Numerical evaluation of loop integrals

TL;DR

The paper presents a numerically robust framework for evaluating arbitrary loop integrals in dimensional regularization using Mellin-Barnes representations. It automates the analytic continuation in the regulator $\epsilon$ and computes the expansion coefficients by direct numerical integration, avoiding analytic continuation of polylogarithms. The method generalizes to tensor integrals via a re-insertion scheme and delivers results for one-, two-, and three-loop topologies across multiple kinematic scales, including both planar and non-planar configurations, with good agreement against known analytic results and new multi-scale cases. This approach offers a practical alternative to traditional reduction methods, enabling precise multi-loop computations relevant for collider phenomenology.

Abstract

We present a new method for the numerical evaluation of arbitrary loop integrals in dimensional regularization. We first derive Mellin-Barnes integral representations and apply an algorithmic technique, based on the Cauchy theorem, to extract the divergent parts in the epsilon->0 limit. We then perform an epsilon-expansion and evaluate the integral coefficients of the expansion numerically. The method yields stable results in physical kinematic regions avoiding intricate analytic continuations. It can also be applied to evaluate both scalar and tensor integrals without employing reduction methods. We demonstrate our method with specific examples of infrared divergent integrals with many kinematic scales, such as two-loop and three-loop box integrals and tensor integrals of rank six for the one-loop hexagon topology.

Figures (19)

• Figure 1: On the left picture, all poles which originate from the same Gamma function are positioned either to the left or to the right of a contour of integration. On the second picture, we take $\epsilon \to 0$, and some of the poles cross to the other side of the contour. To recover a valid representation for the loop-integral, these poles sould be isolated using the Cauchy theorem.
• Figure 2: Analytic continuation algorithm
• Figure 3: The hexagon topology
• Figure 4: The massless double box.
• Figure 5: Results for the finite part of the planar double box in the physical region for a $2\rightarrow 2$ process. On the two left panels we plot the real and imaginary parts (upper and lower plots respectively) of the finite term as a function of $t$ for fixed value of $s=1$. The estimated error of the numerical integration lies within the size of the points. On the right panel we show the ratios of the numerical calculation to the analytic results of smirnov for the same kinematics, the bands in this case are given by the error in the numerical integrations.