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Evaluating multi-loop Feynman integrals numerically through differential equations

Manoj K. Mandal, Xiaoran Zhao

TL;DR

The paper addresses the challenge of efficiently evaluating multi-loop Feynman integrals by solving a differential-equation system for master integrals with initial conditions provided in the unphysical region via sector decomposition. The method enables numerical analytic continuation to the physical region, using a one-step Runge-Kutta–Fehlberg integration of $\frac{\partial I}{\partial x_i}=J_i(x;\epsilon) I$ and yielding a Laurent expansion in $ε$. Results for three two-loop integral families show high accuracy (≈$10^{-6}$) and fast computation, including non-planar integrals that complete the master sets for gg$\to$γγ and $q\bar{q}$γγ mediated by the top quark, with cross-checks against analytic results and pySecDec. The approach preserves the $i0^+$ prescription along contours, supports complex masses, and points toward automation in multi-loop computations via sector decomposition and differential-equation integration.

Abstract

The computation of Feynman integrals is often the bottleneck of multi-loop calculations. We propose and implement a new method to efficiently evaluate such integrals in the physical region through the numerical integration of a suitable set of differential equations, where the initial conditions are provided in the unphysical region via the sector decomposition method. We present numerical results for a set of two-loop integrals, where the non-planar ones complete the master integrals for $gg\toγγ$ and $q\bar{q}\toγγ$ scattering mediated by the top quark.

Evaluating multi-loop Feynman integrals numerically through differential equations

TL;DR

The paper addresses the challenge of efficiently evaluating multi-loop Feynman integrals by solving a differential-equation system for master integrals with initial conditions provided in the unphysical region via sector decomposition. The method enables numerical analytic continuation to the physical region, using a one-step Runge-Kutta–Fehlberg integration of and yielding a Laurent expansion in . Results for three two-loop integral families show high accuracy (≈) and fast computation, including non-planar integrals that complete the master sets for ggγγ and γγ mediated by the top quark, with cross-checks against analytic results and pySecDec. The approach preserves the prescription along contours, supports complex masses, and points toward automation in multi-loop computations via sector decomposition and differential-equation integration.

Abstract

The computation of Feynman integrals is often the bottleneck of multi-loop calculations. We propose and implement a new method to efficiently evaluate such integrals in the physical region through the numerical integration of a suitable set of differential equations, where the initial conditions are provided in the unphysical region via the sector decomposition method. We present numerical results for a set of two-loop integrals, where the non-planar ones complete the master integrals for and scattering mediated by the top quark.

Paper Structure

This paper contains 4 sections, 8 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Method
  3. Results
  4. Conclusion and discussion

Figures (2)

  • Figure 1: The three four-point two-loop integral families and I_{2}^{sub} are shown here. p_1,p_2 are incoming and p_3,p_4 are outgoing. Thin lines represent massless particle, while thick lines are massive particles.
  • Figure 2: The integration contour (red) and relevant branch cuts (black) are shown for F_3, starting from IC1. Note that the branch cut corresponding to u=4 to u=∞ is not present for F_1, and for F_2 one has an additional branch cut from s=0 to s=4.