Discovering heavy neutrino oscillations in rare Bc meson decays at HL-LHCb

In this work, we study the lepton flavor and lepton number violating $B_{c}$ meson decays via two intermediate on-shell Majorana neutrinos $N_j$ into two charged leptons and a charged pion $B_{c}^{\pm} \to \mu^{\pm} \ N_j \to \mu^{\pm} \tau^{\pm} \pi^{\mp}$. We evaluated the possibility to measure the modulation of the decay width along the detector length produced as a consequence of the lepton flavor violating process, in a scenario where the heavy neutrinos masses range between $2.0$ GeV $\leq M_N \leq 6.0$ GeV. We study some realistic conditions which could lead to the observation of this phenomenon at futures $B$ factories such HL-LHCb.

One of the most promising SM extensions based on SSM is the Neutrino-Minimal-Standard-Model (νMSM) [44,45], which introduces two almost degenerate HN's with masses M N 1 ≈ M N 2 ∼ 1GeV, and a third HN with mass M N 3 ∼keV which is a natural candidate for DM. Apart to explain the small active neutrino masses, the νMSM allows to explain sucesfully the BAU by means of leptogenesis from HNs oscillations, also known as Akhmedov-Rubakov-Smirnov (ARS) mechanism [46].
In a previous article [15], we have described the effects of Heavy Neutrino Oscillations (HNOs) in the so-called rare Lepton Number Violating (LNV) and Lepton Flavor Violating (LFV) pseudoscalar B meson decays, via two almost degenerate heavy on-shell Majorana neutrinos (M N i ∼ 1GeV), which can oscillate among themselves. The aim of this article is to develop a more realistic analysis of the experimental conditions needs to detect the aforementioned phenomenon. We will focus specially on the HL-LHCb which due to his excellent detector resolution [47,48] could make possible the observation of the HNOs.
The work is arranged as follows: In Sec. II, we study the production of the heavy neutrinos in B ± c meson decays. In Sec. III, we describe the simulations of the HN production. In Sec. IV, we present the results and a discussion of its and In sec. V we provide a brief summary of the article.

II. PRODUCTION OF THE RHN
As we stated above, we are interested in studying the lepton flavor and leptop number violation processes (B ± c → µ ± N j → µ ± τ ± π ∓ ) which are caracterized by the following Feynman diagrams (Fig. 1). In this work we will consider the scenario where the two heavy neutrino (N 1 and N 2 ) masses fall in the range of a few GeVs and are almost degenerate The mixing coefficient between the standard flavor neutrino ν ( = e, µ, τ ) and the heavy mass eigenstate N i is B N i (i = 1, 2), then the light neutrino flavor state can be defined as where B ν i (i = 1, 2, 3) and B N j (j = 1, 2) are the complex elements of the 5 × 5 PMNS matrix, and will be parameterized as follow The mass difference between HNs is expressed as Y stand to measures the mass difference in terms of Γ N = (1/2)(Γ N 1 + Γ N 2 ) which is the (average of the) total decay width of the intermediate Heavy Neutrino. The decay width It is important to mention that due to the dependency on |B N i | the factors K Ma , then it is not expected a significant impact if one factor dominates over the other. However, in this work we will assume that K Ma 1 ≈ K Ma 2 ≡ K. In adittion, we will consider the mixing 1 In this work we define the light neutrino flavor state as ν = However, other authors also use U N or V N as the heavy-light mixings elements (i.e. As a consequence of the aforementioned, the HN total decay width are almost equals (Γ N 1 ≈ Γ N 2 ) and can be written as In Ref. [15] it was obtained the L-dependent effective differential decay width considering the effect of HNOs (see Eq. 6) and considering the effects of a detector of lenght L, for fixed values 2 of HN velocity (≡ β N ) and HN Lorentz factor (≡ γ N ) where L osc = (2πβ N γ N )/∆M N is the HN oscillation length and the angle θ LV stands for the relative CP-violating phase between N 1 and N 2 , that comes from the B N i elements 3 and is given by It is worth to mention, that, in general, M is moving in the lab frame when it decays into N and 1 , therefore, the product γ N β N is not always fixed, and can be written as where E N is the heavy neutrino energy in the lab frame, depending onp N direction in the M -rest frame (Σ ). The relation among E N , p N and the angle θ N is given by the Lorentz energy transformation (see Fig. 3) where the corresponding factors in the M -rest frame (Σ ) are given by we remarks that β M is the velocity of M in the lab frame, and λ(x, y, z) is Therefore, the Eq. 6 must be re-written in differential form and integrated over all di- where L osc (p N ) adopts the following form and The term d Γ(B + → + 1 N )/dΩp N is gven by  The observation of the studied phenomenon (Eq. 12) depends on the number of produced B c mesons (N Bc ) at the particular experiment. The HL-LCHb is design to reach a luminosity L = 2 · 10 −34 cm −2 s −1 [54], transforming it into one of the most promising B factories. The B mesons production cross-sections is σ B ≈ 86.6 µb [55], however, σ Bc is suppressed by a factor 10 −3 respect to σ B [56], this suppression factor implies that for each 10 6 B mesons we have 10 3 B c mesons. The HNs production has been calculated in detail in Refs. [12][13][14], in addittion, assuming a 50% detector efficency the expected number of Heavy Neutrino events (with HNs masses between 3.5 − 5.5 GeV and |B N | 2 = 10 −5 ) can reach ≈ 3000 for 6 years of operation.

IV. RESULTS AND DISCUSSION
In this article, we have studied the modulation dΓ(B c )/dL for the LNV B ± c meson decays assuming conditions that could be present at LHCb experiment. We focus on a scenario that contains two almost degenerate (on-shell) heavy Majorana neutrinos. This scenario has been studied in previous work Ref. [15] in which we have explored the modulation in a more academic frame, in this paper we consider more realistic conditions that could lead to a discovery in the upcoming years.
The Fig. 5 shows the Differential Decay Width dΓ(B ± c )/dL for fixed values of γ N and β N , which are determined from the average values of γ B ∓ c presented in Fig. 4, for two values of θ LV . The solid lines stand for the processes which include the effects of HNOS, while the dashed lines do not. It could be seen that the effects of HNOS over dΓ(B ± c )/dL could enhance or decrease it near a factor of two in comparison with the case with NO-HNOS, for some regions of L. In addition, for dΓ(B ± c )/dL with NO-HNOS effects there is no modulation and only it is present the damped effect produced due to the probability that the HN decay.
We noticed, that the difference between the process for B + c and B − c is maximized when the CP violation angle is θ LV = π/2 (as expected from Eq. 12). We can also observed that as the distance L grows, both curves tend to converge, this is because as the HN propagates, the cumulative probability that the HN has decayed is greater, this effect is characterized by the exponential factor present in dΓ(B c )/dL (Eq. 12), which specifically accounts for the probability that the HN decays within the detector of length L.  respectively. We remarks that we had simulated the same number of events for processes that involes B + c and its CP conjugate (B − c ), despide that from Eq. 6 we knows that cross sections of B + c and B − c are different if θ LV = 0. Both cases include their respective statistical error and consider γ N and β N distributed according to the result presented in Fig. 4. From   Fig. 8 we can see that there is only a modest difference between B ± c distributions, e.g in L = 0 − 60 mm, based on what we think it won't be possible to distinguish the oscillation for neither, θ LV = π/2 or θ LV = π/4, with more than 5σ's from the "non-oscillation" scenario with only 100 signal events. A more positive scenario is show in Fig. 9 and θ LV = π/4. Here the bin width is ∆L = 20 mm, in addition, it was considered that γ B ± c are distributed according Fig. 4. and θ LV = π/4. Here the bin width is ∆L = 20 mm, in addition, it was considered that γ B ± c are distributed according Fig. 4. and θ LV = π/4. Here the bin width is ∆L = 20 mm, in addition, it was considered that γ B ± c are distributed according Fig. 4. and θ LV = π/4. Here the bin width is ∆L = 20 mm, in addition, it was considered that γ B ± c are distributed according Fig. 4.

V. SUMMARY
In this work we have studied the decay of HN's and their modulation in rare B c meson decays at the HL-LHCb conditions. Here we have found that the modulation produce by the HNO's could be observed if 1000 HN events are detected, this number is consistent with the expected number of HN decays at HL-LHCb.