Decays of charged $B$-mesons into three charged leptons and a neutrino

In the framework of the Standard Model we present predictions for partial widths, double and single differential distributions, and forward--backward lepton asymmetries for four-leptonic decays $B^- \to \mu^+\mu^- {\bar \nu}_e\, e^-$, $B^- \to e^+ e^- {\bar \nu}_\mu\,\mu^-$, $B^- \to \mu^+ \mu^-{\bar \nu}_\mu\,\mu^-$, and $B^- \to e^+ e^- {\bar \nu}_e\, e^-$. We consider the contributions of virtual photon emission from the light and heavy quarks of the $B^-$--meson, and we include bremsstrahlung of a virtual photon from the charged lepton in the final state. We use the model of vector meson dominance for calculation of virtual photon emission by the light quark of the $B^-$--meson and take into account the isotopic correction.


Introduction
Four-leptonic decays of -mesons allow a precise test of Standard Model (SM) predictions in the higher orders of perturbation theory. At the same time these decays may be background processes to the helicity-suppressed ultra-rare decays , Ñ`´, which are under study at the Large Hadron Collider (LHC) [1,2,3]. These studies are motivated by searches for Beyond the Standard Model physics.
Rare four-leptonic decays of -mesons in the SM may be divided into two groups. The decays of the first group are forbidden at the tree level and occur through the higher order loop diagrams of perturbation theory -"penguin" and/or "box". In this way of the SM includes flavor changing neutral currents (FCNC). An example of the first group of decays is the process Ñ`´`´and any other four-leptonic decays of neutral -mesons. In the second group, in order to obtain the given multi-lepton final state, a number of tree level weak and electromagnetic processes are involved. Examples are the decay´Ñ`´¯´and analogous processes involving charged -mesons. Both groups are studied at the LHC and potentially could be investigated at the Belle II experiment. Currently only upper limits for branching ratios of the decays , Ñ`´`´and´Ñ`¯´´are available [4,5,6].
The experimental upper limits [4,5] for the decays , Ñ`´`´are an order of magnitude higher than the corresponding theoretical predictions [7] and estimates [8]. The situation with the decay´Ñ`¯´´is different. The experimental upper limit [6], Br`´Ñ`¯´´˘ă 0.16ˆ10´7, (1) obtained with 95% confidence level (CL) is almost an order of magnitude lower than the theoretical predictions [8,9]. We present here to more detailed calculation of the branching ratios of´Ñ`´¯´,´Ñ`´¯´,´Ñ`´¯´and´Ñ`´¯´, taking into account isotopic effects. Also in the phase space of the decays, a correction to non-zero lepton mass is considered. While this leads to better agreement between theory and experiment, some discrepancy remain. Special attention is given to the predictions of the behavior of differential distributions, e.g. forward-backward lepton asymmetries.
This article is organized as follows. In the "Introduction" we give a task description. In Section 1 we write the effective Hamiltonian and give definite the hadronic form factors. In Section 2 the common dependence of the decay amplitudes´Ñ ℓ`ℓ´¯ℓ1 ℓ 1´o n di-lepton 4-momenta is studied. Section 3 contains the exact formulae for amplitudes of the decaý Ñ ℓ`ℓ´¯ℓ1 ℓ 1´f or ℓ ‰ ℓ 1 , and Section 4 provides analogous formulae for ℓ " ℓ 1 . In Section 5 we present numerical results for the decays of charged -mesons into three charged leptons and neutrino and discuss the precision of the predictions. The "Conclusion" contains the main outcome of the work. Some details of the four-leptonic decay kinematics are given in Appendix A.

Effective Hamiltonian and hadronic matrix elements
In terms of fundamental quark and lepton fields, the Hamiltonian for calculation of the amplitudes of four-lepton decays´Ñ ℓ`ℓ´¯ℓ1 ℓ 1´h as the form: The Hamiltonian of the transitions Ñ´Ñ ℓ´¯ℓ is written as: where p q and p q are quark fields, ℓp q and ℓ p q are lepton fields, is the Fermi constant, is the corresponding matrix element of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, and the matrix 5 is defined as 5 " 0 1 2 3 .
The Hamiltonian of the electromagnetic interaction has the form: where the unitary charge " | | is normalized by 2 " 4 ; « 1{137, the fine structure constant, is the charge of the fermion of flavor in units of the unitary charge, p q is the fermionic field of flavor , and p q is the four-potential of the electromagnetic field.
We define the following non-zero hadronic matrix elements, which are needed for the subsequent calculations: where 1 -´is the meson mass, is the its four-momentum,˚is the˚´-meson mass, 2 -mass of the light ( 0 p770q or p782q) mesons, " t 2 ,˚u are masses of the intermediate vector mesons, and are their polarizations. Four-vectors , , and satisfy the conservation law: "`. The components of the fully antisymmetric tensor are fixed by the condition 0123 "´0 123 "´1, and is the metric tensor in Minkovsky space with diag " p1,´1,´1,´1q.
2 Generic structure of the amplitudes for the decayś Ñ ℓ`ℓ´¯ℓ1 ℓ 1´w ith the zero lepton mass approximation There are three main types of diagrams needed for description of the decays´p q Ñ p q´p q Ñ ℓ`p 1 q ℓ´p 2 q¯ℓ1p 3 q ℓ 1´p 4 q, when the flavor of lepton ℓ is different from the flavor of lepton ℓ 1 . The first type arises in the situation when a virtual photon is emitted by light a -quark (see Fig. (1)). The second type corresponds to the emission of a virtual photon from a -quark (see Fig. (2)). The third type is related to bremsstrahlung, when a virtual photon is emitted by the lepton ℓ 1´i n the final state (see Fig. (3) below). The four-momenta and are defined in Appendix A.
The structure of the amplitude, corresponding to diagrams on Figs. (1), (2) and (3) may be presented as where p , q " The lepton currents are In the amplitude ℳ p 2 , 2 q, the pole 1{ 2 of the photon propagator is evident. For calculations with 2 Ñ 0 it is necessary to take into account non-zero lepton masses. This is done using the exact formula (31) for four-particle phase space, and by introducing an effective cut for some value 2 min . If ℓ " for 2 min it makes sense to choose the natural kinematical cut 4 2 . For the case when ℓ " it is better to use the kinematical limits of an experimental device, which are definitely higher than 4 2 .

Tensor
p , q satisfies the condition: p , q " 0. According to this condition, and taking into account the result of Ref. [11], tensor p , q has the form: where "´"´1, is the electric charge of the´-meson in units of | |. The functions p 2 , 2 q, . . . , p 2 , 2 q are dimensionless form factors which depend on two variables, the squares of the transferred four-momenta, 2 and 2 . From (5) it follows that p0, 0q " { 1 .
Using the equations of motion, in the limit of massless leptons, one can obtain the following generic structure for the ampliude ℳ : The exact calculation of the form factors p 2 , 2 q, . . ., and p 2 , 2 q is quite complicated. In the current work we will take into account only the leading singular factors to the corresponding form factors.
Let us start with a study of tensor p q p , q, which describes the contribution of diagram from Fig.1 to the tensor p , q. The main contribution to the structure of tensor p q p , q is given by the lightest intermediate vector resonances that contain a¯-pair. For such states tensor p q p , q has Breit-Wigner poles for variable 2 . Taking into account only the contributions from 0 p770q and p782q mesons, we can write where 2 and Γ 2 are the masses and widths, respectively, of the intermediate vector resonances.
For the zero leptonic mass approximation, the range of values of the variable 2 is 0 ď 2 ď 2 1 . The closest pole in 2 is related to the appearance of the intermediate vector state´. As˚´ą 1 , this pole lies outside of the kinematically allowed range of the decaý Ñ ℓ`ℓ´¯ℓ1 ℓ 1´. The existence of the pole at the mass of the˚´-meson is taken into account when choosing the pole parametrisation of the form factors of the transitions Ñ and Ñ [10]. For non-zero leptonic masses, 2 ℓ 1 ď 2 ď p 1´2 ℓ q 2 . Hence all the remarks above on the poles of tensor p q for variable 2 are still valid.
As the contribution from 0 and resonances is dominant, it is possible to use the following estimate for the branching ratio of´Ñ`´¯´: where the necessary experimental values for the branching ratios are taken from [12]. The estimate (6) does not take into account the fact that the 0 p770q-meson is a wide resonance, i.e., in the case of the 0 p770q-meson, the naive factorization approximation should lead to a lower branching ratios. Also in estimate (6), the photon pole, which should also lead to lower results, is not taken into account. Does estimate (6) contradict the experimental upper limit in (1)? We do not think so, because we attribute to the factor of two 2 accuracy. But the estimate of (6) does point to the possibility that the minimum of the possible theoretical predictions may be above the experimental limit [6]. Now consider tensor p q p , q, which is related to diagram from Fig. (2). In the limit of massless leptons there are no poles for the variable 2 in the kinematically allowed range 0 ď 2 ď 2 1 for the tensor p q p , q. The closest pole outside the allowed range corresponds to the¯quark composition. This is the ϒp1 q meson, whose mass is almost two times higher than the mass of the´-meson. The dominant contribution to emission of a virtual photon by a heavy quark is described using the process´Ñ˚´˚. In this case  Note that the imaginary addition´˚Γ˚does not exist in the propagator, as 2 ă 2˚, i.e., the pole of the˚-meson is not reached. The contribution from the ϒp1 q is taken into account effectively when introducing pole parametrization for the form factor p 2 q. For the variable 2 in the kinematically allowed range, the tensor p q p , q does not have any other poles.
Numerically the contribution of the process on the Fig. (2) to the branching ratio associated with the four-leptonic decay is suppressed comparing to the contribution of the process on the Fig. (1) by factor pΛ{ q 2 , where " 5 GeV, the mass of -quark, and the parameter Λ « 300´´500 MeV. This follows from the exact equations for the form factors of the rare leptonic radiative decays of -mesons [13,14]. Due to the interference between diagrams (1) and (2) near the photonic pole it is necessary, however to take into account the contribution of the diagram (2) to the full branching ratio.
The bremsstrahlung contribution is described by diagram on the Fig. (3). The bremsstrahlung amplitude has a single pole by 2 from the photon propagator. Hence the tensor p q p , q does not have poles by 2 and 2 . It is important to take into account the bremsstrahlung contribution near the pole by 2 , where the zero-mass approximation may not be fully correct.  (7)) using the decay´Ñ´¯´a s an example. The emission of the virtual photon by a light quark is described by the Vector Meson Dominance model. This contribution should be calculated for non-zero lepton masses.
3 Formulae for the decay´Ñ ℓ`ℓ´¯ℓ1 ℓ 1Ć onsider the decays´Ñ`´¯´and´Ñ`´¯´, for the case when the lepton flavors in the final state are different. Generally these decays may be written aś Ñ ℓ`ℓ´¯ℓ1 ℓ 1´f or ℓ ‰ ℓ 1 .
The contribution to the full decay amplitude´p q Ñ ℓ`p 1 q ℓ´p 2 q¯ℓ1p 3 q ℓ 1´p 4 q from Fig. (1) may be calculated using the Vector Meson Dominance model (VMD) (see Fig. (4)). Assuming ℓ " ℓ 1 " 0 and using the effective Hamiltonian (2), one finds, that for VMD the contribution from process (1) is described by diagram (4), and the corresponding amplitude may be written as: where, using the motion equations, For the calculation of the resonances sum in the (7), only the contributions from the lightest 0 and mesons, containing¯-pairs, are taken into account. Because the 0 and mesons are linear combinations of¯and¯-pairs, in order to extract the contributions only¯-pair alone, an isotopic coefficients are introduced. By definition 0 " @ 0ˇ¯D " 1{ ? 2 Figure 5: Diargam for the calculation of ℳ p q (see the equation (8)) using the decay´Ñ´¯´a s an example.  (9)) for the decay´Ñ`´¯´.
and " The contribution from process (2) is given by diagram on Fig. (5), which is the cross-channel of the decay˚Ñ˚of a heavy vector meson into a heavy pseudoscalar meson and a virtual photon, and is represented by There is no imaginary correction in the propagator, as 2 ă 2˚.
Finally, the contribution of the bremsstrahlung process (3) of the virtual photon is described by diagram from Fig. (6). In the case when ℓ ‰ 0 and ℓ 1 ‰ 0, for the amplitude of the bremsstrahlung is: As p2 ℓ`ℓ 1 q 2 ď p´3q 2 ď 2 1 , the second summand does not contain any poles in the whole kinematically allowed range. The second summand may be compatible with the first one only in the range where p´3q 2 " p2 ℓ`ℓ 1 q 2 . But this range is suppressed by the phase space (31) integration. For this the reason we assume that the bremsstrahlung amplitude may be written as In formulas (7), (8) and (9) we denote " ? 2 4 .
The full decay amplitude´p q Ñ ℓ`p 1 q ℓ´p 2 q¯ℓ1p 3 q ℓ 1´p 4 q may be written as The benefit of choosing the notation ℳ p1234q for the full amplitude will be apparent while considering decays with identical charged fermions in the final state (see Section 4). Dimensionless functions p 2 , 2 q " p 12 , 34 q, p 2 , 2 q " p 12 , 34 q, and p 2 , 2 q " p 12 , 34 q are defined as: where the dimensionless variables 12 " 2 { 2 1 , and 34 " 2 { 2 1 are defined in Appendix A, and the dimensionless constants are defined as^" Form factors p 2 q, p q p 2 q, p q 1 p 2 q, and p q 2 p 2 q are also dimensionless functions.
The differential branching ratio of the decay´Ñ ℓ`ℓ´¯ℓ1 ℓ 1´i s calculated as where´is the lifetime of the´-meson, four-particle phase space Φ p1234q 4 is defined by Equation (31), and the summation is performed over the spins of the final fermions. In formula (12) the integration over the angular variables 12 , 34 , and may be performed analytically. After the integration, 2 Br p´Ñ ℓ`ℓ´¯ℓ1ℓ 1´q Because in the decay of the´-meson, all the leptons in the final state are different, it makes sense to define two forward-backward leptonic asymmetries p´q p 12 q and p´q p 34 q as p´q p 12 q " and p´q p 34 q " where˜1 2 is the angle between the propagation directions of the ℓ´and´in the rest frame of the ℓ`ℓ´-pair, and˜3 4 is the angle between the propagation directions of ℓ 1´a nd´in the rest frame of the ℓ 1´¯ℓ 1 pair. It is obvious that˜1 2 "´1 2 and˜3 4 "´3 4 . Equations (14) and (15) are chosen such that they correspond to the notions of Ref. [11].
4 Exact formulae for the decay´Ñ ℓ`¯ℓ ℓ´ℓÍ n practice, the muonic tracks are registered with a much higher efficiency at almost all contemporary experiments. That is why from the experimental point of view the decay´Ñ¯´´i s of the most interest. In this decay the final state contains two identical muons of negative charge. Hence the Fermi antisymmetry should be taken into account.
Consider the full amplitude of the decay´p q Ñ ℓ`p 1 q¯ℓp 3 q ℓ´p 2 q ℓ´p 4 q. In the approximation of zero leptonic masses, the calculation below is applicable to the decay´Ñ `¯´´a s well as to the decay´Ñ`¯´´. The full amplitude of the decay may be written as where the amplitude ℳ p1234q is set by equation (10), and the amplitude ℳ p1432q can be obtained from ℳ p1234q by exchanging 2 Ø 4 . This leads to the necessity of replacing Ñ˜, Ñ˜, 12 Ñ 14 , and 34 Ñ 23 (see Appendix A) in the calculation of ℳ p1432q .
The differential branching ratio of the decay is given by and Φ are set by equations (31) and (32).The common factor of 1{2 is due to by Fermi antisymmetry.
The first and the second summands in (17) are equal. Hence for the branching ratio, it is possible to write Br`´Ñ ℓ`¯ℓ ℓ´ℓ´˘" Br`´Ñ ℓ`ℓ´¯ℓ1 ℓ 1´˘´B r`´Ñ ℓ`¯ℓ ℓ´ℓ´˘, where Br`´Ñ ℓ`¯ℓ ℓ´ℓ´˘" From (19) it follows that in the calculation of the interference contribution it is necessary to perform five-dimension of numerical integration. It is necessary to use the replacements (33) in the matrix element ℳ p1432q .

Numerical results
To calculate the branching ratio, differential distributions, and asymmetries, we use numerical values of the masses, lifetimes and decay widths of the pseudoscalar and vector mesons, and matrix elements of the CKM matrix from Ref. [12]. The constants p770q " 154 MeV and p782q " 46 MeV were calculated in [14].
Suitable parametrizations of the hadronic formfactors (3), except the electromagnetic form factor p 2 q, were obtained in [10]. Using the generic formulae from [15,16] it is possible to find the following parametrization for the form factor p 2 q, calculated in the framework of the Dispersion Quark Model: .81 where ϒ mass of the ϒp1 q meson. The same method allows us to obtain the values of the leptonic constants " 191 MeV and˚" 183 MeV.
We now calculate the branching ratio of the decay´Ñ`´¯´. The natural kinematical cut of the pole by 12 is 12 min " p2 { 1 q 2 « 0.0016. In this case, the numerical integration of the equation (13) by 12 and 34 gives: The value of the branching of the´Ñ`´¯´decay given in (21) is approximately two times less than the corresponding value of 1.3ˆ10´7 from Refs. [8,9]. This difference is mostly due to the isotopic coefficients 0 and in (7), while decreases the contribution from the intermediate vector 0 p770q and p782q resonances to the total branching ratio by a factor 2. This contribution is dominant, so the Br p´Ñ`´¯´q increase by almost the same factor. Also the mean value of is changed from 4.09ˆ10´3 [17] to 3.94ˆ10´3 [12]. A decrease of the branching by 10% is due to the use of the exact formula (31) for the phase space.
The result in equation (21) is compatible with the naive estimate of (6) up to an expected factor of two. The difference between the estimate of (6) and the exact calculation (21) is mostly due to the fact that the estimate does not take into account the pole contribution when 12 Ñ 12 min . The importance of the pole contribution becomes obvious when analysing the double differential distribution 2 Br p´Ñ`´¯´q { 12 34 , which is presented in Fig. (7). The figure features the pole when 12 Ñ 12 min " 4 2 { 2 1 and the ridge of the narrow p782q resonance, the contribution of which defines the maximum of the matrix element. The wide 0 p770q-meson also gives a significant contribution to the branching ratio, but in the distribution of Fig. (7) is not a prominent as the narrow p782q resonance.
The uncertainty on the numerical value of the branching ratio of the decay´Ñ`´¯d epends on the uncertainty on the calculation of the hadronic form factors of the transitions Ñ p770q and Ñ p782q, but does not exceed 20% [10]. Some uncertainty is related to the use of the Vector Meson Dominance model. This uncertainty is mostly due to the choice of a non-perturbative phase between the summands in the (7). In the VMD model this phase is equal to zero. If the relative phase between the contributions from the 0 p770q and p782q mesons into the amplitude (7)  , calculated according to formula (13). On Figure b) the range 12 P r0.00, 0.04s is highlighted, which corresponds to the area of applicability of the model considered in the present work.
0.2ˆ10´7. This dependence points to the importance of a future model-independent study of non-perturbative and non-factorized contributions of the strong interaction to the amplitudes of the decays´Ñ ℓ`ℓ´¯ℓ1 ℓ 1´. Similar issue of generation of additional relative phases between the contributions of different charmonia by nonfactorizable gluons was discussed in [18].
In the model used for the result of (21) the non-resonant contribution, which is not related to the tails from the 0 p770q and p782q resonances, is not taken into account. This contribution may be estimated by using the results from Ref. [19], in this work the branching ratio of the decay Ñ ℓ was predicted, omitting the contributions from 0 and resonances. An estimation of the non-resonant contribution gives Br`´Ñ`´¯´˘N RC "ˆBr p Ñ ℓ q Beneke " 0.1ˆ10´7, which is about 15% of the value of the branching ratio of (21) and is comparable to the uncertainty of the form factors calculation. Note that numerically the contributions to (21) from the processes in Fig. (5) and Fig. (6), which were taken into account, are also comparable to the non-resonant contribution, which was not taken into account.
It seems that the approximation of using only the contributions from the lightest p770q and p782q resonances, which is used in this work, is not applicable if the branching ratio of the decay´Ñ`´¯´will be measured in the range of a 2 ą 1 GeV. In this range it is necessary to take into account the contributions from the p1420q, p1450q, p1650q, and p1700q resonances. These contributions should not affect the branching ratio of the decaý Ñ`´¯´for a 2 ď 1 GeV but will define the behavior in the range a 2 ą 1 GeV. However in the experimental procedure [6], the variable a 2 is chosen to be less than 980 MeV, in order to remove a potential background from the decay Ñ ℓ`ℓ´. So the experimental data are available only in the range of applicability of the current work. This fact allows as to exclude from consideration resonances heavier than the 0 p770q and p782q.
We calculate the branching ratio of the decay´Ñ`´¯´. Formal integration in the range around the photon pole by 12 leads to the rough dependence of the branching on 12 min : Br " Because the efficiency of detection of the muonic pairs for a 2 below 80-100 MeV is low, this range is not suitable for the an experimental observation. On the other hand, if we choose 12 min " pΛ{ 1 q 2 " 0.0002 for Λ " 80 MeV, then The Br p´Ñ`´¯´q will decrease with increasing 12 min .
The decays´Ñ`´¯´and´Ñ`´¯´may be suitable for tests of the hypothesis of leptonic universality, if one measures the branching ratio for the fixed value of If the hints for [20,21,22,23,24] violation of the leptonic universality are true, then the equation (23) may be violated.
We consider predictions for the branching ratio of the decay´Ñ`¯´´, which is the more suitable for experimental observation [6], as the efficiency of muon detection is higher than the efficiency of electron detection. Numerical integration of the interference contribution (19) for 12 min " p2 { 1 q 2 " 0.0016 gives Br`´Ñ`¯´´˘«´0. 11 .638ˆ10´1 2 s which is comparable due to uncertainty of the strong non-perturbative effects, the contributions from equations (5) and (6) and the result with the non-resonant contribution omitted. So we may state that in the limit of massless leptons, with a 30% precision from equations (18) and (21), it follows that: This is obtained for 12 min " p2 { 1 q 2 " 0.0016. This prediction is almost four times higher than the experimental upper limit (1), obtained in Ref. [6]. What may explain the discrepancy between the experimental result and the theoretical prediction? First, there is quite high uncertainty of the theoretical prediction (24). Second, the value of Br p´Ñ`¯´´q depends on the relative phase between the contributions of the 0 p770q and p782q resonances.
In the framework of VMD it is zero, however various non-perturbative contributions may lead  (12) and (17) respectively.
to non-zero value. All the other contribution, which were omitted in the current work could not significantly influence the numerical result of equation (24). It seems unlikely that the discrepancy between the prediction and measured result may be attributed to Beyond the Standard Model physics.
We consider single differential distributions for the decays´Ñ`´¯´and´Ñ¯´´.
One-dimensional projections of the double differential distribution 2 Br 12 34 by 12 and 34 are given in Fig. (8) and (9) respectively. The distributions by 12 are given in the range r0, 0.04s, which corresponds to the area of applicability of the model. Fig. (8) features a photon pole for 12 Ñ 12 min " p2 { 1 q 2 " 0.0016 and a peak from the p782q resonance for 12 Ñ p { 1 q 2 « 0.023. Due to the fact that the 0 p770q meson has a width of about 150 MeV, the contribution from this meson in Fig. (8) appears as a wide background to the narrow peak of the p782q resonance. The distributions by 34 in Fig. (9) does not have poles, in agreement with the analysis from Section 2, and demonstrates the importance of taking into account the Fermi antisymmetry in the decay´Ñ`¯´´, because due to the additional contribution from Fermi antisymmetry the shapes of the distributions by 34 in the decayś Ñ`´¯´and´Ñ`¯´´are significantly different. An analogous difference may be seen in the distributions by 12 " cos 12 and 34 " cos 34 , which are presented in Fig. (10) and (11)  for the decays a)´Ñ`´¯á nd b)´Ñ`¯´´, obtained by integration by 12 12 34 of formulae (12) and (17) (12) and (17) accordingly.
square of the corresponding mass is defined as: where the are four-momenta of charged leptons in the final state. The distributions by 124 are presented in Fig. (12). One can see from the figure that the shape of the distribution by 12 is not very sensitive to the procedure of Fermi antisymmetrization.
It is well known that forward-backward lepton asymmetries are very sensitive to BSM physics. For the decay´Ñ`´¯´it is possible to define forward-backward lepton asymmetries p´q p 12 q and p´q p 34 q according to equations (14) and (15). These asymmetries are shown in Fig. (13). The asymmetry p´q p 12 q is shown only for the interval 12 P r0, 0.04s, which corresponds to the area of applicability of the current model. In this interval, excluding the area of the p782q resonance, the contributions to p´q p 12 q come from electromagnetic and strong processes;thus this asymmetry is close to zero in almost all of the considered range. The shape of the asymmetry p´q p 34 q is very similar to the shape of the asymmetries in three-body semileptonic decays of -mesons.
One cannot to study forward-backward lepton asymmetries in the decay´Ñ`¯´´, as in this case there are two identical negative muons in the final state. Experimentally it is not possible to distinguish which of the negatively charged muons should be attributed to thè´pair, and which to the¯´-pair.
All the above that is related to the differential distributions for the decays´Ñ`´¯á nd´Ñ`¯´´is also related to the differential distributions for the decays´Ñ´¯´a nd´Ñ`¯´´. In this model the lepton universality holds, so the differential distributions of the two latter decays are not needed. a) b) Figure 13: Forward-backward lepton asymmetries a) p´q p 12 q and b) p´q p 34 q for the decay´Ñ`´¯´, calculated using equations (14) and (15) respectively.

Conclusion
In the present work, • theoretical predictions for the branching ratios of the decays´Ñ`´¯´and Ñ`¯´´are obtained in the framework of Standard Model: Br`´Ñ`´¯´˘« 0.6ˆ10´7 and Br`´Ñ`¯´´˘« 0.7ˆ10´7, and uncertainties for every prediction are discussed; • the difference between the obtained predictions and the predictions from Ref. [8] is discussed, as well as the compatibility with the recent experimental result [6] by the LHCb collaboration; • the possibility to test the hypothesis of lepton universality in rare four-leptonic decays of -mesons with three charged leptons in the final state is analysed; • double and single differential distributions for the decays´Ñ`´¯´and´Ñ¯´´a re considered, and some recommendations for searches for Beyond the Standard Model physics in these decays are given. Figure 14: Kinematics of the decay´p q Ñ ℓ`p 1 q ℓ´p 2 q¯ℓ1p 3 q ℓ 1´p 4 q. Angle 12 is defined in the rest frame of ℓ`p 1 q ℓ´p 2 q-pair; angle 34 is defined in the rest frame of¯ℓ1p 3 q ℓ 1´p 4 qpair; angle is defined in the rest frame of´-meson.
For the pair 14 and 23 , the analogous condition holds: 14 P r0, 1s and for a fixed 14 , Consider the kinematics of the decay´p q Ñ ℓ`p 1 q ℓ´p 2 q¯ℓ1p 3 q ℓ 1´p 4 q, when the flavor of negatively charged lepton ℓ´p 2 q is different from the flavor of the negatively charged lepton ℓ 1´p 4 q. Let the positively charged lepton have the momentum k 1 , and let the antineutrino have the momentum k 3 . We define an angle 12 between the momentum of the positively charged lepton and the direction of the -meson ( -axis) in the rest frame of the ℓ`ℓ´pair, and another angle 34 between the direction of the antineutrino and the direction of the -meson ( -axis) in the rest frame of ℓ 1´¯ℓ 1 -pair. Then 12 " cos 12 " 1 1{2 p1, 12 , 34 q p 23`24´13´14 q , 34 " cos 34 " 1 1{2 p1, 12 , 34 q p 14`24´13´23 q , where p , , q " 2`2`2´2´2´2 , the triangle function. Angles 12 P r0, s and 34 P r0, s. Hence 12 P r´1, 1s and 34 P r´1, 1s. Angles are measured relative to -axis. Also let us define an angle P r0, 2 q in the rest frame of the -meson between the planes which are set by the pairs of vectors pk 1 , k 2 q and pk 3 , k 4 q. Introduce a vector a 1 " k 1ˆk2 , perpendicular to the plane pk 1 , k 2 q, and vector a 3 " k 4ˆk 3 , perpendicular to the plane pk 3 , k 4 q. Then cos "´a 1 , a 3| a 1 | |a 3 | .
where˜is the angle of planes pk 1 , k 4 q and pk 2 , k 3 q, measured relative to plane pk 1 , k 4 q. The equation (32) may be obtained in a fully analogous way to (31). The cos˜can be found by exchanging indices in equations (29) and (30) as 2 Ø 4. Also in order to perform numerical integration it is necessary to have all the definitions of using the set of variables 12 , 34 ,