Abstract
Using the PDG 22 compilation of the $e^+e^-\to$ Hadrons data $\oplus$ the recent CMD3 data for the pion form factor and the value of gluon condensate $\langle\alpha_s G^2\rangle$ from heavy quarkonia, we extract the value of the four-quark condensate: $\rho\alpha_s\langle\bar\psi\psi\rangle^2= (5.9\pm 0.4)\times 10^{-4}$ GeV$^6$ from the ratio ${\cal R}_{10}$ of Laplace sum rules. We also show the inconsistency in using at the same time the standard SVZ value of the gluon and the vacuum saturation of the four-quark condensates. The dimension eight condensate is found to be : $d_8= (11.3\pm 1.1)\times 10^{-2}$ GeV$^8$. Using the previous QCD condensates, we extract from the lowest $\tau$-like decay moment ${\cal R}_\tau^{ee}$ the value of the QCD coupling : $\alpha_s(M^2_\tau)=0.329(10)$ within fixed order perturbation theory (FO) and the standard Operator Product Expansion (OPE). The corresponding value of the sum of the non-perturbative contribution is : $\delta_{NP}(M_\tau)=(2.32\pm 0.15)\times 10^{-2}$. Reciprocically, using $\alpha_s(M_\tau)$, $\alpha_s G^2$ and $d_8$ as inputs, we test the stability of the value of the four-quark condensate obtained from the lowest $\tau$-like moment. We complete our analysis by updating our previous determinations of the lowest order hadronic vacuum polarization contributions to the lepton anomalies and to $\alpha(M^2_Z)$. We obtain in Table \ref{tab:amu1}: $a_\mu\vert^{hvp}_{l.o}= (7036.5\pm 38.9)\times10^{-11}, \, a_\tau\vert^{hvp}_{l.o}= (3494.8\pm 24.7)\times10^{-9} $. and $\alpha(M^2_Z)=(2766.3\pm 4.5)\times 10^{-5}$. This new value of $a_\mu$ leads to : $\Delta a_\mu\equiv a_\mu^{exp}-a_\mu^{th} = (142\pm 42_{th}\pm 41_{exp})\times 10^{-11}$. which reduces the tension between the SM prediction and experiment.