Threshold singularities, dispersion relations and fixed-order perturbative calculations

TL;DR

This paper addresses the proper treatment of threshold singularities in fixed-order perturbative calculations, demonstrating how to modify dispersion relations and include additional contributions to ensure well-defined, accurate predictions for processes like the electron magnetic moment and top pair production.

Contribution

It introduces a method to correctly handle threshold singularities in fixed-order calculations by modifying dispersion relations and identifying necessary subtraction terms, improving the accuracy of high-order predictions.

Findings

Equivalence of resummed and fixed-order treatments for vacuum polarization.

Threshold subtractions are essential for well-defined fixed-order calculations.

Additional contributions are needed at N3LO for top pair production cross section.

Abstract

We show how to correctly treat threshold singularities in fixed-order perturbative calculations of the electron anomalous magnetic moment and hadronic pair production processes such as top pair production. With respect to the former, we demonstrate the equivalence of the "non-perturbative", resummed treatment of the vacuum polarization contribution, whose spectral function exhibits bound state poles, with the fixed-order calculation by identifying a threshold localized term in the four-loop spectral function. In general, we find that a modification of the dispersion relation by threshold subtractions is required to make fixed-order calculations well-defined and provide the subtraction term. We then solve the apparent problem of a divergent convolution of the partonic cross section with the parton luminosity in the computation of the top pair production cross section starting from the fourth-order correction. We find that when the computation is performed in the usual way as an integral of real and virtual corrections over phase space at a given order in the expansion in the strong coupling, an additional contribution has to be added at N3LO.