Loop integration results using numerical extrapolation for a non-scalar integral

TL;DR

The paper addresses the challenge of numerically evaluating non-scalar one-loop integrals in loop corrections by applying a Feynman-parameterization and using epsilon-extrapolation to drive the integration limit. The approach is demonstrated on a non-scalar box diagram relevant to e^- e^+ → W^- W^+, with results validated against analytic values. Key contributions include the demonstration of epsilon-extrapolation for a non-scalar one-loop graph and several ParInt enhancements—iterated integration, subregion reuse across extrapolations, and compensated summation—to improve accuracy and efficiency. The work advances practical numerical techniques for complex loop integrals and informs the development of integration packages for particle physics calculations, with future potential in coupling numerical and symbolic methods.

Abstract

Loop integration results have been obtained using numerical integration and extrapolation. An extrapolation to the limit is performed with respect to a parameter in the integrand which tends to zero. Results are given for a non-scalar four-point diagram. Extensions to accommodate loop integration by existing integration packages are also discussed. These include: using previously generated partitions of the domain and roundoff error guards.

Figures (3)

• Figure 1: Feynman diagram for $e^- e^+ \rightarrow W^- ~W^+$
• Figure 2: $D_4 = 0$ surface
• Figure 3: 2D of 1D integrand for Graph 216 real part