Discovering intermediate mass sterile neutrinos through \tau^{-}\to\pi^{-}\mu^{-}e^{+}\nu(\text{or }\overline{\nu}) decay

# Discovering intermediate mass sterile neutrinos through τ−→π−μ−e+ν(or ¯¯¯ν) decay

## Abstract

Distinguishing the Dirac and Majorana nature of neutrinos remains one of the most important tasks in neutrino physics. By assuming that the decay is resonantly enhanced by the exchange of an intermediate mass sterile neutrino , we show that the energy spectrum of emitted pions and muons can be used to easily distinguish between the Dirac and Majorana nature of . This method takes advantage of the fact that the flavor of light neutrinos is not identified in the tau decay under consideration. We find that it is particularly advantageous, because of no competing background events, to search for in the mass range , where denotes the mass of particle .

###### pacs:
11.30.Fs, 13.35.Dx, 13.35.Hb, 14.60.St

## I Introduction

Lepton number is an absolutely conserved property of the standard model of particle physics. However, observations of flavor oscillations in neutrinos have revealed that neutrinos are massive as well as mixed with one another, opening the interesting possibility that lepton number violation (LNV) occurs. Being electrically neutral fermions, neutrinos can get their observed masses from the well known Yukawa couplings (Dirac neutrinos) or from their self-conjugated fields (Majorana neutrinos), in other words, they can be different or identical to their own antiparticles, respectively. Another interesting consequence of massive neutrinos is that the occurrence of lepton-flavor violation (LFV) in decays of charged leptons is possible due to the mixing mechanism, although with unobservably suppressed rates. Their observation at proposed flavor factories experiments would indicate that mechanisms of LFV beyond the standard neutrino mixing are necessary.

Among the important implications that neutrinos turn out to be Majorana particles is that their masses originate beyond the Yukawa couplings via the Higgs mechanism. The presence of Majorana mass terms for neutrinos entail total lepton number non-conservation by two units (), which would manifest, for instance, in neutrinoless double-beta decays of nuclei, hadrons and leptons. So far, violating processes have not been observed, leading to strong constrains on their mass and mixing parameters Rodejohann:2011mu (). In recent years, many studies have been reported on the possibility of observing LNV by units in decays of mesons mesons () and tau leptons tau () mediated by the exchange of a heavy Majorana neutrino. The upper bounds on branching fractions obtained at flavor factories tau-exp (); mesons-exp () have allowed to set constraints on the square of mixing of light ordinary neutrinos and heavy sterile neutrinos ranging from (where ) as the neutrino mass increases from a few hundreds of MeV to a few GeV.

The question of whether neutrinos are Dirac or Majorana particles remains one of the most important questions for particle physics experiments. The existence of heavy Majorana neutrinos would open the possibility to find mechanisms to explain the smallness of active neutrino masses, as well as the viability of theories to explain the baryon asymmetry or dark matter abundance in the universe Asaka:2005an (); Canetti:2014dka (); Drewes:2015iva (). Different processes that may be sensitive to the effects of light or heavy Majorana neutrinos have been proposed and several experimental searches are underway. Beyond the standard searches of nuclear neutrinoless double-beta decay experiments, several studies have been suggested to distinguish between the Dirac and Majorana nature of neutrinos, among others, precise measurements of neutrino-electron scattering Kayser:1981nw (); Barranco:2014cda (), the pseudo-Dalitz plot of sequential weak decay processes involving pair as final products Kim:2016bxw (), and the spectrum of charged leptons from decays of intermediate on-shell heavy neutrinos Cvetic:2015naa ().

The Dirac-type neutrinos can mediate processes with LFV but not LNV. Conversely, Majorana neutrinos can intervene in processes with both LFV and LNV. In this paper we use these properties of neutrinos to investigate how the study of decays can help to disentangle the nature of the intermediate mass sterile neutrino mediating these decays. Since the flavor of the light active neutrino in the final state is not identified, it is not straightforward to distinguish scenarios with LFV () from those with LNV (). Among the four channels possible in our case, the choice of a mono-energetic pion allows one to single out two specific contributions. We show how, in the context of these two chosen channels, the energy spectrum of the emitted muon can be useful to distinguish between the Majorana and Dirac nature of the intermediate mass sterile neutrino.

This paper is organized as follows: In Sec. II we compute the partial decay width of decays assuming it to be facilitated by an on-shell intermediate mass sterile neutrino. Here we discuss how by choosing mono-energetic pions we can pick out the Feynman diagrams relevant to our methodology. Then in Sec. III we discuss about the muon energy spectrum and show how it helps in distinguishing between the Dirac and Majorana possibilities for the intermediate mass neutrino. In Sec. IV we compare our chosen modes with which has been already searched for experimentally. Here we note that there exists a mass range for in which we have no contamination from background events. Finally we conclude in Sec. V reiterating the salient features of our methodology.

## Ii Partial decay width of the τ−→π−μ−e+ν(or ¯¯¯ν) decays

Let us consider the decay1, which we will assume to be mediated by the exchange of a single on-shell (intermediate mass) sterile neutrino , see Fig. 1. This decay violates lepton flavor irrespective of the specific flavor identity of the neutrino (or anti-neutrino) in the final state. If in the final state we have a muon antineutrino (), the decay under consideration is a LNV process and must be mediated by an intermediate mass Majorana neutrino as shown in Figs. (b)b and (c)c; otherwise, the intermediate neutrino can be either Dirac or Majorana. Since, the flavor of the neutrino (or anti-neutrino) in our final state can not be determined, it is not straightforward to distinguish the Dirac and Majorana cases. Nevertheless, this flavor ambiguity forces us to find some observables that can be used to distinguish between the Dirac and Majorana neutrinos.

It is interesting to observe that the contributions in Figs. (a)a and (b)b can be differentiated from those in Figs. (c)c and (d)d by looking at the pion kinematics2. Owing to the on-shell nature of the intermediate mass neutrino , the pion energy spectrum (in the rest frame of the tau lepton) is mono-energetic for Figs. (a)a and (b)b and has a continuous energy distribution for Figs. (c)c and (d)d. Conversely, the muon spectrum is continuous in all cases. In the following we will focus in the case of a mono-energetic pion emission and show how measurements of the branching ratio and muon energy spectrum in this case allows us to distinguish between the effects of Dirac and Majorana neutrinos.

Thus, let us consider only the Feynman diagrams shown in Figs. (a)a and (b)b. For given a mass of the sterile neutrino , the distinctive feature of Figs. (a)a and (b)b is the mono-energetic pion with energy . The intermediate mass Majorana neutrino can be produced on-shell for mass values in the range , i.e. . Using the narrow-width approximation for , the partial decay width of tau decay in our case is given by:

 Γ(τ−→π−μ−e+ν)=Γ(τ−→π−N)Γ(N→μ−e+ν(or ¯¯¯ν))ΓN, (1)

where is the full width of the sterile neutrino, and the partial decay widths appearing in Eq. (1) are given, respectively, by:

 Γ(τ−→π−N) (2) Γ(N→μ−e+ν(or ¯¯¯ν)) =Γ(N→μ−e+νe)+Γ(N→μ−e+¯¯¯νμ)=G2Fm5N∣∣VμN∣∣2192π3(1−8rμ+8r3μ−r4μ−12r2μlnrμ)(1+α Reμ), (3) (4)

where  MeV PDG2016 (), , , with denoting the mixing of active neutrino of flavor with the sterile neutrino . In the above expressions, is a parameter that allows to distinguish the sterile intermediate Dirac () and Majorana () neutrinos3, , and is the Källén function. It is easy to check that Eq. (2) gives the usual expression for the decay width of when one takes and . Similarly, when and (the case of Dirac neutrino) Eq. (4) becomes identical to the well known rate of muon decay in the crossed channel.

In order to provide an estimate of the branching fraction we need an input for the total width . From the sum over all the exclusive decay channels that open below Cvetic:2014nla () we get the following expressions for total decay width of for two typical values of the neutrino mass:

 ΓN ≈G2Fm5N96π3(15|VeN|2+8∣∣VμN∣∣2+2|VτN|2), (formN=0.25 GeV) (5) ΓN ≈G2Fm5N96π3(7|VeN|2+7∣∣VμN∣∣2+2|VτN|2). (formN=1 GeV) (6)

Using the upper bounds for the mixing elements that were reported in Ref. Cvetic:2014nla (), we evaluate the decay width of the intermediate mass neutrino. The results are given in Table 1.

Finally, the branching fraction for the decay can be obtained by dividing Eq. (1) by the measured width of the lepton PDG2016 (). The estimated upper limits for the partial widths involved Eqs. (2) and (4) are shown in Table 2 for the two values of , and the branching fraction for the decay are shown in the last column.

We conclude that, with current bounds on the mixing elements , the effects the branching fraction of the tau decay under consideration may be more sensitive to the effects of Majorana neutrinos for larger values of heavy neutrino mass. Those branching fractions lie at the edge of Belle II capabilities, which is expected to produce about tau lepton pairs in the full dataset Inami:2016aba (). Conversely, from an experimental upper limit on the decay at future flavor factories, one can constrain the parameter space under the assumption that the exchanged sterile neutrino is a Dirac or Majorana particle.

## Iii Muon energy spectrum in the τ−→π−μ−e+ν(or ¯¯¯ν) decays

From Eq. (4) it is clear that the decay rate for (more precisely, the decay rate of the intermediate mass neutrino ) is the same for both Dirac and Majorana cases up to an overall normalization factor that depends on the nature of the exchanged neutrino. Therefore, in order to distinguish between the Majorana and Dirac cases, we have to look at another observable like the muon energy spectrum.

The normalized muon energy distribution for the decay in the rest frame of is given by,

 1Γ(N→μ−e+ν(or ¯¯¯ν))dΓ(N→μ−e+ν(or ¯¯¯ν))dEμ=96m3N(−13mNm2μ+(12+αReμ)(m2N+m2μ)Eμ−(23+2αReμ)mNE2μ)√E2μ−m2μ((m4N−m4μ)(m4N+m4μ−8m2μm2N)−24m4μm4Nlog(mμmN))(1+αReμ). (7)

The muon energy distributions for  GeV and  GeV and for various values of are shown in Fig. 2. It is clear from these plots that the shape of the muon spectrum can help to differentiate between the contributions of the intermediate mass Dirac and Majorana neutrinos. A peak of the muon energy spectrum at the end of the maximum allowed muon energy would signal that the intermediate mass neutrino is a Dirac particle. Conversely, a peaked spectrum towards lower energies (i.e. away from the kinematic end-point of ) would indicate its Majorana nature.

## Iv Comparison with τ−→μ−π+π− mode, and discussion on some possible new physics contributions

Fig. 3 summarizes the procedure that needs to be followed to establish the nature of the intermediate mass sterile neutrino exchanged in decays (here flavor of the final light neutrino or anti-neutrino is unknown) with anticipation that this three-prong decay is observed at future colliders, such as Belle II. In this context it is important to note that the lepton flavor violating decay can also contribute to our decay channels via the sequential decay . Such lepton flavor violating decays have already been searched for in experiments Aubert:2005tp (); Miyazaki:2012mx (). To make a quantitative comparison between the decay modes and we define the ratio, . Assuming that both the decay modes are facilitated by the exchange of an intermediate mass neutrino and applying the narrow-width approximation for it, we get the result as shown in Fig. 4. Note that in both the decay modes we consider the to be mono-energetic in the rest frame of . From Fig. 4 it is very clear that for lower values of the has much larger branching ratio than . In the region there is no contribution from the mode due to phase space considerations, however, can still give contributions as long as (see the dashed line in Fig. 4 which goes to infinity in the mass region ). Due to lack of any background events in the mass range , it is the experimentally clean region to study the decay mode . It is thus expected that observing the decay modes ought to be feasible with future colliders and following the flowchart of Fig. 3 it would be possible to decipher the nature of the heavy intermediate on-shell sterile neutrino.

Note that there can be some other generic new physics contributions, other than the intermediate mass sterile neutrinos, to our decay mode (or ). Some such possibilities are shown in Fig. 5. As is clear from Fig. 5 there can be some lepton-flavor violating new physics via some scalar () or vector () particle (e.g. lepton flavor violating modes of Higgs, or , or some ), as well as some lepto-quark (, with ) contributions. It must be noted that in all these new physics possibilities, the pion is not mono-energetic. Hence, following the flowchart of Fig. 3, we can avoid these new physics possibilities completely. Furthermore, the lepton flavor violating modes of Higgs, , as well as those of are severely constrained by experimental data. The lepto-quarks in Figs. 5 also facilitate lepton-flavor violation, and must therefore be severely constrained by searches. Thus, other new physics possibilities for (or ) decay are not only very constrained by existing data, but their presence, if any, does not affect our analysis as they are easily discarded by considering mono-energetic pions as emphasized in our methodology here.

## V Conclusion

From the four possible contributions to the decay as shown in Fig. 1, choosing a mono-energetic in the rest frame of should allow us to isolate two of them (Figs. (a)a and (b)b). The spectrum of muons produced from the decay of the intermediate mass neutrino in Figs. (a)a and (b)b, would indicate that a peak observed in the muon energy spectrum below its kinematic endpoint corresponds to an intermediate mass Majorana neutrino. Thus by a clever analysis of the pion and muon energy spectra, we can easily distinguish between the Dirac and Majorana possibilities for the intermediate on-shell sterile neutrino.

Note added: A related idea to the one presented in this paper was reported in Ref. Yuan:2017xdp (). Our work differs in several ways. First, contrary to us, Ref. Yuan:2017xdp () considers the decay , where the lepton and meson have opposite charges (or same-sign charged leptons in the final state) which requires that two separate vertices are detected to avoid the exchange of identical fermions. Second, the authors of Ref. Yuan:2017xdp () compute the pion instead of the muon spectrum. This procedure, however, will not allow to distinguish the nature of the exchanged neutrino since the pion spectra for Dirac and Majorana cases differ only in normalization but not in shape.

###### Acknowledgements.
The work of C.S.K. was supported by the NRF grant funded by the Korean government of the MEST (No. 2016R1D1A1A02936965). G.L.C. is grateful to Conacyt for financial support under Projects No. 236394, No. 250628 (Ciencia Básica), and No. 296 (Fronteras de la Ciencia). C.S.K and D.S. would like to thank CINVESTAV, Mexico, where some initial part of this work was done, for hospitality.

### Footnotes

1. The analogous decay follows a similar discussion under the exchange of flavor labels.
2. Of course, if the lifetime of is large enough, those set of diagrams are also differentiated by the pion (muon) identification at the primary (secondary) or secondary (primary) vertices, respectively. Our analysis assumes no finite propagation of the intermediate mass neutrino with the displaced vertices.
3. Note that a Majorana neutrino () with can not be distinguished here from a Dirac neutrino (). However, if we instead consider the decay and look at the pion and muon energy distributions as discussed in this paper, the above problem does not appear.

### References

1. W. Rodejohann, Int. J. Mod. Phys. E 20, 1833 (2011)
2. C. Dib, V. Gribanov, S. Kovalenko and I. Schmidt, Phys. Lett. B 493, 82 (2000); C. Dib, V. Gribanov, S. Kovalenko and I. Schmidt, Part. Nucl. Lett. 106, 42 (2001); D. Delepine, G. Lopez Castro and N. Quintero, Phys. Rev. D 84, 096011 (2011), Erratum: [Phys. Rev. D 86, 079905(E) (2012)]; G. Lopez Castro and N. Quintero, Phys. Rev. D 87, 077901 (2013); D. Milanes, N. Quintero and C. E. Vera, Phys. Rev. D 93, no. 9, 094026 (2016); H. Yuan, T. Wang, G. L. Wang, W. L. Ju and J. M. Zhang, JHEP 1308, 066 (2013); G. Cvetic, C. Dib, S. K. Kang and C. S. Kim, Phys. Rev. D 82, 053010 (2010); G. Cvetic, C. Dib and C. S. Kim, JHEP 1206, 149 (2012); C. Dib and C. S. Kim, Phys. Rev. D 89, no. 7, 077301 (2014); G. Cvetic, C. S. Kim, Y. J. Kwon and Y. M. Yook, Phys. Rev. D 93, no. 1, 013003 (2016); G. Cvetic and C. S. Kim, Phys. Rev. D 94, no. 5, 053001 (2016), Erratum: [Phys. Rev. D 95, no. 3, 039901(E) (2017)]; G. Cvetic, F. Halzen, C. S. Kim and S. Oh, arXiv:1702.04335 [hep-ph]; G. Cvetic and C. S. Kim, arXiv:1705.09403 [hep-ph]; S. Mandal and N. Sinha, Phys. Rev. D 94, no. 3, 033001 (2016); B. Shuve and M. E. Peskin, Phys. Rev. D 94, no. 11, 113007 (2016); C. O. Dib, C. S. Kim and K. Wang, arXiv:1703.01936 [hep-ph] (to be published in Chinese Physics C.); C. O. Dib, C. S. Kim and K. Wang, Phys. Rev. D 95, no. 11, 115020 (2017).
3. C. Dib, J. C. Helo, M. Hirsch, S. Kovalenko and I. Schmidt, Phys. Rev. D 85, 011301 (2012); G. L. Castro and N. Quintero, Phys. Rev. D 85, 076006 (2012), Erratum: [Phys. Rev. D 86, 079904(E) (2012)]; D. Gomez Dumm and P. Roig, Nucl. Phys. Proc. Suppl. 253-255, 12 (2014); J. Zamora-Saa, JHEP 1705, 110 (2017); G. Cvetic, C. Dib, C. S. Kim and J. D. Kim, Phys. Rev. D 66, 034008 (2002), Erratum: [Phys. Rev. D 68, 059901(E) (2003)].
4. J. P. Lees et al. [BaBar Collaboration], Phys. Rev. D 85, 071103 (2012); J. P. Lees et al. [BaBar Collaboration], Phys. Rev. D 89, no. 1, 011102 (2014); O. Seon et al. [BELLE Collaboration], Phys. Rev. D 84, 071106 (2011); R. Aaij et al. [LHCb Collaboration], Phys. Rev. D 85, 112004 (2012); R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. 112, no. 13, 131802 (2014); J. Harrison [LHCb and BaBar and Belle Collaborations], Nucl. Part. Phys. Proc. 260, 143 (2015).
5. Y. Miyazaki et al. [Belle Collaboration], Phys. Lett. B 719, 346 (2013); Y. Miyazaki et al. [Belle Collaboration], Phys. Lett. B 682, 355 (2010); B. Aubert et al. [BaBar Collaboration], Phys. Rev. Lett. 95, 191801 (2005).
6. M. Drewes and B. Garbrecht, Nucl. Phys. B 921, 250 (2017).
7. L. Canetti, M. Drewes and B. Garbrecht, Phys. Rev. D 90, no. 12, 125005 (2014).
8. T. Asaka, S. Blanchet and M. Shaposhnikov, Phys. Lett. B 631, 151 (2005); T. Asaka and M. Shaposhnikov, Phys. Lett. B 620, 17 (2005); L. Canetti, M. Drewes, T. Frossard and M. Shaposhnikov, Phys. Rev. D 87, 093006 (2013).
9. B. Kayser and R. E. Shrock, Phys. Lett. 112B, 137 (1982).
10. J. Barranco, D. Delepine, V. Gonzalez-Macias, C. Lujan-Peschard and M. Napsuciale, Phys. Lett. B 739, 343 (2014); W. Rodejohann, X. J. Xu and C. E. Yaguna, JHEP 1705, 024 (2017).
11. C. S. Kim and D. Sahoo, arXiv:1612.00607 [hep-ph].
12. G. Cvetic, C. Dib, C. S. Kim and J. Zamora-Saá, Symmetry 7, 726 (2015).
13. C. Patrignani et al. [Particle Data Group], Chin. Phys. C 40, no. 10, 100001 (2016).
14. G. Cvetic, C. S. Kim and J. Zamora-Saá, Phys. Rev. D 89, no. 9, 093012 (2014).
15. K. Inami [Belle II Collaboration], PoS ICHEP 2016, 574 (2016).
16. B. Aubert et al. [BaBar Collaboration], Phys. Rev. Lett. 95, 191801 (2005).
17. Y. Miyazaki et al. [Belle Collaboration], Phys. Lett. B 719, 346 (2013).
18. H. Yuan, Y. Jiang, T. h. Wang, Q. Li and G. L. Wang, arXiv:1702.04555 [hep-ph].
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