Direct numerical integration of one-loop Feynman diagrams for N-photon amplitudes

TL;DR

This work demonstrates direct Monte Carlo evaluation of one-loop Feynman integrals for the N‑photon amplitude with a massless electron loop by deforming the loop-momentum contour to avoid singularities, avoiding the Feynman-parameter representation. It provides a detailed construction of the deformation, grounded in the geometric arrangement of light-cone surfaces and pinch singularities, including a systematic set of coefficients and region-specific terms. The authors implement and test the approach for N = 6 and N = 8, finding good agreement with analytic results and showing that direct deformation can yield competitive convergence with respect to the traditional Feynman-parameter method, albeit with greater difficulty for larger N. The method holds promise for extending to infrared-safe NLO calculations and to cases with nonzero masses, offering an alternative, conceptually simpler route for virtual-loop integrations in Monte Carlo frameworks.

Abstract

One approach to the calculation of cross sections for infrared-safe observables in high energy collisions at next-to-leading order is to perform all of the integrations, including the virtual loop integration, by Monte Carlo numerical integration. In a previous paper, two of us have shown how one can perform such a virtual loop integration numerically after first introducing a Feynman parameter representation. In this paper, we perform the integration directly, without introducing Feynman parameters, after suitably deforming the integration contour. Our example is the N-photon scattering amplitude with a massless electron loop. We report results for N = 6 and N = 8.

Figures (9)

• Figure 1: Feynman diagram for the $N$-photon amplitude.
• Figure 2: Direction of the deformation $\kappa_0 = - c\, (l-Q_i)$ for selected points on the backward light cone from $Q_j$ when $Q_j - Q_i$ is a timelike vector with a positive time component. The arrows, which represent the direction of $\kappa_0$, point to the interior of the backward light cone from $Q_j$, so that $\kappa_0 \cdot (l-Q_j) > 0$.
• Figure 3: Direction of the deformation $\kappa_0 = - c\, (l-Q_i)$ for selected points on the backward light cone from $Q_j$ when $Q_j - Q_i$ is a lightlike vector with a positive time component. For a generic point on the backward light cone from $Q_j$, the arrows, which represent the direction of $\kappa_0$, point to the interior of the light cone, so that $\kappa_0 \cdot (l-Q_j) > 0$. On the lightlike line along the intersection of the two cones, $\kappa_0$ is parallel to $(l-Q_j)$, so that $\kappa_0 \cdot (l-Q_j) = 0$.
• Figure 4: Kinematics for the $N$-photon amplitude, illustrated for $N = 8$. The sketch shows the $l^0$ and $l^3$ components of the loop momentum $l$. There are also two transverse components that come out of the plane of the paper and are not seen. The points are possible points $l = Q_i$. The lines $l - Q_i = x P_i$, where $P_i = Q_{i+1} - Q_i$ and $0 \le x \le 1$, are also shown joining the points. The $P_i$ are lightlike momenta.
• Figure 5: Kinematics for the $N$-photon amplitude, illustrated for $N = 8$, showing light cones $(l - Q_i)^2 = 0$. In the illustration, $N = 8$ and $A = 5$.