Calibrating the Na\"ive Cornell Model with NRQCD

Along the years, the Cornell Model has been extraordinarily successful in describing hadronic phenomenology, in particular in physical situations for which an effective theory of the strong interactions such as NRQCD cannot be applied. As a consequence of its achievements, a relevant question is whether its model parameters can somehow be related to fundamental constants of QCD. We shall give a first answer in this article by comparing the predictions of both approaches. Building on results from a previous study on heavy meson spectroscopy, we calibrate the Cornell model employing NRQCD predictions for the lowest-lying bottomonium states up to N$^3$LO, in which the bottom mass is varied within a wide range. We find that the Cornell model mass parameter can be identified, within perturbative uncertainties, with the MSR mass at the scale $R = 1\,$GeV. This identification holds for any value of $\alpha_s$ or the bottom mass, and for all perturbative orders investigated. Furthermore, we show that: a) the "string tension" parameter is independent of the bottom mass, and b) the Coulomb strength $\kappa$ of the Cornell model can be related to the QCD strong coupling constant $\alpha_s$ at a characteristic non-relativistic scale. We also show how to remove the $u=1/2$ renormalon of the static QCD potential and sum-up large logs related to the renormalon subtraction by switching to the low-scale, short-distance MSR mass, and using R-evolution. Our R-improved expression for the static potential remains independent of the heavy quark mass value and agrees with lattice QCD results for values of the radius as large as $0.8\,$fm, and with the Cornell model potential at long distances. Finally we show that for moderate values of $r$, the R-improved NRQCD and Cornell static potentials are in head-on agreement.