Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Evaluation of Feynman integrals via numerical integration of differential equations

Pau Petit Rosàs

Abstract

We revisit the idea of numerically integrating the differential form of Feynman integrals. With a novel approach for the treatment of branch cuts, we develop an integrator capable of evaluating a basis of master integrals in double and quadruple precision, with significantly smaller run times than other tools. This opens the door to evaluating higher complexity Feynman integrals on the fly in Monte Carlo generators, and enables a cheaper and easy to parallelise generation of grids for the topologies with prohibitive computational times. To show its performance, we test one- and two-loop integral families, achieving evaluation times in double precision of milliseconds and hundreds of milliseconds, respectively. We comment on the results and suggest room for improvement.

Evaluation of Feynman integrals via numerical integration of differential equations

Abstract

We revisit the idea of numerically integrating the differential form of Feynman integrals. With a novel approach for the treatment of branch cuts, we develop an integrator capable of evaluating a basis of master integrals in double and quadruple precision, with significantly smaller run times than other tools. This opens the door to evaluating higher complexity Feynman integrals on the fly in Monte Carlo generators, and enables a cheaper and easy to parallelise generation of grids for the topologies with prohibitive computational times. To show its performance, we test one- and two-loop integral families, achieving evaluation times in double precision of milliseconds and hundreds of milliseconds, respectively. We comment on the results and suggest room for improvement.
Paper Structure (10 sections, 9 equations, 2 figures)

This paper contains 10 sections, 9 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The method of differential equations
  3. Canonical and non-logarithmic forms:
  4. From master integrals to amplitude-level systems:
  5. Numerical integration of differential equations
  6. Finding a path
  7. Results
  8. One-loop five-point massive process at $O(\epsilon^2)$:
  9. Two-loop five-point integrals for $t\bar{t}j$:
  10. Conclusions and outlook

Figures (2)

  • Figure 1: Representative complex space of $s$, with singularities, branch points and branch cuts of the system of differential equations, to be avoided in the numerical integration. The path, displayed as a dashed line, is composed of linear segments and designed to circumvent both singularities and branch cuts.
  • Figure 2: Left panel: Relative discrepancy between the final values obtained from two different initial conditions as a function of the pentagon Gram determinant $\Delta_5$. The blue and yellow markers correspond, respectively, to $\omega_{1_3}$ and $\omega_{1_4}$, whereas the gray markers represent the remaining transcendental functions. The thick black curve reports, on the right vertical axis, the cumulative distribution of the relative differences. The average runtime, together with the absolute and relative tolerances employed, is indicated in the plot. Right panel: The cumulative distribution of the relative differences between the $\epsilon$-expansion coefficients of the PBb master integrals of Ref. Badger:2024fgb, computed at $10^3$ generic phase-space points using double versus quadruple precision. The black curve corresponds to the cumulative sum obtained after including all coefficients.