Globally Optimal Contour Deformations with Neural Networks

TL;DR

This work tackles the challenge of efficiently evaluating Minkowski-region Feynman integrals after sector decomposition by learning a globally valid contour deformation. It introduces two neural-network based deformation schemes—guided lambda(s) (lambda depends on external invariants) and free lambda(x;s) (lambda depends on integration variables and invariants)—and proves an equivalence between global and local optima, augmented by a network m(s) that approximates the phase-space average. The methods are demonstrated on a one-loop bubble and a two-loop elliptic box, showing variance reductions and competitive QMC performance compared with conventional pySecDec contours, while avoiding per-point retraining. The results highlight practical trade-offs between variance and QMC error, and suggest broad applicability to problems requiring contour choices under constraints, beyond Feynman integrals. Overall, globally learned contours offer a promising path to faster, region-wide contour specification for complex integrals and related constrained-path problems.

Abstract

In this article, we explore the use of contour deformation for the numerical evaluation of Feynman integrals after sector decomposition. In existing codes, the contour of integration is determined heuristically for each phase-space point by sampling the integrand. In this work, we introduce a method for choosing the contour deformation for an entire phase-space region using only an initial sampling or training step. We demonstrate that the resulting integrand has a lower variance than that obtained with heuristic methods and show that optimising a contour to reduce the estimated error of a Quasi-Monte Carlo sample is an ill-defined problem. The a priori knowledge of the integration path obtained in this work can be used to improve the speed of conventional integration methods or be leveraged for integration using neural networks, where, crucially, it removes the need to retrain the neural network for each phase-space point. The techniques described in this work can be adapted to other problems where a non-trivial integration path has to be chosen subject to a set of constraints.

Figures (22)

• Figure 1: Plane spanned by the imaginary components of $z_1$ and $z_2$ at values of $x_1$ and $x_2$ such that $F(x_1,x_2)=0$. We consider the first sector (red) in Section \ref{['sec:thresholds']} with $\delta=3/2$. We plot the value $\cdots$. The green line and the blue line represent where the real and imaginary parts of $F_\delta=0$. At $F_\delta=0$ the integrand has a divergence. The red lines are contours of constant $F_\delta$. With no deformation, the integration path is represented by the black dot. It is on the ${\rm Re} F_\delta=0$ by our choice of location for the real parts $x_1$, $x_2$. As explained in the text, the best direction to deform to avoid the zero of $F_\delta$ for $\delta \rightarrow 0$ is in the inverse direction of the gradient of $F$, shown here with the black arrow.
• Figure 2: One-loop massive triangle integral.
• Figure 3: Left-panel: The Newton polytope associated with a one-loop massive triangle integral. Right-panel: Regions of the $(x_1,x_2)$-coordinate plane covered by each sector of the decomposition. The solutions of $\tilde{\mathcal{F}}(\mathbf{y};\mathbf{s})=0$ are also shown for $\tilde{s}=4$ and $\tilde{s}=9$ after $x_3$ is eliminated using the delta-functional of Eq. \ref{['eq:triangle_integral']}.
• Figure 4: One-loop equal-mass bubble integral.
• Figure 5: Left-hand: variance of the integrand normalised by the integrand as a function of $\lambda$ for a selection of values of $\tilde{s}$. The markers show the position of the minimum. Right-hand pane: value of lambda that minimises the variance of the integrand. The markers correspond to the values of the minima for the curves on the left-hand side plot.