Pole position of the $a_1(1260)$ from $\tau$-decay

TL;DR

This paper analyzes the three-pion system in tau decay to precisely determine the pole position of the $a_1(1260)$ meson using a unitarity-based model and experimental data, addressing complex resonance structures.

Contribution

The study introduces a model based on approximate three-body unitarity to extract the $a_1(1260)$ pole position from tau decay data, providing a detailed analysis of resonance structures.

Findings

Extracted the $a_1(1260)$ pole mass as approximately 1209 MeV.

Determined the width of the $a_1(1260)$ as approximately 576 MeV.

Developed a method to analyze three-body decay amplitudes with unitarity constraints.

Abstract

We perform an analysis of the three-pion system with quantum numbers $J^{PC}=1^{++}$ produced in the weak decay of $\tau$ leptons. The interaction is known to be dominated by the axial meson $a_1(1260)$. We build a model based on approximate three-body unitarity and fix the free parameters by fitting it to the ALEPH data on $\tau^-\to \pi^-\pi^+\pi^-\,\nu_\tau$ decay. We then perform the analytic continuation of the amplitude to the complex energy plane. The singularity structures related to the $\pi\pi$ subchannel resonances are carefully addressed. Finally, we extract the $a_1(1260)$ pole position $m_p^{(a_1(1260))}-i\Gamma_p^{(a_1(1260))}/2$ with $m_p^{(a_1(1260))} = (1209 \pm 4^{+12}_{-9})\text{MeV}$, $\Gamma_p^{(a_1(1260))} = (576 \pm 11 ^{+89}_{-20})\text{MeV}$}.