# Skewed Sudakov Regime, Harmonic Numbers, and Multiple Polylogarithms

## TL;DR

This paper investigates the skewed Sudakov regime in massless QED, deriving a linear evolution equation for the form factor, revealing a spectrum of harmonic numbers, and expressing the asymptotic expansion using multiple polylogarithms.

## Contribution

It introduces the skewed Sudakov regime, derives its linear evolution equation, and characterizes the spectrum and polylogarithmic expansion of the form factor.

## Key findings

- The asymptotic is described by a single form factor with a linear evolution equation.
- The spectrum of the linear operator consists of shifted harmonic numbers.
- The expansion in the fine structure constant is expressed via multiple polylogarithms.

## Abstract

On the example of massless QED we study an asymptotic of the vertex when only one of the two virtualities of the external fermions is sent to zero. We call this regime the skewed Sudakov regime. First, we show that the asymptotic is described with a single form factor, for which we derive a linear evolution equation. The linear operator involved in this equation has a discrete spectrum. Its eigenfunctions and eigenvalues are found. The spectrum is a shifted sequence of harmonic numbers. With the spectrum found, we represent the expansion of the asymptotic in the fine structure constant in terms of multiple polylogarithms. Using this representation, the exponentiation of the doubly logarithmic corrections of the Sudakov form factor is recovered. It is pointed out that the form factor of the skewed Sudakov regime is growing with the virtuality of a fermion decreasing at a fixed virtuality of another fermion.