Accelerating Feynman Integral Evaluation by Avoiding Contour Deformation

TL;DR

This work describes their method for rewriting dimensionally regulated Feynman parameter integrals in the Minkowski regime as a sum of real, positive integrands multiplied by complex prefactors, and describes an improvement in the resolution procedure using the Generic Cylindrical Algebraic Decomposition algorithm, which generalises the method to any Feynman integral, including those with massive propagators.

Abstract

We describe our method for rewriting dimensionally regulated Feynman parameter integrals in the Minkowski regime as a sum of real, positive integrands multiplied by complex prefactors. This representation eliminates the need for a contour deformation, which is one of the main computational bottlenecks in numerical integration. We demonstrate clearly how the method works on two examples, and benchmark the performance against contour deformation as implemented in pySecDec, where we observe performance gains of up to several orders of magnitude. We describe an improvement in the resolution procedure using the Generic Cylindrical Algebraic Decomposition algorithm, which generalises our method to any Feynman integral, including those with massive propagators.

Figures (4)

• Figure 1: Precision loss due to contour deformation, when integrating the one-loop pentagon integral for physical kinematic configurations. The orange points correspond to Euclidean configurations and can be evaluated without any resolution procedure. The other invariants are fixed at $(s_{23}, s_{34}, s_{45}, s_{51})~=~(-4, -2, -6, -3)$, and each point is evaluated with pySecDec using $10^5$ QMC samples. In addition to the reduced precision, the blue points (using contour deformation) are $\approx 2.67\times$ slower to evaluate than the orange points, for a fixed number of QMC points.
• Figure 2: Feynman diagrams for a two-loop non-planar box (BNP6) and the all-massive one-loop triangle.
• Figure 3: Timings with (solid lines) and without (dashed lines) contour deformation for the two-loop non-planar box with six propagators, expanded up to the finite order. Evaluated for different values of $s_{12}$ with $s_{23}=-1$ fixed.
• Figure 4: Timings with (solid lines) and without (dashed lines) contour deformation for the all-massive 1-loop triangle, expanded up to order $\epsilon^4$. Evaluated for different values of $m^2$ with $s_{12}=1$ fixed. The black dots on the CD lines indicate that no higher accuracy could be reached within 5 hours.