On renormalons of static QCD potential at $u=1/2$ and $3/2$

TL;DR

This paper analyzes the $u=1/2$ and $u=3/2$ renormalons in the static QCD potential using effective field theory, clarifying their suppression and implications for perturbative series and renormalon subtraction methods.

Contribution

It provides a formal framework for analyzing renormalons in the static QCD potential and examines the suppression of the $u=3/2$ renormalon in momentum space.

Findings

Clarifies the suppression of $u=3/2$ renormalon in momentum space.

Highlights difficulties in simultaneous IR renormalon and divergence subtraction.

Supports recent renormalon subtraction methods in $\a_s(M_Z)$ determination.

Abstract

We investigate the $u=1/2$ [$\mathcal{O}(\Lambda_{\rm QCD})$] and $u=3/2$ [$\mathcal{O}(\Lambda_{\rm QCD}^3)$] renormalons in the static QCD potential in position space and momentum space using the OPE of the potential-NRQCD effective field theory. This is an old problem and we provide a formal formulation to analyze it. In particular we present detailed examinations of the $u=3/2$ renormalons. We clarify how the $u=3/2$ renormalon is suppressed in the momentum-space potential in relation with the Wilson coefficient $V_A(r)$. We also point out that it is not straightforward to subtract the IR renormalon and IR divergences simultaneously in the multipole expansion. Numerical analyses are given, which clarify the current status of our knowledge on the perturbative series. The analysis gives a positive reasoning to the method for subtracting renormalons used in recent $\alpha_s(M_Z)$ determination from the QCD potential.