Borel summation and momentum-plane analyticity in perturbative QCD

TL;DR

This paper develops a new method to analyze the analyticity of QCD Green functions using Borel summation and Mellin transforms, linking asymptotic Borel behavior to Landau singularities.

Contribution

It introduces a compact expression for Borel sums in QCD and explores their momentum-plane analyticity, connecting Borel transform asymptotics with Landau singularities.

Findings

Established a connection between Borel transform asymptotics and Landau singularities.

Derived a compact expression for Borel sums using inverse Mellin transforms.

Applied the method to quark polarization functions and renormalon chains.

Abstract

We derive a compact expression for the Borel sum of a QCD amplitude in terms of the inverse Mellin transform of the corresponding Borel function. The result allows us to investigate the momentum-plane analyticity properties of the Borel-summed Green functions in perturbative QCD. An interesting connection between the asymptotic behaviour of the Borel transform and the Landau singularities in the momentum plane is established. We consider for illustration the polarization function of massless quarks and the resummation of one-loop renormalon chains in the large-$\beta_0$ limit, but our conclusions have a more general validity.