On factorisation schemes for the electron parton distribution functions in QED

TL;DR

This paper compares different factorisation schemes for electron PDFs in QED, deriving analytical solutions and analyzing the impact of running coupling effects on the PDFs at large momentum fractions.

Contribution

It introduces a DIS-inspired factorisation scheme for electron PDFs in QED and analyzes the effects of including all-order running coupling effects on their behavior.

Findings

Large-$z$ PDFs depend dramatically on running-$eta$ effects.

Analytical solutions are derived for the large-$z$ region.

Next-to-leading logarithmic PDFs match leading-logarithmic forms at large-$z$.

Abstract

The electron, positron, and photon Parton Distribution Functions (PDFs) of the unpolarised electron have recently been computed at the next-to-leading logarithmic accuracy in QED, by adopting the $\overline{\rm MS}$ factorisation scheme. We present here analogous results, obtained by working in a different framework that is inspired by the so-called DIS scheme. We derive analytical solutions relevant to the large-$z$ region, where we show that the behaviour of the PDFs depends in a dramatic way on whether running-$\alpha$ effects are included to all orders, as opposed to being truncated to some fixed order. By means of suitable initial and evolution conditions, next-to-leading logarithmic accurate PDFs are obtained whose large-$z$ functional forms are identical to those of their leading logarithmic counterparts.