Resonance Chiral Lagrangian analysis of ${\tau }^{-}\to {\eta }^{(\prime )}{\pi }^{-}{\pi }^0{\nu }_{\tau }$

The hadronization structure of ${\tau }^{-}\to \eta {\pi }^{-}{\pi }^0{\nu }_{\tau }$ decays is analyzed using chiral perturbation theory with resonances, considering only the contribution of the lightest meson resonances at leading order in the $1/N_C$ expansion. After imposing the asymptotic behavior of vector spectral functions ruled by QCD, unknown effective couplings are determined by fitting the ${\tau }^{-}\to \eta {\pi }^{-}{\pi }^0{\nu }_{\tau }$ branching ratio and decay spectrum to recent data. Predictions for the partner decay ${\tau }^{-}\to {\eta }^{\prime }{\pi }^{-}{\pi }^0{\nu }_{\tau }$ and the low-energy behavior of the cross section $\sigma (e^{+}{e}^{-}\to \eta {\pi }^{+}{\pi }^{-})$ are also discussed.

Figure 1

Topologies contributing to the final hadron state in ${\tau }^{-}\to {\eta }^{(\prime )}{\pi }^{-}{\pi }^0{\nu }_{\tau }$ decays in the $N_C\to \infty$ limit. Crossed circles indicate QCD vector current insertions. Single lines represent pseudoscalar mesons ($\eta$, $\pi$) while double lines stand for $\rho$-resonance intermediate states.

Figure 2

Contour in the $c_3-d_2$ plane compatible with the branching ratio of the decay ${\tau }^{-}\to \eta {\pi }^{-}{\pi }^0{\nu }_{\tau }$ at the level of one standard deviation. The black strips highlight the two regions which yield lowest values of ${\chi }^2/\mathrm{dof}$ (within one unit) from a fit to Belle data [43] on the corresponding spectral function.

Figure 3

Theoretical curves fitting the spectral function for ${\tau }^{-}\to \eta {\pi }^{-}{\pi }^0{\nu }_{\tau }$ decay, compared to experimental data [43].

Figure 4

Prediction for the branching ratio of the decay ${\tau }^{-}\to {\eta }^{\prime }{\pi }^{-}{\pi }^0{\nu }_{\tau }$ consistent (within one sigma) with the ${\tau }^{-}\to \eta {\pi }^{-}{\pi }^0{\nu }_{\tau }$ branching ratio quoted by the PDG. The horizontal line, corresponding to $\mathrm{BR}({\tau }^{-}\to {\eta }^{\prime }{\pi }^{-}{\pi }^0{\nu }_{\tau })=0.8\times {10}^{-4}$, represents the current PDG bound. The notation for the black strip is the same as in Fig. 2.

Figure 5

Prediction for $\sigma (e^{+}{e}^{-}\to \eta {\pi }^{+}{\pi }^{-})$ low-energy behavior from our analysis of ${\tau }^{-}\to \eta {\pi }^{-}{\pi }^0{\nu }_{\tau }$ decays, in comparison with DM1 [59], ND [60], DM2 [61], CMD-2 [62], and BABAR [63] data.