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 $\pi^+\pi^-$, $\pi^0\gamma$, $\eta \gamma$, $\pi^+\pi^-\pi^0$, $K^+K^-$, $K_L K_S$, the dipion spectrum in the $\tau$ 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 $\Phi$ 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 $a_\mu$ over the range of validity of BHLS$_2$ is shown to strongly reduce the former BHLS systematics. The uncertainty on $a_\mu^{ \rm th}(\sqrt{ s}< 1.05$ 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 $a_\mu^{ HVP-LO}= 686.65 \pm 3.01 +(+1.16,-0.75)_{ syst.}$ in units of $10^{ -10}$. Using a conservative estimate for the light--by--light contribution, our evaluation is $a_\mu^{ \rm th}=\left [11\,659\,175.96 \pm 4.17 +(+1.16,-0.75)_{ syst.}\right]\times 10^{ -10}$. The $\rho^0\gamma$ mixing in dispersive and LQCD approaches is discussed and may amount to a shift $\delta a_\mu[\pi\pi]_{ \rho\gamma} = +(3.10\pm0.31)\times 10^{ -10}$ at LO+NLO, presently treated as additional systematics. Taking this shift into account, $a_\mu^{ \rm th}-a_\mu^{ \rm BNL}$ exhibits a significance not smaller than $3.8 \sigma$.
Categories: hep-ph