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Nuclear Physics B 905 (2016) 337-358

www.elsevier.com/locate/nuclphysb

A predictive 3-3-1 model with A4 flavor symmetry

A.E. Cárcamo Hernándeza*, R. Martinezb

a Universidad Técnica Federico Santa María and Centro Científico-Tecnológico de Valparaíso, Casilla 110-V,

Valparaíso, Chile

b Universidad Nacional de Colombia, Departamento de Física, Ciudad Universitaria, Bogotá D.C., Colombia Received 1 December 2015; received in revised form 19 February 2016; accepted 23 February 2016 Available online 27 February 2016 Editor: Tommy Ohlsson

Abstract

We propose a predictive model based on the SU(3)c ® SU(3)l ® U(1)x gauge group supplemented by the a4 ® Z3 ® Z4 ® Zg ® Zj6 discrete group, which successfully describes the SM fermion mass and mixing pattern. The small active neutrino masses are generated via inverse seesaw mechanism with three very light Majorana neutrinos. The observed charged fermion mass hierarchy and quark mixing pattern are originated from the breaking of the Z4 ® Zg ® Zjg discrete group at very high scale. The obtained values for the physical observables for both quark and lepton sectors are in excellent agreement with the experimental data. The model predicts a vanishing leptonic Dirac CP violating phase as well as an effective Majorana neutrino mass parameter of neutrinoless double beta decay, with values mpp = 2 and 48 meV for the normal and the inverted neutrino mass hierarchies, respectively.

© 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction

Despite the great success of the Standard Model (SM), recently confirmed by the discovery of the 126 GeV Higgs boson by LHC experiments [1-4], there are many aspects not yet explained such as the origin of the fermion mass and mixing hierarchy as well as the mechanism responsible for stabilizing the electroweak scale [5,6]. This discovery of the Higgs scalar field allows to

* Corresponding author.

E-mail address: aecarcamoh2005@gmail.com (A.E. Cárcamo Hernández).

http://dx.doi.Org/10.1016/j.nuclphysb.2016.02.025

0550-3213/© 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

consider extensions of the SM with additional scalar fields that can be useful to explain the existence of Dark Matter [7].

The Standard Model is a theory with many phenomenological achievements. However in the Yukawa sector of the SM there are many parameters related with the fermion masses with no clear dynamical origin. Because of this reason, it is important to study realistic models that allow to set up relations among all these parameters of the Yukawa sector. Discrete flavor symmetries allow to establish ansatz that explains the flavor problem, for recent reviews see Refs. [8-10]. These discrete flavor symmetries may be crucial in building models of fermion mixing that address the flavor problem. Non-abelian discrete flavor symmetries arise in string theories due to the discrete features of the fixed points of the orbifolds [11]. For instance, the discrete D4 group is originated in the S1/Z2 orbifold [11].

Besides that, another of the greatest mysteries in particle physics is the existence of three fermion families at low energies. The quark mixing angles are small whereas the leptonic mixing angles are large. Models based on the gauge symmetry SU(3)c ® SU(3)l ® U(1)x have the feature of being vectorlike with three families of fermions and are therefore anomaly free [12-16]. When the electric charge is defined in the linear combination of the SU(3)l generators T3 and T8, it is a free parameter, independent of the anomalies (j3). The choice of this parameter defines the charge of the exotic particles. Choosing j3 = --—=, the third component of the weak

lepton triplet is a neutral field vC, which allows to build the Dirac matrix with the usual field vL of the weak doublet. If one introduces a sterile neutrino NR in the model, then it is possible to generate light neutrino masses via inverse seesaw mechanism. The 3-3-1 models with j3 = ——3 have the advantage 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 makes 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. Furthermore, the 3-3-1 models can provide an explanation for the 750 GeV diphoton excess recently reported by ATLAS and CMS [17] as well as for the 2 TeV diboson excess found by ATLAS [18].

Neutrino oscillation experiments [6,19-23] indicate that there are at least two massive active neutrinos and at most one massless active neutrino. In the mass eigenstates, it is necessary for the solar neutrinos oscillations that &m2sun = m2x = m2 — m^ where m2 — m\ > 0. For the atmospheric neutrinos oscillations it is required that Sm^tm = = m3 — m2 where the difference can be positive (normal hierarchy) or negative (inverted hierarchy). Neutrino oscillations do not give information neither on the absolute value of the neutrino mass nor on the Majorana or Dirac nature of the neutrino. However there are neutrino mass bounds arising from cosmology [24], tritium beta decay [25] and double beta decay [26-32,34,33].

The neutrino masses and mixings are known from neutrino oscillations, which depend on the squared neutrino mass differences and not on the absolute value of the neutrino masses. The global fits of the available data from the Daya Bay [19], T2K [20], MINOS [21], Double CHOOZ [22] and RENO [23] neutrino oscillation experiments, constrain the neutrino mass squared splittings and mixing parameters [35]. The current neutrino data on neutrino mixing parameters can be very well accommodated in the approximated tribimaximal mixing matrix,

Utbm =

f [2 1 0

V 3 Vs

V6 V3 V2

•J6 V3 V2 )

which is consistent with two large mixing angles and one very small one of order zero. Specfi

cally, the mixing angles predicted by the tribimaximal mixing matrix satisfy (sin2 012) TBM = 3,

(sin2 023) TBM = 1, and (sin2 013) TBM = 0. However, the 3-3-1 model is not able to generate the tribimaximal matrix structure. Because of this reason, discrete symmetry groups [36-38,41,39, 40,42-45] that act on the fermion families are imposed with the aim to generate ansatz that reproduce these matrices. One of the most promising discrete flavor groups is A4, 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. Another approach to describe the fermion mass and mixing pattern consists in postulating particular mass matrix textures (see Ref. [46] for some works considering textures). Besides that, models with Multi-Higgs sectors, Grand Unification, Extradimensions and Superstrings as well as with horizontal symmetries have been proposed in the literature [8,47-50] to explain the observed pattern of fermion masses and mixings.

In this paper we propose a version of the SU(3)c x SU(3)l x U(1)x model with an additional flavor symmetry group A4 ® Z3 ® Z4 ® Z6 ® Zi6 and an extended scalar sector needed in order to reproduce the specific patterns of mass matrices in the fermion sector that successfully account for fermion masses and mixings. The particular role of each additional scalar field and the corresponding particle assignments under the symmetry group of the model are explained in detail in Sec. 2. The model we are building with the aforementioned discrete symmetries, preserves the content of particles of the minimal 3-3-1 model, but we add additional very heavy scalar fields with quantum numbers that allow to build Yukawa terms invariant under the local and discrete groups. This generates the predictive and viable textures that explain the 18 physical observables in the quark and lepton sectors, i.e., the 9 charged fermion masses, 2 neutrino mass squared splittings, 3 lepton mixing parameters, 3 quark mixing angles and 1 CP violating phase of the CKM quark mixing matrix. Our model successfully describes the prevailing pattern of the SM fermion masses and mixing.

The content of this paper is organized as follows. In Sec. 2 we outline the proposed model. In Sec. 3 we discuss lepton masses and mixings and show our corresponding results. Our results for the masses and mixings in the quark sector followed by a numerical analysis are presented in Sec. 4. Finally in Sec. 5, we state our conclusions. In Appendix A we present a brief description of the A4 group.

2. The model

We extend the SU(3)c ® SU(3)l ® U(1)x group of the minimal 3-3-1 model by adding an extra flavor symmetry group A4 ® Z3 ® Z4 ® Z6 ® Z16, in such a way that the full symmetry G experiences a three-step spontaneous breaking, as follows:

G = SU(3)c ® SU (3)l ® U (1)x ® A4 ® Z3 ® Z4 ® Z6 ® Z16 (2)

——*SU(3)c ® SU (3)l ® U(1)x ® Z3 -XSU(3)c ® SU (2)l ® U (1)y SU(3)c ® U(1)q,

where the different symmetry breaking scales satisfy the following hierarchy vn, vp ^ vx < Atnt.

We define the electric charge in our 3-3-1 model in terms of the SU(3) generators and the identity, as follows:

Q = T3 -4= T8 + XI,

with I = Diag(1, 1, 1), T3 = 2Diag(1, —1, 0) and T8 = (2—3)Diag(1, 1, —2).

The anomaly cancellation of SU(3)l requires that the two families of quarks be accommodated in 3* irreducible representations (irreps). Besides that, the number of 3* irreducible representations is six, as follows from the quark colors. We accommodate the other family of quarks into a 3 irreducible representation. Furthermore, we have six 3 irreps taking into account the three families of leptons. Thus, the SU(3)l representations are vector like and free of anomalies. Having anomaly free U(1)x representations requires that the quantum numbers for the fermion families be assigned in such a way that the combination of the U(1)x representations with other gauge sectors cancels anomalies. Consequently, to avoid anomalies, the fermions have to be accommodated into the following (SU(3)c, SU(3)l, U(1)x) left- and right-handed representations:

L1/2,3 =

D1,2 -U1,2 J1,2

(y1,2,3)c

: (3, 3*, 0),

: (3, 3, 1/3),

: (1, 3,-1/3),

D¿2 : (3, 1,-1/3), U^2 : (3, 1, 2/3), j1/ : (3, 1,-1/3), eR : (1, 1,-1),

n1 : (1, 1, 0),

U3 : (3, 1, 2/3), D3 : (3, 1,-1/3), Tr : (3, 1, 2/3),

(1, 1,-1), : (1, 1, 0),

(1, 1, -1), .: (1, 1, 0),

where UlL and DlL for i = 1, 2, 3 are three up- and down-type quark components in the flavor basis, while vlL and elL (eL, ¡xL, tl) are the neutral and charged lepton families. The right-handed fermions are assigned as SU(3)l singlets representations having U(1)x quantum numbers equal to their electric charges. Furthermore, the fermion spectrum of the model includes as heavy fermions: a single flavor quark T with electric charge 2/3, two flavor quarks J1,2 with charge — 1/3, three neutral Majorana leptons (v1,2,3)L and three right-handed Majorana leptons N^2,3 (see Ref. [51] for a recent discussion about neutrino masses via double and inverse see-saw mechanism for a 3-3-1 model).

The 3-3-1 models extend the scalar sector of the SM into three 3's irreps of SU(3)l, where one heavy triplet x acquires a vacuum expectation value (VEV) at the TeV scale, vx, breaking the SU(3)l x U(1)x symmetry down to the SU(2)l x U(1)y electroweak group of the SM and then giving masses to the non SM fermions and gauge bosons; and two lighter triplet fields n and p that get VEVs vn and vp, respectively, at the electroweak scale thus generating the mass for the fermion and gauge sector of the SM. We enlarge the scalar sector of the minimal 3-3-1 model by introducing 14 SU(3)l scalar singlets, namely, , Zj, Sj, V, A, 0, t and a (j = 1, 2, 3).

The scalars of our model are accommodated into the following [SU(3)l, u(1)x] representations:

xU (ux + ^X ± -zx)) j = 1, 2, 3,

—(vp + ± iZp)

j = 1, 2, 3,

(-2+ Hn ± iZn)\

: (3,-1/3), j : (1, 0), r : (1, 0), V : (1, 0),

: (3, 2/3), zj : (1, 0), $ : (1, 0), A : (1, 0),

(3,-1/3), Sj : (1, 0), a - (1, 0),

j = 1, 2, 3.

The scalar fields are grouped into triplet and singlet representations of A4. The scalar fields of our model have the following assignments under A4 ® Z3 ® Z4 ® Z6 ® Z16:

/ 2in \ / 2in \

(1,e-2r, 1, 1, n , p - 1, 1, n , / - (1, 1, 1, 1, 1),

/ in \ / 2in in \

f - (3, 1, 1, 1,-1), Z - (3,1, 1, 1,e~n\, S - (3,e-T, 1, 1,e"nj,

/ i^ \ / in \ / in \

a - (1,1, 1, 1,e-_nj, p - (1,1, 1,e-", 1j , A - (l,1,-1,e-", 1j ,

0 - (1',1,i, 1, 1) , r - (1",1,i, 1, 1) ,

where the numbers in boldface are dimensions of the A4 irreducible representations. The leptons transform under A4 ® Z3 ® Z4 ® Z6 ® Z16 as:

Ll - (3,e2jin, 1, 1,-1) , eR - (1, 1, 1, 1,e1J8r^j, ^R - (1', 1, 1, 1,i) ,

rR - (V',1, 1, 1,e'-4-y Nr - (3,e2j3r, 1, 1,-1^ . (7)

Note that left handed leptons are unified into a A4 triplet representation 3, whereas the right handed charged leptons are assigned into different A4 singlets, i.e, 1, 1' and 1''. Furthermore, the right handed Majorana neutrinos are unified into a A4 triplet representation. The A4 ® Z3 ® Z4 ® Z6 ® Z16 assignments for the quark sector are:

eL - (1,1, 1, 1,e-t), Ql - (1',1, 1, 1, 1), Ql - (1'',1, 1, 1, 1),

1 / 2ni Tin \

1 ~ (l,e-"3~, 1, 1,e~8~\

2n i in

1,e 3 , 1, -1,e 8

(l',e-2JT, 1, 1,i) , U3R - (l",e-, 1, 1, ^ , ), D2r - (l,e2j3-, 1,-1, 1) , DR - (l",e^, 1,-1,

TR - (l",1, 1, 1, 1) , jR - (l',1, 1, 1, 1) , JR - (l",1, 1, 1, 1) .

With the above particle content, the following relevant Yukawa terms for the quark and lepton sector arise:

-lq=y(U QlPU a+y222 QlPU a+y3U QlU

+ y(D)Ol n*Dl T(ph?a2 I y{D)Ol n*D2 TlpA3a I y{D)Ol n*D3 *2A3g + yll QLn DR A7 + y 12 OLn DR A6 + y 13 OLn DR a 6

+y2D QLnD ^+y2D Qw Dr ^ + y2D QLnD ^AA3 +y3D QlPDr ^+y3D QPDi +y3D qLpdR A

+ y{T)Q3LxTR + yJaLx *4 + y(2J)Q2LX JR (9)

a7 ) a4 , a2

-4L) = hL LlP^ 1 eR~A& + hL (LlPS) v, /R^ + h{L (LlPS) r Tr a3 +hL llXNr A+2 hiN (ÑrNc i (nt ■^+x (pT ■ p )

2 iN V R R)i A

+ h2N {nrNr^^ Saa + hpSabc (l"l (p*)cAa + H.c., (10)

where the dimensionless couplings yiU), yD (i, j = 1,2, 3), y(T), y[J), y^J), ^^^^, ^^^, hpT, hL, h\N, x, h2N and hp are 0(1) parameters. Here we assumed that all Yukawa couplings are real, excepting y( D), y<3D) and hpT which are assumed to be complex.

Although the flavor discrete groups in Eq. (2) look rather sophisticated, each discrete group factor plays its own role in generating predictive fermion textures that successfully account for the low energy fermion flavor data. To describe the pattern of fermion masses and mixing angles, one needs to postulate particular Yukawa textures. As we will see in the next sections, the predictive textures for the lepton and quark sectors will give rise to the experimentally observed deviation of the tribimaximal mixing pattern and to quark mixing emerging only from the down type quark sector, respectively. A candidate for generating specific Yukawa textures is the A4 flavor symmetry, which needs to be supplemented by the Z3 ® Z4 ® Z6 ® Z1 6 discrete group. As we will see in the next sections, this predictive setup can successfully account for fermion masses and mixings. The inclusion of the A4 discrete group reduces the number of parameters in the Yukawa and scalar sector of the SU(3)c ® SU(3)l ® U(1)x model making it more predictive. We choose A4 since it is the smallest discrete group with a three-dimensional irreducible representation and 3 distinct one-dimensional irreducible representations, which allows to naturally accommodate the three fermion families. We unify the left-handed leptons in the A4 triplet representation and the right-handed leptons are assigned to A4 singlets. Regarding the quark sector, we assign quarks into A4 singlet representations. In what follows we describe the role of each discrete cyclic group factor introduced in our model. The Z3 symmetry separates the A4 scalar triplets participating in the Yukawa interactions for charged leptons from those ones participating in the neutrino Yukawa interactions. Besides that, the Z3 symmetry avoids mixings between SM quarks and exotic quarks since the right handed exotic quarks are neutral under this symmetry whereas the right handed SM quarks have non-trivial Z3 charges. Thus the Z3 symmetry decouples the SM quarks from the exotic quarks resulting in a reduction of quark sector model parameters. Furthermore, the Z4 symmetry is also important for reducing the number of quark sector model parameters, since due to this symmetry, the SU(3)l scalar singlets A4 non-trivial

singlets only appear in the down type quark Yukawa terms. Consequently this Z4 symmetry together with the A4 assignments for quarks described in Eq. (8), results in a diagonal up type quark mass matrix, thus giving rise to a quark mixing only emerging from the down type quark sector. The Z6 symmetry is crucial for explaining the hierarchy between the SM down and SM

up type quarks without tuning the SM down type quark Yukawa couplings, since it is the small-

est cyclic symmetry that allows j in the Yukawa term that generates the bottom quark mass, which is X3 — (X = 0.225 is one of the Wolfenstein parameters) times a 0(1) parameter. The Z16 symmetry gives rise to the observed hierarchy among charged fermion masses and quark mixing angles. It is worth mentioning that the properties of the ZN groups imply that the Z16 symmetry is the smallest cyclic symmetry that allows to build the Yukawa term qQLp*UR j of

dimension twelve from a j insertion on the QlP*uR operator, crucial to get the required X8 suppression (where X = 0.225 is one of the Wolfenstein parameters) needed to naturally explain the smallness of the up quark mass. Regarding the charged lepton sector, let us note that the five dimensional Yukawa operators j (lLpf) 1 eR, -j(LLpf)v, /xR and j(LLpf)v rR are A4 invariant but do not preserve the Z16 symmetry, as follows from the charges assignments given by Eqs. (6) and (7).

In what follows we comment about the possible VEVs patterns for the A4 scalar triplets f, Z and S. Here we assume a hierarchy between the VEVs of the A4 scalar triplets f, Z and S, i.e., vS <<vz << vf, which implies that the mixing angles of these scalar triplets are very small since they are suppressed by the ratios of their VEVs, which is a consequence of the method of recursive expansion proposed in Ref. [52]. Consequently, we can neglect the mixing between the A4 scalar triplets f, Z and S, and treat their corresponding scalar potentials independently. The relevant terms determining the VEV directions of any A4 scalar triplet are:

V (E) = -4 (SS*)1 + Ke,1 (EE*)1 (EE*)1 + KE,2 (EE)1 (E*E*)1 + ke,3 (EE*)1, (££%

+ ke,4 ((EE)r (£*£% + h.c.] + ke,5 ((EE)r, (E*E*)r + h.c.]

+ E (EE*k (EE% + E (EEhs (E*E% , (11)

where E = f, Z, S.

The part of the scalar potential for each A4 scalar triplet has 8 free parameters: 1 bilinear and 7 quartic couplings. The minimization conditions of the scalar potential for a A4 triplet yield the following relations:

d (V (x)} 2 / 2 2 2 A Í

= ~2v^1 Vx + + vx2 + vx3J + 2K^,3V^1 (2^ - v^2 - v^3/

+ 4k£,2v£,

x + v|2 cos (20S1 - 20^2) + v|3 cos (20S1 - 20x3) + 8kx,7VX1 (v|2 + v|3)

+ 4 (kx,4 + vx1 [2v|1 - v|2 cos (20S1 - 20x2) - v|3cos (20x1 - 20x3)]

+ 4kx,6vx1 vX2 {1 + cos (20x1 - 20^)} + vX3 {1 + cos (20x1 - 20x3)}

d (V (E)) 2 / 222 \ / 222 \

= "2vE2/E + 4kE,1 VE2 (vVE1 + VE2 + VE3) + 2kE,3 VE2 [2vE2 - VE1 - VE3)

+ 4KE,2VE2 [vE2 + vE1 cos (20E2 - 20eO + vE3 cos (20E2 - 20e3)

+ 8KE,7VE2 (VE1 + VE3)

+ 4 (ke,4 + KE,s) VE^2vE2 - vE1 cos(20E2 - 20eO - vE3cos (20E2 - 20e3)]

+ 4KE,6VE2 [vE1 {1 + cos (20E2 - 20E1)} + vE3 {1 + cos (20E2 - 20e3)} = 0,

= -2VE3/e + 4KE,1VE3 (vE1 + vE2 + vE3) + 2KE,3VE3 (2vE3 - vE1 - vE2)

+ 4ke,2vE3

E + vE1 cos (20E1 - 20e2) + vE2 cos (20e3 - 20e2)

+ 8ke,7ve3 (VE1 + vl2)

+ 4 (ke,4 + KE,s) VE3 [2v|3 - vE1 cos (20E1 - 20e2) - v|2cos (20E3 - 20E2)]

+ 4ke,6VE3 vE1 {1 + cos (20E1 - 20e2) } + vE2 {1 + cos (20e3 - 20e2) } = 0, (12)

where (E) = (vE1 el0E1 ,VE2ei0E2 ,VE3 el0E^. Here in order to simplify the analysis, we restrict to the simplest case of zero phases in the VEV patterns of the A4 triplet scalars, i.e., 0E1 = 0E2 = 0E3 = 0. Then, from the scalar potential minimization equations given by Eq. (12), the following relations are obtained:

[3ke,3 - 4 (Ke,6 + KeJ) + 6 (Ke,4 + kE,5)] (vE1 - v|2) = 0,

[3ke,3 - 4 (ke,6 + KE,7) + 6 (ke,4 + ke,5)] (vE1 - v|3) = 0,

[3ke,3 - 4 (kE,6 + KE,7) + 6 (kE,4 + kE,5)] (vE2 - v|3) = 0. (13)

From the relations given by Eq. (13) and setting k^,3 = 4 (kj,6 + k^,7) - 2 (k^,4 + ^,5), we obtain the following VEV pattern:

(1,1,1), (z)=^2 (1,0,1), (S)=73 (1,1,-1). (14)

In the case of f, this is a vacuum configuration preserving a Z3 subgroup of A4, which has been extensively studied in many A4 flavor models (for recent reviews see Refs. [8-10]). The VEV pattern for the A4 triplet scalar Z is similar to the one previously studied in an A4 and T7 flavor SU(5) GUT models [38,44] and in a 6HDM with A4 flavor symmetry [37]. As we will see in the next section, the VEV patterns for the A4 triplets f, Z and S given in Eq. (14) are crucial to get a predictive model that successfully reproduces the experimental values of the physical observables in the lepton sector.

Furthermore we assume that these SU(3)l scalar singlets get VEVs at a scale Aint much larger than vx (which is of the order of the TeV scale), with the exception of Sj (j = 1, 2, 3), which get VEVs much smaller than the electroweak symmetry breaking scale v = 246 GeV. The VEVs of the fj (j = 1, 2, 3), p, A, 0, t and a scalar singlets break the SU(3)c ® SU(3)l ® U(1)x ® A4 ® Z3 ® Z4 ® Z6 ® Z16 symmetry down to SU(3)c ® SU(3)l ® U(1)x ® Z3 at the scale Aint.

From the expressions given above, and using the vacuum configuration for the A4 scalar triplets given in Eq. (14), we find the following relations:

= 3 [3 (*f,1 + Kf,2) + 4 (/Cf,6 + Kf,7)] v|,

2 ( ( ) ( )] = 3 [3 (iCZ,1 + ^,2) + 4 (kz,6 + ^,7)] v^,

2 ( ( ) ( )]

H2S = 3 [3 (ks,1 + KS,2) + 4 (kS,6 + KS,7)] vS. (15)

These results show that the VEV directions for the three A4 triplets, i.e., f, Z and S scalars in Eq. (14), are consistent with a global minimum of the scalar potential (11) of our model for a large region of parameter space.

Besides that, as the hierarchy among charged fermion masses and quark mixing angles emerges from the breaking of the Z4 ® Z6 ® Z16 discrete group, we set the VEVs of the SU(3)l singlet scalar fields f, p, A, 0, t and a, with respect to the Wolfenstein parameter X = 0.225 and the model cutoff A, as follows:

vp ~ vt ~ v0 ~ va ~ vf ~ va ~ Aint = XA. (16)

Furthermore, we assume that the A4 scalar triplets participating in the neutrino Yukawa interactions have VEVs much smaller than the electroweak symmetry breaking scale. Besides that, as previously mentioned, we assume a hierarchy among the VEVs of the two A4 scalar triplets participating in the neutrino Yukawa terms. Consequently, as we will see in the next section, the Majorana neutrinos acquire very small masses and thus an inverse seesaw mechanism for the generation of light active neutrino masses, takes place. Therefore, we have the following hierarchy among the VEVs of the scalar fields in our model:

vs << vz << vp ~ vn ~ v << vx << Aint. (17)

In what follows, we briefly comment about the low energy scalar sector of our model. The renormalizable low energy scalar potential of the model is given by:

Vh = IJ2X(Xfx) + ^(«f«) + V2p(.PfP) + f (xmjPk£ijk + H.c.) + X1(xfx)(xfx)

+ X2 (pip)(pip) + X3(nin)(nin) + X4(x ix)(pip) + X5(x fx)(nfn) + X6 (pfp)(nfn) + X7(x fn)(nfx) + X8(x fp)(p fx) + Xg(p fn)(nfp). (18)

After the symmetry breaking takes place, it is found that the scalar mass eigenstates are related with the weak scalar states by [14,15]:

^ ^=r*(t) (HI)=■ (19)

hv-U)' W=Q)=R • <20>

cos aT(fiT) sin aT(fiT)\ - sinaT(fiT) cosaT(fiT) )

r<xt(pt)

where tan ¡3T = vn/vp, and tan2aT = M\/(M2 - M3) with:

-1 0 0 1

mi = 4k6VvVp + 2v2fvx, M2 = 4X2vp -V2fvx tan fiT,

M3 = 4^3 v2 -v2fvx / tan fij-

It is noteworthy to mention that the our model has the following scalar states at low energies: 4 massive charged Higgs (H±, one CP-odd Higgs (A^), 3 neutral CP-even Higgs

(h0, H0, H30) and 2 neutral Higgs (H0, ) bosons. We identify the scalar h0 with the SM-like 126 GeV Higgs boson discovered at the LHC. Let us note that the neutral Goldstone bosons G0, G0, G0, g2 correspond to the longitudinal components of the Z, Z', K° and K0 gauge bosons, respectively. Besides that, the charged Goldstone bosons G± and G± are associated to the longitudinal components of the W± and K± gauge bosons, respectively [12,15].

3. Lepton masses and mixings

From Eq. (10) and taking into account that the VEV pattern of the A4 triplet, SU(3)l singlet scalar field f satisfies Eq. (14) with the nonvanishing VEVs of the SU(3)l singlet scalars f and a, set to be equal to XA (being A the cutoff of our model) as indicated by Eq. (16), we find that the charged lepton mass matrix is given by:

Ml = rJlPidiag (me,m^,mz

1 0 0 Pi = I 0 1 0

0 0 eia

1 M M2 , M

1 «2 M

being a the complex phase of hpT, and the charged lepton masses are:

(i'h 8 me = a1 a

(i'U 5 V = a2 )A

(iU 3 mT = a3 A

where X = 0.225 is one of the Wolfenstein parameters, v = 246 GeV the scale of electroweak symmetry breaking and ajl) (i = 1, 2, 3) are 0(1) parameters. Let us note that the charged lepton masses are linked with the scale of electroweak symmetry breaking through their power dependence on the Wolfenstein parameter X = 0.225, with 0(1) coefficients. Furthermore, from the lepton Yukawa terms given in Eq. (10) it is easy to see that our model does not feature flavor changing leptonic neutral Higgs decays. Consequently, our model cannot explain the recently reported anomaly in the h ^ \xt decay, implying that a measurement of its branching fraction will be decisive for its exclusion.

Regarding the neutrino sector, we can write the neutrino mass terms as:

-¿mL = 2( vC VR Nr )Mv

+ H.c.,

where the neutrino mass matrix is constrained from the A4 flavor symmetry and has the following form:

/ 03x3 Md 03x3 Mv = MD 03x3 M-x \03x3 MT Mr

and the submatrices are given by:

hpVpVz Va

/h1NVAV2 -h2NVit

-1 0 -1

Va - ~h2N

V3A v2+xv2

h2NVS3A

h2NVS3A

V h2NVV3A

V 1 0 0'

My = hL V I 0 1 0

h2N VA

V^j+XV

A^ Va /

As previously mentioned, we assume in our model that the SU(3)l scalar singlet, A4 triplet S interacting with the right handed Majorana neutrinos gets a very small vacuum expectation value, much smaller than the electroweak symmetry breaking scale, which results in very small masses for these Majorana neutrinos. Consequently, this setup can generate small active neutrino masses through an inverse seesaw mechanism.

As shown in detail in Ref. [51], the full rotation matrix is approximately given by:

(B¡+B¡)

(B¡-B¡)

B3Ux V2 Ux

(-1-S) V2

B2UR \

^Ur ^Ur)

2V2h(xL)vx and the physical neutrino mass matrices are:

M(1) = Md (Mt)-1 MRM-1MT,

B2 ~ B3 ~

hkrMD,

M(2) = -MT + 1 Mr,

M(3) = MT + - Mr,

where mV1 corresponds to the active neutrino mass matrix whereas M(2) and mV3) are the exotic Dirac neutrino mass matrices. Note that the physical eigenstates include three active neutrinos and six exotic neutrinos. The exotic neutrinos are pseudo-Dirac, with masses ~ and a small splitting MR. Furthermore, Rv, UR and Ux are the rotation matrices which diagonalize M(1), M(2) and M(3), respectively [51].

From Eq. (30) it follows that the light active neutrino mass matrix is given by:

m(1) h»2V

'A 0 A

hiNA 0

2 2 hiNVA + ^s + hiNAt

hiN —

hlN~t)

A = —

hlv2pvlvlv0

2 h(L)v\A4 '

hlv2p v2 v° 2hf)v2 A3

hiN A +

2h2N v2

- vs + hiN A

The neutrino mass matrix given in Eq. (32) only depends on two effective parameters: A and B. These effective parameters include the dependence on the various model parameters. It is noteworthy that A and B are suppressed by inverse powers of the high energy cutoff A of our model.

The light active neutrino mass matrix M according to:

is diagonalized by a unitary rotation matrix Rv

RT,M(l)Rv =

mi 0 0 j ( cos 0 0

0 m2 0, with Rv = 0 i

0 0 m3 sin 0 0

0 = ±-, 4

where the upper sign corresponds to normal (0 = +n/4) and the lower one to inverted (0 = —n/4) hierarchy, respectively. The light active neutrino masses for the normal (NH) and inverted (IH) mass hierarchies are given by:

NH : 0 = +— : 4 n

IH : 0 =--:

mv, = 0,

mv1 = 2A,

mv2 = B,

mv2 = B,

mv3 = 2A,

mv3 = 0.

We also find that the PMNS leptonic mixing matrix is given by: /

U = RJLPRV -

sin 0 ia

sin0 eia+

sin0 eia-

4= e-~

1 2in e 3

cos 0 ia i sin0

V3 e + V3

cos 0 eia+ I sin0 -3 e + -3

cos 0 „ia-

It is worth commenting that the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix depends only on the parameter a, while the neutrino mass squared splittings are controlled by parameters A and B.

The standard parametrization of the leptonic mixing matrix implies that the lepton mixing angles satisfy [6]:

IU I2 1 1

sin2 012 = ' ,f' ,2 = 7-, sin2 013 = | Ue3 |2 = - (1 + cos a), (37)

1 -|Ue3|2 2 - cos a 3

• 2n lUM3|2 1 l , V3sina \

sin2 023 = ' = - 1 +--.

23 1 | Ue312 2\ cos a - 2)

The resulting PMNS matrix (36) reduces to the trimaximal mixing matrix (1) in the limit a = n, for the inverted and normal hierarchies of the neutrino mass spectrum. Let us note that the lepton mixing angles are controlled by a single parameter (a), whereas the neutrino mass squared splittings only depend on the parameters A and B.

The Jarlskog invariant and the CP violating phase are [6]:

J = Im (UeiU^U^U^) =--^ cos 20, sin5 =

6^/3 ' cos 0i3 sin20i2sin2023 sin20i3

Taking into account that 0 = ±f, our model predicts J = 0 and 8 = 0, which results in a vanishing leptonic Dirac CP violating phase.

In what follows we adjust the three free effective parameters a, A and B of the lepton sector of our model to reproduce the experimental values of the five physical observables in the neutrino sector, i.e., three leptonic mixing parameters and two neutrino mass squared splittings, reported in Tables 1, 2, for the normal (NH) and inverted (IH) hierarchies of the neutrino mass spectrum, respectively. We fit only a to adjust the values of the leptonic mixing parameters sin2 0ij, whereas A and B for the normal (NH) and inverted (IH) mass hierarchies are given by:

NH : mV1 = 0, mV2 = B =JAm21 ^ 9 meV, mV3 = 2A = JAm\x « 51 meV; (39)

IH : mV2 = B Am21 + Am23 ^ 50 meV, mV1 = 2A = y Am23 ^ 49 meV, mV3 = 0,

as resulting from Eqs. (35), (34) and the definition Am2j = m2 - m2-. The best fit values of Am^jj have been taken from Tables 1 and 2 for the normal and inverted mass hierarchies, respectively.

We vary the model parameter a in Eq. (37) to fit the leptonic mixing parameters sin2 0ij to the experimental values reported in Tables 1, 2. We obtain the following best fit result:

NH : a = -0.88n, sin2 012 « 0.34, sin2 023 « 0.61, sin2 013 « 0.0232; (41)

IH : a = 0.12 n, sin2 012 « 0.34, sin2 023 « 0.61, sin2 013 « 0.0238. (42)

From the comparison of Eqs. (42), (41) with Tables 1, 2, it follows that sin2 013 and sin2 023 are in excellent agreement with the experimental data, for both normal and inverted mass hierarchies, whereas sin2 012 is deviated 2a away from its best fit values. This shows that the physical observables in the lepton sector obtained in our model are consistent with the experimental data. Furthermore, as previously mentioned, our model predicts a vanishing leptonic Dirac CP violating phase.

Table 1

Range for experimental values of neutrino mass squared splittings and leptonic mixing parameters, taken from Ref. [35], for the case of normal hierarchy.

Parameter

Am^i (10"

~5 eV2)

Am31 (10"

~3 eV2)

sin2 012

sin2 023

sin2 013

Best fit Iff range 2a range 3ff range

7.42-7.79 7.26-7.99 7.11-8.11

2.41-2.53 2.35-2.59 2.30-2.65

0.307-0.339 0.292-0.357 0.278-0.375

0.439-0.599 0.413-0.623 0.392-0.643

0.0234

0.0214-0.0254 0.0195-0.0274 0.0183-0.0297

Table 2

Range for experimental values of neutrino mass squared splittings and leptonic mixing parameters, taken from Ref. [35], for the case of inverted hierarchy.

Parameter

Am21 (10"

~5 eV2)

Am23 (10"

~3 eV2)

sin2 012

sin2 023

sin2 013

Best fit 1ff range 2ff range 3ff range

7.42-7.79 7.26-7.99 7.11-8.11

2.32-2.43 2.26-2.48 2.20-2.54

0.307-0.339 0.292-0.357 0.278-0.375

0.530-0.598 0.432-0.621 0.403-0.640

0.0240

0.0221-0.0259 0.0202-0.0278 0.0183-0.0297

In the following we proceed to determine the effective Majorana neutrino mass parameter, which is proportional to the amplitude of neutrinoless double beta (Ovfifi ) decay. This effective Majorana neutrino mass parameter has the form:

mßß =

J2 Uekmvk

where U2 and mVk are the PMNS mixing matrix elements and the Majorana neutrino masses, respectively.

Using Eqs. (36), (39), (40) and (43), it follows that the effective Majorana neutrino mass parameter, for both Normal and Inverted hierarchies, acquires the following values:

2 meV mßß = 1 47 meV

for NH for IH.

Our results for the effective Majorana neutrino mass parameter given above, are beyond the reach of the present and forthcoming Ovfifi decay experiments. The EXO-200 experiment [26] sets the current best upper limit on the effective neutrino mass parameter equal to mpp < 160 meV, corresponding to (136Xe) > 1.6 x 10 yr at 90% C.L. This bound will be improved within the not too distant future. The GERDA "phase-II" experiment [27,28] is expected to reach

i/2 (76Ge) > 2 x 10 yr, which corresponds to mßß < 100 meV. A bolometric CUORE exper-

iment, using 130Te [29], is currently under construction. This experiment features an estimated sensitivity of about T°^(130Te) ~ 1026 yr, corresponding to an effective Majorana neutrino mass parameter mpp < 50 meV. Besides that, there are proposals for ton-scale next-to-next generation experiments using 136Xe [30,33] and 76Ge [27,32], which claim sensitivi-

ties over T

0vßß

[i/2 " 10 yr' corresponding to an effective Majorana neutrino mass parameter mßß ~ 12-30 meV. For a recent review, see for example Ref. [34]. Consequently, Eq. (44)

indicates that our model predicts T^J^ at the level of sensitivities of the next generation or next-to-next generation Ovfifi experiments.

4. Quark masses and mixing

From the quark Yukawa terms of Eq. (9) and the relation given by Eq. (16), we find that the SM quarks do not mix with the exotic quarks and that the SM quark mass matrices are:

fa1U)x8 o 0

ai, ) h4

a(i)h7 a2i)h6

ali à al3 he q

Va3D )h6e-i'q

a(D) h 5 a22 h

a3d )h5

aD )h5

a(D) h 3

where X = 0.225 is one of the Wolfenstein parameters, v = 246 GeV the scale of electroweak symmetry breaking and aj,D) (i, j = 1, 2, 3) are o(1) parameters. Moreover, we find that the exotic quark masses are:

mj = y

— y(T) vx

IJ1 = yl

(J) vx

J2 = y2

(J) vx

Since the charged fermion mass and quark mixing pattern emerges from the breaking of the Z4 ® Z6 ® Zi6 discrete group and in order to simplify the analysis, the following scenario is considered:

arg (ci1D^ = arg (a3D))

(D) (D) a• • = a •• aij aji ,

i,j = 1,2, 3.

Besides that, to show that the quark textures given above can fit the experimental data, and

in order to simplify the analysis, we adopt a benchmark where we set a^ = ax3u' = 1 and

^22 — , as suggested by naturalness arguments and by the relation mc ■

(D) (D) (D) (D) (D) (D)

' mb, respectively.

Then, we proceed to fit the parameters a^ ) a22 ), a33), ai2 ), a\3), °23) and the phase Sq, to reproduce the 10 physical observables of the quark sector, i.e., the six quark masses, the three mixing angles and the CP violating phase. The obtained values for the quark masses, the three quark mixing angles and the CP violating phase S in Table 3 correspond to the best fit values:

(D) hi (D) 13

~ 1.11, ~ 0.43,

(D) a22

(D) a23

~ 0.59, ~ 1.13,

a(D) ~ 0.54, a(3D) ~ 1.42,

Sq ~ 66°

The obtained quark masses, quark mixing angles and CP violating phase exhibit an excellent agreement with the experimental data. Let us note, that despite the aforementioned simplifying assumptions that allow us to eliminate some of the free parameters, a good fit with the low energy quark flavor data is obtained, showing that our model is indeed capable of a very good fit to the experimental data of the physical observables for the quark sector. The obtained and experimental values for the physical observables of the quark sector are reported in Table 3. We use the experimental values of the quark masses at the MZ scale, from Ref. [53] (which are

Table 3

Model and experimental values of the quark masses and CKM parameters.

Observable Model value Experimental value

mu (MeV) 1.14 , 45+0.56 1.45—0.45

mc (MeV) 635 635 ± 86

mt (GeV) 173.9 172.1 ± 0.6 ± 0.9

md (MeV) 2.9 2 9+0.5 2.9—0.4

ms (MeV) 57.7 57 7+16.8 5/.'—15.7

mb (GeV) 2.82 2.82+0.09 2.82—0.04

sin 012 0.225 0.225

sin 023 0.0412 0.0412

sin 013 0.00352 0.00351

S 66° 68°

similar to those in [54]), whereas the experimental values of the CKM parameters are taken from Ref. [6].

In what follows we briefly comment about the phenomenological implications of our model in the flavor changing processes involving quarks. As previously mentioned, the different Z3 charge assignments for SM and exotic right handed quark fields imply the absence of mixing between them. Due to the absence of mixings between SM and exotic quarks, the exotic T, J1 and J2 quarks do not exhibit flavor changing neutral decays into SM quarks and gauge bosons, SM light 126 GeV Higgs boson and SM quarks. Thus, assuming that the H0 and H2 neutral Higgs bosons are heavier than the exotic T, J1 and J2 quarks, it follows that the flavor changing neutral exotic quark decays are absent in our model. Consequently these exotic quarks can be searched at the LHC via their flavor changing charged decays into SM quarks and gauge bosons, specifically in their dominant decay modes T ^ bW and J1,2 ^ tW. These exotic quarks can be produced at the LHC via Drell-Yan processes mediated by charged gauge bosons, where the final states will include the exotic T quark with a SM down type quark as well as any of the exotic J1 or J2 quarks with a SM up type quark. Furthermore, from the quark Yukawa terms, one can easily see that the our model predicts the absence of flavor changing top quark decays t ^ hc and t ^ hu at tree level. The flavor changing top quark decays t ^ hc and t ^ hu only arise at one loop level and will involve virtual charged gauge bosons and exotic quarks running in the loops. Thus, a measurement of the branching fraction for the t ^ hc and t ^ hu decays at the LHC will be crucial for confirming or ruling out our model. It would be interesting to perform a detailed study of the exotic quark production at the LHC, the exotic quark decay modes and the flavor changing top quark decays. This is beyond the scope of this work and is left for future studies.

5. Conclusions

We constructed a predictive SU(3)c ® SU(3)l ® U(\)X model with j = — based on the A4 flavor symmetry supplemented by the Z3 ® Z4 ® Z6 ® Z16 discrete group. Our model successfully accounts for the observed fermion masses and mixing angles. The obtained values for the physical observables in both quark and lepton sectors exhibit an excellent agreement with the experimental data. The A4, Z4 and Z3 symmetries allow to reduce the number of parameters in the

Yukawa terms, increasing the predictivity power of the model. The breaking of the Z4 ® Z6 ® Z16 discrete group at high energy, gives rise to the observed charged fermion mass pattern and quark mixing hierarchy. In our model the Majorana neutrinos acquire very small masses, much smaller than the Dirac neutrino masses, thus giving rise to an inverse seesaw mechanism for the generation of the light active neutrino masses. In this scenario, the spectrum of neutrinos includes very light active neutrinos and TeV scale pseudo Dirac nearly degenerate sterile neutrinos. Our model predicts a vanishing leptonic Dirac CP violating phase as well as an effective Majorana neutrino mass, relevant for neutrinoless double beta decay, with values mpp = 2 and 48 meV, for the normal and the inverted hierarchies, respectively. For the inverted hierarchy neutrino mass spectrum, our obtained value of 48 meV for the effective Majorana neutrino mass is within the declared reach of the next generation bolometric CUORE experiment [29] or, more realistically, of the next-to-next generation tone-scale Ovfifi-decay experiments. Under the assumption that the exotic T, J1 and J2 quarks are lighter than the H20 and H° neutral Higgs bosons, our model predicts the absence of the flavor changing neutral exotic quark decays, which implies that they can be searched at the LHC via their dominant flavor changing charged decay modes T ^ bW and J1,2 ^ iW. Furthermore, our model predicts the absence of flavor changing neutral top quark decays at tree level, implying that they occur at one loop level. Possible directions for future work along these lines would be to study the constraints on the heavy charged gauge boson masses in our model arising from the upper bound on the branching fraction for the flavor changing top quark decays, the oblique parameters, the Zbb vertex and the Higgs diphoton signal strength. The heavy exotic quark decays and their production at the LHC may be useful to study. All these studies require careful investigations that we left outside the scope of this work.

Acknowledgements

A.E.C.H. was supported by FONDECYT (Chile), Grant No. 11130115 and by DGIP internal Grant No. 111458. R.M. was supported El Patrimonio Autónomo Fondo Nacional de Finan-ciamiento para la Ciencia, la Tecnología y la Innovación Fransisco José de Caldas programme of COLCIENCIAS in Colombia.

Appendix A. The product rules for A4

The A4 group has one three-dimensional 3 and three distinct one-dimensional 1, 1' and 1" irreducible representations, satisfying the following product rules:

1 ® 1 = 1, 1'® 1" = 1, 1'® 1' = 1", 1"® 1" = 1'.

Considering (xi,y\,z\) and (x2,y2,z2) as the basis vectors for two A4-triplets 3, the following relations are fulfilled:

3 ® 3 = 3, © 3a © 1 © 1'® 1'',

(3 ® 3)1 = xiyi + X2y2 + X3y3,

(3 ® 3)3s = (X2y3 + X3y2,X3y1 + x\y3,x\y2 + X2y\),

(3 ® 3)1' = Xxyx + MX2y2 + 0J2X3y3,

(3 ® 3)3a = (X2y3 - X3y2,X3yx - X1y3,X1y2 - X2y1),

(3 ® 3)1« = X1y1 + M2X2y2 + 0JX3y3,

where m = el 3. The representation 1 is trivial, while the non-trivial 1' and 1'' are complex conjugate to each other. Some reviews of discrete symmetries in particle physics are found in Refs. [8-10,55].

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