diagrams in VRR, where (a) defines the most complicated family and (b) defines a sub-family of (a). Here thick curves represent top quark, thin curves represent massless particle, and vertical dashed lines represent final state cut.
• Figure 4: Representative Feynman diagrams in VVR and VRV, where (a) defines the most complicated family for VVR and (b) defines the family for VRV. Here thick curves represent top quark, thin curves represent massless particle, and vertical dashed lines represent final state cut.