BHLS$_2$ Upgrade : $\tau$ spectra, muon HVP and the [$\pi^0,~\eta,~\eta^\prime$] System

TL;DR

This paper extends the BHLS$_2$ model to better describe $ au$ dipion spectra and annihilation channels, providing improved fits, evidence for CVC violation, and updated muon HVP estimates with significant tension between KLOE and BaBar data.

Contribution

The paper introduces an additional symmetry breaking term in BHLS$_2$ to accurately model $ au$ spectra and annihilation channels, and updates muon HVP calculations considering experimental tensions.

Findings

Improved fit to Belle $ au$ dipion spectra.

Evidence for CVC violation in $ au$ decay.

Two distinct solutions for muon HVP from KLOE and BaBar data.

Abstract

The generic Hidden Local Symmetry (HLS) model has given rise to the BHLS$_2$ variant which introduces symmetry breaking 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 Belle $\tau$ dipion spectra is not satisfactory whereas the dealing with its annihilation sector ($e^+ e^- \rightarrow \pi^+ \pi^-/\pi^+ \pi^-\pi^0/ \pi^0 \gamma/\eta \gamma/K^+ K^-/K_L K_S$) is optimum. This issue is solved by an additional breaking term allowing to include the $[\pi^0,\eta,\eta^\prime]$ system. This extension of the usual 't~Hooft determinant term only affects the BHLS$_2$ kinetic energy term. One obtains a fair account for the $\tau$ dipion spectra as well as for the annihilation channels. The Belle dipion spectrum provides evidence for a violation of CVC in the $\tau$ lepton decay; this evidence is strengthened by imposing the conditions $<0|J_\mu^q |[q^\prime \overline{q^\prime}](p)>=ip_\mu f_q \delta_{q q^\prime }, \{ [q \overline{q}], q=u,d,s\}$ on BHLS$_2$ axial current matrix elements. BHLS$_2$ is found to recover the usual (completed) formulae for the [$\pi^0,~\eta,~\eta^\prime$] mixing parameters and the global fits return mixing parameter values in agreement with expectations and better uncertainties. Updating the muon HVP, one argues that the strong tension between the KLOE and BaBar pion form factors imposes to provide two solutions : $a_\mu^{HVP-LO}({\rm KLOE})=687.31 \pm 2.93$ and $a_\mu^{HVP-LO}({\rm BaBar})=692.36 \pm 2.95$, in units of $10^{-10}$, rather than some combination of these. Taking into account common systematics, their differences with the experimental BNL-FNAL average value exhibit significances $> 5.4\sigma$ (KLOE) and $> 4.1\sigma$ (BaBar) with fit probabilities favoring the former.