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 $m_p^{(a_1(1260))} = (1209 \pm 4^{+12}_{-9})\text{MeV}$, $\Gamma_p^{(a_1(1260))} = (576 \pm 11 ^{+89}_{-20})\text{MeV}$}.
Highlights
The internal dynamics of the quantum chromodynamics (QCD) degrees of freedom manifests itself in the spectrum of hadron resonances
We perform an analysis of the three-pion system with quantum numbers JPC 1⁄4 1þþ produced in the weak decay of τ leptons
Despite the fact that the corresponding JPC 1⁄4 1þþ partial wave dominates the hadronic weak decay of τ leptons as well as diffractive reactions of high-energy pions, the parameters of the a1ð1260Þ are still poorly known. While the latter reactions suffer from an irreducible background due to nonresonant processes, the system of three pions produced in τ decay provides a very clean access to axial-vector resonances
Summary
The internal dynamics of the quantum chromodynamics (QCD) degrees of freedom manifests itself in the spectrum of hadron resonances. For a majority of states, ab initio QCD calculations of their decay properties, e.g., decay widths, branching ratios, are not yet available. The problem may be related to the presence of a large, coherent, nonresonant background, known as the Deck process in pion diffraction [28,29,30,31] This process happens to dominate in the JPC 1⁄4 1þþ partial wave and directly influences the extraction of the a1ð1260Þ resonance parameters in pion diffraction. An independent determination of the a1ð1260Þ resonance properties is relevant for a better understanding of this state and to constrain the Deck process, which contributes significantly to other partial waves including the ones with the exotic quantum numbers 1−þ [31].
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