Exploring CP-violation, via heavy neutrino oscillations, in rare B meson decays at Belle II

In this article we study the rare B-meson decay via two on-shell almost-degenerate Majorana Heavy Neutrinos, into two charged leptons and two pseudoscalar mesons ($B^{\pm} \to D^0 \ell^{\pm}_1 \ell^{\pm}_2 \pi^{\mp}$). We consider the scenario where the heavy neutrino masses are $\sim 2$ GeV and the heavy-light mixing coefficients are $|B_{\ell N}|^2 \sim 10^{-5}$, and evaluate the possibility to measure the CP-asymmetry at Belle II. We present some realistic conditions under which the asymmetry could be detected.

The nature of Dirac neutrinos only allows them to appear in processes that are lepton number conserving (LNC). Majorana neutrinos can induce both lepton number conserving and lepton number violating (LNV) processes, which allows a wider spectrum of physics to take place. An important example of this is baryogenesis via leptogenesis, where the LNV and CP-violating processes can lead to a generation of a lepton number asymmetry in the early universe, which is then converted (through sphaleron processes [83][84][85]) to the baryon number asymmetry observed in the universe [86]. There are many different models that try to explain this asymmetry. However, two standard approaches that use Majorana neutrinos for successful Leptogenesis are out-of-equilibrium HN decays (or Thermal Leptogenesis) and leptogenesis from oscillations. Both of them use sterile neutrinos as an extension to the standard model, with their masses being calculated with the seesaw type-I mechanism. This mechanism allows us to have heavy neutrinos using the fact that the SM neutrinos have very low masses. These HNs satisfy the Sakharov conditions [87] in order to produce the asymmetry dynamically. Consequently, thermal leptogenesis [88][89][90] takes into account the lepton number asymmetry generated by the decay of a massive Majorana neutrino in a thermal bath, while the latter, known as Akhmedov-Rubakov-Smirnov (ARS) mechanism [91], leads to a lepton number asymmetry by means of HN oscillations. The main difference between the two mechanisms comes from the fact that the first case is a freeze-out situation while the ARS mechanism can be seen as a freeze-in one.
The range of the HN masses for thermal leptogenesis is dictated by the amount of CP violation that can be generated 1 . In the most simple scenarios leptogenesis is constructed with masses M N 10 10 GeV, or M N 1 TeV if one takes into account resonant effects [93], whereas the ARS mechanism allows neutrinos to reach masses as low as ∼ 1 GeV. The HN mass scale for thermal leptogenesis cannot be reached in modern experiments, while ARS leptogenesis allows a variety of experiments to try and probe not only the nature of neutrinos, but also leptogenesis [94].
This article is organized in the following way: In Sec. II we present the effective CP-violating B meson decay width for the LNV process B ± → D 0 ± 1 ± 2 π ∓ , and in Appendices A-D more details are given. In Sec. III we present the numerical results for this effective branching ratio (with 1 = 2 = µ) and for the related CP asymmetry ratio, for different values of the detector length, of the ratio of the HN mass difference and the HN total decay width, and for different values of the CP-violating phase. In Sec. IV we discuss the possibility for the detection of various such signals within the detector at Belle II and summarize our results.

II. CP VIOLATION IN HEAVY NEUTRINO DECAY
The simplest extension of the SM that explains the smallness of the active neutrino masses is the addition of right-handed neutrinos (ν R ). Then, the relevant terms of the new Lagrangian L N will read where M R is the mass of the right-handed neutrino. The coupling to the Dirac neutrinos (Y ν ) will give rise to small Dirac masses. After diagonalizing the mass matrix, three very light neutrinos are obtained, as well as three heavy ones, this is the well known seesaw mechanism [8][9][10]. The mass of the light neutrinos will be given by where M N = M R is a 3 × 3 mass matrix of the heavy neutrino and φ is the electroweak vacuum expectation value of the Higgs field. By tuning the parameters in the above equation one can reach neutrino masses ∼ 1 GeV, resulting in Yukawa couplings ∼ 10 −5 . This type of scenario is well discussed in the νMSM model [20,21]. Two key ingredients in the νMSM model is the CP violation that occurs in the mixing of the heavy neutrinos and a resonant effect when the masses of two of them satisfy the condition In previous articles we explored the HN CP-violating decays: i) considering only resonant CP violation without HN oscillation effects [97,98,100,104] and ii) nonresonant HN oscillation effects [59,[101][102][103]. In this article, we will considerer the decay B ± → D 0 ± 1 ± 2 π ∓ (see Fig. 1) extending the previous analysis, by considering simultaneously both of the aforementioned CP-violating sources, in order to explore these signals at Belle II experiment.
In this work, we will assume the existence of several (three) Heavy Neutrino states N j (j = 1, 2, 3), with respective masses M Nj . In addition, we will assume that the first two heavy neutrinos are almost degenerate and with masses in the range of ∼ 1 GeV, and the third neutrino is much heavier The first three active neutrinos ν (where = e, µ, τ ) will have, in general, admixtures of the mentioned heavy mass eigenstates where the heavy-light mixing elements B Nj are in general small complex numbers The heavy neutrino N 3 will not enter our considerations because in the considered decays it is off-shell, in contrast to N 1 and N 2 . We will consider CP-violating decays of B mesons into two light leptons ( 1 2 ) and a pion, mediated by heavy on-shell neutrinos N j (j = 1, 2). It turns out that (effective) branching ratios for the decays of the type B → D 1 2 π (cf. Fig. 1) are significantly larger than the decays B → 1 2 π, by about a factor of 30-40 when M N ≈ 2 GeV, cf. Ref. [105] (Figs. 19a and 20a there), 2 the main reason being the different CKM matrix elements |V cb | ∼ 10 |V ub |. For this reason, we will consider the decay channels B → D 1 2 π, Fig. 1. Furthermore, in order to avoid the kinematic suppression from heavy leptons, we exclude from our consideration the case of produced τleptons. In addition, to avoid the present stringent upper bounds on the heavy-light mixing B eNj , we exclude from our consideration the case of produced = e leptons. Thus we will take 1 2 = µµ. The N 1 -N 2 oscillation effects in such decays ( 1 2 = µµ) turn out to disappear in LNC decays but survive in LNV decays [101]. Hence, we will consider the LNV decays B ± → D 0 µ ± µ ± π ∓ , Fig. 1. The CP-violating B meson decay width for such a process, which accounts for the fact that the process will be detected only if the HN decays during its crossing through the detector (effective Γ), and includes both the overlap (resonant) [98,100] and the HN-oscillation CP-violating sources [101][102][103], is given by where L stands for the distance between the two vertices of the process (the flight length of the on-shell neutrino and θ LV is the CP-violating phase 4 which, according to notation of Eq. (5) can be written as In Eq. (6), the HN Lorentz kinematical parameters in the lab frame β N and γ N = 1/ 1 − β 2 N are assumed to be constant. This can be extended to the realistic case of variable β N [106], and this extension is explained in Appendix C. 6 We also assumed that |B N1 | = |B N2 (≡ |B N |) ( = µ, e, τ ).
Furthermore, the expression (6), in addition to the aforementioned approximations (fixed β N and common |B N |'s), is obtained in an approximation of combining the overlap (resonant) and oscillation effects, which is valid when Y is significantly larger than one, e.g. Y 5. This is explained in more detail in Appendix D, where several steps of derivation of the expression (6) are given.
In general, Γ N = (Γ N1 + Γ N2 )/2 where Γ Nj is the total decay width of HN N j (j = 1, 2). However, due to our mentioned assumption of |B N1 | = |B N2 | (≡ |B N |), we have Γ N1 = Γ N2 = Γ N . This is so because the total decay width of the heavy neutrino N j is [97,98] where N Ma j are the effective mixing coefficients whose range is ∼ 1-10 and account for all possible HN decay channels. The N j coefficients are presented in Fig. 2. 5 The numerical values of η(Y ) and δ(Y ) were obtained in [98,100], and the explicit expression for η(Y ) was obtained in [97]. Based on the mentioned numerical values of δ(Y ), we observe that they can be reproduced with high precision by the explicit expression for δ(Y ) given here. 6 We denote in the Appendices A-C the lab frame as Σ , and hence the Lorentz kinematical parameters are denoted as β " N and γ N , cf. Eq. (C3). From now on, as mentioned earlier we will consider only the case 1 = 2 = µ. We notice that |B µNj | 2 ≈ |B τ Nj | 2 10 −5 and |B eNj | 2 < 10 −7 , so that the K Ma j can receive significant contribution only from µ and τ decay channels (and we note that N Ma µj + N Ma τ j ≈ 10). We note that the mixings B µN1 and B µN2 can be, in principle, significantly different for the two HNs, and therefore, the two mixing factors K Ma j (j = 1, 2) may differ significantly from each other. However, as mentioned earlier, in this work we will assume that |B N1 | = |B N2 | (≡ |B N |). We will take K Ma 1 ≈ K Ma 2 = 10 |B µN | 2 and the HN total decay width then reads We note also that the HN masses are almost equal, i.e. M Nj M N .
The usual measure of the relative CP violation effect is given by the CP asymmetry ratio

III. RESULTS
In this Section we show the numerical results for the effective branching ratio Br eff (B ± ) = Γ eff (B ± → D 0 µ ± µ ± π ∓ )/Γ(B → all) and the CP asymmetry ratio A CP in (12) for different values of the Y parameter and the maximal displaced vertex length L, which can be interpreted as the (effective) detector length (L ≤ L det ). The calculations were performed by numerical integration with the VEGAS algorithm [109] in each step of L and Y . All integrations were performed using M N = 2 GeV and heavy-light mixings |B µN | 2 = |B τ N | 2 = 10 −5 . The selected mixing values are consistent with the present experimental constraints given in Ref. [47] and references therein. Moreover, two different values (scenarios) were chosen for the CP-violating phase: θ LV = π/2, π/4. The kinematical Lorentz factor γ N and β N in Eq. (6) in reality are not fixed, but vary and are obtained as explained in Appendix C, where the general expression Γ eff for the case of only one HN N is given in Eq. (C4). In the case of two (almost degenerate) HNs N j (j = 1, 2) the expression (C4) gets extended by the overlap (resonant) and oscillation terms as those appearing in Eq.(6), leading to our main formula

IV. DISCUSSION OF THE RESULTS AND SUMMARY
In this work we have studied the CP-violating effects in the rare B meson decays mediated by two on-shell HNs. Unlike previous works, our calculations include both overlap (resonant) and oscillating effects. The variation of the values of the parameter Y ≡ ∆M N /Γ N shows that there exists a mass-difference regime in which the CP-violating effects can be noticeable. Our formulas are approximations which are good if Y is not too small (Y 5), because we do not know (and do not include) the terms which are simultaneously overlap and oscillation effects. On the other hand, if Y < 1, i.e., the mass difference ∆M N is smaller than the decay width Γ N , the CP-violating effects are expected to be highly suppressed and A CP → 0 as Y → 0. We set the maximum value of the displaced vertex length (effective detector length) L to L = 1000 mm in order to obtain a realistic prediction of the number of events that can take place at Belle II experiment.
From figure 3 we can see that the biggest difference from B + and B − effective branching ratios occurs between the 200 and 400 mm. Furthermore, the channel difference changes with the CP violating phase θ LV , where the biggest CP violation appears at π/2 and the smallest occurs at π/4. For values of θ LV = 0, π, there will be no difference between the channels. If the parameter Y increases from 5 to 10 ( Figure 4) one can notice that now the biggest CP violation moves to the left and occurs between 50 and 200 mm, while the maximum occurs at θ LV = π/2. The effect produced by the parameter Y can be read from figures 5 and 6. Values of Y > 15 shows little difference between the channels, this is well expected as for larger Y the resonant and oscillating regimes will disappear when ∆M ∝ Y 1. The maximum CP violation is strongly dependent on the length L, as seen from L = 300 mm in figure  5 and L = 1000 mm in figure 6. Figure 7 shows the asymmetry as a function of the length L. Although, the biggest value of the CP asymmetry appears for small values of the length (L ∼ 50 − 300 mm), the branching ratios increase as L → 1000 mm. Thus, biggest values of CP asymmetry are not enough to detect events. Therefore, the branching ratios must also be taken into account in order to have a signal in the detector. Figure 8 shows the asymmetry as a function of Y . The biggest values of CP asymmetry appear for Y = 1 − 20, and will disappear for Y > 50.
Moreover, table I presents the expected number of events N e (B ± ) = N B × Br eff (B ± ), considering that the number of B meson expected at Belle II is N B = 5 × 10 10 .  In summary, in this work we studied the B-mesons decays B ± → D 0 µ ± 1 µ ± 2 π ∓ at Belle II, considering a 1000 mm effective detector length. We focused in a scenario with two almost-degenerate heavy neutrinos with masses around M N ∼ 2 GeV. The effective branching ratios were calculated by considering that the heavy neutrino total decay width is equal for both, as a consequence of the assumption that the heavy-light mixing coefficients satisfy |B N1 | = |B N2 | (≡ |B N | 2 ) for = µ, τ . Further, we considered |B µN | 2 ∼ |B τ N | 2 ∼ 10 −5 |B eN | 2 . The calculations were performed in a scenario that contains both the overlap (resonant) and oscillating CP-violating sources. We observed that the biggest difference of detectable events occurs for Y = 5 and θ = π/2 (Table I).
We established that for certain presently allowed regime of values of |B µN | 2 , Y (≡ ∆M N /Γ N ) and θ LV , and with M N ≈ 2 GeV, the aforementioned effects can be observed at Belle II. The differential decay width of the process B → D 1 N (see Fig. 9) was obtained in Ref. [106] and has the following form: 7 Here, q 2 is the squared four-momentum of the W * boson,q is the unitary direction vector of q in the B-rest frame Σ ,p 1 is the unitary direction of p 1 of 1 in the W * -rest ( 1 N -rest) frame Σ. The expression | T | 2 stands for the squared decay amplitude and is given by where The expression (A3) is defined in terms of two form factors, F 1 and F 0 . The form factor F 1 (q 2 ) is presented in [110] and is expressed in terms of w(q 2 ) and z(w) Therefore, from Ref. [110], F 1 (q 2 ) can be expressed as In the last equation the free parameters ρ 2 and F 1 (w = 1) have been determined by the Belle Collaboration [111] ρ 2 = 1.09 ± 0.05 , (A7a) |V cb |F 1 (w = 1) = (48.14 ± 1.56) × 10 −3 . (A7b) The form factor F 0 (q 2 ) is given as [110] 8 where f 0 (w = 1) ≈ 1.02 and ρ 2 0 ≈ 1.102. The decay width for B → D 1 N decays is For the effective decay width, which takes into account only those decays in which the exchanged on-shell N decays within the detector, we refer to Appendix C.
Here, the canonical decay width Γ is where f π (≈ 0.1304 GeV) is the decay constant of pion, and the other factors are These results can be combined with the result (A9) to obtain the decay width for the decay where the expressions (A1) and (A9) are used for the first factor, and (B1) and (B2) for the second factor of the integrand. For Γ N we refer to Eq. (10).

Appendix C: Lorentz factors of on-shell N in laboratory frame
In this Appendix we follow the presentation given in Ref. [106]. The expression (B4) refers to the decay width for all the decays of the type B ± → D 0 ± 1 N → ± 1 ± 2 π ∓ , including those where the on-shell N decays outside the detector. However, if we realistically consider that only those decays are detected in which the on-shell N decays within the detector (of length L), we need to multiply the integrand in Eq. (B4) with the probability P N of decaying of the produced on-shell N within the length L.
where τ N = 1/Γ N is the lifetime of N in its rest frame. The velocity β N and the Lorentz factor γ N = 1/ 1 − (β N ) 2 are those of the N neutrino in the lab frame Σ . 9 At Belle II, the kinetic energy of the produced Υ(4S) is 0.421 GeV, and this implies that its Lorentz factor in the lab frame Σ is γ Υ = 1.0398 and β Υ = 0.274. When Υ(4S) produces a pair of B mesons, the kinetic energy of B mesons in the Υ-rest frame is 0.010 GeV, which is negligible. Thus the velocity of the B mesons in the lab frame Σ is equal to the velocity of Υ(4S) Then, the factor γ N β N appearing in the probability (C1) can be calculated by calculating the energy E N of the N neutrino in the lab frame (see below) and this leads to the effective decay width for the considered process which is as the expression (B4) but with inclusion of the N decay probability within the effective detector length L. 10 The energy E N of the produced heavy neutrino N in the lab frame and is given by (cf. App. B of Ref. [106]) The factors, as a function of the squared invariant mass of W * − , q 2 (see Fig. 9), are where the angles θ q , θ 1 and φ 1 range as follow: For a more detailed explanation of the aforemention expressions we refer to Ref. [106].
Appendix D: Effective width of the LNV B decay channel with overlap and oscillation effects Here we will explain how the expression (6) is obtained. We work in the case when the Lorentz factors in the lab frame β N and γ N ≡ 1/ 1 − β 2 N are considered to be fixed. 11 In addition, we use the assumption made throughtout this work that the heavy-light mixing elements satisfy |B N1 | = |B N2 | (≡ |B N |), where = µ, e, τ . When no oscillation is assumed [i.e., only the overlap (resonant) effects included], the effective decay width for the considered LNV decay channnel is [98] [cf. also [101] Eq. (13) there] We recall that L here is the length of flight of the on-shell N j in the detector before it decays (within the detector), and the parameter Y and the N 1 -N 2 overlap functions δ(Y ) and η(Y ) are given in Eqs. (7) and (9. The differential decay rate dΓ eff /dL for this decay width is then On the other hand, when Y 1 and thus the overlap contributions ∼ δ(Y ) and ∼ η(Y )/Y can be neglected, we obtained in Ref. [101] the corresponding differential decay width with N 1 -N 2 oscillation effects included 12 where L osc is the HN oscillation length If we now combine the overlap (resonant) contributions contained in the expression (D2) with the oscillation contributions contained in the expression (D3), we obtain The expression (D3) was obtained in Ref. [101] from the expression (D2) under the assumption that the overlap contributions (∼ δ(Y ), η(Y )/Y ) there were negligible, i.e., that Y 1. Combination of these two expressions into the expression (D5) thus involves an approximation of neglecting oscillation terms which involve overlap effects, i.e., terms of the type ∼ (η(Y )/Y ) cos(2πL/L osc ± θ LV ) or similar (we do not know these terms). This approximation is also reflected in the fact that the expression (D5) is negative for some flight lengths L, which should not happen. However, if Y is significantly larger than one (say, Y 5), these negative contributions are small in absolute value and appear only in very short intervals of L, and consequently the expression (D5) can be regarded as a reasonably good approximation containing simultaneously both the overlap (resonant) and oscillation contributions, especially when it is integrated over L.
Integration of the partial decay width (D5) from L = 0 to L then gives us the expression (6) in Sec. II.