Abstract
We explore the collinear limit of final-state quark splittings at order {alpha}_s^2 . While at general NLL level, this limit is described simply by a product of leading-order 1 → 2 DGLAP splitting functions, at the NNLL level we need to consider 1 → 3 splitting functions. Here, by performing suitable integrals of the triple-collinear splitting functions, we demonstrate how one may extract {mathrm{mathcal{B}}}_2^q(z) , a differential version of the coefficient {mathrm{mathcal{B}}}_2^q that enters the quark form factor at NNLL and governs the intensity of collinear radiation from a quark. The variable z corresponds to the quark energy fraction after an initial 1 → 2 splitting, and our results yield effective higher-order splitting functions, which may be considered as a step towards the construction of NNLL parton showers. Further, while in the limit z → 1 we recover the standard soft limit results involving the CMW coupling with scale kt, the z dependence we obtain also motivates the extension of the notion of a physical coupling beyond the soft limit.
Highlights
One encounters observables sensitive to widely disparate scales and large logarithms in scale ratios, which require resummation
Given the recent spurt in activity in the context of next-to-leading logarithmic (NLL) showers, which demonstrates that showers can be designed to achieve broad NLL accuracy, it is legitimate to think about whether further advances in resummation can be brought to bear on the construction of NNLL accurate showers
The fact that one can obtain one parent gluon kinematical distribution from the other at order αs2, by using a relationship valid in the limit of a massless gluon, implies that the effect of the gluon virtuality has effectively been absorbed into the structure of eq (3.36). This is reminiscent of the fact that in the soft limit and to NLL accuracy for global observables one can replace the emission of a massive gluon by a massless gluon, with the effect of the gluon branching included in the argument of the coupling and the CMW factor K
Summary
This result is equivalent to that for fixed θg and z so that we have ρ d2σ(1) σ0 dρ dz θg d2σ(1) σ0 dθg dz CF αs 2π
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