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

Abstract

The $b\overline{b}$ spectrum is calculated with the use of a relativistic Hamiltonian where the gluon-exchange potential between a quark and an antiquark is taken as in background perturbation theory. We observed that the splittings ${\ensuremath{\Delta}}_{1}=\ensuremath{\Upsilon}(1\mathrm{D})\ensuremath{-}{\ensuremath{\chi}}_{b}(1\mathrm{P})$ and other splittings between low-lying states are very sensitive to the QCD constant ${\ensuremath{\Lambda}}_{\mathrm{V}}{(n}_{\mathrm{f}})$ which occurs in the vector scheme, and good agreement with the experimental data is obtained for ${\ensuremath{\Lambda}}_{\mathrm{V}}(2\ensuremath{-}\mathrm{loop}, {n}_{\mathrm{f}}=5)=325\ifmmode\pm\else\textpm\fi{}10\mathrm{MeV}$ which corresponds to the conventional ${\ensuremath{\Lambda}}_{\overline{\mathrm{MS}}}(2\ensuremath{-}\mathrm{loop},{n}_{\mathrm{f}}=5)=238\ifmmode\pm\else\textpm\fi{}7\mathrm{MeV},$ ${\ensuremath{\alpha}}_{s}(2\ensuremath{-}\mathrm{loop}, {M}_{Z})=0.1189\ifmmode\pm\else\textpm\fi{}0.0005,$ and to a large freezing value of the background coupling: ${\ensuremath{\alpha}}_{\mathrm{crit}}(2\ensuremath{-}\mathrm{loop}, {q}^{2}=0)={\ensuremath{\alpha}}_{\mathrm{crit}}(2\ensuremath{-}\mathrm{loop}, \stackrel{\ensuremath{\rightarrow}}{r}\ensuremath{\infty})=0.58\ifmmode\pm\else\textpm\fi{}0.02.$ If the asymptotic freedom behavior of the coupling is neglected and an effective freezing coupling ${\ensuremath{\alpha}}_{\mathrm{static}}=\mathrm{const}$ is introduced, as in the Cornell potential, then precise agreement with ${\ensuremath{\Delta}}_{1}(\mathrm{exp})$ and ${\ensuremath{\Delta}}_{2}(\mathrm{exp})$ can be reached for the rather large Coulomb constant ${\ensuremath{\alpha}}_{\mathrm{static}}=0.43\ifmmode\pm\else\textpm\fi{}0.02.$ We predict a value for the mass $M(2\mathrm{D})=10451\ifmmode\pm\else\textpm\fi{}2\mathrm{MeV}.$