NUPHB12989S0550-3213(14)00038-810.1016/j.nuclphysb.2014.02.002The AuthorsHigh Energy Physics – PhenomenologyTable 1The particle contents and their transformation property under the family symmetry S4×Z7 and U(1)R, where ω7=e2πi7.Fieldlνcecμcτchu,dφTϕφSηξφT0ζ0φS0η0

S43′3′11113′23213′132

Z7ω71ω73ω74ω751ω7ω7ω76ω76ω76ω75ω75ω72ω72

U(1)R111110000002222

Table 2The predictions for the leptonic CP phases, light neutrino masses mi(i=1,2,3) and the effective mass |mββ| of the neutrinoless doublet-beta decay, where the unit of mass is meV.(x,y)δCPθ23/°θ12/°α21α31m1m2m3|mββ|Mass order

(−1.898,−0.316)π32.49634.3090π128.020128.311137.136122.038NO

(−1.898,0.316)057.504

(0.139,−0.612)π32.496π024.23325.72454.7479.423NO

(0.139,0.612)057.504

(0.101,0.340)π32.4960π49.66950.41411.15948.507IO

(0.101,−0.340)057.504

(−0.120,0.535)π32.496π054.86655.54125.97719.931IO

(−0.120,−0.535)057.504

(−0.050,0.233i)π/2450057.28457.93075.48857.901NO

(−0.050,−0.233i)−π/2

Table 3The transformation properties of the fields under the family symmetry S4×Z4×Z5 and U(1)R, where ω5=e2πi5.Fieldlνcecμcτchu,dφTϕφSηχξφT0ζ0φS0ξ0η0ρ0σ0

S43′3′11113′23′23′13′131211

Z411i−1−i1ii1111−1−111111

Z5ω53ω52ω52ω52ω52111ω5ω5ω53ω5311ω53ω53ω54ω54ω5

U(1)R1111100000002222222

Table 4The predictions for the Majorana phases, the light neutrino masses |mi| (i=1,2,3) and the effective mass |mββ| of the neutrinoless double-beta decay at LO.xα21α31|m1| (meV)|m2| (meV)|m3| (meV)|mββ| (meV)Mass order

−0.5173π010.89113.89750.3552.628NO

1.00790055.91356.57628.12156.134IO

Table 5The representation matrices of the generators S, T and U for the five irreducible representations of S4 in the chosen basis, where ω=e2πi/3.STU

1, 1′11±1

2(1001)(ω00ω2)(0110)

3, 3′13(−1222−1222−1)(1000ω2000ω)∓(100001010)

Table 6Character table of the group S4, where G denotes the representative element of each conjugacy class.Classes1C13C26C2′8C36C4

G1SUTSTU

111111

1′11−11−1

2220−10

33−1−101

3′3−110−1

Generalised CP and trimaximal TM1 lepton mixing in S4 family symmetryCai-ChangLilcc0915@mail.ustc.edu.cnGui-JunDing⁎dinggj@ustc.edu.cnDepartment of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, ChinaDepartment of Modern PhysicsUniversity of Science and Technology of ChinaHefeiAnhui230026China⁎Corresponding author.AbstractWe construct two flavor models based on S4 family symmetry and generalised CP symmetry. In both models, the S4 family symmetry is broken down to the Z2SU subgroup in the neutrino sector, as a consequence, the trimaximal TM1 lepton mixing is produced. Depending on the free parameters in the flavon potential, the Dirac CP is predicted to be either conserved or maximally broken, and the Majorana CP phases are trivial. The two models differ in the neutrino sector. The flavon fields are involved in the Dirac mass terms at leading order in the first model, and the neutrino mass matrix contains three real parameters such that the absolute neutrino masses are fixed. Nevertheless, the flavon fields enter into the Majorana mass terms at leading order in the second model. The leading order lepton mixing is of the tri-bimaximal form which is broken down to TM1 by the next to leading order contributions.1IntroductionIn the past years, the Daya Bay [1], RENO [2] and Double Chooz [3] experiments, together with the long-baseline experiments T2K [4] and MINOS [5], have provided an accurate determination of the last unknown lepton mixing angle θ13, with the latest central value measured by Daya Bay being θ13≃8.7° [6]. The measurement of the reactor angle excluded many neutrino mass models, and led to new model building strategies based on family symmetries [7–10]. So far all the three lepton mixing angles and both mass-squared differences Δmsol2 and Δmatm2 have been measured to reasonably good accuracy. However, barely nothing is known on the leptonic CP phases, which contain one Dirac phase δCP and two Majorana phases α21 and α31. The global analysis of the current neutrino oscillation data gives that the 3σ range of δCP is [0,2π) [11–13], although there is some indications for non-zero δCP. Therefore we still don't know whether CP violation occurs in the lepton sector and how large it is if the CP symmetry is really violated. Measuring the leptonic CP violation is one of important goals of future long-baseline neutrino oscillation experiments [14].Family symmetry and its spontaneous breaking have turned out to be able to naturally derive some mass independent textures, please see Ref. [15] for a review. In order to explain the observed lepton mixing angles and predict CP phases at the same time, it is natural to extend the family symmetry to include a generalised CP symmetry [16–22]. In this setup, the symmetries are spontaneously broken by the flavon vacuum expectation values (VEVs) which take specific discrete complex phases. As in the paradigm of family symmetry, the whole symmetry including both family and CP symmetries are generally broken into different subgroups in the neutrino and charged lepton sectors, and the mismatch between the two remnant subgroups gives rise to particular predictions for lepton mixing angles and CP phases.Combining family symmetry with generalised CP symmetry is a promising framework to predict the values of CP violating phases. It has arisen some interesting discussions in the past years. Imposing generalised CP symmetry within the context of simple μ–τ interchange symmetry [23], A4 [22], S4 [19–21,24,25] and T′ [26] family symmetries have been explored (other approaches to discrete symmetry and CP violation can be found in Refs. [27–29]). In such scenario, the mixing angles and CP phases are generally predicted to be strongly correlated with each other because of the constraint of the family and CP symmetries. The so-called trimaximal TM2 neutrino mixing, whose second column of the mixing matrix is of the form (1,1,1)T/3, is frequently produced. In Refs. [19,22], the TM2 mixing is a natural consequence of the preserved Z2S family symmetry in the neutrino sector. In the present work, we shall focus on the trimaximal TM1 mixing whose first column of the mixing matrix takes the form (2,−1,−1)T/6, since the TM1 mixing leads to better agreement of solar mixing angle θ12 with the measured value than the TM2 pattern. We shall construct two typical models based on S4 family symmetry and the corresponding generalised CP symmetry. Both models predict TM1 mixing due to the remnant Z2SU symmetry in the neutrino sector and the Dirac CP is conserved or maximally broken. In the first model (Model 1), the flavon fields enter into the neutrino Dirac couplings instead of the Majorana mass terms of the right-handed neutrinos at leading order (LO), and the TM1 mixing is generated at LO. After taking into account the measured solar and atmospheric neutrino mass squared differences and the reactor mixing angle θ13, the absolute neutrino masses and the effective mass |mββ| for the neutrinoless double beta decay are fixed completely. For the second model (Model 2), the lepton mixing is of the tri-bimaximal form at LO with the remnant Z2S×Z2SU family symmetry in the neutrino sector, and the next-to-leading order (NLO) corrections further breaks Z2S×Z2SU into Z2SU such that TM1 pattern is obtained. Since the non-zero θ13 arises from the NLO contributions, the relative smallness of the reactor angle with respect to the solar and atmospheric mixing angles are explained.The paper is organized as follows. In Section 2, we briefly review the concept of generalised CP symmetry and the generalised CP transformation compatible with S4 family symmetry. Moreover, the possible residual CP symmetries consistent with the remnant Z2SU family symmetry in the neutrino sector and the corresponding phenomenological predictions for the lepton mixing parameters are investigated. In Section 3, we present the first model, and we show that the desired vacuum configuration with their phase structure can be realized in a supersymmetric context. In Section 4, we specify our second model, the LO structure of the model, the vacuum alignment and the NLO corrections induced by higher dimensional operators are discussed. We summarize and conclude in Section 5. The details of the group theory of S4 are given in Appendix A, where the explicit representation matrices and the Clebsch–Gordan coefficients are listed.2General analysis of lepton mixing with residual Z2SU family symmetry and CP symmetry2.1Generalised CP transformations consistent with S4It is highly non-trivial to combine a family symmetry Gf with the generalised CP symmetry together [17–19]. Let us consider a generic multiplet of fields φ(x) in the irreducible representation r of Gf. Under the action of Gf, φ(x) transforms as(2.1)φ⟶Gfρr(g)φ(x),g∈Gf, where ρr(g) is the representation matrix for the element g in the irreducible representation r. The generalised CP transformation should leave the kinetic term |∂φ|2 invariant and it acts on φ(x) as(2.2)φ(x)⟶CPXrφ⁎(x′), where Xr is a unitary matrix, x′=(t,−x) and we have omit the action of CP on spinor indices for the case that φ is a spinor. Notice that we are considering the “minimal” theory in which the generalised CP transformation maps the field φ∼r into its complex conjugate φ⁎∼r⁎. The generalised CP transformation Xr has to be consistently defined to be compatible with the family symmetry Gf. Hence the so-called consistency condition [17–19,30] must be satisfied(2.3)Xrρr⁎(g)Xr−1=ρr(g′),g,g′∈Gf. Note that Eq. (2.3) should be fulfilled for all the irreducible representations of Gf. Moreover, Eq. (2.3) implies that the generalised CP transformation Xr maps the group element g into g′, and this mapping preserves the family symmetry group structure [18,19]. Therefore Eq. (2.3) defines a homomorphism of the family symmetry group Gf. It is now established that there is one to one correspondence between the generalised CP transformations and the automorphism group of the family symmetry group [30].In the present work, we shall concentrate on the family symmetry Gf=S4, which can be generated by three generators S, T and U. It is convenient to work in the T generator diagonal basis, the representation matrices for the three generators in different S4 irreducible representations are summarized in Table 5. The corresponding Clebsch–Gordan coefficients are listed in Appendix A. The automorphism structure of S4 is rather simple, since it doesn't have non-trivial outer automorphism. Therefore the automorphism of S4 is exactly its inner automorphism, and the automorphism group of S4 is isomorphic to S4 itself. For the representative automorphism element conj(U):(S,T,U)→(S,T2,U), where conj(h) denotes a group conjugation with an element h, i.e. conj(h):g→hgh−1 with h,g∈S4, the associated generalised CP transformation Xr0 is determined by the consistency equations(2.4)Xr0ρr⁎(S)(Xr0)−1=ρr(S),Xr0ρr⁎(T)(Xr0)−1=ρr(T2),Xr0ρr⁎(U)(Xr0)−1=ρr(U). From the explicit form of the representation matrices shown in Table 5, we see that for any irreducible representations r of S4, the following relations are fulfilled(2.5)ρr⁎(S)=ρr(S),ρr⁎(U)=ρr(U),ρr⁎(T)=ρr(T2). Therefore Xr0 is fixed to be equal to identity (up to an arbitrary overall phase), i.e.(2.6)Xr0=1. Including the family symmetry transformation, the generalised CP transformation consistent with the S4 family symmetry is given by(2.7)Xr=ρr(g)Xr0=ρr(g),g∈S4. Hence the generalised CP transformation consistent with an S4 family symmetry is of the same form as the family group transformation in the chosen basis. We confirm the results in Refs. [18,19] that the generalised CP transformation group is the identity up to inner automorphism. Since we have found all generalised CP transformations consistent with the S4 family symmetry, we turn to investigate their phenomenological implications on lepton masses and flavor mixings in the following.2.2Lepton mixing from S4⋊HCP breaking into GCPl≅Z3T⋊HCPl and GCPν≅Z2SU×HCPνIn this work we shall introduce the family symmetry S4 together with the corresponding generalised CP symmetry HCP at high energy scale, where HCP is the collection of the generalised CP transformations Xr. Hence the original symmetry of the theory is S4⋊HCP. To obtain phenomenologically acceptable lepton masses and mixings, the original symmetry should be broken in both charged lepton and neutrino sectors. The mismatch between the symmetry breaking patterns in the neutrino and charged lepton sectors leads to particular predictions for lepton mixing angles and CP phases. In Ref. [19], the symmetry is broken down to Z2S×HCPν in the neutrino sector, and the residual family symmetry Z2S={1,S} enforces that the lepton mixing is the trimaximal TM2 pattern [31–33], where the second column of the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix is proportional to (1,1,1)T. In the present work, we shall investigate another case that S4⋊HCP is broken to GCPl≅Z3T⋊HCPl and GCPν≅Z2SU×HCPν in the charged lepton and the neutrino sectors respectively, where Z3T={1,T,T2} and ZSU={1,SU}. The remnant Z2SU symmetry would lead to the trimaximal TM1 mixing pattern [21,34,35], where the first column of the PMNS matrix is proportional to (2,−1,−1)T. General phenomenology analysis has shown that TM1 mixing can lead to excellent agreement with the present data [33]. Furthermore, if the residual family symmetry in the neutrino sector chosen to be Z2U={1,U}, the third column of the mixing matrix would be proportional to (0,1,−1)T. The reactor mixing angle would be predicted to be zero, and it is not consistent with both the experimental measurements [1–6] and the global data fitting [11–13]. Hence we don't consider this scenario.In the charged lepton sector, the full symmetry S4⋊HCP is broken to GCPl≅Z3T⋊HCPl. For GCPl to be a well-defined symmetry, the consistency condition of Eq. (2.3) should be satisfied for the residual family symmetry subgroup Z3T, i.e. the element Xrl of HCPl should fulfill(2.8)Xrlρr⁎(T)Xrl−1=ρr(T′),T′∈Z3T={1,T,T2}. It is easy to check that the remnant CP symmetry HCPl can take the value(2.9)HCPl={ρr(1),ρr(T),ρr(T2),ρr(U),ρr(TU),ρr(T2U)}. Without loss of generality, we assume that the three generations of the left-handed lepton doublets are unified into the three-dimensional representation 3. The same results would be obtained if the lepton doublets were assigned to 3′ of S4, since the representation 3′ differs from 3 only in the overall sign of the generator U. The charged lepton mass matrix ml is constrained by the remnant family symmetry Z3T and the remnant CP symmetry HCPl as(2.10a)ρ3†(T)mlml†ρ3(T)=mlml†,(2.10b)X3l†mlml†X3l=(mlml†)⁎, where the charged lepton mass matrix ml is given in the convention in which the left-handed (right-handed) fields are on the left-hand (right-hand) side of ml. Since the representation matrix ρ3(T) is diagonal, the invariant condition Eq. (2.10a) under Z3T implies that mlml† is diagonal with(2.11)mlml†=diag(me2,mμ2,mτ2), where me, mμ and mτ denote the electron, muon and tau masses, respectively. For the case of Xrl={ρr(1),ρr(T),ρr(T2)}, the conditions of Eq. (2.10b) is satisfied automatically, and therefore no additional constraints are required. For the remaining values Xrl={ρr(U),ρr(TU),ρr(T2U)}, the residual CP invariant condition of Eq. (2.10b) implies mμ=mτ. Hence this case is not viable phenomenologically. We note that in the models constructed in Sections 3 and 4, the Z3T remnant symmetry is broken by the flavon VEVs in order to facilitate the generation of the charged lepton mass hierarchies without fine tuning. However, we properly arrange the breaking such that the resulting charged lepton mass matrix remains diagonal. As a consequence, the hermitian product mlml† is invariant under the action of Z3T elements and the generalised CP transformations Xrl={ρr(1),ρr(T),ρr(T2)}, i.e. Eqs. (2.10a), (2.10b) are satisfied. Therefore the following general analysis is still meaningful and valid, and in particular it guides our model building.Now we turn to the neutrino sector. In order to reproduce the TM1 mixing pattern, the symmetry S4⋊HCP is spontaneously broken to GCPν=Z2SU×HCPν. The residual CP symmetry HCPν should be consistent with the residual family symmetry Z2SU, and therefore its element Xrν has to fulfill the consistency equation(2.12)Xrνρr⁎(SU)Xrν−1=ρr(SU). One can easily check that there are only 4 possible choices for Xrν, i.e.(2.13)HCPν={ρr(1),ρr(S),ρr(U),ρr(SU)}. The light neutrino mass matrix mν is constrained by the residual family symmetry Z2SU and residual CP symmetry HCPν as [19](2.14a)ρ3T(SU)mνρ3(SU)=mν,(2.14b)X3νTmνX3ν=mν⁎. The most general neutrino mass matrix which satisfies Eq. (2.14a) is of the form(2.15)mν=α(2−1−1−12−1−1−12)+β(100001010)+γ(011110101)+δ(01−1120−10−2), where the four parameters α, β, γ and δ are generally complex, and the remnant CP invariant condition of Eq. (2.14b) would further constrain these parameters to be real or purely imaginary.In order to diagonalize light neutrino mass matrix mν in Eq. (2.15), it is useful to first perform a tri-bimaximal transformation UTB(2.16)mν′=UTBTmνUTB=(3α+β−γ000β+2γ−6δ0−6δ3α−β+γ), with(2.17)UTB=(23130−1613−12−161312). Then we investigate the implication of the remnant CP invariant condition of Eq. (2.14b). Two distinct phenomenological predictions arise for Xrν={ρr(1),ρr(SU)} and Xrν=ρr(1),ρr(SU)}. We shall discuss the two cases in detail in the following.(1)Xrν=ρr(1),ρr(SU)In this case, we can straightforwardly find that all the four parameters α, β, γ and δ are constrained to be real. As a result, mν′ becomes a real symmetry matrix and can be diagonalized by a rotation matrix R(θ) in the (2,3) sector with(2.18)R(θ)=(1000cosθsinθ0−sinθcosθ), where(2.19)tan2θ=−26δ3α−2β−γ. Hence we have(2.20)Uν′Tmν′Uν′=diag(m1,m2,m3),Uν′=R(θ)P, where P is a unitary diagonal matrix with entries ±1 or ±i, which encode the CP parity of the neutrino state. Furthermore, the light neutrino masses m1,2,3 are determined to be(2.21)m1=|3α+β−γ|,m2=12|3(α+γ)−sign((3α−2β−γ)cos2θ)24δ2+(3α−2β−γ)2|,m3=12|3(α+γ)+sign((3α−2β−γ)cos2θ)24δ2+(3α−2β−γ)2|. We see that the three neutrino masses depend on four real parameters, and therefore any neutrino mass spectrum can be realized in this scenario. Since the charged lepton mass matrix is diagonal, the lepton mixing matrix UPMNS is fixed by neutrino sector completely, and we have(2.22)UPMNS=UTBUν′=(23cosθ3sinθ3−16cosθ3+sinθ2−cosθ2+sinθ3−16cosθ3−sinθ2cosθ2+sinθ3)P. The three lepton mixing angles θ13, θ12 and θ23 are predicted to be(2.23)sin2θ13=13sin2θ,sin2θ12=cos2θ2+cos2θ=13−23tan2θ13,sin2θ23=12−6sinθcosθ3−sin2θ=12±tanθ132(1−2tan2θ13). For the best fitting value of the reactor angle θ13=8.71° [13], the remaining two mixing angles are determined to be θ12≃34.31° and θ23≃32.49° or θ23≃57.51°, which are compatible with the preferred values from global fits. Adopting the PDG parameterization [36], the Dirac CP violating phase δCP and two Majorana CP violating phases α21 and α31 take the values(2.24)sinδCP=sinα21=sinα31=0, which implies(2.25)δCP,α21,α31=0,π. Hence there is no CP violation in this case.(2)Xrν=ρr(S),ρrν(U)Solving the residual CP invariant equation of Eq. (2.14b), we find the three parameters α, β and γ are real, and δ is purely imaginary. The unitary transformation Uν′ diagonalizing the neutrino mass matrix mν′ is of the form(2.26)Uν′=(1000cosθsinθ0−isinθicosθ)P, with(2.27)tan2θ=2i6δ3(α+γ). The resulting PMNS matrix is(2.28)UPMNS=UTBUν′=(23cosθ3sinθ3−16cosθ3+isinθ2−icosθ2+sinθ3−16cosθ3−isinθ2icosθ2+sinθ3)P. The lepton mixing angles and CP phases are determined to be(2.29)|sinδCP|=1,sinα21=sinα31=0,sin2θ13=13sin2θ,sin2θ12=cos2θ2+cos2θ=13−23tan2θ13,sin2θ23=12. The predictions for both the solar and reactor mixing angles are the same as the ones in case (I), and the atmospheric mixing angle is maximal. Moreover, we have maximal Dirac CP violation δCP=±π2, and Majorana phases are trivial with α21,α31=0,π. Finally, the light neutrino masses are given by(2.30)m1=|3α+β−γ|,m2=12|−3α+2β+γ+sign((α+γ)cos2θ)9(α+γ)2−24δ2|,m3=12|−3α+2β+γ−sign((α+γ)cos2θ)9(α+γ)2−24δ2|. Notice that the above results are exactly the same as those of Ref. [17], although we use a different basis in which the generator T is diagonal. The chosen basis in the present paper is particularly suitable to build TM1 model, since the charged lepton mass matrix is diagonal in this basis and the lepton mixing completely comes from the neutrino sector. Now that we have finished the general analysis, we proceed to construct models to realize these model independent results. Two typical models would be proposed in the following sections. In the first model, the lepton mixing is the TM1 pattern at LO. In the second model, tri-bimaximal mixing is produced at LO, and it is broken to TM1 mixing by the NLO corrections. As a consequence, the relative smallness of θ13 with respect to θ12 and θ23 is explained.3Model 1In this section, we shall present the first TM1 model (Model 1) based on S4⋊HCP with the extra symmetry Z7×U(1)R. We shall formulate the model in the framework of type I see-saw mechanism and supersymmetry (SUSY). Both the three generations of left-handed lepton doublets l and the right-handed neutrinos νc are assigned to transform as S4 triplet 3′, while the RH charged leptons ec, μc and τc are all invariant under S4. The involved fields and their transformation rules under the family symmetry S4×Z7×U(1)R are summarized in Table 1. Notice that the auxiliary Z7 symmetry separates the flavon fields entering the charged lepton sector at LO from those entering the neutrino sector, and it is also helpful to achieve the charged lepton mass hierarchies and suppress the NLO corrections. Compared with most flavor models in which the flavon fields generally couple to the right-handed neutrinos at LO, the flavons are involved in the neutrino Dirac mass term instead of the Majorana mass term of right-handed neutrino in the present model.3.1Vacuum alignmentWe adopt the now-standard F-term alignment mechanism to arrange the vacuum [37]. A continuous U(1)R symmetry related to R-parity is generally introduced under which the matter fields carry a +1 R-charge while the electroweak Higgs and flavon fields are uncharged. In addition, one needs the so-called driving fields carrying two unit of R-charge, and hence each term in the superpotential can contain at most one driving field. In the SUSY limit, the minimization of the flavon potential can be achieved simply by ensuring that the F-terms of the driving fields vanish at the minimum. The required driving fields and their transformation rules are listed in Table 1. The LO driving superpotential wd invariant under the family symmetry S4×Z7 can be written as(3.1)wd=wdl+wdν, where wdl is the flavon superpotential which contains the flavons only entering into the charged lepton at LO, i.e.(3.2)wdl=f1(φT0(φTφT)3′)1+f2(φT0(ϕφT)3′)1+f3ζ0(φTφT)1+f4ζ0(ϕϕ)1. wdν is the superpotential associated with the flavons in the neutrino sector(3.3)wdν=g1(φS0(φSφS)3)1+g2(φS0(ηφS)3)1+g3ξ(φS0φS)1+g4(η0(φSφS)2)1+g5(η0(ηη)2)1+g6ξ(η0η)1, where (…)r denotes the contraction into S4 irreducible representation r according to the Clebsch–Gordan coefficients presented in Appendix A. We note that the first term vanishes automatically due to the anti-symmetric property of the contraction (φSφS)3. Since we require the theory to be invariant under the generalised CP transformation, then all the couplings fi and gi in Eqs. (3.2), (3.3) are constrained to be real. We start from the charged lepton sector, and the F-term conditions obtained from the driving fields φT0 and ζ0 read(3.4)∂wd∂φT10=2f1(φT12−φT2φT3)+f2(ϕ1φT2+ϕ2φT3)=0,∂wd∂φT20=2f1(φT22−φT1φT3)+f2(ϕ1φT1+ϕ2φT2)=0,∂wd∂φT30=2f1(φT32−φT1φT2)+f2(ϕ1φT3+ϕ2φT1)=0,∂wd∂ζ0=f3(φT12+2φT2φT3)+2f4ϕ1ϕ2=0. We find two possible solutions for the vacuum (up to S4 transformations). The first one is given by(3.5)〈φT〉=(111)vT,〈ϕ〉=(1−1)vϕ,withvT2=2f43f3vϕ2. The second solution is(3.6)〈φT〉=(010)vT,〈ϕ〉=(01)vϕ,with vT=−f22f1vϕ. We shall choose the second solution in this work. We note that the phase of vϕ can be absorbed into lepton fields, and therefore we can take both vϕ and vT to be real, since f1 and f2 are real. Furthermore, vϕ and vT are expected to be of the same order of magnitude without fine tuning among the parameters f1 and f2. The vacuum expectation values of φS, η and ξ, which give rise to TM1 mixing in the neutrino sector, are determined by the F-terms of the associated driving fields as follows:(3.7)∂wdν∂φS10=g2(η1φS2+η2φS3)+g3ξφS1=0,∂wdν∂φS20=g2(η1φS1+η2φS2)+g3ξφS3=0,∂wdν∂φS30=g2(η1φS3+η2φS1)+g3ξφS2=0,∂wdν∂η10=g4(φS32+2φS1φS2)+g5η12+g6ξη2=0,∂wdν∂η20=g4(φS22+2φS1φS3)+g5η22+g6ξη1=0. There are two independent solutions to this set of equations up to S4 family symmetry transformations. The first solution is(3.8)〈φS〉=(111)vS,〈η〉=(11)vη,〈ξ〉=vξ. The VEVs vS, vη and vξ are related with each other via(3.9)vS2=g3(2g2g6−g3g5)12g4g22vξ2,vη=−g32g2vξ, where vξ is undetermined and generally complex. With the representation matrix given in Appendix A, we can straightforwardly check that the S4 family symmetry is broken down to Z2S×Z2SU subgroup by the vacuum alignment of Eq. (3.8). The second solution reads as(3.10)〈φS〉=(2−1−1)vS,〈η〉=(11)vη,〈ξ〉=vξ, with(3.11)vS2=g3(g3g5+g2g6)3g4g22vξ2,vη=g3g2vξ. We find the S4 family symmetry is spontaneously broken down to Z2SU subgroup in this case. In order to reproduce the TM1 pattern, we choose the second solution in the following. Since all the couplings gi are real due to the generalised CP invariance, Eq. (3.11) implies that the VEVs vη and vξ have the same phase up to π, and the phase difference between vS and vξ is 0, π or ±π2 depending on the sign of g3g4(g3g5+g2g6). In addition, it is natural to expect that the three VEVs vξ, vη and vS are of the same order of magnitudes. As shall be shown below, the phase of vξ turns out to be an overall phase of the light neutrino mass matrix, and hence it can be absorbed into the neutrino fields. That is to say we can take vξ to be real without loss of generality. As a consequence, the VEV vη would be real as well and the VEV vS is real for the product g3g4(g3g5+g2g6)>0 or purely imaginary for g3g4(g3g5+g2g6)<0.Regarding the order of magnitude of the different VEVs, as we shall find in the following, the charged lepton mass hierarchies can be naturally reproduced if vϕ/Λ and vT/Λ are of order λ2, i.e.(3.12)vϕΛ∼vTΛ∼λ2, where λ≃0.23 is the Cabibbo angle. In order to guarantee the stability of the successful LO results under the inclusion of higher dimensional terms, we choose all the VEVs in the model are of the same order λ2Λ, i.e.(3.13)vSΛ∼vηΛ∼vξΛ∼λ2. This assumption is frequently used in the family symmetry model building.3.2The lepton masses and mixingThe most general superpotential for the charged lepton masses, which is invariant under the family symmetry, is of the form(3.14)wl=yτΛ(lφT)1τchd+yμ1Λ2(l(φTφT)3′)1μchd+yμ2Λ2(l(ϕφT)3′)1μchd+ye1Λ3(lφT)1(φTφT)1echd+ye2Λ3((lφT)2(φTφT)2)1echd+ye3Λ3((lφT)3′(φTφT)3′)1echd+ye4Λ3((lφT)3(φTφT)3)1echd+ye5Λ3((lϕ)3′(φTφT)3′)1echd+ye6Λ3((lϕ)3(φTφT)3)1echd+ye7Λ3((lφT)2(ϕϕ)2)1echd+ye8Λ3(lφT)1(ϕϕ)1echd+⋯, where dots represent the higher dimensional operators which will be commented later. Generalised CP symmetry enforces the Yukawa couplings to be real. Due to the constraint of the Z7 symmetry, the electron, muon and tau mass terms are suppressed by 1/Λ, 1/Λ2 and 1/Λ3 respectively. With the vacuum alignment of Eq. (3.6), we find the resulting charged lepton mass matrix is diagonal with(3.15)me=(ye2−2ye3+2ye5vϕvT+ye7vϕ2vT2)vT3Λ3vd,mμ=(2yμ1+yμ2vϕvT)vT2Λ2vd,mτ=yτvTΛvd, in which vd=〈hd〉 is the VEV of the electroweak Higgs field hd. We see that the observed mass hierarchies among the charged leptons can be generated for vϕ/Λ∼vT/Λ∼λ2. For the vacuum of φT and ϕ in Eq. (3.6), we can check that the S4 family symmetry is broken completely in the charged lepton sector, since T〈φT〉=ω2〈φT〉 and T〈ϕ〉=ω2〈ϕ〉. However, the lepton flavor mixing is associated with the hermitian combination mlml†, which is obviously invariant under the action of T, i.e., T†mlml†T=mlml†. Consequently there is still a remnant Z3T symmetry in the charged lepton sector if we concentrate on lepton flavor mixing. Furthermore, we can check that only three of the 24 generalised CP symmetries are preserved by mlml† and HCPl={ρr(1),ρr(T),ρr(T2)}.Neutrino masses are generated by type I see-saw mechanism. The LO superpotential is given by(3.16)wν=y1Λ(lνc)1ξhu+y2Λ((lνc)2η)1hu+y3Λ((lνc)3φS)1hu+M(νcνc)1, where the first three terms contribute to the neutrino Dirac mass whereas the last one is the Majorana mass terms for the right-handed neutrinos. All the couplings are again real because of the imposed generalised CP symmetry. Given the vacuum configuration of Eq. (3.10), we can read out the Dirac and Majorana mass matrices as follows(3.17)mD=y1vuvξΛ[(100001010)+x(011110101)+y(0−11102−1−20)],mM=M(100001010), where vu=〈hu〉 and the parameters x, y are(3.18)x=y2vηy1vξ,y=y3vSy1vξ. After extracting the common phase of the VEVs vS, vη and vξ, the parameter x is real, while y is real or purely imaginary. The light neutrino mass matrix is given by the see-saw formula(3.19)mν=−mDmM−1mDT=α(2−1−1−12−1−1−12)+β(100001010)+γ(011110101)+δ(01−1120−10−2). It is the most general neutrino mass matrix consistent with the residual Z2SU flavor symmetry, as is shown in Eq. (2.15). The four parameters α, β, γ and δ are given by(3.20)α=−y2m0,β=(4y2−2x2−1)m0,γ=(y2−x2−2x)m0,δ=−3xym0, where m0=y12vu2vξ2MΛ2 is the overall scale of the light neutrino masses. We see that α, β and γ are real parameters, δ is real or imaginary for vS being real or imaginary, respectively. Furthermore, the effective mass parameter |mββ| for the neutrinoless double-beta decay is given by(3.21)|mββ|=m0|2α+β|.As shown in Eq. (3.11), if the combination g3g4(g3g5+g2g6)>0, which leads to real vS and δ parameters, the vacuum alignments of the flavons φS, η and ξ in Eq. (3.10) are invariant under the action of both ρr(1) and ρr(SU) elements of HCP. Therefore the generalised CP symmetry is broken to HCPν={ρr(1),ρr(SU)} in the neutrino sector. This case is identical to case (I) of the general analysis inspired by symmetry arguments. The corresponding light neutrino mass matrix of Eq. (3.19) is real, the lepton mixing is exactly the TM1 pattern with conserved CP, and the predictions for light neutrino masses and mixing angles are given in Eq. (2.21) and Eq. (2.23). Notice that the light neutrino mass matrix of Eq. (3.19) depends on three real parameters x, y and m0, their values can be fixed by the measured values of the mass squared differences Δmsol2 and Δmatm2 and the reactor neutrino mixing angle θ13. As a result, both the absolute scale of the neutrino masses and the lepton mixing angles are fixed. For the best fitting values of Δmsol2=7.45×10−5 eV2, Δmatm2=2.417(2.410)×10−3 eV2 and sin2θ13=0.0229 from Ref. [13], we find there are 8 solutions to the values of x and y in the case that both x and y are real. The corresponding predictions for the light neutrino masses and the lepton mixing parameters are summarized in Table 2. It is obvious that the former 4 solutions correspond to a normal ordering (NO) neutrino mass spectrum, and the latter 4 correspond to inverted ordering (IO) spectrum. Moreover, we see that the predicted values for the atmospheric mixing angle θ23 (32.496° and 57.504°) are slightly beyond the 3σ range of the current global data fitting [11–13]. We note that the NLO corrections and the renormalization group evolution effects could bring the model to agree with the experimental data. However, in these scenarios a value of θ23 very close to the maximal mixing value of 45° would be unnatural. The next generation neutrino oscillation experiments, in particular those exploiting a high intensity neutrino beam, will reduce the experimental error on θ23 to few degrees. If no significant deviations from maximal atmospheric mixing will be detected, these 8 solutions will be ruled out.Another possibility of g3g4(g3g5+g2g6)<0 gives rise to an imaginary vS such that the parameter δ in the neutrino mass matrix of Eq. (3.19) is purely imaginary as well. The remnant CP symmetry in the neutrino sector is HCPν={ρr(S),ρr(U)}. This corresponds to the case (II) discussed in the general analysis of Section 2.2. The predictions for the mixing parameters and the light neutrino masses are given in Eq. (2.29) and Eq. (2.30). The lepton mixing is of the TM1 form, and maximal Dirac CP violation |δCP|=π/2 and maximal atmospheric mixing θ23=45° are produced in this case. Analogously, the light neutrino sector is also controlled by three real parameters, and hence the model is quite predictive, as shown in the last two lines of Table 2. The neutrino mass spectrum can only be normal ordering in this case.Generally the LO results are modified by the subleading terms invariant under the imposed symmetry. Because of the auxiliary Z7 symmetry in the present model, all the subleading corrections can be obtained by inserting the combination ΦlΦν into the LO terms of wd, wl and wν in Eqs. (3.1), (3.14), (3.16),11All possible S4 contractions should be considered here, and only the correction to the electron mass terms is an exception with the form (lΦν4)echd/Λ4. where Φl={φT,ϕ} and Φν={φS,η,ξ} denote the flavons in the charged lepton and neutrino sectors respectively. As a result, the corresponding corrections to the lepton masses and mixing angles are suppressed by 〈Φl〉〈Φν〉/Λ2∼λ4 with respect to the LO contributions and therefore can be negligible.4Model 2In this section, we shall try to improve the previous model by generating the reactor mixing angle at the next-to-leading order (NLO) such that the correct order of magnitude of θ13 is produced. In this model, the LO lepton mixing is the well-known tri-bimaximal mixing pattern which is broken to TM1 mixing by NLO contributions. Analogous to the previous model, the present model is based on the symmetry S4⋊HCP with the extra symmetry Z4×Z5×U(1)R in order to eliminate unwanted operators. The matter fields, flavon fields, driving fields and their transformation rules under the family symmetry are summarized in Table 3. As previous model of Section 3, the remnant symmetry of the hermitian combination mlml† is Z3T⋊HCPl with HCPl={ρr(1),ρr(T),ρr(T2)}, and the original symmetry S4⋊HCP is broken down to GCPnu=Z2SU×HCPν. As a consequence, the model-independent analysis results of Section 2.2 are realized within one model, and the Dirac CP phase δCP is predicted to be trivial or maximal. In the following, we firstly discuss the vacuum alignment of the model, then specify the structure of the model at LO and NLO.4.1Vacuum alignmentThe most general driving superpotential wdl associated with the charged lepton sector, which is invariant under the family symmetry S4×Z4×Z5, can be written as(4.1)wdl=f1(φT0(φTφT)3′)1+f2(φT0(ϕφT)3′)1+f3ζ0(φTφT)1+f4ζ0(ϕϕ)1. It is exactly the same as Eq. (3.2). Hence the vacuum of the flavon fields φT and ϕ is of the same form as shown in Eq. (3.6), i.e.(4.2)〈φT〉=(010)vT,〈ϕ〉=(01)vϕ,with vT=−f22f1vϕ. We see that vϕ and vT carry the same phase up to π. Since the phase of vϕ can be absorbed by leptons, we can take vϕ and vT to be real without loss of generality. From the following predictions for charged lepton masses in Eq. (4.10), we note that the mass hierarchies between the charged leptons can be produced for(4.3)vϕΛ∼vTΛ∼O(λ2). The driving superpotential wdν involving the flavons of the neutrino sector reads(4.4)wdν=g1(φS0(φSφS)3)1+g2(φS0(ηφS)3)1+g3ξ0(φSφS)1+g4ξ0(ηη)1+Mη(η0η)1+g5(η0(χχ)2)1+g6ρ0(χχ)1+g7ρ0ξ2+g8σ0(χφS)1, where all the coupling gi and mass parameter Mη are real because of the imposed generalised CP symmetry. Since the contraction (φSφS)3 vanishes due to the antisymmetry of the associated S4 Clebsch–Gordan coefficients, the first term proportional to g1 gives null contribution. In the SUSY limit, the vacuum configuration is determined by the vanishing of the derivative of the driving superpotential wdν with respect to each component of the driving fields. The minimization equations for the vacuum take the following form:(4.5)∂wdν∂φS10=g2(η1φS2−η2φS3)=0,∂wdν∂φS20=g2(η1φS1−η2φS2)=0,∂wdν∂φS30=g2(η1φS3−η2φS1)=0,∂wdν∂ξ0=g3(φS12+2φS2φS3)+2g4η1η2=0,∂wd∂η10=Mηη2+g5(χ32+2χ1χ2)=0,∂wd∂η20=Mηη1+g5(χ22+2χ1χ3)=0,∂wd∂ρ0=g6(χ12+2χ2χ3)+g7ξ2=0,∂wd∂σ0=g8(χ1φS1+χ2φS3+χ3φS2)=0. The solution to these equation are(4.6)〈φS〉=(111)vS,〈η〉=(11)vη,〈χ〉=(01−1)vχ,〈ξ〉=vξ. The VEVs vS, vη, vχ and vξ are related by(4.7)vS2=−g4g52g726g3g62Mη2vξ4,vη=−g5g72g6Mηvξ2,vχ2=g72g6vξ2, where vξ parameterizes a flat direction in the driving superpotential wdν, and it is in general complex. It is straightforward to check that the VEVs of the flavon fields φS, η and ξ preserve the remnant K4 subgroup generated by Z2S and Z2SU, while the VEV of χ is invariant only under the action of Z2SU. In our model presented below, φS and η couple with the right-handed neutrino at LO, as shown in Eq. (4.12). The resulting lepton mixing is of the tri-bimaximal form. The flavons χ and ξ enter into the neutrino sector at NLO, and the LO residual K4 symmetry is further broken down to Z2SU. As a result, the NLO contributions modify the LO tri-bimaximal mixing into TM1 pattern. In order to achieve the measured size of θ13≃λ/2 [38,39], we could choose(4.8)vSΛ∼vηΛ∼vχΛ∼vξΛ∼O(λ). Consequently the NLO corrections are suppressed by a factor λ with respect to the LO contributions, and therefore the reactor angle is of the correct order λ. Note that the VEVs of the flavon fields in the neutrino and the charged lepton sectors are chosen to be of different order of magnitude: λΛ v.s. λ2Λ, please see Eq. (4.3) and Eq. (4.8). This mild hierarchy can be accommodated because these two sets of VEVs depend on different model parameters.4.2Leading order resultsThe superpotential for the charged lepton masses, which is allowed by the symmetry, is given by(4.9)wl=yτΛ(lφT)1τchd+yμ1Λ2(l(φTφT)3′)1μchd+yμ2Λ2(l(ϕφT)3′)1μchd+ye1Λ3(lφT)1(φTφT)1echd+ye2Λ3((lφT)2(φTφT)2)1echd+ye3Λ3((lφT)3′(φTφT)3′)1echd+ye4Λ3((lφT)3(φTφT)3)1echd+ye5Λ3((lϕ)3′(φTφT)3′)1echd+ye6Λ3((lϕ)3(φTφT)3)1echd+ye7Λ3((lφT)2(ϕϕ)2)1echd+ye8Λ3(lφT)1(ϕϕ)1echd+⋯, which is identical to the corresponding superpotential of Model 1 shown in Eq. (3.14). After electroweak and flavor symmetry breaking in the way of Eq. (4.2), we obtain a diagonal charged lepton mass matrix:(4.10)ml=(yevT2Λ2000yμvTΛ000yτ)vTΛvd, where ye and yμ are the results of the different contributions of the yei and yμi respectively with(4.11)yμ=2yμ1+yμ2vϕvT,ye=ye2−2ye3+2ye5vϕvT+ye7vϕ2vT2. Now we turn to the neutrino sector, The LO superpotential relevant to the neutrino masses is of the form(4.12)wν=y(lνc)1hu+y1((νcνc)3′φS)1+y2((νcνc)2η)1, where all the three couplings y, y1 and y2 are real because of the generalised CP symmetry. We can easily read out the Dirac neutrino mass matrix as(4.13)mD=yvu(100001010). Given the vacuum of the flavons φS and η shown in Eq. (4.6), the mass matrix for the right-handed neutrino takes the form(4.14)mM=a(2−1−1−12−1−1−12)+b(011110101), where a=y1vS and b=y2vη. The light neutrino mass matrix is given by the see-saw formula, yielding(4.15)mν=−mDmM−1mDT=UTBdiag(m1,m2,m3)UTBT. That is to say the LO lepton flavor mixing is the tri-bimaximal pattern. The reason is that the VEVs of φS and η break the S4 family symmetry into a residual K4≅Z2S×Z2SU subgroup, i.e. the vacuum of φS and η in Eq. (4.6) is invariant under both Z2S and Z2SU. Furthermore, the light neutrino masses m1,2,3 in Eq. (4.15) are given by(4.16)m1=y2vu2−3y1vS+y2vη,m2=−y2vu22y2vη,m3=−y2vu23y1vS+y2vη. It is interesting to note that the following sum rule is satisfied(4.17)1m3−1m1=1m2. Since the VEVs vS and vη are related through Eq. (4.7), the phase different between vS and vη is fixed to discrete values 0, π or ±π/2 for the product g3g4<0 or g3g4>0, respectively. Moreover, the phase of vξ can be absorbed by redefining the right-handed neutrino fields, therefore we can set vξ to be real, and then another VEV vS would be real or purely imaginary. For the case of vS being imaginary, Eq. (4.16) implies that the light neutrino masses are degenerate, i.e. |m1|=|m3|. Therefore this case is not phenomenologically viable, and we shall choose vS to be real (or vS and vη have the same phase up to relative sign) in the following. Then the neutrino mass-squared differences are predicted to be(4.18)Δmsol2≡|m2|2−|m1|2=3(3x+1)(x−1)4(3x−1)2(y2vu2y2vη)2,Δmatm2≡|m3|2−|m1|2=−12x(9x2−1)2(y2vu2y2vη)2,for NO,Δmatm2≡|m2|2−|m3|2=3(3x−1)(x+1)4(3x+1)2(y2vu2y2vη)2,for IO, where x=y1vSy2vη is a real parameter. Furthermore, the effective mass parameter |mββ| for the neutrinoless doublet beta is given by(4.19)|mββ|=|x+12(3x−1)||y2vu2y2vη|. Since the solar neutrino mass squared difference Δmsol2 is positive, we have x>1 or x<−13 from Eq. (4.18). By further inspecting the atmospheric neutrino mass squared difference Δmatm2, we find that neutrino spectrum is normal ordering (NO) for x<−13 and inverted order (IO) for x>1. Taking the best fit values Δmsol2=7.45×10−5 eV2 and Δmatm2=2.417(2.410)×10−3 eV2 for NO (IO) spectrum from Ref. [13], we get two solutions for the ratio x (one for normal ordering and another for inverted ordering):(4.20)x=−0.5173,1.0079. The corresponding predictions for the Majorana phases, the light neutrino masses and |mββ| are presented in Table 4.4.3Next-to-leading-order correctionsSince the LO tri-bimaximal mixing pattern leads to a vanishing reactor angle θ13 which has been definitely excluded by the experimental measurements, NLO corrections are needed to achieve agreement with the present data. In this section, we shall address the NLO corrections indicated by higher dimensional operators compatible with all the symmetries of the model. As we shall show, the NLO contributions break the remnant family K4≅Z2S×Z2SU in the neutrino sector down to Z2SU. As a result, a non-zero θ13 is generated and it is naturally smaller than θ12 and θ23 which arise at LO.In the following, we first discuss the NLO corrections to the charged lepton sector. For the driving superpotential wdl, the most relevant subleading operators can be written as(4.21)δwdl=(φT0Ψl2Ψν2Ψν′)1/Λ3+(ζ0Ψl2Ψν2Ψν′)1/Λ3, where we have suppressed all dimensionless coupling constants, and all the possible S4 contractions should be considered with Ψl={φT,ϕ}, Ψν={φS,η} and Ψν′={χ,ξ}. These operators are suppressed by 〈Ψν〉2〈Ψν′〉/Λ3∼λ3 compared to LO terms in wdl of Eq. (4.1). Hence the subleading corrections to the VEVs of the φT and ϕ appear at the relative order λ3 such that their vacuum configurations at NLO can be parameterized as(4.22)〈φT〉=vT(ϵ1λ31+ϵ2λ3ϵ3λ3),〈ϕ〉=vϕ(ϵ4λ31) where the coefficients ϵi (i=1,2,3,4) have absolute value of order one and are generally complex due to the undetermined phase of vξ. Note that the shift of the second component of ϕ has been absorbed into the redefinition of the undetermined parameters vϕ. The subleading corrections to the charged lepton superpotential wl take the form(4.23)δwl=(lΨlΨν2Ψν′)1hdτc/Λ4+(lΨl2Ψν2Ψν′)1hdμc/Λ5+(lΨl3Ψν2Ψν′)1hdec/Λ6, where the dimensionless coupling constants are omitted. The charged lepton mass matrix is obtained by adding the contributions of this set of high dimensional operators evaluated with the insertion of the LO VEVs of Eqs. (4.2), (4.6), to those of the LO superpotential in Eq. (4.9) evaluated with the NLO vacuum configuration in Eq. (4.22). We find that each element of the charged lepton mass matrix receives corrections from both the subleading operators δwl in Eq. (4.23) and the shifted vacuum alignment of Eq. (4.22). As a consequence, its off-diagonal elements become non-zero and are all suppressed by λ3 with respect to the diagonal ones. Therefore the charged lepton mass matrix including subleading corrections can be written as(4.24)mlNLO=(meλ3mμλ3mτλ3memμλ3mτλ3meλ3mμmτ). Its contribution to the lepton mixing angles is of order λ3 and can be safely neglected. Since the off-diagonal elements are quite small in particular the (2,1) and (3,1) entries, perturbatively diagonalizing the above NLO charged lepton mass matrix mlNLO reveals that the NLO corrections to the charged lepton masses are of relative order λ6, and hence they are negligible as well.Next, we turn to discuss the NLO corrections in the neutrino sector. The NLO contributions to the driving superpotential wdν is suppressed by one power of 1/Λ with respect to the LO terms in Eq. (4.4), and it takes the form22The subleading corrections to the terms proportional to η0 and ρ0 are of the form (η0Ψν3Ψν′)1/Λ2 and (ρ0Ψν3Ψν′)1/Λ2, which are suppressed by 1/Λ2 instead of 1/Λ.(4.25)δwdν=h1Λ((φS0φS)2(χχ)2)1+h2Λ((φS0φS)3(χχ)3)1+h3Λ((φS0φS)3′(χχ)3′)1+h4Λξ((φS0φS)3′χ)1+h5Λ((φS0η)3(χχ)3)1+h6Λ((φS0η)3′(χχ)3′)1+h7Λ((φS0η)3′χ)1ξ+h8Λξ0(φS(χχ)3′)1+h9Λξ0ξ(φSχ)1+h10Λξ0(η(χχ)2)1+h11Λσ0(χ(χχ)3′)1+h12Λσ0ξ(χχ)1+h13Λσ0ξ3, where all the couplings hi are again real because of the generalised CP symmetry. Repeating the minimization procedure of Section 4.1, we find that the LO vacuum configuration is modified into(4.26)〈φS〉=vS′(111)+δvS(01−1),〈χ〉=vχ(01−1)+δvχ(111), with(4.27)vS′−vS=−(h8g3+h103g3vηvS)vχ2Λ,δvS=(h4g2vSvη−h7g2)vχvξΛ,δvχ=−h13vξ3g8vSvξ2Λ+2vξ3vS(h12g8−h7g2+h4vSg2vη)vχ2Λ, and the vacuum of η doesn't acquires non-trivial shifts at this order. Obviously the shifts vS′−vS, δvS and δvχ are suppressed with respect to the LO VEVs vS and vχ by a factor λ. Notice that the shifted vacuum of φS and χ in Eq. (4.26) is the most general form of VEV invariant under the Z2SU subgroup. The reason is that the NLO terms δwdν of Eq. (4.25) only involve the neutrino flavons φS, η, χ and ξ whose LO VEVs leave Z2SU invariant.From Section 4.2, we know that the VEVs vS, vη and vξ2 have to share the same phase, i.e. the product g3g4<0 is needed otherwise the light neutrino mass spectrum would be partially degenerate. Furthermore, Eq. (4.6) implies that the phase different between vχ and vξ is 0, π or ±π2 for g6g7>0 or g6g7<0, respectively. As a result, vS′ and vS carry the sane phase. Since it is always possible to absorb the phase of vξ by a redefinition of the matter fields, we can take vξ to be real without loss of generality. Then vS′, vη and vχ2 would be real, while vχ and δvS can be real or purely imaginary depending on g6g7>0 or g6g7<0.Now we come to the NLO corrections to the LO neutrino superpotential wν in Eq. (4.12). The higher order corrections to the neutrino Dirac mass are of the form(4.28)(lνcΨν2Ψν′)1hu/Λ3. The corresponding contributions are suppressed by λ3 compared to the LO term y(lνc)1hu. Such small corrections have a tiny impact for the neutrino mass matrix and lepton mixing parameters, and therefore can be neglected. The NLO corrections to the RH neutrino Majorana mass terms are(4.29)δwν=s1(νcνc)1(χχ)1/Λ+s2((νcνc)2(χχ)2)1/Λ+s3((νcνc)3(χχ)3)1/Λ+s4((νcνc)3′(χχ)3′)1/Λ+s5ξ((νcνc)3′χ)1/Λ+s6ξ2(νcνc)1/Λ. The resulting corrections to the RH neutrino mass matrix mM can be obtained by inserting the LO vacuum of χ and ξ in Eq. (4.6) into these operators. Another source of corrections to mM arises from the LO superpotential wν in Eq. (4.12) evaluated with the NLO VEVs of Eq. (4.26). Adding the two contributions, we obtained the corrected RH neutrino mass matrix as(4.30)mMNLO=a(2−1−1−12−1−1−12)+b(011110101)+c(100001010)+d(01−1120−10−2), with(4.31)a=y1vS′+2s4vχ2/Λ,b=y2vη+s2vχ2/Λ,c=s6vξ2/Λ−2s1vχ2/Λ,d=y1δvS+s5vχvξ/Λ=[y1(h4g2vSvη−h7g2)+s5]vχvξΛ, where parameters a and b have been redefined to include the NLO contributions. Note that c and d arise from the NLO contributions, and they are suppressed by a factor λ with respect to a and b, i.e.(4.32)a,b∼λΛ,c,d∼λ2Λ. Applying the see-saw relation, the light neutrino mass matrix at NLO takes the form(4.33)mνNLO=−mD(mMNLO)−1mDT,mνNLO=α(2−1−1−12−1−1−12)+β(100001010)+γ(011110101)+δ(01−1120−10−2). It is the most general neutrino mass matrix invariant under residual family symmetry Gν=Z2SU={1,SU}, as shown in Eq. (2.15). The parameters α, β, γ and δ are given by(4.34)α=−a(2b+c)+d2(3a−b+c)[(3a+b−c)(2b+c)−6d2],β=−3a2−b2+c2+2d2(3a−b+c)[(3a+b−c)(2b+c)−6d2],γ=−3a2+b(c−b)+d2(3a−b+c)[(3a+b−c)(2b+c)−6d2],δ=−d(3a+b−c)(2b+c)−6d2, where the overall factor y2vu2 is omitted here. Because the theory is required to be invariant under the generalised CP transformations, the phases of the model parameters are strongly constrained. The vacuum alignment of Eq. (4.7) implies that the phase different between vχ and vξ is 0, π or π/2 for g6g7>0 and g6g7<0 respectively. Further recalling that vs and vξ2 should have a common phase (up to relative sign) to avoid degenerate light neutrino masses at LO. Therefore, a, b and c are real while d is real or imaginary after the unphysical phase of vξ is extracted. As a result, α, β and γ in Eq. (4.30) are real parameters whereas δ can be real or purely imaginary. In the following, we discuss the two cases one after another.Firstly, we consider the case that vχ is real, which corresponds to the parameter domain of g6g7>0. We can check that the remnant CP symmetry in the neutrino sector is HCPν={ρr(1),ρr(SU)} in this case. All the four parameters α, β, γ and δ are real. This is exactly the case (I) of model-independent analysis in Section 2. Remembering that the subleading operators in the charged lepton sector induce corrections to the lepton mixing angles as small as λ3. Hence, the lepton flavor mixing is determined by the neutrino sector. From Section 2, we know that the resulting lepton mixing matrix is(4.35)UPMNS=(23cosθ3sinθ3−16cosθ3+sinθ2−cosθ2+sinθ3−16cosθ3−sinθ2cosθ2+sinθ3), with(4.36)tan2θ=−26δ3α−2β−γ=26d3a−b−2c∼O(λ). The lepton mixing angles are given by(4.37)sinθ13=|sinθ3|≃|2d3a−b−2c|∼O(λ),sin2θ12≃13+O(λ2),sin2θ23≃12±2d3a−b−2c. We see that the reactor angle θ13 is predicted to be of the correct order of λ, and thus experimentally preferred value could be achieved. The solar mixing angle retains its tri-bimaximal value to the first order of λ, and the atmospheric angle can deviate from its maximal mixing value of 45°. As a consequence, the deviation of the atmospheric angle from maximal mixing, indicated by the latest global fits, can be produced. In addition, we find a simple sum rule sin2θ23≃0.5±2sinθ13. This relation might be testable in the near future as soon as the experimental uncertainties for θ23 are reduced. Furthermore, since the light neutrino mass matrix is real, there is no CP violation in this case, both the Dirac CP phase and the Majorana CP phases are 0 or π.Then we consider the remaining case of vχ being purely imaginary, i.e. the phase different between vχ and vξ is ±π2. This scenario can be realized in the parameter domain g6g7<0. The generalised CP symmetry is broken down to HCPν={ρr(S),ρr(U)} in the neutrino sector. This corresponds to the case (II) of Section 2. The resulting parameters α, β, γ are real and δ is imaginary. The lepton mixing matrix is of the form(4.38)UPMNS=(23cosθ3sinθ3−16cosθ3+isinθ2−icosθ2+sinθ3−16cosθ3−isinθ2icosθ2+sinθ3), with(4.39)tan2θ=2i6δ3(α+γ)=2i6d3(a+b)∼O(λ). Consequently the three mixing angles θ13, θ12 and θ23 are modified to(4.40)sinθ13≃|2d3(a+b)|∼O(λ),sin2θ12=13+O(λ2),sin2θ23=12. It is noteworthy that we obtain maximal Dirac CP violation δCP=±π/2 in this case while the Majorana CP phases are still trivial with sinα21=sinα13=0. In short summary, our model produces the tri-bimaximal mixing at LO, which is further broken down to trimaximal TM1 mixing by NLO contributions. Depending on the coupling product g6g7 being positive or negative, the two cases arising from the model independent analysis can be realized.5ConclusionsThe measurement of sizable reactor mixing angle θ13 has opened up the possibility of measuring leptonic CP violations. In particular, the measurement of Dirac CP phase is one of the primary goals of next generation neutrino oscillation experiments. On the theoretical side, the origin of CP violation remains a mystery. Extending family symmetry to include generalised CP symmetry together with its spontaneous breaking is a promising framework to predict both mixing angles and CP phases.In this work, we analyse the interplay of generalised CP symmetry and the S4 family symmetry. Firstly we perform a model independent analysis of the possible lepton mixing matrices and the corresponding lepton mixing parameters, which arise from the symmetry breaking of S4⋊HCP into Z3T⋊HCPl in the charged lepton sector and Z2SU⋊HCPν in the neutrino sector. We find that the lepton flavor mixing is of the TM1 form and the Dirac CP can be vanishing or maximally broken while the Majorana CP is trivial with sinα21=sinα31=0.Furthermore, we construct two models to realize the above model independent results based on S4 family symmetry and the generalised CP symmetry. The two models differ in the neutrino sectors. In the first model, the flavon fields enter in the neutrino Dirac mass term instead of the Majorana mass term for right-handed neutrinos at LO. The resulting light neutrino mass matrix is predicted to depend on three real parameters, and therefore the absolute neutrino masses and the effective mass |mββ| for neutrinoless double beta decay are completely fixed after considering the constraints from the measured values of the neutrino mass squared differences Δmsol2 and Δmatm2 and the reactor angle θ13. The lepton mixing matrix is the TM1 pattern, and the subleading corrections are small enough to be negligible. In the case of g3g4(g3g5+g2g6)>0, the Dirac CP phase δCP is 0 or π, and neutrino mass spectrum can be normal ordering or inverted ordering. For the case of g3g4(g3g5+g2g6)<0, the Dirac CP is maximal δCP=±π/2, and the neutrino mass spectrum can only be normal ordering.In the second model, the S4 family symmetry is broken down to Z2S×Z2SU in the neutrino sector at LO, and therefore the LO lepton mixing is of the tri-bimaximal form. NLO correction terms break the remnant symmetry Z2S×Z2SU into Z2SU, as a result, the TM1 mixing is produced and the relative smallness of θ13 with respect to θ12 and θ23 is explained. Depending on the product g6g7 being positive or negative, the Dirac CP is predicted to be conserved or maximally broken. Moreover, we have shown that the desired vacuum alignment together with their phase structure can be achieved.AcknowledgementsOne of the author (G.J.D.) is grateful to Stephen F. King, Christoph Luhn and Alexander J. Stuart for stimulating discussions on generalised CP symmetry. G.J.D. would also like to thank Stephen F. King and the School of Physics and Astronomy at the University of Southampton for hospitality during his visit, where part of this work was done. The research was partially supported by the National Natural Science Foundation of China under Grant Nos. 11275188 and 11179007. Appendix AGroup Theory of S4S4 is the permutation group of order 4 with 24 elements, and it has been widely used as a family symmetry. In this work, we shall follow the conventions and notations of Refs. [19,40], where S4 is expressed in terms of three generators S, T and U. These three generators satisfy the multiplication rules:(A.1)S2=T3=U2=(ST)3=(SU)2=(TU)2=(STU)4=1. Note that the generators S and T alone generate the group A4, while the generated group by T and U is S3. The S4 group elements can be divided into 5 conjugacy classes(A.2)1C1={1},3C2={S,TST2,T2ST},6C2′={U,TU,SU,T2U,STSU,ST2SU},8C3={T,ST,TS,STS,T2,ST2,T2S,ST2S},6C4={STU,TSU,T2SU,ST2U,TST2U,T2STU}, where the conjugacy class is denoted by kCn, k is the number of elements belonging to it, and the subscript n is the order of the elements contained in it. As a result of these conjugacy classes and the theorems that prove that the number of inequivalent irreducible representations is equal to the number of conjugacy classes and the sum of the squares of the dimensions of the inequivalent irreducible representations must be equal to the order of the group, it is easy to see that S4 has two singlet irreducible representations 1 and 1′, one two-dimensional representation 2 and two three-dimensional irreducible representations 3 and 3′. In this work, we shall work in the basis where the representation matrix of the generator T is diagonal. As a result, the charged lepton mass matrix would be diagonal if the remnant subgroup Z3T≡{1,T,T2} is preserved in the charged lepton sector. The explicit forms of the representation matrix for the three generators are listed in Table 5, and hence the chosen basis coincides with that of Ref. [19]. The character table of S4 group follows immediately, as shown in Table 6. Moreover, the Kronecker products between different irreducible representations are as follows(A.3)1⊗R=R,1′⊗1′=1,1′⊗2=2,1′⊗3=3′,1′⊗3′=3,2⊗2=1⊕1′⊕2,2⊗3=2⊗3′=3⊗3′,3⊗3=3′⊗3′=1⊕2⊕3⊕3′,3⊗3′=1′⊕2⊕3⊕3′ where R stands for any irreducible representation of S4.In the end, we present the Clebsch–Gordan (CG) coefficients in the chosen basis. All the CG coefficients can be reported in the form of α⊗β, αi denotes the element of the left base vectors α, and βi is the element of the right base vectors β. For the product of the singlet 1′ with a doublet or a triplet, we have(A.4)1′⊗2=2=α(β1−β2),1′⊗3=3′=α(β1β2β3),1′⊗3′=3=α(β1β2β3). The CG coefficients for the products involving the doublet representation 2 are found to be2⊗2=1⊕1′⊕2,with{1=α1β2+α2β11′=α1β2−α2β12=(α2β2α1β1)2⊗3=3⊕3′,with {3=(α1β2+α2β3α1β3+α2β1α1β1+α2β2)3′=(α1β2−α2β3α1β3−α2β1α1β1−α2β2)2⊗3′=3⊕3′,with {3=(α1β2−α2β3α1β3−α2β1α1β1−α2β2)3′=(α1β2+α2β3α1β3+α2β1α1β1+α2β2) Finally, for the products of the triplet representations 3 and 3′, we find3⊗3=3′⊗3′=1⊕2⊕3⊕3′,with{1=α1β1+α2β3+α3β22=(α2β2+α1β3+α3β1α3β3+α1β2+α2β1)3=(α2β3−α3β2α1β2−α2β1α3β1−α1β3)3′=(2α1β1−α2β3−α3β22α3β3−α1β2−α2β12α2β2−α3β1−α1β3)3⊗3′=1′⊕2⊕3⊕3′,with {1=α1β1+α2β3+α3β22=(α2β2+α1β3+α3β1−(α3β3+α1β2+α2β1))3=(2α1β1−α2β3−α3β22α3β3−α1β2−α2β12α2β2−α3β1−α1β3)3′=(α2β3−α3β2α1β2−α2β1α3β1−α1β3) We note that the CG coefficients presented above are in accordance with the results of Refs. [19,40,41].References[1]F.P.AnDAYA-BAY CollaborationPhys. Rev. Lett.1082012171803arXiv:1203.1669 [hep-ex]F.P.AnChin. Phys. C372013011001arXiv:1210.6327 [hep-ex][2]J.K.AhnRENO CollaborationPhys. Rev. 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