1 Introduction
###### Abstract

We make a scalar extension of the B-L gauge model where the non-abelian discrete group drives mainly the Yukawa sector. Motived by the large and small hierarchies among the quark and active neutrino masses respectively, the quark and lepton families are not treated on the same footing under the assignment of the discrete group. As a consequence, the Nearest Neighbor Interactions (NNI) textures appear in the quark sector, leading to the CKM mixing matrix, whereas in the lepton sector, a soft breaking of the symmetry in the effective neutrino mass that comes from type I see-saw mechanism, provides a non-maximal atmospheric angle and a non-zero reactor angle.

B-L Model with Symmetry

Nearest Neighbor Interaction Textures and Broken Symmetry

Juan Carlos Gómez-Izquierdo***email: jcgizquierdo1979@gmail.com, M  Mondragónemail: myriam@fisica.unam.mx

Instituto de Física, Universidad Nacional Autónoma de México, México 01000, D.F., México.

Departamento de Física, Centro de Investigación y de Estudios Avanzados del I. P. N.,

Apdo. Post. 14-740, 07000, Ciudad de México, México.

## 1 Introduction

How to explain and understand tiny neutrino masses and the fermion mixings respectively in and beyond the Standard Model (SM) is still an open question. Up to now, it is not clear if there is an organizing principle in the Yukawa sector that explains the almost diagonal CKM mixing matrix and its counterpart PMNS one, that has large mixing values.

The pronounced hierarchy among the quark masses, and , could be behind the small mixing angles that parametrize the CKM, which depend strongly on the mass ratios [1, 2, 3]. From a phenomenological point of view, hierarchy among the fermion masses may be understood by means of textures (zeros) in the fermion mass matrices [1, 2, 3, 4]. On the theoretical side, mass textures can be generated dynamically by non-abelian discrete symmetries [5, 6, 7, 8, 9]. The Fritzsch [10, 11, 12] and the NNI [13, 14, 15, 16] textures are hierarchical, however, only the latter one can accommodate with good accuracy the CKM matrix.

In the lepton sector, the hierarchy seems to work differently in the mixings since the charged lepton masses are hierarchical, , but the active neutrino masses exhibit a weak hierarchy [17, 18] that may be responsible for the large mixing values. If the neutrinos obey a normal mass ordering, large mixings can also be obtained by the Fritzsch and NNI textures [17, 19, 18]. It is worth mentioning that non-hierarchical fermion mass matrices could also accommodate the lepton mixing angles [20, 21]. Nevertheless, the hierarchy might have nothing to do with the mixing, since large mixings might be explained by discrete symmetries which were motivated mainly by the experimental values, and . The symmetry [22, 23, 24, 25, 26, 27] was proposed to be behind the atmospheric and reactor mixing angle values. This symmetry predicts exactly that and , which were consistent with the experimental data many years ago. The Tri-Bimaximal (TB) mixing pattern [28, 29, 8] was suggested for obtaining the above angles plus , for the solar angle which coincides approximately with the experimental value. An intriguing fact is that the above mixing pattern does not depend on the lepton masses up to corrections to the lower order in the mixing matrices.

Fermion masses and mixings may be studied under the same flavor symmetry in the baryon number minus lepton number (B-L) gauge model where, apart from the SM fields, three right-handed neutrinos (RHN’s) are included to cancel anomalies, as well as a scalar particle to give them mass à la Higgs. RHN’s are essential to explain the tiny neutrino masses by means of the type I see-saw mechanism [30, 31, 32, 33, 34, 35, 36] (for other mechanism in B-L see [37]). Leptogenesis, dark matter and inflation have been explored widely in this appealing extension [38, 39, 40, 41, 42, 43, 44]. Moving on to the mixing, the non-abelian group has been proposed as the underlying flavor symmetry in different frameworks [45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73]. One motivation to use this discrete symmetry in the lepton sector is to generate the symmetry  [23, 24, 25, 26, 27, 22] or the TB mixing matrix [74, 75, 52]. In the quark sector, the symmetry can give rise to the Fritzsch and generalized Fritzsch mass textures [53, 76]. More recently, it was shown that the NNI mass textures are hidden in the flavor symmetry [60].

Therefore, we make a scalar extension of the B-L gauge model where the non-abelian discrete group drives mainly the Yukawa sector. Motived by the large and small hierarchies among the quark and active neutrino masses respectively, the quark and lepton families are not treated on the same footing under the assignment of the discrete group. As a consequence, NNI textures appear in the quark sector, leading to the CKM mixing matrix, whereas in the lepton sector, a soft breaking of the symmetry in the effective neutrino mass that comes from type I see-saw mechanism, provides a non-maximal atmospheric angle and a non-zero reactor angle.

The plan of this paper is as follows: the B-L gauge model and the flavor symmetry are described briefly in Section 2, the fermion masses and mixings will be discussed in Section 3. In section 4 some conclusions are drawn.

## 2 Flavored B-L Model

The B-L gauge model is based on the gauge group where, apart from the SM fields, three RHN’s and a singlet scalar field are added to the matter content. Under B-L, the quantum numbers for quarks, leptons and Higgs () are , and (), respectively. The allowed Lagrangian is

 LB−L=LSM−yD¯L~HN−12yN¯NcϕN−V(H,ϕ) (1)

with

 V(H,ϕ)=μ2B−Lϕ†ϕ+λB−L2(ϕ†ϕ)2−λHϕ(H†H)(ϕ†ϕ). (2)

where . The spontaneous symmetry breaking of happens usually at high energies so the breaking scale is larger than the electroweak scale, . In this first stage, the RHN’s become massive particles, along with this, an extra gauge boson, , appears as a result of breaking the gauge group. The rest of the particles turn out massive when the Higgs scalars acquire their vacuum expectation value (vev’s) and tiny active neutrino masses are explained by the type I see-saw mechanism.

 ⟨H⟩=1√2(0v),⟨ϕ⟩=ϕ0√2. (3)

On the other hand, let us describe briefly the non-Abelian group , which is the permutation group of three objects and this has three irreducible representations: two 1-dimensional, and , and one 2-dimensional representation, (for a detailed study see [5]). So, the three dimensional real representation can be decomposed as: or . The multiplication rules among the irreducible representations are

 1S⊗1S=1S,1S⊗1A=1A,1A⊗1S=1A,1A⊗1A=1S, 1S⊗2=2,1A⊗2=2,2⊗1S=2,2⊗1A=2; (a1a2)2⊗(b1b2)2=(a1b1+a2b2)1S⊕(a1b2−a2b1)1A⊕(a1b2+a2b1a1b1−a2b2)2. (4)

Having introduced the theoretical framework and the flavor symmetry that will play an important role in the present work, it is worthwhile to point out that the present idea has been developed in the framework of left-right symmetric model (LRSM) [77]. However, the B-L gauge model has a simpler Yukawa term than the LRSM model, which allows us to work with fewer couplings in the fermionic mass matrices. Along with this, the quark sector makes the difference between the present work and that developed in [77].

Now, let us remark important points about the scalar sector and the family assignment under the flavor symmetry in our model. Due to the flavor symmetry, three Higgs doublets have been added in this model to obtain the CKM mixing matrix. At the same time, as we will see, the charged leptons and Dirac neutrinos mass matrices are built to be diagonal, so the mixing will come from the RHN mass matrix. Then, three singlets scalars fields, , are needed to accomplish this. Along with this, in order to try explaining naively the contrasting values between the CKM and PMNS mixing matrices, let us point out a crucial difference in the way the quark and lepton families have been assigned under the irreducible representations of . Hierarchy among the fermion masses makes suggests that both in the quark and Higgs sector, the first and second family are put together in a flavor doublet and the third family in a singlet . On the contrary, for the leptons, the first family has been assigned to a singlet and the second and third families to a doublet . As consequence of this assignment, the hierarchical NNI textures are hidden in the quark mass matrices. In the lepton sector, on the other hand, the lepton mixings can be understood from an approximated symmetry in the effective neutrino mass matrix [77].

In Table 1, the full assignment for the matter content is shown. The symmetry has been added in order to prohibit some Yukawa couplings in the lepton sector, but this is not enough to obtain diagonal mass matrices. So, an extra symmetry will be imposed below.

Thus, the most general form for the Yukawa interaction Lagrangian that respects the flavor symmetry and the gauge group, is given as

 −LY =yd1[¯Q1L(H1d2R+H2d1R)+¯Q2L(H1d1R−H2d2R)]+yd2[¯Q1LH3d1R+¯Q2LH3d2R] +yd3[¯Q1LH1+¯Q2LH2]d3R+yd4¯Q3L[H1d1R+H2d2R]+yd5¯Q3LH3d3R +yui(H→~H,dR→uR)+ye1¯L1H3e1R+ye2[(¯L2H2+¯L3H1)e2R+(¯L2H1−¯L3H2)e3R] +ye3[¯L2H3e2R+¯L3H3e3R]+yDi(H→~H,eR→N)+yN1¯Nc1ϕ3N1 +yN2[¯Nc1(ϕ1N2+ϕ2N3)+(¯Nc2ϕ1+¯Nc3ϕ2)N1]+yN3[¯Nc2ϕ3N2+¯Nc3ϕ3N3]+h.c. (5)

At this stage, an extra symmetry is used to obtain diagonal charged and neutrinos Dirac mass matrices. This symmetry does not modify the Majorana mass matrix form. Explicitly, in the above Lagrangian, we demand that

 L3↔−L3,e3R↔−e3R,N3↔−N3,ϕ2↔−ϕ2. (6)

so the off-diagonal entries and in the lepton sector are absent. Then, this allows to identify properly the charged lepton masses, at the same time, we can speak strictly about the symmetry in the effective neutrino mass.

We should comment that the scalar potential will not be analyzed for the moment, since we are interested in masses and mixings for fermions. The scalar potential of the SM with three families of Higgs and the representation has been studied in [45, 78, 79, 80, 81, 82, 83, 84]. So, in the model, a similar study should follow that direction.

In the standard basis, we have

 −LY=¯qiL(Mq)ijqjR+¯ℓiL(Mℓ)ijℓjR+12¯νiL(Mν)ijνcjL+12¯Nci(MR)ijNj+h.c. (7)

where the type I see-saw mechanism has been realized, . From Eq.(5), the mass matrices have the following form

 Mq=⎛⎜⎝aq+b′qbqcqbqaq−b′qc′qfqf′qgq⎞⎟⎠,Mℓ=⎛⎜⎝aℓ000bℓ+cℓ000bℓ−cℓ⎞⎟⎠,MR=⎛⎜⎝aRbRb′RbRcR0b′R0cR⎞⎟⎠, (8)

where the and . Explicitly, the matrix elements for the quarks and lepton sectors are given as

 aq=yq2⟨H3⟩,b′q=yq1⟨H2⟩,bq=yq1⟨H1⟩,cq=yq3⟨H1⟩,c′q=yq3⟨H2⟩,fq=yq4⟨H1⟩; f′q=yq4⟨H2⟩,gq=yq5⟨H3⟩,aℓ=yℓ1⟨H3⟩,bℓ=yℓ3⟨H3⟩,cℓ=yℓ2⟨H2⟩,aR=yN1⟨ϕ3⟩; bR=yN2⟨ϕ1⟩,b′R=yN2⟨ϕ2⟩,cR=yN3⟨ϕ3⟩. (9)

It is convenient to remark the number of Yukawa couplings that appear in the flavored B-L model in comparison to the flavored LRSM scenario [77].

## 3 Masses and Mixings

### 3.1 Quark Sector: NNI Textures

The quark mass matrix, , has already been obtained in the context of this model [46, 47, 48, 49, 50, 51, 52, 57, 58, 59, 60]. It is important to point out, as it was shown [60], that this mass matrix possesses implicitly a kind of NNI textures, but with more free parameters than the canonical NNI [13, 14, 15, 16] ones that only contain four. This is relevant since it shows that NNI textures are hidden in the flavor symmetry [60], so it may not be necessary to use larger discrete groups, for example the symmetry [85, 86, 87, 88, 89, 90, 91], to understand the mixing by hierarchical mass matrices, although extending the symmetry group may be necessary in other contexts.

Let us comment briefly how to get to the modified NNI textures. For more details see the original paper [60]. Taking the quark mass matrix, , this can written as

 Mq=aq1+⎛⎜⎝b′qbqcqbq−b′qc′qfqf′qgq−aq⎞⎟⎠=aq1+M′q. (10)

is diagonalized by unitary matrices, then one obtains, where . Therefore, we just have to diagonalize to get the mixing matrices. That means that . Having done that, let and , one can get easily where

 mq=UTθM′qUθ=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝02√3b′q02√3b′q02√3c′q02√3f′qgq−aq⎞⎟ ⎟ ⎟ ⎟ ⎟⎠,Uθ=⎛⎜⎝cosθsinθ0−sinθcosθ0001⎞⎟⎠, (11)

with the following conditions

 tanθ=cqc′q=fqf′q=⟨H1⟩⟨H2⟩andtan2θ=b′qbq=⟨H2⟩⟨H1⟩ , (12)

which give us the relation , then as was shown in [60]. Notice that possesses the NNI textures but displaced by which has to be small to obtain the CKM mixing. If is absent in the quark mass matrix, then the well known NNI textures would appear in a straightforward way. In order to not include extra discrete symmetries to prohibit the second term in the Yukawa mass term (see Eq.(5)), which give rise to , let us adopt the benchmark where which means that .

In this framework, we find out the and unitary matrices that diagonalize . Then, we must build the bilineal forms: and , however, in this work we will only need to obtain the left-handed matrix which take places in the CKM matrix. This is given by where the former matrix contains the CP-violating phases, , that comes from . is a real orthogonal matrix and it is parametrized as

 OqL=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝− ⎷~mq2(ρq−−Rq)Kq+4yqδq1κq1− ⎷~mq1(σq+−Rq)Kf+4yqδq2κq2 ⎷~mq1~mq2(σq−+Rq)Kq+4yqδq3κq3− ⎷~mq1κq1Kq−δq1(ρq−−Rq) ⎷~mq2κq2Kq−δq2(σq+−Rq) ⎷κq3Kq−δq3(σq−+Rq) ⎷~mq1κq1(ρq−−Rq)2yqδq1− ⎷~mq2κq2(σq+−Rq)2yqδq2 ⎷κq3(σq−+Rq)2yqδq3⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ (13)

with

 ρq± ≡1+~m2q2±~m2q1−y2q,σq±≡1−~m2q2±(~m2q1−y2q),δq(1,2)≡(1−~m2q(1,2))(~m2q2−~m2q1); δq3 ≡(1−~m2q1)(1−~m2q2),κq1≡~mq2−~mq1yq,κq2≡~mq2yq−~mq1,κq3≡yq−~mq1~mq2; Rq ≡√ρq2+−4(~m2q2+~m2q1+~m2q2~m2q1−2~mq1~mq2yq),Kq±≡yq(ρq+±Rq)−2~mq1~mq2. (14)

In the above expressions, all the parameters have been normalized by the heaviest physical quark mass, . Along with this, from the above parametrization, , is the only dimensionless free parameter that cannot be fixed in terms of the physical masses, but is is constrained by, . Therefore, the left-handed mixing matrix that takes places in the CKM matrix is given by where . Finally, the CKM mixing matrix is written as

 VPMNS=OTuLPqOdL,Pq=Q†uQd=diag.(1,eiηq1,eiηq2). (15)

This CKM mixing matrix has four free parameters, namely , , and two phases and which could be obtained numerically; in this work, the physical quark masses (at scale) will be taken (just central values) as inputs: , , and , , [92]. In the following, a naive analysis will be performed to tune the free parameters. Then, we define

 χ2(yu,yd,ηq1,ηq2)=(∣∣Vthud∣∣−Vexud)2σ2ud+(∣∣Vthus∣∣−Vexus)2σ2us+(∣∣Vthub∣∣−Vexub)2σ2ub+(∣∣Jth∣∣−Jex)2σ2J. (16)

where the experimental values are given as [93]

 Vexud=0.97434+0.00011−0.00012,Vexus=0.22506±0.00050,Vexub=0.00357±0.00015,

and

 Jth=Im[Vthus Vthcb V∗thcs V∗thub],Jex=3.04+0.21−0.20×10−5.

Then, we obtain the following values for the free parameters that fit the mixing values up to

 yu=0.996068,yd=0.922299,ηq1=4.48161,ηq2=3.64887, (17)

with these values, one obtains

 ∣∣VthCKM∣∣=⎛⎜⎝0.974330.225050.003560.224900.973590.039260.009010.038310.99922⎞⎟⎠,Jth=3.04008×10−5. (18)

As can be seen, these values are in good agreement with the experimental data, this is not a surprise since the NNI textures work quite well in the quark sector.

### 3.2 Lepton Sector: Broken μ↔τ Symmetry

As we already mentioned, the lepton mass matrices have been already diagonalized in the framework of the LRSM [77], where a systematic study was realized on the mixing angles. So, we will just mention the relevant points and comment on the results.

The mass matrix is complex and diagonal, then one can identify straightforwardly the physical masses; since the mass matrix is diagonalized by and , this is, with . After factorizing the phases, we have   where

 me=Diag.(me,mμ,mτ),S23=⎛⎜⎝100001010⎞⎟⎠,Pe=diag.(eiηe/2,eiημ/2,eiητ/2) (19)

As a result, one obtains , and .

On the other hand, the effective neutrino mass matrix is given by

Now as hypothesis, we will assume that is larger than , in this way the effective mass matrix can be written as

 Mν≡⎛⎜ ⎜⎝m0ee−m0eμ(1+ϵ)−m0eμ(1−ϵ)−m0eμ(1+ϵ)m0μμ(1+ϵ)2m0μτ(1−ϵ2)−m0eμ(1−ϵ)m0μτ(1−ϵ2)m0μμ(1−ϵ)2⎞⎟ ⎟⎠, (21)

where , , and are complex. In here, is a complex parameter which will be considered as a perturbation to the effective mass matrix such that . In order to break softly the symmetry, we demand that so we will neglect the quadratic terms in the above matrix hereafter and a perturbative diagonalization will be carried out.

In order to cancel the contribution that comes from the charged lepton sector, we proceed as follows. We know that , then where the latter mixing matrix will be obtained below. Then, with

 Mν≈⎛⎜ ⎜⎝m0ee−m0eμ−m0eμ−m0eμm0μμm0μτ−m0eμm0μτm0μμ⎞⎟ ⎟⎠+⎛⎜ ⎜⎝0m0eμ ϵ−m0eμ ϵ−m0eμ ϵ−2m0μμ ϵ0−m0eμ ϵ02m0μμ ϵ⎞⎟ ⎟⎠=M0ν+Mϵν. (22)

Notice that possesses the symmetry and this is diagonalized by

 U0ν=⎛⎜ ⎜ ⎜ ⎜⎝cosθν eiηνsinθν eiην0−sinθν√2cosθν√2−1√2−sinθν√2cosθν√21√2⎞⎟ ⎟ ⎟ ⎟⎠, (23)

where the matrix elements are written as

 m0ee =(m0ν1cos2θν+m0ν2sin2θν)e2iην,−m0eμ=1√2cosθνsinθν(m0ν2−m0ν1)eiην; m0μμ =12(m0ν1sin2θν+m0ν2cos2θν+m0ν3),m0μτ=12(m0ν1sin2θν+m0ν2cos2θν−m0ν3). (24)

Including the perturbation, , applying one gets . Explicitly

 Mν=Diag.(m0ν1,m0ν2,m0ν3)+⎛⎜ ⎜⎝00−sinθν(m0ν3+m0ν1) ϵ00cosθν(m0ν3+m0ν2) ϵ−sinθν(m0ν3+m0ν1) ϵcosθν(m0ν3+m0ν2) ϵ0⎞⎟ ⎟⎠. (25)

The contribution of second matrix to the mixing one is given by

 Uϵν≈⎛⎜⎝N10−N3sinθ r1 ϵ0N2N3cosθν r2 ϵN1sinθν r1 ϵ−N2cosθν r2 ϵN3⎞⎟⎠ (26)

where the , and are the normalization factors which are given as

 N1=1√1+sin2θν|r1ϵ|2,N2=1√1+cos2θν|r2ϵ|2,N3=1√1+sin2θν|r1ϵ|2+cos2θν|r2ϵ|2, (27)

with . Finally, the effective mass matrix given in Eq.(21) is diagonalized approximately by . Therefore, the theoretical PMNS mixing matrix is written as . Explicitly,