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

Abstract

We study the universal static potential $V_{\mathrm{st}}(r)$ 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 ${\Lambda }_{\overline{\mathrm{MS}}}(n_f)$, taken from pQCD, while the infrared (IR) regulator $M_{\mathrm{B}}$ is taken from the background perturbation theory and expressed via the string tension. The vector couplings ${\alpha }_{\mathrm{V}}(r)$ in the static potential and ${\alpha }_{\mathrm{F}}(r)$ 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 $r_0{\Lambda }_{\overline{\mathrm{MS}}}(n_f=3)=0.77\pm 0.03$, which agrees with the lattice data, is obtained for $M_{\mathrm{B}}=(1.15\pm 0.02)$ GeV. However, better agreement with the bottomonium spectrum is reached for a smaller ${\Lambda }_{\overline{\mathrm{MS}}}(n_f=3)=(325\pm 15)$ 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 $M(1{}^3{D}_3)=10169(2)$ MeV and $M(1{}^3{D}_1)=10155(3)$ MeV are predicted.

Comments: 21 pages, 5 figures, revtex4