Resonant CP violation in rare tau decay

In this work, we study the lepton number violating tau decays via intermediate on-shell Majorana neutrinos $N_j$ into two scalar mesons and a lepton $\tau^{\pm} \to M_1{}^{\pm} N_j \to M_1^{\pm} M_2^{\pm} \ell^{\mp}$. We calculate the Branching ratios $Br(\tau^{\pm})$ and the CP asymmetry $(\Gamma(\tau^+) - \Gamma(\tau^-))/(\Gamma(\tau^+) + \Gamma(\tau^-))$ for such decays, in a scenario that contain at least two heavy Majorana neutrinos. The results show that the CP asymmetry is small, but becomes comparable with the branching ratio ${\rm Br}(\tau^{\pm})$ when their mass difference is comparable with their decay width, $\Delta M_N \sim \Gamma_N$. On the other hand, we present regions of the heavy-light neutrino mixing elements, in which the $CP$ asymmetry could be explore in future tau factories.


Introduction
During the last decades, neutrino experiments that have shown that neutrinos have nonzero masses [1,2], also suggest that the first three mass eigenstates are very light with masses ∼ 1 eV, and the mixing between flavour and mass eigenstates is characterized by the Pontecorvo-Maki-Nakagawa-Sakata Matrix, U PMNS [3]. Therefore, if these light masses are produced by means of some see-saw mechanism [4,5], the existence of one or more heavier neutrinos is needed. The current experimental uncertainties in the B PMNS matrix elements allow introduce these new heavy neutral leptons called sterile neutrinos (SN) [6][7][8][9][10], however the small values of these uncertainties imply a strongly suppressed interaction between standard model (SM) particles and SN. In addition, due to the fact that neutrinos are massive particles, a fundamental question arises: are neutrinos Dirac or Majorana particles?, If neutrinos are Dirac particles, the reactions in which they participate must preserve the lepton number (∆L = 0). On the contrary, if neutrinos are Majorana particles, they are indistinguishable from their antiparticles, and the lepton number can be violated in two units (∆L = 2). On the other hand, Neutrino oscillations (NOs) experiments have confirmed that θ 13 angle of B PMNS is non zero [11,12], thus, the possibility of CP violation in the light neutrino sector is still open; nevertheless, extra sources of CP violation are needed in order to explain Baryogenesis via Leptogenesis [13]. Recent studies explored the CP violation and the phenomenology of SN neutrinos in the context of rare meson decays [14][15][16][17][18][19][20][21], however, in this work we will focus in the phenomenology of the rare tau decays [22][23][24] in the framework of tau factories, such as Super Charm-Tau Factory (CTF) in the Budker Institute of Nuclear Physics (Novosibirsk, Russia), [25,26] making it possible to extend the SN searches to tau decay processes. In this letter we focus in the rare decays of tau leptons into two scalar mesons and one charged lepton ( = e, µ), via two on-shell intermediate neutrinos N j , and look for the possibility of detection of CP asymmetries in such decays. The relevant processes are the lepton number violating channels τ ± → M ± 1 M ± 2 ∓ where M 1 ,M 2 = π,K and = e,µ. We also show that the branching ratios are very small 1 , but could be appreciable enough and could be measured in future τ factories where huge numbers of taus will be produced [26,27], if the heavylight neutrino mixing elements are sufficiently large but still lower than the present upper bounds. The program of this paper is the following: in Section 2 we present the notation and formalism for the rare tau decay; in Sections 3 we present the relevant expression for the branching ratio calculations; in Sections 4 we present the relevant expression for the CP asymmetries calculations; in Sections 5 we present the results of the relevant parameters for the future searches; finally, in section 6 we present the summary and conclusions.

Process and Formalism
As we stated above, we are interested in studying the ∆L = 2 rare tau decays mediated by two on-shell heavy (0.140 ≤ M N ≤ 1.638 GeV) Majorana neutrinos with the expectation of obtaining CP violating signal in the neutrino sector. The relevant Feynman diagrams of the studied processes are presented in Fig. 1 and Fig. 2   In order to write down the amplitude and all the relevant quantities, we first define the neutrino flavor state as: where B N j are the elements of the P M N S matrix 2 (heavy-light neutrino mixings elements) which are define as follow the left side of Eq. (2.1) stand for light neutrino sector and the right side for the heavy neutrino sector. The amplitude for a general process involving n sterile neutrinos is 3 where f 1 and f 2 are the meson decay constants of M ± 1 and M ± 2 , and V M 1 , V M 2 are the mixings elements of CKM matrix corresponding to mesons M 1 and M 2 , respectively. The factors / L D ± and / L C ± contain the information related to the kinematics and are given by and finally the factors P j (D) and P j (C) are the heavy Majorana neutrino propagators

6)
2 Experimental limits for |B N j | 2 in our mass range of interest are presented in figure Fig. 7. 3 The definitions M D ± and M C ± can be understood as the amplitude for the direct channel and for the crossed one, respectively. Furthermore, the squared amplitude probability for the process will be here Γ N j is the total decay width of the intermediate neutrinos, and can be approximated as follow the factors N j being effective mixing coefficients and are presented in Fig. 3 for our mass range of interest.  The decay with of the process is given as follow is the symmetry factor that counts for identical particles in the final states, d 3 denotes the number of states available per unit of energy in the 3-body final state 4 .
here, p 1 and p 2 denote the momenta of M 1 and M 2 respectively, and p the momentum of the charged lepton (see Fig. 1 and Fig. 2).

Branching ratio of
In a scenario with n = 2 sterile neutrinos, the decay widths presented in Eq.(2.9) can be written as the double sum of the contributions of N i and N j (i, j = 1, 2), with the mixing elements factored out here Γ's are the canonical decay widths (without heavy-light explicit mixing), and k (±) j are parameters which contain the corresponding mixing factors and are presented in Eq. (3.2).
Due to the fact that |/ L D we can omit the subscripts ± in the contribution terms Γ τ (DD * ) ij and Γ τ (CC * ) ij in Eq. (3.1). The canonical decay widths Γ τ ± (XY * ) ij , where X, Y stand for direct and crossed channel (X, Y = C, D) and (i, j = 1, 2), are given by From now on, we will pay our attention in a scenario where both mesons are equal, then ud when the mesons are pions and us when they are kaons. The canonical decay width has been evaluated numerically by means of Monte-Carlo integrations using Vegas algorithm [28] 5 . Furthermore, the evaluation were implemented using small Γ N j = 10 −3 in the heavy neutrino propagators. The numerical results can be summarized as follows: i) The contribution of (DD * ) jj and (CC * ) jj channels are approximately equal, thus ii) The contribution of (DC * ) ij and (CD * ) ij channels are approximately equal, thus ii are suppressed by a factor ∼ 10 −3 , besides taking into account the latter point iii), the terms Γ τ ± (DC * ) jj and Γ τ ± (DC * ) ij are negligible in all cases, in comparison with Γ τ (DD * ) jj and Γ τ (CC * ) jj . 5 The integration were performed in two different languages Pyhton and Fortran in order to reduce the uncertainties. 6 It is important to note that the dependence Γτ (DD * )jj ∝ 1/ΓN j is in agreement with the fact that sterile neutrino are weakly interacting particles and therefore the narrow width approximation v) The contribution of (DD * ) ij and (CC * ) ij channels are approximately equal, and can reach the same order of magnitude than the (DD * ) jj and (CC * ) jj contributions 7 .
Thus, under the above considerations and taking into account that M 1 = M 2 = M π , M K , we rewrite the Eq. (3.1) only in terms of the dominant contributions, as follows measures the effect of N 1 − N 2 overlap 8 , the factor η(y) y will be discussed later, however, their values are presented in Fig. 4 and The diagonal canonical decay widths, presented in Eq. (3.5b), can be implemented by means of the narrow width approximation where the functions Z(a, b, c, d) and λ(x, y, z) are kinematical functions, which are defined in Appendix B. The branching ratio for the process where Γ(τ ± → all) is the total decay width for τ ± lepton and is given by In order to have a more realistic discussion, we must consider the acceptance factor, which is defined as the probability of the neutrino N j decay inside of a detector of length L where γ N j is the Lorentz time dilation factor in the Laboratory frame and β is the neutrino speed 9 . Therefore, the effective branching ratio 10 is .
(3.10) 7 The effect of this kind of interference will be studied later in detail. 8 stand for the real part. 9 In this work, we will provide γN j ∼ 2, β ∼ 1 and L = 1 mts. 10 The Br eff (τ ± ) correspond to the real branching ratio, while Γ eff (τ ± ) correspond to the effective decay with, whose can be measured in an experiment.

CP Asymmetry of τ
In this section we will calculate the size of CP asymmetry A CP , which is defined as follows The CP violation comes from the complex phases in the transition amplitudes Eq. (2.3a), and the observable effects only arise due to interference of at least two amplitudes. The CP-odd phases are those that come from the Lagrangian of the theory, in other words from the heavy-light mixing elements (B N ); these phases change sign between a process and its conjugate. On the other hand, the CP-even phases appear as absorptive parts in the propagators Eq. (2.6) and do not change sign for the conjugate process. In order to have a more phenomenological discussion about CP violation, it is useful define a new quantity A CP Br eff (τ + ) which is the corresponding branching ratio for the CP-violating asymmetry 11 The CP-violating difference Γ(τ + )−Γ(τ − ) is proportional to the imaginary part of Γ τ (DD * ) 12 and can be written as 12 where we have neglected all the (DC * ) and (CD * ) interference contributions , due to fact that numerical simulation shows that they are strongly suppressed in comparison with (DD * ) and (CC * ). The imaginary part of Eq. (4.3) correspond to the imaginary part of the off-diagonal elements in Eq. (3.5) The imaginary part of the product of propagators (see Eq. (A.7b) in Appendix. A) can be expressed using the narrow width approximation as the validity of Eq. (4.5b) strongly depends on the assumption
In Eq. (4.6) the subscripts N W A and N U M stand for "Narrow Width Approximation" and "Numerical", respectively. The values of η(y) were evaluated numerically using finite ∆M N and their values are presented in Fig. 4 as a function of y ≡ ∆M N /Γ N . The general expression of Eq. (4.4) including the η(y) parameter and under the assumptions (4.7) finally, the CP-violating difference becomes  From Eq. (4.1), Eq. (4.8) and Fig. 4 we can conclude that the best scenario for simultaneous maximization of ACP and Br(τ ), occurs when y = 1. From now on, we will focus in a scenario where heavy neutrinos are almost degenerate ∆M N ∼ Γ N ; within this context we have assumed |B N 1 | ≈ |B N 2 | ≡ |B N |, where = e, µ, τ and the mixing elements are consequently There is just one caveat in the expressions above: we have disregarded the effect of N 1 − N 2 oscillation, these type of oscillations have been studied in detail in Ref. [19] and it is straightforward to show that the L dependent effective differential decay width is 14 where Γ(τ + → π + N ) and Γ(N → π + µ − ) are kinematical functions presented in appendix A. In Eq. (4.11) it is also possible to notice that the oscillation length is L osc = 2πβ N γ N ∆M N . Then, the argument of cosine in Eq. 4.11 can be written as 2π L Losc + θ 12 , therefore, in order to integrate out there are two possible scenarios: 1. L L osc : In this regime we recover the main contributions of the L-independent effective decay width (Eq. (3.10)), because the oscillation term ∼ cos f (L) + θ 12 gives a relatively negligible contribution when integrated over several L osc .
2. L L osc : In this scenario the integration of expression 4.11 is 12) in 4.12 we can see, immediately, that when L osc L and L osc = L the oscillation effect disappear and we recover the L-independent main contributions of the Eq. (3.10). On the other hand, when L ∼ L osc neutrinos have traveled enough to have a welldefined oscillation, which means that neutrinos have not decayed yet (i.e. P N 1). Moreover, L ∼ L osc means y ≡ ∆M N Γ N ≈ 2π P N 1 and then from Fig. 4 we notice that y 1 destroy the effect of resonant CP violation. Therefore, the fact that disregard the N 1 − N 2 oscillation when we have chosen η(y) ∼ 1 is valid.
It is important to note that the oscillation effect is present when L ∼ L osc , therefore, in general CP violating scenarios (i.e. when we are off CP resonant region) this must be taken into account.

Results
In this section the main results obtained in this work will be applied in order to provide a clue for future searches in tau factories. The result for the effective branching ratios presented in Eq. (3.10) are shown in Fig. 5 and Fig. 6 Figure 5. Effective branching ratios per unit of |B eN | 2 |B τ N | 2 . Here we use the following input parameters: cos θ 12 = 1/ √ 2, overlap factor δ 12 = 0.5, detector length L = 1 mts, neutrino speed β = 1 and Lorentz factor γ N = 2. Figure 6. Effective branching ratios per unit of |B µN | 2 |B τ N | 2 . Here we use the following input parameters: cos θ 12 = 1/ √ 2, overlap factor δ 12 = 0.5, detector length L = 1 mts, neutrino speed β = 1 and Lorentz factor γ N = 2.
The difference between the cases with M M = π and M M = K in the final states is mainly due to the elements of CKM matrix, whereas for pions V π ≈ 0.97 and V K ≈ 0.22, respectively. Moreover, the values of meson decay constant are f π ≈ 0.13 GeV and f K ≈ 0.15 GeV, therefore K 2 π /K 2 K ≈ 2 × 10 2 . In order to estimate the region of heavy-light mixings elements |B N | 2 |B τ N | 2 which can be explored in future experiment 15 we define the following relation , (5.1) here N τ is the number of τ lepton produced in an experiment and S(M N ) is given by The actual experimental limits for heavy-light mixing elements are given in Ref. [29], and we have summarized them in Fig. 7(a) for the range of mass of interest. On the other hand, and due to the fact that our results depend on |B τ N | 2 |B N | 2 , we present in Fig. 7(b) the product of the experimental limits of interest. The CTF in Novosibirsk, Russia is expected to collect 10 10 pairs of τ ± leptons after few years of operation [26], therefore under the latter considerations we can estimate the mixing region that can be explored in such experiment, this region is presented in Fig. 8. It is important to point out that due to the CKM elements suppresion only channels with pions in the final state offer real possibilities to constrain the heavy-light mixings parameters.

Summary and Conclusions
In this letter we studied the (∆L = 2) rare tau decays τ ± → M ± 1 M ± 2 ∓ , where M 1 and M 2 are pseudo scalar mesons (M 1 , M 2 = π, K) and the charged lepton can be = e, µ, also we studied the possibility of CP violation detection in future tau factories. We have assumed that the decays occur via the exchange of two on-shell sterile neutrinos N j at tree level, and we have shown that the amplitude of these processes is suppressed by the mixing elements of the PMNS matrix |B τ N | 2 |B N | 2 . The aforementioned CP violation effects come from the interference between the N 1 and N 2 propagators and the complex phases (CP-odd phases φ N j , see Eq. (2.2)) in the PMNS mixing matrix. Our results shows that these signals of CP violation could be detected in future tau factories for τ ± → π ± π ± ∓ tau decays, where = e, µ if there exist, at least, two sterile neutrinos in the on-shell mass range, their masses are almost degenerate ∆M N ∼ Γ N , the CP odd phases sin θ 12 1 and the mixing parameters are in the allowed region of Fig. 7. In such a case, the CP-violating difference Γ(τ + ) − Γ(τ − ) becomes large and comparable with Γ(τ + ) + Γ(τ − ) and the corresponding CP asymmetry A CP becomes A CP ∼ 1. In addition, there exist several models with quasidegeneracy ∆M N ∼ Γ N , between them it is worth to mention the well-know νMSM model [30,31], where the quasi-degeneracy of the two heavy neutrinos (with mass M N j ∼ 1 GeV) is fundamental in order to get a successful dark matter candidate. However, our results can be framed in the context of the νMSM model or more general models [21,32] with at least two quasi-degenerate neutrinos.

Acknowledgments
This work was supported by Fellowship Grant Becas Chile No. 74160012, CONICYT (J.Z.S). Also, the author want to thank for valuable discussions with David Alvarez-Castillo.