Dissecting the collinear structure of quark splitting at NNLL

2021/09/30 by Mrinal Dasgupta, Basem Kamal El-Menoufi
Mathematics · Physics and Astronomy · #Algorithm #Black Holes and Theoretical Physics #Mathematics #Order (exchange) #Particle physics #Particle physics theoretical and experimental studies #Physics #Quantum Chromodynamics and Particle Interactions #Quark #hep-ph

paper · pdf · doi:10.1007/jhep12(2021)158

33 pages, 5 figures. Some typographical errors are corrected. Matches published version

openalex publication_date 2021/12/22 · arxiv created 2022/02/16 · arxiv updated 2022/02/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

A bstract We explore the collinear limit of final-state quark splittings at order α_s² <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>α</mml:mi> <mml:mi>s</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> . 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 B₂^q(z) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>ℬ</mml:mi> <mml:mn>2</mml:mn> <mml:mi>q</mml:mi> </mml:msubsup> <mml:mfenced> <mml:mi>z</mml:mi> </mml:mfenced> </mml:math> , a differential version of the coefficient B₂^q <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>ℬ</mml:mi> <mml:mn>2</mml:mn> <mml:mi>q</mml:mi> </mml:msubsup> </mml:math> 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 k t , the z dependence we obtain also motivates the extension of the notion of a physical coupling beyond the soft limit.

Citations

• A complete (ααs)calculation of the photon + 1 jet rate in e+ e− annihilation
• Automated NNLL + + NLO resummation for jet-veto cross sections
• HypExp, a Mathematica package for expanding hypergeometric functions around integer-valued parameters
• Analysis of PSLQ, an integer relation finding algorithm
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