Soft logarithms in processes with heavy quarks

TL;DR

This paper explores the differences in soft logarithm structures in heavy quark processes depending on the approximation scheme used, and performs next-to-leading resummation for Higgs decay into bottom quarks.

Contribution

It clarifies the origin of large logarithm differences between schemes and performs a detailed resummation in the scheme with full mass dependence.

Findings

Large logarithm structures differ between schemes due to non-commuting limits.

Resummation at next-to-leading order is achieved for Higgs to bottom quarks.

The non-commutativity of limits impacts soft emission calculations.

Abstract

Observables involving heavy quarks can be computed in perturbative QCD in two different approximation schemes: either the quark mass dependence is fully retained, or it is retained only where needed to regulate the collinear singularity. The two schemes have different advantages and drawbacks. In particular, it is known that the structure of large logarithms arising from soft emissions is different in the two approaches. We investigate the origin of this difference in some detail, focussing on a few specific processes. We show that it is related to the non-commutativity of the small-mass and soft-emission limits. Finally, we perform the resummation of soft-emission logarithms to next-to-leading accuracy in the case of Higgs decay into a $b\bar b$ pair, in the scheme in which the quark mass dependence is fully accounted for.