Violation of Casimir Scaling for Static QCD Potential at Three-loop Order

TL;DR

This paper calculates high-order corrections to the static QCD potential and demonstrates for the first time a violation of Casimir scaling at three-loop order, with implications for lattice QCD and potential tests.

Contribution

It provides the first calculation of Casimir scaling violation at ${ m O}(\alpha_s^4)$ in the static QCD potential, including detailed analysis for various representations.

Findings

Casimir scaling violation appears at ${ m O}(\alpha_s^4)$.

Violations are within current lattice bounds.

Potential for experimental testing of the violation.

Abstract

We compute the full ${\cal O}(\alpha_s^4)$ and ${\cal O}(\alpha_s^4\log\alpha_s)$ corrections to the potential $V_R(r)$ between the static color sources, where $V_R(r)$ is defined from the Wilson loop in a general representation $R$ of a general gauge group $G$. We find a violation of the Casimir scaling of the potential, for the first time, at ${\cal O}(\alpha_s^4)$. The effect of the Casimir scaling violation is predicted to reduce the tangent of $V_R(r)/C_R$ proportionally to specific color factors dependent on $R$. We study the sizes of the Casimir scaling violation for various $R$'s in the case $G=SU(3)$. We find that they are well within the present bounds from lattice calculations, in the distance region where both perturbative and lattice computations of $V_R(r)$ are valid. We also discuss how to test the Casimir scaling violating effect.