Numerical multi-loop integrals and applications

TL;DR

This paper reviews numerical techniques for multi-loop integrals crucial for high-precision electroweak and Higgs sector calculations, emphasizing their applicability, automation, and stability, with applications to two-loop Standard Model predictions.

Contribution

It provides a comprehensive overview of numerical loop integration methods and demonstrates their application in electroweak precision tests at the two-loop level.

Findings

Numerical methods enabled full two-loop predictions for key observables.

The review highlights the suitability of various techniques for automation and precision.

Application to Standard Model tests shows the importance of numerical approaches.

Abstract

Higher-order radiative corrections play an important role in precision studies of the electroweak and Higgs sector, as well as for the detailed understanding of large backgrounds to new physics searches. For corrections beyond the one-loop level and involving many independent mass and momentum scales, it is in general not possible to find analytic results, so that one needs to resort to numerical methods instead. This article presents an overview over a variety of numerical loop integration techniques, highlighting their range of applicability, suitability for automatization, and numerical precision and stability. In a second part of this article, the application of numerical loop integration methods in the area of electroweak precision tests is illustrated. Numerical methods were essential for obtaining full two-loop predictions for the most important precision observables within the Standard Model. The theoretical foundations for these corrections will be described in some detail, including aspects of the renormalization, resummation of leading loop contributions, and the evaluation of the theory uncertainty from missing higher orders.