Study of the heavy-lepton decayτ→ν3π

Abstract

A construction is made of the hadronic matrix element of the axial-vector current $〈3\ensuremath{\pi}|{A}_{\ensuremath{\lambda}}|0〉$ which enters in the amplitude for the decay $\ensuremath{\tau}\ensuremath{\rightarrow}\ensuremath{\nu}+3\ensuremath{\pi}$. The matrix element has ${1}^{+}$ and ${\mathrm{O}}^{\ensuremath{-}}$ parts, and both are expanded in terms of three-body $\ensuremath{\rho}\ensuremath{\pi}$ and $\ensuremath{\epsilon}\ensuremath{\pi}$ isobar channels. A modular picture of the process has evolved in which one module describes the effects of rearrangement in $3\ensuremath{\pi}$ states having these quantum numbers while the other module pertains to the origin of the $3\ensuremath{\pi}$ system and is peculiar to the process at hand. The first factor has been calculated in a previous paper and exhibits a noteworthy ${1}^{+}$ enhancement effect at 1.1 GeV. The second factor is modeled here in detail and is shown to be fully parametrizable by using the methods of hard-pion current algebra; its ${1}^{+}$ part is presumed to be dominated by the ${A}_{1}$ resonance. The entire composition contains only three parameters, two of which are the mass and width of the ${A}_{1}$, ${m}_{A}$ and ${\ensuremath{\Gamma}}_{A}$. As such it provides a suitable means of examining $\ensuremath{\tau}\ensuremath{\rightarrow}\ensuremath{\nu}3\ensuremath{\pi}$ phenomenology in the presence of several interesting effects and may be employed in conjunction with suitably refined data in order to obtain the values of ${m}_{A}$ and ${\ensuremath{\Gamma}}_{A}$. The meager experimental data currently available are shown to suggest tentative values of these parameters around ${m}_{A}=1.10$ GeV and ${\ensuremath{\Gamma}}_{A}=0.45$ GeV. The ${1}^{+}$ enhancement effect due to final-state rearrangement is assessed relative to the effect of the resonance alone and the extent to which it acts as a peak shifter or peak sharpener is evaluated. Finally, attention is drawn to the possible need for special care and interest in handling the generally neglected ${0}^{\ensuremath{-}}$ state, in view of recent results from diffraction experiments in the pseudoscalar $3\ensuremath{\pi}$ channel.