Erratum to: $\hbox {BHLS}_{2}$, 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 mechanisms, provides an Effective Lagrangian (BHLS) able to encompass a large number of processes within a unified framework. This allowed one to design a global fit procedure which provides a fair simultaneous description of the $e^+ e^-$ annihilation into six final states ($\pi ^+\pi ^-$, $\pi ^0\gamma $, $\eta \gamma $, $\pi ^+\pi ^-\pi ^0$, $K^+K^-$, $K_L K_S$), the dipion spectrum in the $\tau $ decay and some more light meson decay partial widths. In this paper, additional breaking schemes are defined which improve the BHLS working and extend its scope so as to absorb spacelike processes within a new framework (${\hbox {BHLS}}_2$). The phenomenology previously explored with BHLS is fully revisited in the ${\hbox {BHLS}}_2$ context with special emphasis on the $\phi $ mass region using all available data samples. It is shown that ${\hbox {BHLS}}_2$ addresses perfectly the close spacelike region covered by NA7 and Fermilab data; it is also shown that the recent lattice QCD (LQCD) information on the pion form factor are accurately predicted by the ${\hbox {BHLS}}_2$ fit functions derived from fits to only annihilation data. The contribution to the muon anomalous magnetic moment $a_\mu ^{\mathrm{th}}$ of these annihilation channels over the range of validity of ${\hbox {BHLS}}_2$ (up to $\simeq $ 1.05 GeV) is updated within the new ${\hbox {BHLS}}_2$ framework and shown to strongly reduce the former BHLS systematics. The uncertainty on $a_\mu ^{\mathrm{th}}(\sqrt{s}< 1.05 \, \hbox {GeV}$) is much improved compared to standard approaches relying on direct integration methods of measured spectra. Using the ${\hbox {BHLS}}_2$ results, the leading-order HVP contribution to the muon anomalous moment is $a_\mu ^{\mathrm{HVP-LO}}= 686.65 \pm 3.01 +(+1.16,-0.75)_{\mathrm{syst}}$ in units of $10^{-10}$. Using a conservative estimate for the light-by-light contribution, our evaluation for the muon anomalous magnetic moment is $a_\mu ^{\mathrm{th}}=\left[ 11\,659\,175.96 \pm 4.17 +(+1.16,-0.75)_{\mathrm{syst}}\right] \times 10^{-10}$. The relationship between the dispersive and LQCD approaches to the $\rho ^0$–$\gamma $ mixing is also discussed which may amount to a shift of $\delta a_\mu [\pi \pi ]_{\rho \gamma }=+(3.10\pm 0.31) \times 10^{-10}$ at LO+NLO, presently treated as additional systematics. Taking also this shift into account, the difference $a_\mu ^{\mathrm{th}}-a_\mu ^{\mathrm{BNL}}$ exhibits a significance not smaller than $3.8 \sigma $