# Lepton masses and mixings in an multi-Higgs model with radiative seesaw mechanism

###### Abstract

We propose a renormalizable multi-Higgs model with symmetry, accounting for the experimental deviation from the tribimaximal mixing pattern of the neutrino mixing matrix. In this framework we study the charged lepton and neutrino masses and mixings. The light neutrino masses are generated via a radiative seesaw mechanism, which involves a single heavy Majorana neutrino and neutral scalars running in the loops. The obtained neutrino mixings and mass squared splittings are in good agreement with the neutrino oscillation experimental data for both normal and inverted hierarchy. The model predicts an effective Majorana neutrino mass 4 meV and 50 meV for the normal and the inverted neutrino spectrum, respectively. The model also features a suppression of CP violation in neutrino oscillations, a low scale for the heavy Majorana neutrino (few TeV) and, due to the unbroken symmetry, a natural dark matter candidate.

## I Introduction

The existence of three generations of fermions, as well as their particular pattern of masses and mixing cannot be understood within the Standard Model (SM), and makes it appealing to consider a more fundamental theory addressing these issues. This problem is especially challenging in the neutrino sector, where the striking smallness of neutrino masses and large mixing between generations suggest a different kind of underlying physics than what should be responsible for the masses and mixings of the quarks. Unlike in the quark sector, where the mixing angles are very small, two of the three neutrino mixing angles, the atmospheric and the solar are large, while the reactor angle is comparatively small PDG ; Abe:2011sj ; Adamson:2011qu ; Abe:2011fz ; An:2012eh ; Ahn:2012nd ; Tortola:2012te ; Fogli:2012ua ; GonzalezGarcia:2012sz .

In the literature there has been a formidable amount of effort to understand the origin of the leptonic flavor structure, with various proposed scenarios and models of neutrino mass generation. Among those approaches to understand the pattern of neutrino mixing, models with discrete flavor symmetries are particularly popular (for recent reviews see Refs. King:2013eh ; Altarelli:2010gt ; Ishimori:2010au ). There is a great variety of such models, some with Multi-Higgs sectors textures ; Ma:2006km ; Ma:2006fn ; Hernandez:2013mcf ; Machado:2010uc ; Ma:2001dn ; Babu:2002dz ; Altarelli:2005yx ; Altarelli:2009gn ; Bazzocchi:2009pv ; He:2006dk ; Fukuyama:2010mz ; Fukuyama:2010ff ; Holthausen:2012wz ; Ahn:2012tv ; Ahn:2013mva ; Chen:2012st ; Toorop:2010ex ; Memenga:2013vc ; Ferreira:2013oga ; Felipe:2013vwa ; Machado:2007ng ; Machado:2011gn ; Felipe:2013ie ; Varzielas:2012ai ; Ishimori:2012fg ; Bhattacharya:2013mpa ; Teshima:2005bk ; Mohapatra:2006pu ; Ma:2013zca ; Canales:2013cga ; Canales:2012dr ; Dong:2011vb ; Kajiyama:2013sza ; Hernandez:2013hea ; Mohapatra:2012tb ; Varzielas:2012pa ; Ding:2013hpa ; Cooper:2012bd ; King:2013hj ; Morisi:2013qna ; Morisi:2013eca ; Varzielas:2012nn ; Bhattacharyya:2012pi ; Ma:2013xqa ; Nishi:2013jqa ; Varzielas:2013sla ; Aranda:2013gga , Extra Dimensions Rius:2001dd ; Dobrescu:1998dg ; Altarelli:2005yp ; Ishimori:2010fs ; Kadosh:2010rm ; Kadosh:2013nra ; Ding:2013eca ; CarcamoHernandez:2012xy , Grand Unification GUT or Superstrings String . Another approach attempts to describe certain phenomenological features of the fermion mass hierarchy by postulating particular zero-texture Yukawa matrices textures .

In this context, the groups explored recently in the literature include He:2006dk ; Fukuyama:2010mz ; Fukuyama:2010ff ; Ahn:2012tv ; Ahn:2013mva ; Chen:2012st ; Toorop:2010ex ; Memenga:2013vc ; Holthausen:2012wz ; Ferreira:2013oga ; Felipe:2013vwa ; Machado:2007ng ; Machado:2011gn ; Felipe:2013ie ; Varzielas:2012ai ; Ishimori:2012fg ; Bhattacharya:2013mpa ; King:2013hj ; Morisi:2013qna ; Morisi:2013eca ; Altarelli:2005yp ; Kadosh:2010rm ; Kadosh:2013nra , Varzielas:2012nn ; Bhattacharyya:2012pi ; Ma:2013xqa ; Nishi:2013jqa ; Varzielas:2013sla ; Aranda:2013gga , Teshima:2005bk ; Mohapatra:2006pu ; Ma:2013zca ; Canales:2013cga ; Canales:2012dr ; Dong:2011vb ; Kajiyama:2013sza ; Hernandez:2013hea , Altarelli:2009gn ; Bazzocchi:2009pv ; Mohapatra:2012tb ; Varzielas:2012pa ; Ding:2013hpa ; Ding:2013eca ; Ishimori:2010fs and Cooper:2012bd . These models can be implemented in a supersymmetric framework Babu:2002dz ; Altarelli:2005yx ; Altarelli:2009gn ; Bazzocchi:2009pv ; Morisi:2013eca , or in extra dimensional scenarios with Ishimori:2010fs ; Ding:2013eca or Altarelli:2005yp ; Kadosh:2010rm ; Kadosh:2013nra .

The popular tribimaximal (TBM) ansatz for the leptonic mixing matrix

(1) |

can originate, in particular, from . TBM corresponds to mixing angles with , , and . On the other hand the T2K Abe:2011sj , MINOS Adamson:2011qu , Double Chooz Abe:2011fz , Daya Bay An:2012eh and RENO Ahn:2012nd experiments have recently measured a non-vanishing mixing angle , ruling out the exact TBM pattern. The global fits of the available data from neutrino oscillation experiments Tortola:2012te ; Fogli:2012ua ; GonzalezGarcia:2012sz give experimental constraints on the neutrino mass squared splittings and mixing parameters. We use the values from Tortola:2012te , shown in Tables 1 and 2, for the cases of normal and inverted hierarchy, respectively. It can be seen that the data deviate significantly from the TBM pattern.

Parameter | (eV) | (eV) | |||

Best fit | |||||

range | |||||

range | |||||

range |

Parameter | (eV) | (eV) | |||

Best fit | |||||

range | |||||

range | |||||

range |

Here we present a renormalizable model with discrete flavor symmetry, which is consistent with the current neutrino data for the neutrino masses and mixings shown in Tables 1,2 and which has less effective model parameters than other similar models, as discussed in section IV. We choose since it is the smallest symmetry with one three-dimensional and three distinct one-dimensional irreducible representations, where the three families of fermions can be accommodated rather naturally. Thereby we unify the left-handed leptons in the triplet representation and assign the right-handed leptons to singlets. This type of setup was proposed for the first time in Ref. Ma:2001dn . In our model there is only one right-handed SM singlet Majorana neutrino , and the scalar sector includes three triplets, one of which is a SM singlet while the other two are doublets. We further impose on the model a discrete symmetry, in order to separate the two triplets transforming as doublets, so that one of them participates only in those Yukawa interactions which involve right-handed singlets , while the other one participates only in those with the right-handed SM sterile neutrino . Finally, a (spontaneously broken) symmetry is introduced to forbid terms in the scalar potential with odd powers of the SM singlet scalar field , the only one transforming non-trivially under . We assume that the symmetry is not affected by the Electroweak Symmetry Breaking. Therefore the scalar fields coupled to the neutrinos have vanishing vacuum expectation values, which implies that the light neutrino masses are not generated at tree level via the usual seesaw mechanism, but instead are generated through loop corrections in a variant of the so-called radiative seesaw mechanism. The loops involve a heavy Majorana neutrino and neutral scalars, which in turn couple through quartic interactions with other neutral scalars in the external lines. The smallness of neutrino masses generated via a radiative seesaw mechanism is attributed to the smallness of the loop factor and to the quadratic dependence on the small neutrino Yukawa coupling. The scale of new physics can therefore be kept low, with the heavy Majorana neutrino mass of a few TeV. The radiative seesaw mechanism has been discussed in Refs. Ahn:2012tv ; Ahn:2013mva in the context of a similar model, but with a field content quite different from ours: we introduce only one SM singlet Majorana neutrino instead of an triplet, with the lepton doublets as triplets, as in Ref. He:2006dk and many other models, but not as in Ref. Ahn:2012tv , where they are assigned to singlets. Our scalar content is also distinct, with one additional triplet (and no singlets), which acquires a VEV in a different direction of the group space.

The paper is organized as follows. In section II we outline the proposed model. The results, in terms of neutrino masses and mixing, are presented in section III. This is followed by a numerical analysis in section IV. We conclude with discussions and a summary in V. Several technical details are presented in appendices: appendix A collects some necessary facts about the group, appendix B contains a discussion of the full invariant scalar potential, and appendix C deals with the mass spectrum for the physical scalars that enter in the radiative seesaw loops.

## Ii The Model

Our model is a multi-Higgs doublet extension of the Standard Model (SM), with the full symmetry experiencing a two-step spontaneous breaking

(2) | |||

We extend the fermion sector of the SM by introducing only one additional field, a SM singlet Majorana neutrino, . The scalar sector is significantly enlarged and contains the six doublets and three singlets . We group them in triplets of . The complete field content and its assignments is given below:

(3) | |||

(4) | |||

(5) | |||

(6) |

Here the numbers in bold face are dimensions of representations of the corresponding group factor in Eq. (2), the third number from the left is the weak hypercharge and the last two numbers are and parities, respectively. The three families of the left-handed SM doublet leptons are unified in a single triplet while the right-handed SM singlet charged leptons are accommodated in the three distinct singlets . The only right-handed SM singlet neutrino introduced in our model is assigned to of in order for its Majorana mass term be invariant under this symmetry. The presence of this term is crucial for our construction as explained below. Note that neither the nor singlet representations of satisfy this condition as can be seen from the multiplication rules in Eq. (41).

The two SM doublet triplet scalars are distinguished by their parities . We require that this symmetry remains unbroken after the electroweak symmetry breaking. Therefore, , which transforms non-trivially under , does not acquire a vacuum expectation value. The preserved discrete symmetry also allows for stable dark matter candidates, as in Ma:2006km ; Ma:2006fn . In our model they are either the lightest neutral component of or the Majorana neutrino . We do not address this question in the present paper. We introduce two SM doublet triplets, in order to ensure that one scalar triplet gives masses to the charged leptons, while the other one , with vanishing VEV, couples to the SM singlet neutrino . Thus neutrinos do not receive masses at tree level. The SM singlet triplet is introduced in order to generate a neutrino mass matrix texture compatible with the experimentally observed deviation from the TBM pattern. As we will explain in the following, the neutrino mass matrix texture generated via the one loop seesaw mechanism is mainly due to the VEV of the SM singlet triplet scalar , which is assumed to be much larger than the scale of the electroweak symmetry breaking GeV. In this way, the contribution associated with the direction in -space that shapes the charged lepton mass matrix is suppressed and effectively absent in the neutrino mass matrix, leading to a mixing matrix that is TBM to a good approximation. The discrete symmetry is also an important ingredient of our approach, as will be shown below. Once it is imposed it forbids the terms in the scalar potential involving odd powers of the SM singlet triplet scalar . This results in a reduction of the number of free model parameters and selects a particular direction of symmetry breaking in the group space. The symmetry is broken after the field acquires a non vanishing vacuum expectation value.

With the field content of Eqs. (3)-(6), the Yukawa part of the model Lagrangian for the lepton sector takes the form

(7) |

with (). The subscripts denote projecting out the corresponding singlet in the product of the two triplets.

Note that the assignment of the charged right-handed leptons (5) to different singlets leads, as can be seen in Eq. (7), to different Yukawa couplings of the electrically neutral components of the to the different charged leptons . The lightest of the should be interpreted as the SM-like 125 GeV Higgs observed at the LHC LHC-H-discovery , and the mentioned non-universality of its couplings to the charged leptons is in agreement with the recent ATLAS result ATLAS-CONF-2013-010 , strongly disfavoring the case of coupling universality.

As can be seen from the Appendix C, the masses of all the neutral scalar states from the triplets and , except for the SM-like Higgs , are proportional to GeV and consequently are very heavy. Our model is not predictive in the scalar sector, having numerous free uncorrelated parameters in the scalar potential. We simply choose the scale such that the heavy scalars are pushed outside the LHC reach. The loop effects of the heavy scalars contributing to certain observables can be suppressed by the appropriate choice of the other free parameters. All these adjustments, as will be shown in Sec. IV, do not affect the neutrino sector, which is totally controlled by three effective parameters, depending in turn on the scalar potential parameters and the lepton-Higgs Yukawa couplings.

The scalar fields can be decomposed as:

(8) |

with

(9) |

The Higgs doublets and the singlet fields can acquire vacuum expectation values:

(10) |

Our requirement (see (2)) that is preserved implies, according to the field assignment of (3), that

(11) |

This can be achieved by having a positive squared mass term of in the scalar potential. As a consequence of (7) and (11) neutrinos do not acquire masses at tree level. As will be discussed in more detail in section III, their masses are radiatively generated through loop diagrams involving virtual neutral scalars and the heavy Majorana neutrino in the internal lines. The aforementioned virtual scalars couple to real scalars due to the scalar quartic interactions, leading to the radiative seesaw mechanism of neutrino mass generation Ma:2006km ; Ma:2006fn .

We assume the following VEV pattern for the neutral components of the SM Higgs doublets () and for the components of the triplet SM singlet scalar :

(12) |

Here and . This choice of directions in the space is justified by the observation that they describe a natural solution of the scalar potential minimization equations. Indeed, in the single-field case, invariance readily favors the direction over e.g. the solution for large regions of parameter space. The vacuum is a configuration that preserves a subgroup of , which has been extensively studied by many authors (see for example Refs. Altarelli:2005yp ; Altarelli:2005yx ; He:2006dk ; Toorop:2010ex ; Ahn:2012tv ; Mohapatra:2012tb ; Chen:2012st ). In our model we have more fields, but there are also classes of the invariants favoring respective VEVs of two fields in orthogonal directions, as desired for our analysis. Therefore our assumption is essentially that the quartic couplings in the potential involving and are within the range of the parameter space where these directions are the global minimum. More details are presented in Appendix B, where the minimization conditions of the full scalar potential of our model are considered, showing that the vacuum (12), together with the vacuum (12), are consistent.

As follows from Eqs. (7) and (8), the neutrino Yukawa interactions are described by the following Lagrangian:

(13) |

We consider the scenario where . A moderate hierarchy in the VEVs is quite natural, given that is a SM gauge singlet and its VEV does not have to be related to the electroweak scale. The scale of is ultimately controlled by the squared mass term in the potential. From Eq. (7) and Eq. (76) it follows (for details see Appendix C) that the neutrino Yukawa interactions, in terms of the physical scalar fields, can be approximately written as:

(14) | |||||

When the subleading effects are considered, there is some mixing between the scalar states, so that , will appear in the Yukawa couplings of , , and the other scalars will also appear in the Yukawa couplings to . As described in more detail in Appendix C, the parameter is given by:

(15) |

in terms of the masses () of the neutral CP-odd scalar fields.

## Iii Lepton masses and mixing

From Eq. (7), and by using the product rules for the group given in Appendix A, it follows that the charged lepton mass matrix is given by

(16) |

The neutrino mass term does not appear at tree-level due to vanishing v.e.v. of field (11). It arises in the form of a Majorana mass term

(17) |

from radiative corrections at 1-loop level. The leading 1-loop contributions to the complex symmetric Majorana neutrino mass matrix are derived from Eqs. (14) and (77). The corresponding diagrams are shown in Fig. 1.

In the approximation discussed in Appendix C we obtain

(18) |

where:

(19) | |||||

(20) |

Here is a dimensionless parameter, which takes into account the difference between a pair of quartic couplings of the scalar potential (see Appendix C for details). We introduced the function Hernandez:2013mcf :

(21) |

Since depends only on the square of the VEVs, even a moderate hierarchy in the VEVs significantly suppresses contributions related to . Furthermore, because , and are only generated through and are then much smaller than and (). Consequently the zero entries in Eq. (18) become

(22) |

and are strongly suppressed in comparison to the other entries if , as assumed in our model. Note that a similar neutrino mass matrix texture was obtained in Ref.Memenga:2013vc from higher dimensional operators.

The neutrino mass matrix given in Eq. (18) depends effectively only on three parameters: , and . As seen from Eqs. (19), (20), the parameters and contain the dependence on various model parameters. It is relevant that and are loop suppressed and are approximately inverse proportional to . As seen from Eqs. (19) and (20), a non-vanishing mass splitting between the CP even and CP odd neutral scalars is crucial. Its absence would lead to massless neutrinos at one loop level. Note also that universality in the quartic couplings of the scalar potential, which would correspond to , would imply and lead to only one massive neutrino. For simplicity, we parametrize the non-universality of the relevant couplings through the parameter , defined in Eq. (53). As will be shown below, the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix depends only on the parameter , while the neutrino mass squared splittings are controlled by parameters and .

A complex symmetric Majorana mass matrix , as in Eq. (17), can be diagonalized by a unitary rotation of the neutrino fields so that

(23) |

where are real and positive. The rotation matrix has the form

(24) |

We identify the Majorana neutrino masses and Majorana phases for the two possible solutions with with the normal (NH) and inverted (IH) mass hierarchies, respectively. They are

NH | (25) | ||||

IH | (26) |

Note that the nonvanishing Majorana phases are and for normal and inverted mass hierarchies, respectively.

With the rotation matrices in the charged lepton sector , given in Eq. (16), and in the neutrino sector , given in Eq. (24), we find the PMNS mixing matrix:

(27) |

From the standard parametrization of the leptonic mixing matrix, it follows that the lepton mixing angles are PDG :

(28) | |||

where the upper sign corresponds to normal () and the lower one to inverted () hierarchy, respectively. The PMNS matrix (27) of our model reproduces the magnitudes of the corresponding matrix elements of the TBM ansatz (1) in the limit and for the inverted and the normal hierarchy, respectively. In both cases the special value for implies that the physical neutral scalars originating from are degenerate in mass. Notice that the lepton mixing angles are controlled by the Majorana phases , where the plus and minus signs correspond to normal and inverted mass hierarchy, respectively.

The Jarlskog invariant and the CP violating phase are given by PDG :

(29) |

Since , we predict and for , implying that in our model CP violation is suppressed in neutrino oscillations.

## Iv Phenomenological implications

In the following we adjust the free parameters of our model to reproduce the experimental values given in Tables 1, 2 and discuss some implications of this choice of the parameters.

As seen from Eqs. (25), (26) and (27), (28) we have only three effective free parameters to fit: , and . It is noteworthy that in our model a single parameter () determines all three neutrino mixing parameters , and as well as the Majorana phases . The parameters and control the two mass squared splittings . Therefore we actually fit only to adjust the values of , while and for the NH and the IH hierarchies are simply

(30) | |||

(31) |

as follows from Eqs. (25), (26) and the definition . In Eqs. (30), (31) we assumed the best fit values of from the Tables 1, 2.

Varying the model parameter in Eq. (28) we have fitted the to the experimental values in Tables 1, 2. The best fit result is:

(32) | |||

(33) |

Comparing Eqs. (32), (33) with Tables 1, 2 we see that and are in excellent agreement with the experimental data, for both NH and IH, with within a deviation from its best fit values.

The effective parameters , and depend on various model parameters: the SM singlet neutrino Majorana mass , the quartic and bilinear couplings of the model Lagrangian (7), (45), as well as on the scale of symmetry breaking . It is worth checking that the solution in Eqs. (30)-(33) does imply neither fine-tuning or very large values of dimensionful parameters. For this purpose consider a point in the model parameter space with all the relevant dimensionless quartic couplings in Eqs. (45), (53) given by

(34) |

compatible with the perturbative regime (). Absence of fine-tuning in this sector favors . Using Eqs. (68)-(74) one may derive an order of magnitude estimate

(35) |

where the function for the values chosen in Eq. (34). In this estimation we assumed and . Let us also assume that the neutrino and electron Yukawa couplings in Eq. (7) are comparable . From Eqs. (7), (8), (12) and the value of the electron mass we estimate

(36) |

Then from Eqs. (30), (31) and (35) we roughly estimate

(37) |

Therefore for any value GeV and without special tuning of the model parameters we are in the ballpark of the neutrino mass squared splittings measured in neutrino oscillation experiments (Tables 1, 2). Both the scale of new physics , related to the symmetry, and the SM singlet Majorana neutrino mass could be comparatively low, around a few TeV.

With the values of the model parameters given in Eqs. (30)-(33), derived from the oscillation experiments, we can predict the amplitude for neutrinoless double beta () decay, which is proportional to the effective Majorana neutrino mass

(38) |

where and are the PMNS mixing matrix elements and the Majorana neutrino masses, respectively.

Then, from Eqs. (24)-(27) and (30)-(33), we predict the following effective neutrino mass for both hierarchies:

(39) |

This is beyond the reach of the present and forthcoming neutrinoless double beta decay experiments. The presently best upper limit on this parameter meV comes from the recently quoted EXO-200 experiment Auger:2012ar yr at the 90 % CL. This limit will be improved within the not too distant future. The GERDA experiment Abt:2004yk ; Ackermann:2012xja is currently moving to “phase-II”, at the end of which it is expected to reach yr, corresponding to MeV. A bolometric CUORE experiment, using Alessandria:2011rc , is currently under construction. Its estimated sensitivity is around yr corresponding to meV. There are also proposals for ton-scale next-to-next generation experiments with Xe KamLANDZen:2012aa ; Auty:2013:zz and Ge Abt:2004yk ; Guiseppe:2011me claiming sensitivities over yr, corresponding to meV. For recent experimental reviews, see for example Ref. Barabash:1209.4241 and references therein. Thus, according to Eq. (39) our model predicts at the level of sensitivities of the next generation or next-to-next generation experiments.

## V Conclusions

We have presented a simple renormalizable model that successfully accounts for the charged lepton and neutrino masses and mixings. The neutrino masses arise from a radiative seesaw mechanism, which explains their smallness, while keeping the scale of new physics at the comparatively low values, which could be about a few TeV (for the single SM singlet neutrino ). The neutrino mixing is approximately tribimaximal due to the spontaneously broken symmetry of the model. The experimentally observed deviation from the TBM pattern is implemented by introducing the SM singlet triplet . Its VEV breaks symmetry and properly shapes the neutrino mass matrix at 1-loop level. CP violation in neutrino oscillations is suppressed.

The model has only 3 effective free parameters in the neutrino sector, which, nevertheless, allowed us to reproduce with good accuracy the mass squared splittings and all mixing angles measured in neutrino oscillation experiments for both normal and inverted neutrino spectrum.

The model predicts the effective Majorana neutrino mass for neutrinoless double beta decay to be 4 meV and 50 meV for the normal and the inverted neutrino spectrum, respectively.

The lightest neutral scalar of our model, , interpreted as the SM-like 125 GeV Higgs boson observed at the LHC, has non-universal Yukawa couplings to the charged leptons . This is in agreement with the recent ATLAS result ATLAS-CONF-2013-010 , strongly disfavoring the case of Yukawa coupling universality.

## Acknowledgments

This work was partially supported by Fondecyt (Chile) under grants 1100582 and 1100287. IdMV was supported by DFG grant PA 803/6-1 and by the Swiss National Science Foundation. HP was supported by DFG grant PA 803/6-1. AECH thanks Dortmund University for hospitality where part of this work was done. The visit of AECH to Dortmund University was supported by Dortmund University and DFG-CONICYT grant PA-803/7-1.

## Appendix A The product rules for

The following product rules for the group were used in the construction of our model Lagrangian:

(40) | |||

(41) |

Denoting and as the basis vectors for two -triplets , one finds:

(42) | |||

(43) | |||

(44) |

where . The representation is trivial, while the non-trivial and are complex conjugate to each other. Comprehensive reviews of discrete symmetries in particle physics can be found in Refs. King:2013eh ; Altarelli:2010gt ; Ishimori:2010au ; Discret-Group-Review .

## Appendix B Scalar Potential

The scalar potential of the model is constructed of the three triplet fields and in the way invariant under the group in Eq. (2).

For convenience we separate its terms into the three different groups as

(45) |

where

(46) | |||||

(47) | |||||

(48) |

Where all parameters of the scalar potential have to be real.

Now we are going to determine the conditions under which the VEV pattern for the components of the triplet , given in Eq. (12), is a solution of the scalar potential, assuming that the vacuum preserves the appropriate subgroup of as in Eq. (12). Then, from the previous expressions and from the minimization conditions of the scalar potential, the following relations are obtained:

(49) | |||||

(50) | |||||