$\tau$ Vector Spectral Function from a Chirally Invariant Hadronic Model with Weak Interaction

TL;DR

This paper develops a gauge-invariant linear sigma model describing scalar, pseudoscalar, vector, and axial-vector mesons at around 1 GeV, incorporating electroweak interactions and analyzing the vector spectral function.

Contribution

It introduces a chirally invariant hadronic model with electroweak symmetry, providing a detailed description of the vector spectral function including decay contributions.

Findings

Vector channel is well described

Decay W^- → π^- π^0 is necessary for accurate line shape

Model achieves good phenomenological agreement

Abstract

The linear sigma model we present here describes the vacuum phenomenology of the $N_F=2$ scalar, pseudoscalar, vector and axial-vector mesons at energies $\simeq 1\text{ GeV}$. Together with a local $SU(2)_L\times U(1)_Y$ symmetry obtained from a local transformation of the fields we obtain a gauge invariant effective description of electroweak interaction with hadrons in the vacuum. We show that the vector channel is described well and that the contributions from the direct decay of $W^- \rightarrow \pi^- \pi^0$ are, although quantitatively small, necessary to reproduce the line shape of the vector spectral function.