Abstract
The Hidden Local Symmetry (HLS) Model provides a framework able to encompass several physical processes and gives a unfied description of these in an energy range extending up to the $\phi$ mass. Supplied with appropriate symmetry breaking schemes, the HLS Model gives a broken Effective Lagrangian (BHLS). The BHLS Lagrangian gives rise to a fit procedure in which a simultaneous description of the $e^+ e^-$ annihilations to $\pi^+\pi^-$, $\pi^0 \gamma$, $\eta \gamma$, $\pi^+\pi^- \pi^0$, $K^+ K^-$, $K_L K_S$ and of the dipion spectrum in the decay $\tau^\pm \rightarrow \pi^\pm \pi^0 \nu$ can be performed. Supplemented with a few pieces of information on the $\rho^0-\omega-\phi$ system, the $\tau$ dipion spectrum is shown to predict accurately the pion form factor in $e^+ e^-$ annihilations. Physics results derived from global fits involving or excluding the $\tau$ dipion spectra are found consistent with each others. Therefore, no obvious mismatch between the $\tau$ and $e^+e^- $ physics properties arises and the $\tau-e^+e^- $ puzzle vanishes within the broken HLS Model.
Highlights
The pion form factor in the e+e− → π+π− annihilation (Fπee(s)) and in the the τ± → π±π0ν decay (Fπτ(s)) are expected to differ only by isospin symmetry breaking (IB) terms
The Hidden Local Symmetry Model (HLS) Model is a framework which encompasses simultaneously several different physics processes covered by a large number of already available data samples
In order to substantiate this specific breaking of the HLS model, let us quote a result derived from a global fit involving all the channels listed in the Introduction; the ratio fρ0γ(s)/ fρ±W shown in Figure 1 exhibits significant variations over the HLS energy range of interest
Summary
The pion form factor in the e+e− → π+π− annihilation (Fπee(s)) and in the the τ± → π±π0ν decay (Fπτ(s)) are expected to differ only by isospin symmetry breaking (IB) terms. Understanding the relationship between Fπee(s) and Fπτ(s) is important as it can allow for 2 different evaluations of the dipion contribution to aμ(ππ), the muon Hadronic Vacuum Polarization (HVP) which could be merged together if consistent with each other. This relationship supposes a good understanding of isospin symmetry breaking and an appropriate modelling. For a long time [1, 2], the comparison between |Fπee(s)|2 and |Fπτ(s)|2 was not satisfactory and the mismatch [3] was severe enough that one started to speak of a "e+e− vs τ" puzzle.
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