In this talk, I summarize the results obtained recently in Ref.\,\cite{ SNe} using the PDG 22 compilation of the $e^+e^-\to$ Hadrons $\oplus$ the recent CMD3 data for the pion form factor. Using the gluon condensate $\langle \alpha_s G^2\rangle=(6.49\pm 0.35)\times 10^{ -2}$ GeV$^4$ from heavy quark sum rules, the extracted QCD four-quark and dimension eight condensate condensates values are: $\rho\alpha_s\langle\bar\psi\psi\rangle^2= (5.98\pm 0.64)\times 10^{ -4}$ GeV$^6$ and $d_8= (4.3\pm 3.0)\times 10^{ -2}$ GeV$^8$ from the ratio ${ \cal R}_{ 10}$ of Laplace sum rules to order $\alpha_s^4$. Inversely using these estimated values of the condensates, we obtain from ${ \cal R}_{ 10}$: $\langle \alpha_s G^2\rangle=(6.12\pm 0.61)\times 10^{ -2}$ GeV$^4$ which leads to the average $(6.40\pm 0.30)\times 10^{ -2}$ GeV$^4$. %from light and heavy quark systems. Using the lowest $\tau$-like decay moment, the mean result of Fixed Order (FO) and Contour Improved (CI) PT series within the standard OPE is : $\alpha_s(M_\tau)=0.3385(50)(136)_{ syst}$ [resp. $0.3262(37)(78)_{ syst}$] to order $\alpha_s^4$ [resp. $\alpha_s^5$] leading to $\alpha_s(M_Z)$=0.1207(17)(3) [resp. 0.1193(11)(3)], while the sum of the non-perturbative contribution at $M_\tau$ is\,: $\delta^V_{ NP}(M_\tau)=(2.3\pm 0.2)\times 10^{ -2}$. Using the same data, one also obtains the LO hadronic vacuum polarization to the muon and $\tau$ anomalous magnetic moments: $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} $ which leads to : $\Delta a_\mu\equiv a_\mu^{ exp}-a_\mu^{ th} = (143\pm 42_{ th}\pm 22_{ exp})\times 10^{ -11}$ indicating about 3$\sigma$ discrepancy between the SM predictions and experiment. One also finds: $\alpha^{ (5)}(M_Z)\vert_{ had}=(2766.3\pm 4.5)\times 10^{ -5}$.
Categories: hep-ph hep-ex hep-lat