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

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 $d=4-2\epsilon$ 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.

48 pages + appendices, 11 figures. v2: minimal modifications, journal version

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