Branching ratio estimates of τ− → π−η(′)ντ decays

We study the rare τ− → πηντ decays. These processes occur via isospin violation and belong to the so-called second-class currents unseen in Nature so far. Our analysis is based on the framework of resonance chiral theory supplemented by dispersion relations. In this contribution we discuss the prospects for their discovery at forthcoming B-factories such Belle-II. While we find a total branching fraction for the π−η decay mode of ∼ 1.7 × 10−5, well within the reach of Belle-II, the π−η′ channel might be one or two orders of magnitude more suppressed.


Introduction
The τ is the only lepton heavy enough to decay into hadrons. Actually, its partial decay width involving hadrons in the final state is of ∼ 65%. At the exclusive level, hadronic τ decays represent a clean laboratory to access the non-perturbative regime of QCD, i.e. they are useful to understand the hadronization of QCD currents, to study form factors and to extract resonance parameters. Rare decay channels of the τ are a very promising area to pursue at forthcoming B-factories since the interaction is suppressed and the sensitivity to new physics might eventually be enhanced. In the Standard Model (SM), the suppression of rare τ decays might be due to several reasons e.g.: -Cabbibo suppression: strange hadronic final states are suppressed with respect to non-strange ones since the V us element of the CKM matrix enters the description instead of V ud . Examples of Cabbibo suppressed decays are τ − → K − ην τ and τ − → K − η ν τ that we have analyzed in Refs. [1,2].
-Phase-space: decays involving kaons, η or η mesons in the final state lead, usually, to phase-space suppression because of the large mass of these states.
-Second-class currents: decays occurring via G-parity violation i.e. the G-parity of the vector current is opposite to the G-parity of the hadronic system. In the isospin limit, G-parity is exact and these processes are forbidden in the SM. However, isospin is an S U(2) approximate symmetry broken both by the up and down quark mass and electric charge differences leading the suppression.
In this contribution we address the study of the rare second-class currents τ − → π − η ( ) ν τ decays [3]. Our goal is two-fold: to describe the participating hadronic form factors, predict the decay spectra and estimate the corresponding branching ratios in order to stimulate the experimental collaborations to measure these decays for the first time. The expression for the invariant mass distribution has been derived in detail in Ref. [3] and reads where q PQ (s) is a kinematical factor and F π − η ( ) +,0 (s) are the vector (+) and scalar (0) form factors normalized to unity at the origin. Notice that F π − η ( ) + (0), an isospin-violating term (cf. Eq. (2)), factorizes in Eq. (1) explaining the smallness of these decays.
• Scalar form factor: we have considered several parameterizations ordered according to their increasing fulfillment of analyticity and unitarity (See Ref. [3] for details and expressions).
-Breit-Wigner: we start again with the RChT framework by considering first the exchange of the a 0 (980) resonance and then including the a 0 (1450) into the description. We have taken into account some (elastic and inelastic) final state hadronic interactions effects by resuming the imaginary part of the self-energy loop function into the propagator. However, this description does not incorporate the real part of the loop function and, therefore, it violates analyticity. -Elastic dispersion relation: to fulfill analyticity and elastic unitarity we rely on the Omnès integral.
Unfortunately, we lack of any kind of experimental data either for the phase shift or for the decays spectra. Therefore, for our analysis we get a model for the phase from the elastic scattering amplitudes computed in Ref. [5] in RChT at one-loop unitarized through the N/D method. -Coupled channels: in Ref. [3] we have provided an alternative expression equivalent to the Omnès solution with one subtraction, given as a closed-form expression, for the single elastic case. We have also shown that one might accommodate inelasticities and obtain form factors in coupledchannels reasoning in the same way. In particular, for our analysis we have considered two-and three-coupled channels problem for the π − η, K − K 0 and π − η scalar form factors.

Branching ratio predictions
In Fig. 1 we show the distribution for the τ − → π − ην τ decay depending on the scalar form factor we have employed. The vector contribution dominates the distribution at low-energies with a clear  [3]. Three coupled channels for the scalar contribution.