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

TL;DR

This paper presents a comprehensive formula for heavy quarkonium energy levels in perturbative QCD up to high order, including detailed mathematical components and derivations, advancing theoretical understanding in quantum chromodynamics.

Contribution

It provides the first complete formula for heavy quarkonium energy levels at ${ m N}^3{ m LO}$, incorporating complex constants and integrals, with a detailed derivation.

Findings

Explicit formula for energy levels up to ${ m N}^3{ m LO}$

Inclusion of QCD Bethe logarithm in integral form

Expression involving rational, transcendental numbers, and sums

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.