Semi-numerical resummation of event shapes

Abstract

For many event-shape observables, the most difficult part of a resummation in the Born limit is the analytical treatment of the observable's dependence on multiple emissions, which is required at single logarithmic accuracy. We present a general numerical method, suitable for a large class of event shapes, which allows the resummation specifically of these single logarithms. It is applied to the case of the thrust major and the oblateness, which have so far defied analytical resummation and to the two-jet rate in the Durham algorithm, for which only a subset of the single logs had up to now been calculated.

Soft evolution of multi-jet final states

“…Event shapes in electron-positron, electron-proton and hadron-hadron collisions have been studied for a long time (see for instance [45,46] and references therein) and a general framework for resumming event shapes at next-to-leading logarithmic (NLL) accuracy was developed in refs. [47][48][49][50]. Very high logarithmic accuracy (N 3 LL) was achieved using Soft Collinear Effective Theory (SCET) for particular event shapes in e + e − collisions [51,52].…”

“…In this case, Q 12 = Q 34 = √ s, Q 13 = Q 24 = √ −t and Q 14 = Q 23 = √ −u, and the soft anomalous dimension becomes (see e.g. [47][48][49][50])…”

“…Caesar [47][48][49][50]. Within this method, observables defined on Born configurations with an arbitrary number of hard partons can in principle be considered.…”

“…[110]. 3 For our discussion, we follow the formalism developed in the context of the program Caesar [47][48][49][50], which allows one to resum global event shapes in a semi-automated way. With a couple of generalizations, the Caesar framework is sufficient for our purposes.…”

“…Following refs. [47][48][49][50] we consider positive-definite observables V that measure the difference in the energy-momentum flow of an event with respect to the Born configuration, where V ({p}) = 0. For a single emission with momentum k, which is soft and collinear to leg l, the observable V is parametrized as follows 4…”