Calculation of Quarkonium Spectrum and $m_b,m_c$ to Order ${\alpha }_s^4$

Abstract

We include two loop, relativistic one loop and second order relativistic tree level corrections, plus leading nonperturbative contributions, to obtain a calculation of the lower states in the heavy quarkonium spectrum correct up to, and including, $O({\alpha }_s^4)$ and leading ${\Lambda }^4/m^4$ terms. This allows us, in particular, to obtain a model independent determination of the pole masses of the $b,c$ quarks, $$m_b=5015_{-70}^{+110}mev;m_c=1884_{-133}^{+222}mev$$ to which correspond the $\overline{MS}$ masses, $${\bar{m}}_b({\bar{m}}_b^2)=4453_{-32}^{+50}mev;{\bar{m}}_c({\bar{m}}_c^2)=1547_{-102}^{+169}mev.$$ The decay $\Gamma ({\rm Y}\to e^{+}{e}^{-})$ is found in agreement with experiment, $$\Gamma ({\rm Y}\to e^{+}{e}^{-})=1.135_{-0.29}^{+0.27}kev(exp.=1.320\pm 0.04kev),$$ and the hyperfine splitting is predicted to be $$M({\rm Y})-M(\eta )=48.5_{-12.2}^{+15.7}mev.$$

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