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 $\Lambdav^4/m^4$ terms. The results were obtained with, in particular, the value of the two loop static coefficient due to Peter; this been recently challenged by Schr\"oder. In our previous paper we used Peter's result; in the present one we now give results with Schr\"oder'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 Schr\"oder's expression we now have, $$m_b=5\,001^{+104}_{-66}\;\mev;\quad \bar{m}_b(\bar{m}_b^2)=4\,440^{+43}_{-28}\;\mev,$$ $$m_c=1\,866^{+190}_{-154}\;\mev;\quad \bar{m}_c(\bar{m}_c^2)=1\,531^{+132}_{-127}\;\mev.$$ Moreover, $$\Gammav(\Upsilonv\rightarrow e^+e^-)=1.07\pm0.28\;\kev \;(\hbox{exp.}=1.320\pm0.04\,\kev)$$ and the hyperfine splitting is predicted to be $$M(\Upsilonv)-M(\eta)=47^{+15}_{-13}\;\mev.$$