Cornell Model Calibration with NRQCD at N$^3$LO

Abstract

The typical binding energy of heavy hadron spectroscopy makes the system accessible to perturbative calculations in terms of non-relativistic QCD. Within NRQCD the predictions of heavy quarkonium energy levels rely on the accurate description of the static QCD potential $V_{\rm QCD}(r)$. Historically, heavy quarkonium spectroscopy was studied using phenomenological approaches such as the Cornell model $V_{\rm Cornell}=-\kappa/r+\sigma\, r$, which assumes a short-distance dominant Coulomb potential plus a liner rising potential that emerges at long distances. Such model works reasonably well in describing the charmonium and bottomonium spectroscopy. However, even when there are physically-motivated arguments for the construction of the Cornell model, there is no conection a priori with QCD parameters. Based on a previous work on heavy meson spectroscopy, we calibrate the Cornell model with NRQCD predictions for the lowest lying bottomonium states at N$^3$LO, in which the bottom mass is varied within a wide range. We show that the Cornell model mass parameter can be identified with the low-scale short-distance MSR mass at the scale $R = 1$ GeV. This identification holds for any value of $\alpha_s$ or the bottom mass. For moderate values of $r$, the NRQCD and Cornell static potentials are in head-on agreement when switching the pole mass to the MSR scheme, which allows to simultaneously cancel the renormalon and sum up large logarithms.