Full Formula for Heavy Quarkonium Energy Levels at Next-to-next-to-next-to-leading Order

Abstract

We derive a full formula for the energy level of a heavy quarkonium state identified by the quantum numbers $n$, $\ell$, $s$ and $j$, up to ${\cal O}(\alpha_s^5 m)$ and ${\cal O}(\alpha_s^5 m \log \alpha_s)$ in perturbative QCD. The QCD Bethe logarithm is given in a one-parameter integral form. The rest of the formula is given as a combination of rational numbers, transcendental numbers ($\pi$, $\zeta(3)$, $\zeta(5)$) and finite sums (besides the 3-loop constant $\bar{a}_3$ of the static potential whose full analytic form is still unknown). A derivation of the formula is given.