Dirac or Inverse Seesaw Neutrino Masses with B−L Gauge Symmetry and S 3 Flavour Symmetry
aa r X i v : . [ h e p - ph ] D ec Dirac or Inverse Seesaw Neutrino Masses with B − L Gauge Symmetry and S Flavour Symmetry
Ernest Ma ∗ and Rahul Srivastava † Department of Physics and Astronomy, University of California,Riverside, California 92521, USA The Institute of Mathematical Sciences, Chennai 600 113, India
Abstract
Many studies have been made on extensions of the standard model with B − L gauge symmetry. The addition of three singlet (right-handed) neutrinos renders itanomaly-free. It has always been assumed that the spontaneous breaking of B − L isaccomplished by a singlet scalar field carrying two units of B − L charge. This resultsin a very natural implementation of the Majorana seesaw mechanism for neutrinos.However, there exists in fact another simple anomaly-free solution which allows Diracor inverse seesaw neutrino masses. We show for the first time these new possibilitiesand discuss an application to neutrino mixing with S flavour symmetry. ∗ [email protected] † [email protected] I. INTRODUCTION
The standard model (SM) of quarks and leptons is free of gauge anomalies. If it isextended to include an extra U (1) gauge group, the conditions for the absence of gaugeanomalies impose nontrivial constraints. Nevertheless, interesting discoveries of such pos-sibilities have been made. One example [1] adds a lepton triplet per family to the SM.Another [2] adds a number of new superfields to the minimal supersymmetric standardmodel (MSSM) of three families. On the other hand, the most studied extension is that of U (1) B − L . Using the particle content of the standard model, all triangle gauge anomaliesare zero except for X U (1) B − L = − − − = +3. Note that the gauge-gravitational anomaly is also zero because − −
1) = +3. Numerous studies have been made regarding this model of U (1) B − L .In this paper, we point out that there is another simple choice for three families [3–5].Let ν Ri ∼ n i under B − L , then n , , = (+5 , − , −
4) yields − (+5) − ( − − ( − = +3 (2)as well. In fact, since − (5) − ( − − ( −
4) = +3, the mixed gauge-gravitational anomalyis also zero. Now the standard-model Higgs doublet ( φ + , φ ) T does not connect ν L with ν R and there is no neutrino mass. Consider then three heavy Dirac singlet fermions N L,R , alltransforming as − B − L . They do not change the anomaly conditions, but now inthe case of ν R and ν R , (¯ ν L , ¯ N L ) is linked to ( ν R , N R ) through the 2 × M ν,N = m m M (3)where m comes from h φ i and m comes from h χ i , assuming the Yukawa coupling ¯ N L ν R χ ,i.e. χ ∼ +3 under B − L . Now, the invariant mass M is naturally large, so the Diracseesaw [6] yields a small neutrino mass m m /M . In the conventional U (1) B − L model, χ ∼ +2 under B − L , is chosen to break the gauge symmetry, so that ν R gets a Majoranamass and lepton number L is broken to ( − L . Here, χ ∼ +3 means that L remains aconserved global symmetry, with ν L,R and N L,R all having L = 1. Since ν R ∼ +5 does notconnect with ν L or N L directly, there is one massless neutrino in this case. However, thedimension-five operator ¯ N L ν R χ ∗ χ ∗ / Λ is allowed by U (1) B − L and would give it a very smallDirac mass. Alternatively, a second scalar χ ∼ B − L = +3 for the scalar field χ ; is the new idea of this paper,which was not recognized anywhere before. Because the pairing of two neutral fermions ofthe same chirality in this model always results in a nonzero B − L charge not divisible by 3units, it is impossible to construct an operator of any dimension for a Majorana mass termwhich violates B − L . Hence the statement that B − L is reduced to a conserved global U (1) symmetry is definitely robust.Another possible outcome of the above scenario is obtained with the choice of two complexscalar fields χ ∼ χ ∼ U (1) B − L . In this case, ν L is not connected to ν R ,R ∼ −
4. It is connected however to N L,R through the mass matrix spanning (¯ ν L , N R , ¯ N L )as follows: M ν,N = m m m ′ M M m (4)where m and m ′ come from χ . This is then an inverse seesaw [7–9], i.e. m ν ≃ m m /M .In the case of ν R ∼ +5, the corresponding mass matrix spanning (¯ ν L , N R , ¯ N L , ν R ) is givenby M ν,N = m m m ′ M M m m m (5)where m comes from χ . Thus ν R also gets an inverse seesaw mass ≃ m m ′ /M . Notethat Eq. (5) is the 4 × × L is broken to ( − L for ν L and ν R , and N , , are heavypseudo-Dirac fermions. However, ν R ,R remain massless and may only be produced inpairs. They are thus good dark-matter candidates if they become massive. This may beaccomplished with a third scalar χ ∼ II. THE S FLAVOUR SYMMETRY: µ − τ SECTOR
In this section we expand upon the first scenario discussed in the previous section, us-ing two singlet scalar fields transforming as +3 under the B − L symmetry, in analogy tohaving ν R and ν R transforming as −
4. This allows us to use the doublet representationsof the non-Abelian discrete symmetry group S to understand the leptonic family structureas indicated by present neutrino oscillation data and other experimental constraints. Ourapplication of S will be different from those of Ref. [4, 5] since we have the three N L,R fieldsin addition to ν R ,R ,R . We will first work out the details of our model for the simpler caseof µ − τ sector and show how one can use S symmetry to obtain maximal mixing. Thenwe will discuss the implications of imposing S symmetry to the full 3 × S symmetry group.The S group is the smallest non-Abelian discreet symmetry group and is the group ofthe permutation of three objects. It consists of six elements and is also isomorphic to thesymmetry group of the equilateral triangle. It admits three irreducible representations 1, 1 ′ and 2 with the tensor product rules1 × ′ = 1 ′ , ′ × ′ = 1 , × , × ′ = 2 , × ′ + 2 (6)In this work, following the earlier works [11, 12], we choose to work with the complexrepresentation of the S group. In the complex representation, if φ φ , ψ ψ ∈ ⇒ φ † φ † , ψ † ψ † ∈ (7)such that φ ψ + φ ψ , φ † ψ + φ † ψ ∈ φ ψ − φ ψ , φ † ψ − φ † ψ ∈ ′ φ ψ φ ψ , φ † ψ φ † ψ ∈ (8)With this brief summary of S group and its irreducible representations, we now move onto constructing an S invariant µ − τ sector. In later section we will generalize the discussionof this section to the full 3 × A. The S invariant µ − τ sector We denote the left handed lepton doublets by L α = ( ν αL , l αL ) T where α = µ, τ ; the righthanded charged leptons are denoted as µ R , τ R and the right handed neutrinos as ν µR , ν τR .Also, let us denote the heavy singlet fermions as N iL,R ; i = 2 ,
3. The “Standard Model like”scalar doublet is denoted by Φ = ( φ + , φ ) T and the singlet scalars are denoted by χ , .Let the B − L charge and S assignment of the above fields be as shown in Table I. Fields B − L S Fields B − L S L µ − ′ L τ − µ R − ′ τ R − N L − ′ N L − N R − ′ N R − ν µR ν τR − χ χ B − L and S charge assignment for the fields. The S and B − L invariant Yukawa L (2) Y interaction can be written as L (2) Y = L (2) L α l R + L (2) L α N R + L (2) N L N R + L (2) N L ν R (9)where L (2) L α l R = y µ ¯ L µ Φ µ R + y τ ¯ L τ Φ τ R L (2) L α N R = g ¯ L µ ˆΦ ∗ N R + g ¯ L τ ˆΦ ∗ N R L (2) N L N R = M ¯ N L N R + M ¯ N L N R L (2) N L ν R = f ¯ N L ⊗ ν µR ν τR ⊗ χ χ ′ + f ¯ N L ⊗ ν µR ν τR ⊗ χ χ (10)In writing (10), we used the notation ˆΦ ∗ = iτ Φ ∗ = ( φ , φ − ) T . Moreover, y µ , y τ are theYukawa couplings of the charged lepton sector whereas f i , g i and M i denote the dimensionlesscoupling constants between the leptons and the heavy fermions.After symmetry breaking the scalar fields get vacuum expectation values (VEVs) h φ i = v , h χ i i = u i ; i = 2 ,
3. Then the mass matrix relevant to charged leptons is given by M l = v y µ y τ (11)Also, the 4 × ν µL , ¯ ν τL , ¯ N L , ¯ N L ) and ( ν µR , ν τR , N R , N R ) T of neutrinosand the heavy fermions is given by M ν,N = g v ∗
00 0 0 g v ∗ f u − f u M f u f u M (12)As remarked earlier, the mass terms M i between the heavy fermions can be naturallylarge, so we can block diagonalize (12) assuming that f i , g i << M i . The block diagonalizedmass matrix of light neutrinos is given by M ν = m (2) N L ν R M (2) N L N R m (2) L α N R = v ∗ f g M u − f g M u f g M u f g M u (13)where m (2) L α N R , M (2) N L N R and m (2) N L ν R are the 2 × L (2) L α N R , L (2) N L N R and L (2) N L ν R terms of (10). This light neutrino mass matrix can be furtherdiagonalized by the bi-unitary transformation. The neutrino masses and the mixing an-gles obtained from (13) will be dependent on the specific values of the coupling constants f i , g i , M i as well as the VEVs v, u i ; i = 2 ,
3. For sake of illustration we explicitly computethem for two simplifying scenario leading to maximal mixing.
Case I: f = f = f , g = g = g , M = M = M , u = u .In this case the neutrino mass matrix becomes M Iν = f gv ∗ M u − u u u (14) Case II: f = f = f , u = u = u , g M = g M .In this case the neutrino mass matrix becomes M IIν = f uv ∗ g M − g M g M g M (15)Both the mass matrices in (14) and (15) can be written as M ν = κ a − ba b (16)where κ = fgv ∗ M f uv ∗ , a = u g M , b = u In Case I g M In Case II (17)The mass matrix in (16) can be easily diagonalized by a bi-unitary transformation leadingto the neutrino masses ν = √ κa and ν = √ κb with mixing angle θ = π where κ, a, b for both cases are given in (17). Therefore, S allows us to understand maximal mixing inthe µ − τ sector. III. THE COMPLETE S INVARIANT LEPTON SECTOR
The B − L charge and S assignment of the fields for the full lepton sector is as shownin Table II. Fields B − L S Fields B − L S L e − ′ e R − ′ L µ − ′ µ R − ′ L τ − τ R − N L − ′ N R − ′ N L − ′ N R − ′ N L − N R − ν eR ′ ν µR ν τR − χ χ B − L and S charge assignment for the fields. The S and B − L invariant Yukawa L Y interaction can be written as L Y = L L α l R + L L α N R + L N L N R + L N L ν R (18)where L L α l R = y ′ e ¯ L e Φ e R + y ′ ¯ L e Φ µ R + y ′ ¯ L µ Φ e R + y ′ µ ¯ L µ Φ µ R + y τ ¯ L τ Φ τ R L L α N R = g ′ ¯ L e ˆΦ ∗ N R + g ′ ¯ L e ˆΦ ∗ N R + g ′ ¯ L µ ˆΦ ∗ N R + g ′ ¯ L µ ˆΦ ∗ N R + g ¯ L τ ˆΦ ∗ N R L N L N R = M ′ ¯ N L N R + M ′ ¯ N L N R + M ′ ¯ N L N R + M ′ ¯ N L N R + M ¯ N L N R L N L ν R = f ′ Λ (cid:0) ¯ N L ν eR (cid:1) ⊗ χ ∗ χ ∗ ⊗ χ ∗ χ ∗ + f ′ Λ (cid:0) ¯ N L ν eR (cid:1) ⊗ χ ∗ χ ∗ ⊗ χ ∗ χ ∗ + f ′ ¯ N L ⊗ ν µR ν τR ⊗ χ χ ′ + f ′ ¯ N L ⊗ ν µR ν τR ⊗ χ χ ′ + f ¯ N L ⊗ ν µR ν τR ⊗ χ χ (19)As before, in writing (19), we used the notation ˆΦ ∗ = iτ Φ ∗ = ( φ , φ − ) T . Moreover, y α are the Yukawa couplings of the charged leptons whereas f ij , g ij and M ij denote thedimensionless coupling constants between the leptons and the heavy fermions.At this point we like to remark that in (19) there is still a freedom to redefine a fewfields (i.e. the pairs N L − N L , N R − N R and e R − µ R ) in a way that certain couplings canbe made equal to zero. For sake of later convenience we choose to use this freedom of fieldredefinition to make f ′ = M ′ = y ′ = 0. Moreover, we relabel the remaining non-zerocouplings of these redefined fields as f ′ ij → f ij , g ′ ij → g ij , M ′ ij → M ij .Now, after symmetry breaking the scalar fields get VEVs h φ i = v , h χ i i = u i ; i = 2 , M l = v y e y y µ
00 0 y τ (20)The mass matrix (20) can be readily diagonalized by bi-unitary transformation. In thelimit of y e << y µ we get0 θ l ≈ tan − (cid:18) − y y µ (cid:19) ; m e ≈ v y e cos θ l12 m µ ≈ v (cid:0) y µ cos θ l12 − y sin θ l12 (cid:1) ; m τ ≈ vy τ (21)If y = y µ then maximal mixing is achieved i.e. θ l = − π , with m µ = √ vy µ .Also, the 6 × ν eL , ¯ ν µL , ¯ ν τL , ¯ N L , ¯ N L , ¯ N L ) and ( ν eR , ν µR , ν τR , N R , N R , N R ) T of neutrinos and the heavy fermions is given by M ν,N = g v ∗ g v ∗
00 0 0 g v ∗ g v ∗
00 0 0 0 0 g v ∗ f u − f u M f Λ u ∗ u ∗ f u − f u M M f u f u M (22)As remarked earlier, the mass terms M ij between the heavy fermions can be naturallylarge, so we can block diagonalize (22) assuming that f ij , g ij << M ij . The block diagonalizedmass matrix of light neutrinos is given by M ν = m N L ν R M N L N R m L α N R = v ∗ ( g M − g M ) f u M M ( g M − g M ) f u M M − f g u M ( g M − g M ) f u + f g M u M M ( g M − g M ) f u + f g M u M M − f g u M ( g M − g M ) f u M M ( g M − g M ) f u M M f g u M (23)where we have written u = u ∗ u ∗ Λ . Also, the 3 × m L α N R , M N L N R and m N L ν R are obtained from the terms L L α N R , L N L N R and L N L ν R of (19), respectively. Thislight neutrino mass matrix can be further diagonalized by the bi-unitary transformation.The neutrino masses and the mixing angles obtained from (23) will be dependent on thespecific values of the coupling constants f ij , g ij , M ij as well as the VEVs v, u i ; i = 2 , g ij = g and M ij = M we get M ν = gv ∗ M − f u f u f u − f u f u (24)Diagonalizing the mass matrix (24) we have θ ν ≈ θ ν ≈ tan − (cid:18) f f (cid:19) ; θ ν ≈ tan − f p f + f ! m ν ≈ m ν ≈ p f + f ) f g | v | M p f + f + f | u | ; m ν ≈ p f + f + f g | v | M | u | (25)Since, u << u , we have a normal hierarchy pattern with two nearly massless neutrinosand one relatively heavy neutrino. Moreover, the massless neutrino will also gain smallmass, if any of the M ij ’s or g ij ’s are not equal to M or g respectively. Also, if they deviatesignificantly from these values then one can possibly recover degenerate or inverted hierarchypatterns also.Now, if U l and U ν are the mixing matrices of the charged leptons and neutrinos respec-tively, then the PMNS mixing matrix is given by U PMNS = U † l U ν (26)Taking y = y µ , f = − f and f = p f + f in (21) and (25) we get θ ν = − θ l = π and θ ν = tan − ( − ) which gives PMNS mixing angles consistent with present 3 − σ limitsof global fits obtained from experiments [13]. With θ ν = − θ l = π and θ ν = tan − ( − ) we get the value of mixing angles as θ = 30 . ◦ , θ = 7 . ◦ and θ = 50 . ◦ . The values of θ and θ are slightly below the lower limits (30 . ◦ , 7 . ◦ respectively)quoted in [13]. A much better fit can be obtained if we take slightly different values, for instance taking θ l = − ◦ , θ ν = − . ◦ and θ ν = π gives us θ = 33 . ◦ , θ = 9 . ◦ and θ = 51 . ◦ . The Quark Sector
In our minimal model with only one doublet scalar, the quark sector can be accommo-dated in a simple way if both the left handed quark doublets Q iL = ( u iL , d iL ) T , i = 1 , , u iR , d iR ; i = 1 , , S .A better understanding of the quark sector can be obtained if, to our minimal model, weadd more doublet scalars transforming non-trivially under S . One such example for quarksector, albeit in context of a different model for lepton sector, has already been worked outin [11, 12]. We plan to present a similar extension of our minimal model in a future work. IV. CONCLUSIONS
The idea that B − L should be a gauge symmetry has been around for a long time. Theminimal version of three ν R transforming as − ν R ’s transforming as +5 , − , −
4. Weshow how these assignments may be used to obtain seesaw Dirac neutrino masses, as well asinverse seesaw Majorana neutrino masses. We then apply S symmetry to the first case, andobtain realistic neutrino and charged-lepton mass matrices with a mixing pattern consistentwith experiments. ACKNOWLEDGMENTS
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