Principal Value Resummation

TL;DR

The paper tackles large threshold corrections in hadron-hadron processes by reformulating resummation through a principal value prescription that defines a finite exponent $E(n,Q^2)$, thereby regularizing infrared renormalon ambiguities. This PV approach connects to a Borel-transform intuition and to an evolution equation, yielding a real, finite resummed exponent while isolating higher-twist corrections that begin at $O(\Lambda_{QCD}/Q)$. Numerical analysis shows the leading exponent can be well approximated by a finite asymptotic series up to moderate moment values, after which higher-twist effects damp the growth, ensuring finiteness as $z\to1$. The method provides a principled way to organize perturbative corrections and to incorporate nonperturbative contributions, with potential applications to other inclusive hadronic processes.

Abstract

We present a new resummation formula for the Drell-Yan cross section. The formal resummation of threshold corrections in Drell-Yan hard-scattering functions produces an exponent with singularities from the infrared pole of the QCD running coupling. Our reformulation treats such `infrared renormalons' by a principal value prescription, analogous to a modified Borel transform. The resulting expression includes all large threshold corrections to the hard scattering function as an asymptotic series in $α_s$, but is a finite function of $Q^2$. We find that the ambiguities of the resummed perturbation theory imply the presence of higher twist corrections to quark-antiquark hard-scattering functions that begin at $Λ_{QCD}/Q$. This suggests an important role for higher twist in the phenomonolgy of hadron-hadron inclusive cross sections. We also discuss the numerical evaluation of the exponent and its asymptotic perturbation series for representative values of $Q^2$.