I Introduction
###### Abstract

A model for neutrino masses and mixing is devised appointing the see-saw mechanism. The proffered model is fabricated with a combination of Type -I and Type-II see-saw contributions of which the latter dominates. The scalars and the leptons in the model are assigned charges conducive to obtain the mass matrices viable for the scheme. The Type -II see-saw mass matrix accommodates atmospheric mass splitting and maximal mixing in the atmospheric sector (). It is characterized by vanishing solar mass splitting and whereas the third neutrino mixing angle is free to acquire any value of . Particular alternatives of corresponding to the popular lepton mixings viz. (tribimaximal), (bimaximal), (golden ratio) are accounted for. Another choice of (no solar mixing) is reckoned. The subdominant Type-I see-saw constituent of the model propels all the neutrino oscillation parameters into the ranges allowed by the data which in its turn get interrelated owing to their common origin. This makes the model testable in the light of future experimental data. As an example, emerges in the first (second) octant for normal (inverted) ordering. CP-violation is governed by phases present in the right-handed Majorana neutrino mass matrix, . Only normal ordering is allowed if these phases are absent. If is complex the Dirac CP-violating phase , is capable of being large, i.e., , and inverted ordering of neutrino masses is also permitted. T2K and NOVA preliminary data favouring normal ordering and predicts lightest neutrino mass to be 0.05 eV or more within the framework of this model.

Key Words:  Neutrino mixing, , Solar splitting, A4, see-saw, Leptonic CP-violation

Ameliorating the popular lepton mixings with A4 symmetry: A see-saw model for realistic neutrino masses and mixing

Soumita Pramanick***email: soumita509@gmail.com

Department of Physics, University of Calcutta, 92 Acharya Prafulla Chandra Road, Kolkata 700009, India

## I Introduction

Intensive investigations with proficient experimental set-ups worldwide has determined neutrino masses and mixing to a great extent. In spite of these the elusive neutrinos continue to allure us with certain mysteries in their characteristics including the ordering of their masses, their absolute mass scale, their Dirac or Majorana nature, the octant of the atmospheric mixing angle and CP-violation in lepton sector. While future experiments illuminate our understanding of these riddles, an endeavor of composing a model of neutrino masses and mixing in concord with the experimental observations is portrayed. Such enterprise had been the subject of [1] where the two small quantities and the ratio, got interrelated while both were derived from a single perturbation to a dominant scenario devoid of them. In [2] the larger mixing parameters like and were ascribed to the dominant fundamental structure of neutrino masses and mixing whereas the other oscillation parameters i.e., , the deviation of from , and originated from a smaller see-saw [3] generated perturbation 111Earlier attempts on neutrino mass models with some oscillation parameters much smaller than the others can be located in [4].. This evidently manifests constraints on the measured parameters. Certain symmetries can give rise to vanishing rather easily and new models based on perturbations of such structures are also common in literature [5, 6]. Here, a schematic layout of the current enterprise is delineated. The following standard parametrization form of the lepton mixing matrix – the Pontecorvo, Maki, Nakagawa, Sakata (PMNS) matrix – has been used

 U=⎛⎜⎝c12c13s12c13s13e−iδ−c23s12+s23s13c12eiδc23c12+s23s13s12eiδs23c13s23s12+c23s13c12eiδ−s23c12+c23s13s12eiδc23c13⎞⎟⎠, (1)

where and . Neutrino masses and mixing are generated by a two-component Lagrangian formalism, one of the dominant Type-II see-saw kind while the subdominant contribution comes from a Type-I see-saw constituent. The larger atmospheric mass splitting, and maximal atomspheric mixing () is embedded within the Type-II see-saw structure whereas the solar splitting, and are kept to be zero. The solar mixing angle on the other hand is free to vary continuously and acquire any desired value of by suitably tuning the model parameters. Needless to mention that neither nor are vanishing [7]. Evidences of non-maximal yet large exist. The solar mixing angle is also constrained by experiments. The Type-I see-saw amends the dominant Type-II see-saw contribution and alleviates these issues present in it in order to harmonize the model with the oscillation observations. The starting structure is associated with several lepton mixing patterns such as tribimaximal (TBM), bimaximal (BM), and golden ratio (GR) mixings to which we are conversant with as particular choices of . All these mixing options have and , being the only discriminating factor as specified in Table. 1. In this Table, the fourth option corresponds to no solar mixing (NSM) which being bestowed with the virtue of the mixing angles to be either maximal, i.e., () or vanishing ( and ) serves as an interesting alternative. An -based model with identical objectives only for the NSM case was the prime intent of [8]. This attempt along with [8] differ from the other earlier works on [9, 10, 11] in similar directions as in most of them neutrino mass matrix was derived as an outcome of an underlying Type-II see-saw mechanism and obtaining TBM was of chief importance. Recent activities directed towards more realistic mixing patterns [12] often leading to breaking of symmetry can be found in [13]. A few distinctive aspects of this model are worth noting at this point. Firstly, a combination of Type-I and Type-II see-saw is considered. Secondly, the model is designed in a generalized manner enabled to accommodate many popular mixing patterns. Further, soft symmetry breaking terms are prohibited. All symmetry conserving terms are included in the Lagrangian. The symmetries are broken spontaneously only. Scalars and leptons involved in the model are assigned suitable charges to successfully implement this feature. An analogous pursuit based on resulted in [14].

A concise description of the discrete symmetry initiates the discourse followed by a vivid analysis of the model. In the next section, the operational strategy is described. The results so obtained are compared to the experimental data in the following section. The concluding segment comprises of the inferences of this undertaking. A detailed study of the rich scalar sector associated with this model to the extent of local minimization of the scalar potential is furnished in the Appendix.

## Ii The group A4

is the even permutation group of four objects having 12 elements and two generators and satisfying the property . It has four inequivalent irreducible representations viz. one 3 dimensional representation and three 1 dimensional representations namely, and . These three dimension-1 representations are singlets under whereas they transform as 1, , and respectively under the action of , being a cube root of unity. Therefore it is apparent that . The pertinent form of the generators and acting on the 3 dimensional representations are given by222This choice of basis has the generator diagonal. One can equivalently perform an analogous analysis in a basis in which the generator is diagonal. Needless to mention that the two bases are related by some unitary basis transformation.,

 S=⎛⎜⎝1000−1000−1⎞⎟⎠    and    T=⎛⎜⎝010001100⎞⎟⎠. (2)

It is imperative to note the product rule for the three dimensional representation is:

 3⊗3=1⊕1′⊕1′′⊕3⊕3. (3)

When two triplets of given by and , with ; are combined according to Eq. (3), then the resultant triplets can be represented by and where,

 ci = (a2b3+a3b22,a3b1+a1b32,a1b2+a2b12),or,ci≡αijkajbk, di = (a2b3−a3b22,a3b1−a1b32,a1b2−a2b12),or,di≡βijkajbk,(i,j,k,are cyclic). (4)

and the , and so obtained can be scripted as:

 1 = a1b1+a2b2+a3b3≡ρ1ijaibj, 1′ = a1b1+ω2a2b2+ωa3b3≡ρ3ijaibj, 1′′ = a1b1+ωa2b2+ω2a3b3≡ρ2ijaibj. (5)

The group is studied in extensive details in [9, 10].

## Iii The Mass Model

The model portrayed in this section comprises of scalars and leptons with specific charges in such a fashion that the mass matrices convenient for the adopted see-saw scheme follows as a natural consequence of spontaneous symmetry breaking. All terms allowed by the symmetries under consideration are included in the Lagrangian. No soft symmetry-breaking term is allowed.

The model propounded possess the right-handed charged leptons transforming as , , and under . The left-handed lepton doublets of the three flavours constitute an triplet, so does the right-handed neutrinos333The notation followed closely resembles that of [9].. Table. 2 shows the lepton constituents of the model together with their transformation properties under and . The hypercharge and lepton number assignments are also shown. It is worth mentioning that the lepton number assignment of the right-handed neutrinos is opposite in sign compared to that of the other leptons. Such a choice is responsible for retaining the form of the neutrino Dirac mass matrix proportional to the identity matrix by forbidding contributions from additional scalars which spoils that form as will be revealed in course of our discussion. Such choices of properties of the fields as shown in Table. 2 are not unique. A list of all possible options can be found in [15] of which this model adopts class B of [15]. The model is restricted to leptons only444 Quark models based on has been explored in [16] and [17]..

Masses of all leptons originate from -invariant Yukawa couplings. Several scalars have to be appointed for this purpose555 Models addressing this issue by separating the breaking of and are widely studied in literature [10]. The former is mediated by the usual doublet and triplet scalars of that are invariant under . The breaking of is induced by the vev of ‘flavon’ scalar fields that are singlets of but their transformations under is non-trivial. Though such models are economic effective dimension-5 interactions comes into play in order to connect the fermions with the two types of scalar fields simultaneously leading to an interpretation as an effective theory. that acquire suitable vacuum expectation values (vevs). The charged leptons acquire their Dirac-type masses through the doublet scalar fields forming an triplet. The neutrino Dirac mass matrix is generated by an invariant doublet , having lepton number666 Opposite lepton numbers are assigned to and in order to prohibit their coupling with so that the Dirac mass matrix could perpetuate its proportionality to the identity matrix. 2. triplet scalars are required for the Type-II see-saw for left-handed neutrino mass matrix that include triplet fields and along with transforming as , , of . These are used to construct the dominant Type-II see-saw neutrino mass matrix. Effects of the subdominant Type-I see-saw contribution is included perturbatively. conserving Yukawa couplings produce the right-handed neutrino mass matrix as well777The right-handed neutrinos are singlets. Thus it is possible to include dimension three direct Majorana mass terms for these fields that softly break . Throughout this model is conserved. No soft symmetry breaking of is done. It is broken only spontaneously when the scalars acquire their vevs.. Several singlet scalars are involved in generation of the Majorana masses for the right-handed neutrinos viz. () transforming as triplets and () transforming as , and under . Table. 3 evinces transformation properties of the model scalars under and together with their hypercharge, lepton number and vev configurations. The vevs of the doublet scalars are of while that of the triplets are several orders of magnitude smaller than the doublet vevs in concord with the small neutrino masses as well as the parameter of electroweak symmetry breaking. As expected, the vevs of the singlets responsible for right-handed neutrino mass lies much above the electroweak scale. The mass terms of the neutrinos (both Type-I and Type-II see-saw) and that of the charged leptons are generated by a conserving Lagrangian that preserves as well888Lepton number is also conserved for the mass terms of Dirac kind.:

 Lmass = yjρjik¯lLilRjΦ0k  (charged lepton mass) (6) + fρ1ik¯νLiNRkη0  (neutrino Dirac mass) + 12(∑n=a,b^YLn αijkνTLiC−1νLj^ΔL0nk+YLζ ρζijνTLiC−1νLjΔL0ζ)  (neutrino Type−II see−saw mass) + 12(∑p=a,b,c^YRp αijkNTRiC−1NRj^ΔR0kp+YRγ ργijNTRiC−1NRjΔR0γ)  (rh neutrino mass)+h.c.

The scalars acquire the following vevs ( part is suppressed):

 ⟨Φ0⟩=v√3⎛⎜⎝111⎞⎟⎠,⟨η0⟩=u,⟨^ΔL0a⟩=vLa⎛⎜⎝100⎞⎟⎠,⟨^ΔL0b⟩=vLb⎛⎜⎝111⎞⎟⎠,⟨ΔL01⟩=⟨ΔL02⟩=⟨ΔL03⟩=uL, (7)
 ⟨^ΔR0a⟩=vRa⎛⎜⎝111⎞⎟⎠,⟨^ΔR0b⟩=vRb⎛⎜⎝1ωω2⎞⎟⎠,⟨^ΔR0c⟩=vRc⎛⎜⎝1ω2ω⎞⎟⎠, (8)
 ⟨ΔR01⟩=u1R,⟨ΔR02⟩=u2R,⟨ΔR03⟩=u3R. (9)

An elaborate study of the conserving scalar potential involving the fields listed in Table. 3 is the prime content of the Appendix of this paper. Local minimization is performed and the conditions corresponding to the particular vev structures as indicated in Eqs. (7-9) are obtained. The mass matrix for the charged leptons and the left-handed Majorana neutrinos so ensued are:

 Meμτ=v√3⎛⎜⎝y1y2y3y1ωy2ω2y3y1ω2y2ωy3⎞⎟⎠,MνL=⎛⎜ ⎜ ⎜ ⎜⎝(YL1+2YL2)uL12^YLbvLb12^YLbvLb12^YLbvLb(YL1−YL2)uL12(^YLavLa+^YLbvLb)12^YLbvLb12(^YLavLa+^YLbvLb)(YL1−YL2)uL⎞⎟ ⎟ ⎟ ⎟⎠. (10)

where the choice of is made. The Yukawa couplings involved in the charged lepton mass matrix satisfies . The Type-II see-saw dominant component of the neutrino mass matrix, , gives rise to the atmospheric splitting and maximal atmospheric mixing but is devoid of solar splitting and is therefore characterized by two masses and . It is useful to define . Needless to mention, is positive (negative) for normal (inverted) ordering. For the desired structure to eventuate certain identifications of the vev and Yukawa products are essential viz. , , and .

The neutrino mass matrix of Dirac nature and the right-handed neutrino mass matrix of Majorana kind acquires the following structures:

 MD=fu I,MνR=mR⎛⎜⎝χ1χ6χ5χ6χ2χ4χ5χ4χ3⎞⎟⎠. (11)

where,

 mRχ1 ≡ (YR1u1R+YR2u2R+YR3u3R) mRχ2 ≡ (YR1u1R+ωYR2u2R+ω2YR3u3R) mRχ3 ≡ (YR1u1R+ω2YR2u2R+ωYR3u3R) mRχ4 ≡ 12(^YRavRa+^YRbvRb+^YRcvRc) mRχ5 ≡ 12(^YRavRa+ω^YRbvRb+ω2^YRcvRc) mRχ6 ≡ 12(^YRavRa+ω2^YRbvRb+ω^YRcvRc). (12)

The scale of the right-handed Majorana neutrino masses is set by and in Eqs. (11) and (12) are dimensionless quantities of . Certain identifications of the vev and Yukawa couplings products are indispensable to achieve the mass matrices of the proper form:

 YR1u1R=mR(r11+2r23),  YR2u2R=mR(r22+2r13),  YR3u3R=mR(r33+2r12) ^YRavRa=2mR(r11−r23),  ^YRbvRb=2mR(r22−r13)  and  ^YRcvRc=2mR(r33−r12). (13)

The in Eq. (13) are given by :

 r11 ≡ √2bsin2θ012+asin2θ012, r22 ≡ −√2bsinθ012−b2sin2θ012−acosθ012+a2cos2θ012+a2, r33 ≡ −b√2sin2θ012−√2bsinθ012+acosθ012+a2cos2θ012+a2, r12 ≡ bcos2θ012+a2√2sin2θ012+bcosθ012−a√2sinθ012, r13 ≡ −bcos2θ012−a2√2sin2θ012+bcosθ012−a√2sinθ012, r23 ≡ b2sin2θ012−a2cos2θ012+a2  . (14)

where and are dimensionless quantities of . The mass matrices depicted in Eq. (10) could be expressed in a more convenient form by applying a couple of transformations viz. on the left-handed lepton doublets and on the right-handed neutrino singlets of keeping the right-handed charged leptons unaltered. The transformation matrices are expressed as:

 UL=1√3⎛⎜⎝1111ω2ω1ωω2⎞⎟⎠=VR. (15)

Such a transformation diagonalizes the charged lepton mass matrix. This basis in which the charged lepton mass matrix is diagonal and the entire lepton mixing is governed by the neutrino sector is termed as the flavour basis in which the mass matrices acquire the following forms:

 Mflavoureμτ=⎛⎜⎝me000mμ000mτ⎞⎟⎠,MflavourνL=12⎛⎜ ⎜⎝2m(0)1000m+m−0m−m+⎞⎟ ⎟⎠, (16)

Demanding the neutrino Dirac mass matrix preserves its proportionality to the identity matrix necessitates certain transformations of the right-handed neutrino mass matrix. Thus we get,

 MD=fu I,MflavourνR=mR4ab⎛⎜⎝r11r12r13r12r22r23r13r23r33⎞⎟⎠. (17)

It is apparent to identify where sets the scale of Dirac masses of the neutrinos. Needless to mention that the matrices in Eq. (17) will take part in Type-I see-saw mechanism.

## Iv Modus Operandi

The four mass matrices in the flavour basis obtained from the model are given in Eq. (16) and (17). In this basis the entire lepton mixing and CP-violation is controlled solely by the neutrino sector that happens to be the cardinal point of our discussion in this section. The Type-II see-saw derived is the dominant component to which the subdominant contribution attributed by the Type-I see-saw is incorporated by perturbation theory. The flavour basis mass matrices have to undergo one more basis transformations for successful implementation of this scheme. More precisely they ought to be expressed in the mass basis of the neutrinos which by definition has the left-handed neutrino mass matrix diagonal in it. Thus,

 M0=MmassνL=U0TMflavourνLU0=⎛⎜ ⎜ ⎜⎝m(0)1000m(0)1000m(0)3⎞⎟ ⎟ ⎟⎠, (18)

where,

 U0=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝cosθ012sinθ0120−sinθ012√2cosθ012√21√2sinθ012√2−cosθ012√21√2⎞⎟ ⎟ ⎟ ⎟ ⎟⎠. (19)

It immediately follows from Eqs. (18), (1) and (19) that in the Type-II see-saw component solar splitting is absent, and . The columns of are the unperturbed flavour basis. Once again we demand that the neutrino Dirac mass matrix remains proportional to identity. In order to satisfy this the reverse transformation has to be applied on the right-handed neutrino fields. This leads to changes in form of right-handed neutrino mass matrix. The matrices contributing in Type-I see-saw are now given by:

 MD=mDI  and  MmassνR=mR2√2ab⎛⎜ ⎜ ⎜⎝0bbba√2−a√2b−a√2a√2⎞⎟ ⎟ ⎟⎠  . (20)

It is imperative to note that and can in general be complex. One can in principle trade off and in terms of complex numbers and respectively, where and are dimensionless real quantities of . The Type-I see-saw contribution so obtained is given by

 M′=[MTD(MνR)−1MD]=m2DmR⎛⎜ ⎜ ⎜ ⎜ ⎜⎝0y eiϕ1y eiϕ1y eiϕ1x eiϕ2√2−x eiϕ2√2y eiϕ1−x eiϕ2√2x eiϕ2√2⎞⎟ ⎟ ⎟ ⎟ ⎟⎠. (21)

Here the Dirac mass matrix is proportional to identity. It was checked that the same results can follow as long as is diagonal. It is noteworthy that exhibits a discrete symmetry. The results remain intact even if that choice is relaxed. Now onwards the entire procedure is carried on in the mass basis of the neutrinos using the mass matrices expressed in Eqs. (18) and (21).

## V Results

The neutrino mass matrices derived as a consequence of the Type-I and Type-II see-saw mechanism has been discussed in the previous section, of which the former is considered significantly smaller than the latter. The leptonic mixing matrix is characterized by , , and free to vary in absence of the Type-I see-saw contribution. Consequences for four choices of the value of corresponding to TBM, BM, GR, and NSM cases together with vanishing solar splitting are examined. This along with the atmospheric mass splitting allowed by the data depict the Type-II see-saw structure. Inclusion of Type-I see-saw corrections perturbatively up to first order is capable of modulating the neutrino oscillation parameters into the ranges preferred by data. The global best-fit of these parameters are displayed in the next section.

### v.1 Data

The current 3 global fits of the neutrino oscillation parameters are: [18, 19]

 Δm221 = (7.02−8.08)×10−5eV2,θ12=(31.52−36.18)∘, |Δm231| = (2.351−2.618)×10−3eV2,θ23=(38.6−53.1)∘, θ13 = (7.86−9.11)∘,δ=(0−360)∘. (22)

These numbers are taken from NuFIT2.1 of 2016 [18]. Needless to mention, , such that for normal ordering (NO) and for inverted ordering (IO). Two best-fit points of are evinced by the data in the first and in the second octants. Consonance of the model with the recent T2K and NOVA hints [20, 21] of close to - has been discussed towards the end of the paper.

### v.2 Real MνR (ϕ1=0 or π,ϕ2=0 or π)

As a warm-up exercise let us consider the simpler case of real. No CP-violation eventuate in such a scenario as the phases of Eq. (21) are 0 or . This leads to four different alternatives available for choosing and that are captured compactly by taking and real and allowing them to assume both signs for notational convenience. It will be soon clear how the experimental observations prefer one or the other of these four alternatives. Thus for real the Type -I see-saw contribution appears like:

 M′=m2DmR⎛⎜ ⎜ ⎜⎝0yyyx√2−x√2y−x√2x√2⎞⎟ ⎟ ⎟⎠. (23)

The degeneracy of the two neutrino masses in the Type-II see-saw ensuring the vanishing solar splitting necessitates the application of degenerate perturbation theory to obtain the corrections for the solar sector mixing parameters. The entire dynamics of this sector is dictated by the upper submatrix of given by:

 M′2×2=m2DmR(0yyx/√2). (24)

This gives rise to:

 θ12=θ012+ζ,tan2ζ=2√2(yx). (25)

For functional ease it is useful to define a quantity, as:

 sinϵ=y√y2+x2/2and cosϵ=x/√2√y2+x2/2,i.e., tanϵ=12tan2ζ. (26)

Once a mixing pattern is selected the corresponding gets fixed and the experimental bounds of determines the ranges and by means of Eq. (22) and Eq. (26) as featured in Table. 4. The ratio is positive (negative) when </