QCD parameters and $f_{B_c}$ from heavy quark sum rules

We report results of our recent works [1,2] where we where the correlations between the c,b-quark running masses{m}_{c,b}, the gluon condensate<\alpha_s G^2> and the QCD coupling \alpha_s in the MS-scheme from an analysis of the charmonium and bottomium spectra and the B_c-meson mass. We use optimized ratios of relativistic Laplace sum rules (LSR) evaluated at the \mu-subtraction stability point where higher orders PT and D< 6-8-dimensions non-perturbative condensates corrections are included. We obtain [1] \alpha_s(2.85)=0.262(9) and \alpha_s(9.50)=0.180(8) from the (pseudo)scalar M_{\chi_{0c(0b)}}-M_{\eta_{c(b)}} mass-splittings at \mu=2.85(9.50) GeV. The most precise result from the charm channel leads to \alpha_s(M_\tau)=0.318(15) and \alpha_s(M_Z)=0.1183(19)(3) in excellent agreement with the world average: \alpha_s(M_Z)=0.1181(11)[3,4]. Updated results from a global fit of the (axial-)vector and (pseudo)scalar channels using Laplace and Moments sum rules @ N2LO [1] combined with the one from M_{B_c} [2] lead to the new tentative QCD spectral 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 <\alpha_s G^2> from the (axial)-vector charmonium channels combined with previous determinations in Table 1, leads to the new QSSR average [1]: <\alpha_s G^2>_average=(6.35\pm 0.35)x 10^{-2} GeV^4. Our results clarify the (apparent) discrepancies between different estimates of <\alpha_s G^2> from J/\psi sum rule but also shows the sensitivity of the sum rules on the choice of the \mu-subtraction scale. As a biproduct, we deduce the B_c-decay constants f_{B_c}=371(17) MeV and f_{B_c}(2S)< 139(6) MeV.