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

We compute the full ${\cal O}(α_s^4)$ and ${\cal O}(α_s^4\logα_s)$ 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 ${\cal O}(α_s^4)$. 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.