# On sterile neutrino mixing with

###### Abstract

Matrix element of sterile neutrino mixing with is the least constrained in the literature among the three () mixing parameters characterizing the sterile neutrino phenomenology. We study the contribution of massive dominantly sterile neutrinos to purely leptonic -decays and semileptonic decays of and K, D mesons. We consider some decays allowed in the Standard Model (SM) as well as Lepton Flavor and Lepton Number Violating (LFV, LNV) decays forbidden in the SM. From the existing experimental data on the branching ratios of these processes we derived new limits on more stringent than the ones existing in the literature. These limits are extracted in a model independent way without any ad hoc assumptions on the relative size of the three different sterile neutrino mixing parameters.

###### pacs:

13.35.Hb, 13.15.+g, 13.20.-v, 13.35.Dx## I Introduction

Lepton flavors are conserved in the Standard Model (SM) due to the presence of an accidental lepton flavor symmetry, which, however, is broken by non-zero neutrino masses. Neutrino oscillation experiments have proven that neutrinos are massive, although very light, particles mixing with each other. Moreover, neutrino oscillations is the first and so far the only observed phenomenon of lepton flavor violation (LFV). In the sector of charged leptons LVF is strongly suppressed by the smallness of neutrino square mass differences compared to the characteristic momentum scale, , of an LFV process which is typically of the order of the charged lepton mass . If neutrinos are Majorana particles there can also occur lepton number violating (LNV) processes. They are also suppressed by the smallness of the absolute value of . However, the situation may dramatically change if there exist either heavy neutrinos , known as sterile, mixed with the active flavors or if there are some new LFV and LNV interactions beyond the SM.

Here we study the former possibility and consider an extension of the SM with right-handed neutrinos. In the case of species of the SM singlet right-handed neutrinos , besides the three left-handed weak doublet neutrinos the neutrino mass term can be written as

(1) | |||||

(2) |

Here are and symmetric Majorana mass matrices, and is a Dirac type matrix. Rotating the neutrino mass matrix to the diagonal form by a unitary transformation

(3) |

one ends up with Majorana neutrinos with masses . The matrix is a neutrino mixing matrix. In special cases among neutrino mass eigenstates there may appear pairs with masses degenerate in absolute values. Each of these pairs can be collected into a Dirac neutrino field. This situation corresponds to conservation of certain lepton numbers assigned to these Dirac fields. Generically in this setup neutrino mass eigenstates can be of any mass. For consistency with neutrino phenomenology (for recent review, cf. rev-nu-phen ()) among them there must be the three very light neutrinos with different masses and dominated by the active flavors (). The remaining states may also have certain admixture of the active flavors and, therefore, participate in charged and neutral current interactions of the SM contributing to LNV and LFV processes. Explanation of the presence in the neutrino spectrum of the three very light neutrinos requires additional physically motivated assumptions on the structure of the mass matrix in (1). The celebrated “see-saw” mechanism see-saw (), presently called type-I see-saw, is implemented in this framework assuming that . Then, there naturally appear light neutrinos with masses of the order of dominated by . Also, there must be present heavy Majorana neutrinos with masses at the scale of . Their mixing with active neutrino flavors is suppressed by a factor which should be very small. In particular scenarios this generic limitation of the see-saw mechanism can be relaxed relax (). Then the heavy neutrinos could be, in principle, observable at LHC, if their masses are within the kinematical reach the corresponding experiments. Very heavy or moderately heavy Majorana entry of the the neutrino mass matrix naturally appears in various extensions of the SM. The well known examples are given by the -based supersymmetric SUSY-GUT () and ordinary GUT () grand unification models. The supersymmetric versions of see-saw are also widely discussed in the literature (see, for instance, SUSY-see-saw () and references therein).

In the present paper we study the above mentioned generic case of the neutrino mass matrix in (1) without implying a specific scenario of neutrino mass generation. We assume there is at least one moderately heavy neutrino in the MeV-GeV domain or even lighter. The presence or absence of these neutrino states, conventionally called sterile neutrinos, is a question for experimental searches. If exist, they may contribute to some LNV and LFV processes as intermediate nearly on-mass-shell states. This would lead to resonant enhancement of their contributions to these processes. As a result, it may become possible to either observe the LNV, LFV processes or set stringent limits on sterile neutrino mass and mixing with active neutrino flavors () from non-observation of the corresponding processes.

On the other hand the sterile neutrinos in this mass range are motivated by various phenomenological models Mohapatra (), in particular, by the recently proposed electroweak scale see-saw models EWSS1 (), EWSS2 (). They may also play an important astrophysical and cosmological role. The sterile neutrinos in this mass range may have an impact on Big Bang nucleosynthesis, large scale structure formation nuclsyn (), supernovae explosions supernovae (). Moreover, the keV-GeV sterile neutrinos are good dark matter candidates DM-Bar-1 (); DM-Bar-2 (); DM-Bar-3 () and offer a plausible explanation of baryogenesis Barg (). Dark Matter sterile neutrinos, having small admixture of active flavors, may suffer radiative decays and contribute to the diffuse extragalactic radiation and x-rays from galactic clusters galclast (). This is, of course, an incomplete list of cosmological and astrophysical implications of sterile neutrinos. More details on this subject can be found in Refs. Dolgov1 (), Smirnov1 ().

The phenomenology of sterile neutrinos in the processes, which can be searched for in laboratory experiments have been studied in the literature in different contexts and from complementary points of view (for earlier studies see Shrock ()). Their resonant contributions to and meson decays have been studied in Refs. K-decPaper (); tau-decPaper (); Ivanov:2004ch (); M-decPaper (); Cvetic:2010rw (); Atre:2009rg (). Another potential process to look for sterile Majorana neutrinos is like-sign dilepton production in hadron collisions Almeida:2000pz (); Panella:2001wq (); Han:2006ip (); Kovalenko:2009td (). Possible implications of sterile neutrinos have been also studied in LFV muonium decay and high-energy muon-electron scattering Cvetic:2006yg (). An interesting explanation of anomalous excess of events observed in the LSND LSND () and MiniBooNE MiniBooNE () neutrino experiments has been recently proposed Gninenko () in terms of sterile neutrinos with masses from 40 MeV to 80 MeV. An explanation comes out of their possible production in neutral current interactions of and subsequent radiative decay to light neutrinos.

Here we study a scenario with only one sterile neutrino state . Phenomenology of a single sterile neutrino is specified by its mass and three mixing matrix elements , , . In the present paper we focus on the derivation of limits on the matrix element , which is currently least constrained in the literature. Towards this end we use the results of experimental measurements of branching ratios of purely leptonic decays and semileptonic decays of and mesons PDG (). One of the key points of our derivation is its model independent character, in the sense that we do not apply any additional assumptions on the relative size of the three mixing parameters . Such ad hoc assumptions are typical in the literature and stem from the fact that all these three parameters enter in the decay rate formulas of any decay, potentially receiving contribution from as an intermediate state. Therefore, in order to extract individual limits on each mixing parameter one may need additional information on them. We will show that in purely leptonic decays it is unnecessary and in the other cases this sort of information can be procured by a joint analysis of certain sets of leptonic and semileptonic decays of and .

The paper is organized as follows. In the next Section II we present decay rate formulas for and pseudoscalar meson LFV and LNV decays in the resonant domains of sterile neutrino mass . In Section III we derive upper limits on from the existing experimental data on purely leptonic 5-body decays, semileptonic and decays, considering sterile neutrino contribution as an intermediate state and in some cases as one of the final state particles. Section IV contains summary and discussion of our main results.

## Ii Decay Rates

Neutrino interactions are represented by the SM Charged (CC) and Neutral Current (NC) Lagrangian terms. In the mass eigenstate basis they read

(4) |

where and . We consider the case with a single sterile neutrino and, therefore, we choose and identify .

In what follows we study sterile neutrino contribution to the following decays

(5) | |||||

(6) |

where and . In the first decay of Eq. (5) both denote the standard neutrino or antineutrino dominated by any of the neutrino flavors . These reactions include lepton number and flavor conserving as well as LFV and LNV decays. In the first case they receive the SM contributions, which alone give good agreement with the experimental data.

The LFV and LNV decays (5) are only possible beyond the SM. In the present framework they proceed according to the diagrams shown in Fig.1 with sterile neutrino as a virtual particle. Considering LNV decays we assume that sterile neutrino is a Majorana particle . When the intermediate sterile neutrino in these diagrams is off-shell their contribution to the processes (5) is negligibly small tau-decPaper (), being far away from experimental reach. On the other hand there exist specific domains of sterile neutrino mass where N comes, for kinematical reasons, close to its mass-shell leading to resonant enhancement K-decPaper (); tau-decPaper (); M-decPaper () of the diagrams in Fig.1. These domains of will be specified below.

The decay rate formulas for the reactions in Eq. (5) can be directly calculated from the diagrams in Fig.1 and Lagrangian (4) for arbitrary mass of sterile neutrino. We focus on the regions of where the sterile neutrino contribution is resonantly enhanced K-decPaper (); tau-decPaper (); M-decPaper (). In these mass domains the intermediate sterile neutrino in Fig. 1 can be treated as nearly on-mass-shell state. This is to say, the sterile neutrino is produced in the left vertices of the diagrams in Fig.1, propagate as a free unstable particle and then finally decays in the right vertices. Thus the decay rate formulas for the reactions can be represented in the form of products of the two factors: or meson decay rate to the sterile neutrino and a branching ratio of the sterile neutrino decay , where represent final state particles of (5). This representation is approximate and valid in the “narrow width approximation” , where is the total decay width of sterile neutrino. As seen from Fig.6, this condition is satisfied in the region of studied in our analysis where MeV . Below we list the decay rate formulas in this approximation for the reactions in Eq. (5) specifying the corresponding resonant regions of where they are applicable. These formulas are readily derived from the diagrams in Fig.1, considering the two vertices as the two independent processes of sterile neutrino production and its subsequent decay.

For semileptonic decays of mesons and -lepton the decay rate formulas are

(7) | |||||

(8) | |||||

(9) |

valid in ,

(10) |

valid in . Studying in subsection III.1 purely leptonic -decays shown in Eq. (5), we will need the decay rates summed over all the standard light neutrino and antineutrino in the final state. The corresponding formulas take the form

valid in . Here for Dirac and Majorana case of sterile neutrino , respectively. Summation in (II) and (II) runs over . The partial decay rates and and the total decay rate of sterile neutrino involved in Eqs. (7)-(II) are specified in Appendix. Implicitly all the partial decay rates include the corresponding threshold step-functions. For further convenience we rewrite Eq.(II), (II) in the form

where

(15) | |||||

(16) |

In Eqs. (II)-(16) we used notations introduced in Eqs. (45)-(A).

As we already mentioned, in the resonant regions of the sterile neutrino mass , specified in Eqs. (7)-(II), the intermediate sterile neutrino , produced in and meson decays (see Fig.1), propagates as a real particle and decays at certain distance from the production point. If this distance is larger than the size of the detector, the sterile neutrino escapes from it before decaying and the signature of , or cannot be recognized. In this case in order to calculate the rate of or meson decay within a detector one should multiply the theoretical expressions in (7)-(II) by the probability of sterile neutrino decay within a detector of the size . Within reasonable approximations it takes the form Atre:2009rg ()

(17) |

where is the total decay rate of sterile neutrino calculated in (61).

Then, the rates of and meson decays within detector volume should be estimated according to

(18) |

where are decay rates given by Eqs. (7)-(II). In our numerical analysis we take for concreteness which is typical for this kind of experiments. In Fig. 2 we plotted v.s. sterile neutrino mass for several values of mixing matrix elements . For illustration of typical tendencies we assumed in this plot . We do not use this assumption in our analysis. As seen, becomes small for MeV even for rather large values of . Thus, in this region of the effect of finite size of detector, described by , significantly affects the decay rates of the studied processes and should be taken into account.

## Iii Limits on Sterile Neutrino Mixing

In the literature there are various limits on the mixing parameters (with ) extracted from direct and indirect experimental searches PDG () for this particle, in a wide region of its mass. A recent summary of these limits, extracted from the corresponding experimental data, can be found in Ref. Atre:2009rg (). In the present paper we focus on the least constrained mixing parameter . In Fig. 3 we show the exclusion plots for existing in the literature Atre:2009rg () together with our exclusion curves derived in the present section. For derivation of these curves we will analyze sterile neutrino contribution to the decays listed in (5)-(6).

As seen from Eqs. (7)-(II) the decay rates of the processes (5) depend on all the three (with ) mixing matrix elements. In the literature it is common practice to adopt some ad hoc assumptions on their relative size in order to extract limits on them from the experimental bounds on the corresponding decay rates. In particular, limits from CHARM CHARM () and NOMAD NOMAD () plotted in Fig. 3 assume . These assumptions may reduce reliability of the obtained limits. Below we derive analytic expressions for limits on in different mass ranges of without any kind of such assumptions.

### iii.1 Purely leptonic decays

First we exploit for extraction of the following experimental results for the branching ratios of purely leptonic -decays PDG ()

(19) | |||||

(20) |

The first decay has been observed experimentally and its experimentally measured branching ratio agrees with the SM prediction within the standard deviation . Neutrino assignment in the final states of the decays (19)-(20) corresponds to what is suggested by the SM. However, in the experiments, measuring these decays, the final state neutrinos cannot be actually identified. Therefore, considering beyond the SM mechanisms with LFV one should take into account the possibility that all the light neutrinos , and may contribute to the final state of the decays (19)-(20). Formulas (II)-(II) were derived for the very this case. They describe the sterile neutrino resonant contribution (diagram Fig.1(a)) to the decays (19)-(20) and will be used in the analysis of this subsection.

We also assume that the sterile neutrino contribution to the process (19), if exists, should be less than . For the decay (20), not yet observed experimentally, there exists only the above indicated upper bound and the sterile neutrino contribution has to obey this bound.

Taking into account the finite detector size effect according to Eq. (18) we write for decay rate within detector volume

(21) |

with given by (II), (II). As we discussed in the previous section, the probability of sterile neutrino decay within detector becomes rather small for MeV. Therefore, in this mass range we may approximate the expression in (17) by . This is a reasonable approximation for this part of our analysis since the limits, which will be obtained here, correspond to the exclusion curve (a) in Figs. 3 and curves in Figs. 4, 5 located in the region MeV, where .

In this approximation we find from (II) and (21)

(22) | |||||

According to our assumption, discussed after Eqs. (19)-(20), we require

(23) |

where s is the -lepton mean life PDG (). Then we obtain the following upper limits

(24) |

(25) |

Similarly, we derive limits based on the experimental bound (20). Using Eq. (II), we find

(26) | |||||

(27) | |||||

(28) |

Here, denotes left-hand side of the experimental bound in (20). The limits (24)-(28) are plotted in Fig. 3-5 for the case of sterile Majorana neutrino. Drawing the exclusion curves, we selected the most stringent limit among (24)-(28) for each mass value within the studied mass range. As seen, the present experimental data (19)-(20) on purely leptonic -decays set rather weak constraints on and on , in the mass region MeV. Our limits on , corresponding to the curve (a) in Fig 3, are significantly weaker than the limitations from other searches shown in Fig. 3. However, our limits for and in Figs. 4, 5 to our best knowledge are new in this mass region.

### iii.2 Leptonic and semileptonic decays

Now we combine the purely leptonic -decays considered in the previous subsection with the semileptonic decays of and -mesons using the experimental data (19), (20) and the experimental limits on the following branching ratios PDG ():

(29) | |||||

(30) | |||||

(31) |

Assuming that in all these decays sterile neutrino N contributes resonantly we should limit ourselves to the mass domain:

(32) |

Within this mass domain the experimental bounds (30) contribute to our analysis only up to MeV corresponding to the mass range of the resonant contribution of sterile neutrino to these decays of K-meson. In the above list (29)-(31) one could also include the existing experimental bounds on the other LNV and LFV decays of and mesons. However, they have negligible impact on our results presented below.

In this part of our analysis we put for the probability (see Eq. (17)) of decay of nearly on-mass-shell sterile neutrino, resonantly contributing to the analyzed processes. Thus we assume that these processes occur completely within a detector volume. This is a good approximation for the case of the limits on , which will be derived here and displayed in Fig. 3 as curve (b). To see this one can check the plot for shown in Fig. 2.

In the mass domain (32) we can use Eqs. (7)-(II) for the corresponding decay rates. Below we combine these formulas in a system of equations. Solving them with respect to and applying the experimental bounds (19), (20) and (29)-(31) we find upper limits on this mixing parameter. For our purpose it is sufficient to use ether of the two experimental bounds (19), (20). We select (19) which leads to a bit more stringent limits on .

Let us introduce the following notations

(33) | |||||

where are mean lives of and ; the right-hand sides of the experimental bounds in (29)-(31) are denoted by ; the quantity was introduced after Eqs. (19) and (20).

Now we can rewrite the experimental limits on (19) and (29)-(31) in the form

(34) | |||

(35) | |||

(36) |

Here . Solving (34)-(36) we find

(37) |

where

(38) |

We have checked that in the mass region (32) all the coefficients . The parameter is defined in (15). We plotted the corresponding exclusion curve in Fig.3 labeled by (b) for the case of Majorana sterile neutrino. As seen, our limits are more stringent than the existing ones from CHARM CHARM () and DELPHI DELPHI () experiments in the sterile neutrino mass region 300 MeV900 MeV. Note that in difference from the existing limits on our limits are model independent in the sense that we have not made any assumptions on the other two mixing parameters and . Instead, we excluded them combining the experimental limits on the branching ratios of different processes (19), (20) and (29)-(31).

### iii.3 Sterile neutrino in the final state

Other experimental data which we apply for deriving limits on are PDG ()

(39) | |||||

(40) |

where in the first line the central value 17.85 corresponds to and 17.36 to . Both these experimental results agree with the SM predictions within the standard deviations and . We already commented in subsection III.1 (after Eqs. (19), (20)), that in the reported experimental results like in Eqs. (19)-(20) and (39)-(40) the final state neutrino assignment is made according to what is suggested by the SM. However, in the experiments, measuring these decays, the final state neutrinos cannot be actually identified and are observed as a missing energy signature. Therefore, it is liable to imagine that instead of one or even both of the standard light neutrinos in the final states of decays in (39)-(40) there may occur some other neutral particles such as sterile neutrinos. We assume that in these modes of -decay appears one sterile neutrino accompanied by any of . Its mass must satisfy to and for the decays (39) and (40) respectively. We also assume that this contribution, if exists, should be less than the corresponding standard deviation .

The contribution of sterile neutrino to (39), (40) in the form

(41) |

should be less than the corresponding since (39), (40) are in agreement with the SM.

Therefore, using (46) and (47) we find the limits

(42) |

where the minimal of the two values in the curl brackets are selected for each value of . The corresponding exclusion curve is shown in Fig.3 and comprises the two parts (c) and (e). The part (c) is dominated by the constraints on purely leptonic -decay mode while the part (e) is mainly due to the semileptonic mode shown in (41). The exclusion curve (c), (e) cover a mass region MeV. This curve sets new limits on for 0 MeV and 300 MeV 700 MeV. In the region 500 MeV 700 MeV they are less stringent than our limits derived in the previous subsection from the data (19), (20), (29)-(31) and corresponding to the curve (b) in Fig. 3. For 100 MeV the part (c) of our exclusion curve is nearly constant and our limits for this mass range can be displayed as

(43) |

As we discussed previously, sterile neutrino produced in (41) can decay within a detector with a probability defined in (17). This would result in appearance of a displaced vertex attributed to this sort of decay in addition to the production vertex (41). The limit in (42) does not take into account such a possibility and sum up the event rates of sterile neutrino decay both within and outside a detector. However, one can imagine an experiment where the displaced vertices of the above mentioned type are looked for and are either observed or, more probably, excluded at certain confidence level. For the latter case our limits in the region 100 MeV would drastically change. In order to illustrate the influence of this additional criterium of event selection on our limits we impose on the processes (41) a condition that sterile neutrino decays outside detector. This results in multiplication of the corresponding decay rate formulas (46), (47) by the probability factor . The modified limits take the form

(44) |

Here we used an inequality , where with defined in (61), (62). In this case our exclusion curve for in Fig. 3 in comparison to the case of (42) changes its part (e) to (d) leaving the part (c) intact. Now the exclusion curve (c)-(d) covers a mass region MeV. Note again that this is just an illustration of an impact of as yet non-existing experimental data allowing discrimination of the events with the displaced vertices associated with the sterile neutrino decay.

## Iv Summary and Conclusions

We studied resonant contribution of sterile neutrino to leptonic and semileptonic decays of as well as to some semileptonic decays of and mesons. Comparison of our predictions with the corresponding experimental data on these decays allowed us to extract new limits on the mixing matrix element shown in Fig. 3 as curves (b), (c), (e). In the two domains of the sterile neutrino mass 0 MeV and 300 MeV 900 MeV our limits on are more stringent than the limits existing in the literature. For 0 100 MeV our limit to a good approximation is