Comment on "Calculation of Quarkonium Spectrum and m_b, m_c to Order alpha^4"

Abstract

In a recent paper, we included 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 ${\text{Lambdav}}^4/m^4$ terms. The results were obtained with, in particular, the value of the two loop static coefficient due to Peter; this been recently challenged by Schröder. In our previous paper we used Peter's result; in the present one we now give results with Schröder's, as this is likely to be the correct one. The variation is slight as the value of $b_1$ is only one among the various $O({\alpha }_s^4)$ contributions. With Schröder's expression we now have, $$m_b=5001_{-66}^{+104}\text{mev};{\bar{m}}_b({\bar{m}}_b^2)=4440_{-28}^{+43}\text{mev},$$ $$m_c=1866_{-154}^{+190}\text{mev};{\bar{m}}_c({\bar{m}}_c^2)=1531_{-127}^{+132}\text{mev}.$$ Moreover, $$\text{Gammav}(\text{Upsilonv}\to e^{+}{e}^{-})=1.07\pm 0.28\text{kev}(exp.=1.320\pm 0.04\text{kev})$$ and the hyperfine splitting is predicted to be $$M(\text{Upsilonv})-M(\eta )=47_{-13}^{+15}\text{mev}.$$

Comments: 7 pages, TeX