# $\tau \to \ell + \eta/ \eta^{'}/ \pi^0$

###### tags: `tau` `belleII` `marseille`

## Previous searches

| channel | experiment | luminosity [/fb]|BF upper limit | link to paper|
| -------- | -------- | -------- | -------- | -------- |
| $\tau \to \ell \eta$ | Belle | 401|| [link to paper](https://arxiv.org/pdf/hep-ex/0703009v1.pdf)|
| $\tau \to \ell \eta$ | Babar | 339| $(1.1−2.4) \times 10^{−7}$| [link to paper](https://arxiv.org/pdf/hep-ex/0610067.pdf)|

## Motivation
* BF of LFV tau decays involving pseudo scalar mesons could be enehanced up to $10^ {-6}$ level by new mediators (LFV Z' bosons), higgs-mediated LFV interaction, new mediator in SUSY model, heavy singlet Dirac fermion
* sensitivity already at first generation of B factory

## Belle simulation
* KORALB/TAUOLA [15] for τ + τ − , ( LZ: but TAUOLA handles only tau decays, generation with KKMC?)
* QQ [16] for B B̄ and continuum, http://www.lns.cornell.edu/public/CLEO/soft/QQ/
* BHLUMI for Bhabha, S. Jadach et al., Comp. Phys. Commun. 79, 305 (1992).
* KKMC (S. Jadach et al., Comp. Phys. Commun. 130, 260 (2000)) for e+ e− → µ+ µ−
* AAFH [19] for two-photon processes.
* Signal MC is generated by KORALB/TAUOLA.

## Event reconstruction
* main handle: large used of (likelihood based) PID
* 1-prong tag
* tag track oppositely charged wrt signal lepton * any number of neutrals on tag side
* signal side:
1. $\eta$:
a.$\gamma\gamma$,
b.$\pi^+\pi^-\pi^0(\gamma\gamma)$
2. $\eta^{\prime}$:
a.$\rho(\pi^+\pi^-)\gamma$,
b.$\eta(\gamma\gamma)\pi^+\pi^-$
3. $\pi^0: \gamma\gamma$
* tracks in fiducial volume
* minimum transvers momentum of 0.1 GeV
* minimum cluster Energy 0.1(0.05) GeV for 1x1(3x1) topology
* 1.a and 4 : 1x1 topology, nGoodTracks ==2
* other final states: 3x1, nGoodTracks ==4

* signal lepton requires el(muon)ID >0.9 correpsonding to 92%(88%) efficiencie and p_T >0.7 GeV
* no FSR correction, reject radiative electrons to reduce fake mesons rate: $cos\theta_{e\gamma}>0.99$

#### neutral meson reconstruction
* $\pi^0$: two photons with energy > 0.1 GeV within invariant mass window [0.115 , 0.152] and $p_{\gamma\gamma}>0.1$
* $\eta$ candidates:
* $\gamma\gamma$ mode, $0.515 < m_{\gamma\gamma} < 0.570$ ( $ -3\sigma$, +2.5\sigma interval) + $\pi^0$ veto: any photons with E>0.05 GeV and $m_{inv}$ in the range [0.1,0.16] GeV
* $\pi^+\pi^-\pi^0$: $0.532 < m_{\gamma\gamma} < 0.562$ ( $\pm 3\sigma$ interval)

* $\eta^{\prime}$ (* not included in MC generic, very rare decays):
* $\rho\gamma$ mode: $0.550 < m_{\pi^+\pi^-} <0.900$ and $0.930 < m_{\rho\gamma} <0.970$ corresponding to $-3\sigma, +2.5\sigma$ + pi0 veto formed by photons from eta_prime cand + any gamma with energy > 0.05
* dominant background from $\tau \to h\rho^0 \nu_{\tau} (+\pi^0)$ with h kaon or pion with a photon from $\pi^0$ decay, beam background or initial state radiation (ISR).

* $\eta(\gamma\gamma)\pi^+\pi^-$ mode: (LZ: how are tracks consistent with being pions defined? How is sigma computed? MC resoutions from fitted simulation?) apply electron veto on pion cand elID <0.1, inv mass constraint $0.92 < m_{\eta\pi^+\pi^-}< 0.98$ corresponding to 3$\sigma$ wide window.

* Neutrinos presence on tag side by asking $p_miss >0.4$ and within fiducial volume and pointing to the tag side, $cos\theta_{miss-thurst} < -0.55$
* for 1x1 modes: beam bkg and ISr photons suppression with tighter energy photon cuts, $E_{\gamma1} > 0.6 (0.9)$ GeV and $E_{\gamma2} > 0.25 (0.2)$ for $\eta(\pi^0)$ modes
* against radiative continuum limit $N_{\gamma}^{tag(sig)} \le 2(1)$
* against dilepton final states ( Bhabha, $\mu\mu$) limit lepton and track momenta <4.5
* for $\pi^0$ mode only also lower limit $p_{lepton}>1.5$ GeV
* define $m_{tag}$ : invariant mass of tag tack ( with pion mass hypo) + photons in tag side, reuire $m_{tag}<1.777$
* constrain visible energy define as sum of M0 candidate, lepton, tag track with pion hypo, and photons energies:
* constrain opening nalge between lepto and reconstructed M0 momentum
* exploit correlation between the momentum of the tag track and the cosine of the opening angle between the thrust and missing moemntum in CM, $cos\theta_{miss−thrust}$:

* exploit correlation between missing momentum and missing mass square in lab frame, separately for leponic (A) and hadronic (B) tags ( different neutrinos multiplicities):

* 3x1 has less bkg, do not apply the p_tagVs cosTheta_thrust-miss selection
### Signal resolution and bkg extimate
* signal resolutions fitted after all selections with asymmetric Gaussian to model radiative tails
* elliptical regions retaining 90% of selected signal blind in the data

## neutral meson reco at Belle II
* steering files for eta, eta_prime reco: https://stash.desy.de/users/lacaprar/repos/etaprime/browse/steering_files

# Belle II 1x1 approach
* previous study at MPP, Master Thesis https://docs.belle2.org/record/3748/files/Master%20Thesis%20Marton%20Nemeth-Csoka.pdf
* pg.31, 32 Babar and Belle comparison
* 2021 report https://docs.belle2.org/record/2269/files/DPG-Spring-Conference.pdf?
* internal WG report https://indico.belle2.org/event/5815/contributions/29193/attachments/14434/21759/Tau_MuPi0_WorkingGroup.pdf
*