The infrared behaviour of the static potential in perturbative QCD

TL;DR

The paper addresses how to define and compute the static quark–antiquark potential in perturbative QCD within an effective field theory framework. By integrating out hard and soft scales, it introduces NRQCD and pNRQCD, where V_s(r) is a matching coefficient that captures ultrasoft dynamics separate from the Wilson loop. The authors show that at relative order α_s^3 the potential acquires an explicit μ-dependent ln(r μ) term with a calculable coefficient, arising from ultrasoft contributions, and that proper matching ensures cancellation of infrared logs with Wilson-loop contributions. This clarifies longstanding ambiguities in defining the static potential and provides a robust basis for using it in Schrödinger-type analyses of heavy-quark systems, while highlighting IR renormalon considerations. The work thus distinguishes the perturbative static potential from the Wilson loop and lays out a consistent scale-separated framework for heavy-quark dynamics in QCD.

Abstract

The definition of the quark-antiquark static potential is given within an effective field theory framework. The leading infrared divergences of the static singlet potential in perturbation theory are explicitly calculated.

Figures (2)

• Figure 1: The matching of the static potential. On the right side are the pNRQCD fields: simple lines are singlet propagator, double lines are octet propagators, circled-crosses are the singlet-octet vertices of Eq. (\ref{['pnrqcd0']}) and the wavy line is the US gluon propagator.
• Figure 2: Graphs contributing to $\delta \langle W_\Box \rangle$. Dashed lines represent Coulomb exchanges.