Probing Majorana Neutrinos and their CP Violation in Decays of Charged Scalar Mesons π, K, D, D_s, B, B_c

Abstract

Some of the outstanding questions of particle physics today concern the neutrino sector, in particular whether there are more neutrinos than those already known and whether they are Dirac or Majorana particles. There are different ways to explore these issues. In this article we describe neutrino-mediated decays of charged pseudoscalar mesons such as π^±, K^± and B^±, in scenarios where extra neutrinos are heavy and can be on their mass shell. We discuss semileptonic and leptonic decays of such kinds. We investigate possible ways of using these decays in order to distinguish between the Dirac and Majorana character of neutrinos. Further, we argue that there are significant possibilities of detecting CP violation in such decays when there are at least two almost degenerate Majorana neutrinos involved. This latter type of scenario fits well into the known neutrino minimal standard model (νMSM) which could simultaneously explain the Dark Matter and Baryon Asymmetry of the Universe.

1. Introduction

To date it is unclear whether the neutrinos we know are Dirac of Majorana fermions. Unlike Dirac fermions, Majorana fermions cannot be distinguished from their own antiparticles. As a consequence, processes involving Dirac neutrinos conserve charges such as Lepton Number, while processes involving Majorana neutrinos will not conserve them. There exist several processes which may clarify the Majorana or Dirac nature of neutrinos. Among such processes the most prominent are neutrinoless double beta decays (0νββ) in nuclei [1–10]. Other such processes are specific scattering processes [11–21], and rare meson decays [22–40].

A related issue in neutrinos physics is the absolute mass values of the known neutrinos. While the experimental evidence of neutrino oscillations within the known three flavor states [41–46] clearly shows that these particles cannot all be massless, the oscillations are only sensitive to mass differences, not to their absolute values. In contrast, 0νββ decays are sensitive to the absolute mass and may help in their determination, if neutrinos turn out to be Majorana particles. So far the best bounds on the absolute masses of the light neutrinos come from Cosmology m_ν ≳ 0.23 eV [47].

A pending question is then why light neutrinos are so light, specifically so much lighter than all other Standard Model (SM) fermions. Interesting enough, the existence of such very light neutrinos can be explained via the seesaw mechanism [48–52] where more neutrinos are required and where all of them are, in general, Majorana particles. In the simplest form of this mechanism, the masses of the light neutrinos are $~{\mathcal{M}}_D^2/{\mathcal{M}}_R(\lesssim 1\mathrm{eV})$, where ${\mathcal{M}}_D$ is an electroweak scale or lower. At the same time, additional neutrinos, usually much heavier (with masses ${\mathcal{M}}_R\gg 1$ TeV) and sterile under electroweak interactions except through small mixing with the SM flavors, are required. This mixing is suppressed as $~{\mathcal{M}}_D/{\mathcal{M}}_R(\ll 1)$. Besides the simplest scenario, there are other seesaw scenarios in which the heavy neutrinos may have lower masses, namely near or below 1 TeV [53–59] and even near the 1 GeV scale or below [15,60–65], and at the same time their mixing with the SM flavors may not be so extremely suppressed as in the original scenarios.

In the first part of this work we discuss lepton number violating (LNV) semileptonic decays of charged pseudoscalar mesons such as K^± and B^±, mediated by a heavy Majorana neutrino on its mass shell, cf. Ref. [26]. The pions, which are the lightest mesons, can have only leptonic decays; we discuss hypothetical leptonic decays that could be mediated by on-shell heavy neutrinos. Such decays could be either lepton number conserving (LNC) or lepton number violating (LNV) when the mediating neutrinos are Majorana particles, cf. Ref. [37], while only LNC decays occur when the neutrino is of Dirac type. We present ways of determining the nature of neutrinos using the differential decay rates of these processes.

Yet another interesting issue in neutrino physics is the possible existence of CP violation in the lepton sector, a phenomenon that could be measured, for example, in neutrino oscillations [66]. Alternatively, as we present in the second part of this work, leptonic CP violation may show in leptonic LNC and LNV decays of charged pions, cf. Ref. [38], or in semileptonic LNV decays of K^± and B^±, cf. Refs. [39,40]. It turns out that such CP violation becomes appreciable and possibly detectable in these decays if the scenario contains at least two on-shell heavy neutrinos that are almost degenerate. Interestingly, this scenario fits well into the neutrino minimal standard model (νMSM) [60,61,67–72] which contains two almost degenerate Majorana neutrinos of mass ∼ 1 GeV and another lighter neutrino of mass ∼ 10¹ keV, besides the three light neutrinos of mass ≲1 eV. This model can explain simultaneously the existence of neutrino oscillations, dark matter and baryon asymmetry of the Universe. Furthermore, in more general frameworks of low-scale seesaw, baryon asymmetry (but not dark matter) is explained while keeping even larger values of the heavy-light mixing [73] than in the νMSM, and in such frameworks the case of almost degenerate Majorana neutrinos is preferred [74] since it allows larger mixings. CP violation effects in the neutrino sector in scenarios with nearly degenerate heavy neutrinos have also been investigated earlier [75,76] using a more detailed formalism, although it amounts to the same effect described here which is the interference of amplitudes with two slightly different dispersive and absorptive parts in the neutrino self energy.

In Section 2 we discuss the LNV semileptonic decays of mesons $M^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }{M^{\prime }}^{\mp }$, mediated by an on-shell Majorana neutrino (which henceforth we call N) where M is a heavy pseudoscalar (M = K, D, D_s, B, B_c), M′ is a lighter pseudoscalar, and ℓ_j (j = 1, 2) are charged leptons, and we present there the corresponding branching ratios. In Section 3 we present the expressions and values of the branching ratios for the LNC and LNV leptonic decays of charged pions mediated by on-shell N sterile neutrino, π^± → e^±N → e^±e^±µ^∓ν, as well as the differential branching ratio dBr/dE_µ for these decays. We discuss the possibilities of detecting such branching ratios and to discern from them the Majorana or Dirac nature of neutrinos. In Section 4 we then extend the analysis of the mentioned leptonic and semileptonic decays to a scenario where we have at least two heavy on-shell sterile neutrinos involved (N₁, N₂), and we present an analysis of CP-violating asymmetries ${\mathcal{A}}_{\mathrm{CP}}\equiv [\mathrm{\Gamma }(M^{-})-\mathrm{\Gamma }(M^{+})]/[\mathrm{\Gamma }(M^{-})+\mathrm{\Gamma }(M^{+})]$ for such processes. In Appendices A.1 and A.2 we present explicit formulas for our LNV semileptonic decays, and in Appendices A.4 and A.5 explicit formulas needed for the analysis of our LNC and LNV leptonic decays of the charged pion. Appendix A.3 contains formulas needed for evaluation of the decay width of the heavy neutrino N, and in Appendix A.6 we derive an identity relevant for CP violation asymmetry. In Section 5 we discuss and summarize our results.

2. Lepton Number Violating Semileptonic Decays of Scalar Mesons

If there is a Majorana sterile neutrino N with mass M_N ~ 1 GeV, its existence could be discerned by detecting semileptonic LNV decays of heavy mesons mediated by on-shell N. Here we will consider such LNV decays $M^{\pm }\to {\ell }_1^{\pm }N\to {\ell }_1^{\pm }{\ell }_2^{\pm }{M^{\prime }}^{\mp }$, where M and M′ are pseudoscalar mesons (M = K, D, D_s, B, B_c; M′ = π, K, D, D_s) while ℓ₁ and ℓ₂ are charged leptons (e, µ or τ), cf. Figure 1.

Large part of this Section uses the results of Refs. [26,37], and for certain general formulas, those of Refs. [39,40].

We consider a scenario where we have at least one heavy sterile neutrino N, which has (small) mixing B_ℓN with the three known neutrino flavors ν_ℓ (ℓ = e, µ, τ)

$${\nu }_{\ell }={\displaystyle \sum _{k=1}^3{B}_{\ell {\nu }_k}{\nu }_k+B_{\ell N}}N+\dots$$

(1)

Here, ν_k (k = 1, 2, 3) are the three light mass neutrino eingenstates. The considered decays may be appreciable only if N can go on its mass shell (in the s-type channel, Figure 1), creating a very large resonant enhancement of order m_N/Γ_N, condition which is fulfilled if

$$\begin{array}{l}(M_{M^{\prime }}+M_{{\ell }_2})<M_N<(M_M-M_{{\ell }_1}),\mathrm{or}/\mathrm{and}\\ (M_{M^{\prime }}+M_{{\ell }_1})<M_N<(M_M-M_{{\ell }_2})\end{array}$$

(2)

Consequently, only tree level resonant amplitudes need to be considered. The mixing matrix B appearing in Equation (1) should be unitary, implying that the PMNS 3 × 3 block $B_{\ell {\nu }_k}(\ell =e,\mu ,\tau \mathrm{and}k=1,2,3)$ is in general not unitary. If one adds extra and heavy neutrinos—as in most seesaw models—the unitarity of B thus provides upper limit constraints on the heavy-to-light mixing elements [34,77–79]. This is in part one of the reasons for the high suppression suffered by all lepton flavour violating processes involving heavy neutrinos.

2.1. Branching Ratio for$M^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }{M^{\prime }}^{\mp }$

The decay width of the considered decays can be written as final particles’ phase space integral of the square of the reduced decay amplitude $\mathcal{T}(M^{\pm })$ (summed over helicities of charged leptons)

$$\mathrm{\Gamma }(M^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }{M^{\prime }}^{\mp })=(2-{\delta }_{{\ell }_1{\ell }_2})\frac{1}{2!}\frac{1}{2M_M}\frac{1}{{(2\pi )}^5}{\displaystyle \int d_3|\mathcal{T}(M^{\pm }){|}^2}$$

(3)

Factor 1/2! above is the symmetry factor when the two produced leptons are equal; d₃ is the integration differential of the final three-particle phase space

$$d_3\equiv \frac{d^3{\overrightarrow{p}}_1}{2E_{{\ell }_1}({\overrightarrow{p}}_1)}\frac{d^3{\overrightarrow{p}}_2}{2E_{{\ell }_2}({\overrightarrow{p}}_2)}\frac{d^3{\overrightarrow{p}}_{M^{\prime }}}{2E_{M^{\prime }}({\overrightarrow{p}}_{M^{\prime }})}{\delta }^{(4)}(p_M-p_1-p_2-p_{M^{\prime }})$$

(4)

The resulting decay width can be written as

$$\mathrm{\Gamma }(M^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }{M^{\prime }}^{\mp })=(2-{\delta }_{{\ell }_1{\ell }_2})|k{|}^2\left[\tilde{\mathrm{\Gamma }}(DD*)+\tilde{\mathrm{\Gamma }}(CC*)+{\tilde{\mathrm{\Gamma }}}_{\pm }(DC*)+{\tilde{\mathrm{\Gamma }}}_{\pm }(CD*)\right]$$

(5)

where k is the mixing factor

$$k=B_{{\ell }_{1}N}{B}_{{\ell }_{2}N}$$

(6)

and ${\tilde{\mathrm{\Gamma }}}_{\pm }(XY*)$ are the normalized (i.e., without the explicit mixing) contributions from the X channel and the complex-conjugate of the Y channel (X, Y = D, C, where D and C stand for the direct and crossed channels)

$${\tilde{\mathrm{\Gamma }}}_{\pm }(XY*)\equiv K^2\frac{1}{2!}\frac{1}{2M_M}\frac{1}{{(2\pi )}^5}{\displaystyle \int d_{3}P(X)P(Y)*M_N^2{T}_{\pm }(X)T_{\pm }(Y)*}$$

(7)

The expressions for T_±(X)T_±(Y)* (X, Y = D, C) are given in Appendix A.1. T_±(X) is the relevant part of the amplitude in the X channel and forms part of the total decay amplitude $\mathcal{T}(M^{\pm })$, cf. Appendix A.1. In Equation (5), notice that subscripts ± for the contributions $\tilde{\mathrm{\Gamma }}(DD*)$ and $\tilde{\mathrm{\Gamma }}(CC*)$ are unnecessary because |T₊(D)|² = |T₋(D)|² and |T₊(C)|² = |T₋(C)|². P (X) (X = D, C) are the propagator functions of the intermediate neutrino N in the two channels

$$P(D)=\frac{1}{[{(p_M-p_1)}^2-M_N^2+i{\mathrm{\Gamma }}_N{M}_N]}$$

(8a)

$$P(C)=\frac{1}{[{(p_M-p_2)}^2-M_N^2+i{\mathrm{\Gamma }}_N{M}_N]}$$

(8b)

The overall constant K² in Equation (7) is

$$K^2=G_F^4{f}_M^2{f}_{M^{\prime }}^2|V_{Q_u{Q}_d}{V}_{q_u{q}_d}{|}^2$$

(9)

Here, f_M and f_M′ are the decay constants of M^± and M′^∓, and $V_{Q_u{Q}_d}$ and $V_{q_u{q}_d}$ are the corresponding CKM matrix elements. We denote the valence quark content of M⁺ as $Q_u{\overline{Q}}_d$; of M′⁺ as $q_u{\overline{q}}_d$.

When the intermediate neutrino N has such a mass that it is on mass shell, Equation (2), the squares of the propagators (8) are reduced to Dirac delta functions because Γ_N ≪ M_N

$$\begin{array}{l}|P(X){|}^2={\left|\frac{1}{{(p_M-p_k)}^2-M_N^2+i{\mathrm{\Gamma }}_N{M}_N}\right|}^2\\ =\frac{\pi }{M_N{\mathrm{\Gamma }}_N}\delta ({(p_M-p_k)}^2-M_N^2({\mathrm{\Gamma }}_N\ll M_N)\end{array}$$

(10)

where p_k = p₁, p₂ for X = D, C. In this on-shell case, the DD* and CC* contributions in Equation (5) are large, and the interference contributions DC* and CD* are negligible in comparison (cf. Ref. [39] for details on this point), leading to

$$\mathrm{\Gamma }(M^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }{M^{\prime }}^{\mp })=(2-{\delta }_{{\ell }_1{\ell }_2})|k{|}^2\left[\tilde{\mathrm{\Gamma }}(DD*)+\tilde{\mathrm{\Gamma }}(CC*)\right]$$

(11a)

$$\equiv \mathrm{\Gamma }(M^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }{M^{\prime }}^{\mp };DD*)+\mathrm{\Gamma }(M^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }{M^{\prime }}^{\mp };CC*)$$

(11b)

when ℓ1 = ℓ2, we even have $\tilde{\mathrm{\Gamma }}(DD*)=\tilde{\mathrm{\Gamma }}(CC*)$ The normalized decay width $\tilde{\mathrm{\Gamma }}(DD*)$ can be calculated explicitly, and it turns out to be

$$\tilde{\mathrm{\Gamma }}(DD*)=\frac{K^2{M}_M^5}{128{\pi }^2}\frac{M_N}{{\mathrm{\Gamma }}_N}{\lambda }^{1/2}(1,y_N,y_{{\ell }_1}){\lambda }^{1/2}\left(1,\frac{y^{\prime }}{y_N},\frac{y_{{\ell }_2}}{y_N}\right)Q(y_N;y_{{\ell }_1},y_{{\ell }_2},y^{\prime })$$

(12)

and $\tilde{\mathrm{\Gamma }}(CC*)$ is obtained by the simple exchange $y_{{\ell }_1}\leftrightarrow y_{{\ell }_2}$

$$\tilde{\mathrm{\Gamma }}(CC*)=\tilde{\mathrm{\Gamma }}(DD*)(y_{{\ell }_1}\leftrightarrow y_{{\ell }_2})$$

(13)

The notations used in Equations (12) and (13) are

$$\lambda (y_1,y_2,y_3)=y_1^2+y_2^2+y_3^2-2y_1{y}_2-2y_2{y}_3-2y_3{y}_1$$

(14a)

$$y_N=\frac{M_N^2}{M_M^2},y_{{\ell }_s}=\frac{M_{{\ell }_s}^2}{M_M^2},y^{\prime }=\frac{M_{M^{\prime }}^2}{M_M^2},({\ell }_s={\ell }_1,{\ell }_2)$$

(14b)

and the function $Q(y_N;y_{{\ell }_1},y_{{\ell }_2},y^{\prime })$ is given in Appendix A.2. In the limit of massless charged leptons $(y_{{\ell }_1}=y_{{\ell }_2}=0)$, the expression (12) reduces to

$$\tilde{\mathrm{\Gamma }}(DD*)|{}_{M_{{\ell }_1}=M_{{\ell }_2}=0}=\frac{K^2{M}_M^5}{256{\pi }^2}\frac{M_N}{{\mathrm{\Gamma }}_N}{y}_N^2{(1-y_N)}^2{\left(1-\frac{y^{\prime }}{y_N}\right)}^2$$

(15)

We note that the expression (12), although having the explicit mixing dependence factored out [cf. Equation (11)], contains the dependence on the mixing coefficients B_ℓN in the denominator due to the N-decay width there Γ_N ∝ |B_ℓN|² (ℓ = e, µ, τ, see below). This factor 1/Γ_N in $\tilde{\mathrm{\Gamma }}(DD*)$ of Equation (12) represents the N-on-shell effect Equation (10). As a result, the considered width $\mathrm{\Gamma }(M^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }{M^{\prime }}^{\mp })$ is by many orders of magnitude larger when N is on-shell than it would be if N were off-shell. For more quantitative analyses, it is thus important to have an expression for Γ_N as a function of mass M_N. Using the results of Ref. [39], we can write this decay width as

$${\mathrm{\Gamma }}_N=\tilde{\mathcal{K}}{\overline{\mathrm{\Gamma }}}_N(M_N)$$

(16)

where the corresponding canonical (i.e., without any mixing dependence) decay width is

$${\overline{\mathrm{\Gamma }}}_N(M_N)\equiv \frac{G_F^2{M}_N^5}{96{\pi }^3}$$

(17)

and the factor $\tilde{\mathcal{K}}$ contains the dependence on the heavy-light mixing factors

$$\tilde{\mathcal{K}}(M_N)\equiv \tilde{\mathcal{K}}={\mathcal{N}}_{eN}|B_{eN}{|}^2+{\mathcal{N}}_{\mu N}|B_{\mu N}{|}^2+{\mathcal{N}}_{\tau N}|B_{\tau N}{|}^2$$

(18)

In this expression, ${\mathcal{N}}_{\ell N}(M_N)\equiv N_{\ell N}(\ell =e,\mu ,\tau )$ are the effective mixing coefficients; these are numbers ∼ 10⁰–10¹ which depend on the mass M_N. In Appendix A.3 we write down the relevant formulas for the evaluation of these coefficients. The results of these evaluations are presented in Figure 2, for the case of Majorana and Dirac neutrino N, in the entire neutrino mass interval 0.1 GeV < M_N < 6.3 GeV which will be of interest in this work. For further clarifying remarks we refer to Appendix A.3. Equations (11) and (12) imply that $\mathrm{\Gamma }(M^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }{M^{\prime }}^{\mp })$ is proportional to $1/\tilde{\mathcal{K}}(\propto 1/|B_{\ell N}{|}^2)$. Hence, we can define a canonical branching ratio $\overline{\mathrm{Br}}$, being the part of the branching ratio $\mathrm{Br}(DD*)\equiv \mathrm{\Gamma }(M^{\pm }\to {\ell }_1^{\pm }{\ell }_2\pm {M^{\prime }}^{\mp };DD*)/\mathrm{\Gamma }(M^{\pm }\to \mathrm{all})$ with no explicit or implicit heavy-light mixing factors

$$\mathrm{Br}(M^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }{M^{\prime }}^{\mp };DD*)\equiv \frac{\mathrm{\Gamma }(M^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }{M^{\prime }}^{\mp };DD*)}{\mathrm{\Gamma }(M^{\pm }\to \mathrm{all})}=\frac{(2-{\delta }_{{\ell }_1}{\delta }_{{\ell }_2})|k{|}^2}{\mathrm{\Gamma }(M^{\pm }\to \mathrm{all})}\tilde{\mathrm{\Gamma }}(DD*)$$

(19a)

$$=(2-{\delta }_{{\ell }_1{\ell }_2})\frac{|k{|}^2}{\tilde{\mathcal{K}}}2\overline{\mathrm{Br}}(y_N;y_{{\ell }_1},y_{{\ell }_2};y^{\prime })$$

(19b)

Use of the expressions (12) and Equations (16) and (17) then gives for the canonical branching ratio the following expression:

$$\begin{array}{l}\overline{\mathrm{Br}}(DD*)\equiv \overline{\mathrm{Br}}(y_N;y_{{\ell }_1},y_{{\ell }_2},y^{\prime })\\ =\frac{3\pi }{8}\frac{K^2{M}_M}{G_F^2\mathrm{\Gamma }(M^{\pm }\to \mathrm{all})}\frac{1}{y_N^2}{\lambda }^{1/2}(1,y_N,y_{{\ell }_1}){\lambda }^{1/2}\left(1,\frac{y^{\prime }}{y_N},\frac{y_{{\ell }_2}}{y_N}\right)Q(y_N;y_{{\ell }_1},y_{{\ell }_2},y^{\prime })\end{array}$$

(20)

where the notations (14) and (9) are used. In the limit of massless charged leptons $(M_{{\ell }_1}=M_{{\ell }_2}=0)$ this expression becomes simpler

$$\overline{\mathrm{Br}}(y_N;0,0,y^{\prime })=\frac{3\pi }{16}\frac{K^2{M}_M}{G_F^2\mathrm{\Gamma }(M^{\pm }\to \mathrm{all})}{(1-y_N)}^2{\left(1-\frac{y^{\prime }}{y_N}\right)}^2$$

(21)

2.2. Effect of the Long Neutrino Lifetime on the Observability of $M^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }{M^{\prime }}^{\mp }$

In the mentioned branching ratios, an often important effect of suppression due to the decay (i.e., nonsurvival) probability was not included. Namely, if the detector for the considered decays has a certain length L, the produced (on-shell) massive neutrino N could survive during its flight through the detector, and would decay later outside it. Such decays are thus not detected and should be eliminated from the width and the branching ratio of the considered process $M^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }{M^{\prime }}^{\mp }$, by introducing a suppression factor (nonsurvival probability) P_N = 1 − exp[−t/(τ_Nγ_N)], where t ≈ L/β_N is the time of flight of N through the detector (β_N is the velocity of N in the lab frame), and ${\gamma }_N={(1-{\beta }_N^2)}^{-1/2}$ is the Lorentz time dilation factor. Hence, the suppression factor, which should multiply the branching ratio, is

$$P_N=1-\exp \left[-\frac{L}{{\tau }_N{\gamma }_N{\beta }_N}\right]\approx 1-\exp \left[-\frac{L{\mathrm{\Gamma }}_N}{{\gamma }_N}\right]$$

(22)

In the last relation, we used β_N ≈ 1 and τ_N = 1/Γ_N [≡ 1/Γ(N → all)], in the units used here (c = 1 = ħ). This decay-within-the-detector probability P_N has been discussed and presented for the processes with intermediate on-shell particle (such as N) in Refs. [16,37–39,80–82]. In this respect, here we follow mostly the notations of Ref. [39]. Usually, the quantity P_N is small and is then written as

$$P_N\approx L/({\tau }_N{\gamma }_N{\beta }_N)(\approx L/({\tau }_N{\gamma }_N))\mathrm{if}{P}_N\ll 1$$

(23)

which agrees with Equation (22) in the limit of small P_N. The suppression factor (22) can be rewritten as

$$P_N=1-\exp \left(-\frac{L}{L_N}\right)=1-\exp \left(-\frac{L}{{\overline{L}}_N}\tilde{\mathcal{K}}\right)$$

(24a)

$$\approx \frac{L}{{\overline{L}}_N}\tilde{\mathcal{K}}\mathrm{if}{P}_N\ll 1$$

(24b)

Here, L_N is the decay length, and ${\overline{L}}_N$ is the canonical decay length (independent of the mixing parameters B_ℓ′N)

$$L_N^{-1}={\overline{L}}_N^{-1}\tilde{\mathcal{K}}$$

(25a)

$$L_N^{-1}=\frac{{\overline{L}}_N(M_N)}{{\gamma }_N}=\frac{1}{{\gamma }_N}\frac{G_F^2{M}_N^5}{96{\pi }^3}$$

(25b)

where $\tilde{\mathcal{K}}$ and ${\overline{\mathrm{\Gamma }}}_N(M_N)$ are from Equations (16)–(18), cf. also Figure 2. Equation (24b) suggests that it is convenient to define a canonical (i.e., independent of mixing) probability ${\overline{P}}_N$ for the decay of N within the detector as

$${\overline{P}}_N=\frac{1\mathrm{m}}{{\overline{L}}_N}\Rightarrow P_N\approx {\overline{P}}_N\left(\frac{L}{1\mathrm{m}}\right)\tilde{\mathcal{K}}$$

(26)

We present the inverse canonical decay length, ${\overline{L}}_N^{-1}$, for γ_N = 2, in Figure 3 as a function of M_N. We note that ${\overline{L}}_N^{-1}$ increases very fast (as $M_N^5$) when M_N increases. Therefore, the supression due to the factor P_N may not necessarily be strong (i.e., $P_N\cancel{\ll }1$) for semileptonic LNV decays of heavier mesons M^±, such as B^±.

If we use eyeball estimates for the coefficients ${\mathcal{N}}_{\ell N}$ of the left-hand Figure 2, approximate expressions for the factor $\tilde{\mathcal{K}}$ of Equation (18) for Majorana neutrinos can be written

$$\tilde{\mathcal{K}}\approx 15|B_{e{N}_j}{|}^2+8|B_{\mu N_j}{|}^2+2|B_{\tau N_j}{|}^2(K\mathrm{decays})$$

(27a)

$$\tilde{\mathcal{K}}\approx 7(|B_{e{N}_j}{|}^2+|B_{\mu N_j}{|}^2)+2|B_{\tau N_j}{|}^2(D,D_s\mathrm{decays})$$

(27b)

$$\tilde{\mathcal{K}}\approx 8(|B_{e{N}_j}{|}^2+|B_{\mu N_j}{|}^2)+3|B_{\tau N_j}{|}^2(B,B_c\mathrm{decays})$$

(27c)

In order to estimate better the values of $\tilde{\mathcal{K}}$ Equations (27) and thus the suppression factor P_N Equation (24), we need to know the present upper bounds for the squares |B_ℓN|² as a function of $M_N^2$. These upper bounds we take from compilation of values of Ref. [29], based in turn on upper bound values obtained in Refs. [83–96]. We present them in

Table 1. Presently known upper bounds for the squares |B_ℓN|² of the heavy-light mixing matrix elements, for various specific values of M_N.

MN [GeV]	|BeN|2	|BµN|2	|BτN|2
0.1	(1.5 ± 0.5) × 10−8	(6.0 ± 0.5) × 10−6	(8.0 ± 0.5) × 10−4
0.3	(2.5 ± 0.5) × 10−9	(3.0 ± 0.5) × 10−9	(1.5 ± 0.5) × 10−1
0.5	(2.0 ± 0.5) × 10−8	(6.5 ± 0.5) × 10−7	(2.5 ± 0.5) × 10−2
0.7	(3.5 ± 0.5) × 10−8	(2.5 ± 0.5) × 10−7	(9.0 ± 0.5) × 10−3
1.0	(4.5 ± 0.5) × 10−8	(1.5 ± 0.5) × 10−7	(3.0 ± 0.5) × 10−3
2.0	(1.0 ± 0.5) × 10−7	(2.5 ± 0.5) × 10−5	(3.0 ± 0.5) × 10−4
3.0	(1.5 ± 0.5) × 10−7	(2.5 ± 0.5) × 10−5	(4.5 ± 0.5) × 10−5
4.0	(2.5 ± 0.5) × 10−7	(1.5 ± 0.5) × 10−5	(1.5 ± 0.5) × 10−5
5.0	(3.0 ± 0.5) × 10−7	(1.5 ± 0.5) × 10−5	(1.5 ± 0.5) × 10−5
6.0	(3.5 ± 0.5) × 10−7	(1.5 ± 0.5) × 10−5	(1.5 ± 0.5) × 10−5

, for specific chosen values of M_N in the mass range of interest. We remark that the upper bounds have in some cases strong dependence on the precise values of M_N, see Ref. [29] for further details. In order to use only rough estimates for the values of $\tilde{\mathcal{K}}$, we present in

Table 2. Rough estimates of upper bounds for |B_ℓN|² (ℓ = e, µ, τ), for M_N in three different ranges around the values 0.25, 1, 3 GeV; and the inverse of the canonical decay length, ${\overline{L}}_N^{-1}$ (in units of m⁻¹ and for γ_N = 2).

MN [GeV]	|BeN|2	|BµN|2	|BτN|2	${\overline{L}}_N^{-1}[m^{-1}]$
≈ 0.25	10−8	10−7	10−4	0.11
≈ 1.0	10−7	10−7	10−2	1.1 × 102
≈ 3.0	10−6	10−4	10−4	3 × 104

order of magnitude values for upper bounds of |B_ℓN|². These rough upper bounds are given for three typical ranges of our interest: M_N around 0.25; 1; 3 GeV. They are relevant for the decays of K; (D, D_s); (B, B_c), respectively. The corresponding values of the inverse of the canonical decay length, ${\overline{L}}_N^{-1}$, are included. As seen in Tables 1 and 2, the upper bounds for |B_τN|² are at present significantly less stringent and are expected to become more stringent in the future. When we combine Equations (24b) with (27) and Table 2, we obtain for the decay-within-the-detector probability $P_N\equiv {\overline{P}}_N\tilde{\mathcal{K}}$ the following estimates and upper bounds, relevant for the K decays (M_N ≈ 0:25 GeV), D and D_s decays (M_N ≈ 1 GeV), and B and B_c decays (M_N ≈ 3 GeV), all when L = 1 m and γ_N = 2:

$$\begin{array}{c}{P}_N(M_N\approx 0.25\mathrm{Gev})\approx 1.7|B_{e{N}_j}{|}^2+0.9|B_{\mu N_j}{|}^2(+0.2|B_{\tau N_j}{|}^2)\\ \lesssim {10}^{-8}+{10}^{-7}(+{10}^{-5})\end{array}$$

(28a)

$$\begin{array}{c}{P}_N(M_N\approx 1\mathrm{Gev})\approx 0.8\cdot {10}^3|B_{e{N}_j}{|}^2+0.8\cdot {10}^3|B_{\mu N_j}{|}^2(+2\cdot {10}^2|B_{\tau N_j}{|}^2)\\ \lesssim {10}^{-4}+{10}^{-4}(+{10}^0)\end{array}$$

(28b)

$$\begin{array}{c}{P}_N(M_N\approx 3\mathrm{Gev})\approx 3\cdot {10}^5|B_{e{N}_j}{|}^2+3\cdot {10}^5|B_{\mu N_j}{|}^2(+1\cdot {10}^5|B_{\tau N_j}{|}^2)\\ \lesssim {10}^0+{10}^0(+{10}^0)\end{array}$$

(28c)

In order to have the analysis and the formulas simpler, in the rest of this Section we will assume that one mixing parameter, |B_ℓN|, dominates over the other two mixing parameters:

$$|B_{\ell N}|\gg |B_{{\ell }^{\prime }N}|({\ell }^{\prime }\ne \ell )$$

(29)

For example, it may well be that ℓ = µ, i.e., that |B_µN| ≫ |B_eN|, |B_τN|. Then, of the branching ratios $\mathrm{Br}(M^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }{M^{\prime }}^{\mp })$ the largest will be $M^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }{M^{\prime }}^{\mp }$ which, according to Equations (29) and (19) (note that DD* and CC* give the same contribution since ℓ₁ = ℓ₂ now), is:

$$\mathrm{Br}(M^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }{M^{\prime }}^{\mp })=4\frac{|B_{\ell N}{|}^4}{\tilde{\mathcal{K}}}\overline{\mathrm{Br}}$$

(30)

Multiplying this expression by the probability P_N of the decay in the detector, Equation (26), we obtain the effective branching ratio Br_eff

$${\mathrm{Br}}_{\mathrm{eff}}(M^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }{M^{\prime }}^{\mp })=P_N\mathrm{Br}(M^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }{M^{\prime }}^{\mp })=\left[{\overline{P}}_N\left(\frac{L}{1\mathrm{m}}\right)\tilde{\mathcal{K}}\right]\times \left[4\frac{|B_{\ell N}{|}^4}{\tilde{\mathcal{K}}}\overline{\mathrm{Br}}\right]$$

(31a)

$$=\left(\frac{L}{1\mathrm{m}}\right)4|B_{\ell N}{|}^4{\overline{P}}_N\overline{\mathrm{Br}}\equiv |B_{\ell N}{|}^4\left(\frac{L}{1\mathrm{m}}\right){\overline{\mathrm{Br}}}_{\mathrm{eff}}$$

(31b)

We see that in the effective branching ratio, Br_eff, the complicated dependence on mixing parameters encoded in $\tilde{\mathcal{K}}$[cf. Equation (18)] disappeared because factors $\tilde{\mathcal{K}}$ cancel here. All the mixing effects in Br_eff are in the simple factor |B_ℓN|⁴. Unfortunately, this factor represents a strong suppression, in comparison with Br of Equations (19) where $\mathrm{Br}\propto |k|{}^2/\tilde{\mathcal{K}}=|B_{{\ell }_{1}N}{B}_{{\ell }_{2}N}|{}^2/\tilde{\mathcal{K}}~|B_{\ell N}{|}^2$.

In the last identity (31b) we introduced the canonical (i.e., without any mixing dependence) effective branching ratio ${\overline{\mathrm{Br}}}_{\mathrm{eff}}$

$${\overline{\mathrm{Br}}}_{\mathrm{eff}}\equiv 4{\overline{P}}_N\overline{\mathrm{Br}}=4\left(\frac{1\mathrm{m}}{{\overline{L}}_N}\right)\overline{\mathrm{Br}}$$

(32)

where we recall that $\overline{\mathrm{Br}}$ was defined in Equations (19) and (20). Here $\overline{{\mathrm{Br}}_{\mathrm{eff}}}$ is half the value of $\overline{{\mathrm{Br}}_{\mathrm{eff}}}$ in Ref. [39] because the latter expression referred to the exchange of two Majorana neutrinos instead of one.

Only when M^± = B^± or $B_c^{\pm }$, i.e., when the mass of the on-shell N can be high (M_N ≳ 1 GeV), would it be possible to have P_N ∼ 1 [Equation (28c)]; and in such a case Equations (31) do not apply, but rather Equations (30), i.e., Br_eff = Br in this case. Figures 4–7 show the effective canonical branching ratios (32) as a function of the neutrino mass M_N, for various considered LNV decays of the type M^± → ℓ^±ℓ^±M′^∓: Figure 4 for M = K; Figure 5a, b for M = D, D_s, respectively; Figures 6a and 7a for M = B, B_c, respectively. We took ℓ = e, µ, and L = 1 m and γ_N = 2. For the case when P_N ∼ 1 and hence the estimates Equations (30) apply, Figures 6b and 7b present the branching ratios $\overline{\mathrm{Br}}(x)$ as a function of M_N, for M^± = B^± and $B_c^{\pm }$, respectively. For the meson decay constants and CKM matrix elements, needed for the evaluation of K² factor of Equation (9), and for the masses and lifetimes of the mesons, we used the values of Ref. [97]. The values of the decay constants f_B and $f_{B_c}$ were taken from Ref. [98]: f_B = 0.196 GeV, $f_{B_c}=0.322\mathrm{Gev}$. We note that the presented formulas for ${\overline{\mathrm{Br}}}_{\mathrm{eff}}$ and $\overline{\mathrm{Br}}$ can be evaluated also for the decays $M^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }{M^{\prime }}^{\mp }$ when ℓ₁ ≠ ℓ₂. Furthermore, when the final leptons are τ leptons (and M^± = B^± or $B_c^{\pm }$), the values of branching ratios turn out to be similar to those in Figures 6 and 7, but the range of M_N in this case is shorter: M_M′ +M_τ < M_N <M_M – M_τ.

Table 3. Values of the factor ${\overline{\mathrm{Br}}}_{\mathrm{eff}}$, Equation (32), with L = 1 m and γ_N = 2, for some of the LNV decays M^± → ℓ^±ℓ^±π^∓. The value of M_N is chosen such that the maximal value of ${\overline{\mathrm{Br}}}_{\mathrm{eff}}$ is obtained (the value of M_N is given in parentheses, in GeV). For M^± = K^±, two different values are given, for ℓ = e and ℓ = µ. For all other cases, ℓ = µ is taken (when ℓ = e the values are similar).

M±	K± (ℓ = e)	K± (ℓ = µ)	D±	$D_s^{\pm }$	B±	$B_c^{\pm }$
${\overline{\mathrm{Br}}}_{\mathrm{eff}}$	6.8 (0.38)	3.8 (0.35)	3.9 (1.39)	70. (1.47)	0.96 (3.9)	199. (4.7)

displays values of ${\overline{\mathrm{Br}}}_{\mathrm{eff}}$ for representative values of M_N in the decays M^± → ℓ^±ℓ^±M′^∓.

As an illustrative example, let us consider the decays $D_s^{\pm }\to {\mu }^{\pm }{\mu }^{\pm }{\pi }^{\pm }$, which is one of the preferred decay modes proposed at CERN-SPS [80,81], and, in addition, let us assume that |B_μN|² is the dominant mixing. In such a case, Equations (31) and Table 3 imply for the experimentally measurable branching fraction Br_eff

$${\mathrm{Br}}_{\mathrm{eff}}(D_s^{\pm }\to {\mu }^{\pm }{\mu }^{\pm }{\pi }^{\mp })\equiv P_N\mathrm{Br}(D_s^{\pm }\to {\mu }^{\pm }{\mu }^{\pm }{\pi }^{\mp })~{10}^2|B_{\mu N}{|}^4$$

(33)

The present rough upper bound on the mixing for M_N ≈ 1 GeV is |B_µN|² ≲ 10⁻⁷, cf. Table 2. Equation (33) then implies that Br^(eff) ≲ 10⁻¹² for such decays. The proposed experiment at CERN-SPS [80,81] could produce D and D_s mesons in numbers by several orders of magnitude higher than 10¹², which would open the possibility to explore whether there is a production of sterile Majorana neutrinos N in the mass range M_N ∼ 1 GeV.

If the decays $B_c^{\pm }\to {\mu }^{\pm }{\mu }^{\pm }{\pi }^{\mp }$ are considered (we do not use B^± decays as they are CKM-suppressed compared to $B_c^{\pm }$), the results of Figure 7a and Equation (31b) imply an effective branching ratio

$${\mathrm{Br}}_{\mathrm{eff}}(B_c^{\pm }\to {\mu }^{\pm }{\mu }^{\pm }{\pi }^{\mp })~{10}^2|B_{\mu N}{|}^4$$

(34)

which is similar to the case of D_s, Equation (33) in a detector of the same length (L = 1 m, in our example). Since D_s is significantly lighter than B_c, the relevant neutrinos that give a sizeable effect are also lighter and thus longer living, implying a smaller P_N factor for the same detector length. On the other hand, D_s decays have no CKM suppression compared to B_c (|V_cs| ≈ 1 while |V_cb| ≈ 0.04) and the B_c channels are more numerous, so that the true branching ratios of the latter are smaller. These two effects compensate in a given detector, as shown in Equations (33) and (34). However for a longer detector the observable D_s branching ratio increases considerably [80,81].

3. Charged Pion Decays Mediated by On-Shell Massive Neutrinos

In the previous Section we considered semileptonic decays of mesons heavier than the pion. For pion decays, there are purely leptonic modes only, since the pion is the lightest meson. In this Section we present and discuss the branching ratios for the LNC decay π^± → e^±N → e^±e^±µ^∓ν_e and the LNV decay ${\pi }^{\pm }\to e^{\pm }N\to e^{\pm }{e}^{\pm }{\mu }^{\mp }{\overline{\nu }}_{\mu }$. We also consider the differential branching ratios dBr/dE_µ, where E_µ is the energy of the produced µ^∓. In contrast to the previous Section where the intermediate N neutrino has to be Majorana, here N can be either Majorana or Dirac. In the Dirac case, only the LNC mode is possible, while both LNC and LNV modes occur in the Majorana case. However, the experimental distinction between these two modes cannot be resolved by simply examining the final state, since the produced neutrino (ν_e or ${\overline{\nu }}_{\mu }$) is not detectable. The major part of this Section refers to Ref. [37], and for certain details we use here results of Refs. [38,39]. The formalism is somewhat more complicated now because we have four final particles (in the previous Section there were three). Nonetheless, several features turn out to be similar as in the previous Section.

The considered processes are presented in Figures 8 and 9. As in the previous Section, we consider a scenario with at least one heavy sterile neutrino N, which has suppressed heavy-light mixing coefficients B_ℓN with the first three neutrino flavors ν_ℓ (ℓ = e, µ, τ), cf. Equation (1). The mentioned decay rates may be nonnegligible only if the intermediate neutrino N is on-shell, i.e.,

$$(M_{\mu }+M_e)<M_N<(M_{\pi }-M_e)$$

(35)

and the process is of the s-type, Figures 8 and 9. Specifically, this means 106.2 MeV < M_N < 139 MeV.

3.1. Branching Ratios for π^± → e^±e^±µ^∓ν

The decay widths Γ^(X)(π^± → e^±e^±µ^∓ν) (X=LNC, LNV) can be written in terms of the corresponding reduced decay amplitudes ${\mathcal{T}}_{\pi ,\pm }^{(\mathrm{X})}$

$${\mathrm{\Gamma }}^{(\mathrm{X})}({\pi }^{\pm }\to e^{\pm }{e}^{\pm }{\mu }^{\mp }\nu )=\frac{1}{2!}\frac{1}{2M_{\pi }}\frac{1}{{(2\pi )}^8}{\displaystyle \int d_4}|{\mathcal{T}}_{\pi ,\pm }^{(\mathrm{X})}{|}^2$$

(36)

Here, 1/2! represents the symmetry factor from two final state electrons, and d₄ is the integration element of the phase space of the four final particles

$$d_4=\left({\displaystyle \prod _{j=1}^2\frac{d^3{\overrightarrow{p}}_j}{2E_e({\overrightarrow{p}}_j)}}\right)\frac{d^3{\overrightarrow{p}}_{\mu }}{2E_{\mu }({\overrightarrow{p}}_{\mu })}\frac{d^3{\overrightarrow{p}}_{\nu }}{2|{\overrightarrow{p}}_{\nu }|}{\delta }^{(4)}(p_{\pi }-p_1-p_2-p_{\mu }-p_{\nu })$$

(37)

Here, p₁ and p₂ are the momenta of e; in the direct channel, the e momentum at the first (left-hand) vertex is p₁, and in the crossed channel it is p₂, cf. Figures 8 and 9. When we use the expressions for the amplitudes ${\mathcal{T}}_{\pi ,\pm }^{(\mathrm{X})}$ of Appendix A.4, for the specific considered case N₁ = N (and no N₂), the decay width (36) can be written as

$${\mathrm{\Gamma }}^{(\mathrm{X})}({\pi }^{\pm }\to e^{\pm }{e}^{\pm }{\mu }^{\mp }\nu )=|k^{(\mathrm{X})}{|}^2[{\tilde{\mathrm{\Gamma }}}_{\pi }^{(\mathrm{X})}(DD*)+{\tilde{\mathrm{\Gamma }}}_{\pi }^{(\mathrm{X})}(CC*)+{\tilde{\mathrm{\Gamma }}}_{\pi ,\pm }^{(\mathrm{X})}(DC*)+{\tilde{\mathrm{\Gamma }}}_{\pi ,\pm }^{(\mathrm{X})}(CD*]$$

(38)

where X = LNC, LNV; $k_{\pm }^{(\mathrm{X})}$ are the corresponding mixing factors

$$k^{(\mathrm{LNV})}=B_{eN}^2,k^{(\mathrm{LNC})}=B_{eN}{B}_{\mu N}^{*}$$

(39)

and ${\tilde{\mathrm{\Gamma }}}_{\pi }^{(\mathrm{X})}(YZ*)$ are the normalized (i.e., without explicit mixing dependence) decay widths

$${\tilde{\mathrm{\Gamma }}}_{\pi ,\pm }^{(\mathrm{X})}(YZ*)=K_{\pi }^2\frac{1}{2!}\frac{1}{2M_{\pi }}\frac{1}{{(2\pi )}^8}{\displaystyle \int d_4}{P}^{(\mathrm{X})}(Y)P^{(\mathrm{X})}(Z)*T_{\pi ,\pm }^{(\mathrm{X})}(YZ*)$$

(40)

The expressions for $T_{\pi ,\pm }^{(\mathrm{X})}(YZ*)$ are given in Appendix A.4, for the direct (YZ* = DD*), crossed (YZ* = CC*) and direct-crossed interference contributions (YZ* = DC*, CD*). We have $T_{\pi ,+}^{(\mathrm{X})}(DD*)=T_{\pi ,-}^{(\mathrm{X})}(DD*)$ and $T_{\pi ,+}^{(\mathrm{X})}(CC*)=T_{\pi ,-}^{(\mathrm{X})}(CC*)$, hence the terms ${\tilde{\mathrm{\Gamma }}}_{\pi }^{(\mathrm{X})}(DD*)$ and ${\tilde{\mathrm{\Gamma }}}_{\pi }^{(\mathrm{X})}(CC*)$ in Equation (38) have no subscripts ±. Further, in Equation (40)P^(X) (Y) represent the N propagator functions of the direct and crossed channels (Y = D,C)

$$P^{(\mathrm{LNC})}(D)=\frac{1}{[{(p_{\pi }-p_1)}^2-M_N^2+i{\mathrm{\Gamma }}_N{M}_N]},P^{(\mathrm{LNV})}(D)=M_N{P}^{(\mathrm{LNC})}(D)$$

(41a)

$$P^{(\mathrm{LNC})}(C)=\frac{1}{[{(p_{\pi }-p_2)}^2-M_N^2+i{\mathrm{\Gamma }}_N{M}_N]},P^{(\mathrm{LNV})}(C)=M_N{P}^{(\mathrm{LNC})}(C)$$

(41b)

where Γ_N ≡ Γ(N → all), and $K_{\pi }^2$ is the following constant:

$$K_{\pi }^2=G_F^4{f}_{\pi }^2|V_{ud}{|}^2\approx 2.983\times {10}^{-22}{\mathrm{Gev}}^{-6}$$

(42)

It turns out that in the case when the intermediate N neutrino is on-shell, i.e., when its mass is in the interval of Equation (35), the squares of the propagators (41) reduce to simple delta functions due to the inequality Γ_N ≪ M_N

$$\begin{array}{l}|P^{(\mathrm{LNC})}(\mathrm{X}){|}^2={\left|\frac{1}{{(p_{\pi }-p_k)}^2-M_N^2+i{\mathrm{\Gamma }}_N{M}_N}\right|}^2\\ =\frac{\pi }{M_N{\mathrm{\Gamma }}_N}\delta ({(p_{\pi }-p_k)}^2-M_N^2)({\mathrm{\Gamma }}_N\ll M_N)\end{array}$$

(43)

where p_k = p₁, p₂ for X = D, C. In this on-shell case, the DD* and CC* contributions in Equation (38) are large and equal, and the interference contributions DC* and CD* are negligible in comparison; we refer to [38] for details on this point. Hence the decay width (38) can be written in the on-shell case as

$${\mathrm{\Gamma }}^{(\mathrm{X})}({\pi }^{\pm }\to e^{\pm }{e}^{\pm }{\mu }^{\mp }\nu )=2|k^{(\mathrm{X})}{|}^2\tilde{\mathrm{\Gamma }}({\pi }^{\pm }\to e^{\pm }{e}^{\pm }{\mu }^{\mp }\nu )$$

(44)

Here, the normalized decay width ${\tilde{\mathrm{\Gamma }}}_{\pi }^{(\mathrm{X})}(DD*)={\tilde{\mathrm{\Gamma }}}_{\pi }^{(\mathrm{X})}(CC*)\equiv \tilde{\mathrm{\Gamma }}({\pi }^{\pm }\to e^{\pm }{e}^{\pm }{\mu }^{\mp }\nu )$ turns out to be the same for X = LNC and X = LNV,

$$\begin{array}{l}\tilde{\mathrm{\Gamma }}({\pi }^{\pm }\to e^{\pm }{e}^{\pm }{\mu }^{\mp }\nu )\equiv {\tilde{\mathrm{\Gamma }}}_{\pi }^{(\mathrm{X})}(DD*)\equiv {\tilde{\mathrm{\Gamma }}}_{\pi }(CC*)\\ =\frac{K_{\pi }^2}{192{(2\pi )}^4}\frac{M_N^{11}}{M_{\pi }^3{\mathrm{\Gamma }}_N}{\lambda }^{1/2}(x_{\pi },1,x_e)[x_{\pi }-1+x_e(x_{\pi }+2-x_e)]\mathcal{F}(x_{\mu },x_e)\end{array}$$

(45)

where the following notations are used:

$$\lambda (y_1,y_2,y_3)=y_1^2+y_2^2+y_3^2-2y_1{y}_2-2y_2{y}_3-2y_3{y}_1$$

(46a)

$$x_{\pi }=\frac{M_{\pi }^2}{M_N^2},x_e=\frac{M_e^2}{M_N^2},x_{\mu }=\frac{M_{\mu }^2}{M_N^2}$$

(46b)

and the function $\mathcal{F}(x_{\mu },x_e)$ is given in Appendix A.5. In the approximation M_e = 0, the above result becomes simpler

$$\underset{M_e\to 0}{\lim }\tilde{\mathrm{\Gamma }}({\pi }^{\pm }\to e^{\pm }{e}^{\pm }{\mu }^{\mp }\nu )=\frac{K_{\pi }^2}{192{(2\pi )}^4}\frac{M_N^{11}}{{\mathrm{\Gamma }}_N{M}_{\pi }^3}{(x_{\pi }-1)}^{2}f(x_{\mu })$$

(47)

where the function $f(x_{\mu })=\mathcal{F}(x_{\mu },0)$ is

$$f(x_{\mu })=1-8x_{\mu }+8x_{\mu }^3-x_{\mu }^4-12x_{\mu }^2\ln x_{\mu }$$

(48)

It can be checked that the decay rate (44), with N on shell, coincides with the factorized expression

$$\mathrm{\Gamma }({\pi }^{+}\to e^{+}{e}^{+}{\mu }^{-}\nu )=\mathrm{\Gamma }({\pi }^{+}\to e^{+}N)\mathrm{Br}(N\to e^{+}{\mu }^{-}\nu )$$

(49)

In Appendix A.5 we also provide the differential decay rates dΓ^(X)(π^± → e^±e^±µ^∓ν)/dE_µ with respect to the final muon energy in the rest frame of N neutrino. They turn out to have quite different forms for X = LNC and X = LNV cases (we will return to this later in this Section).

In order to obtain the branching fractions of the considered processes, we need to divide the decay width by the total decay width of the charged pion, Γ(π^± → all)

$$\mathrm{\Gamma }({\pi }^{+}\to \mathrm{all})=2.529\times {10}^{-17}\mathrm{Gev}$$

(50a)

$$\approx \frac{1}{8\pi }{G}_F^2{f}_{\pi }^2{M}_{\mu }^2{M}_{\pi }|V_{ud}{|}^2{\left(1-\frac{M_{\mu }^2}{M_{\pi }^2}\right)}^2(1+\delta g_{\pi })$$

(50b)

where the expression (50b) represents the by far most dominant decay mode π^± → µ^±ν_µ, and

$$\delta g_{\pi }=\frac{M_e^2}{M_{\mu }^2}\frac{{(1-M_e^2/M_{\pi }^2)}^2}{M_{\mu }^2{(1-M_{\mu }^2/M_{\pi }^2)}^2}$$

(51)

represents a (very small) relative correction coming from the π^± → e^±ν_e decay (δg_π. ≈ 1.3 × 10⁻⁴)

As we can see in Equations (44) and (45), the decay width Γ(π^± → e^±e^±µ^∓ν) are inversely proportional to the (very small) decay width Γ(N → all) ≡ Γ_N, which in turn is proportional to the mixings $\tilde{\mathcal{K}}~|B_{\ell N}{|}^2(\ell =e,\mu ,\tau )$, cf. Equations (16)–(18) and Figure 2. As in Section 2.1, this effect represents the N-on-shell effect Equation (43) and it makes the considered width Γ(π^± → e^±e^±μ^∓ν) by many orders of magnitude larger than it would be in the case of off-shell N. In the (narrow) mass interval (35) for on-shell N in the considered pion decays, the factor $\tilde{\mathcal{K}}$ in the width Γ_N Equation (16) has the following approximate form (cf. Appendix A.3 and Figure 2):

$$\tilde{\mathcal{K}}({\pi }^{\pm }\to e^{\pm }{e}^{\pm }{\mu }^{\mp }\nu )\approx 1.6|B_{eN}{|}^2+1.1(|B_{\mu N}{|}^2+|B_{\tau N}{|}^2)$$

(52)

This expression, within the precision given here, is valid equally for the Majorana and for the Dirac N and agrees with that given in Ref. [38] for the Majorana case. However, the affirmation in Ref. [38] that $\tilde{\mathcal{K}}$ for Dirac N is smaller by a factor of two is not correct.

In certain analogy with Section 2.1, we can define here the canonical branching ratio ${\overline{\mathrm{Br}}}_{\pi }$ as the part of the branching ratio ${\mathrm{Br}}_{\pi }^{(\mathrm{X})}\equiv \mathrm{\Gamma }({\pi }^{\pm }\to e^{\pm }{e}^{\pm }{\mu }^{\mp }\nu )/\mathrm{\Gamma }(\pi \to \mathrm{all})$ which contains no explicit or implicit heavy-light mixing dependence (and is independent of X = LNC or LNV)

$${\overline{\mathrm{Br}}}_{\pi }\equiv 2\frac{\tilde{\mathcal{K}}}{{\mathrm{\Gamma }}_{\pi }}\tilde{\mathrm{\Gamma }}({\pi }^{\pm }\to e^{\pm }{e}^{\pm }{\mu }^{\mp }\nu )=\frac{\tilde{\mathcal{K}}}{|k^{(\mathrm{X})}{|}^2}{\mathrm{Br}}^{(\mathrm{X})}({\pi }^{\pm }\to e^{\pm }{e}^{\pm }{\mu }^{\mp }\nu )$$

(53a)

$$=\frac{1}{16\pi }\frac{K_{\pi }^2{M}_{\pi }^3}{G_F^2\mathrm{\Gamma }({\pi }^{+}\to \mathrm{all})}\frac{1}{x_{\pi }^3}{\lambda }^{1/2}(x_{\pi },1,x_e)[x_{\pi }-1+x_e(x_{\pi }+2-x_e)]\mathcal{F}(x_{\mu },x_e)$$

(53b)

$$=\frac{1}{2}\frac{1}{x_{\mu }{(x_{\pi }-x_{\mu })}^2(1+\delta g_{\pi })}{\lambda }^{1/2}(x_{\pi },1,x_e)[x_{\pi }-1+x_e(x_{\pi }+2-x_e)]\mathcal{F}(x_{\mu },x_e)$$

(53c)

where the notations (46) are used, and in Equation (53c) we used the expression (125) in Appendix A.5. We note that $\tilde{\mathcal{K}}~|B_{\ell N}{|}^2$ and |k^(X)|² ∼ |B_ℓN|⁴, and ${\mathrm{Br}}_{\pi }^{(\mathrm{X})}\propto |k^{(\mathrm{X})}{|}^2/\tilde{\mathcal{K}}~|B_{\ell N}{|}^2$, where B_ℓN stands for a generic heavy-light mixing coefficient (|B_ℓN|~|B_eN|~|B_μN|). Naively, we should have very strong suppression ${\mathrm{Br}}_{\pi }^{(\mathrm{X})}\propto |k^{(X)}{|}^2\propto |B_{\ell N}{|}^4$; however, the on-shellness (43) brings in factor $1/{\mathrm{\Gamma }}_N\propto 1/\tilde{\mathcal{K}}~1/|B_{\ell N}{|}^2$, reducing the suppression to ${\mathrm{Br}}_{\pi }^{(\mathrm{X})}\propto |B_{\ell N}{|}^2$. In Figure 10 we present the canonical branching fraction ${\overline{\mathrm{Br}}}_{\pi }$ as a function of M_N in the entire interval of on-shellness (35). In Figure 11 this canonical branching fraction is presented at the lower edge of the on-shellness interval, where the effects of M_e ≠ 0 turn out to be appreciable. We can see that the branching ratio is the largest when M_N ≈ 0:130 GeV.

The differential branching ratios dBr^(X)/dE_µ, where E_µ is the final muon energy in the N-rest frame, are obtained directly from the differential branching ratios $d{\tilde{\mathrm{\Gamma }}}_{\pi }^{(\mathrm{X})}/d{E}_{\mu }$, and the latter quantity for X = LNV is given explicitly in Appendix A.5. The canonical differential branching ratios, free of any mixing dependence, can be defined in analogy with the definition of the canonical total branching ratio (53a), and, in contrast to canonical branching ratios (53) they do depend on whether X = LNC or X = LNV

$$\frac{d{\overline{\mathrm{Br}}}_{\pi }^{(\mathrm{X})}}{d{E}_{\mu }}\equiv 2\frac{\tilde{\mathcal{K}}}{{\mathrm{\Gamma }}_{\pi }}\frac{d\tilde{\mathrm{\Gamma }}({\pi }^{\pm }\to e^{\pm }{e}^{\pm }{\mu }^{\mp }\nu )}{d{E}_{\mu }}=\frac{\tilde{\mathcal{K}}}{|k^{(\mathrm{X})}{|}^2}\frac{d{\mathrm{Br}}^{(\mathrm{X})}({\pi }^{\pm }\to e^{\pm }{e}^{\pm }{\mu }^{\mp }\nu )}{d{E}_{\mu }}$$

(54)

Explicit expressions for these quantities, when X = LNV and X = LNC, are given in Appendix A.5 in Equations (126) and (128), and in the limit M_e = 0 in Equation (129).

The differential (and full) branching ratios for the process π^± → e^±e^±µ^∓ν differ in the cases when the on-shell N is Majorana or Dirac. When N is Dirac, only the X = LNC process contributes. When N is Majorana, both LNC and LNV processes contribute. We can write the differential and full branching ratios in these cases in terms of their canonical counterparts, by defining first the combined canonical differential branching ratios

$$\frac{d{\overline{\mathrm{Br}}}_{\pi }(\alpha )}{d{E}_{\mu }}\equiv \alpha \frac{d{\overline{\mathrm{Br}}}_{\pi }^{(\mathrm{LNV})}}{d{E}_{\mu }}+(1-\alpha )\frac{d{\overline{\mathrm{Br}}}_{\pi }^{(\mathrm{LNC})}}{d{E}_{\mu }}$$

(55)

where 0 ≤ α ≤ 1. Then it is straightforward to check that in the cases of Dirac and Majorana N neutrino, the branching ratios can be expressed in terms of the above quantites (55)

$$\frac{d{\mathrm{Br}}^{(\mathrm{Dir}.)}({\pi }^{\pm }\to e^{\pm }{e}^{\pm }{\mu }^{\mp }\nu )}{d{E}_{\mu }}=\frac{|k^{(\mathrm{LNC})}{|}^2}{{\tilde{\mathcal{K}}}^{(\mathrm{Dir}.)}}\frac{d{\overline{\mathrm{Br}}}_{\pi }(\alpha =0)}{d{E}_{\mu }}$$

(56a)

$$\frac{d{\mathrm{Br}}^{(\mathrm{Maj}.)}({\pi }^{\pm }\to e^{\pm }{e}^{\pm }{\mu }^{\mp }\nu )}{d{E}_{\mu }}=\frac{(|k^{(\mathrm{LNC})}{|}^2+|k^{(\mathrm{LNC})}{|}^2)}{{\tilde{\mathcal{K}}}^{(\mathrm{Maj}.)}}\frac{d{\overline{\mathrm{Br}}}_{\pi }({\alpha }_M)}{d{E}_{\mu }}$$

(56b)

where we recall the definition (39) of the coefficients k^(X), and the “Majorana LNV admixture” parameter α_M appearing in Equation (56b) is defined as

$${\alpha }_M=\frac{|k^{(\mathrm{LNV})}{|}^2}{(|k^{(\mathrm{LNC})}{|}^2+|k^{(\mathrm{LNC})}{|}^2)}=\frac{|B_{eN}{|}^2}{(|B_{eN}{|}^2+|B_{\mu N}{|}^2)}$$

(57)

Integration of the relations (56) over E_µ leads to the full branching ratios

$${\mathrm{Br}}^{(\mathrm{Dir}.)}({\pi }^{\pm }\to e^{\pm }{e}^{\pm }{\mu }^{\mp }\nu )=\frac{|k^{(\mathrm{LNC})}{|}^2}{{\tilde{\mathcal{K}}}^{(\mathrm{Dir}.)}}{\overline{\mathrm{Br}}}_{\pi }=\frac{|B_{eN}{|}^2|B_{\mu N}{|}^2}{{\displaystyle \sum {}_{\ell =e,\mu ,\tau }{\mathcal{N}}_{\ell N}^{(\mathrm{Dir}.)}|B_{\ell N}{|}^2}}{\overline{\mathrm{Br}}}_{\pi },$$

(58a)

$${\mathrm{Br}}^{(\mathrm{Maj}.)}({\pi }^{\pm }\to e^{\pm }{e}^{\pm }{\mu }^{\mp }\nu )=\frac{(|k^{(\mathrm{LNC})}{|}^2+|k^{(\mathrm{LNC})}{|}^2)}{{\tilde{\mathcal{K}}}^{(\mathrm{Maj}.)}}{\overline{\mathrm{Br}}}_{\pi }=\frac{|B_{eN}{|}^2(|B_{eN}{|}^2+|B_{\mu N}{|}^2)}{{\displaystyle \sum {}_{\ell =e,\mu ,\tau }{\mathcal{N}}_{\ell N}^{(\mathrm{Maj}.)}|B_{\ell N}{|}^2}}{\overline{\mathrm{Br}}}_{\pi }$$

(58b)

where we used the fact that the E_µ-integrated expression ${\overline{\mathrm{Br}}}_{\pi }(\alpha )$ is independent of α, cf. Equations (53). In Figures 12 we present these canonical differential branching ratios as a function of the muon energy E_µ in the rest frame of N, for four different values of mass M_N in the on-shell interval (35), and for five different values of the admixture parameter (α_M = 1, 0.8, 0.5, 0.2 and 0), The value α_M = 0 corresponds to the case of Dirac N. From the curves of Figure 12 we can conclude: if N is Majorana neutrino with a significant value of the admixture parameter α_M and with mass in the interval (35), then the measurement of such differential branching ratios may be able to confirm the Majorana nature of N.

If the differential branching ratios are studied with respect to the muon energy ${E^{\prime }}_{\mu }$ in the pion rest frame, the distinction between the Dirac and Majorana case is more difficult, cf. Ref. [37].

3.2. Effect of the Long Neutrino Lifetime on the Observability of π^± → e^±e^±µ^∓ν

The branching ratios presented in this Section so far, Equations (58) in conjunction with Equations (53), should be multiplied by the probability P_N for the on-shell neutrino N to decay within the detector (of length L), as explained in Section 2.2, Equations (22)–(26) and Figure 3. As a consequence, the effective (true, measurable) branching ratios are not Br (∼ |B_ℓN|²) of Equations (58), but Br_eff = P_NBr

$$\begin{array}{l}{\mathrm{Br}}_{\mathrm{eff}}^{(\mathrm{Dir}.)}({\pi }^{\pm }\to e^{\pm }{e}^{\pm }{\mu }^{\mp }\nu )=P_N^{(\mathrm{Dir}.)}{\mathrm{Br}}^{(\mathrm{Dir}.)}({\pi }^{\pm }\to e^{\pm }{e}^{\pm }{\mu }^{\mp }\nu )\\ =\left[{\overline{P}}_N\left(\frac{L}{1\mathrm{m}}\right){\tilde{\mathcal{K}}}^{(\mathrm{Dir}.)}\right]\left[\frac{|B_{eN}{|}^2|B_{\mu N}{|}^2}{{\tilde{\mathcal{K}}}^{(\mathrm{Dir}.)}}{\overline{\mathrm{Br}}}_{\pi }\right]\end{array}$$

(59a)

$$=|B_{eN}{|}^2|B_{\mu N}{|}^2\left(\frac{L}{1\mathrm{m}}\right){\overline{P}}_N{\overline{\mathrm{Br}}}_{\pi }=|B_{eN}{|}^2|B_{\mu N}{|}^2\left(\frac{L}{1\mathrm{m}}\right){\overline{\mathrm{Br}}}_{\pi ,\mathrm{eff}}$$

(59b)

$$\begin{array}{l}{\mathrm{Br}}_{\mathrm{eff}}^{(\mathrm{Maj}.)}({\pi }^{\pm }\to e^{\pm }{e}^{\pm }{\mu }^{\mp }\nu )=P_N^{(\mathrm{Maj}.)}{\mathrm{Br}}^{(\mathrm{Maj}.)}({\pi }^{\pm }\to e^{\pm }{e}^{\pm }{\mu }^{\mp }\nu )\\ =\left[{\overline{P}}_N\left(\frac{L}{1\mathrm{m}}\right){\tilde{\mathcal{K}}}^{(\mathrm{Maj}.)}\right]\left[\frac{|B_{eN}{|}^2(|B_{eN}{|}^2+|B_{\mu N}{|}^2)}{{\tilde{\mathcal{K}}}^{(\mathrm{Maj}.)}}{\overline{\mathrm{Br}}}_{\pi ,\mathrm{eff}}\right]\end{array}$$

(60a)

$$=|B_{eN}{|}^2(|B_{eN}{|}^2+|B_{\mu N}{|}^2)\left(\frac{L}{1\mathrm{m}}\right){\overline{P}}_N{\overline{\mathrm{Br}}}_{\pi }=|B_{eN}{|}^2(|B_{eN}{|}^2+|B_{\mu N}{|}^2)\left(\frac{L}{1\mathrm{m}}\right){\overline{\mathrm{Br}}}_{\pi ,\mathrm{eff}}$$

(60b)

In Equations (59a) and (60a) we used the expressions (58) for Br and Equation (26) for P_N. In Equations (59b) and (60b) we introduced canonical (i.e., without any mixing dependence) effective branching ratio ${\overline{\mathrm{Br}}}_{\pi ,\mathrm{eff}}$

$${\overline{\mathrm{Br}}}_{\pi ,\mathrm{eff}}\equiv {\overline{P}}_N{\overline{\mathrm{Br}}}_{\pi }$$

(61)

with ${\overline{\mathrm{Br}}}_{\pi }$ given in Equations (53), and the canonical nonsurvival probability ${\overline{P}}_N$ presented in Figure 3 in Section 2.2 for a wide range of N neutrino masses. In Figure 13 we present ${\overline{P}}_N$ in the here relevant narrower mass interval (35).

As in the case of semileptonic LNV decays of Section 2.2, we notice that also here in the effective branching ratios, Equations (59) and (60), the complicated dependence on the mixing parameters entailed by the factors $\tilde{\mathcal{K}}={\displaystyle {\sum }_{\ell }{\mathcal{N}}_{\ell N}|B_{\ell N}{|}^2}$ of Γ_N [cf. Equation (18)], cancels out, and there remains only simple dependence on the mixing parameters, in the form $|B_{eN}{|}^2|B_{\mu N}{|}^2$ or $|B_{eN}{|}^2(|B_{eN}{|}^2+|B_{\mu N}{|}^2)$|B |². Further, in comparison with the branching ratios Br of Equations (58), which are $~|B_{\ell N}{|}^2$, the effective (true) branching ratios Br_eff of Equations (59) and (60) are unfortunately significantly more suppressed by the mixing parameters, namely ${\mathrm{Br}}_{\mathrm{eff}}~|B_{\ell N}{|}^4$.

The presently known experimental bounds on the mixing parameters $|B_{lN}{|}^2(l=e,\mu ,\tau )$ in the here relevant narrow mass range (35), are: $|B_{eN}{|}^2\lesssim {10}^{-8}$[99]; $|B_{\mu N}{|}^2\lesssim {10}^{-6}$ [100–102]; $|B_{\tau N}{|}^2\lesssim {10}^{-4}$ [103]; cf. also Refs. [29,74,104,105].

The future pion factories, such as the Project X at Fermilab, will be designed to produce charged pions with lab energies E_π of a few GeV and luminosities ∼ 10²² cm⁻²s⁻¹ [_106,107], and ∼ 10²⁹ charged pions could be expected per year.

The canonical effective branching ratio (61) can be estimated as

$${\overline{\mathrm{Br}}}_{\pi ,}{}_{\mathrm{eff}}\lesssim 10^{-}{}^6$$

(62)

as can be inferred from Equation (61) and Figures 13 and 10. Equations (59) and (60) then imply that the effective (true) branching ratios for the considered reactions are (we assume L = 1m for the detector length)

$${\mathrm{Br}}_{\mathrm{eff}}^{(\mathrm{Dir}.)}\lesssim |B_{eN}{|}^2|B_{\mu N}{|}^2{10}^{-6},$$

(63a)

$${\mathrm{Br}}_{\mathrm{eff}}^{(\mathrm{Maj}.)}\lesssim |B_{eN}{|}^2(|B_{eN}{|}^2+|B_{\mu N}{|}^2)|{10}^{-6}$$

(63b)

If the larger among the mixing elements $(|B_{\ell N}{|}^2,\ell =e,\mu )$is $|B_{\mu N}{|}^2\lesssim {10}^{-6}$, the LNC processes dominate, and the effective branching ratios (63) have the common upper bounds

$${\mathrm{Br}}_{\mathrm{eff}}^{(\mathrm{Dir}.,\mathrm{Maj}.)}\lesssim |B_{eN}{|}^2|B_{\mu N}{|}^2{10}^{-6}\lesssim |B_{eN}{|}^2{10}^{-12}$$

(64)

If in this case $|B_{lN}{|}^2$ is close to its present upper bound, $|B_{lN}{|}^2~{10}^{-8}$, we obtain ${\mathrm{Br}}_{\mathrm{eff}}^{(\mathrm{Dir}.,\mathrm{Maj}.)}\lesssim 10^{-}{}^{20}$. This implies that up to 10⁹ events π^± → e^±e^±µ^∓ν could be detected per year in such a scenario.

On the other hand, if the larger among the mixing elements $(B_{\ell N}{|}^2,\ell =e,\mu )$ is $|B_{eN}{|}^2(\lesssim {10}^{-8})$, the LNV processes dominate, and the effective branching ratios (63) have the following upper bounds:

$${\mathrm{Br}}_{\mathrm{eff}}^{(\mathrm{Dir}.)}\lesssim |B_{\mu N}{|}^2{10}^{-14}$$

(65a)

$${\mathrm{Br}}_{\mathrm{eff}}^{(\mathrm{Maj}.)}\lesssim {10}^{-22}$$

(65b)

In such a case we have, $|B_{\mu N}{|}^2<|B_{eN}{|}^2\le {10}^{-8}$, and up to 10⁷ events could be detected per year.

The present upper bounds on $|B_{\ell N}{|}^2$ suggest that the first scenario, Equation (64), is more likely.

The measurement of the effective branching ratios alone cannot distinguish between the Dirac and the Majorana character of intermediate neutrino N. However, as argued in Section 3.1 and presented in Figure 12, the measurement of the differential branching ratios for the considered processes is a promising way to discern the character of the neutrinos. However, the differential branching ratios (56) must be multiplied by the nonsurvival probability P_N, in order to obtain the effective (true, measurable) differential branching ratios dBr_eff/dE_µ. In analogy with Equations (59) and (60) we obtain, using Equation (56)

$$\begin{array}{l}\frac{d{\mathrm{Br}}_{\mathrm{eff}}^{(\mathrm{Dir}.)}({\pi }^{\pm }\to e^{\pm }{e}^{\pm }{\mu }^{\mp }v)}{d{E}_{\mu }}=P_N\frac{d{\mathrm{Br}}^{(\mathrm{Dir}.)}({\pi }^{\pm }\to e^{\pm }{e}^{\pm }{\mu }^{\mp }v)}{d{E}_{\mu }}\\ =|B_{eN}{|}^2|B_{\mu N}{|}^2\left(\frac{L}{1m}\right)\frac{d{\overline{\mathrm{Br}}}_{\pi ,eff}(\alpha =0)}{d{E}_{\mu }}\end{array}$$

(66a)

$$\begin{array}{l}\frac{d{\mathrm{Br}}_{\mathrm{eff}}^{(\mathrm{Dir}.)}({\pi }^{\pm }\to e^{\pm }{e}^{\pm }{\mu }^{\mp }v)}{d{E}_{\mu }}=P_N\frac{d{\mathrm{Br}}^{(\mathrm{Dir}.)}({\pi }^{\pm }\to e^{\pm }{e}^{\pm }{\mu }^{\mp }v)}{d{E}_{\mu }}\\ =|B_{eN}{|}^2(|B_{eN}{|}^2+|B_{\mu N}{|}^2)\left(\frac{L}{1m}\right)\frac{d{\overline{\mathrm{Br}}}_{\pi ,eff}({\alpha }_M)}{d{E}_{\mu }}\end{array}$$

(66b)

where the canonical differential effective branching ratios were introduced, in analogy with Equation (55)

$$\frac{d{\overline{\mathrm{Br}}}_{\pi ,\mathrm{eff}}(\alpha )}{d{E}_{\mu }}={\overline{P}}_N\frac{d{\overline{\mathrm{Br}}}_{\pi }(\alpha )}{d{E}_{\mu }}$$

(67)

with $d{\overline{\mathrm{Br}}}_{\pi }(\alpha )/d{E}_{\mu }$ given in Equation (55). The parameter α_M appearing in Equation (66b) was defined in Equation (57). In analogy with Figure 12, and using the values of ${\overline{P}}_N$ from Figure 13, we deduce that the values of the y-axes of Figure 12a–d, must be multiplied by ${\overline{P}}_N\approx 2.0\times 10^{-}{}^3$, 2.8×10⁻³, 3.7×10⁻³, and 4.8 × 10⁻³, respectively (assuming L = 1 m and γ_N = 2), in order to obtain the representation for the curves of the canonical differential effective ratios $d{\overline{\mathrm{Br}}}_{\mathrm{\pi },\mathrm{eff}}(\alpha )/d{E}_{\mu }$.

4. CP Violation in Charged Meson Decays Mediated by Massive Sterile Neutrinos

CP violation in the lepton sector could be measured by neutrino oscillations [66]. Here we consider the possibilities of measuring CP violation in meson decays mediated by sterile neutrinos N, such as the semileptonic LNV decays of charged heavy pseudoscalar mesons considered in Section 2, or the leptonic (LNV and LNC) decays of charged pions considered in Section 3.

It turns out that CP violation in all such decays is possible in scenarios with at least two massive sterile neutrinos N_j (j = 1, 2). CP violation in the neutrino sector is expected whether neutrinos are Dirac or Majorana particles. However, in the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) mixing matrix [108–110], the number of possible CP-violating phases is larger when the neutrinos are Majorana particles. If $n=3+\mathcal{N}$ is the total number of neutrinos (N is the number of sterile neutrinos), the number of CP-violating phases is n(n − 1)/2 if neutrinos are Majorana, and (n − 1)(n − 2)/2 if neutrinos are Dirac, cf. Ref. [111].

CP violation in the decays ${\pi }^{\pm }\to e^{\pm }{N}_j\to e^{\pm }{e}^{\pm }{\mu }^{\mp }\nu$ was investigated in Ref. [38], and in the decays $M^{\pm }\to {\ell }_1{N}_j\to {\ell }_1{\ell }_2{M^{\prime }}^{\mp }$ in Refs. [39,40]. In both cases, it turns out that, even though the rates are extremely small, the CP violation asymmetry in these charged decays may become appreciable, even close to order unity, when the two intermediate neutrinos can go on shell and are almost degenerate in mass $M_{N_1}\approx M_{N_2}$. This is to be contrasted with CP violation in charged meson decays due to the standard CKM mechanism in the quarks sector, where the rates are larger but the asymmetries are much smaller, e.g., of order 10⁻⁴ in K^± → 3π decays [112].

4.1. CP Violation in Semileptonic LNV Decays $M^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }{M^{\prime }}^{\mp }$

As mentioned above, we will consider the scenario with at least two sterile neutrinos, N_j (j = 1, 2), both of which can go on shell in the intermediate state, i.e., with masses $M_{N_j}$ satisfying the condition shown in Equation (2). The processes of interest are again those of Figure 1, except that now in both the direct (D) and crossed (C) channels there are two possible neutrinos exchanged: N₁ or N₂. The relative measure of CP violation for these processes will be the asymmetry:

$${\mathcal{A}}_{CP}(M)\equiv \frac{\mathrm{\Gamma }(M^{-}\to {\ell }_1^{-}{\ell }_1^{-}{M^{\prime }}^{+})-\mathrm{\Gamma }(M^{+}\to {\ell }_1^{+}{\ell }_1^{+}{M^{\prime }}^{-})}{\mathrm{\Gamma }(M^{-}\to {\ell }_1^{-}{\ell }_1^{-}{M^{\prime }}^{+})+\mathrm{\Gamma }(M^{+}\to {\ell }_1^{+}{\ell }_1^{+}{M^{\prime }}^{-})}$$

(68)

The corresponding LNV decay widths $\mathrm{\Gamma }(M^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }{M^{\prime }}^{\mp })$ will be obtained now, in the scenario of two sterile neutrinos $(\mathcal{N}=2)$, in close analogy with the calculation in Section 2.1 which was performed for the case of one neutrino $N(\mathcal{N}=1)$. The relations (3) and (4) and (9) there hold without change now, but the relations (5)–(8) obtain the following slightly more general form when $\mathcal{N}=2$:

$$\begin{array}{l}\mathrm{\Gamma }(M^{\pm }\to {\ell }_1{\ell }_2{M^{\prime }}^{\mp })=\\ (2-{\delta }_{{\ell }_1{\ell }_2}){\displaystyle \sum _{i=1}^2{\displaystyle \sum _{j=1}^2{k}_i^{(\pm )}{k}_j^{(\pm )}{}^{*}[\tilde{\mathrm{\Gamma }}{(D{D}^{*})}_{ij}+\tilde{\mathrm{\Gamma }}{(C{C}^{*})}_{ij}+{\tilde{\mathrm{\Gamma }}}_{\pm }{(D{C}^{*})}_{ij}+{\tilde{\mathrm{\Gamma }}}_{\pm }{(C{D}^{*})}_{ij}]}}\end{array}$$

(69)

Here, $k_j^{(\pm )}$are the mixing coefficients

$$k_j^{(-)}=B_{{\ell }_1{N}_j}{B}_{{\ell }_2{N}_j},k_j^{(+)}={(k_j^{(-)})}^{*}$$

(70)

and ${\tilde{\mathrm{\Gamma }}}_{\pm }{(X{Y}^{*})}_{ij}$ are 2 × 2 matrices, and represent the normalized (i.e., without the explicit mixing dependence) contributions of N_i exchange in the X channel and complex-conjugate of the N_j exchange in the Y channel (X, Y = C, D)

$${\tilde{\mathrm{\Gamma }}}_{\pm }{\left(X{Y}^{*}\right)}_{ij}\equiv K^2\frac{1}{2!}\frac{1}{2M_M}\frac{1}{{\left(2\pi \right)}^5}{\displaystyle \int d_3{P}_i\left(X\right)P_j{(Y)}^{*}{M}_{N_i}{M}_{N_j}{T}_{\pm }\left(X\right)T_{\pm }{\left(Y\right)}^{*}}$$

(71)

The expressions for T_±(X)T_±(Y)^∗ (where X, Y = D, C) are the same as in Equation (7) and are given in Appendix A.1, and P_j(X) (X = D, C) are the propagators of the exchanged neutrinos N_j in the direct and crossed channels

$$P_j(D)=\frac{1}{\left[{\left(p_M-p_1\right)}^2-M_{N_j}^2+i{\mathrm{\Gamma }}_{N_j}{M}_{N_j}\right]}$$

(72a)

$$P_j(C)=\frac{1}{\left[{\left(p_M-p_2\right)}^2-M_{N_j}^2+i{\mathrm{\Gamma }}_{N_j}{M}_{N_j}\right]}$$

(72b)

We will disregard effects due to non-diagonal neutrino widths in their mass basis. For these details we refer to Ref. [40]. The total decay width of N_j, ${\mathrm{\Gamma }}_{N_j}$, is given by Equations (16)–(18), where each N_j has its own mixing parameter ${\tilde{\mathcal{K}}}_j$ Equation (18), i.e., ${\tilde{\mathcal{K}}}_j={{\displaystyle {\sum }_{\ell }{\mathcal{N}}_{\ell N_j}|B_{\ell N_j}|}}^2$, where the coefficients ${\mathcal{N}}_{\ell Nj}$ as a function of $M_{N_j}$ are given by the left-hand Figure 2 in Section 2.1.

As we will see below, the CP asymmetry parameter ${\mathcal{A}}_{\mathrm{CP}}(M)$, Equation (68), may acquire a significant nonzero value if simultaneously: (a) the phases ${\varphi }_{\ell j}$of the PMNS heavy-light mixing elements $B_{\ell Nj}=|B_{\ell Nj}|\exp (i{\varphi }_{\ell j})$ fulfill certain conditions: $|\sin {\theta }_{21}|\equiv |\sin ({\varphi }_{{\ell }_{1}2}+{\varphi }_{{\ell }_{2}2}-{\varphi }_{{\ell }_{1}1}-{\varphi }_{{\ell }_{2}1})|\cancel{\ll }1$; and (b) the mass difference $\mathrm{\Delta }{M}_N\equiv M_{N_2}-M_{N_1}$ is sufficiently small $(|\mathrm{\Delta }{M}_N|\cancel{\gg }{\mathrm{\Gamma }}_{N_j})$.

First let us calculate the quantities

$$S_{\mp }(M)\equiv \mathrm{\Gamma }(M^{-}\to {\ell }_1^{-}{\ell }_2^{-}{M^{\prime }}^{+})\mp \mathrm{\Gamma }(M^{+}\to {\ell }_1^{+}{\ell }_2^{+}{M^{\prime }}^{-})$$

(73)

appearing in the numerator and the denominator of the CP violation parameter ${\mathcal{A}}_{\mathrm{CP}}(M)$. Equation (68). We introduce the following notations which will be needed below:

$${\kappa }_{{\ell }_1}=\frac{|B_{{\ell }_1{N}_2}|}{|B_{{\ell }_1{N}_1}|},{\kappa }_{{\ell }_2}=\frac{|B_{{\ell }_2{N}_2}|}{|B_{{\ell }_2{N}_1}|}$$

(74a)

$$B_{{\ell }_{\kappa }{N}_j}\equiv |B_{{\ell }_{\kappa }{N}_j}|e^{i\varphi {\ell }_{\kappa }j}(\kappa ,j=1,2)$$

(74b)

$${\theta }_{ij}\equiv ({\varphi }_{{\ell }_{1}i}+{\varphi }_{{\ell }_{2}i}-{\varphi }_{{\ell }_{1}i}-{\varphi }_{{\ell }_{2}i})(i,j=1,2)$$

(74c)

For example, in the specific case when ${\ell }_1={\ell }_2=\mu$, we have ${\theta }_{21}=2({\varphi }_{\mu }{}_2-{\varphi }_{\mu }{}_1)=2(\arg (B_{\mu N_2})-\arg (B_{\mu N_1}))$. As in Section 2.1, when both N_j are on-shell, it turns out that the interference contributions to the quantities S_∓(M) from the direct (D) and crossed (C) channels (DC^∗ and CD^∗) are suppressed by several orders of magnitude in comparison with the contributions from the direct (DD^∗) and crossed (CC^*) channels, and we will neglect them (we refer to [39] for details on this point). Then it follows from the expression (69)

$$\begin{array}{l}{S}_{-}(M)\equiv \mathrm{\Gamma }(M^{-}\to {\ell }_1^{-}{\ell }_2^{-}{M^{\prime }}^{+})-\mathrm{\Gamma }(M^{+}\to {\ell }_1^{+}{\ell }_2^{+}{M^{\prime }}^{-})\\ =4(2-{\delta }_{{\ell }_1{\ell }_2})|B_{{\ell }_1{N}_1}||B_{{\ell }_2{N}_1}||B_{{\ell }_1{N}_2}||B_{{\ell }_2{N}_2}|\left\{\sin {\theta }_{21}\left[\mathrm{Im}\tilde{\mathrm{\Gamma }}{(D{D}^{*})}_{12}+\mathrm{Im}\tilde{\mathrm{\Gamma }}{(C{C}^{*})}_{12}\right]\right\}\end{array}$$

(75)

and

$$\begin{array}{l}{S}_{+}(M)\equiv \mathrm{\Gamma }(M^{-}\to {\ell }_1^{-}{\ell }_2^{-}{M^{\prime }}^{+})+\mathrm{\Gamma }(M^{+}\to {\ell }_1^{+}{\ell }_2^{+}{M^{\prime }}^{-})\\ =2(2-{\delta }_{{\ell }_1{\ell }_2})|B_{{\ell }_1{N}_1}{|}^2|B_{{\ell }_2{N}_1}{|}^2\{\tilde{\mathrm{\Gamma }}{(D{D}^{*})}_{11}[1+{\kappa }_{{\ell }_1}^2{\kappa }_{{\ell }_2}^2\frac{\tilde{\mathrm{\Gamma }}{(D{D}^{*})}_{22}}{\tilde{\mathrm{\Gamma }}{(D{D}^{*})}_{11}}+2{\kappa }_{{\ell }_1}{\kappa }_{{\ell }_2}\cos {\theta }_{21}{\delta }_1]\\ +\tilde{\mathrm{\Gamma }}{(C{C}^{*})}_{11}[1+{\kappa }_{{\ell }_1}^2{\kappa }_{{\ell }_2}^2\frac{\tilde{\mathrm{\Gamma }}{(C{C}^{*})}_{22}}{\tilde{\mathrm{\Gamma }}{(C{C}^{*})}_{11}}+2{\kappa }_{{\ell }_1}{\kappa }_{{\ell }_2}\cos {\theta }_{21}{\delta }_1]\}\end{array}$$

(76)

In the sum (76), the coefficient δ₁ represents the effect of N₁-N₂ overlap contributions

$${\delta }_j\equiv \frac{\mathrm{Re}\tilde{\mathrm{\Gamma }}{(X{X}^{*})}_{12}}{\tilde{\mathrm{\Gamma }}{(X{X}^{*})}_{jj}},(X=D;C;j=1;2)$$

(77)

We expect δ₁ ≈ 0 when $\mathrm{\Delta }{M}_N\gg {\mathrm{\Gamma }}_{N_j}$ (where: $\mathrm{\Delta }{M}_N\equiv M_N{}_{{}_2}-M_N{}_{{}_1}>0$); numerical calculations (see later) confirm this expectation and show that δ_j is practically independent of the channel X = D, C. The normalized decay widths $\tilde{\mathrm{\Gamma }}{(D{D}^{*})}_{jj}$and $\tilde{\mathrm{\Gamma }}{(C{C}^{*})}_{jj}$ are those of Equations (12) and (13) of Section 2.1, with the substitutions $M_N\mapsto M_{N_j}$, $y_N\mapsto y_{N_j}\equiv M_{N_j}^2/M_M^2$ [cf. Equation (14b)] and ${\mathrm{\Gamma }}_N\mapsto {\mathrm{\Gamma }}_{N_j}(={\tilde{\mathcal{K}}}_j{\overline{\mathrm{\Gamma }}}_N)$

$$\tilde{\mathrm{\Gamma }}{(D{D}^{*})}_{jj}=\frac{K^2{M}_M^5}{128{\pi }^2}\frac{M_{N_j}}{{\mathrm{\Gamma }}_{N_j}}{\lambda }^{1/2}(1,y_{N_j},y_{{\ell }_j}){\lambda }^{1/2}\left(1,\frac{y^{\prime }}{y_{N_j}},\frac{y_{{\ell }_2}}{y_{N_j}}\right)Q(y_{N_j};y_{{\ell }_j},y_{{\ell }_2},y^{\prime })$$

(78)

where j = 1 or j = 2; $\tilde{\mathrm{\Gamma }}{(C{C}^{*})}_{jj}$is obtained from $\tilde{\mathrm{\Gamma }}{(D{D}^{*})}_{jj}$ by the simple exchange $y_{{\ell }_1}\leftrightarrow y_{{\ell }_2}$[cf. Equation (13)].

For evaluation of the CP-violating difference S₋(M), Equation (75), the quantity $\mathrm{Im}\tilde{\mathrm{\Gamma }}{(X{X}^{\ast })}_{12}(X=D;C)$ is of central importance. In the integrand of $\mathrm{Im}\tilde{\mathrm{\Gamma }}{(X{X}^{\ast })}_{12}$ appears as a factor the following combination of the propagators of N₁ and N₂ [cf. Equation (71)]:

$$\mathrm{Im}(P_1(D)P_1{(D)}^{*})=\frac{(p_N^2-M_{N_1}^2){\mathrm{\Gamma }}_{N_2}{M}_{N_2}-{\mathrm{\Gamma }}_{N_1}{M}_{N_1}(p_N^2-M_{N_2}^2)}{[{(p_N^2-M_{N_1}^2)}^2+{\mathrm{\Gamma }}_{N_1}^2{M}_{N_1}^2][{(p_N^2-M_{N_2}^2)}^2+{\mathrm{\Gamma }}_{N_2}^2{M}_{N_2}^2]}$$

(79a)

$$=\eta \times \frac{\pi }{M_{N_2}^2-M_{N_1}^2}\left[\delta (p_N^2-M_{N_2}^2)+\delta (p_N^2-M_{N_1}^2)\right]$$

(79b)

In Equation (79b) we used the narrow width approximation:

$$\frac{{\mathrm{\Gamma }}_{N_j}{M}_{N_j}}{\left[{\left(p_N^2-M_{N_j}^2\right)}^2+{\mathrm{\Gamma }}_{N_j}^2{M}_{N_j}^2\right]}=\pi \delta (p_N^2-M_{N_j}^2)(for{\mathrm{\Gamma }}_{N_j}\ll M_{N_j})$$

(80)

and in Equation (79b) the parameter η was introduced which parametrizes any deviation from the naive expectation η = 1. We expect η ≈ 1 when $\mathrm{\Delta }{M}_N^2\gg {\mathrm{\Gamma }}_{N_1},{\mathrm{\Gamma }}_{N_2}$, where $\mathrm{\Delta }{M}_N^2\equiv M_{N_2}^2-M_{N_1}^2>0$. In Appendix A.6 we argue that this parameter η, in the case of near degeneracy $\mathrm{\Delta }{M}_N\ll M_{N_1}$, is a simple function of only one variable y ≡ΔM/Γ_N where $\mathrm{\Delta }{M}_N\equiv M_{N_2}-M_{N_1}>0$ and ${\mathrm{\Gamma }}_N\equiv (1/2)({\mathrm{\Gamma }}_{N_1}+{\mathrm{\Gamma }}_{N_2})$

$$\eta (y)=\frac{y^2}{(y^2+1)}{|}_{y=\mathrm{\Delta }{M}_N/{\mathrm{\Gamma }}_N}$$

(81)

when $\mathrm{\Delta }{M}_N\ll M_{N_1}\equiv M_N(\Rightarrow \mathrm{\Delta }{M}_N^2=2M_N\mathrm{\Delta }{M}_N)$, and where

$${\mathrm{\Gamma }}_N\equiv \frac{1}{2}({\mathrm{\Gamma }}_{N_1}+{\mathrm{\Gamma }}_{N_2}),y\equiv \frac{\mathrm{\Delta }{M}_N}{{\mathrm{\Gamma }}_N}$$

(82)

This implies that in the case of two almost degenerate sterile neutrinos $(\mathrm{\Delta }{M}_N\ll M_{N_1})$ we have for the factor in Equation (79b) the following identities:

$$\eta \times \frac{1}{\mathrm{\Delta }{M}_N^2}=\frac{1}{2M_N{\mathrm{\Gamma }}_N}\frac{\eta (y)}{y}=\frac{\mathrm{\Delta }{M}_N^2}{{(\mathrm{\Delta }{M}_N^2)}^2+4M_N^2{\mathrm{\Gamma }}_N^2}$$

(83)

We note that the factor 4 in the denominator on the right-hand side of Equation (83) is nontrivial, because a somewhat different result would have been obtained by a more simple and direct consideration of the expression (79a) in the limit ${\mathrm{\Gamma }}_{N_j}\ll M_{N_j}(j=1,2)$. The result (81) [or equivalently, Equation (83)] has been confirmed also by numerical evaluation of $\mathrm{Im}\tilde{\mathrm{\Gamma }}{(X{X}^{*})}_{12}$, Refs. [38,39] (see below). The mechanism (79) [with the identity (83)] is of central importance for the CP violation in the processes considered here. The quantity $\eta /\mathrm{\Delta }{M}_N^2$, at fixed Γ_N and fixed $M_N\equiv M_{N_1}$, achieves its maximum when $\mathrm{\Delta }{M}_N^2=2M_N{\mathrm{\Gamma }}_N$, i.e., y = 1 [⇔ ΔM_N = Γ_N (≪ M_N)], i.e., when the two sterile neutrinos are almost degenerate. If ΔM_N ≁Γ_N (i.e., y ≁1), then the quantity η(y)/y = y/(y² + 1) in Equation (83) is very small and CP violation effects disappear.

The mechanism (79) was used in Ref. [38] in the context of CP violation of leptonic decays of charged pions, and in Refs. [39,40] in the here presented context of CP violation of semileptonic LNV decays of heavy pseudoscalars.

We recall that the expression (79) has the same structure with Dirac delta functions as Equation (10) in Section 2.1; however, the factors in front of these Dirac delta functions are different now. Therefore, integration over the final particles’ phase space can be performed now in the same way as in Section 2.1, i.e., analytically. This leads, in analogy with Equation (12), to the result

$$\begin{array}{l}\mathrm{Im}\tilde{\mathrm{\Gamma }}{(DD*)}_{12}=\eta (y)\times \frac{1}{\mathrm{\Delta }{M}_N^2}\frac{K^2{M}_M^5}{128{\pi }^2}{M}_{N_1}{M}_{N_2}\\ \times {\displaystyle \sum _{j=1}^2{\lambda }^{1/2}(1,y_{N_j},y_{{\ell }_1})}{\lambda }^{1/2}\left(1,\frac{y^{\prime }}{y_{N_j}},\frac{y_{{\ell }_2}}{y_{N_j}}\right)Q(y_{N_j};y_{{\ell }_1},y_{{\ell }_2},y^{\prime })\end{array}$$

(84a)

$$\mathrm{Im}\tilde{\mathrm{\Gamma }}{(CC*)}_{12}=\mathrm{Im}\tilde{\mathrm{\Gamma }}{(DD*)}_{12}(y_{{\ell }_1}\leftrightarrow y_{{\ell }_2})$$

(84b)

where we recall the notations Equations (14) and (82), $y_{N_j}\equiv M_{N_j}^2/M_M^2$, the function Q is presented in Appendix A.2, and we denoted $\mathrm{\Delta }{M}_N\equiv M_{N_2}-M_{N_1}>0$ and $\mathrm{\Delta }{M}_N^2\equiv M_{N_2}^2-M_{N_1}^2>0$. We note that in Equation (84) we have not yet assumed the near degeneracy of the two sterile neutrinos.

From here on in this Section, we will consider the case of near degeneracy of the two on-shell sterile neutrinos (ΔM_N ≪ M_N, where $M_N\equiv M_{N_1}$), in which case Equations (81) and (82) hold and the quantities Equations (83) and (84) become appreciable and CP violation can thus become significant. Therefore, we have

$$y_{N_2}\approx y_{N_1}\equiv y_N\equiv \frac{M_N^2}{M_M^2}$$

(85)

where $M_N\equiv M_{N_1}\approx M_{N_2}$. In this case the identity (81) holds, cf. Appendix A.6, and the expression (84a) becomes simpler

$$\mathrm{Im}\tilde{\mathrm{\Gamma }}{(DD*)}_{12}=\frac{\eta (y)}{y}\times \frac{K^2{M}_M^5{M}_N}{128{\pi }^2{\mathrm{\Gamma }}_N}{\lambda }^{1/2}(1,y_N,y_{{\ell }_1}){\lambda }^{1/2}\left(1,\frac{y^{\prime }}{y_N},\frac{y_{{\ell }_2}}{y_N}\right)Q(y_N;y_{{\ell }_1},y_{{\ell }_2},y^{\prime })$$

(86a)

$$=\frac{\eta (y)}{y}\tilde{\mathrm{\Gamma }}{(DD*)}_{11}\frac{2{\tilde{\mathcal{K}}}_1}{({\tilde{\mathcal{K}}}_1+{\tilde{\mathcal{K}}}_2)}$$

(86b)

where in the last identity we used the expression (12), the identity (82), and the fact that ${\mathrm{\Gamma }}_{N_2}/{\mathrm{\Gamma }}_{N_1}={\tilde{\mathcal{K}}}_2/{\tilde{\mathcal{K}}}_1$ [cf. Equation (18) for N₁ and N₂].

The normalized decay matrix elements ${\tilde{\mathrm{\Gamma }}}_{\pm }{(XY*)}_{ij}$ Equation (71), were evaluated in Ref. [39] also numerically, by Monte Carlo integration and using finite small widths ${\mathrm{\Gamma }}_{N_j}$ in the propagators. The numerical calculations confirmed the presented formulas, among them the expressions (78), (86), and (81)–(83). The form (81) of η(y) was confirmed numerically with a precision better than a few per mille. The numerical evaluations also confirmed that the direct-crossed interference terms (DC* and CD*) are really negligible. We refer to [39] for details.

Further, the mentioned numerical evaluations gave us values of the N₁-N₂ overlap parameter δ₁ as defined in Equation (77), i.e., the parameter which appears in the expression (76) and represents the N₁-N₂ overlap effects. It turned out that the numerical values of the parameters δ_j (j = 1, 2), as well as of η, are practically independent of: the channel contribution considered (DD* or CC*), of the type of pseudoscalar mesons (M^±, M′^∓), and of the light leptons (ℓ₁, ℓ₂ = e, μ) involved in the considered decays. The numerical results show that the parameter δ ≡ (1/2)(δ₁ + δ₂) is a function of only one variable, namely $y\equiv \mathrm{\Delta }{M}_N/{\mathrm{\Gamma }}_N$ (the same is true for η)

$$\delta =\delta (y),\delta \equiv \frac{1}{2}({\delta }_1+{\delta }_2)$$

(87a)

$$\frac{{\delta }_1}{{\delta }_2}=\frac{\tilde{\mathrm{\Gamma }}{(DD*)}_{22}}{\tilde{\mathrm{\Gamma }}{(DD*)}_{11}}=\frac{{\tilde{\mathrm{\Gamma }}}_{N_1}}{{\tilde{\mathrm{\Gamma }}}_{N_2}}=\frac{{\tilde{\mathcal{K}}}_1}{{\tilde{\mathcal{K}}}_2}$$

(87b)

The numerical values of the parameter δ are given in

Table 4. Values of the N₁-N₂ overlap parameter δ(y) as a function of y ≡ ΔM_N/Γ_N.

$y\equiv \frac{\mathrm{\Delta }{M}_N}{{\mathrm{\Gamma }}_N}$	log10 y	δ(y)
0.10	−1.000	0.989 ± 0.001
0.30	−0.523	0.917 ± 0.001
0.50	−0.301	0.800 ± 0.001
0.70	−0.155	0.673 ± 0.001
0.80	−0.097	0.610 ± 0.001
0.90	−0.046	0.551 ± 0.002
1.00	0.000	0.499 ± 0.002
1.25	0.097	0.390 ± 0.003
1.67	0.222	0.264 ± 0.003
2.50	0.398	0.138 ± 0.001
5.00	0.699	0.038 ± 0.001
10.0	1.000	0.0098 ± 0.0010

as a function of y. It is not clear whether there exists a simple analytic expression for δ as a function of y. Further, in Figure 14 we present the quantities η/y and δ as a function of y. The values for δ in Table 4 are practically equal to the values of the corresponding δ parameter in the rare leptonic decays of the charged pions π^± → e^±N → e^±e^±µ^∓ν, cf. next Section 4.1 and Ref. [38].

Branching ratios of experimental significance here can be defined by dividing the expressions S_∓(M), Equations (73) and (75) and (76), by the corresponding sum of total decay widths [Γ(M⁺ → all) + Γ(M⁻ → all)], which is practically equal to 2Γ(M^± → all)

$${\mathrm{Br}}_{+}(M)\equiv \frac{S_{+}(M)}{2\mathrm{\Gamma }(M^{\pm }\to \mathrm{all})}$$

(88a)

$${\mathrm{Br}}_{-}(M)\equiv {\mathcal{A}}_{\mathrm{CP}}(M){\mathrm{Br}}_{+}(M)=\frac{S_{-}(M)}{2\mathrm{\Gamma }(M^{\pm }\to \mathrm{all})}$$

(88b)

If employing the canonical (independent of mixing) branching ratio $\overline{\mathrm{Br}}(y_N;y_{{\ell }_1},y_{{\ell }_2},y^{\prime })\equiv \overline{\mathrm{Br}}(DD*)$ defined in Equation (20) of Section 2.1, and its CC^* analog, $\overline{\mathrm{Br}}(CC*)\equiv \overline{\mathrm{Br}}(y_N;y_{{\ell }_2},y_{{\ell }_1},y^{\prime })$, then it turns out that the branching ratios Br_±(M) of Equation (88) can be rewritten in terms of $\overline{\mathrm{Br}}$, of the heavy-light mixing parameters $|B_{\ell N_j}|$, and ${\tilde{\mathcal{K}}}_j={\displaystyle {\sum }_{\ell }{\mathcal{N}}_{\ell N_j}|B_{\ell N_j}{|}^2}$ of function η(y)/y = y/(y² + 1) and the overlap function δ(y) tabulated in Table 4

$$\begin{array}{l}{\mathrm{Br}}_{+}(M)\equiv \frac{\mathrm{\Gamma }(M^{-}\to {\ell }_1^{-}{\ell }_2^{-}{M^{\prime }}^{+})+\mathrm{\Gamma }(M^{+}\to {\ell }_1^{+}{\ell }_2^{+}{M^{\prime }}^{-})}{2\mathrm{\Gamma }(M^{\pm }\to \mathrm{all})}=2(2-{\delta }_{{\ell }_1{\ell }_2})[{\displaystyle \sum _{j=1}^2\frac{B_{{\ell }_1{N}_j}{|}^2|B_{{\ell }_2{N}_j}{|}^2}{{\tilde{\mathcal{K}}}_j}}\\ +4\delta (y)\frac{|B_{{\ell }_1{N}_2}||B_{{\ell }_2{N}_1}||B_{{\ell }_1{N}_2}||B_{{\ell }_2{N}_2}|}{{(\tilde{\mathcal{K}}}_1+{\tilde{\mathcal{K}}}_2)}\cos {\theta }_{21}](\overline{\mathrm{Br}}(CC*))\end{array}$$

(89a)

$$\begin{array}{l}{\mathrm{Br}}_{-}(M)\equiv {\mathcal{A}}_{\mathrm{CP}}(M){\mathrm{Br}}_{+}(M)\equiv \frac{\mathrm{\Gamma }(M^{-}\to {\ell }_1^{-}{\ell }_2^{-}{M^{\prime }}^{+})-\mathrm{\Gamma }(M^{+}\to {\ell }_1^{+}{\ell }_2^{+}{M^{\prime }}^{-})}{2\mathrm{\Gamma }(M^{\pm }\to \mathrm{all})}\\ =8(2-{\delta }_{{\ell }_1{\ell }_2})\frac{|B_{{\ell }_1{N}_2}||B_{{\ell }_2{N}_1}||B_{{\ell }_1{N}_2}||B_{{\ell }_2{N}_2}|}{{(\tilde{\mathcal{K}}}_1+{\tilde{\mathcal{K}}}_2)}\sin {\theta }_{21}\frac{\eta (y)}{y}(\overline{\mathrm{Br}}(DD*)+\overline{\mathrm{Br}}(CC*))\end{array}$$

(89b)

This leads to an expression for the CP violation parameter ${\mathcal{A}}_{\mathrm{CP}}(M)$ defined in Equation (68), which now involves only the heavy-light mixing parameters $|B_{\ell N_j}|$ and ${\tilde{\mathcal{K}}}_j$ [cf. Equation (18)], the function η(y)/y = y/(y² + 1), where y ≡ ΔM_N/Γ_N, and the overlap function δ(y) tabulated in Table 4:

$$\begin{array}{l}{\mathcal{A}}_{\mathrm{CP}}(M)\equiv \frac{{\mathrm{Br}}_{-}(M)}{{\mathrm{Br}}_{+}(M)}\equiv \frac{\mathrm{\Gamma }(M^{-}\to {\ell }_1^{-}{\ell }_2^{-}{M^{\prime }}^{+})-\mathrm{\Gamma }(M^{+}\to {\ell }_1^{+}{\ell }_2^{+}{M^{\prime }}^{-})}{\mathrm{\Gamma }(M^{-}\to {\ell }_1^{-}{\ell }_2^{-}{M^{\prime }}^{+})+\mathrm{\Gamma }(M^{+}\to {\ell }_1^{+}{\ell }_2^{+}{M^{\prime }}^{-})}\\ =\frac{4\sin {\theta }_{21}}{\left[{\displaystyle {\sum }_{j=1}^2\frac{|B_{{\ell }_1{N}_j}{|}^2|B_{{\ell }_2{N}_j}{|}^2}{|B_{{\ell }_1{N}_1}||B_{{\ell }_2{N}_1}||B_{{\ell }_1{N}_2}||B_{{\ell }_2{N}_2}|}\frac{({\tilde{\mathcal{K}}}_1+{\tilde{\mathcal{K}}}_2)}{{\tilde{\mathcal{K}}}_j}}+4\delta (y)\cos {\theta }_{21}\right]}\frac{y}{(y^2+1)}\end{array}$$

(90a)

$$=\frac{4\sin {\theta }_{21}}{\left\{\left[{\kappa }_{{\ell }_1}{\kappa }_{{\ell }_2}\left(1+\frac{{\tilde{\mathcal{K}}}_1}{{\tilde{\mathcal{K}}}_2}\right)+\frac{1}{{\kappa }_{{\ell }_1}{\kappa }_{{\ell }_2}}\left(1+\frac{{\tilde{\mathcal{K}}}_2}{{\tilde{\mathcal{K}}}_1}\right)\right]+4\delta (y)\cos {\theta }_{21}\right\}}\frac{y}{\left(y^2+1\right)}$$

(90b)

In the usually considered case ℓ₁ = ℓ₂ (≡ℓ), i.e., when the considered decays are M^± → ℓ^±ℓ^±M′^∓, the formulas (89) and (90) get simpler, because in such a case $\overline{\mathrm{Br}}(CC*)=\overline{\mathrm{Br}}(DD*)\equiv \overline{\mathrm{Br}}$, and $B_{{\ell }_2{N}_j}=B_{{\ell }_1{N}_j}\equiv B_{\ell N_j}$, and ${\kappa }_{{\ell }_2}={\kappa }_{{\ell }_1}={\kappa }_{\ell }$

$${\mathrm{Br}}_{+}(M)=4\left[{\displaystyle \sum _{j=1}^2\frac{|B_{\ell N_j}{|}^4}{{\tilde{\mathcal{K}}}_j}+4\delta (y)\frac{|B_{\ell N_1}{|}^2|B_{\ell N_2}{|}^2}{({\tilde{\mathcal{K}}}_1+{\tilde{\mathcal{K}}}_2)}\cos {\theta }_{21}}\right]\overline{\mathrm{Br}}$$

(91a)

$${\mathrm{Br}}_{-}(M)\equiv {\mathcal{A}}_{\mathrm{CP}}(M){\mathrm{Br}}_{+}(M)=16\frac{|B_{\ell N_1}{|}^2|B_{\ell N_2}{|}^2}{({\tilde{\mathcal{K}}}_1+{\tilde{\mathcal{K}}}_2)}\cos {\theta }_{21}\frac{\eta (y)}{y}\overline{\mathrm{Br}}$$

(91b)

$${\mathcal{A}}_{\mathrm{CP}}(M)=\frac{4\sin {\theta }_{21}}{\left\{\left[{\kappa }_{\ell }^2\left(1+\frac{{\tilde{\mathcal{K}}}_1}{{\tilde{\mathcal{K}}}_2}\right)+\frac{1}{{\kappa }_{\ell }^2}\left(1+\frac{{\tilde{\mathcal{K}}}_2}{{\tilde{\mathcal{K}}}_1}\right)\right]+4\delta (y)\cos {\theta }_{21}\right\}}\frac{y}{(y^2+1)}$$

(91c)

These formulas become even simpler if the absolute values of the heavy-light mixings of N₁ and N₂ are equal (but not their phases), i.e., when

$$|B_{{\ell }^{\prime }{N}_2}{|}^2\approx |B_{{\ell }^{\prime }{N}_1}{|}^2\equiv |B_{{\ell }^{\prime }N}{|}^2({\ell }^{\prime }=e,\mu ,\tau )$$

(92)

In such a case, we have all ${\kappa }_{{\ell }^{\prime }}\approx 1$, and ${\tilde{\mathcal{K}}}_2\approx {\tilde{\mathcal{K}}}_1\equiv \tilde{\mathcal{K}}$ and therefore the expressions (91) reduce to

$${\mathrm{Br}}_{+}(M)\approx 8\frac{|B_{\ell N}{|}^4}{\tilde{\mathcal{K}}}\overline{\mathrm{Br}}(1+\mathcal{O}(\delta ))$$

(93a)

$${\mathrm{Br}}_{-}(M)\equiv {\mathcal{A}}_{\mathrm{CP}}(M){\mathrm{Br}}_{+}(M)\approx 8\frac{|B_{\ell N}{|}^4}{\tilde{\mathcal{K}}}\sin {\theta }_{21}\frac{\eta (y)}{y}\overline{\mathrm{Br}}$$

(93b)

$${\mathcal{A}}_{\mathrm{CP}}(M)\approx \sin {\theta }_{21}\frac{y}{y^2+1}(1+\mathcal{O}(\delta )\le \frac{1}{2}\sin {\theta }_{21}(1+\mathcal{O}(\delta ))$$

(93c)

In these expressions, we assumed, in addition, that the N₁-N₂ overlap terms are small $(\mathcal{O}(\delta ))$.

In order to obtain the corresponding effective (true) branching ratios, we have to multiply the above branching ratios Br_± by the decay-within-the-detector probability $P_{N_j}\equiv {\overline{P}}_N{\tilde{\mathcal{K}}}_j(L/1\mathrm{m})$, as in Section 2.2 (L is the length of the detector). Again, the complicated mixing dependence entailed in the parameters ${\tilde{\mathcal{K}}}_j$ gets cancelled in this multiplication. When adopting the simplifying assumption Equation (92), i.e., the validity of Equations (93), we obtain the following effective branching ratio Br_eff and the CP violation effective branching ratio ${\mathcal{A}}_{\mathrm{CP}}{\mathrm{Br}}_{\mathrm{eff}}$:

$$\begin{array}{l}{\mathrm{Br}}_{\mathrm{eff}}(M^{\pm }\to {\ell }^{\pm }{\ell }^{\pm }{M^{\prime }}^{\mp })=P_N{\mathrm{Br}}_{+}(M)\approx \left[{\overline{P}}_N\left(\frac{L}{1\mathrm{m}}\right)\tilde{\mathcal{K}}\right]\left[8\frac{|B_{\ell N}{|}^4}{\tilde{\mathcal{K}}}\overline{\mathrm{Br}}\right]\\ =\left(\frac{L}{1\mathrm{m}}\right)8|B_{\ell N}{|}^4{\overline{P}}_{N}Br\equiv 2|B_{\ell N}{|}^4\left(\frac{L}{1\mathrm{m}}\right){\mathrm{Br}}_{\mathrm{eff}}\end{array}$$

(94a)

$$\begin{array}{l}{\mathcal{A}}_{\mathrm{CP}}{\mathrm{Br}}_{\mathrm{eff}}(M^{\pm }\to {\ell }^{\pm }{\ell }^{\pm }{M^{\prime }}^{\mp })=\sin {\theta }_{21}\frac{y}{y^2+1}\times 2|B_{\ell N}{|}^4\left(\frac{L}{1m}\right){\overline{\mathrm{Br}}}_{\mathrm{eff}}\\ \lesssim |B_{\ell N}{|}^4\left(\frac{L}{1m}\right)\sin {\theta }_{21}{\overline{\mathrm{Br}}}_{\mathrm{eff}}\end{array}$$

(94b)

In these formulas, we used the canonical effective branching ratio ${\overline{\mathrm{Br}}}_{\mathrm{eff}}$ as defined via Equations (32) and (20) and depicted in Figures 4–7 as a function of $M_N(=M_{N_1}\approx M_{N_2})$. The values of the Lorentz factors in the lab system are taken to be γ_N = 2 for both N₁ and N₂, keeping in mind that ${\overline{\mathrm{Br}}}_{\mathrm{eff}}$ scales as 1/γ_N. We recall that ${\theta }_{21}=2({\varphi }_{\ell 2}-{\varphi }_{\ell 1})=2(\arg (B_{\ell N_2})-\arg (B_{\ell N_1}))$. We notice that on the right-hand side of Equation (94a) there is an additional factor two in comparison with Equation (31b) of Section 2.2; this factor two comes from the fact that we now have contributions of two intermediate neutrinos N₁ and N₂, and we neglected the contributions from the N₁-N₂ overlap $(\mathcal{O}(\delta ))$.

As at the end of Section 2.2, let us consider now as an illustrative example the decays $D_s^{\pm }\to {\mu }^{\pm }{\mu }^{\pm }{\pi }^{\mp }$. In addition, let us assume that |B_µN|² is the dominant mixing. In such a case, the estimate Equation (33) is still valid, and the CP-violating difference of the effective branching ratios, ${\mathcal{A}}_{\mathrm{CP}}{\mathrm{Br}}_{\mathrm{eff}}$, is obtained by comparison of Equations (94a) and (94b)

$${\mathcal{A}}_{\mathrm{CP}}{\mathrm{Br}}_{\mathrm{eff}}(D_s^{\pm }\to {\mu }^{\pm }{\mu }^{\pm }{\pi }^{\mp })\sim {10}^2|B_{\mu N}{|}^4\sin {\theta }_{21}\frac{y}{y^2+1}{|}_{y\equiv \mathrm{\Delta }{M}_N/{\mathrm{\Gamma }}_N}\lesssim {10}^2|B_{\mu N}{|}^4\sin {\theta }_{21}$$

(95)

Since for M_N ≈ 1 GeV we have at present |B_µN|² ≲ 10⁻⁷, cf. Table 2, Equation (95) means that ${\mathcal{A}}_{\mathrm{CP}}{\mathrm{Br}}^{(\mathrm{eff})}\lesssim {10}^{-12}$ for such decays. As already mentioned at the end of Section 2.2, the proposed CERN-SPS experiment [80,81] could produce D and D_s mesons in numbers by several orders of magnitude higher than 10¹², and production of the sterile Majorana neutrinos N_j could be explored. Further, if there exist two almost degenerate sterile neutrinos of mass M_N ∼ 1 GeV (this is so in the νMSM model [60,61,67–72]), such that y ≡ ΔM_N/Γ_N ∼ 1, then we would have η(y)/y ≡ y/(y²+1) ∼ 1. In such a case the estimate (95) would imply that the CP-violating difference of effective branching ratios, ${\mathcal{A}}_{\mathrm{CP}}{\mathrm{Br}}_{\mathrm{eff}}(D_s)$, is of the same order as the effective branching ratio Br_(eff)(D_s) (if the phase difference $|{\theta }_{21}|\cancel{\ll }1$). Therefore, if experiments can discover the mentioned νMSM-type Majorana neutrinos, they will possibly detect also CP violation effects coming from the Majorana neutrinos.

The case of $B_c^{\pm }\to {\mu }^{\pm }{\mu }^{\pm }{\pi }^{\mp }$ is similar to the case of $D_s^{\pm }\to {\mu }^{\pm }{\mu }^{\pm }{\pi }^{\mp }$ described above, cf. Equations (33) and (34) at the end of Section 2.2. Therefore, Equation (95) is valid also for CP violation in such decays of $B_c^{\pm }$. For the relative advantages and disadvantages of $D_s^{\pm }$and $B_c^{\pm }$ decays, we refer to the comments at the end of Section 2.2.

4.2. CP Violation in Pion Decays ${\pi }^{\pm }\to e^{\pm }{e}^{\pm }{\pi }^{\mp }\nu$

In this Section, we will only briefly outline the calculation of the CP violation asymmetry in the (LNC and LNV) semileptonic decays ${\pi }^{\pm }\to e^{\pm }{e}^{\pm }{\pi }^{\mp }\nu$ as described in Section 3. We will assume the presence of at least two nearly degenerate sterile neutrinos N_j (j = 1, 2) that can go on shell in the intermediate state, as in Section 4.1. The present Section is a similar extension of the analysis of the decays ${\pi }^{\pm }\to e^{\pm }{e}^{\pm }\nu$ of Section 3 to two sterile neutrinos. The results of the present Section are largely based on Ref. [38]. Only few details will be presented here, for further details we refer to Ref. [38].

Similarly to the previous Section 4.1, the quantities relevant for the CP violation will be

$${\mathrm{Br}}_{{}^{\pi ,\pm }}^{(\mathrm{X})}=\frac{S_{\pm }^{(\mathrm{X})}(\pi )}{2\mathrm{\Gamma }({\pi }^{\pm }\to \mathrm{all})}\equiv \frac{{\mathrm{\Gamma }}^{(\mathrm{X})}({\pi }^{-}\to e^{-}{e}^{-}{\mu }^{+}\nu )\pm {\mathrm{\Gamma }}^{(\mathrm{X})}({\pi }^{+}\to e^{+}{e}^{+}{\mu }^{-}\nu )}{2\mathrm{\Gamma }({\pi }^{\pm }\to \mathrm{all})}$$

(96a)

$${\mathcal{A}}_{{}^{\pi ,\mathrm{CP}}}^{(\mathrm{X})}=\frac{{\mathrm{Br}}_{\pi ,-}^{(\mathrm{X})}}{{\mathrm{Br}}_{\pi ,+}^{(\mathrm{X})}}=\equiv \frac{{\mathrm{\Gamma }}^{(\mathrm{X})}({\pi }^{-}\to e^{-}{e}^{-}{\mu }^{+}\nu )-{\mathrm{\Gamma }}^{(\mathrm{X})}({\pi }^{+}\to e^{+}{e}^{+}{\mu }^{-}\nu )}{{\mathrm{\Gamma }}^{(\mathrm{X})}({\pi }^{-}\to e^{-}{e}^{-}{\mu }^{+}\nu )+{\mathrm{\Gamma }}^{(\mathrm{X})}({\pi }^{+}\to e^{+}{e}^{+}{\mu }^{-}\nu )}$$

(96b)

where X = LNC, LNV. The total branching ratios are ${\mathrm{Br}}_{\pm }={\mathrm{Br}}_{\pm }^{(\mathrm{LNV})}+{\mathrm{Br}}_{\pm }^{(\mathrm{LNC})}$ when N_j are Majorana neutrinos, and ${\mathrm{Br}}_{\pm }={\mathrm{Br}}_{\pm }^{(\mathrm{LNC})}$^{) ±} when N_j are Dirac neutrinos. We adopt the same conventions and the same notations as in the previous Section 4.1. In addition, since we have now LNV and LNC processes, we introduce the additional notations

$${\theta }^{(\mathrm{LNV})}=2({\varphi }_{e2}-{\varphi }_{e1})$$

(97a)

$${\theta }^{(\mathrm{LNC})}=({\varphi }_{e2}-{\varphi }_{e1})-({\varphi }_{\mu 2}-{\varphi }_{\mu 1})$$

(97b)

As in Section 4.1, the requirement that the quantities sin θ^(X) (here: X = LNV, LNC) be nonzero, and the requirement of the near degeneracy of the two neutrinos $(\mathrm{\Delta }{M}_N\ll M_{N_1}\equiv M_N)$ in conjunction with the expressions Equations (79)–(83) for Im(P₁(D)P₂(D)^∗), are needed in order that the CP violation parameters ${\mathcal{A}}_{{}^{\pi ,\mathrm{CP}}}^{(\mathrm{X})}\ne 0$ acquire nonnegligible values. Analysis similar to that of the previous Section 4.1 (but algebraically more complicated) leads then to the results for the quantities defined in Equations (96). More specifically, the results for the Dirac case, ${\mathrm{Br}}_{\pi ,+}^{(\mathrm{Dir}.)}\equiv {\mathrm{Br}}_{\pi ,+}^{(\mathrm{LNC})}$ and ${\mathcal{A}}_{\pi ,\mathrm{CP}}^{(\mathrm{Dir}.)}\equiv {\mathcal{A}}_{\pi ,\mathrm{CP}}^{(\mathrm{LNC})}$, are the following:

$$\begin{array}{l}{\mathrm{Br}}_{\pi ,+}^{(\mathrm{Dir}.)}\equiv \frac{{\mathrm{\Gamma }}^{(\mathrm{LNC})}({\pi }^{-}\to e^{-}{e}^{-}{\mu }^{+}\nu )+{\mathrm{\Gamma }}^{(\mathrm{LNC})}({\pi }^{+}\to e^{+}{e}^{+}{\mu }^{-}\nu )}{2\mathrm{\Gamma }({\pi }^{\pm }\to \mathrm{all})}\\ =\left[{\displaystyle \sum _{j=1}^2\frac{|B_{e{N}_j}{|}^2|B_{\mu N_j}{|}^2}{{\tilde{\mathcal{K}}}_j}}+4\delta (y)\frac{|B_{e{N}_1}||B_{e{N}_2}||B_{\mu N_1}||B_{\mu N_2}|}{{(\tilde{\mathcal{K}}}_1{+\tilde{\mathcal{K}}}_2)}\cos {\theta }^{(\mathrm{LNC})}\right]{\overline{\mathrm{Br}}}_{\pi }\end{array}$$

(98a)

$$\begin{array}{l}{\mathcal{A}}_{\pi ,\mathrm{CP}}^{(\mathrm{Dir}.)}\equiv \frac{{\mathrm{\Gamma }}^{(\mathrm{LNC})}({\pi }^{-}\to e^{-}{e}^{-}{\mu }^{+}\nu )-{\mathrm{\Gamma }}^{(\mathrm{LNC})}({\pi }^{+}\to e^{+}{e}^{+}{\mu }^{-}\nu )}{{\mathrm{\Gamma }}^{(\mathrm{LNC})}({\pi }^{-}\to e^{-}{e}^{-}{\mu }^{+}\nu )+{\mathrm{\Gamma }}^{(\mathrm{LNC})}({\pi }^{+}\to e^{+}{e}^{+}{\mu }^{-}\nu )}\\ =\frac{4\sin {\theta }^{(\mathrm{LNC})}}{\begin{array}{l}\left[\frac{|B_{e{N}_1}||B_{\mu N_1}|}{|B_{e{N}_2}||B_{\mu N_2}|}\left(1+\frac{{\tilde{\mathcal{K}}}_2}{{\tilde{\mathcal{K}}}_1}\right)+\frac{|B_{e{N}_2}||B_{\mu N_2}|}{|B_{e{N}_1}||B_{\mu N_1}|}\left(1+\frac{{\tilde{\mathcal{K}}}_1}{{\tilde{\mathcal{K}}}_2}\right)+4\delta (y)\cos {\theta }^{(\mathrm{LNC})}\right]\\ \end{array}}\frac{\eta (y)}{y}\end{array}$$

(98b)

The expression for the canonical quantity ${\overline{\mathrm{Br}}}_{\pi }$, appearing in Equation (98a), is given in Equation (53) in conjunction with the notation (46) in Section 3.1 The results for the Majorana case ${\mathrm{Br}}_{\pi ,+}^{(\mathrm{Maj}.)}\equiv {\mathrm{Br}}_{\pi ,+}^{(\mathrm{LNV})}+{\mathrm{Br}}_{\pi ,+}^{(\mathrm{LNC})}$ and ${\mathcal{A}}_{\pi ,\mathrm{CP}}^{(\mathrm{Maj}.)}$ are the following:

$$\begin{array}{l}{\mathrm{Br}}_{\pi ,+}^{(\mathrm{Maj}.)}\equiv \frac{{\displaystyle \sum {}_{\mathrm{X}=\mathrm{LNV},\mathrm{LNC}}\left({\mathrm{\Gamma }}^{(\mathrm{X})}({\pi }^{-}\to e^{-}{e}^{-}{\mu }^{+}\nu )+{\mathrm{\Gamma }}^{(\mathrm{X})}({\pi }^{+}\to e^{+}{e}^{+}{\mu }^{-}\nu )\right)}}{2\mathrm{\Gamma }({\pi }^{\pm }\to \mathrm{all})}\\ =[{\displaystyle \sum _{j=1}^2\frac{|B_{e{N}_j}{|}^2(|B_{e{N}_j}{|}^2+|B_{\mu N_j}{|}^2}{{\tilde{\mathcal{K}}}_j}+4\delta (y)\frac{|B_{e{N}_1}||B_{e{N}_2}|}{({\tilde{\mathcal{K}}}_1+{\tilde{\mathcal{K}}}_2)}}\\ \times (|B_{e{N}_1}||B_{e{N}_2}|\cos {\theta }^{(\mathrm{LNV})}+|B_{\mu N_1}||B_{\mu N_2}|\cos {\theta }^{(\mathrm{LNC})}){\overline{\mathrm{Br}}}_{{}^{\pi }}]\end{array}$$

(99a)

$$\begin{array}{l}{\mathcal{A}}_{\pi ,\mathrm{CP}}^{(\mathrm{Maj}.)}\equiv \frac{{\displaystyle \sum {}_{\mathrm{X}=\mathrm{LNV},\mathrm{LNC}}\left({\mathrm{\Gamma }}^{(\mathrm{X})}({\pi }^{-}\to e^{-}{e}^{-}{\mu }^{+}\nu )-{\mathrm{\Gamma }}^{(\mathrm{X})}({\pi }^{+}\to e^{+}{e}^{+}{\mu }^{-}\nu )\right)}}{{\displaystyle \sum {}_{\mathrm{X}=\mathrm{LNV},\mathrm{LNC}}\left({\mathrm{\Gamma }}^{(\mathrm{X})}({\pi }^{-}\to e^{-}{e}^{-}{\mu }^{+}\nu )+{\mathrm{\Gamma }}^{(\mathrm{X})}({\pi }^{+}\to e^{+}{e}^{+}{\mu }^{-}\nu )\right)}}\\ =4\left(\sin {\theta }^{(\mathrm{LNV})}+\frac{|B_{\mu N_1}||B_{\mu N_2}|}{|B_{e{N}_1}||B_{e{N}_2}|}\sin {\theta }^{(\mathrm{LNC})}\right)\\ \times [\frac{(|B_{e{N}_1}{|}^2+|B_{\mu N_1}{|}^2)}{|B_{e{N}_2}{|}^2}\left(1+\frac{{\tilde{\mathcal{K}}}_2}{{\tilde{\mathcal{K}}}_1}\right)+\frac{|B_{e{N}_2}{|}^2+|B_{\mu N_2}{|}^2}{|B_{e{N}_1}{|}^2}\left(1+\frac{{\tilde{\mathcal{K}}}_1}{{\tilde{\mathcal{K}}}_2}\right)\\ {+4\delta (y)\cos {\theta }^{(\mathrm{LNV})}+\frac{|B_{\mu N_1}||B_{\mu N_2}|}{|B_{e{N}_2}||B_{e{N}_2}|}\cos {\theta }^{(\mathrm{LNC})}]}^{-1}\times \frac{\eta (y)}{y}\end{array}$$

(99b)

The function η(y)/y ≡ y/(y²+1). is the same as in Section 4.1 (with: y ≡ ΔM_N/Γ_N). Even more so numerical evaluations give for the N₁-N₂ overlap parameter δ(y) the same values as in the semileptonic decays of Section 4.1, cf. Table 4 and Figure 14 there.

If we assume that $|B_{{}^{\ell N_2}}|\approx |B_{{}^{\ell N_1}}|$ (for ℓ = e, μ, τ) i.e., Equation (92), then we have ${\tilde{\mathcal{K}}}_1\approx {\tilde{\mathcal{K}}}_2\equiv \tilde{\mathcal{K}}$, and the expressions for ${\mathcal{A}}_{\pi ,\mathrm{CP}}$ simplify significantly

$${\mathcal{A}}_{\pi ,\mathrm{CP}}^{(\mathrm{Dir}.)}=\frac{\sin {\theta }^{(\mathrm{LNC})}}{(1+\delta (y)\cos {\theta }^{(\mathrm{LNC})})}\frac{\eta (y)}{y}=\sin {\theta }^{(\mathrm{LNC})}\frac{\eta (y)}{y}(1+\mathcal{O}(\delta ))$$

(100a)

$${\mathcal{A}}_{\pi ,\mathrm{CP}}^{(\mathrm{Maj}.)}=\left(\frac{|B_{e{N}_1}{|}^2\sin {\theta }^{(\mathrm{LNV})}+|B_{\mu N_1}{|}^2\sin {\theta }^{(\mathrm{LNC})}}{|B_{e{N}_1}{|}^2+|B_{\mu N_1}{|}^2}\right)\frac{\eta (y)}{y}(1+\mathcal{O}(\delta ))$$

(100b)

As in the case of semileptonic LNV decays of the previous Section 4.1, we see that the CP asymmetry parameter ${\mathcal{A}}_{\pi ,\mathrm{CP}}$ can become appreciable and even of order one if the following two conditions are fulfilled simultaneously: (a) at least one of the angles θ^(X) (X=LNC,LNV), defined in Equation (97), is appreciable; (b) the quantity y ≡ ΔM_N/Γ_N is y ∼ 1 (near degeneracy). In such cases, the estimates for the effective (true) branching ratios ${\mathrm{Br}}_{\mathrm{eff}}^{(\mathrm{X})}$ of Equations (63) and (64) would apply also to the CP-violating difference of effective branching ratios, ${\mathcal{A}}_{\pi ,\mathrm{CP}}^{(\mathrm{X})}{\mathrm{Br}}_{\mathrm{eff}}^{(\mathrm{X})}$, where (X) = (Dir.),(Maj.).

5. Conclusions

We have studied lepton number violating (LNV) semileptonic decays of charged pseudoscalar mesons, specifically π^±, K^±, D^±, $D_s^{\pm }$, B^± and $B_c^{\pm }$, mediated by heavy neutrinos that can go on their mass shell.

We first presented the LNV semileptonic decays of charged Kaons and of the heavier mesons D^±, $D_s^{\pm }$, B^± and $B_c^{\pm }$, in processes of the form $M^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }{M^{\prime }}^{\mp }$, mediated by on-shell massive neutrinos, where M is the decaying meson and M′ a correspondingly lighter meson. We estimated the branching ratios as functions of the neutrino masses and mixing parameters, and found the scenarios where upper limits on the mixing parameters can be obtained. We also studied the effect on the observability of these decays due to the long neutrino lifetime, as the secondary decay vertex is likely to fall outside the detector for the range of neutrino masses that are relevant to these processes.

We then presented our corresponding study of charged pion decays, which in this case are purely leptonic since pions are the lightest mesons. Here we can have modes that conserve lepton number (LNC) as well as modes that violate lepton number (LNV), if the intermediate neutrinos are of Majorana type, while only the former modes occur if the intermediate neutrino is of Dirac type. However, these modes are not distinguished by the final state because the latter involves a standard neutrino, which is not experimentally observable. We find that it could be possible to discern the Majorana or Dirac nature of neutrinos if one is able to observe features in the final state distribution.

We finally explored the possibility of observing CP violation in the lepton sector using these meson decays mediated by massive neutrinos on shell. The CP signal in charged meson decays is the usual asymmetry between the decays of opposite charge mesons. We found that leptonic CP violation may show in semileptonic LNV decays of charged Kaons and charged B mesons, as well as in LNC and LNV decays of charged pions, depending on the mass of the intermediate neutrinos. It turns out that such CP violation becomes appreciable and possibly detectable if there are at least two heavy neutrinos almost degenerate in mass that can go on their mass shell. The neutrino mass splitting that gives maximal CP asymmetries is close to the neutrino decay width. This type scenario fits well into the so called neutrino minimal standard model (νMSM), which contains two almost degenerate Majorana neutrinos of mass near 1 GeV and another lighter neutrino of mass of order 10¹ keV, a model that can explain simultaneously neutrino oscillations, the dark matter and the baryon asymmetry of the Universe.