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 $Δ_1= Υ({\rm 1D})-χ_b({\rm 1P})$ and other splittings are very sensitive to the QCD constant $Λ_V(n_f)$ which occurs in the Vector scheme, and good agreement with the experimental data is obtained for $Λ_V(2$-loop, $n_f=5)= 325\pm 10$ MeV which corresponds to the conventional $Λ_{\bar{MS}} (2-$loop, $n_f=5)= 238\pm 7$ MeV, $α_s(2-$loop, $M_Z)=0.1189\pm 0.0005,$ and a large freezing value of the background coupling: $α_{\rm crit} (2$-loop, $q^2=0)=α_{\rm crit} (2$-loop, $r\to \infty)=0.58\pm 0.02$. If the asymptotic freedom behavior of the coupling is neglected and an effective freezing coupling $α_{\rm static}=const$ is introduced, as in the Cornell potential, then precise agreement with $Δ_1({\rm exp})$ and $Δ_2({\rm exp})$ can be reached for the rather large value $α_{\rm static} =0.43\pm 0.02$. We predict a value for the mass M(2D) = 10451\pm2 MeV.

17 pages, no figures, 6 tables