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

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.


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 (ν) are very light massive particles M ν ∼ 1 eV [1,2] and, consequently, that the Standard Model must be extended.
Among the well-know SM extensions based on SSM, we can mention the Neutrino-Minimal-Standard-Model, νMSM [37,38], which introduces two almost degenerate HN with masses M N 1 ≈ M N 2 ∼ 1GeV, leading to a successful BAU and a third HN with mass M N 3 ∼keV to be a natural candidate for DM.
Recently, NOs experiments have shown that the mixing-angle θ 13 is non zero [39] and also suggest the possibility of CP violation in the light neutrino sector [40]. However, extra sources of CP violation are needed in order to explain the BAU via leptogenesis (see [41] 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 [42].
In a previous article [34] 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 (τ ± → π ± π ± µ ∓ ) via two almost degenerate heavy on-shell Majorana neutrinos (M N i ∼ 1GeV), 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 [34], 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 = µ. The heavy neutrinos N 1 and N 2 studied in this letter are almost degenerate (M N 1 ≈ M N 2 ) and the mass difference 1 FIG. 1. Heavy neutrino production in tau lepton decay. Left Panel: Feynman diagrams for the LNV process τ + → π + π + − . Right Panel: Feynman diagrams for the LNV process τ − → π − π − + The relevant expressions for the aforementioned processes were presented in [29,34] as a function of the distance between production and detection vertices, called L. Therefore, the L dependent effective differential decay width is given by Here, |B N i | are the mixing coeficients ( = µ, τ and i = 1, 2); the angle θ LV stands for the CP-violating phase; the factors γ N β N are the Lorentz factors and the heavy neutrino velocity 2 , respectively (see Appendix I for more details). The factors Γ(τ + → π + N ) and Γ(N → π + µ − ) are the canonical partial decay widths (without mixing factors), which can 1 The neutrino (N i ) total decay width is expressed as Γ N i , the factor Γ N stand for Γ N = (Γ N 1 + Γ N 2 )/2 and Y represent a parameter which allows us to express the mass difference in terms of Γ N . 2 Ref. [34] considers γ N β N of the produced N j 's (in the laboratory frame) as fixed parameters γ N β N = 2.
However, the product γ N β N is in general not fixed, because τ is moving in the lab frame when it decays into N and π. , where G F = 1.166 × 10 −5 GeV −2 is the Fermi coupling constant, |V ud | = 0.974 is a CKM matrix element and f π ≈ 130.4 M eV the pion decay constant. The total heavy neutrino decay width Γ Ma (M N i ) is given by where K Ma i accounts for the mixings elements and reads as Here, N Ma i are the effective mixing coefficients, which account for all possible decay channels of N i (see Refs. [43,44]) and are presented in Fig. 2. We note that for our mass range of interest N Ma i ∼ 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 N i | 2 1. It is important to note that the mixings |B N 1 | 2 and |B N 2 | 2 can be different for the two heavy neutrinos, and consequently, the factors K Ma i (i = 1, 2) might be dissimilar from each other. However, in this letter we will assume that |B N 1 | 2 = |B N 2 | 2 and thus K Ma In addition, we focus on the scenario in which |B τ N | mixing parameter is much larger . We have chosen this scenario since methods for constraining |B τ N | are lacking, so that it is much less constrained than |B µN | and |B eN |, particularly in our mass range of interest M N < m τ (see Refs. [43,45,46] and references therein). Furthermore, according to Fig. 2 we will assume that N Ma τ i ≈ 2.5 and N Ma ei ≈ N Ma µi ≈ 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 We note that CP violating phase (θ LV ) can be extracted by means of the difference between the L-dependent effective differential decay width for τ + and τ − 3 According |B τ Ni | 2 |B µNi | 2 ∼ |B eNi | 2 and N Ma i ∼ 1 the factor K is approximated as We have simulated the τ ± production via the e + e − → τ + τ − process and its subsequent bution, here qq stand for any light quark (q = u, d, s andq =ū,d,s). We have carried out the simulation using MadGraph5 aMC@NLO [47] for τ + and τ − individually, assuming Belle II kinematical parameters 4 . The τ + and τ − do not show significant differences in their γ N β N distributions, which are presented in the left panel of Fig. 4. The Universal FeynRules Output (UFO) [48] files were generated by means of FeynRules libraries [49].
It is important to point out that for our mass range of interest most of the heavy neutrinos The factor γ N β N is model by sampling a distribution obtained from the simulation (ex- In this work we have studied the modulation dΓ/dL for the LNV process τ ± → π ± π ± ∓ 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 [34] 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 allowed 5 (HNO case) and may be relevant for explanations of BAU via leptogenesis [41,42,50,51].
We note that the simulation of the production of on-shell heavy neutrinos, N , gave the same distribution of γ N β N for both τ + and τ − , and when it is considered, the modulation dΓ/dL is smeared due to the fact that we have a distribution of small values of γ N β N Fig. 4 instead of a fixed (average) value (cf. Fig. 5). In addition in studying the modulation for fixed and variable γ N β N (Fig. 5) we can observe deviation of dΓ/dL when HNOs are considered (blue and red lines) and when they are neglected (green lines).
We have studied the modulation dΓ/dL for τ ± decays and for different values of the parameters M N , Y , and the CP violating phase θ 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 [52]. We find the modulation shape strongly dependent on the CP-violation phase θ LV , which supports the possibility of obtaining the value of θ LV from measurements of dΓ/dL.  Here we used |B τ N | 2 = 5 · 10 −4 and |B µN | 2 = 5 · 10 −8 .
In Fig. 7 (left panel) when M N = 1.0 GeV and θ LV = π/2, we observed that the number of expected τ + decays in the region 0 ≤ L ≤ 600 mm are larger for Y = 5 than Y = 10.
In Fig. 8 (left panel) when M N = 0.5 GeV and θ LV = π/2, we observed that the expected number of τ + decays inside the whole region 0 ≤ L ≤ 1000 mm are larger for Y = 5 than Y = 10. The same can be observed when M N = 0.5 GeV and θ LV = π/4 (Fig. 8, right   panel). For the τ − decays, the situation is different: for M N = 0.5 GeV and θ LV = π/2, the number of expected τ − decays inside the whole region 0 ≤ L ≤ 1000 mm is larger for Y = 10 than Y = 5. On the other hand, when M N = 0.5 GeV and θ LV = π/4, we observed from Fig. 8 (right panel), that the difference between Y = 5 and Y = 10 is negligible for τ − decays in the full range of 0 ≤ L ≤ 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 θ LV = π/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 [52]. We notice for the case of guishing them in experiment. However, in the case of 500 simulated events, we have enough statistical significance to separe the τ + and τ − modulation, in the range 50 ≤ L ≤ 500 mm. Here we used |B τ N | 2 = 10 −2 and |B µN | 2 = 10 −7 . In summary, in this work we have considered the heavy neutrino oscillations of τ ± decays in a scenario which contains two heavy, almost-degenerate neutrinos (N j ), with masses in the range 0.5 GeV ≤ M N ≤ 1.5 GeV. We have explored the feasibility to measure CPviolating HNOs processes in such a scenario where the modulation of dΓ/dL for the process τ ± → π ± N → π ± π ± µ ∓ at Belle II can be resolved inside the detector. We have established some realistic conditions for |B τ N | 2 , |B µN | 2 and Y (≡ ∆M N /Γ N ) where the aforementioned effect can be observed. APPENDIX I In Ref. [34] we have considered the parameters β N (HN velocity) and γ N (HN Lorentz factor) of the produced heavy neutrinos N in the laboratory frame (Σ) as fixed values (β N γ N = 2). However, the factor β N γ N is in general not fixed, due to the τ lepton is moving in the lab frame when it decays into N 's and π's. The factor β N γ N can be written as follow where the energy of the HN in the lab frame, E N , depends on its directionp N in the τ -rest frame (Σ ). The HN energy E N can be written in terms of the angle θ N and momentump N as follow where the corresponding quantities in the τ -rest frame (Σ ) are fixed β τ is the velocity of τ in the lab frame, and λ is given by Therefore, the Eq. 1 can be written as where now the oscillation length L osc , appearing in the last term, also depends on the It is important to remarks that in a real experiment the produced τ 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.