BHLS$_2$ Upgrade : $τ$ spectra, muon HVP and the [$π^0, η, η^′$] System

The generic Hidden Local Symmetry (HLS) model has recently given rise to its BHLS2variant, defined by introducing symmetry breaking mechanisms mostly in the vector mesonsector; the central piece is a modification of the covariant derivative at the root of the HLSapproach. However, the description of the τ dipion spectra, especially the Belle one, is notfully satisfactory whereas the simultaneous dealing with its annihilation sector (e+e− →π+π−/π+π−π0/π0γ/ηγ/K+K−/KLKS) is optimum. The present study shows thatthis issue is solved by means of an additional breaking term which also allows to consistently include the mixing properties of the [π0, η, η′] system within this extended BHLS2(EBHLS2) scope. This additional breaking piece which affects only the kinetic energypart of the BHLS2 Lagrangian is an extension of the usual ’t Hooft determinant term. Onethus obtains a fair account for the τ dipion spectra which comes complementing the fairaccount of the annihilation channels already reached. The Belle dipion spectrum is foundto provide evidence in favor of a violation of CVC in the τ lepton decay; this evidence isenforced by imposing the conditions = ipµfqδqq′, {[qq], q = u, d, s} onaxial current matrix elements accessible to EBHLS2. The EBHLS2 Lagrangian is foundto provide the usual (completed) formulae for the [π0, η, η′] mixing parameters and theglobal fits return mixing parameter values in agreement with expectations and carrying better uncertainties. On the other hand, one updates the evaluation for the leading order muonHVP aHV P −LOµ . One argues that the strong tension between the BaBar and KLOE pionform factors imposes to provide the two solutions : aHV P µ −LO(KLOE) = 687.31 ± 2.93and aHV P −LOµ (BaBar) = 692.36 ± 2.95, in units of 10−10, rather than some combinationof these. Taking into account a common systematics, their differences with the experimental BNL-FNAL average value exhibit significances > 5.4σ (KLOE) and > 4.1σ (BaBar)with fit probabilities favoring the former