Exploring CP-Violating heavy neutrino oscillations in rare tau decays at Belle II

TL;DR

This paper investigates the potential to observe CP-violating heavy neutrino oscillations in rare tau decays at Belle II, focusing on lepton number violation and decay width modulation within a specific neutrino mass range.

Contribution

It introduces a detailed analysis of heavy neutrino oscillations and their CP violation effects in tau decays at Belle II, considering realistic experimental conditions.

Findings

Decay width modulation could be measurable at Belle II.

Heavy neutrino oscillations may reveal CP violation.

Feasibility depends on neutrino mass and detector sensitivity.

Abstract

In this work, we study the lepton number violating tau decays via two intermediate on-shell Majorana neutrinos $N_j$ into two charged pions, and a charged lepton $\tau^{\pm} \to \pi^{\pm} N_j \to \pi^{\pm} \pi^{\pm} \ell^{\mp}$. We consider the scenario where the heavy neutrino masses are within $0.5$ GeV $\leq M_N \leq 1.5$ GeV. We evaluated the possibility to measure the modulation of the decay width along the detector length for these processes at taus factories, such as Belle II. We study some realistic conditions which could lead to the observation of this phenomenon at futures $\tau$'s factories.

Equations

$$\frac{d}{dL}$$

$$\times \left\{\sum _{j=1}^2|B_{\mu N_j}{|}^2|B_{\tau N_j}{|}^2+2|B_{\mu N_1}||B_{\tau N_1}||B_{\mu N_2}||B_{\tau N_2}|\cos \left(L\frac{\Delta M_N}{{\gamma }_N{\beta }_N}+{\theta }_{LV}\right)\right\}.$$

$$\overline{\Gamma }({\tau }^{\pm }\to {\pi }^{\pm }N)$$

$$\overline{\Gamma }(N\to {\pi }^{\pm }{\ell }^{\mp })$$

$${\Gamma }_{\mathrm{Ma}}(M_{N_i})\equiv {\Gamma }_{N_i}\approx {\mathcal{K}}_i^{\mathrm{Ma}}\frac{G_F^2{M}_{N_i}^5}{96{\pi }^3}$$

$${\mathcal{K}}_i^{\mathrm{Ma}}={\mathcal{N}}_{ei}^{\mathrm{Ma}}|B_{e{N}_i}{|}^2+{\mathcal{N}}_{\mu i}^{\mathrm{Ma}}|B_{\mu N_i}{|}^2+{\mathcal{N}}_{\tau i}^{\mathrm{Ma}}|B_{\tau N_i}{|}^2.$$

$${\Gamma }_{\mathrm{Ma}}(M_{N_i})\equiv {\Gamma }_N(M_N)\approx 2.5|B_{\tau N}{|}^2\frac{G_F^2{M}_N^5}{96{\pi }^3}.$$

$$\frac{d}{dL}\Gamma ({\tau }^{+})-\frac{d}{dL}\Gamma ({\tau }^{-})=$$

$$\displaystyle\times 4\ |B_{\mu N_{1}}||B_{\tau N_{1}}||B_{\mu N_{2}}||B_{\tau N_{2}}|\sin\Big{(}L\;\frac{\Delta M_{N}}{\gamma_{N}\beta_{N}}\Big{)}\sin\Big{(}\theta_{LV}\Big{)}\ .$$

$${\beta }_N{\gamma }_N=\sqrt{(E_N({\hat{p}}_N^{\prime })/M_N{)}^2-1},$$

$$E_N={\gamma }_{\tau }(E_N^{\prime }+\cos {\theta }_N{\beta }_{\tau }|{\vec{p}}_N^{\prime }|),$$

$$E_N^{\prime }=\frac{M_{\tau }^2+M_N^2-M_{\pi }^2}{2M_{\tau }},|{\vec{p}}_N^{\prime }|=\frac{1}{2}{M}_{\tau }{\lambda }^{1/2}\left(1,\frac{M_{\pi }^2}{M_{\tau }^2},\frac{M_N^2}{M_{\tau }^2}\right),$$

$$\lambda (a,b,c)=a^2+b^2+c^2-2ab-2ac-2cb.$$

$$\frac{d}{dL}$$

$$\displaystyle\times\Bigg{(}\sum_{i=1}^{2}|B_{\mu N_{i}}|^{2}|B_{\tau N_{i}}|^{2}+2|B_{\mu N_{1}}||B_{\tau N_{1}}||B_{\mu N_{2}}||B_{\tau N_{2}}|\cos\Big{(}2\pi\;\frac{L}{L_{\rm osc}({\hat{p}}^{\prime}_{N})}+\theta_{LV}\Big{)}\Bigg{)}\ ,$$

$$L_{\mathrm{osc}}({\hat{p}}_N^{\prime })=\frac{2\pi {\beta }_N{\gamma }_N}{M_N}=\frac{2\pi }{M_N^2}|{\vec{p}}_N({\hat{p}}_N^{\prime })|=\frac{2\pi }{M_N}{\left[(E_N({\hat{p}}_N^{\prime })/M_N{)}^2-1\right]}^{1/2}.$$

Full text

$\tau^{\pm}\to\pi^{\pm}N_{j}\to\pi^{\pm}\pi^{\pm}\ell^{\mp}$. We consider the scenario where the heavy neutrino masses are within $0.5$ GeV $\leq M_{N}\leq 1.5$ GeV. We evaluated the possibility to measure the modulation of the decay width along the detector length for these processes at tau factories, such as Belle II. We study some realistic conditions which could lead to the observation of this phenomenon at futures $\tau$ factories.

Heavy Neutrinos, Lepton Number Violation, Tau Factory, Heavy Neutrino Oscillations, Belle II.

I Introduction

The first indications of physics beyond the standard model (SM) come from neutrino oscillations (NOs), baryonic asymmetry of the universe (BAU) and dark matter (DM). In the recent years NOs experiments have confirmed that active neutrinos ($\nu$) are very light massive particles $M_{\nu}\sim 1$ eV Fukuda et al. (1998); Eguchi et al. (2003) and, consequently, that the Standard Model must be extended. One of the most popular SM extensions, which explains very small neutrino masses, among others unknowns, is the See-Saw Mechanism (SSM) Mohapatra et al. (2007); Mohapatra and Smirnov (2006). The SSM introduces a new Majorana particle (SM-singlet) called heavy neutrinos (HN), which induces a dimension-5 operator Weinberg (1979) and leads to a very light active Majorana neutrino. Due to the fact that heavy neutrinos are singlet under the $SU(2)_{L}$ symmetry group, their interaction with gauge bosons ($Z,W^{\pm}$) and other leptons ($e,\mu,\tau$) must be highly suppressed. Despite this suppression, they can be searched via colliders Das et al. (2019a); Das and Okada (2017, 2013); Antusch et al. (2019); Das et al. (2019b, 2017); Chakraborty et al. (2018); Cvetic and Kim (2019); Antusch et al. (2017); Cottin et al. (2018); Duarte et al. (2019); Drewes and Hajer (2019); Bhupal Dev et al. (2019); Cvetič et al. (2019a, b); Das (2018); Das et al. (2016, 2018), rare meson decays Dib et al. (2000); Cvetic et al. (2012, 2014a, 2014b, 2015a, 2015b); Dib et al. (2015); Moreno and Zamora-Saa (2016); Milanes and Quintero (2018); Mejia-Guisao et al. (2018) and tau factories Zamora-Saa (2017); Kim et al. (2017); Dib et al. (2019). Among the well-know SM extensions based on SSM, we can mention the Neutrino-Minimal-Standard-Model, $\nu$MSM Asaka et al. (2005); Asaka and Shaposhnikov (2005), which introduces two almost degenerate HN with masses $M_{N1}\approx M_{N2}\sim 1$GeV, leading to a successful BAU and a third HN with mass $M_{N3}\sim$keV to be a natural candidate for DM.

Recently, NOs experiments have shown that the mixing-angle $\theta_{13}$ is non zero An et al. (2012) and also suggest the possibility of $CP$ violation in the light neutrino sector Abe et al. (2018). However, extra sources of $CP$ violation are needed in order to explain the BAU via leptogenesis (see Chun et al. (2018) for a review). In addition, when heavy neutrino masses are below the electroweak scale ($M_{N}<246$ GeV), the BAU is generated via CP-violating heavy neutrino oscillations (HNOs) during their production Drewes et al. (2016).

In a previous article Zamora-Saa (2017) we have studied the resonant CP-violation and described the effects of HNOs on it. The study was carried out in the context of lepton number violating (LNV) tau lepton decay ($\tau^{\pm}\to\pi^{\pm}\pi^{\pm}\mu^{\mp}$) via two almost degenerate heavy on-shell Majorana neutrinos ($M_{N_{i}}\sim 1$GeV), which can oscillate among themselves. The purpose of this letter is to explore more realistic experimental conditions in order to observe such HNOs, extending the analysis beyond the resonant CP-violating scenario.

The work is arranged as follows: In Sec. II, we study the production of the heavy neutrinos in tau’s decays. In Sec. III, we present the results of the simulation of the HN production. In Sec. IV, we present the results and shows conclusions.

II Production of the RHN

As established in the previous article Zamora-Saa (2017), we are interested in studying the LNV processes which are represented by the Feynman diagrams shown in Fig. 1, and from this point on, we will focus on the case of $\ell=\mu$. The heavy neutrinos $N_{1}$ and $N_{2}$ studied in this letter are almost degenerate ($M_{N1}\approx M_{N2}$) and the mass difference111The neutrino ($N_{i}$) total decay width is expressed as $\Gamma_{Ni}$, the factor $\Gamma_{N}$ stand for $\Gamma_{N}=(\Gamma_{N1}+\Gamma_{N2})/2$ and $Y$ represent a parameter which allows us to express the mass difference in terms of $\Gamma_{N}$. ($|\Delta M_{N}|=M_{N2}-M_{N1}\equiv Y\Gamma_{N}$) is in the range $5~{}\Gamma_{N}\leq|\Delta M_{N}|\leq 10~{}\Gamma_{N}$.

The relevant expressions for the aforementioned processes were presented in Cvetic et al. (2015b); Zamora-Saa (2017) as a function of the distance between production and detection vertices, called $L$. Therefore, the $L$ dependent effective differential decay width is given by

[TABLE]

Here, $|B_{\ell N_{i}}|$ are the mixing coeficients ($\ell=\mu,\tau$ and $i=1,2$); the angle $\theta_{LV}$ stands for the CP-violating phase; the factors $\gamma_{N}\beta_{N}$ are the Lorentz factors and the heavy neutrino velocity222Ref. Zamora-Saa (2017) considers $\gamma_{N}\beta_{N}$ of the produced $N_{j}$’s (in the laboratory frame) as fixed parameters $\gamma_{N}\beta_{N}=2$. However, the product $\gamma_{N}\beta_{N}$ is in general not fixed, because $\tau$ is moving in the lab frame when it decays into $N$ and $\pi$., respectively (see Appendix I for more details). The factors $\overline{\Gamma}(\tau^{+}\to\pi^{+}N)$ and $\overline{\Gamma}(N\to\pi^{+}\mu^{-})$ are the canonical partial decay widths (without mixing factors), which can be written as

[TABLE]

where $G_{F}=1.166\times 10^{-5}\ GeV^{-2}$ is the Fermi coupling constant, $|V_{ud}|=0.974$ is a CKM matrix element and $f_{\pi}\approx 130.4\ MeV$ the pion decay constant. The total heavy neutrino decay width $\Gamma_{\rm Ma}(M_{N_{i}})$ is given by

[TABLE]

where $\mathcal{K}_{i}^{\rm Ma}$ accounts for the mixings elements and reads as

[TABLE]

Here, ${\cal N}_{\ell i}^{\rm Ma}$ are the effective mixing coefficients, which account for all possible decay channels of $N_{i}$ (see Refs. Atre et al. (2009); Bondarenko et al. (2018)) and are presented in Fig. 2. We note that for our mass range of interest ${\cal N}_{\ell i}^{\rm Ma}\sim 1$. In Eq. 4, the first two terms include both charged current and neutral current decays, whereas the third term arises purely due to neutral current decays. The neutral current decays are calculated in the approximation $|B_{\ell N_{i}}|^{2}\ll 1$.

It is important to note that the mixings $|B_{\ell N_{1}}|^{2}$ and $|B_{\ell N_{2}}|^{2}$ can be different for the two heavy neutrinos, and consequently, the factors $\mathcal{K}^{\rm Ma}_{i}$ $(i=1,2)$ might be dissimilar from each other. However, in this letter we will assume that $|B_{\ell N_{1}}|^{2}=|B_{\ell N_{2}}|^{2}$ and thus $\mathcal{K}^{\rm Ma}_{1}\approx\mathcal{K}^{\rm Ma}_{2}$ $(\equiv\mathcal{K})$. In addition, we focus on the scenario in which $|B_{\tau N}|$ mixing parameter is much larger than the other mixings (i.e. $|B_{\tau N_{i}}|^{2}\gg|B_{\mu N_{i}}|^{2}\sim|B_{eN_{i}}|^{2}$). We have chosen this scenario since methods for constraining $|B_{\tau N}|$ are lacking, so that it is much less constrained than $|B_{\mu N}|$ and $|B_{eN}|$, particularly in our mass range of interest $M_{N}<{m_{\tau}}$ (see Refs. Atre et al. (2009); Abreu et al. (1997); Vaitaitis et al. (1999) and references therein). Furthermore, according to Fig. 2 we will assume that ${\cal N}_{\tau i}^{\rm Ma}\approx 2.5$ and ${\cal N}_{ei}^{\rm Ma}\approx{\cal N}_{\mu i}^{\rm Ma}\approx 7.5$. With these assumptions, we infer from Eq. (4) and Figs. 2, 3 that both heavy neutrinos have approximately the same total decay width. Additionally, since the $\tau$ channel333According $|B_{\tau N_{i}}|^{2}\gg|B_{\mu N_{i}}|^{2}\sim|B_{eN_{i}}|^{2}$ and ${\cal N}_{\ell i}^{\rm Ma}\sim 1$ the factor $\mathcal{K}$ is approximated as $\mathcal{K}={\cal N}_{ei}^{\rm Ma}\;|B_{eN_{i}}|^{2}+{\cal N}_{\mu i}^{\rm Ma}\;|B_{\mu N_{i}}|^{2}+{\cal N}_{\tau i}^{\rm Ma}\;|B_{\tau N_{i}}|^{2}\approx{\cal N}_{\tau i}^{\rm Ma}\;|B_{\tau N_{i}}|^{2}$. dominates $\Gamma_{\rm Ma}(M_{N_{i}})$ , we can write

[TABLE]

We note that CP violating phase ($\theta_{LV}$) can be extracted by means of the difference between the $L$-dependent effective differential decay width for $\tau^{+}$ and $\tau^{-}$

[TABLE]

III Heavy neutrino simulations and results

We have simulated the $\tau^{\pm}$ production via the $e^{+}e^{-}\to\tau^{+}\tau^{-}$ process and its subsequent decay to HN ($e^{+}e^{-}\to\tau^{+}\tau^{-}\to q\bar{q}N$) in order to get a realistic $\gamma_{N}\beta_{N}$($\equiv|{\vec{p}}_{N}|/M_{N}$) distribution, here $q\bar{q}$ stand for any light quark ($q=u,d,s$ and $\bar{q}=\bar{u},\bar{d},\bar{s}$). We have carried out the simulation using MadGraph5_aMC@NLO Alwall et al. (2014) for $\tau^{+}$ and $\tau^{-}$ individually, assuming Belle II kinematical parameters444The beam energies for $e^{+}$ and $e^{-}$ are $4$ GeV and $7$ GeV, respectively.. The $\tau^{+}$ and $\tau^{-}$ do not show significant differences in their $\gamma_{N}\beta_{N}$ distributions, which are presented in the left panel of Fig. 4. The Universal FeynRules Output (UFO) Degrande et al. (2012) files were generated by means of FeynRules libraries Alloul et al. (2014).

It is important to point out that for our mass range of interest most of the heavy neutrinos $(N)$ tend to decay outside of the detector’s radial acceptance $L_{D}\approx 1000$ mm (Fig. 4 right-panel), introducing a strong suppression factor.

The factor $\gamma_{N}\beta_{N}$ is model by sampling a distribution obtained from the simulation (explained above) and used to evaluate $\frac{d\Gamma}{dL}(M_{N},Y,\theta_{LV},L)$ as shown in Eq. II. For each value of $M_{N}$ ,$Y$ ,$\theta_{LV}$ and $L$ Eq. II is sample with $10000$ different values of $\gamma_{N}\beta_{N}$ to calculate the expected value of $\frac{d\Gamma}{dL}(M_{N},Y,\theta_{LV},L)$.

IV Discussion of the results and summary

In this work we have studied the modulation $d\Gamma/dL$ for the LNV process

$\tau^{\pm}\to\pi^{\pm}\pi^{\pm}\ell^{\mp}$ under Belle II conditions, in a scenario which contains two almost degenerate (on-shell) Majorana neutrinos $(N_{j})$. This scenario has been studied in a previous work Zamora-Saa (2017) in which we explored the resonant CP-violation in rare tau decays. In that work, we found that when $Y=1$ the CP-violation is maximized and the heavy neutrino oscillation effects are negligible (NO HNO case). However, small deviations ($Y=5,10$) from $Y=1$ are allowed555In section 4 of the Ref. Zamora-Saa (2017) this is calculated and explained in detail. (HNO case) and may be relevant for explanations of BAU via leptogenesis Strumia (2006); Drewes and Eijima (2016); Drewes et al. (2016); Chun et al. (2018).

We note that the simulation of the production of on-shell heavy neutrinos, $N$, gave the same distribution of $\gamma_{N}\beta_{N}$ for both $\tau^{+}$ and $\tau^{-}$, and when it is considered, the modulation $d\Gamma/dL$ is smeared due to the fact that we have a distribution of small values of $\gamma_{N}\beta_{N}$ Fig. 4 instead of a fixed (average) value (cf. Fig. 5). In addition in studying the modulation for fixed and variable $\gamma_{N}\beta_{N}$ (Fig. 5) we can observe deviation of $d\Gamma/dL$ when HNOs are considered (blue and red lines) and when they are neglected (green lines).

We have studied the modulation $d\Gamma/dL$ for $\tau^{\pm}$ decays and for different values of the parameters $M_{N}$, $Y$, and the CP violating phase $\theta_{LV}$. In addition, we remark that in Figs. 5 - 8 the number of events considered were almost infinite and the vertex resolution considered was $0.03$ mm Abe et al. (2010). We find the modulation shape strongly dependent on the CP-violation phase $\theta_{LV}$, which supports the possibility of obtaining the value of $\theta_{LV}$ from measurements of $d\Gamma/dL$.

In Fig. 6 (left panel), when $M_{N}=1.5$ GeV and $\theta_{LV}=\pi/2$, we observe that the number of expected $\tau^{+}$ decays inside the region $100\leq L\leq 650$ mm is larger for $Y=10$ than $Y=5$; with difference in the remaining regions being negligible. On the other hand, when $M_{N}=1.5$ GeV and $\theta_{LV}=\pi/4$, we observe from Fig. 6 (right panel) that for $\tau^{+}$ decays inside the region $50\leq L\leq 200$ mm the number of expected events is larger for $Y=5$ than $Y=10$. Conversely, inside the region $200\leq L\leq 800$ mm, $Y=10$ dominates over $Y=5$, while for $L\geq 800$ mm the differences are negligible. For the $\tau^{-}$ decays, the situation changes drastically: for $M_{N}=1.5$ GeV and $\theta_{LV}=\pi/2$, the number of expected $\tau^{-}$ decays inside the region $50\leq L\leq 100$ mm is larger for $Y=10$ than $Y=5$. Conversely inside $100\leq L\leq 650$ mm $Y=5$ dominates over $Y=10$. When $M_{N}=1.5$ GeV and $\theta_{LV}=\pi/4$ inside the region $50\leq L\leq 500$ mm, $Y=5$ dominates over $Y=10$, and for $500\leq L\leq 1000$ mm, the opposite is true.

In Fig. 7 (left panel) when $M_{N}=1.0$ GeV and $\theta_{LV}=\pi/2$, we observed that the number of expected $\tau^{+}$ decays in the region $0\leq L\leq 600$ mm are larger for $Y=5$ than $Y=10$. Conversely, from $600\leq L\leq 1000$ mm, $Y=10$ dominates over $Y=5$. Futhermore, when $M_{N}=1.0$ GeV and $\theta_{LV}=\pi/4$, we observed from Fig. 7 (right panel) that the number of expected $\tau^{+}$ decays in the entire region $0\leq L\leq 1000$ mm is larger for $Y=5$ than $Y=10$. For the $\tau^{-}$ decays, the situation is different: for $M_{N}=1.0$ GeV and $\theta_{LV}=\pi/2$, the number of expected $\tau^{-}$ decays inside the region $0\leq L\leq 600$ mm is larger for $Y=10$ than $Y=5$. Conversely, from $600\leq L\leq 1000$ mm, $Y=5$ dominates over $Y=10$. When $M_{N}=1.0$ GeV and $\theta_{LV}=\pi/4$, in the region $200\leq L\leq 1000$ mm $Y=5$ dominates over $Y=10$. In other regions the differences are negligible.

In Fig. 8 (left panel) when $M_{N}=0.5$ GeV and $\theta_{LV}=\pi/2$, we observed that the expected number of $\tau^{+}$ decays inside the whole region $0\leq L\leq 1000$ mm are larger for $Y=5$ than $Y=10$. The same can be observed when $M_{N}=0.5$ GeV and $\theta_{LV}=\pi/4$ (Fig. 8, right panel). For the $\tau^{-}$ decays, the situation is different: for $M_{N}=0.5$ GeV and $\theta_{LV}=\pi/2$, the number of expected $\tau^{-}$ decays inside the whole region $0\leq L\leq 1000$ mm is larger for $Y=10$ than $Y=5$. On the other hand, when $M_{N}=0.5$ GeV and $\theta_{LV}=\pi/4$, we observed from Fig. 8 (right panel), that the difference between $Y=5$ and $Y=10$ is negligible for $\tau^{-}$ decays in the full range of $0\leq L\leq 1000$ mm.

In Fig. 9, we present results for a finite number of detected events including the statistical uncertainties, when $M_{N}=1.5$ GeV; $Y=5$ and $\theta_{LV}=\pi/2$. In the left panel, we present results for 100 simulated events and in the right panel, for 500 simulated events. Furthermore, the considered vertex-position resolution was $0.03$ mm Abe et al. (2010). We notice for the case of 100 simulated events, the two distributions are quite similar. Which may preclude distinguishing them in experiment. However, in the case of 500 simulated events, we have enough statistical significance to separe the $\tau^{+}$ and $\tau^{-}$ modulation, in the range $50\leq L\leq 500$ mm.

In summary, in this work we have considered the heavy neutrino oscillations of $\tau^{\pm}$ decays in a scenario which contains two heavy, almost-degenerate neutrinos $(N_{j})$, with masses in the range $0.5$ GeV $\leq M_{N}\leq 1.5$ GeV. We have explored the feasibility to measure CP-violating HNOs processes in such a scenario where the modulation of $d\Gamma/dL$ for the process $\tau^{\pm}\to\pi^{\pm}N\to\pi^{\pm}\pi^{\pm}\mu^{\mp}$ at Belle II can be resolved inside the detector. We have established some realistic conditions for $|B_{\tau N}|^{2}$, $|B_{\mu N}|^{2}$ and $Y(\equiv\Delta M_{N}/\Gamma_{N})$ where the aforementioned effect can be observed.

V Acknowledgments

This work is supported in part by FONDECYT Grant No. 3180032 (J.Z.S.). The work of S.T.A. is supported by the National Science Foundation (NSF) grant 1812377.

Appendix I

In Ref. Zamora-Saa (2017) we have considered the parameters $\beta_{N}$ (HN velocity) and $\gamma_{N}$ (HN Lorentz factor) of the produced heavy neutrinos N in the laboratory frame ($\Sigma$) as fixed values ($\beta_{N}\gamma_{N}=2$). However, the factor $\beta_{N}\gamma_{N}$ is in general not fixed, due to the $\tau$ lepton is moving in the lab frame when it decays into $N$’s and $\pi$’s. The factor $\beta_{N}\gamma_{N}$ can be written as follow

[TABLE]

where the energy of the HN in the lab frame, $E_{N}$, depends on its direction ${\hat{p}}^{\prime}_{N}$ in the $\tau$-rest frame ($\Sigma^{\prime}$).

The HN energy $E_{N}$ can be written in terms of the angle $\theta_{N}$ and momentum ${\hat{p}}^{\prime}_{N}$ as follow

[TABLE]

where the corresponding quantities in the $\tau$-rest frame ($\Sigma^{\prime}$) are fixed

[TABLE]

$\beta_{\tau}$ is the velocity of $\tau$ in the lab frame, and $\lambda$ is given by

[TABLE]

Therefore, the Eq. II can be written as

[TABLE]

where now the oscillation length $L_{\rm osc}$, appearing in the last term, also depends on the direction ${\hat{p}}^{\prime}_{N}$

[TABLE]

It is important to remarks that in a real experiment the produced $\tau$ leptons can have a wide range of momenta, these momenta can be well described by a distribution, which was simulated and obtained in the present work.