Lepton-pair production in dipion $\tau$ lepton decays

We study the ${\tau }^{-}\to {\nu }_{\tau }{\pi }^{-}{\pi }^0{\ell }^{+}{\ell }^{-}$ ($\ell =e$, $\mu$) decays, which are $O({\alpha }^2)$ suppressed with respect to the dominant di-pion tau decay channel. Both the inner bremsstrahlung and the structure- (and model-) dependent contributions are considered. In the $\ell =e$ case, structure-dependent effects are $\mathcal{O}(1\%)$ in the decay rate, yielding a clean prediction of its branching ratio, $2.3\times {10}^{-5}$, measurable with BABAR or Belle(-II) data. For $\ell =\mu$, both contributions have a similar magnitude, and we get a branching fraction of $(1.6\pm 0.3)\times {10}^{-7}$, reachable by the end of Belle-II operation. These decays allow one to study the dynamics of strong interactions with simultaneous weak and electromagnetic probes; their knowledge will contribute to reducing backgrounds in lepton flavor and lepton number violating searches.

Figure 1

Feynman diagrams contributing to the ${\tau }^{-}\to {\nu }_{\tau }{\pi }^{-}{\pi }^0{\ell }^{+}{\ell }^{-}$ decay: (a) Inner bremsstrahlung (IB) off the tau lepton; (b) IB off the ${\pi }^{-}$ meson, (c) contact term.

Figure 2

Contributions to $V^{\mu \nu }$ in RChT of the hadronic vertex $W^{*-}\to {\pi }^{-}{\pi }^0{\gamma }^{*}$ vertices. Those involving resonances are highlighted with a thick dot. Insertion of the weak charged current is represented by the square dot. Resonances ($\rho$ unless specified) are represented with a double line, and ${\gamma }^{*}$ stands for ${\gamma }^{*}\to {\ell }^{+}{\ell }^{-}$.

Figure 3

Contributions from the Wess-Zumino-Witten to the $W^{*-}\to {\pi }^{-}{\pi }^0{\gamma }^{*}$ vertex [59, 60]. The weak charged current is represented by the square dot.

Figure 4

Contributions to the normalized invariant mass distribution for $e^{+}{e}^{-}$ pair production (interferences are not displayed). A double logarithmic scale was used. The second peak is due to the $\rho (770)$ dominance of the virtual photon propagator.

Figure 5

Contributions to the normalized invariant mass distribution for ${\mu }^{+}{\mu }^{-}$ pair production (interferences are not displayed). The second peak is due to the $\rho (770)$ dominance of the virtual photon propagator.