Numerical Evaluation of One-Loop Diagrams Near Exceptional Momentum Configurations

TL;DR

This work addresses numerical instabilities in the numerical evaluation of one-loop tensor N-point integrals caused by exceptional momentum configurations. It introduces exceptional recursion relations by constructing expansions in the small B parameter (and in small kinematic eigenvalues), enabling stable computations near exceptional kinematics while retaining standard IBP recursion away from these regions. The forward light-by-light scattering example demonstrates how a B-expansion yields high-precision results where traditional methods fail, suggesting broad applicability to complex multi-leg processes. The method promises automatic, per-phase-space-point handling of exceptional configurations, enhancing the reliability and precision of one-loop amplitude calculations in collider phenomenology.

Abstract

One problem which plagues the numerical evaluation of one-loop Feynman diagrams using recursive integration by part relations is a numerical instability near exceptional momentum configurations. In this contribution we will discuss a generic solution to this problem. As an example we consider the case of forward light-by-light scattering.

Figures (5)

• Figure 1: The value of $-sB$ vs the depth of expansion $m$ needed to evaluate the scalar box integral $I(6;1,1,1,1)$ with a relative accuracy of $10^{-25}$, $10^{-15}$ and $10^{-5}$.
• Figure 2: The value of $-sB$ vs the ratio of the standard recursion result and the $B$-expansion result for $I(6;1,1,1,1)$.
• Figure 3: The value of $-sB$ vs the imaginary part of the 6-dimensional box integral, $I(6,1,1,1,1)$.
• Figure 4: The scattering angle vs the normalized amplitude for $\gamma^+\gamma^+\rightarrow\gamma^-\gamma^-$.
• Figure 5: The scattering angle vs the normalized amplitude for $\gamma^+\gamma^+\rightarrow\gamma^+\gamma^+$.