Seesaw scale discrete dark matter and two-zero texture Majorana neutrino mass matrices

# Seesaw scale discrete dark matter and two-zero texture Majorana neutrino mass matrices

## Abstract

In this paper we present a scenario where the stability of dark matter and the phenomenology of neutrinos are related by the spontaneous breaking of a non-Abelian flavor symmetry (). In this scenario the breaking is done at the seesaw scale, in such a way that what remains of the flavor symmetry is a symmetry, which stabilizes the dark matter. We have proposed two models based on this idea, for which we have calculated their neutrino mass matrices achieving two-zero texture in both cases. Accordingly, we have updated this two-zero texture phenomenology finding an interesting correlation between the reactor mixing angle and the sum of the light neutrino masses. We also have a correlation between the lightest neutrino mass and the neutrinoless double beta decay effective mass, obtaining a lower bound for the effective mass within the region of the nearly future experimental sensitivities.

Neutrino Masses and Mixing, Dark Matter Stability

## I Introduction

Neutrino masses, the existence of dark matter (DM), and the baryon asymmetry in the Universe (BAU) are the most important evidences of physics beyond the Standard Model (SM). Here, we propose that the same symmetry explaining neutrino mixing angles is also responsible for the dark matter stability in the context of the discrete dark matter (DDM) mechanism Hirsch:2010ru (). Under certain conditions it would be possible also to account for the BAU via leptogenesis. The DDM is based upon the fact that the breaking of a discrete non-Abelian flavor symmetry into one of its subgroups by means of the electroweak symmetry breaking mechanism1, accounts for the neutrino masses and mixing pattern and for the dark matter stability.

In the original model Hirsch:2010ru (), the group of even permutation of four objects, , was considered as the flavor symmetry.2 contains one tridimensional irreducible representation, , and three one-dimensional irreducible representations, , , and , whose algebra can be reviewed in the Appendix, see, for instance, Ma:2001dn (); Babu:2002dz (); Altarelli:2005yx (); Altarelli:2006kg (). The particle content includes four Higgs doublets, three of them transforming as an triplet and the SM Higgs as the singlet ; four right-handed (RH) neutrinos, three of them in a triplet representation of , ; and a singlet of , . The lepton doublets and the right charged leptons transform as the three different singlets under , in such a way that the mass matrix for the charged lepton is diagonal. Breaking the symmetry into a subgroup through the electroweak symmetry breaking, provides the stability mechanism for the DM, arising from the -odd part of the triplet and at the same time accounts for the neutrino masses and mixing patterns by means of the type I seesaw Minkowski:1977sc (); Yanagida:1979as (); Glashow (); GellMann:1980vs (); Mohapatra:1980yp (); schechter:1980gr (). The predictions for the neutrino sector are an inverse mass hierarchy spectrum with a massless neutrino, 3, and a vanishing reactor neutrino mixing angle, , that nowadays is ruled out by the current experimental data Tang:2015vug (); Forero:2014bxa (); Gonzalez-Garcia:2015qrr (); Capozzi:2016rtj (). Even though, there is an issue with the reactor mixing angle, the original model has a quite interesting neutrino and DM phenomenology, as can be seen in Boucenna:2011tj ().

In a subsequent paper, this model has been modified Meloni:2010sk (), by adding a fifth RH neutrino transforming as and changing the representation of to . This new model gives as predictions: a normal mass spectrum, a lower bound for the neutrinoless double beta decay effective mass, , and a nonzero reactor neutrino mixing angle. Nevertheless, even if this mixing angle were nonzero at its maximum value, it is again ruled out by the current experimental data Tang:2015vug ().

There are some other works in this direction, where other flavor symmetry groups have been used, for instance, a model based on the dihedral group where some flavor changing neutral currents are present and constrain the DM sector Meloni:2011cc () and a model based on the  Boucenna:2012qb (). Finally, it is worth mentioning that there have been works tackling the problem of the vanishing reactor mixing angle within the DDM model Hamada:2014xha (), but in such a case the symmetry has to be explicitly broken in the scalar potential. For models in which dark matter transforms nontrivially under a non-Abelian flavor symmetry, see, for instance, Adulpravitchai:2011ei (); Varzielas:2015joa (); Varzielas:2015sno ().

The paper is organized as follows: In Sec. II we explain our models giving their matter content and derive the neutrino mass matrices. In Sec. III we discuss the phenomenology, and in Sec. IV we draw our conclusions.

## Ii Reactor mixing angle and the DDM mechanism

We will consider two extensions of the model in Ref. Hirsch:2010ru (), hereafter referred as model A and model B, where in addition to the original model matter content, we have added one extra RH neutrino , in a singlet representation of ( or ), and three real scalar singlets of the SM transforming as a triplet under , . The relevant particle content and quantum numbers of model A and model B are summarized in Tables 1 and 2, respectively. The RH neutrino is assigned to the representation of in model A and to the representation in model B. The flavon fields, , acquire a vacuum expectation value around the seesaw scale, such that is broken into a at this scale instead of at the electroweak scale as in the original model. In this way, the flavon fields contribute to the RH neutrino masses.

### Model A

If we consider the matter content in Table  1, the lepton Yukawa Lagrangian is given by4

 L(A)Y =yeLelceH+yμLμlcμH+yτLτlcτH +yν1Le[NTη]1+yν2Lμ[NTη]1′′+yν3Lτ[NTη]1′+yν4LeN4H+yν5LτN5H (1) +M1NTNT+M2N4N4+yN1[NTϕ]3NT+yN2[NTϕ]1N4+yN3[NTϕ]1′′N5+H.c.

where stands for the product of the two triplets contracted into the representation of . In this way, is responsible for quark (considering the quarks as singlet of ) and charged lepton masses, the latter automatically diagonal, . The Dirac neutrino mass matrix arises from and . The flavon fields will contribute to the RH neutrino mass matrix. Once the flavon fields acquire a vev, will be broken. In order to preserve a symmetry, the alignment of the vevs will be of the form

 ⟨H0⟩=vh≠0,⟨η01⟩=vη≠0,⟨η02,3⟩=0,⟨ϕ1⟩=vϕ≠0,⟨ϕ2,3⟩=0. (2)

Therefore, is the vacuum alignment for the scalar triplets, which is a way to break spontaneously into a subgroup, in the basis where the generator is diagonal, see the Appendix.

From Eqs. (II) and (2), the Dirac neutrino mass matrix is given by

 m(A)D=⎛⎜ ⎜⎝yν1vη00yν4vh0yν2vη0000yν3vη000yν5vh⎞⎟ ⎟⎠, (3)

and the Majorana neutrino mass matrix is

 MR=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝M100yN2vϕyN3vϕ0M1yN1vϕ000yN1vϕM100yN2vϕ00M20yN3vϕ0000⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (4)

With these mass matrices, the light neutrinos get Majorana masses through the type I seesaw relation, , taking the form

 m(A)ν≡⎛⎜⎝a0b00cbcd⎞⎟⎠, (5)

where

 a=(yν4vh)2M2,b=yν1yν5vηvhyN3vϕ−yN2yν4yν5v2hyN3M2,c=yν2yν5vηvhyN3vϕ,d=(yN2yν5vh)2(yN3)2M2−(yν5vh)2M1(yN3vϕ)2+2 yν3yν5vηvhyN3vϕ. (6)

The mass matrix in Eq. (5) has the form of the two-zero neutrino mass matrix Frampton:2002yf (), which phenomenology has been extensively studied in the literature, see, for instance, Frampton:2002yf (); Xing:2002ta (); Frampton:2002rn (); Kageyama:2002zw (); Merle:2006du (); Fritzsch:2011qv (); Ludl:2011vv (); Meloni:2014yea (); Zhou:2015qua (); Kitabayashi:2015jdj (). This matrix is consistent with both neutrino mass hierarchies and can accommodate the experimental value for the reactor mixing angle,  Frampton:2002yf (); Xing:2002ta (); Frampton:2002rn (); Kageyama:2002zw (); Merle:2006du (); Fritzsch:2011qv (); Ludl:2011vv (); Meloni:2014yea (); Zhou:2015qua (); Kitabayashi:2015jdj (). The phenomenological implications of this scenario are studied in Sec. III.

### Model B

The lepton Yukawa Lagrangian for the matter content and assignments of model B, in Table 2, is given by

 L(B)Y =yeLelceH+yμLμlcμH+yτLτlcτH +yν1Le[NTη]1+yν2Lμ[NTη]1′′+yν3Lτ[NTη]1′+yν4LeN4H+yν5LμN5H (7) +M1NTNT+M2N4N4+yN1[NTϕ]3NT+yN2[NTϕ]1N4+yN3[NTϕ]1′N5+H.c.

As in model A, the mass matrix of the charged leptons is diagonal, due to the flavor symmetry, while the Dirac neutrino mass matrix takes the form

 m(B)D=⎛⎜ ⎜⎝yν1vη00yν4vh0yν2vη000yν5vhyν3vη0000⎞⎟ ⎟⎠. (8)

The Majorana neutrino mass matrix is of the same form as Eq. (4). The light neutrinos mass matrix after the type I seesaw is

 m(B)ν≡⎛⎜⎝ab0bdc0c0⎞⎟⎠, (9)

where

 a=(yν4vh)2M2,b=yν1yν5vηvhyN3vϕ−yN2yν4yν5v2hyN3M2,c=yν3yν5vηvhyN3vϕ,d=(yN2yν5vh)2(yN3)2M2−(yν5vh)2M1(yN3vϕ)2+2 yν2yν5vηvhyN3vϕ, (10)

which correspond, as before, to another two-zero texture flavor neutrino mass matrix,  Frampton:2002yf (), which also is consistent with both neutrino mass hierarchies and can also accommodate the reactor mixing angle,  Frampton:2002yf (); Xing:2002ta (); Frampton:2002rn (); Kageyama:2002zw (); Merle:2006du (); Fritzsch:2011qv (); Ludl:2011vv (); Meloni:2014yea (); Zhou:2015qua (); Kitabayashi:2015jdj ().

## Iii Results

In the previous section, we obtained the two-zero texture neutrino mass matrices and for models A and B, respectively. We performed the analysis using four independent constraints, coming from the two complex zeroes, to correlate two of the neutrino mixing parameters: the neutrino masses and mixing angles, the two Majorana phases and the Dirac CP violating phase.5 We took the experimental values of the three mixing angles and the two squared mass differences as inputs and numerically scanned within their regions and determined the regions allowed by two correlated variables of interest. We have used in the analysis the data from three different groups that perform the neutrino global fits Forero:2014bxa (); Gonzalez-Garcia:2015qrr (); Capozzi:2016rtj ().

In Figs. 13, and 5 we show the correlation between the atmospheric mixing angle, , and the sum of light neutrino masses, , for model A on the left panels and model B on the right ones. In these graphics, the allowed regions in vs. for the normal hierarchy (NH) is plotted in magenta and for the inverse hierarchy (IH) in cyan. The in the atmospheric angle are represented by the horizontal blue and red shaded regions for the inverted and normal mass hierarchy, respectively, and the best fit values correspond to the horizontal blue and red dashed lines for the inverse and normal hierarchies respectively. In Forero et al. Forero:2014bxa (), they have a local minimum in the atmospheric mixing angle for the IH analysis; that we represent as a red pointed line in Fig. 1. In addition, in the analysis by Capozzi et al. Capozzi:2016rtj (), they have obtained two different and separated regions in the atmospheric angle also for the IH; that we show as the double blue shaded horizontal bands in Fig. 5. The grey vertical band represents a disfavored region in the sum of light neutrino masses, eV, by the Planck Collaboration Ade:2015xua ().

From the plots in Figs. 1, 3, and 5, it can be seen that in model A both hierarchies have an overlap within the 1 region for the atmospheric mixing angle, while in model B depending on what data is used the overlap not always exists. For data from Forero et al. Forero:2014bxa () and Gonzalez-Garcia et al. Gonzalez-Garcia:2015qrr (), only the NH the atmospheric mixing angle overlaps with the 1 region (even though for data from Gonzalez-Garcia:2015qrr () it happens for large neutrino masses disfavored by Planck). For data from Capozzi et al. Capozzi:2016rtj (), only in the IH case there is the 1 overlap in the second octant for the atmospheric mixing angle. Finally, it is worth mentioning that the NH and IH regions in model A are the same but interchanged in model B.

The other correlation we obtained in the models is the neutrinoless double beta decay effective mass parameter, , with the lightest neutrino mass, , where in the normal hierarchy and in the inverted hierarchy. Figures 24, and 6 show versus for model A () on the left panels and model B () on the right ones. The region for the NH within are in dark magenta and the overlap for the atmospheric mixing angle of 1 in magenta; similarly, the region corresponding to the IH within are in dark cyan and within in cyan. The horizontal red shaded region corresponds the current experimental limit on neutrinoless double beta decay Guiseppe:2008aa (); the red and blue lines are the forthcoming experimental sensitivities on  Agostini:2013mzu (); Albert:2014awa (); Gando:2012zm (); CUORE () and  Bornschein:2003xi (), respectively. The vertical blue shaded region is disfavored by the current Planck data Ade:2015xua (). In the graphics, we also show in yellow and green the bands corresponding to the “flavor-generic” inverse and normal hierarchy neutrino spectra, respectively.

It can be seen from Figs. 2 and 4 that for model B there is no overlap between the prediction and the experimental data for the atmospheric mixing angle and therefore, we only show the data for the regions in the IH. In Fig. 6, it can be seen that also the results in model B do not overlap with the 1 region for the NH case, as we mentioned before. The models predict Majorana phases giving a minimal cancelation for the , as can bee seen in Figs. 24, and 6. The allowed regions for the are in the upper lines for NH and IH generic bands. The two-zero textures and are sensitive to the value of the atmospheric mixing angle. In the cases in which the atmospheric mixing angle prediction overlaps with the experimental value at , it translates to a localized region for neutrinoless double beta decay within the near future experimental sensitivity, which is a desirable feature. A better measurement of the atmospheric mixing angle would be crucial for this kind of scenarios.

The dark matter phenomenology arising from the models (A and B) is different from that in the original DDM model, where the limit for large masses ( GeV) was not allowed. The DM phenomenology is similar to the one in the inert Higgs doublet model Deshpande:1977rw () with two active and two inert Higgses. What can be said about the DM phenomenology is that there is no inconvenient in generating the correct relic abundance even if the mass of the DM candidate is bigger than the mass of the gauge bosons. The limits presented in the minimal dark matter model Cirelli:2005uq () apply, and for those masses, it annihilates mainly into gauge bosons.

Finally, it is worth mentioning that neutrino phenomenology and the dark matter phenomenology are related by the way is broken into the symmetry. This breaking dictates the pattern of masses and the mixing of the neutrinos, and at the same time, this is responsible for the DM stability. This is the connection between DM and neutrinos in the presented models.

## Iv Conclusions

We have constructed two models based on the discrete dark matter mechanism where the non-Abelian flavor symmetry is spontaneously broken at the seesaw scale, into a remanent . In these models, we have a total of five RH neutrinos. In this case, two RH neutrinos are in the odd sector, and the other three RH are even under . These three RH neutrinos are responsible for giving the light neutrino masses via type I seesaw. Additionally, we have added flavon scalar fields leading to the breaking in such a way that we obtained two-zero textures for the light Majorana neutrinos. These textures give rise to rich neutrino phenomenology: the results are in agreement with the experimental data of the reactor mixing angle and accommodate the two possible neutrino mass hierarchies, NH and IH.

Another consequence of the way is broken, in addition to dictating the neutrino phenomenology, is that these models contain a DM candidate stabilized by the remnant symmetry. The DM phenomenology in this case will be different than the original DDM Boucenna:2011tj (), where the limit for large DM masses ( GeV) was not allowed, and will be similar to the inert Higgs doublet model Deshpande:1977rw () with extra scalar fields. A detailed discussion of the DM phenomenology is beyond the scope of the present work and will be presented in a further work inprep ().

Additionally, we have updated the analysis for the two-zero textures mass matrix obtained for both models and . We presented the correlation between the atmospheric mixing angle and the sum of the light neutrino masses as well as the lower bounds for neutrinoless double beta decay effective mass parameter; the latter being in the region of sensitivity of the near future experiments. Finally, if the flavon fields acquire vevs at a scale slightly higher than the seesaw scale, the remaining symmetry at the seesaw scale is the , and this would imply a mixing of the three even RH neutrinos, which could be crucial if we want to have a scenario for leptogenesis, since in the symmetric case this was not possible.6

## Acknowledgements

This work has been supported in part by Grants No. PAPIIT IA101516, No. PAPIIT IN111115, No. CONACYT 132059 and SNI. J.M.L. would like to thank CONACYT (México) for financial support.

## Appendix A The A4 Product Representation

The group has four irreducible representations: three singlets , , and and one triplet and two generators: and following the relations . The one-dimensional unitary representations are

 1:S=1,T=1,1′:S=1,T=ω,1′′:S=1,T=ω2, (11)

where . In the basis where is real diagonal,

 S=⎛⎜⎝1000−1000−1⎞⎟⎠andT=⎛⎜⎝010001100⎞⎟⎠. (12)

The product rule for the singlets are

 1×1=1′×1′′=1,1′×1′=1′′,1′′×1′′=1′, (13)

and triplet multiplication rules are

 (ab)1=a1b1+a2b2+a3b3,(ab)1′=a1b1+ωa2b2+ω2a3b3,(ab)1′′=a1b1+ω2a2b2+ωa3b3,(ab)31=(a2b3,a3b1,a1b2),(ab)32=(a3b2,a1b3,a2b1), (14)

where and .

### Footnotes

1. It is possible to break the flavor symmetry with flavon field at other scale other than the electroweak, as we will discuss later.
2. The motivation for choosing as the flavor group is because it is the smallest non-Abelian discrete group with triplet irreducible representation. Therefore, it is possible to have in the same multiplet some inert and active particles after the flavor symmetry breaking and at the same time a reduced number of couplings for these triplets. Later we will see that this reduced number of couplings is the reason why we got correlations between the observables in the neutrino sector.
3. This is because only two of the RH neutrinos participate in the seesaw mechanism.
4. The contribution accounts for the symmetric part of how the two triplets can be contracted, namely and .
5. The method we have used is known and can be reviewed, for instance, in Ludl:2011vv (); Meloni:2014yea ().
6. This is studied somewhere else inprep ()

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