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BHLS$_2$, a New Breaking of the HLS Model and its Phenomenology

Previous studies have shown that the Hidden Local Symmetry (HLS) Model, supplied with appropriate symmetry breaking, provides an Effective Lagrangian (BHLS) able to encompass a large number of processes within a unified framework. A global fit provides a simultaneous description of the $e^+ e^-$ annihilation into $π^+π^-$, $π^0γ$, $ηγ$, $π^+π^-π^0$, $K^+K^-$, $K_L K_S$, the dipion spectrum in the $τ$ decay and other light meson partial widths. Additional breaking schemes extend its scope to spacelike processes within a new framework (BHLS$_2$). The phenomenology previously explored with BHLS is revisited in the BHLS$_2$ context with emphasis on the $Φ$ mass region. It is shown that BHLS$_2$ addresses the close spacelike region covered by NA7 and Fermilab data; also,the Lattice QCD pion form factor is accurately predicted by BHLS$_2$ using fits to annihilation data only. These channels contribution to $a_μ$ over the range of validity of BHLS$_2$ is shown to strongly reduce the former BHLS systematics. The uncertainty on $a_μ^{\rm th}(\sqrt{s}< 1.05$ GeV) is much improved compared to standard approaches relying on direct integration methods of measured spectra. Using the BHLS$_2$ results, the leading order HVP contribution to the muon anomalous moment is $a_μ^{HVP-LO}= 686.65 \pm 3.01 +(+1.16,-0.75)_{syst.}$ in units of $10^{-10}$. Using a conservative estimate for the light--by--light contribution, our evaluation is $a_μ^{\rm th}=\left [11\,659\,175.96 \pm 4.17 +(+1.16,-0.75)_{syst.}\right]\times 10^{-10}$. The $ρ^0γ$ mixing in dispersive and LQCD approaches is discussed and may amount to a shift $δa_μ[ππ]_{ργ} = +(3.10\pm0.31)\times 10^{-10}$ at LO+NLO, presently treated as additional systematics. Taking this shift into account, $a_μ^{\rm th}-a_μ^{\rm BNL}$ exhibits a significance not smaller than $3.8 σ$.