A new S_{4} flavor model based on SU(3)C⊗SU(3)L⊗U(1)X gauge symmetry responsible for fermion masses and mixings is constructed. The neutrinos get small
masses from only an antisextet of SU(3)_{L} which is in a doublet under S_{4}. In this
work, we assume the VEVs of the antisextet differ from each other under S_{4} and the
difference of these VEVs is regarded as a small perturbation, and then the model can fit the experimental data on neutrino masses and mixings. Our results show
that the neutrino masses are naturally small and a deviation from the tribimaximal
neutrino mixing form can be realized. The quark masses and mixing matrix are also
discussed. The number of required Higgs multiplets is less and the scalar potential
of the model is simpler than those of the model based on S_{3} and our previous S_{4} model. The assignation of VEVs to antisextet leads to the mixing of the new gauge bosons and those in the standard model. The mixing in the charged gauge bosons
as well as the neutral gauge bosons is considered.
1. Introduction
The experiments on neutrino oscillation confirm that neutrinos are massive particles [1–6]. The parameters of neutrino oscillations such as the squared mass differences and mixing angles are now well constrained. The data in PDG2012 [7–11] imply
(1)sin2(2θ12)=0.857±0.024(t12≃0.6717),sin2(2θ13)=0.098±0.013(s13≃0.1585),sin2(2θ23)>0.95,Δm212=(7.50±0.20)×10-5eV2,Δm322=(2.32-0.08+0.12)×10-3eV2.
These large neutrino mixing angles are completely different from the quark mixing ones defined by the CKM matrix [12, 13]. This has stimulated work on flavor symmetries and non-Abelian discrete symmetries are considered to be the most attractive candidate to formulate dynamical principles that can lead to the flavor mixing patterns for quarks and lepton. There are many recent models based on the non-Abelian discrete symmetries, such as A4 [14–29], S3 [30–65], and S4 [66–93].
An alternative to extend the standard model (SM) is the 3-3-1 models, in which the SM gauge group SU(2)L⊗U(1)Y is extended to SU(3)L⊗U(1)X which is investigated in [94–109]. The extension of the gauge group with respect to SM leads to interesting consequences. The first one is that the requirement of anomaly cancelation together with that of asymptotic freedom of QCD implies that the number of generations must necessarily be equal to the number of colors, hence giving an explanation for the existence of three generations. Furthermore, quark generations should transform differently under the action of SU(3)L. In particular, two quark generations should transform as triplets, one as an antitriplet.
A fundamental relation holds among some of the generators of the group:
(2)Q=T3+βT8+X,
where Q indicates the electric charge, T3 and T8 are two of the SU(3) generators, and X is the generator of U(1)X. β is a key parameter that defines a specific variant of the model. The model thus provides a partial explanation for the family number, as also required by flavor symmetries such as S4 for 3-dimensional representations. In addition, due to the anomaly cancelation one family of quarks has to transform under SU(3)L differently from the two others. S4 can meet this requirement with the representations 1_ and 2_.
There are two typical variants of the 3-3-1 models as far as lepton sectors are concerned. In the minimal version, three SU(3)L lepton triplets are (νL,lL,lRc), where lR are ordinary right-handed charged leptons [94–98]. In the second version, the third components of lepton triplets are the right-handed neutrinos, (νL,lL,νRc) [99–105]. To have a model with the realistic neutrino mixing matrix, we should consider another variant of the form (νL,lL,NRc) where NR are three new fermion singlets under SM symmetry with vanishing lepton numbers [110–113].
In our previous works we have also extended the above application to the 3-3-1 models [110–113]. In [112] we have studied the 3-3-1 model with neutral fermions based on S4 group, in which most of the Higgs multiplets are in triplets under S4 except that χ is in a singlet, and the exact tribimaximal form [114–117] is obtained, in which θ13=0. As we know, the recent considerations have implied θ13≠0, but small as given in (1). This problem has been improved in [111] by adding a new triplet ρ and another antisextet s′, in which s′ is regarded as a small perturbation. Therefore the model contains up to eight Higgs multiplets, and the scalar potential of the model is quite complicated.
In this paper, we propose another choice of fermion content and Higgs sector. As a consequence, the number of required Higgs is fewer and the scalar potential of the model is much simpler. The resulting model is near that of our previous work in [111] and includes those given in [112] as a special case and the physics is also different from the former. With the similar analysis as in [111], S4 contains two triplets irreducible representation, one doublet and two singlets. This feature is useful to separate the third family of fermions from the others which contains a 2_ irreducible representation which can connect two maximally mixed generations. Besides the facilitating maximal mixing through 2_, it provides two inequivalent singlet representations 1_ and 1_′ which play a crucial role in consistently reproducing fermion masses and mixing as a perturbation. We have pointed out that this model is simpler than that of S3 and our previous S4 model, since fewer Higgs multiplets are needed in order to allow the fermions to gain masses and to break the gauge symmetry. Indeed, the model contains only six Higgs multiplets. On the other hand, the neutrino sector is simpler than those of S3 and S4 models [111, 112].
The rest of this work is organized as follows. In Sections 2 and 3 we present the necessary elements of the 3-3-1 model with S4 flavor symmetry as in the above choice, as well as introducing necessary Higgs fields responsible for the charged-lepton masses. In Section 4, we discuss on quark sector. Section 5 is devoted to the neutrino mass and mixing. In Section 6 we discuss the gauge boson pattern of the model. We summarize our results and make conclusions in Section 7. Appendix A is devoted to the Higgs potential and minimization conditions. Appendix B is devoted to S4 group with its Clebsch-Gordan coefficients. Appendix C presents the lepton numbers and lepton parities of model particles.
2. Fermion Content
The gauge symmetry is based on SU(3)C⊗SU(3)L⊗U(1)X, where the electroweak factor SU(3)L⊗U(1)X is extended from those of the SM whereas the strong interaction sector is retained. Each lepton family includes a new fermion singlet carrying no lepton number (NR) arranged under the SU(3)L symmetry as a triplet (νL,lL,NRc) and a singlet lR. The residual electric charge operator Q is therefore related to the generators of the gauge symmetry by [110–112]
(3)Q=T3-13T8+X,
where Ta(a=1,2,…,8) are SU(3)L charges with TrTaTb=(1/2)δab and X is the U(1)X charge. This means that the model under consideration does not contain exotic electric charges in the fundamental fermion, scalar, and adjoint gauge boson representations.
The particles in the lepton triplet have different lepton numbers (1 and 0), so the lepton number in the model does not commute with the gauge symmetry unlike the SM. Therefore, it is better to work with a new conserved charge ℒ commuting with the gauge symmetry and related to the ordinary lepton number by diagonal matrices [110–112, 118]
(4)L=23T8+ℒ.
The lepton charge arranged in this way (i.e., L(NR)=0 as assumed) is in order to prevent unwanted interactions due to U(1)ℒ symmetry and breaking (due to the lepton parity as shown below), such as the SM and exotic quarks, and to obtain the consistent neutrino mixing.
By this embedding, exotic quarks U and D as well as new non-Hermitian gauge bosons X0 and Y± possess lepton charges as of the ordinary leptons: L(D)=-L(U)=L(X0)=L(Y-)=1. The lepton parity is introduced as follows: Pl=(-)L, which is a residual symmetry of L. The particles possess L=0, ±2 such as NR, ordinary quarks, and bileptons having Pl=1; the particles with L=±1 such as ordinary leptons and exotic quarks having Pl=-1. Any nonzero VEV with odd parity, Pl=-1, will break this symmetry spontaneously [112]. For convenience in reading, the numbers L and Pl of the component particles are given in Appendix C.
In this paper we work on a basis where 3_ and 3_′ are real representations whereas the two-dimensional representation 2_ of S4 is complex, 2_*(1*,2*)=2_(2*,1*), and
(5)2_⊗2_=1_(12+21)⊕1_′(12-21)⊕2_(22,11).
The lepton content of this model is similar to that of [111] but is different from the one in [112]; namely, in [112] three left-handed leptons are put in one triplet 3_ under S4, whereas in this model we put the first family of leptons in singlets 1_ of S4, while the two other families are in the doublets 2_. In the quark content, the third family is put in a singlet 1_ and the two others in a doublet 2_ under S4 which satisfy the anomaly cancelation in 3-3-1 models. The difference in fermion content leads to the difference between this work and our previous work [112] in physical phenomenon as seen bellow. Under the [SU(3)L,U(1)X,U(1)ℒ,S_4] symmetries as proposed, the fermions of the model transform as follows:
(6)ψ1L=(ν1L,l1L,N1Rc)T~[3,-13,23,1_],l1R~[1,-1,1,1_],ψαL=(ναL,lαL,NαRc)T~[3,-13,23,2_],lαR~[1,-1,1,2_],(α=2,3),Q3L=(u3L,d3L,UL)T~[3,13,-13,1_],u3R~[1,23,0,1_],d3R~[1,-13,0,1_],UR~[1,23,-1,1_],QiL=(diL,-uiL,DiL)T~[3*,0,13,2_],(i=1,2),diR~[1,-13,0,2_],uiR~[1,23,0,2_],DiR~[1,-13,1,2_],
where the subscript numbers on field indicate respective families in order to define components of their S4 multiplets. In the following, we consider possibilities of generating masses for the fermions. The scalar multiplets needed for this purpose would be introduced accordingly.
3. Charged Lepton Mass
In [112], both three families of left-handed fermions and three right-handed quarks are put in a triplet under S4. To generate masses for the charged leptons, we have introduced two SU(3)L scalar triplets ϕ and ϕ′ lying in 3_ and 3_′ under S4, respectively, with VEVs 〈ϕ〉=(vvv)T and 〈ϕ′〉=(v′v′v′)T. From the invariant Yukawa interactions for the charged leptons, we obtain the right-handed charged leptons mixing matrices which are diagonal ones, UlR=1, and the right-handed one given by [112]
(7)UL=13(1111ωω21ω2ω).
Similar to the charged lepton sector, to generate the quark masses, we have additionally introduced the three scalar Higgs triplets χ, η, η′ lying in 1_, 3_, and 3_′ under S4, respectively. Quark masses can be derived from the invariant Yukawa interactions for quarks with supposing that the VEVs of η, η′, and χ are (u,u,u), (u′,u′,u′), and w, where u=〈η10〉, u′=〈η1′0〉, and w=〈χ30〉. The other VEVs 〈η30〉, 〈η3′0〉, and 〈χ10〉 vanish if the lepton parity is conserved. In addition, the VEV w also breaks the 3-3-1 gauge symmetry down to that of the standard model and provides the masses for the exotic quarks U and D as well as the new gauge bosons. The u, u′ as well as v, v′ break the SM symmetry and give the masses for the ordinary quarks, charged leptons, and gauge bosons. To keep consistency with the effective theory, we assume that w is much larger than those of ϕ and η [112]. The unitary matrices which couple the left-handed quarks uL and dL with those in the mass bases are unit ones (ULu=1, ULd=1), and the CKM quark mixing matrix at the tree level is then UCKM=UdL†UuL=1. For a detailed study on charged lepton and quark mass the reader can see [112].
In [112], to generate masses for neutrinos, we have introduced one SU(3)L antisextet lying in 1_ under S4 and one SU(3)L antisextet lying in 3_ under S4 with the VEV of s being set as (〈s1〉,0,0) under S4. The neutrino masses are explicitly separated and the lepton mixing matrix yields the exact tribimaximal form [112] where θ13=0 which is a small deviation from recent neutrino oscillation data [7]. However, this problem will be improved in this work.
Because the fermion content of the model, as given in (6), is the same as that of one in [111] under all symmetries, so the charged-lepton mass is also similar to the one in [111]. Indeed, to generate masses for the charged leptons, we need two scalar triplets:
(8)ϕ=(ϕ1+ϕ20ϕ3+)~[3,23,-13,1_],ϕ′=(ϕ1′+ϕ2′0ϕ3′+)~[3,23,-13,1_′],
with VEVs 〈ϕ〉=(0,v,0)T and 〈ϕ′〉=(0,v′,0)T.
The Yukawa interactions are
(9)-ℒl=h1(ψ-1Lϕ)1_l1R+h2(ψ-αLϕ)2_lαR+h3(ψ-αLϕ′)2_lαR+h.c.=h1(ψ-1Lϕ)1_l1R+h2(ψ-2Lϕl2R+ψ-3Lϕl3R)+h3(ψ-3Lϕ′l3R-ψ-2Lϕ′l2R)+h.c.
The mass Lagrangian of the charged leptons reads
(10)-ℒlmass=(l-1L,l-2L,l-3L)Ml(l1R,l2R,l3R)T+h.c.Ml=(h1v000h2v-h3v′00h2v+h3v′)≡(me000mμ000mτ).
It is then diagonalized, and
(11)UeL+=UeR=I.
This means that the charged leptons l1,2,3 by themselves are the physical mass eigenstates, and the lepton mixing matrix depends on only that of the neutrinos that will be studied in Section 5.
We see that the masses of muon and tauon are separated by the ϕ′ triplet. This is the reason why we introduce ϕ′ in addition to ϕ.
The charged lepton Yukawa couplings h1,2,3 relate to their masses as follows:
(12)h1v=me,2h2v=mτ+mμ,2h3v′=mτ-mμ.
The current mass values for the charged leptons at the weak scale are given by [7]
(13)me=0.511MeV,mμ=105.66MeV,mτ=1776.82MeV.
Thus, we get
(14)h1v=0.511MeV,h2v=941.24MeV,h3v′=835.58MeV.
It follows that if v′ and v are of the same order of magnitude, h1≪h2 and h2~h3. This result is similar to the case of the model based on S3 group [111]. On the other hand, if we choose the VEV of ϕ as v=100GeV, then h1~5×10-6, h3~h2~10-4.
4. Quark Mass
To generate the quark masses with a minimal Higgs content, we additionally introduce the following scalar multiplets:
(15)χ=(χ10,χ2-,χ30)T~[3,-13,23,1_],η=(η10,η2-,η30)T~[3,-13,-13,1_],η′=(η1′0,η2′-,η3′0)T~[3,-13,-13,1_′].
It is noticed that these scalars do not couple with the lepton sector due to the gauge invariance. The Yukawa interactions are then
(16)-ℒq=f3Q-3LχUR+f(Q-iLχ*)2_DiR+h3uQ-3Lηu3R+hu(Q-iLϕ*)2_uiR+h′u(Q-iLϕ′*)2_uiR+h3dQ-3Lϕd3R+hd(Q-iLη*)2_diR+h′d(Q-iLη′*)2_diR+h.c=f3Q-3LχUR+fQ-iLχ*DiR+h3uQ-3Lηu3R+hu(Q-1Lϕ*u1R+Q-2Lϕ*u2R)+h′u(Q-2Lϕ′*u2R-Q-1Lϕ′*u1R)+h3dQ-3Lϕd3R+hd(Q-1Lη*d1R+Q-2Lη*d2R)+h′d(Q-2Lη′*d2R-Q-1Lη′*d1R)+h.c.
Suppose that the VEVs of η, η′, and χ are u, u′, and w, where u=〈η10〉, u′=〈η1′0〉, and w=〈χ30〉. The other VEVs 〈η30〉, 〈η3′0〉, and 〈χ10〉 vanish due to the lepton parity conservation [111]. The exotic quarks therefore get masses mU=f3w and mD1,2=fw. In addition, w has to be much larger than those of ϕ, ϕ′, η, and η′ for a consistency with the effective theory. The mass matrices for ordinary up-quarks and down-quarks are, respectively, obtained as follows:
(17)Mu=(huv-h′uv′000huv+h′uv′000h3uu)≡(mu000mc000mt),Md=(hdu-h′du′000hdu+h′du′000h3dv)≡(md000ms000mb).
Similar to the charged leptons, the masses of u-c and d-s quarks are in pair separated by the scalars ϕ′ and η′, respectively. We see also that the introduction of η′ in addition to η is necessary to provide the different masses for u and c quarks as well as for d and s quarks.
The expressions (17) yield the relations:
(18)h3uu=mt,2huv=mu+mc,2h′uv′=mc-mu,h3dv=mb,2hdu=md+ms,2h′du′=ms-md.
The current mass values for the quarks are given by [7]
(19)mu=(1.8÷3.0)MeV,md=(4.5÷5.5)MeV,mc=(1.25÷1.30)GeV,ms=(90.0÷100.0)MeV,mt=(172.1÷174.9)GeV,mb=(4.13÷4.37)GeV.
Hence
(20)h3uu=(172.1÷174.9)GeV,h3dv=(4.13÷4.37)GeV,huv=(625.9÷651.5)MeV,hdu=(47.25÷52.75)MeV,h′du′=(42.75÷47.25)MeV,h′uv′=(624.1÷648.5)MeV.
It is obvious that if u~v~v′~u′, the Yukawa coupling hierarchies are hu~h′u≪h3u, hd~h′d≪h3d, and the couplings between up-quarks (hu,h′u,h3u) and Higgs scalar multiplets are slightly heavier than those of down-quarks (hd,h′d,h3d), respectively.
The unitary matrices, which couple the left-handed up- and down-quarks with those in the mass bases, are ULu=1 and ULd=1, respectively. Therefore we get the CKM matrix
(21)UCKM=ULd†ULu=1.
This is a good approximation for the realistic quark mixing matrix, which implies that the mixings among the quarks are dynamically small. The small permutations such as a breaking of the lepton parity due to the VEVs 〈η30〉, 〈η3′0〉, and 〈χ10〉 or a violation of ℒ and/nor S4 symmetry due to unnormal Yukawa interactions, namely, Q-3Lχu3R, Q-iLχ*diR, Q-3LχuiR, Q-iLχ*d3R, and so forth, will disturb the tree level matrix resulting in mixing between ordinary and exotic quarks and possibly providing the desirable quark mixing pattern. A detailed study on these problems is out of the scope of this work and should be skipped.
5. Neutrino Mass and Mixing
The neutrino masses arise from the couplings of ψ-αLcψαL, ψ-1Lcψ1L, and ψ-1cψαL to scalars, where ψ-αLcψαL transforms as 3*⊕6 under SU(3)L and 1_⊕2_⊕3_⊕3_′ under S4, ψ-1Lcψ1L transforms as 3*⊕6 under SU(3)L and 1_ under S4, and ψ-1LcψαL transforms as 3*⊕6 under SU(3)L and 2_ under S4. For the known scalar triplets (ϕ,ϕ′,χ,η,η′), only available interactions are (ψ-αLcψαL)ϕ and (ψ-αLcψαL)ϕ′ but explicitly suppressed because of the ℒ-symmetry. We will therefore propose new SU(3)L antisextets instead of coupling to ψ-LcψL responsible for the neutrino masses which are lying in either 1_, 2_, 3_, or 3_′ under S4. In [112], we have introduced two SU(3)L antisextets σ, s which are lying in 1_ and 3_ under S4, respectively. Contrastingly, in this work, with fermion content as proposed, to obtain a realistic neutrino spectrum, the model needs only one antisextet which transforms as follows:
(22)si=(s110s12+s130s12+s22++s23+s130s23+s330)~[6*,23,-43,2_],
where the numbered subscripts on the component scalars are the SU(3)L indices, whereas i=1,2 is that of S4. The VEV of s is set as (〈s1〉,〈s2〉) under S4, in which
(23)〈si〉=(λi0vi000vi0Λi).(i=1,2).
Following the potential minimization conditions, we have several VEV alignments. The first is that 〈s1〉=〈s2〉 and then S4 is broken into an eight-element subgroup, which is isomorphic to D4. The second is that 〈s1〉≠0=〈s2〉 or 〈s1〉=0≠〈s2〉 and then S4 is broken into A4 consisting of the identity and the even permutations of four objects. The third is that 〈s1〉≠〈s2〉≠0 and then S4 is broken into a four-element subgroup consisting of the identity and three double transitions, which is isomorphic to Klein four group [75] (in this paper we denote this group by K4). To obtain a realistic neutrino spectrum, we argue that both the breakings S4→D4 and S4→K4 must take place. We therefore assume that its VEVs are aligned as the former to derive the direction of the breaking S4→D4, and this happens in any case bellow:
(24)λ1=λ2≡λs,v1=v2≡vs,Λ1=Λ2≡Λs,〈s1〉=〈s2〉=〈s〉=(λs0vs000vs0Λs).
The direction of the breaking S4→K4 is equivalent to the breaking D4→{Identity}. In this direction, we set 〈s1〉=〈s〉≠〈s2〉≠0. If D4 is unbroken, we have 〈s1〉=〈s2〉=〈s〉 as in (24), and on the contrary, if D4 is unbroken, we have 〈s〉=〈s2〉≈〈s1〉:
(25)〈s1〉=(λ10v1000v10Λ1).
The difference between 〈s1〉 and 〈s2〉 is very small which is regarded as a small perturbation as considered bellow. It is noteworthy that the derivation in this paper contains a fewer, in comparison with the model based on the S3 group [111], number of Higgs triplets; consequently the Higgs sector and the minimization condition of the potential are much simpler. Moreover, the obtained model, despite the compact in Higgs sector, can fit the current data with θ13≠0, while the old version [112] based on S4 cannot provide nonvanishing θ13.
In general, the Yukawa interactions are
(26)-ℒν=12x(ψ-1LcψL)2_si+12y(ψ-LcψL)2_si+h.c=12x(ψ-1Lcψ2Ls2+ψ-1Lcψ3Ls1)+12y(ψ-2Lcψ2Ls1+ψ-3Lcψ3Ls2)+h.c.
With the alignments of VEVs as in (24) and (25), the mass Lagrangian for the neutrinos is determined by
(27)-ℒνmass=12χ-LcMνχL+h.c.,χL≡(νLNRc),Mν≡(MLMDTMDMR),
where ν=(ν1,ν2,ν3)T and N=(N1,N2,N3)T. The mass matrices are then obtained by
(28)ML,R,D=(0aL,R,DbL,R,DaL,R,DcL,R,D0bL,R,D0dL,R,D),
with
(29)aL=x2λs,aD=x2vs,aR=x2Λs,bL=x2λ1,bD=x2v1,bR=x2Λ1,cL=yλ1,cD=yv1,cR=yΛ1,dL=yλs,dD=yvs,dR=yΛs.
The VEVs Λ1,2 break the 3-3-1 gauge symmetry down to that of the SM and provide the masses for the neutral fermions NR and the new gauge bosons: the neutral Z′ and the charged Y± and X0,0*. The λ1,2 and v1,2 belong to the second stage of the symmetry breaking from the SM down to the SU(3)C⊗U(1)Q symmetry and contribute the masses to the neutrinos. Hence, to keep a consistency we assume that Λ1,s≫v1,s≫λ1,s [105].
Three active neutrinos therefore gain masses via a combination of type I and type II seesaw mechanisms derived from (27) and (28) as
(30)Meff=ML-MDTMR-1MD=(AB1B2B1C1DB2DC2),
where
(31)A=-(aRbD-aDbR)2bR2cR+aR2dR,B1=((aLaR-aD2)bR[aRbDcD+aLbRcR-aD(bRcD+bDcR)]hhhh+aR(aLaR-aD2)dR)×(bR2cR+aR2dR)-1,B2=(-bD2bRcR+bLbR2cR+aDaRbRdD+aR2bLdRhhhh-aRbD(aRdD+aDdR)bD2)×(bR2cR+aR2dR)-1,C1=bR2(cLcR-cD2)+(aR2cL+aD2cR-2aDaRcD)dRbR2cR+aR2dR,C2=-2bDbRcRdD+bR2cRdL+bD2cRdR+aR2(dLdR-dD2)bR2cR+aR2dR,D=(aRcD-aDcR)(bRdD-bDdR)bR2cR+aR2dR.
The following comments of S4 breaking are in order.
If S4 is broken into D4 (D4 is unbroken), we have A=D=0, B1=B2=B, and C1=C2=C, which is presented in Section 5.1.
If S4 is broken into K4 (D4 is broken into {Identity}), we have A≈0, B1≈B2, C1≈C2, and D≠0 but it is very small. In this case the disparity of two VEVs of 〈s〉 is regarded as a small perturbation as shown in Section 5.2.
We next divide our considerations into two cases to fit the data: the first case is S4→D4, and the second one is S4→K4.
5.1. Experimental Constraints in the Case <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M333"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo mathvariant="bold">→</mml:mo><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>
If S4 is broken into D4, λ1=λ2≡λs, v1=v2≡vs, Λ1=Λ2≡Λs, we have A=0, B1=B2≡B, C1=C2≡C, and D=0, and Meff reduces to
(32)Meff=(0BBBC0B0C),
where
(33)B=(λs-vs2Λs)x2,C=(λs-vs2Λs)y.
We can diagonalize the matrix Meff in (32) as follows:
(34)UTMeffU=diag(m1,m2,m3),
where
(35)m1=12(C-C2+8B2)=(λs-vs2Λs)y+y2+2x22,m2=12(C+C2+8B2)=(λs-vs2Λs)y-y2+2x22,m3=C=(λs-vs2Λs)y,
and the neutrino mixing matrix takes the form:
(36)U0=(|K||K|2+2-2|K|2+201|K|2+212|K||K|2+2-121|K|2+212|K||K|2+212)K=-C+C2+8B22B.
Note that m1m2=-2B2. This matrix can be parameterized in three Euler’s angles, which implies
(37)θ13=0,θ23=π4,tanθ12=2|K|.
This case coincides with the data since sin2(2θ13)<0.15 and sin2(2θ23)>0.92 [119, 120]. For the remaining constraints, taking the central values from the data in [119]
(38)sin2(2θ12)≃0.87,(s122=0.32),Δm212=7.59×10-5eV2,Δm322=2.43×10-3eV2,
and we have a solution
(39)m1=0.0280284eV,m2=0.0293347eV,m3=0.0573631eV,
and B=-0.0202757ieV, C=0.0573631eV, K=1.44667, and |x/y|=0.707087. It follows that tanθ12=0.977565, (θ12≃44.350), and the neutrino mixing matrix form is very close to that of bimaximal mixing which takes the form:
(40)U=(0.715083-0.6990400.4942960.50564-120.4942960.5056412.).
Now, it is natural to choose λs, vs2/Λs in eV order, and suppose that λs>vs2/Λs. Let us assume λs-vs2/Λs=0.1, and we have then x=0.399403i and y=-0.573631.
This result is not obviously consistent with the recent data on neutrinos oscillation in which θ13≠0, but small as given in [7]. However, as we will see in Section 5.2, this situation will be improved if the direction of the breaking S4→K4 takes place. This means that, for the model under consideration, both the breakings S4→D4 and S4→K4 (instead of D4→{Identity}) must take place in the neutrino sector.
5.2. Experimental Constraints in the Case <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M374"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo mathvariant="bold">→</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>
In this case S4 is broken into the Klein four group K4, λ1≠λ2, v1≠v2, and Λ1≠Λ2, and the direct consequence is A≈0, B1≈B2, C1≈C2, and D≠0. The general neutrino mass matrix in (30) can be rewritten in the form:
(41)Meff=(0BBBC0B0C)+(a2rapaqapqraqrp),(a=x2y)=(0BBBC0B0C)+(a2r0000r0r0)+(0ap0ap0000p)+(00aq0q0aq00),
where B and C are given by (33), accommodated in the first matrix, which is matched to the case of S4→D4. The three last matrices in (41) are a deviation from the contribution due to the disparity of 〈s1〉 and 〈s2〉, namely, A=a2r, B1-B=ap, B2-B=aq, q=C1-C, p=C2-C, and r=D, with the A, B1,2, C1,2, and D being defined in (31), which correspond to S4→K4.
Substituting (29) into (31) we get
(42)q=([Λs4λ1-λsΛs(Λs3+Λ13)+Λs2Λ1vs2+Λ13vs2hhhh+Λs3vs(vs-2v1)+ΛsΛ12(λ1Λ1-v12)]y)×(Λs(Λs3+Λ13))-1=(λ1-λs)y+([ΛsΛ12vs2+vs2Λs+Λs2Λ13vs2hhhhhh-2Λs2Λ13vsv1-v12Λ1]y),×(1+(ΛsΛ1)3)-1p=Λ1(Λ1vs-Λsv1)2yΛs(Λs3+Λ13)=Λs(vs/Λs-v1/Λ1)2y1+(Λs/Λ1)3,r=-(Λ1vs-Λsv1)2yΛ13+Λs3=Λs(Λs/Λ1)(vs/Λs-v1/Λ1)2y1+(Λs/Λ1)3.
Indeed, if S4→D4, the deviations p, q, r will vanish, therefore the mass matrix Meff in (30) reduces to its first term coinciding with (32). The first term of (41) provides bimaximal mixing pattern, in which θ13=0 as shown in Section 5.1. The other matrices proportional to p, q, r due to contribution from the disparity of 〈s1〉 and 〈s2〉 will take the role of perturbation for such a deviation of θ13. So, in this work we consider the disparity of 〈s1〉 and 〈s2〉 as a small perturbation and terminating the theory at the first order.
Without loss of generality, we consider the case of breaking S4→K4, in which λ1≠λs whereas v1=vs and Λ1=Λs. It is then p=r=0, q=(λ1-λs)y≡ϵy with ϵ=λ1-λs being a small parameter. In this case, the matrix Meff in (41) reduces to
(43)Meff=(0x2yCx2yCx2yCC0x2yC0C)+ϵ(00x20y0x200)≡Meff0+ϵM(1).
At the first order of perturbation, the physical neutrino masses are obtained as
(44)m1′=λ1=m1+ϵ(Kx+yK2+2),m2′=λ2=m2+ϵK(Ky-2x)2(K2+2),m3′=λ3=m3+ϵy2,
where m1,2,3 are the mass values as of the case S4→D4 given by (39). For the corresponding perturbed eigenstates, we put
(45)U⟶U′=U+ΔU,
where U is defined by (36), and
(46)ΔU=(ΔU11ΔU12ΔU13ΔU21ΔU22ΔU23ΔU31ΔU32ΔU33),
with
(47)ΔU11=-ϵ(K2-2)x+2Ky2(K2+2)3/2(m1-m2),ΔU21=-ϵ(Kx-2y)4K2+2(m1-m3)+ϵK[(K2-2)x+2Ky]4(K2+2)3/2(m1-m2),ΔU31=ϵ(Kx-2y)4K2+2(m1-m3)+ϵK[(K2-2)x+2Ky]4(K2+2)3/2(m1-m2),ΔU12=-ϵK[(K2-2)x+2Ky]22(K2+2)3/2(m1-m2),ΔU22=ϵ22Ky+xK2+2(m2-m3)-ϵ22(K2-2)x+2Ky(K2+2)3/2(m1-m2),ΔU32=-ϵ22Ky+xK2+2(m2-m3)-ϵ22(K2-2)x+2Ky(K2+2)3/2(m1-m2),ΔU13=-ϵ22K(Kx-2y)(K2+2)(m1-m3)-ϵ2Ky+x(K2+2)(m2-m3),ΔU23=ΔU33=-ϵ22Kx-2y(K2+2)(m1-m3)+ϵ22K(Ky+x)(K2+2)(m2-m3).
The lepton mixing matrix in this case U′ can still be parameterized in three new Euler’s angles θij′, which are also a perturbation from the θij in the case 1, defined by
(48)s13′=-U13′=ΔU13=-ϵ22K(Kx-2y)(K2+2)(m1-m3)-ϵ2Ky+x(K2+2)(m2-m3)=-ϵy22B,t12′=-U12′U11′=(-[4ϵB2Cx+ϵC2(C+C2+8B2)xhhhhhh+2BC(C+C2+8B2)(2C-ϵy)hhhhhh+8B3(4C+4C2+8B2-ϵy)])×({2[64B4+2C3(C+C2+8B2)hhhhhhhhh-ϵBC(C+C2+8B2)xhhhhhhhhh+2B2(12C2+8CC2+8B2)hhhhhhhhh+ϵCy+ϵyC2+8B2]})-1,t23′=-U23′U33′=4B2+ϵBx-ϵCy4B2-ϵBx+ϵCy.
It is easily to show that our model is consistent since the five experimental constraints on the mixing angles and squared neutrino mass differences can be, respectively, fitted with two Yukawa coupling parameters x, y of the antisextet scalar s with the above mentioned VEVs. Indeed, taking the data in (1) we obtain ϵ≃0.0692, x≃0.0728, y≃-0.1562, and B≃-0.0241eV and C=0.022eV, K=1.943, and t23′=0.9045 [θ23′≃42.13o,sin2(2θ23′)=0.98999 satisfying the condition sin2(2θ23′)>0.95]. The neutrino masses are explicitly given as m1′≃-0.02737eV, m2′≃-0.02870eV, and m3′≃-0.05607eV. The neutrino mixing matrix then takes the form:
(49)U=(0.8251-0.5657-0.15850.33020.6781-0.67160.46970.48880.7426).
6. Gauge Bosons
The covariant derivative of a triplet is given by
(50)Dμ=∂μ-igλa2Wμa-igXXλ92Bμ=∂μ-iPμ,
where λa(a=1,2,…,8) are Gell-Mann matrices, λ9=2/3diag(1,1,1), Trλaλb=2δab, Trλ9λ9=2, and X is X-charge of Higgs triplets.
Let us denote the following combinations:
(51)Wμ′+=Wμ1-iWμ22,Xμ′0=Wμ4-iWμ52,Yμ′-=Wμ6-iWμ72,Wμ′-=(Wμ′+)*,Yμ′+=(Yμ′-)*,
and then Pμ is rewritten in a convenient form as follows:(52)g2(Wμ3+Wμ83+t23XBμ2Wμ′+2Xμ′0,2Wμ′--Wμ3+Wμ83+t23XBμ2Yμ′-2Xμ′0*2Yμ′+-23Wμ8+t23XBμ),with t=gX/g. We note that W4 and W5 are pure real and imaginary parts of X0 and X0*, respectively. The covariant derivative for an antisextet with the VEV part is [121]
(53)Dμ〈si〉=ig2{Wμaλa*〈si〉+〈si〉Wμaλa*T}-igXT9XBμ〈si〉.
The covariant derivative (53) acting on the antisextet VEVs is given by
(54)[Dμ〈si〉]11=ig(23λiWμ3+λi3Wμ8hhhhhh+2313tλiBμ+2viX′0*),[Dμ〈si〉]12=ig2(λiWμ′++viYμ′+),[Dμ〈si〉]13=ig2(viWμ3-vi3Wμ8+2323tviBμhhhhhhh+2λiXμ′0+2ΛiXμ′0*23),[Dμ〈si〉]21=[Dμ〈si〉]12,[Dμ〈si〉]22=0,[Dμ〈si〉]23=ig2(viWμ′++ΛiYμ′+),[Dμ〈si〉]31=[Dμ〈si〉]13,[Dμ〈si〉]32=[Dμ〈si〉]23,[Dμ〈si〉]33=ig(-23ΛiWμ8+2313tΛiBμ+2viXμ′0).
The masses of gauge bosons in this model are defined as follows:
(55)ℒmassGB=(Dμ〈ϕ〉)+(Dμ〈ϕ〉)+(Dμ〈ϕ′〉)+(Dμ〈ϕ′〉)+(Dμ〈χ〉)+(Dμ〈χ〉)+(Dμ〈η〉)+(Dμ〈η〉)+(Dμ〈η′〉)+(Dμ〈η′〉)+Tr[(Dμ〈s1〉)+(Dμ〈s1〉)]+Tr[(Dμ〈s2〉)+(Dμ〈s2〉)],
where ℒmassGB in (55) is different from the one in [122] by the difference of the components of the antisextet s. In [122], 〈s1〉=〈s1〉, namely, λ1=λ2=λs, v1=v2=vs, and Λ1=Λ2=Λs, are taken into account, and the contribution of perturbation has been skipped at the first order. In the following, we consider the general case in which λ1≠λ2, v1≠v2, and Λ1≠Λ2. As a consequence, the fewer number of Higgs multiplets is needed in order to allow the fermions to gain masses and with the simpler scalar Higgs potential as mentioned above.
Substitution of the VEVs of Higgs multiplets into (55) yields
(56)ℒmassGB=v2324[(-9gWμ3+33gWμ8+26gXBμ)281g2(Wμ12+Wμ22)+81g2(Wμ62+Wμ72)hhhhhhhh+(-9gWμ3+33gWμ8+26gXBμ)2]+v′2324[(-9gWμ3+33gWμ8+26gXBμ)281g2(Wμ12+Wμ22)+81g2(Wμ62+Wμ72)hhhhhhhhh+(-9gWμ3+33gWμ8+26gXBμ)2]+ω2108[27g2(Wμ42+Wμ52)+27g2(Wμ62+Wμ72)hhhhhhhhh+36g2Wμ82+122ggxWμ8Bμ+2gX2Bμ2]+u2324[(-9gWμ3-33gWμ8+6gXBμ)281g2(Wμ12+Wμ22)+81g2(Wμ42+Wμ52)hhhhhhhhhh+(-9gWμ3-33gWμ8+6gXBμ)2]+u′2324[(-9gWμ3-33gWμ8+6gXBμ)281g2(Wμ12+Wμ22)+81g2(Wμ42+Wμ52)hhhhhhhhhh+(-9gWμ3-33gWμ8+6gXBμ)2]+g26[2(Λ1v1+Λ2v2)(3Wμ3Wμ4+3Wμ1Wμ6hhhhhhhhhhhhhhhhhhhhhhh-3Wμ2Wμ7-53Wμ4Wμ8)hhhhhhhhh+3(v12+v22+λ12+λ22)Wμ12hhhhhhhhh+3(v12+v22+λ12+λ22)Wμ22hhhhhhhhh+3(v12+v22+2λ12+2λ22)Wμ32hhhhhhhhh+3(+2Λ1λ1+2Λ2λ24v12+4v22+λ12+λ22+Λ12+Λ22hhhhhhhhhhhhhhh+2Λ1λ1+2Λ2λ2)Wμ42hhhhhhhhh+3(-2Λ1λ1-2Λ2λ24v12+4v22+λ12+λ22+Λ12+Λ22hhhhhhhhhhhhhhh-2Λ1λ1-2Λ2λ2)Wμ52hhhhhhhhh+3(v12+v22+Λ12+Λ22)Wμ62hhhhhhhhh+3(v12+v22+Λ12+Λ22)Wμ72hhhhhhhhh+23(-v12-v22+2λ12+2λ22)Wμ3Wμ8hhhhhhhhh+(v12+v22+2λ12+2λ22+8Λ12+8Λ22)Wμ82hhhhhhhhh+18(λ1v1+λ2v2)Wμ3Wμ4hhhhhhhhh+6(λ1v1+λ2v2)Wμ1Wμ6hhhhhhhhh-6(λ1v1+λ2v2)Wμ2Wμ7hhhhhhhhh+23(λ1v1+λ2v2)Wμ4Wμ8]+227t2g2(λ12+λ22+Λ12+Λ22+2v12+2v22)Bμ2-2323tg2(λ12+λ22+v12+v22)Wμ3Bμ-4323tg2[(λ1+Λ1)v1+(λ2+Λ2)v2]Wμ4Bμ-229tg2(λ12+λ22-v12-v22-2Λ12-2Λ22)Wμ8Bμ.
We can separate ℒmassGB in (57) into
(57)ℒmassGB=ℒmassW5+ℒmixCGB+ℒmixNGB,
where ℒmassW5 is the Lagrangian part of the imaginary part W5. This boson is decoupled with mass given by
(58)MW52=g22(ω2+u2+u′2+8v12+8v22+2λ12+2λ22hhhhhh+2Λ12+2Λ22-4Λ1λ1-4Λ2λ2ω2+u2+u′2+8v12+8v22+2λ12+2λ22).
In the limit λ1,λ2,v1,v2→0 we have
(59)MW52=g22(ω2+u2+u′2+2Λ12+2Λ22).ℒmixCGB is the Lagrangian part of the charged gauge bosons W and Y:
(60)ℒmixCGB=g24[v2+v′2+u2+u′2hhhhhh+2(v12+v22+λ12+λ22)v2+v′2+u2+u′2](Wμ12+Wμ22)+g24[v2+v′2+ω2hhhhhhh+2(v12+v22+Λ12+Λ22)](Wμ62+Wμ72)+g2(Λ1v1+λ1v1+Λ2v2+λ2v2)×(Wμ1Wμ6-Wμ2Wμ7).ℒmixCGB in (60) can be rewritten in matrix form as follows:
(61)ℒmixCGB=g24(Wμ′-Yμ′-)MWY2(Wμ′+Yμ′+)T,
where(62)MWY2=2(v2+v′2+u2+u′2+2(v12+v22+λ12+λ22)2(Λ1v1+λ1v1+Λ2v2+λ2v2)2(Λ1v1+λ1v1+Λ2v2+λ2v2)v2+v′2+ω2+2(v12+v22+Λ12+Λ22)).The matrix MWY2 in (62) can be diagonalized as follows:
(63)U2TMWY2U2=diag(MW2,MY2),
where
(64)MW2=g24{+ω2+u2+u′2+2(v2+v′2)-Γ2(λ12+λ22+2v12+2v22+Λ12+Λ22)hhhhhh+ω2+u2+u′2+2(v2+v′2)-Γ},MY2=g24{+ω2+u2+u′2+2(v2+v′2)-Γ2(λ12+λ22+2v12+2v22+Λ12+Λ22)hhhhhh+ω2+u2+u′2+2(v2+v′2)+Γ},
with
(65)Γ=4λ14+4Λ14+(2λ22-2Λ22-ω2+u2+u′2)2-4λ12(2Λ12-2λ22+2Λ22+ω2-u2-u′2-4v12)-4Λ12(2λ22-2Λ22-ω2+u2+u′2-4v12)+32Λ1(λ2+Λ2)v1v2+16(λ2+Λ2)2v22+32λ1v1(Λ1v1+λ2v2+Λ2v2).
With corresponding eigenstates, the charged gauge boson mixing matrix takes the form:
(66)U2=(cosθ-sinθsinθcosθ).
The mixing angle θ is given by
(67)tanθ=4(λ1+Λ1)v1+4(λ2+Λ2)v22λ12-2Λ12+2λ22-2Λ22-ω2+u2+u′2-Γ.
The physical charged gauge bosons are defined
(68)Wμ-=cosθWμ′-+sinθYμ′-,Yμ-=-sinθWμ′-+cosθYμ′-.
In our model, the following limit is often taken into account:
(69)λ1,22,v1,22≪u2,u′2,v2,v′2≪ω2~Λ1,22.
With the help of (69), the Γ in (65) becomes
(70)Γ≃(2Λ12+2Λ22+ω2-u2-u′2)+16Λ1Λ2v1v2+8Λ22v222Λ12+2Λ22+ω2-u2-u′2.
It is then
(71)MW2≃g22(u2+u′2+v2+v′2)-g22ΔMw2,
with
(72)ΔMw2=4(2Λ1Λ2v1v2+Λ22v22)2Λ12+2Λ22+ω2-u2-u′2.
In the limit v1,2→0 the mixing angle θ tends to zero, Γ=2Λ12+2Λ22+ω2-u2-u′2, and one has
(73)MW2=g22(u2+u′2+v2+v′2),MY2=g22(2Λ12+2Λ22+ω2+v2+v′2).
With the help of (69), one can estimate
(74)tanθ≃4Λ1v1+4Λ2v2-2Λ12-2Λ22-ω2-2(Λ12+Λ22)~viΛi,4Λ1v1+4Λ2v2-2Λ12-2Λ22-ω2-2(Λ12+Λ22)(i=1,2).
In addition, from (73), it follows that MW2 is much smaller than MY2. Note that, due to the above mixing, the new gauge boson Y will give a contribution to neutrinoless double beta decay (for details, see [123–125]).
ℒmixNGB is the Lagrangian that describes the mixing among the neutral gauge bosons W3, W8, B, W4. The mass Lagrangian in this case has the form
(75)ℒmixNGB=(v2+v′2)324(-9gWμ3+33gWμ8+26gXBμ)2+ω2108(27g2Wμ42+36g2Wμ82hhhhhhhhhh+122ggxWμ8Bμ+2gX2Bμ2)+(u2+u′2)324[(-9gWμ3-33gWμ8+6gXBμ)281g2Wμ42hhhhhhhhhhhhhh+(-9gWμ3-33gWμ8+6gXBμ)2]+g26[2(Λ1v1+Λ2v2)(3Wμ3Wμ4-53Wμ4Wμ8)hhhhhhhh+3(v12+v22+2λ12+2λ22)Wμ32hhhhhhhh+3(4v12+4v22+λ12+λ22+Λ12+Λ22hhhhhhhhhhhhhh+2Λ1λ1+2Λ2λ2)Wμ42hhhhhhhh+23(-v12-v22+2λ12+2λ22)Wμ3Wμ8hhhhhhhh+(v12+v22+2λ12+2λ22+8Λ12+Λ22)Wμ82hhhhhhhh+18(λ1v1+λ2v2)Wμ3Wμ4hhhhhhhh+23(λ1v1+λ2v2)Wμ4Wμ8]+227t2g2(λ12+λ22+Λ12+Λ22+2v12+2v22)Bμ2-2323tg2(λ12+λ22+v12+v22)Wμ3Bμ-4323tg2[(λ1+Λ1)v1+(λ2+Λ2)v2]Wμ4Bμ-229tg2(λ12+λ22-v12-v22-2Λ12-2Λ22)Wμ8Bμ.
On the basis of (Wμ3,Wμ8,Bμ,Wμ4), the ℒmixNGB in (75) can be rewritten in matrix form:
(76)ℒmixNGB≡12VTM2V,VT=(Wμ3,Wμ8,Bμ,Wμ4),M2=g24(M112M122M132M142M122M222M232M242M132M232M332M342M142M242M342M442,),
where
(77)M112=2(v2+v′2+u2+u′2+2v12+2v22+4λ12+4λ22),M122=-233(v2+v′2-u2-u′2+2v12+2v22-4λ12-4λ22),M132=-2323t(2v2+2v′2+u2+u′2hhhhhhhhhhh+4λ12+4λ22+4v12+4v22),M142=4(Λ1v1+Λ2v2)+12(λ1v1+λ2v2),M222=23(v2+v′2+4ω2+u2+u′2+2v12+2v22hhhhhh+4λ12+4λ22+16Λ12+16Λ22),M232=22t9(2v2+2v′2+2ω2-u2-u′2-4λ12-4λ22hhhhhhhhh+4v12+4v22+8Λ12+8Λ22),M242=43[λ1v1+λ2v2-5(Λ1v1+Λ2v2)],M332=4t227(4v2+4v′2+ω2+u2+u′2+4λ12+4λ22hhhhhhhh+4Λ12+4Λ22+8v12+8v22),M342=-16323t(λ1v1+Λ1v1+λ2v2+Λ2v2),M442=2(ω2+u2+u′2+8v12+8v22+2λ12+2λ22hhhhhh+2Λ12+2Λ22+4Λ1λ1+4Λ2λ2).
The matrix M2 in (76) with elements in (77) has one exact eigenvalue, which is identified with the photon mass:
(78)Mγ2=0.
The corresponding eigenvector of Mγ2 is
(79)Aμ=(3t4t2+18-t4t2+18324t2+180).
Note that in the limit λ1,2,v1,2→0, M142=M242=M342=0, and W4 does not mix with W3μ, W8μ, Bμ. In the general case λ1,2,v1,2≠0, the mass matrix in (76) contains one exact eigenvalues as in (78) with the corresponding eigenstate given in (79).
The mass matrix M2 in (76) is diagonalized via two steps. In the first step, the basic (Wμ3,Wμ8,Bμ′,W4μ) is transformed into the basic (Aμ,Zμ,Zμ′,W4μ) by the matrix:
(80)UNGB=(sW-cW00-cWtW3-sWtW31-tW230cW1-tW23sW1-tW23tW300001).
The corresponding eigenstates are given by
(81)Aμ=sWW3μ+cW(-tW3W8μ+1-tW23Bμ),Zμ=-cWW3μ+sW(-tW3W8μ+1-tW23Bμ),Zμ′=1-tW23W8μ+tW3Bμ.
To obtain (80) and (81) we have used the continuation of the gauge coupling constant g of the SU(3)L at the spontaneous symmetry breaking point, in which
(82)t=32sW3-4sW2.
On this basis, the mass matrix M2 becomes
(83)M′2=UNGB+M2UNGB=g24(00000M22′2M23′2M24′20M23′2M33′2M34′20M24′2M34′2M44′2,),
where
(84)M22′2=2cW2(u2+u′2+v2+v′2+4λ12+4λ22+2v12+2v22),M23′2=(α02[(1-2cW2)(u2+u′2+4λ12+4λ22)hhhhh+v2+v′2+v12+v22]α0)(cW2)-1,M24′2=-4cW(Λ1v1+Λ2v2+3λ1v1+3λ2v2),M33′2=32(Λ12+Λ22)cW2α0+8ω2cW2α0+2cW2(v2+v′2+2v12+2v22)α0+2cW2(2cW2-1)2(u2+u′2)α0+8(2cW2-1)2cW2(λ12+λ22)α0,M34′2=-4αcW[(2-1α0)x0(Λ1v1+Λ2v2)hhhhhhhhh+(2-1α0)(λ1v1+λ2v2)],M44′2=2(u2+u′2+ω2+2λ12+2λ22+2Λ12+2Λ22hhhhhh+4λ1Λ1+4λ2Λ2+8v12+8v22).
In the approximation λ1,22,v1,22≪Λ1,22~ω2, we have
(85)M22′2=2cW2(u2+u′2+v2+v′2),M23′2=2[(1-2cW2)(u2+u′2)+v2+v′2]α0cW2,M24′2=-4cW(Λ1v1+Λ2v2),M33′2=32(Λ12+Λ22)cW2α0+8ω2cW2α0+2cW2(v2+v′2)α0+2cW2(2cW2-1)2(u2+u′2)α0,M34′2=-4x0αcW(Λ1v1+Λ2v2),M44′2=2(u2+u′2+ω2+2Λ12+2Λ22+4λ1Λ1+4λ2Λ2),
with
(86)sW=sinθW,cW=cosθW,tW=tanθW,x0=4cW2+1,α0=(4cW2-1)-1.
From (83), there exist mixings between Zμ, Zμ′ and Wμ4. It is noteworthy that, in the limit v1,2=0, the elements M24′2 and M34′2 vanish. In this case there is no mixing between W4 and Zμ, Zμ′.
In the second step, three bosons gain masses via seesaw mechanism
(87)MZ2=g24[M22′2-(Moff)T(M2×2′2)-1Moff],
where
(88)Moff=(M23′2M24′2),M2×2′2=(M33′2M34′2M34′2M44′2).
Combination of (87), (88), and (85) yields
(89)MZ2=g2(u2+u′2+v2+v′2)2cW2-g22cW2ΔMz2,
where
(90)ΔMz2=4Δz2(4cW4x3-2x0x1+x4)+x12x2x2(x4+4cW4x3)-4Δz2x02,
with
(91)x1=(1-2cW2)(u2+u′2)+v2+v′2,x2=2Λ1(2λ1+Λ1)+2Λ2(2λ2+Λ2)+ω2+u2+u′2,x3=4Λ12+4Λ22+ω2+u2+u′2,x4=(1-4c2)(u2+u′2)+v2+v′2,Δz=Λ1v1+Λ2v2.
The ρ parameter in our model is given by
(92)ρ=MW2MZ2cos2θW=1+δwzMz2≡1+δtree,
where
(93)δwz=g22cW2(ΔMz2-ΔMw2).
Let us assume the relations (A.17) and put v2≡vs, ω=Λ2≡Λs, and then
(94)ΔMz2-ΔMw2≃8(k2+1)vs22k2+3(k2+12cW2-1).
From (92)–(94) we have
(95)δtree=g22cW21Mz28(k2+1)vs22k2+3(k2+12cW2-1).
The experimental value of the ρ parameter and MW are, respectively, given in [7]
(96)ρ=1.0004-0.0004+0.0003(δtree=0.0004-0.0004+0.0003),sW2=0.23116±0.00012,MW=80.358±0.015GeV.
It means
(97)0≤δtree≤0.0007.
From (95) one can make the relations between v, g, and k. Indeed, we have
(98)v=±cW2δtree2k2+3MZg2k2+2k2+1-2cW2.
Figure 1 gives the relation between vs and g, k provided that g=0.5, and k∈(0.9,1.1) in which |vs|∈(0,8.0)Gev.
The relation between vs and g, k with g=0.5 and k∈(0.9,1.1).
Figure 2 gives the relation between g and δtree, vs provided that k=1 and δtree∈(0,0.0007), vs∈(0,8.0)GeV in which |g|∈(0,2.0)GeV. The conditions (69) are satisfied. The Figure 3 gives the relation between k and g, vs provided δtree=0.0005 and g∈(0.4,0.6), vs∈(0,8.0)GeV in which k∈(1,3)GeV (k is a real number, Figure 3(a)) or k=ik1, k1∈(-1.2,-1.05)GeV (k is a pure complex number, Figure 3(b)). The conditions (69) are satisfied. From Figure 3 we see that a lot of values of k that is different from the unit but nearly it still can fit the recent experimental data [7]. It means that the difference of 〈s1〉 and 〈s2〉 as mentioned in this work is necessary.
The relation between g and δtree, vs with k=1 and δtree∈(0,0.0007), vs∈(0,8.0)GeV.
The relation between k and g, vs provided that δtree=0.0005 and g∈(0.4,0.6), vs∈(0,8.0)GeV.
Diagonalizing the mass matrix M2×2′2, we get two new physical gauge bosons
(99)Zμ′′=cosϕZμ′+sinϕWμ4,Wμ4′=-sinϕZμ′+cosϕWμ4.
With the approximation as in (69), the mixing angle ϕ is given by
(100)tanϕ≃2α0cW(Λ1v1+Λ2v2)x0-4α0cW4x3+cW2x2-α0x4~v1Λ1~v2Λ2
provided that v1~v2, Λ1~Λ2.
In the limit λ1,2,v1,2→0 the mixing angle ϕ tends to zero, and the physical mass eigenvalues are defined by
(101)MZμ′′2=g22cW2(x4+4cW4x3),MWμ4′2=g22(u2+u′2+ω2+2Λ12+2Λ22).
From (59) and (101) we see that the Wμ4′ and W5 components have the same mass in the limit λ1,2,v1,2→0. So we should identify the combination of Wμ4′ and Wμ5(102)2Xμ0=Wμ4′-iWμ5,
as physical neutral non-Hermitian gauge boson. The subscript “0” denotes neutrality of gauge boson X. Notice that the identification in (102) only can be acceptable with the limit λ1,2,v1,2→0. In general, it is not true because of the difference in masses of Wμ4′ and Wμ5 as in (58) and (99).
The expressions (74) and (100) show that, with the limit (69), the mixings between the charged gauge bosons W-Y and the neutral ones Z′-W4 are in the same order since they are proportional to vi/Λi (i=1,2). In addition, from (101), MZμ′′2≃g2(4Λ12+4Λ22+ω2) is little bigger than MWμ4′2≃(g2/2)(ω2+2Λ12+2Λ22) (or MXμ02), and |MY2-MXμ02|=(g2/2)(u2+u′2-v2-v′2) is little smaller than MW2=(g2/2)(u2+u′2+v2+v′2). In that limit, the masses of Xμ0 and Y degenerate.
7. Conclusions
In this paper, we have constructed a new S4 model based on SU(3)C⊗SU(3)L⊗U(1)X gauge symmetry responsible for fermion masses and mixing which is different from our previous work in [112]. Neutrinos get masses from only an antisextet which is in a doublet under S4. We argue how flavor mixing patterns and mass splitting are obtained with a perturbed S4 symmetry by the difference of VEV components of the antisextet under S4. We have pointed out that this model is simpler than those of S3 and S4 [111, 112] with the fewer number of Higgs multiplets needed in order to allow the fermions to gain masses but with the simple scalar Higgs potential. Quark mixing matrix is unity at the tree level. The realistic neutrino mixing in which θ13≠0 can be obtained if the direction for breaking S4→K4 takes place. This corresponds to the requirement on the difference of VEV components of the antisextet under S4 group. As a result, the value of θ13 is a small perturbation by |λ1-λ2|. The assignation of VEVs to antisextet leads to the mixing of the new gauge bosons and those in the SM. The mixing in the charged gauge bosons as well as the neutral gauge boson was considered.
AppendicesA. Vacuum Alignment
We can separate the general scalar potential into
(A.1)Vtotal=Vtri+Vsext+Vtri-sext+V¯,
where Vtri and Vsext, respectively, consist of the SU(3)L scalar triplets and sextets, whereas Vtri-sext contains the terms connecting the two sectors. Moreover Vtri,sext,tri-sext conserve ℒ-charge and S4 symmetry, while V¯ includes possible soft terms explicitly violating these charges. Here the soft terms as we meant include the trilinear and quartic ones as well. The reason for imposing V- will be shown below.
The details on the potentials are given as follows. We first denote V(X→X1,Y→Y1,…)≡V(X,Y,…)|X=X1,Y=Y1,… Notice also that (TrA)(TrB)=Tr(ATrB). Vtri is a sum of
(A.2)V(χ)=μχ2χ†χ+λχ(χ†χ)2,V(ϕ)=V(χ⟶ϕ),V(ϕ′)=V(ϕ⟶ϕ′),V(η)=V(ϕ⟶η),V(η′)=V(ϕ⟶η′),V(χ,ϕ)=λ1ϕχ(ϕ†ϕ)(χ†χ)+λ2ϕχ(ϕ†χ)(χ†ϕ),V(χ,ϕ′)=V(ϕ⟶ϕ′,χ),V(χ,η)=V(ϕ⟶η,χ),V(χ,η′)=V(ϕ⟶η′,χ),V(ϕ,ϕ′)=V(ϕ,χ⟶ϕ′)+λ3ϕϕ′(ϕ+ϕ′)(ϕ+ϕ′)+λ4ϕϕ′(ϕ′+ϕ)(ϕ′+ϕ),V(ϕ,η)=V(ϕ,χ⟶η),V(ϕ,η′)=V(ϕ,χ⟶η′),V(ϕ′,η)=V(ϕ⟶ϕ′,χ⟶η),V(ϕ′,η′)=V(ϕ⟶ϕ′,χ⟶η′),V(η,η′)=V(ϕ⟶η,χ⟶η′)+λ3η,η′(η+η′)(η+η′)+λ4η,η′(η′+η)(η′+η),Vχϕϕ′ηη′=μ1χϕη+μ1′χϕ′η′+λ11(ϕ+ϕ′)1_′(η+η′)1_′+λ12(ϕ†η′)1_′(η†ϕ′)1_′+λ13(ϕ+ϕ′)1_′(η′+η)1_′+λ14(ϕ†η)1_(η′†ϕ′)1_+λ15(ϕ+η)1_(ϕ′+η′)1_+λ16(ϕ†η′)1_′(ϕ′†η)1_′+h.c.Vsext is only of V(s),
(A.3)V(s)=μs2Tr(s†s)+λ1sTr[(s†s)1_(s†s)1_]+λ2sTr[(s†s)1_′(s†s)1_′]+λ3sTr[(s†s)2_(s†s)2_]+λ4sTr(s†s)1_Tr(s†s)1_+λ5sTr(s†s)1_′Tr(s†s)1_′+λ6sTr(s†s)2_Tr(s†s)2_,
and Vtri-sext is a sum of
(A.4)V(χ,s)=λ1χs(χ†χ)Tr(s†s)+λ2χs(χ†s†)2_(sχ)2_+λ3χs(χ†s)2_(s†χ)2_,V(ϕ,s)=V(χ⟶ϕ,s),V(ϕ′,s)=V(χ⟶ϕ′,s),V(η,s)=V(χ⟶η,s),V(η′,s)=V(χ⟶η′,s),Vsχϕϕ′ηη′=(λ1′ϕ†ϕ′+λ2′η†η′)Tr(s†s)1_′+λ3′[(ϕ†s†)(sϕ′)]1_+λ4′[(η†s†)(sη′)]1_+h.c.
To provide the Majorana masses for the neutrinos, the lepton number must be broken. This can be achieved via the scalar potential violating U(1)ℒ. However, the other symmetries should be conserved. The violating ℒ potential up to quartic interactions is given as
(A.5)V-=[λ-1Tr(s†s)1_+λ-2η†χ+λ-3η†η+λ-4η′†η′hiiiii+λ-5η†η′+λ-6η′†η+λ-7ϕ†ϕ+λ-8ϕ′†ϕ′iiiiiii+λ-9ϕ†ϕ′+λ-10ϕ′†ϕ]η†χ+[λ-11Tr(s†s)1_′+λ-12η′†χ+λ-13η†η+λ-14η′†η′hhhhh+λ-15η†η′+λ-16η′†η+λ-17ϕ†ϕ+λ-18ϕ′†ϕ′hhhhh+λ-19ϕ†ϕ′+λ-20ϕ′†ϕλ-11Tr(s†s)1_′]1_′η′†χ+λ-21(η†ϕ)(ϕ†χ)+λ-22(η†ϕ′)1_′(ϕ′†χ)1_′+λ-23(η′†ϕ)1_′(ϕ′†χ)1_′+λ-24(η′†ϕ′)1_(ϕ†χ)1_+λ-25(η†s†)2_(sχ)2_+λ-26(η′†s†)2_(sχ)2_+h.c.
We have not explicitly written, but there must additionally exist the terms in V- explicitly violating the only S4 symmetry or both the S4 and ℒ-charge too. In the following, most of them will be omitted, and only the terms of the kind of interest will be provided.
There are the several scalar sectors corresponding to the expected VEV directions. The first direction, 0≠〈s1〉≠〈s2〉≠0, S4, is broken into a subgroup including the elements {1,TS2T2,S2,T2S2T} which is isomorphic to the Klein four-group [75] [S=(1234), T=(123), obeying the relations S4=T3=1, ST2S=T, are generators of S4 group given in [112]]. The second direction, 〈s1〉=〈s2〉=〈s〉≠0, S4, is broken into D4. The third direction, 0=〈s1〉≠〈s2〉, or 0=〈s2〉≠〈s1〉, S4, is broken into A4. As mentioned before, to obtain a realistic neutrino spectrum, we have thus imposed both of the first and the second directions to be performed.
Let us now consider the potential Vtri. The flavons χ, ϕ, ϕ′, η, η′ with their VEVs aligned in the same direction (all of them are singlets) are an automatic solution from the minimization conditions of Vtri. To explicitly see this, in the system of equations for minimization, let us put v*=v, v′*=v′, u*=u, u′*=u′, and vχ*=vχ. Then the potential minimization conditions for triplets reduce to
(A.6)∂Vtri∂ω=4λχω3+2(μχ2+λ1χηu2+λ1χη′u′2+λ1χϕv2hhhhhhhhhhhh+λ1χϕ′v′2)ω-μ1uv-μ1′u′v′=0,(A.7)∂Vtri∂v=4λϕv3+2[μϕ2+λ1ϕηu2+λ1ϕη′u′2hhhhhh+(λ1ϕϕ′+λ2ϕϕ′+λ3ϕϕ′+λ4ϕϕ′)v′2+ω2λ1ϕχ]v+(λ11+λ12+λ13)uu′v′-μ1ωu=0,(A.8)∂Vtri∂v′=4λϕ′v′3+2[μϕ′2+λ1ϕ′ηu2+λ1ϕ′η′u′2hhhhhh+(λ1ϕϕ′+λ2ϕϕ′+λ3ϕϕ′+λ4ϕϕ′)v2+ω2λ1ϕ′χ]v′+(λ11+λ12+λ13)uu′v-μ1′ωu′=0,(A.9)∂Vtri∂u=4ληu3+2[μη2+(λ1ηη′+λ2ηη′+λ3ηη′+λ4ηη′)u′2hhhhhh+λ1ϕηv2+λ1ϕ′ηv′2+ω2λ1ηχ]u+(λ11+λ12+λ13)u′vv′-μ1ωv=0,(A.10)∂Vtri∂u′=4λη′u′3+2[μη′2+(λ1ηη′+λ2ηη′+λ3ηη′+λ4ηη′)u2hhhhhh+λ1ϕη′v2+λ1ϕ′η′v′2+ω2λ1η′χ]u′+(λ11+λ12+λ13)uvv′-μ1′ωv′=0.
It is easily shown that the derivatives of Vtri with respect to the variables u, u′, v, v′ shown in (A.7), (A.8), (A.9), and (A.10) are symmetric to each other. The system of (A.6)–(A.10) always has the solution (u, v, u′, v′) as expected, even though it is complicated. It is also noted that the above alignment is only one of the solutions to be imposed to have the desirable results. We have evaluated that (A.7)–(A.10) have the same structure solution. Consequently, to have a simple solution, we can assume that u=u′=v=v′. In this case, (A.7)–(A.10) reduce a unique equation, and system of (A.6)–(A.10) becomes
(A.11)∂Vtri∂ω=4λχω3+2ω[μχ2+(2λ1χη+2λ1χϕ)v2]-2μ1v=0,∂Vtri∂v=2v[λ4ϕϕ′2ω2(λ1χη+λ1χϕ)+2(μη2+μϕ2)hhhhhh+2(λ11+λ12+λ13+4λ1ϕη+λ1ηη′+λ2ηη′hhhhhhhhhhh+λ3ηη′+λ4ηη′+λ1ϕϕ′+λ2ϕϕ′+λ3ϕϕ′hhhhhhhhhhh+λ4ϕϕ′+2λϕ+2λη)v2-2μ1ω]=0.
This system has a solution as follows:
(A.12)u=u′=v′=v=±ω(μχ2+λχω2)μ1-2ω(λ1χη+λ1χϕ),ω=αμ12(α2-βλχ)-Ω3×22/3(α2-βλχ)(Γ+Γ2+4Ω3)1/3+(Γ+Γ2+4Ω3)1/36×21/3(α2-βλχ),
where
(A.13)Γ=54αβμ1(λχμ12+α2μχ2-βλχμχ2)-108λχμ1βγ(α2-λχβ),Ω=6(α2-βλχ)(2αγ+μ12-βμχ2)-9α2μ12,α=λ1χη+λ1χϕ,β=λ11+λ12+λ13+4λ1ϕη+λϕϕ′+ληη′+2(λη+λϕ),λϕϕ′=λ1ϕϕ′+λ2ϕϕ′+λ3ϕϕ′+λ4ϕϕ′,ληη′=λ1ηη′+λ2ηη′+λ3ηη′+λ4ηη′.
Considering the potentials Vsex and Vtri-sex, we impose that
(A.14)λ1*=λ1,λ2*=λ2,v1*=v1,v2*=v2,Λ1*=Λ1,Λ2*=Λ2,v*=v,v′*=v′,u*=u,u′*=u′,vχ*=vχ,vρ*=vρ,
and we obtain a system of equations of the potential minimization for antisextets:
(A.15)∂V1∂λ1=2{(λ1η′s+λ2η′s+λ3η′s)λ2[(λ1η′s+λ2η′s+λ3η′s)λ1χsω2+μs2+(λ1ηs+λ2ηs+λ3ηs)u2hhhhhhh+(λ1η′s+λ2η′s+λ3η′s)u′2+(λ2′+λ4′)uu′hhhhhhh+λ1ϕsv2+λ1ϕ′sv′2+λ1′vv′hhhhhhh+2(3λ1s+λ2s+λ3s+4λ4s)v1v2+4λ4sΛ1Λ2(λ1η′s+λ2η′s+λ3η′s)(λ1ηs+λ2ηs+λ3ηs)u2]hhhhh+2Λ2(λ1s-λ2s+λ3s)v1v2+2Λ1(λ1s+λ2s)v22hhhhh+2λ1[λ6sΛ22+λ22(2λ1s+λ3s+2λ4s+λ6s)hhhhihhhhhhh+(λ1s-λ2s+λ3s+2λ6s)v22](λ1η′s+λ2η′s+λ3η′s)}=0,∂V1∂λ2=2{(λ1η′s+λ2η′s+λ3η′s)λ1[(λ1η′s+λ2η′s+λ3η′s)λ1χsω2+μs2+(λ1ηs+λ2ηs+λ3ηs)u2hhhhhhh+(λ1η′s+λ2η′s+λ3η′s)u′2+(λ2′+λ4′)uu′hhhhhhh+λ1ϕsv2+λ1ϕ′sv′2+λ1′vv′hhhhhhh+2(3λ1s+λ2s+λ3s+4λ4s)v1v2hhhhhhii+4λ4sΛ1Λ2(λ1η′s+λ2η′s+λ3η′s)(λ1ηs+λ2ηs+λ3ηs)u2]+2Λ1(λ1s-λ2s+λ3s)v1v2hhhhh+2Λ2(λ1s+λ2s)v12hhhhh+2λ2[λ6sΛ12+λ12(2λ1s+λ3s+2λ4s+λ6s)hhhhhhhhhhhh+(λ1s-λ2s+λ3s+2λ6s)v12](λ1η′s+λ2η′s+λ3η′s)λ1ηs}=0,∂V1∂v1=2{(2λ1η′s+λ2η′s+λ3η′s)v2[(2λ1η′s+λ2η′s+λ3η′s)(2λ1χs+λ2χs+λ3χs)ω2+2μs2hhhhhhh+(2λ1ηs+λ2ηs+λ3ηs)u2+(2λ2′+λ4′)uu′hhhhhhh+(2λ1η′s+λ2η′s+λ3η′s)u′2+2λ1ϕsv2+2λ1′vv′hhhhhhh+2λ1ϕ′sv′2+2(λ1Λ2+λ2Λ1)(λ1s-λ2s+λ3s)hhhhhhii+2(λ1λ2+Λ1Λ2)(3λ1s+λ2s+λ3s+4λ4s)(2λ1η′s+λ2η′s+λ3η′s)(2λ1χs+λ2χs+λ3χs)ω2(2λ1χs+λ2χs+λ3χs)]hhhhh+2[2λ2Λ2(λ1s+λ2s)+(λ22+Λ22)hhhhhhhhhh×(λ1s-λ2s+λ3s+2λ6s)2λ2Λ2(λ1s+λ2s)+(λ22+Λ22)]v1hhhhh+4(2λ1s+λ3s+4λ4s+2λ6s)v1v22(2λ1η′s+λ2η′s+λ3η′s)2λ1χs}=0,∂V1∂v2=2{(2λ1η′s+λ2η′s+λ3η′s)v1[(2λ1η′s+λ2η′s+λ3η′s)(2λ1χs+λ2χs+λ3χs)ω2+2μs2hhhhhhh+(2λ1ηs+λ2ηs+λ3ηs)u2+(2λ2′+λ4′)uu′hhhhhhh+(2λ1η′s+λ2η′s+λ3η′s)u′2+2λ1ϕsv2+2λ1′vv′hhhhhhh+2λ1ϕ′sv′2+2(λ1Λ2+λ2Λ1)(λ1s-λ2s+λ3s)hhhhhhii+2(λ1λ2+Λ1Λ2)(3λ1s+λ2s+λ3s+4λ4s)(2λ1η′s+λ2η′s+λ3η′s)(2λ1χs+λ2χs+λ3χs)ω2(2λ1χs+λ2χs+λ3χs)]hhhh+2[2λ1Λ1(λ1s+λ2s)+(λ12+Λ12)hhhhhhhhh×(λ1s-λ2s+λ3s+2λ6s)2λ1Λ1(λ1s+λ2s)+(λ12+Λ12)]v2hhhh+4(2λ1s+λ3s+4λ4s+2λ6s)v2v12(2λ1η′s+λ2η′s+λ3η′s)}=0,∂V1∂Λ1=2{Λ2[(λ1χs+λ2χs+λ3χs)ω2+μs2+λ1ηsu2hhhhhhh+λ2′uu′+λ1η′su′2+λ1′vv′+λ1ϕsv2+λ1ϕ′sv′2hhhhhhii+4λ4sλ1λ2+2(3λ1s+λ2s+λ3s+4λ4s)v1v2(λ1χs+λ2χs+λ3χs)ω2(λ1χs+λ2χs+λ3χs)]hhhhh+2λ2(λ1s-λ2s+λ3s)v1v2+2λ1(λ1s+λ2s)v22hhhhh+2Λ1[λ6sλ22+Λ22(2λ1s+λ3s+2λ4s+λ6s)hhhhhhhhhhhh+(λ1s-λ2s+λ3s+2λ6s)v22]λ1ηs}=0,∂V1∂Λ2=2{Λ1[(λ1χs+λ2χs+λ3χs)ω2+μs2+λ1ηsu2hhhhhhh+λ2′uu′+λ1η′su′2+λ1′vv′+λ1ϕsv2+λ1ϕ′sv′2hhhhhhii+4λ4sλ1λ2+2(3λ1s+λ2s+λ3s+4λ4s)v1v2(λ1χs+λ2χs+λ3χs)ω2(λ1χs+λ2χs+λ3χs)]hhhhi+2λ1(λ1s-λ2s+λ3s)v1v2+2λ2(λ1s+λ2s)v12hhhhi+2Λ2[λ6sλ12+Λ12(2λ1s+λ3s+2λ4s+λ6s)hhhhhhhhhhhh+(λ1s-λ2s+λ3s+2λ6s)v12]λ1ηs}=0,
where V1 is a sum of Vsext and Vtri-sext:
(A.16)V1=Vsext+Vtri-sext.
It is easily shown that (A.15) takes the same form in couples. This system of equations yields the following relations:
(A.17)λ1=κλ2,v1=κv2,Λ1=κΛ2,
where κ is a constant. It means that there are several alignments for VEVs. In this work, to have the desirable results, we have imposed the two directions for breaking S4→D4 and S4→K4 as mentioned, in which κ=1 and κ≠1 but approximates to the unit. In the case that κ=1 or λ1=λ2=λs, v1=v2=vs, and Λ1=Λ2=Λs, the system of (A.15) reduces to system for minimal potential condition consisting of three equations as follows:
(A.18)λs[Aω+μs2+2AsΛs2+2(As+Bs)λs2+Avhhhh+4(As+Bs)vs2]+2BsΛsvs2=0,2(Aω+Bω)+2μs2+Av+Av′+4BsλsΛs+4(As+Bs)(λs2+vs2+Λs2)=0,Λs[Aω+Bω+μs2+2Asλs2+2(As+Bs)Λs2hhhh+Av′+4(As+Bs)vs2]+2Bsλsvs2=0,
where
(A.19)Aω=λ1χsω2,Bω=(λ2χs+λ3χs)ω2,As=2λ4s+λ6s,Bs=2λ1s+λ3s,Av=(λ1′+λ2′+λ4′+λ1ϕs+λ1ϕ′s+λ1ηs+λ2ηshhhh+λ3ηs+λ1η′s+λ2η′s+λ3η′s)v2,Av′=(λ1′+λ2′+λ1ϕs+λ1ϕ′s+λ1ηs+λ1η′s)v2.
The system of (A.18) always has the solution (λs, vs, Λs) as expected, even though it is complicated. It is also noted that the above alignment is only one of the solutions to be imposed to have the desirable results.
B. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M756"><mml:mrow><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> Group and Clebsch-Gordan Coefficients
S4 is the permutation group of four objects, which is also the symmetry group of a cube. It has 24 elements divided into 5 conjugacy classes, with 1_, 1_′, 2_, 3_, and 3_′ as its 5 irreducible representations. Any element of S4 can be formed by multiplication of the generators S and T obeying the relations S4=T3=1, ST2S=T. Without loss of generality, we could choose S=(1234), T=(123) where the cycle (1234) denotes the permutation (1,2,3,4)→(2,3,4,1), and (123) means (1,2,3,4)→(2,3,1,4). The conjugacy classes generated from S and T are
(B.1)C1:1,C2:(12)(34)=TS2T2,(13)(24)=S2,iiiiii(14)(23)=T2S2T,C3:(123)=T,(132)=T2,(124)=T2S2,iiiiiii(142)=S2T,(134)=S2TS2,(143)=STS,iiiiiii(234)=S2T2,(243)=TS2,C4:(1234)=S,(1243)=T2ST,(1324)=ST,iiiiiii(1342)=TS,(1423)=TST2,(1432)=S3,C5:(12)=STS2,(13)=TSTS2,(14)=ST2,hiiiii(23)=S2TS,(24)=TST,(34)=T2S.
The character table of S4 is given as shown in Table 1, where n is the order of class and h is the order of elements within each class. Let us note that C1,2,3 are even permutations, while C4,5 are odd permutations. The two three-dimensional representations differ only in the signs of their C4 and C5 matrices. Similarly, the two one-dimensional representations behave the same.