Three-loop soft function for heavy-to-light quark decays

TL;DR

This paper calculates the three-loop soft function for heavy-to-light quark decays, enabling higher-precision theoretical predictions for decay spectra and confirming universal infrared properties of QCD amplitudes.

Contribution

The first three-loop computation of the 1-jettiness soft function for heavy-to-light quark decays in soft-collinear effective theory, crucial for N$^3$LL$^ extprime$ resummation and future N$^3$LO calculations.

Findings

Provides the three-loop soft function for heavy-to-light decays.

Confirms universal infrared structure predictions of QCD amplitudes.

Enables precise resummation and fixed-order calculations in heavy quark decays.

Abstract

We compute the 1-jettiness soft function for the decay of a heavy quark into a light quark jet plus colorless particles at three-loop order in soft-collinear effective theory. The 1-jettiness measurement fixes the total small light-cone momentum component of the soft radiation with respect to the jet direction. This soft function is a universal ingredient to the factorization of heavy-to-light quark decays in the limit of small 1-jettiness. Our three-loop result is required for resummation at the N$^3$LL$^\prime$ level, e.g. near the endpoint in the photon energy spectrum of the $B \to X_s \gamma$ decay. It is also a necessary ingredient for future calculations of fully-differential heavy-to-light quark decay rates at N$^3$LO using the $N$-jettiness subtraction method, e.g. for semileptonic top decays. Using our result for the soft anomalous dimension we confirm predictions on the universal infrared structure of QCD scattering amplitudes with a massive external quark at three loops.