Résumé
We report results of our recent works [S. Narison, Int. J. Mod. Phys. A33 (2018) no. 10, 1850045; S. Narison. Addendum: Int. J. Mod. Phys. A33 (2018) no. 10, 1850045; S. Narison, Phys. Lett . B802 (2020) 135221.] where the correlations between the c,b -quark running masses m‾c,b , the gluon condensate 〈αsG2〉 and the QCD coupling αs in the MS‾ -scheme from an analysis of the charmonium and bottomium spectra and the Bc -meson mass. We use optimized ratios of relativistic Laplace sum rules (LSR) evaluated at the μ -subtraction stability point where higher orders PT and D≤6−8 -dimensions nonperturbative condensates corrections are included. We obtain [S. Narison, Int. J. Mod. Phys. A33 (2018) no. 10, 1850045; S. Narison. Addendum: Int. J. Mod. Phys. A33 (2018) no. 10, 1850045.] αs(2.85)=0.262(9) and αs(9.50)=0.180(8) from the (pseudo)scalar Mχ0c(0b)−Mηc(b) mass-splittings at μ=2.85(9.50) GeV. The most precise result from the charm channel leads to αs(Mτ)=0.318(15) and αs(MZ)=0.1183(19)(3) in excellent agreement with the world average: αs(MZ)=0.1181(11) [For a review, see e.g: S. Bethke, Nucl. Part. Phys. Proc . 282–284 (2017)149; C. Patrignani et al. (Particle Data Group), Chin. Phys . C40, 100001 (2016) and 2017 update.]. Updated results from a global fit of the (axial-)vector and (pseudo)scalar channels using Laplace and Moments sum rules @ N2LO [S. Narison, Int. J. Mod. Phys. A33 (2018) no. 10, 1850045; S. Narison. Addendum: Int. J. Mod. Phys. A33 (2018) no. 10, 1850045.] combined with the one from MBc [S. Narison, Phys. Lett . B802 (2020) 135221.] lead to the new tentative QCD spetral sum rules (QSSR) average: m‾c(m‾c)|average=1266(6) MeV and m‾b(m‾b)|average=4196(8) MeV. The values of the gluon condensate 〈αsG2〉 from the (axial)-vector charmonium channels combined with previous determinations in Table 1, leads to the new QSSR average: [S. Narison, Int. J. Mod. Phys. A33 (2018) no. 10, 1850045; S. Narison. Addendum: Int. J. Mod. Phys. A33 (2018) no. 10, 1850045.] 〈αsG2〉|average=(6.35±0.35)×10−2 Ge4 . Our results clarify the (apparent) discrepancies between different estimates of 〈αsG2〉 from J/ψ sum rule but also shows the sensitivity of the sum rules on the choice of the μ -subtraction scale. As a biproduct, we deduce the Bc -decay constants fBc=371(17) MeV and fBc(2S)≤139(6) MeV.