The vector coupling $\alpha_{\rm V}(r)$ and the scales $r_0,r_1$ from the bottomonium spectrum

Abstract

We study the universal static potential $V_{\rm st}(r)$ and the force, which are fully determined by two fundamental parameters: the string tension $\sigma=0.18\pm 0.02$ GeV$^2$ and the QCD constants $\Lambda_{\bar{\rm MS}}(n_f)$, taken from pQCD, while the infrared (IR) regulator $M_{\rm B}$ is taken from the background perturbation theory and expressed via the string tension. The vector couplings $\alpha_{\rm V}(r)$ in the static potential and $\alpha_{\rm F}(r)$ in the static force, as well as the characteristic scales, $r_1(n_f=3)$ and $r_0(n_f=3)$, are calculated and compared to lattice data. The result $r_0\Lambda_{\bar{\rm MS}}(n_f=3)=0.77\pm 0.03$, which agrees with the lattice data, is obtained for $M_{\rm B}=(1.15\pm 0.02)$ GeV. However, better agreement with the bottomonium spectrum is reached for a smaller $\Lambda_{\bar{\rm MS}}(n_f=3)=(325\pm 15)$ MeV and the frozen value of $\alpha_V=0.57\pm 0.02$. The mass splittings $\bar M(1D)-\bar M(1P)$ and $\bar M(2P)-\bar M(1P)$ are shown to be sensitive to the IR regulator used. The masses $M(1\,^3D_3)=10169(2)$ MeV and $M(1\,^3D_1)=10155(3)$ MeV are predicted.