Abstract
We propose a method to examine how a parton shower sums large logarithms. In this method, one works with an appropriate integral transform of the distribution for the observable of interest. Then, one reformulates the parton shower so as to obtain the transformed distribution as an exponential for which one can compute the terms in the perturbative expansion of the exponent. We apply this general program to the thrust distribution in electron-positron annihilation, using several shower algorithms. Of the approaches that we use, the most generally applicable is to compute some of the perturbative coefficients in the exponent by numerical integration and to test whether they are consistent with next-to-leading-log summation of the thrust logarithms.
Highlights
Parton shower event generators provide a way to approximately sum large logarithms in QCD
There is a long history of investigations of how well parton shower event generators reproduce the expectation from full QCD for the large logarithm expansion of Eq (3)
Before we continue with the discussion of the parton shower cross section we introduce a more compact notation for operators with renormalization scale dependence
Summary
Parton shower event generators provide a way to approximately sum large logarithms in QCD. The logarithms Lj ðrÞ arise in QCD from the soft and collinear singularities of the theory These same soft and collinear singularities are contained in the splitting functions of a parton shower algorithm. There is a long history of investigations of how well parton shower event generators reproduce the expectation from full QCD for the large logarithm expansion of Eq (3). Marchesini and Webber argued that a parton shower event generator based on angular ordering would better sum large logarithms than alternative formulations [1]. A similar investigation showed analytically that a virtuality ordered dipole shower correctly sums the double logs that appear in the Drell-Yan transverse momentum distribution [12]. XI through XXVIII are quite self-contained, some readers may prefer to jump to the later sections before reading the more general analysis in the earlier sections
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