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

Abstract

We compute the full $\mathcal{O}({\alpha }_s^4)$ and $\mathcal{O}({\alpha }_s^4\log {\alpha }_s)$ 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 $\mathcal{O}({\alpha }_s^4)$. 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.

20 pages, 7 figures, v2: a typo in eq.(13) corrected

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• Three-loop static potential
• Static QCD potential at three-loop order
• Nonperturbative determination of the QCD potential at O(1/m)
• Fermionic contributions to the three-loop static potential
• Spin-dependent potentials from lattice QCD
• Ultrasoft contribution to quarkonium production and annihilation
• Ultrasoft contribution to heavy-quark pair production near threshold
• Casimir scaling in G(2) lattice gauge theory
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