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 $\mathcal{O}({\alpha }_s^{5}m)$ and $\mathcal{O}({\alpha }_s^{5}m\log {\alpha }_s)$ 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.

Comments: 45 pages, 1 figure; v2: minor corrections, version to appear in Nucl. Phys. B; v3: typos corrected, in particular eqs. (194) and (227); v4: corrections in eqs.(42),(44),(49); two equations supplemented below eq.(51); no change in the final results