A fresh look into the heavy quark-mass values

Abstract

Using the recent {\it world average} $\alpha_s(M^2_Z)= 0.118 \pm 0.006$, we give the {\it first direct extraction} from the $\Psi$ and $\Upsilon$ data of the values of the {\it running heavy quark masses} within QCD spectral sum rules to two-loops in the $\overline {MS}$-scheme: $\mr_b(M^{PT2}_b)$ = $(4.23~^{+0.03}_{-0.04} \pm 0.02)$ GeV and $\mr_c(M^{PT2}_c)$ = $(1.23~^{+ 0.02}_{-0.04} \pm 0.03)$ GeV, (the errors are respectively due to $\alf$ and to the gluon condensate), and the corresponding value of the {\it short-distance perturbative pole masses to two-loops}: $M^{PT2}_b=(4.62 \pm 0.02)$ GeV, $M^{PT2}_c= (1.41\pm 0.03)$ GeV, which we compare with the updated values of the {\it non-relativistic pole masses} re-extracted {\it directly} from the two-loop non-relativistic sum rules: $M^{NR}_b= (4.69~^{-0.01}_{+0.02} \pm 0.02)$ GeV and $M^{NR}_c=(1.44\pm 0.02\pm 0.03)$ GeV. It is also informative to compare the {\it three-loop} values of the short-distance pole masses: $M^{PT3}_b=(4.87\pm 0.05 \pm 0.02)$ GeV and $M^{PT3}_c= (1.62\pm 0.07 \pm 0.03)$ GeV, with the {\it dressed mass} $M^{nr}_b = (4.94 \pm 0.10 \pm 0.03)$ GeV, entering into the {\it non-relativistic Balmer formula} including higher order $\alf$ corrections. The $small$ mass-differences $M^{NR}_b-M^{PT2}_b \simeq M^{nr}_b-M^{PT3}_b \simeq 70$ MeV and $M_c^{NR}-M_c^{PT2} \simeq (30 \pm 20)$ MeV {\it can measure the size} of the non-perturbative effect induced by {\it renormalon} type-singularities. An analogous analysis is pursued for the heavy-light mesons, where a simultaneous {\it re-fit} of the $B$ and $B^*$ masses from relativistic sum rules leads to: $M_b^{PT2}= (4.63 \pm 0.08)$ GeV,