Revisiting the Muon Anomaly from $e^+ e^-\to$ Hadrons

In this talk, I revisit and present a more comprehensive estimate of the lowest order Hadronic Vacuum Polarization (HVP) contribution $a_\mu\vert_{hvp}^{lo}$ to the muon anomalous magnetic moment (muon anomaly) from $e^+e^-\to$ Hadrons obtained recently in Ref.[1]. New CMD-3 data on $e^+e^-\to 2\pi$ [2] and precise BABAR [3] and recent BELLE2 [4] $e^+e^-\to 3\pi$ data are usedto update the estimate of the $I=0$ isoscalar channel below the $\phi$-meson mass. Adding the data compiled by PDG22 [5] above 1 GeV and the QCD improved continuum used in Ref. [1], one deduces: $a_\mu\vert^{hvp}_{lo}=(7043\pm 37)\times 10^{-11}$.A comparison with previous data driven ($e^+e^-$ and $\tau$-decays) estimates is done.Including the Higher Order $a_\mu\vert_{hvp}^{ho}$ corrections, the phenomenological estimate of the Hadronic Light by Light scattering up to NLO and the QED and Electroweak (EW) contributions, one obtains: $\Delta a_\mu^{pheno}\equiv a_\mu^{exp}-a_{\mu}^{pheno}= (81\pm 41)\times 10^{-11}$ where the recent experimental value $a_\mu^{exp}$ [6] has been used. This result consolidates the previous one in Ref.[1], after adding the $\pi^0\gamma,\eta\gamma$ contributions, and can be compared with the one from the most precise Lattice result $\Delta a_\mu^{lattice}= (90\pm 56)\times 10^{-11}$. Then, we deduce the (tentative) SM prediction average : $\Delta a_\mu^{SM} = (87\pm 33)\times 10^{-11}$. We complete the paper by revising our predictions on the LO HVP contributions in adding the $\pi^0\gamma,\eta\gamma$ contributions to the ones in Ref.[1]. Then, we obtain: $a_\tau\vert^{hvp}_{lo}=(3516\pm 25)\times 10^{-11}$ and $\Delta \alpha^{(5)}_{had}(M_Z^2)=(2770.7\pm 4.5)\times 10^{-5}$ for 5 flavours.