The restriction on the strong coupling constant in the IR region from the 1D-1P splitting in bottomonium

Abstract

The $b\bar{b}$ spectrum is calculated with the use of a relativistic Hamiltonian where the gluon-exchange between a quark and an antiquark is taken as in background perturbation theory. We observed that the splittings ${\Delta }_1={\rm Y}(1D)-{\chi }_b(1P)$ and other splittings are very sensitive to the QCD constant ${\Lambda }_V(n_f)$ which occurs in the Vector scheme, and good agreement with the experimental data is obtained for ${\Lambda }_V(2$-loop, $n_f=5)=325\pm 10$ MeV which corresponds to the conventional ${\Lambda }_{\overline{MS}}(2-$loop, $n_f=5)=238\pm 7$ MeV, ${\alpha }_s(2-$loop, $M_Z)=0.1189\pm 0.0005,$ and a large freezing value of the background coupling: ${\alpha }_{\mathrm{crit}}(2$-loop, $q^2=0)={\alpha }_{\mathrm{crit}}(2$-loop, $r\to \infty )=0.58\pm 0.02$. If the asymptotic freedom behavior of the coupling is neglected and an effective freezing coupling ${\alpha }_{\mathrm{static}}=const$ is introduced, as in the Cornell potential, then precise agreement with ${\Delta }_1(\exp )$ and ${\Delta }_2(\exp )$ can be reached for the rather large value ${\alpha }_{\mathrm{static}}=0.43\pm 0.02$. We predict a value for the mass M(2D) = 10451\pm2 MeV.

Comments: 17 pages, no figures, 6 tables