QCD parameters and SM-high precisions from e+e- to Hadrons

Using the PDG 22 compilation of the $e^+e^-\to$ Hadrons $\oplus$ the recent CMD3 data for the pion form factor and the value of gluon condensate $<\alpha_s G^2>$ from heavy quarkonia, we extract the value of the four-quark condensate : $\rho\alpha_s<\bar\psi\psi>^2= (5.98\pm 0.64)\times 10^{-4}$ GeV$^6$ and the dimension eight condensate: $d_8= (4.3\pm 3.0)\times 10^{-2}$ GeV$^8$from the ratio ${\cal R}_{10}$ of Laplace sum rules to order $\alpha_s^4$. We 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. Using the previous values of the four-quark and $d_8$ condensates, we re-extract $<\alpha_s G^2>$ from ${\cal R}_{10}$ to be: $(6.12\pm 0.61)\times 10^{-2}$ GeV$^4$ in perfect agreement with the one from heavy quarkonia. We also use the lowest $\tau$-like decay moment ${\cal R}_\tau^{ee}$ to extract the value of the QCD coupling $\alpha_s(M^2_\tau)$=0.3385(145)[resp. 0.3262(86)] (mean of fixed order (FO) and Contour Improved (CI) PT series) to order $\alpha_s^4$ [resp. $\alpha_s^5$] and the standard OPE. The corresponding value of the sum of the non-perturbative contribution is: $\delta_{NP}(M_\tau)=(3.74\pm 0.40)\times 10^{-2}$. Reciprocally, 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 2 : $a_\mu\vert^{hvp}_{l.o}= (7036.5\pm 38.9)\times10^{-11}, a_\tau\vert^{hvp}_{l.o}= (3494.8\pm 24.7)\times10^{-9}$ and $\alpha(M^2_Z)=(2766.3\pm 4.5)\times 10^{-5}$. This new value of $a_\mu$ leads to: $\Delta a_\mu\equiv a_\mu^{exp}-a_\mu^{th} = (142\pm 42_{th}\pm 41_{exp})\times 10^{-11}$.