Correct Definition of The QCD Potential From The Wilson Loop

TL;DR

This paper clarifies the correct definition of the QCD potential energy from Wilson loops, showing it should depend on time and align with classical Yang-Mills theory, correcting previous misconceptions.

Contribution

It introduces a proper definition of the static QCD potential energy from Wilson loops that is consistent with classical Yang-Mills theory at large distances.

Findings

The classical Yang-Mills potential energy is time-dependent at large distances.

Previous literature's static QCD potential energy is inconsistent with classical Yang-Mills theory.

The paper provides a corrected, time-dependent definition of the QCD potential energy.

Abstract

The static Coulomb potential energy between the electron and positron at rest separated by a large distance $R$ obtained from the Wilson loop in QED is same as the Coulomb potential energy obtained in the classical Maxwell theory. Since the Yang-Mills theory was discovered by making analogy with the Maxwell theory by extending U(1) group to the SU(3) group one finds by making analogy with the QED that the QCD potential energy between the quark and antiquark at rest separated by a large distance $R$ obtained from the Wilson loop in QCD is the same potential energy obtained in the classical Yang-Mills theory. This implies that the static QCD potential energy $V(R)$ obtained at the large separation distance $R$ in the literature is not consistent with the classical Yang-Mills theory because the potential energy $V(T,R)$ in the classical Yang-Mills theory is time $T$ dependent even if the quark and antiquark are at rest. In this paper we find the correct definition of the QCD potential energy $V(T,R)$ from the Wilson loop in QCD which, at the large separation distance $R$, is consistent with the classical Yang-Mills theory.