Fermion masses and mixings in the 3-3-1 model with right-handed neutrinos based on the S_{3} flavor symmetry.

# Fermion masses and mixings in the 3-3-1 model with right-handed neutrinos based on the S3 flavor symmetry.

A. E. Cárcamo Hernández    R. Martinez    F. Ochoa Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile,
Universidad Nacional de Colombia, Departamento de Física, Ciudad Universitaria, Bogotá D.C., Colombia.
July 12, 2019
###### Abstract

We propose a 3-3-1 model where the symmetry is extended by and the scalar spectrum is enlarged by extra singlet scalar fields. The model successfully describes the observed SM fermion mass and mixing pattern. In this framework, the light active neutrino masses arise via an inverse seesaw mechanism and the observed charged fermion mass and quark mixing hierarchy is a consequence of the symmetry breaking at very high energy. The obtained physical observables for both quark and lepton sectors are compatible with their experimental values. The model predicts the effective Majorana neutrino mass parameter of neutrinoless double beta decay to be 4 and 48 meV for the normal and the inverted neutrino spectra, respectively. Furthermore, we found a leptonic Dirac CP violating phase close to and a Jarlskog invariant close to about for both normal and inverted neutrino mass hierarchy.

## I Introduction

After the discovery of the GeV Higgs boson by ATLAS and CMS collaborations at CERN Large Hadron Collider (LHC) Aad:2012tfa (); Chatrchyan:2012xdj (), the vacancy of the Higgs boson needed for the completion of the Standard Model (SM) at the Fermi scale has been filled and the weak gauge bosons mass generation mechanism has also been confirmed. Despite LHC experiments indicate that the decay modes of the new scalar state are SM like, there is still room for new extra scalar states, whose search are an essential task of the LHC experiments. Furthermore, despite the great consistency of the SM predictions with the experimental data, there are several aspects that the SM do not explain, some of them are the observed hierarchy among charged fermion masses and quark mixing angles, the tiny neutrino masses and the smallness of the quark mixing angles, which contrast with the sizeable leptonic mixing ones. The global fits of the available data from the Daya Bay An:2012eh (), T2K Abe:2011sj (), MINOS Adamson:2011qu (), Double CHOOZ Abe:2011fz () and RENO Ahn:2012nd () neutrino oscillation experiments, constrain the neutrino mass squared splittings and mixing parameters Forero:2014bxa (). It is a well established experimental fact that the observed hierarchy of charged fermion masses goes over a range of five orders of magnitude in the quark sector and that there are six orders of magnitude between the neutrino mass scale and the electron mass. Accommodating the charged fermion masses in the SM requires an unnatural tunning among its different Yukawa couplings. Furthermore, experiments with solar, atmospheric and reactor neutrinos Agashe:2014kda (); An:2012eh (); Abe:2011sj (); Adamson:2011qu (); Abe:2011fz (); Ahn:2012nd () have brought evidence of neutrino oscillations caused by nonzero mass. All these unexplained issues strongly indicate that new physics have to be invoked to address the fermion puzzle of the SM.

The aforementioned flavour puzzle, not understood in the context of the SM, motivates extensions of the Standard Model that explain the fermion mass and mixing patterns. From the phenomenological point of view, it is possible to describe some features of the mass hierarchy by assuming Yukawa matrices with texture zeroes Fritzsch:1977za (); Fukuyama:1997ky (); Du:1992iy (); Barbieri:1994kw (); Peccei:1995fg (); Fritzsch:1999ee (); Roberts:2001zy (); Nishiura:2002ei (); deMedeirosVarzielas:2005ax (); Carcamo:2006dp (); Kajiyama:2007gx (); CarcamoHernandez:2010im (); Branco:2010tx (); Leser:2011fz (); Gupta:2012dma (); CarcamoHernandez:2012xy (); Hernandez:2013mcf (); Pas:2014bra (); Hernandez:2014hka (); Hernandez:2014zsa (); Nishiura:2014psa (); Frank:2014aca (); Ghosal:2015lwa (); Sinha:2015ooa (); Nishiura:2015qia (); Samanta:2015oqa (); Gautam:2015kya (); Pas:2015hca (); Hernandez:2015hrt (). A very promising approach is the use of discrete flavor groups, which have been considered in several models to explain the fermion masses and mixing (see Refs. Ishimori:2010au (); Altarelli:2010gt (); King:2013eh (); King:2014nza () for recent reviews on flavor symmetries). Models with spontaneously broken flavor symmetries may also produce hierarchical mass structures. Recently, discrete groups such as Ma:2001dn (); He:2006dk (); Chen:2009um (); Ahn:2012tv (); Memenga:2013vc (); Felipe:2013vwa (); Varzielas:2012ai (); Ishimori:2012fg (); King:2013hj (); Hernandez:2013dta (); Babu:2002dz (); Altarelli:2005yx (); Morisi:2013eca (); Altarelli:2005yp (); Kadosh:2010rm (); Kadosh:2013nra (); delAguila:2010vg (); Campos:2014lla (); Vien:2014pta (); Hernandez:2015tna (), Kubo:2003pd (); Kobayashi:2003fh (); Chen:2004rr (); Mondragon:2007af (); Mondragon:2008gm (); Bhattacharyya:2010hp (); Dong:2011vb (); Dias:2012bh (); Meloni:2012ci (); Canales:2012dr (); Canales:2013cga (); Ma:2013zca (); Kajiyama:2013sza (); Hernandez:2013hea (); Ma:2014qra (); Hernandez:2014vta (); Hernandez:2014lpa (); Gupta:2014nba (); Hernandez:2015dga (); Hernandez:2015zeh (); Hernandez:2016rbi (), Mohapatra:2012tb (); BhupalDev:2012nm (); Varzielas:2012pa (); Ding:2013hpa (); Ishimori:2010fs (); Ding:2013eca (); Hagedorn:2011un (); Campos:2014zaa (); Dong:2010zu (); VanVien:2015xha (); Arbelaez:2016mhg (), Frampton:1994rk (); Grimus:2003kq (); Grimus:2004rj (); Frigerio:2004jg (); Babu:2004tn (); Adulpravitchai:2008yp (); Ishimori:2008gp (); Hagedorn:2010mq (); Meloni:2011cc (); Vien:2013zra (), Kawashima:2009jv (); Kaburaki:2010xc (); Babu:2011mv (); Gomez-Izquierdo:2013uaa (), Luhn:2007sy (); Hagedorn:2008bc (); Cao:2010mp (); Luhn:2012bc (); Kajiyama:2013lja (); Bonilla:2014xla (); Vien:2014gza (); Vien:2015koa (); Hernandez:2015cra (); Arbelaez:2015toa (), Ding:2011qt (); Hartmann:2011dn (); Hartmann:2011pq (); Kajiyama:2010sb (), Aranda:2000tm (); Aranda:2007dp (); Chen:2007afa (); Frampton:2008bz (); Eby:2011ph (); Frampton:2013lva (); Chen:2013wba (), Ma:2007wu (); Varzielas:2012nn (); Bhattacharyya:2012pi (); Ma:2013xqa (); Nishi:2013jqa (); Varzielas:2013sla (); Aranda:2013gga (); Ma:2014eka (); Abbas:2014ewa (); Abbas:2015zna (); Varzielas:2015aua (); Bjorkeroth:2015uou (); Chen:2015jta (); Vien:2016tmh (); Hernandez:2016eod () and Everett:2008et (); Feruglio:2011qq (); Cooper:2012bd (); Varzielas:2013hga (); Gehrlein:2014wda (); Gehrlein:2015dxa (); DiIura:2015kfa (); Ballett:2015wia (); Gehrlein:2015dza (); Turner:2015uta (); Li:2015jxa () have been considered to explain the observed pattern of fermion masses and mixings. In particular the flavor symmetry is a very good candidate for explaining the prevailing pattern of fermion masses and mixing. The discrete symmetry is the smallest non-Abelian discrete symmetry group having three irreducible representations (irreps), explicitly two singlets and one doublet irreps. The discrete symmetry was used as a flavor symmetry for the first time in Ref. Pakvasa:1977in (). The different models based on discrete flavor symmetries, have as a common issue the breaking of the flavour symmetry so that the observed data be naturally produced. The breaking of the flavour symmetry takes place when the scalar fields acquire vacuum expectation values.

Besides that, another of the greatest misteries in particle physics is the existence of three fermion families at low energies. The origin of the family structure of the fermions can be addressed in family dependent models where a symmetry distinguish fermions of different families. One explanation to this issue can be provided by the models based on the gauge symmetry , also called 3-3-1 models, which introduce a family non-universal symmetry Georgi:1978bv (); Valle:1983dk (); Pisano:1991ee (); Montero:1992jk (); Foot:1992rh (); Frampton:1992wt (); Ng:1992st (); Duong:1993zn (); Hoang:1996gi (); Hoang:1995vq (); Foot:1994ym (); Martinez:2001mu (); Sanchez:2001ua (); Diaz:2003dk (); Diaz:2004fs (); Dias:2004dc (); Dias:2005yh (); Dias:2005jm (); Ochoa:2005ch (); CarcamoHernandez:2005ka (); Salazar:2007ym (); Benavides:2009cn (); Dias:2010vt (); Dias:2012xp (); Alvarado:2012xi (); Catano:2012kw (); Hernandez:2013mcf (); Hernandez:2014lpa (); Vien:2014pta (); Hernandez:2014vta (); Boucenna:2014ela (); Boucenna:2014dia (); Vien:2014gza (); Phong:2014ofa (); Boucenna:2015zwa (); Hernandez:2015cra (); DeConto:2015eia (); Correia:2015tra (); Dong:2015rka (); Hernandez:2015tna (); Okada:2015bxa (); Binh:2015cba (); Hue:2015fbb (); Benavides:2015afa (); Boucenna:2015pav (); Hernandez:2015ywg (); Dong:2015dxw (); Cao:2015scs (); Martinez:2016ztt (); Borges:2016nne (); Okada:2016whh (); Fonseca:2016xsy (); Fonseca:2016tbn (). These models have a number of phenomenological advantages. First of all, the three family structure in the fermion sector can be understood in the 3-3-1 models from the cancellation of chiral anomalies and asymptotic freedom in QCD. Secondly, the fact that the third family is treated under a different representation, can explain the large mass difference between the heaviest quark family and the two lighter ones. Third, these models contain a natural Peccei-Quinn symmetry, necessary to solve the strong-CP problem Pal:1994ba (); Dias:2002gg (); Dias:2003zt (); Dias:2003iq (). Finally, 3-3-1 models including heavy sterile neutrinos have cold dark matter candidates as weakly interacting massive particles (WIMPs) Mizukoshi:2010ky (); Dias:2010vt (); Alvares:2012qv (); Cogollo:2014jia (). Besides that, the 3-3-1 models can explain the TeV diboson excess found by ATLAS Cao:2015lia (). When the electric charge in the 3-3-1 models is defined in the linear combination of the generators and , it is a free parameter, independent of the anomalies (). The choice of this parameter defines the charge of the exotic particles. Choosing , the third component of the weak lepton triplet is a neutral field which allows to build the Dirac matrix with the usual field of the weak doublet. If one introduces a sterile neutrino in the model, then it is possible to generate light neutrino masses via inverse seesaw mechanism. The 3-3-1 models with have the advantange of providing an alternative framework to generate neutrino masses, where the neutrino spectrum includes the light active sub-eV scale neutrinos as well as sterile neutrinos which could be dark matter candidates if they are light enough or candidates for detection at the LHC, if their masses are at the TeV scale. This interesting feature make the 3-3-1 models very interesting since if the TeV scale sterile neutrinos are found at the LHC, these models can be very strong candidates for unraveling the mechanism responsible for electroweak symmetry breaking.

In the 3-3-1 models, one heavy triplet field with a Vacuum Expectation Value (VEV) at high energy scale , breaks the symmetry into the SM electroweak group , while the another two lighter triplets with VEVs at the electroweak scale and , trigger the Electroweak Symmetry Breaking Hernandez:2013mcf (). Besides that, the 3-3-1 model could possibly explain the excess of events in the decay, recently observed at the LHC, since the heavy exotic quarks, the charged Higges and the heavy charged gauge bosons contribute to this process. On the other hand, the 3-3-1 model reproduces an specialized Two Higgs Doublet Model type III (2HDM-III) in the low energy limit, where both electroweak triplets and are decomposed into two hypercharge-one doublets plus charged and neutral singlets. Thus, like the 2HDM-III, the 3-3-1 model can predict huge flavor changing neutral currents (FCNC) and CP-violating effects, which are severely suppressed by experimental data at electroweak scales. In the 2HDM-III, for each quark type, up or down, there are two Yukawa couplings. One of the Yukawa couplings is for generating the quark masses, and the other one produces the flavor changing couplings at tree level. One way to remove both the huge FCNC and CP-violating effects, is by imposing discrete symmetries, obtaining two types of 3-3-1 models (type I and II models), which exhibit the same Yukawa interactions as the 2HDM type I and II at low energy where each fermion is coupled at most to one Higgs doublet. In the 3-3-1 model type I, one Higgs electroweak triplet (for example, ) provide masses to the phenomenological up- and down-type quarks, simultaneously. In the type II, one Higgs triplet () gives masses to the up-type quarks and the other triplet () to the down-type quarks Hernandez:2013mcf ().

It is noteworthy the flavor symmetry was implemented for the first time in the 3-3-1 model of Ref. Dong:2011vb (). That model introduces a new lepton global symmetry, responsible for lepton number and lepton parity. That lepton parity symmetry suppresses the mixing between ordinary quarks and exotic quarks. Furthermore, the new lepton global symmetry enforces to have different scalar fields in the Yukawa interactions for charged lepton, neutrino and quark sectors. The scalar sector of that model includes six scalar triplets and four scalar antisextets. The assignments of the fermion sector of the the aforementioned model, require that these 6 scalar triplets be distributed as follows, 3 for the quark sector, 2 for the charged lepton sector and 1 for the neutrino sector. Furthermore the 4 scalar antisextets are needed to implement a type II seesaw mechanism. In that model, light active neutrino masses are generated from type-I and type-II seesaw mechanisms, mediated by three heavy right handed Majorana neutrinos and four scalar antisextets, respectively. Since the Yukawa terms of that model are renormalizable, to explain the SM charged fermion mass pattern, one needs to impose a strong hierarchy among the charged fermion Yukawa couplings of the model. Furthermore, the work described in Ref. Dong:2011vb () is mainly focused on the lepton sector, while in the quark sector, the obtained quark mass matrices are diagonal and the quark mixing matrix is trivial.

Recently two of us proposed a model Hernandez:2014vta (), with a scalar sector composed of three scalar triplets and seven scalar singlets, that successfully accounts for quark masses and mixings. In that model, all observables in the quark sector are in excellent agreement with the experimental data, excepting , which turns out to be larger by a factor than its corresponding experimental value, and naively deviated 8 sigma away from it. That model has the following drawbacks: is deviated 8 sigma away from its experimental value, a soft breaking term has to be introduced by hand in the low energy scalar potential in order to fullfill its minimization equations, the top quark mass arises from a five dimensional Yukawa term and lepton masses and mixings are not addressed.

It is interesting to find an alternative and better explanation for the SM fermion mass and mixing hierarchy than the ones considered in Refs. Dong:2011vb (); Hernandez:2014vta (). To this end we propose a multiscalar singlet extension of the model with right handed neutrinos, where and an extra discrete group, extends the symmetry of the model and fifteen very heavy singlet scalar fields are added with the aim to generate viable textures for the fermion sector, that successfully describe the observed SM fermion mass and mixing pattern. Let us note that whereas the scalar sector of our model only has three scalar triplets and fifteen scalar singlets, the scalar sector of the flavour 3-3-1 model of Ref. Dong:2011vb () has six scalar triplets and four scalar antisextets. Whereas in the model of Ref. Dong:2011vb (), the quark mixing matrix is equal to the identity, in our model the quark mixing matrix is in excellent agreement with the low energy quark flavor data. In our model, the obtained physical observables in the quark and lepton sector are consistent with the experimental data. Our model at low energies reduces to the 3-3-1 model with right handed neutrinos, where . Furthermore, our current model does not include the new lepton global symmetry presented in the flavor 3-3-1 model of Ref. Dong:2011vb (). Unlike the flavor 3-3-1 model of Ref. Dong:2011vb (), in our current 3-3-1 model, the charged fermion mass and quark mixing pattern can successfully be accounted for, by having all Yukawa couplings of order unity and arises from the breaking of the discrete group at very high energy, triggered by scalar singlets acquiring vacuum expectation values much larger than the TeV scale. Despite our current model has more scalar singlets than the model that two of us have recently proposed in Ref. Hernandez:2014vta (), our current model addresses both the quark and lepton sectors and does not have the aforementioned drawbacks of the model of Ref. Hernandez:2014vta (). Because of the aforementioned reasons, our current model represents an important improvement over the previously studied scenarios Dong:2011vb (); Hernandez:2014vta (). The particular role of each additional scalar field and the corresponding particle assignments under the symmetry group of the model under consideration are explained in details in Sec. II. The model we are building with the aforementioned discrete symmetries, preserves the content of particles of the 3-3-1 model with , but we add fifteen additional very heavy singlet scalar fields, with quantum numbers that allow to build Yukawa terms invariant under the local and discrete groups. This generates the right textures that successfully account for SM fermion masses and mixings. We assume that the Majorana neutrinos have very small masses, implying that the small active neutrino masses are generated via an inverse seesaw mechanism. This mechanism for the generation of the light active neutrino masses differs from the one implemented in the flavor 3-3-1 model of Ref. Dong:2011vb (), where the light active neutrinos get their masses from type I and type II seesaw mechanisms.

The paper is outlined as follows. In Sec. II we explain some theoretical aspects of the 3-3-1 model with and its particle content, as well as the particle assignments under doublet and singlet representations, in particular in the fermionic and scalar sector. The low energy scalar potential of our model is discussed in Sec II.2. In Sec. III we focus on the discussion of lepton masses and mixing and give our corresponding results. In Sec. IV, we present our results in terms of quark masses and mixing, which is followed by a numerical analysis. Conclusions are given Sec. V. In the appendices we present several technical details: Appendix A gives a brief description of the group; Appendix B shows a discussion of the stability conditions of the low energy scalar potential.

## Ii The model

### ii.1 Particle content

The first 3-3-1 model with right handed Majorana neutrinos in the lepton triplet was considered in Montero:1992jk (). However that model cannot describe the observed pattern of SM fermion masses and mixings, due to the unexplained hierarchy among the large number of Yukawa couplings in the model. Below we consider a multiscalar singlet extension of the 3-3-1 model with right-handed neutrinos, which successfully describes the SM fermion mass and mixing pattern. In our model the full symmetry is spontaneously broken in three steps as follows:

 G=SU(3)C⊗SU(3)L⊗U(1)X⊗S3⊗Z3⊗Z′3⊗Z8⊗Z16Λint−−→ SU(3)C⊗U(1)Q, (1)

where the hierarchy among the symmetry breaking scales is fullfilled.

The electric charge in our 3-3-1 model is defined as Dong:2010zu ():

 Q=T3−1√3T8+XI, (2)

where and are the diagonal generators, is the identity matrix and the charge.

Two families of quarks are grouped in a irreducible representations (irreps), as required from the anomaly cancellation. Furthermore, from the quark colors, it follows that the number of irreducible representations is six. The other family of quarks is grouped in a irreducible representation. Moreover, there are six irreps taking into account the three families of leptons. Consequently, the representations are vector like and do not contain anomalies. The quantum numbers for the fermion families are assigned in such a way that the combination of the representations with other gauge sectors is anomaly free. Therefore, the anomaly cancellation requirement implies that quarks are unified in the following left- and right-handed representations:

 Q1,2L =⎛⎜⎝D1,2−U1,2J1,2⎞⎟⎠L:(3,3∗,0),Q3L=⎛⎜⎝U3D3T⎞⎟⎠L:(3,3,1/3), (3) D1,2,3R:(3,1,−1/3),J1,2R:(3,1,−1/3),U1,2,3R:(3,1,2/3),TR:(3,1,2/3). (4)

Here and () are the left handed up- and down-type quarks in the flavor basis. The right handed SM quarks and () and right handed exotic quarks and are assigned into singlets representations, so that their quantum numbers correspond to their electric charges.

Furthermore, cancellation of anomalies implies that leptons are grouped in the following left- and right-handed representations:

 L1,2,3L =⎛⎜ ⎜⎝\boldmath\mathchar2791,2,3e1,2,3(\boldmath\mathchar2791,2,3)c⎞⎟ ⎟⎠L:(1,3,−1/3), (5) eR:(1,1,−1),N1R:(1,1,0),\boldmath\mathchar278R:(1,1,−1),N2R:(1,1,0),\boldmath\mathchar284R:(1,1,−1),N3R:(1,1,0). (6)

where and () are the neutral and charged lepton families, respectively. Let’s note that we assign the right-handed leptons as singlets, which implies that their quantum numbers correspond to their electric charges. The exotic leptons of the model are: three neutral Majorana leptons and three right-handed Majorana leptons (A recent discussion of double and inverse see-saw neutrino mass generation mechanisms in the context of 3-3-1 models can be found in Ref. Catano:2012kw ()).

The scalar sector the 3-3-1 models includes: three ’s irreps of , where one triplet gets a TeV scale vaccuum expectation value (VEV) , that breaks the symmetry down to , thus generating the masses of non SM fermions and non SM gauge bosons; and two light triplets and acquiring electroweak scale VEVs and , respectively and thus providing masses for the fermions and gauge bosons of the SM.

Regarding the scalar sector of the minimal 331 model, we assign the scalar fields in the following representations:

 \mathchar287 =⎛⎜ ⎜ ⎜⎝\boldmath\mathchar28701\boldmath\mathchar287−21√2(\boldmath\mathchar285\boldmath\mathchar287+\boldmath\mathchar280\boldmath\mathchar287±i\boldmath\mathchar272%\boldmath$\mathchar287$)⎞⎟ ⎟ ⎟⎠:(3,−1/3),\boldmath\mathchar282=⎛⎜ ⎜ ⎜⎝\boldmath\mathchar282+11√2(\boldmath\mathchar285\boldmath\mathchar282+\boldmath\mathchar280\boldmath\mathchar282±i\boldmath\mathchar272%\boldmath$\mathchar282$)\boldmath\mathchar282+3⎞⎟ ⎟ ⎟⎠:(3,2/3), \mathchar273 =⎛⎜ ⎜ ⎜⎝1√2(\boldmath\mathchar285%\boldmath$\mathchar273$+\boldmath\mathchar280\boldmath\mathchar273±i\boldmath\mathchar272%\boldmath$\mathchar273$)\boldmath\mathchar273−2\boldmath\mathchar27303⎞⎟ ⎟ ⎟⎠:(3,−1/3). (7)

We extend the scalar sector of the minimal 331 model by adding the following fifteen very heavy scalar singlets:

 \mathchar283 ∼ (1,0),\boldmath\mathchar286:(1,0),\boldmath\mathchar272:(1,0), (8) \boldmath\mathchar295j : (1,0),\boldmath\mathchar280j:(1,0), \boldmath\mathchar284j : (1,0),Δj:(1,0),j=1,2, Σk : (1,0),k=1,2,3,4.

We assign the scalars into doublet, and singlet representions. The assignments of the scalar fields are:

 \mathchar273 ∼ (1,e2\boldmath\mathchar281i3,1,1,1),% \boldmath\mathchar282∼(1,e−2% \boldmath\mathchar281i3,1,1,1), \mathchar287 ∼ (1,1,1,1,1),\boldmath\mathchar283∼(1′,1,1,1,e−\boldmath\mathchar281i8) \mathchar286 ∼ \mathchar280 ∼ (2,1,1,−1,1),\boldmath\mathchar284∼(2,1,1,i12,1), \boldmath\mathchar2951 ∼ (1,e−2\boldmath\mathchar281i3,1,−i,1),% \boldmath\mathchar2952∼(1′,e−2\boldmath\mathchar281i3,1,−i,1), Δ ∼ (2,e−2\boldmath\mathchar281i3,1,−i,1),Σ1∼(1,1,e2\boldmath\mathchar281i3,−1,e3i\boldmath\mathchar2818), Σ2 ∼ (1,1,e2\boldmath\mathchar281i3,−1,e2i\boldmath\mathchar2818),Σ3∼(1′,1,e2\boldmath\mathchar281i3,−1,e−i\boldmath\mathchar2817), Σ4 ∼ (1,1,e−2\boldmath\mathchar281i3,−1,1).

It has been shown in Ref. Hernandez:2015dga (), that the minimization equations for the scalar potential involving the scalar doublet, imply that the scalar doublets , and can acquire the following VEV pattern:

 (9)

The vacuum configuration of a scalar doublet, pointing either in the or in the directions, has been considered in several flavor models (see for instance Refs. Kubo:2004ps (); Hernandez:2014vta (); Hernandez:2015dga ()). In our model we assume the hierarchy , between the VEVs of the scalar doublets in order to neglect the mixings between these fields and to treat their scalar potentials independently. Let us note that mixing angles between those fields are suppressed by the ratios of their VEVs, as follows from the method of recursive expansion of Ref. Grimus:2000vj ().

In the concerning to the lepton sector, we have the following assignments:

 L1L ∼ (1,e2\boldmath\mathchar281i3,1,i12,1),LL=(L2L,L3L)∼(2,e2\boldmath\mathchar281i3,1,i12,1) eR ∼ \boldmath\mathchar284R ∼ (1′,e−2\boldmath\mathchar281i3,1,1,1)N1R∼(1,e2\boldmath\mathchar281i3,1,i12,1) NR = (N2R,N3R)∼(2,e2% \boldmath\mathchar281i3,1,i12,1), (10)

while the assignments for the quark sector are:

 QL = (Q1L,Q2L)∼(2,1,1,−1,e−i% \boldmath\mathchar2818),Q3L∼(1,1,1,1,1), U1R ∼ (1,e−2\boldmath\mathchar281i3,e2\boldmath\mathchar281i3,1,e6i\boldmath\mathchar2818),U2R∼(1′,e−2\boldmath\mathchar281i3,e2% \boldmath\mathchar281i3,1,e2i% \boldmath\mathchar2818),U2R∼(1,e−2\boldmath\mathchar281i3,1,1,1), D1R ∼ (1,e−2\boldmath\mathchar281i3,1,1,e5i\boldmath\mathchar2818),D2R∼(1,e−2\boldmath\mathchar281i3,e−2% \boldmath\mathchar281i3,−1,e3i% \boldmath\mathchar2818),D3R∼(1′,e−2\boldmath\mathchar281i3,1,−1,1), TR ∼ (11)

In the following we explain the role each discrete group factors of our model. The , , and discrete groups reduce the number of the model parameters. This allow us to get viable textures for the fermion sector that successfully describe the prevailing pattern of fermion masses and mixings, as we will show in sections III and IV. Let us note that we use the discrete group since it is the smallest non-Abelian group that has been considerably studied in the literature. It is worth mentioning that the scalar triplets are assigned to a trivial singlet representation, whereas the scalar singlets are accomodated into three doublets, three trivial singlets and three non trivial singlets. The and symmetries determines the allowed entries of the charged lepton mass matrix. Furthermore, the symmetry distinguishes the right handed exotic quaks, being neutral under from the right handed SM quarks, charged under this symmetry. Note that SM right handed quarks are the only quark fields transforming non trivially under the symmetry. This results in the absence of mixing between SM quarks and exotic quarks. Consequently, the symmetry is crucial for decoupling the SM quarks from the exotic quarks. Besides that, the symmetry selects the allowed entries of the SM quark mass matrices. Besides that, the symmetry separates the scalar doublets participating in the quark Yukawa interactions from those ones participating in the charged lepton and neutrino Yukawa interactions. The symmetry generates the hierarchy among charged fermion masses and quark mixing angles that yields the observed charged fermion mass and quark mixing pattern. It is worth mentioning that the properties of the groups imply that the symmetry is the smallest cyclic symmetry that allows to build the Yukawa term of dimension twelve from a insertion on the operator, crucial to get the required suppression (where is one of the Wolfenstein parameters) needed to naturally explain the smallness of the electron mass.

Now let us briefly comment about a posible large discrete symmetry group that could be used to embed the discrete symmetry of our model. Considering that the discrete group is isomorphic to Ishimori:2010au () and the fact the discrete group is the smallest cyclic group that contains the and symmetries and the symmetry is contained in the group, it follows that the discrete group of our model can be embedded in the discrete group (where ). It would be interesting to implement the discrete symmetry in the 331 model and to study its implications on fermion masses and mixings. This requieres careful studies that are left beyond the scope of the present paper and will be done elsewhere.

With the aforementioned field content of our model, the relevant quark and lepton Yukawa terms invariant under the group , take the form:

 L(Q)Y = y(U)33¯¯¯¯Q3L% \boldmath\mathchar273U3R+y(U)23¯¯¯¯Q2L\boldmath\mathchar282∗U3R\boldmath\mathchar280\boldmath\mathchar283Λ2+y(U)22¯¯¯¯Q2L\boldmath\mathchar282∗UR\boldmath\mathchar280\boldmath\mathchar2833Λ4+y(U)11¯¯¯¯Q1L\boldmath\mathchar282∗UR\boldmath\mathchar280\boldmath\mathchar2837Λ8 (12) +y(D)33¯¯¯¯Q3L% \boldmath\mathchar282D3R\boldmath\mathchar2832Σ2Λ3+y(D)22¯¯¯¯QL\boldmath\mathchar273∗D2R%\boldmath$\mathchar280$Σ3\boldmath\mathchar2833Λ5+y(D)12¯¯¯¯QL\boldmath\mathchar273∗D2R%\boldmath$\mathchar280$Σ4\boldmath\mathchar2834Λ6 +y(D)13¯¯¯¯QL% \boldmath\mathchar273∗D3R\boldmath\mathchar280\boldmath\mathchar2834Σ1Λ6+y(D)11¯QL\boldmath\mathchar273∗D1R\boldmath\mathchar280\boldmath\mathchar2836Λ7 +y(T)¯¯¯¯Q3L% \boldmath\mathchar287TR+y(J)1¯¯¯¯Q1L\boldmath\mathchar287∗J1R+y(J)2¯¯¯¯Q2L\boldmath\mathchar287∗J2R+H.c
 −L(L)Y = h(L)1\boldmath\mathchar282\boldmath\mathchar282e¯¯¯¯L1L\boldmath\mathchar282eR\boldmath\mathchar2838Λ8+h(L)1\boldmath\mathchar282\boldmath\mathchar278(¯LL\boldmath\mathchar282% \boldmath\mathchar284)1\boldmath\mathchar278R\boldmath\mathchar2832Λ3+h(L)2%\boldmath$\mathchar282$\boldmath\mathchar278(¯¯¯¯LL\boldmath\mathchar282\boldmath\mathchar284)1′\boldmath\mathchar278R\boldmath\mathchar2832\boldmath\mathchar272Λ4 (13) +h(L)1\boldmath\mathchar282\boldmath\mathchar282\boldmath\mathchar284(¯LL\boldmath\mathchar282% \boldmath\mathchar284)1% \boldmath\mathchar284R1Λ+h(L)2\boldmath\mathchar282%\boldmath$\mathchar284$(¯¯¯¯LL% \boldmath\mathchar282\boldmath\mathchar284)1′\boldmath\mathchar284R\boldmath\mathchar272Λ2+h(L)1\boldmath\mathchar287(¯¯¯¯LL\boldmath\mathchar287NR)1+h(L)3%\boldmath$\mathchar287$¯¯¯¯L1L% \boldmath\mathchar287N1R +h(1)\boldmath\mathchar282\boldmath\mathchar290abc(¯¯¯¯LaL(LCL)b)1′(\boldmath\mathchar282∗)c\boldmath\mathchar286Λ+h(2)\boldmath\mathchar282% \boldmath\mathchar290abc(¯¯¯¯LaL(L1CL)b(\boldmath\mathchar282∗)cΔ)11Λ +h(3)\boldmath\mathchar282\boldmath\mathchar290abc((¯¯¯¯L1L)a(LCL)b(\boldmath\mathchar282∗)cΔ)11Λ+H.c,

where the dimensionless couplings in Eqs. (12) and (13) are parameters.

Considering that the charged fermion mass and quark mixing pattern arises from the breaking of the discrete group, we set the VEVs of the singlet scalars , , , , , () and () scalar singlets, as follows:

 v\boldmath\mathchar286∼vΔ∼\boldmath\mathchar2775Λ<