BHLS upgrade: spectra, muon HVP and the [ ] system

The generic hidden local symmetry (HLS) model has recently given rise to its $$ variant, defined by introducing symmetry breaking mostly in the vector meson sector; the central mechanism is a modification of the covariant derivative at the root of the HLS approach. However, the description of the $$ dipion spectra, especially the Belle one, is not fully satisfactory, whereas the simultaneous dealing with its annihilation sector ( $$ ) is optimum. We show that this issue is solved by means of an additional breaking term which also allows us to consistently include the mixing properties of the $$ system within this extended $$ ( $$ ) scope. This mechanism, an extension of the usual 't Hooft determinant term, only affects the kinetic energy part of the $$ Lagrangian. One thus obtains a fair account for the $$ dipion spectra which complements the fair account of the annihilation channels already reached. The Belle dipion spectrum is found to provide evidence in favor of a violation of the conserved vector current (CVC) in the $$ lepton decay; this evidence is enforced by imposing the conditions $$ on $$ axial current matrix elements. $$ is found to recover the usual (completed) formulae for the [ $$ ] mixing parameters, and the global fits return mixing parameter values in agreement with expectations and better uncertainties. Updating the muon hadronic vacuum polarization (HVP), one also argues that the strong tension between the KLOE and BaBar pion form factors imposes to provide two solutions, namely $$ and $$ , in units of $$ , rather than some combination of these. Taking into account common systematics, their differences from the experimental BNL-FNAL average value exhibit significance $$ (KLOE) and $$ (BaBar), with fit probabilities favoring the former.