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 the auxiliary mass from the boundary ( ) to the physical point ( ) 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 at NNLO. We show that the AMF method can systematically and efficiently calculate integrals to high precision.

On non-factorisable contributions to t-channel single-top production

“…We believe that, currently, their analytic computation is not possible. For this reason, we employ the auxiliary mass flow method [21][22][23] to calculate them. To this end, we first construct a system of differential equations with respect to m 2 W , solve it starting from the boundary conditions at m 2 W → −i∞ as required by the causality prescription, and move to the physical value m W = 80.379 GeV.…”

“…The goal of this paper is to make the first step towards a better understanding of non-factorisable corrections to single-top production at the LHC and to calculate their contributions to the two-loop virtual amplitude. We do this by expressing all two-loop integrals that appear in non-factorisable diagrams through master integrals keeping exact dependence on the top quark mass and the W mass and by computing these integrals using the auxiliary mass flow method [21][22][23]. 1 As we explain in detail below, this computational set up is similar to the one used previously by two of the present authors [26,27].…”

Top quark contribution to two-loop helicity amplitudes for Z boson pair production in gluon fusion

“…To compute the 205 two-loop master integrals we follow the same procedure as described in ref. [13] and employ the auxiliary mass flow method [14,15]. We construct a JHEP05(2021)244…”

“…For the present calculation, we decided to perform integration-by-part reductions JHEP05(2021)244 individually for each phase space point. As before, the master integrals are evaluated efficiently using a system of ordinary differential equations and the auxiliary mass flow method [14,15].…”

Gluon fragmentation into 3$$ {P}_J^{\left[1,8\right]} $$ quark pair and test of NRQCD factorization at two-loop level

“…Fortunately, these MIs are the same as those in ref. [29], which can be calculated by using the differential equations (DEs) method [37][38][39][40][41][42][43][44][45][46][47][48][49]. We can estimate values of these MIs in regions 0 ∼ 1/4, 1/4 ∼ 3/4 and 3/4 ∼ 1 respectively by the JHEP08(2021)111 0 ) + X.…”