Parton distribution functions in the context of parton showers

When the initial state evolution of a parton shower is organized according to the standard “backward evolution” prescription, ratios of parton distribution functions appear in the splitting probabilities. The shower thus organized evolves from a hard scale to a soft cutoff scale. At the end of the shower, one expects that only the parton distributions at the soft scale should affect the results. The other effects of the parton distributions should have cancelled. This means that the kernels for parton evolution should be related to the shower splitting functions. If the initial state partons can have non-zero masses, this requires that the evolution kernels cannot be the usual MS¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{\mathrm{MS}} $$\end{document} kernels. We work out what the parton evolution kernels should be to match the shower evolution contained in the parton shower event generator Deductor, in which the b and c quarks have non-zero masses.