${\tau }^{-}\to {\eta }^{(\prime )}{\pi }^{-}{\nu }_{\tau }\gamma$ decays as background in the search for second class currents

Observation of ${\tau }^{-}\to {\eta }^{(\prime )}{\pi }^{-}{\nu }_{\tau }$ decays at Belle-II would indicate either a manifestation of isospin symmetry breaking or genuine second class current (SCC) effects. The corresponding radiative ${\tau }^{-}\to {\eta }^{(\prime )}{\pi }^{-}{\nu }_{\tau }\gamma$ decay channels are not suppressed by $G$-parity considerations and may represent a serious background in searches of SCCs in the former. We compute the observables associated to these radiative decays using resonance chiral Lagrangians and conclude that vetoing photons with $E_{\gamma }>100\mathrm{MeV}$ should get rid of this background in the Belle-II environment while searching for the ${\tau }^{-}\to \eta {\pi }^{-}{\nu }_{\tau }$ channel. Similar considerations hold inconclusive for decays involving the ${\eta }^{\prime }$, given the theory’s uncertainties in the prediction of the ${\tau }^{-}\to {\eta }^{\prime }{\pi }^{-}{\nu }_{\tau }$ branching ratio. Still, additional kinematics-based cuts should be able to suppress this background in the ${\eta }^{\prime }$ case to a negligible level.

Figure 1

Effective hadronic vertex (grey blob) that defines the $V_{\mu \nu }$ and $A_{\mu \nu }$ tensors.

Figure 2

Contributions to the effective weak vertex in the MDM model. The wavy line denotes the photon.

Figure 3

Contributions from the Wess-Zumino-Witten functional [56] to ${\tau }^{-}\to {\pi }^{-}\eta \gamma {\nu }_{\tau }$ decays. The cross circle indicates the insertion of the charged weak current.

Figure 4

One-resonance exchange contributions from the $R\chi L$ to the axial-vector form factors of the ${\tau }^{-}\to {\pi }^{-}\eta \gamma {\nu }_{\tau }$ decays. Vertices involving resonances are highlighted with a thick dot.

Figure 5

Two-resonance exchange contributions from the $R\chi L$ to the axial-vector form factors of the ${\tau }^{-}\to {\pi }^{-}\eta \gamma {\nu }_{\tau }$ decays. Vertices involving resonances are highlighted with a thick dot.

Figure 6

One-resonance exchange contributions from the $R\chi L$ to the vector form factors of the ${\tau }^{-}\to {\pi }^{-}\eta \gamma {\nu }_{\tau }$ decays. Vertices involving resonances are highlighted with a thick dot.

Figure 7

Two-resonance exchange contributions from the $R\chi L$ to the vector form factors of the ${\tau }^{-}\to {\pi }^{-}\eta \gamma {\nu }_{\tau }$ decays. Vertices involving resonances are highlighted with a thick dot.

Figure 8

Predictions for the $\mathrm{BR}({\tau }^{-}\to \eta {\pi }^{-}{\nu }_{\tau }\gamma )$ depending on the simulation sample size.

Figure 9

Normalized spectra of the ${\tau }^{-}\to \eta {\pi }^{-}{\nu }_{\tau }\gamma$ decays according to MDM.

Figure 10

Predictions for the $\mathrm{BR}({\tau }^{-}\to \eta {\pi }^{-}{\nu }_{\tau }\gamma )$ as a function of the photon energy cut.

Figure 11

Predictions for the $\mathrm{BR}({\tau }^{-}\to {\eta }^{\prime }{\pi }^{-}{\nu }_{\tau }\gamma )$ depending on the simulation sample size.

Figure 12

Predictions for the normalized spectra of the ${\tau }^{-}\to {\eta }^{\prime }{\pi }^{-}{\nu }_{\tau }\gamma$ decays according to MDM.

Figure 13

$\mathrm{BR}({\tau }^{-}\to \eta {\pi }^{-}{\nu }_{\tau }\gamma )$ represented as a function of the photon energy cut.

Figure 14

Predictions for the ${\tau }^{-}\to \eta {\pi }^{-}{\nu }_{\tau }\gamma$ decays’ branching ratios: 100 normally sampled points in the $R\chi L$ parameter space are plotted.

Figure 15

Normalized spectra of the ${\tau }^{-}\to \eta {\pi }^{-}{\nu }_{\tau }\gamma$ decays according to $R\chi L$.

Figure 16

$\mathrm{BR}({\tau }^{-}\to \eta {\pi }^{-}{\nu }_{\tau }\gamma )$ are represented as a function of the photon energy cut.

Figure 17

Predictions for the ${\tau }^{-}\to {\eta }^{\prime }{\pi }^{-}{\nu }_{\tau }\gamma$ decays’ branching ratios: 100 normally sampled points in the $R\chi L$ parameter space are plotted.

Figure 18

Normalized spectra of the ${\tau }^{-}\to {\eta }^{\prime }{\pi }^{-}{\nu }_{\tau }\gamma$ decays according to $R\chi L$.

Figure 19

$\mathrm{BR}({\tau }^{-}\to \eta {\pi }^{-}{\nu }_{\tau }\gamma )$ as a function of the photon energy cut.

Figure 20

The normalized photon spectra of the leading Low contributions to the $\mathrm{BR}({\tau }^{-}\to {\eta }^{(\prime )}{\pi }^{-}{\nu }_{\tau }\gamma )$.