Abstract
We provide a general method to effectively compute differential and cumulative event-shape distributions to O(αs) precision for massive quarks produced primarily at an e+e− collider. In particular, we show that at this order, due to the screening of collinear singularities by the quark mass, for all event shapes linearly sensitive to soft dynamics, there appear only two distributions at threshold: a Dirac delta function and a plus distribution. Furthermore, we show that the coefficient of the latter is universal for any infra-red and collinear safe event shape, and provide an analytic expression for it. Likewise, we compute a general formula for the coefficient of the Dirac delta function, which depends only on the event-shape measurement function in the soft limit. Finally, we present an efficient algorithm to compute the differential and cumulative distributions, which does not rely on Monte Carlo methods, therefore achieving a priory arbitrary precision even in the extreme dijet region. We implement this algorithm in a numeric code and show that it agrees with analytic results on the distribution for 2-jettiness, heavy jet mass and a massive generalization of C-parameter.
Highlights
Recent years have seen tremendous progress in the understanding and computation of event-shape cross sections for e+e− machines such as LEP or the future linear and circular colliders
We present an efficient algorithm to compute the differential and cumulative distributions, which does not rely on Monte Carlo methods, achieving a priory arbitrary precision even in the extreme dijet region. We implement this algorithm in a numeric code and show that it agrees with analytic results on the distribution for 2-jettiness, heavy jet mass and a massive generalization of C-parameter
This is mainly achieved with the use of factorized expressions derived in the frame of the effective field theory (EFT)1 known as Soft-Collinear Effective Theory (SCET) [8,9,10,11,12]
Summary
Recent years have seen tremendous progress in the understanding and computation of event-shape cross sections for e+e− machines such as LEP or the future linear and circular colliders. While at the time of the “NLO revolution” the O(αs) results seem completely standard, having a dedicated article on those is still useful because: a) known results are mainly numeric and provided as binned distributions, which makes the matching to resummed results unpractical, b) analytic results are faster and easier to implement; c) non-zero quark masses entail that different event-shape schemes can be used, allowing to control the sensitivity to the mass, to the best of our knowledge a possibility never discussed so far; d) new strategies to efficiently compute the cross section are presented, which we believe will be useful for future work; e) our results are an important input for further studies of massive event shapes, which aim to improve our understanding of heavy quark mass determinations in general. Applications of our master formula for a selection of event shapes, analytic formulae for the massive total hadronic cross section and some master integrals are provided in the remaining appendices
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