Ultrasoft Effects in Heavy-Quarkonium Physics

TL;DR

The paper addresses the need for N$^3$LO precision in heavy-quark threshold dynamics by focusing on ultrasoft-gluon effects. It employs the pNRQCD framework to compute leading ultrasoft corrections arising from the chromoelectric dipole interaction, yielding analytic expressions for energy shifts $\Delta E_n$ and wave-function corrections $\Delta \psi^2_n(0)$ in terms of color factors, the Coulomb energy $E^C_n$, and QCD Bethe logarithms $L^E_n$ and $L^_n$. The results reveal sizable, scale-dependent corrections for $b\bar b$ and $t\bar t$ systems and emphasize nonperturbative contributions (e.g., gluon condensates) and the non-Coulombic nature of $J/\psi$, underscoring the necessity to integrate ultrasoft effects with soft and hard contributions for a complete N$^3$LO description. The study thus establishes the ultrasoft sector as a defining element in the threshold dynamics and outlines the steps needed to finalize the N$^3$LO program, including three-loop hard matching and potential calculations.

Abstract

In the framework of nonrelativistic QCD, we consider a new class of radiative corrections, which are generated at next-to-next-to-next-to-leading order through the chromoelectric dipole interaction of heavy quarkonium with ultrasoft virtual gluons. We provide analytical formulae from which the resulting shifts in the quarkonium energy levels and the wave functions at the origin may be calculated. We discuss the phenomenological implications for the top-antitop and Upsilon systems and point out some limitations of describing charmonium using a Coulomb potential.