# BHLS$_2$, a New Breaking of the HLS Model and its Phenomenology

## TL;DR

This paper introduces BHLS$_2$, an improved Hidden Local Symmetry model that better describes light meson processes and reduces uncertainties in the theoretical calculation of the muon anomalous magnetic moment.

## Contribution

BHLS$_2$ extends the BHLS framework with new symmetry breaking schemes, enhancing phenomenological accuracy and predictive power for light meson processes and muon g-2 calculations.

## Key findings

- BHLS$_2$ accurately predicts the lattice QCD pion form factor.
- It significantly reduces systematic uncertainties in muon g-2 theoretical estimates.
- The leading order HVP contribution is precisely evaluated as 686.65 ± 3.01 (stat) + syst.

## Abstract

Previous studies have shown that the Hidden Local Symmetry (HLS) Model, supplied with appropriate symmetry breaking, provides an Effective Lagrangian (BHLS) able to encompass a large number of processes within a unified framework. A global fit provides a simultaneous description of the $e^+ e^-$ annihilation into $\pi^+\pi^-$, $\pi^0\gamma$, $\eta \gamma$, $\pi^+\pi^-\pi^0$, $K^+K^-$, $K_L K_S$, the dipion spectrum in the $\tau$ decay and other light meson partial widths. Additional breaking schemes extend its scope to spacelike processes within a new framework (BHLS$_2$). The phenomenology previously explored with BHLS is revisited in the BHLS$_2$ context with emphasis on the $\Phi$ mass region. It is shown that BHLS$_2$ addresses the close spacelike region covered by NA7 and Fermilab data; also,the Lattice QCD pion form factor is accurately predicted by BHLS$_2$ using fits to annihilation data only. These channels contribution to $a_\mu$ over the range of validity of BHLS$_2$ is shown to strongly reduce the former BHLS systematics. The uncertainty on $a_\mu^{\rm th}(\sqrt{s}< 1.05$ GeV) is much improved compared to standard approaches relying on direct integration methods of measured spectra. Using the BHLS$_2$ results, the leading order HVP contribution to the muon anomalous moment is $a_\mu^{HVP-LO}= 686.65 \pm 3.01 +(+1.16,-0.75)_{syst.}$ in units of $10^{-10}$. Using a conservative estimate for the light--by--light contribution, our evaluation is $a_\mu^{\rm th}=\left [11\,659\,175.96 \pm 4.17 +(+1.16,-0.75)_{syst.}\right]\times 10^{-10}$. The $\rho^0\gamma$ mixing in dispersive and LQCD approaches is discussed and may amount to a shift $\delta a_\mu[\pi\pi]_{\rho\gamma} = +(3.10\pm0.31)\times 10^{-10}$ at LO+NLO, presently treated as additional systematics. Taking this shift into account, $a_\mu^{\rm th}-a_\mu^{\rm BNL}$ exhibits a significance not smaller than $3.8 \sigma$.