The Small x Behaviour of Altarelli-Parisi Splitting Functions

TL;DR

This paper analyzes the small x asymptotic behavior of Altarelli-Parisi splitting functions, revealing enhanced corrections and proposing a reorganization of the series for a well-behaved perturbative expansion.

Contribution

It provides a detailed analysis of the small x behavior of splitting functions and introduces a method to reorganize the series for improved perturbative stability.

Findings

Next-to-leading corrections are asymptotically enhanced by an extra ln(1/x).

A reorganization of the series can satisfy conditions for a well-behaved expansion.

The analysis depends on the choice of factorization scheme.

Abstract

We extract the small x asymptotic behaviour of the Altarelli-Parisi splitting functions from their expansion in leading logarithms of 1/x. We show in particular that the nominally next-to-leading correction extracted from the Fadin-Lipatov kernel is enhanced asymptotically by an extra ln 1\x over the leading order. We discuss the origin of this problem, its dependence on the choice of factorization scheme, and its all-order generalization. We derive necessary conditions which must be fulfilled in order to obtain a well behaved perturbative expansion, and show that they may be satisfied by a suitable reorganization of the original series.