Massive Event-Shape Distributions at N$^2$LL

TL;DR

This paper develops N$^2$LL + $ ext{O}( ext{α}_s)$ resummed expressions for massive event-shape distributions, extending previous work to include non-recoil-sensitive observables and matching with fixed-order QCD for improved accuracy.

Contribution

It provides the first complete resummation at N$^2$LL + $ ext{O}( ext{α}_s)$ for massive event-shapes, including new calculations of jet functions in SCET and bHQET, and generalizes to various observables.

Findings

Resummed expressions for thrust, heavy jet mass, and C-parameter distributions.

Analytic solutions for the P-scheme SCET jet function's RG evolution.

Numerical analysis of mass effects in event-shape observables.

Abstract

In a recent paper we have shown how to optimally compute the differential and cumulative cross sections for massive event-shapes at $\mathcal{O}(\alpha_s)$ in full QCD. In the present article we complete our study by obtaining resummed expressions for non-recoil-sensitive observables to N$^2$LL + $\mathcal{O}(\alpha_s)$ precision. Our results can be used for thrust, heavy jet mass and C-parameter distributions in any massive scheme, and are easily generalized to angularities and other event shapes. We show that the so-called E- and P-schemes coincide in the collinear limit, and compute the missing pieces to achieve this level of accuracy: the P-scheme massive jet function in Soft-Collinear Effective Theory (SCET) and boosted Heavy Quark Effective Theory (bHQET). The resummed expression is subsequently matched into fixed-order QCD to extend its validity towards the tail and far-tail of the distribution. The computation of the jet function cannot be cast as the discontinuity of a forward-scattering matrix element, and involves phase space integrals in $d=4-2\varepsilon$ dimensions. We show how to analytically solve the renormalization group equation for the P-scheme SCET jet function, which is significantly more complicated than its 2-jettiness counterpart, and derive rapidly-convergent expansions in various kinematic regimes. Finally, we perform a numerical study to pin down when mass effects become more relevant.