Discrete flavor symmetry and minimal seesaw mechanism
aa r X i v : . [ h e p - ph ] J a n Discrete flavor symmetry and minimal seesaw mechanism
N. W. Park ∗ Department of Physics, Chonnam National University, Gwangju 500-757, Korea
K. H. Nam † and Kim Siyeon ‡ Department of Physics, Chung-Ang University, Seoul 156-756, Korea (Dated: January 21, 2011)
Abstract
This work proposes a neutrino mass model that is derived using the minimal seesaw mechanismwhich contains only two right-handed neutrinos, under the non-abelian discrete flavor symmetry S ⊗ Z . Two standard model doublets, L µ and L τ , are assigned simultaneously to a representationof S . When the scalar fields introduced in this model, addition to the Standard Model Higgs, andthe leptons are coupled within the symmetry, the seesaw mechanism results in the tri-bi-maximalneutrino mixing. This study examined the possible deviations from TBM mixing related to theexperimental data. PACS numbers: 11.30.Fs, 14.60.Pq, 14.60.StKeywords: ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] . INTRODUCTION
The data of neutrino oscillation experiments revealed a very noticeable form of a mixingmatrix with 0 . ≤ | U e | ≤ . , . ≤ | U µ | ≤ . , and | U e | ≤ . , at the 90%confidence level(CL). The current data also include the mass squared differences which areaccompanied by solar and atmospheric neutrino oscillations, ∆ m sol ≃ (7 +10 − ) × − eV and ∆ m atm ≃ (2 . +1 . − . ) × − eV , respectively[1][2]. The central values of the elements ofthe Pontecorvo-Maki-Nakagawa-Sakata(PMNS) matrix are pointing a unique form of thematrix, which consists of sin θ sol = 1 / √ , sin θ atm = 1 / √ , and sin θ reactor = 0. The three-flavor neutrino mixing matrix specified by those angle sizes is called tri-bi-maximal(TBM)mixing[3], based on the idea that the neutrino mass matrix in a lepton flavor basis is an S group matrix and a canonical subgroup S plays a role of a µ − τ interchange[4]. Therefore,its elegant form suggests that the mass model has originated from a flavor symmetry. Modelswith various types of discrete flavor symmetries have been constructed [5][6][7][8].A previous study considered a minimal seesaw model with only two righthanded neutri-nos, where texture zeros and equalities in the Dirac mass matrix constrain the number ofparameters and all the elements of the 2 × S ⊗ Z symmetry is adopted to construct aminimal seesaw model, which is established with a small content of additional fields beyondthe Standard Model(SM). The Higgs contents are extended to include three SU (2) doubletscalars in one- or two- dimensional representations of S addition to the Standard ModelHiggs, and then the model has a relatively simple Higgs potential, while the models withthree righthanded neutrinos constructed in a group framework of the same kind should haveincluded a rather complicated potential[7][8]. The model naturally represents µ − τ symme-try and the properties of group S ⊗ Z lead the basis to the one in which the charged leptonmass matrix is diagonal. Since the exact S ⊗ Z symmetry for a given field content andgiven charge assignments is necessary, a meaningful issue can be the deviation from TBMmixing as a consequence of the symmetry breaking by a few vevs.The outline is as follows. Section II introduces the framework within the flavor symmetryand the Lagrangian. The aspects that may arise during the symmetry breaking processesand their low energy effects since the seesaw mechanism are reported in Section III. Someextended effects of symmetry breaking, which are plausible for the deviation from TBM,2s well as the experimental accessibility of the model are considered in Section IV. In theconclusion, the theoretical remarks on the model and an analysis on aspects of deviation fromTBM mixing are summarized. The appendix contains the potential of all scalar particlesalong with their vacuum expectation values. II. DISCRETE FLAVOR SYMMETRY S ⊗ Z The 5 irreducible representations of S are , ′ , , , ′ . (1)While the tensor products of , one of the 1-dim representations, are so trivial that they aresimply × r = r × = r , the tensor products of ′ , the other 1-dim representation, are asfollows: ′ × ′ = : ab, (2) ′ × = : ab − ab . (3)In general, the elements of are denoted by a i or b i , for i and j = 1 to 2. According to thegroup theory details for the S -II version in Ref.[10], the product of with another is × = + ′ + , (4) : ( a b + a b ) ′ : ( a b − a b ) : a b − a b a b + a b , (5)The SU(2) lepton doublets L α ≡ ( L e , L µ , L τ ), the righthanded charged lepton singlets E R ≡ ( e R , µ R , τ R ), and the Higgs scalar doublet H in the Standard Model are assigned intosuch representations of S ⊗ Z as follows:Rep . ( , F ( , F ( , F ( , − / G L e ( L µ , L τ ) H ( , − G e R ( µ R , τ R ) (6)3n the above table, the SU(2) representation and hypercharge of a field are denoted by thesubscription ‘ G ’ of the gauge symmetry, and S representation and Z charge of the field aredenoted by the subscription ‘ F ’. Under flavor symmetry, the charge assignment in Eq.(6)does not distinguish the µ flavor from the τ flavor, because L µ and L τ as well as µ R and τ R are in doublets. Therefore, this S ⊗ Z symmetry appears as µ − τ symmetry.To explain the light neutrino masses, additional Higgs scalars, Φ ≡ ( φ , φ ) and χ , andright-handed neutrinos, N R ≡ ( N , N ), are introduced. Their representations under gaugesymmetry and flavor symmetry are given in the following table. Rep. ( r , q ) F ( r , q ) F ( , − F ( , − F ( , / G ( φ , φ ) χ ( , G N N (7)The flavor charges of N and N , ( r , q ) F and ( r , q ) F will be assigned according to themass type, normal hierarchy or inverted hierarchy. In general, the Lagrangian constructedwith the leptons and Higgs scalars in Eq.(6) and Eq.(7) can be written as − L = f ij E iR HL jα + g jk L jα Σ N kR + 12 M kl N kR N lR , (8)where Σ = { H, Φ , χ } . Each i and j runs 1 to 3, whereas each k and l runs 1 to 2.The S ⊗ Z invariant Higgs potential is V = V H + m ϕ Φ † Φ + m χ χ † χ + { Λ(Φ † Φ) + λ ( χ † χ ) } + λ ′ (Φ † Φ)( χ † χ ) + λ ′′ (Φ † χ )( χ † Φ)+ η ′ (Φ † Φ)( H † H ) + η ′′ (Φ † H )( H † Φ)+ κ ( χ † H )( H † χ ) , (9)where V H = m H H † H + 12 η ( H † H ) , (10)and Λ(Φ † Φ) = λ a (Φ † Φ) + λ b (Φ † Φ) ′ + λ c (Φ † Φ) , (11)since the product Φ † Φ can be any of the following representations, ( , , ( ′ ,
1) or ( ,
1) of S ⊗ Z . According to the product rules in Eqs. (2) - (4), (Φ † Φ) = | φ | + | φ | , (Φ † Φ) ′ =4 ∗ φ − φ ∗ φ , and (Φ † Φ) = ( | φ | − | φ | φ ∗ φ + φ ∗ φ ) T . The potential in Eq. (9) issymmetric under the interchange of φ and φ . Therefore, their vacuum expectation values(vevs) are the same, h φ i = h φ i ≡ w . The vevs h χ i and h H i , which are denoted as u and v , respectively, will be defined to minimize the potential in Eq. (9) in Appendix.A.After spontaneous symmetry breaking, the Dirac mass matrix of the charged leptons fromthe Yukawa couplings, f ij E iR HL j , becomes M l − ∼ f v f v
00 0 f v . (12)Thus, this flavor model generates a basis where the matrix of charged lepton masses isdiagonal. All the SM particles are Z -even as shown in Eq.(6), so that their Yukawa couplingsare protected from the Yukawa couplings with a Z -odd Higgs scalar, Φ or χ . III. MINIMAL SEESAW MECHANISM
Minimal Model for Normal Hierarchy : In a minimal model where only two SMsinglet righthanded neutrinos N and N are added to the SM fermions, one zero masseigenvalue is assigned to three active neutrinos. If the charges of right-handed neutrinos N and N are given as ( r , q ) F = ( ′ , −
1) and ( r , q ) F = ( , − S ⊗ Z invariantYukawa couplings give rise to the matrix form: g ij Σ ∼ g χg φ g φ − g φ g φ . (13)The Yukawa coupling N ( φ φ )( L µ L τ ) T is obtained via the tensor product ′ × × = ′ × ′ , whereas the coupling N ( φ φ )( L µ L τ ) T is obtained via the tensor product × × = × . Thus, the different product rules of the same couplings can cause different sizes inthe coupling constants, between a pair of g , g and a pair of g , g . This model assumesthat O ( g , g ) < O ( g , g ).When φ , φ and χ obtain their vacuum expectation values to minimize the potential V inEq.(9), breaking the symmetry S ⊗ Z , the process to acquire the vevs w = h φ i = h φ i and u = h χ i are described in Eq.(A4). With the flavor charges of N and N , the mass matrix5f the right-handed neutrinos is diagonal, so that M kl N kR N lR = M N N + M N N .According to the seesaw mechanism M ν = − m D M − R m TD , the mass matrix of light neutrinos M ν is g u M g g uwM g g uwM g g uwM g w M + g w M − g g w M + g g w M g g uwM − g g w M + g g w M g w M + g w M , (14)where the overall minus sign has been removed by a phase transformation of neutrino fields.If the masses of light neutrinos have a normal hierarchy, then m = 0. If u = w ,and the Yukawa couplings g ij ’s are constrained such that g = g and g = g = g ,the mixing matrix of M ν is exactly TBM. Besides, there are a priori conditions for TBMmixing that should be considered. In Eq.(13), the zero element is a key to θ = 0 in PMNS,and the opposite sign between the 22 element and 32 element is necessary for θ = π/ m = 0, m m <
0, and h φ i = h φ i is derivedfrom S symmetry. In other words, θ = 0 and θ = π/ S symmetry. However, it is unlikely for a Dirac matrix to have exact zero and exact equalitiesas a coincidence. It will be shown that slight discrepancies between those elements causephysically significant deviation from the TBM mixing matrix. The non-zero eigenvaluesof the light neutrino mass matrix in Eq.(14) are given by m = f ( g )( u + 2 w ) /M and m = f ′ ( g ) 2 w /M , where f ( g ) and f ′ ( g ) are the factors determined in terms of Yukawacoupling constants g ’s. Minimal Model for Inverted Hierarchy : When the charges of righthanded neutrinos N and N are given as ( r , q ) F = ( ,
1) and ( r , q ) F = ( , − S ⊗ Z invariantYukawa couplings give rise to g ij Σ ∼ g H g χ g φ g φ . (15)With such charge assignment, the mass matrix of righthanded neutrinos is diagonal; M kl N kR N lR = M N N + M N N . According to the seesaw mechanism M ν = − m D M − R m TD ,6he mass matrix of light neutrinos M ν is g v M + g u M g g uwM g g uwM g g uwM g w M g g w M g g uwM g g w M g w M , (16)where u = h χ i , v = h H i , and w = h φ i = h φ i . When taking the seesaw mechanism, the twozero elements in Eq.(15) are necessary for θ = 0, and g = g is necessary for θ = π/ S ⊗ Z flavor symmetry results in the texture zeros and the equalities,which are the necessary conditions for TBM mixing in PMNS. Non-zero eigenvalues of lightneutrino mass matrix in Eq.(16), m and m , are given by m + m = f ( g ) (cid:18) v M + u + 2 w M (cid:19) (17) m − m = f ′ ( g ) (cid:18) v M + u + 2 w M (cid:19) − u w M M ! / , where f ( g ) and f ′ ( g ) can be determined in terms of Yukawa coupling constants g ’s. Inaddition, possible deviations from those specific elements in Dirac mass matrix can cause aphenomenological deviation from TBM mixing in the PMNS matrix. IV. SYMMETRY BREAKING AND DEVIATION FROM TRI-BI MAXIMALMIXING
It is necessary to examine the possible aspects that can appear when the zeros and equal-ities generated by the flavor symmetry are collapsed during symmetry breaking. Therefore,some simple cases will be discussed. In a trial model, it can be assumed to have a right-handed mass matrix as follows: M R = M , (18)and f ( g ) and f ′ ( g ), in normal hierarch case or in inverted hierarch case Eq.(17), are assumedto be one, for simplicity. It is unlikely to have an exact equality u = w , or to have an exactzero element, as shown in Eq.(13) and Eq.(15), after a series of spontaneous symmetrybreaking mechanisms. In order to examine the effects of symmetry breaking to a deviation7rom low energy TBM mixing matrix, the Dirac mass matrix for normal hierarchy is assumedto have the following form: m D = t uγw w − γw w , (19)where t < u or w , and t is a small value that can occur from S ⊗ Z -violating Yukawacoupling. The factor γ is a Yukawa coupling ratio that affects the mass ratio at low energy,which can arise from a ′ × ′ product weaker than × . The curve (a) in Fig.1 describesthe change in | U e | as the ratio u/w varies from 0 to 1.5 for t = 0, even though the changein | U e | turns out to be independent of t . On the other hand, the variation in u/w does notcause any change in | U e | or | U µ | . The curves (b) and (c) in Fig.1 describe the changes in | U e | and the change in | U µ | , respectively, as the ratio t/w varies from 0 to 1, while keeping u/w = 1. Under the given conditions, a range of u/w , 0 . ≤ u/w ≤ .
07, can lead to | U e | at the 90% CL. The range of | U e | at the 90% CL matches with t/w ≤ .
35, while the rangeof | U µ | at 90% CL matches with t/w ≤ .
46. The size of γ does not affect the mixing anglesin PMNS, but does affect the mass ratios.For inverted hierarchy, the Dirac mass matrix can be supposed to have the following form: m D = v ug t wg t w , (20)where t < u, or w . Yukawa couplings g and g are assumed to be very small so that g t and g t are much smaller than v . The curve (a) in Fig.2 describes the change in | U e | asthe ratio v/w varies from 0 to 1.5 for u = w and t = 0. Under the given conditions, arange of v/w , 0 . ≤ v/w ≤ . | U e | at the 90% CL. Since v is the vacuumexpectation value of the Higgs scalar in the Standard Model, this model does not causeany new physics below approximately 1 TeV. The curves (b) and (c) in Fig.1 describe thechanges in | U e | and | U µ | , respectively, when the 2-1 and 3-1 elements obtain a non-zerovalue due to S ⊗ Z -violating coupling, and their relative ratio g /g varies from 0 to 10.The range of | U e | at the 90% CL matches with g /g ≤ .
8, whereas the range of | U µ | atthe 90% CL matches with g /g ≤ .
7. 8
IG. 1: Unshaded regions exhibit the allowed ranges in the elements of PMNS matrix at the 90%CL. The curve (a) | U e | is obtained at t = 0 in Eq.(19), and its intersections with the bounds areat u/w = 0 .
82 and at u/w = 1 .
07. The curve (b) | U e | is obtained at u/w = 1, and its intersectionwith the bound is at t/w = 0 .
35, whereas the curve (c) | U µ | has an intersection with the boundat t/w = 0 . V. CONCLUSION
A model of lepton masses was constructed in the framework of a discrete flavor symmetry S ⊗ Z . Each family of SM leptons or each of the right-handed neutrinos, is distinguishedfrom each other by their flavor charges. However, a pair of L µ and L τ are assigned to therepresentation of S so that they cannot be distinguished under the Yukawa interaction,and so are a pair of µ R and τ R , indicating µ − τ symmetry. When the symmetry is accompa-nied with Z symmetry, we can obtain the basis where the mass matrix of charged leptonsis diagonal, as shown in Eq.(12). Since the SM particles are all Z -even and the additionalscalar particles are Z -odd, there are no extra Yukawa couplings of charged leptons found.The model contains only two righthanded neutrinos so that the couplings with three scalarfields and three lefthanded neutrinos generate a 2 × S givesrise to µ − τ symmetry, accordingly TBM mixing, and the diagonal mass matrix of charged9 IG. 2: Unshaded regions exhibit the allowed ranges in the elements of the PMNS matrix at the90% CL. The curve (a) | U e | is obtained at t = 0 in Eq.(20), and its intersections with the boundsare at v/w = 0 .
135 and at v/w = 0 . | U e | is obtained at v/w = 0 .
15, and itsintersection with the bound is at g /g = 3 .
8, whereas the curve (c) | U µ | has an intersection withthe bound at g /g = 4 . leptons. Second, the type of neutrino mass spectrum, normal or inverted hierarchy, is re-sulted in by the flavor symmetry. The flavor charges of righthanded neutrinos play a keyrole in determining the order of light neutrino masses. Third, it is possible to build the massmatrices with a significantly small content of scalar particles under a non-abelian discreteflavor symmetry S ⊗ Z .The texture zeros or the equalities due to the symmetry in a Yukawa matrix appear tobe barely protected, once the symmetry undergoes a chain of breakdown. It is natural forPMNS to obtain a deviation from TBM mixing when the symmetry of the Yukawa matrixis broken. Fig.1 and Fig.2 show a schematic diagram of some aspects that can appear in thecorrelation between the order of symmetry breaking and the deviation from TBM. Acknowledgments
KS wishes to thank the members in the Physics Department at Chonnam National Uni-versity for their warm hospitality. 10 ppendix A: Higgs Potential
The S ⊗ Z invariant Higgs potential in Eq.(9) can be rephrased in terms of { φ i , φ † i } with i = 1 and 2, χ , and H . V ( φ i , φ † i , χ, H )= m H | H | + 12 η | H | + m ϕ ( | φ | + | φ | ) + m χ | χ | + 12 ( λ a + λ c )( | φ | + | φ | ) + ( λ a + λ b ) | φ | | φ | + 12 ( λ c − λ b )( φ ∗ φ + φ ∗ φ ) (A1)+ 12 λ | χ | + ( λ ′ + λ ′′ )( | φ | + | φ | ) | χ | +( η ′ + η ′′ )( | φ | + | φ | ) | H | + κ | χ | | H | . When the Higgs particles obtain their vacuum expectation values such that h χ i = u , h H i = v , and h φ i = h φ i = w where u, v, and w are real, the minimal potential can be expressedas follow: V min ( u, v, w ) = m H v + m χ u + 2 m ϕ w + 12 λu + 12 ηv + 2( λ a + λ c ) w (A2)+2( λ ′ + λ ′′ ) u w + 2( η ′ + η ′′ ) v w + κu v , which satisfies ∂ L ∂v = 2 v { m H + κu + ηv + Λ c w } = 0 ∂ L ∂u = 2 u { m χ + λu + κv + Λ b w } = 0 (A3) ∂ L ∂w = 2 w { m ϕ + 2Λ b u + Λ c v + Λ a w } = 0 , where Λ a = 2( λ a + λ b ), Λ b = 2( λ ′ + λ ′′ ) and Λ c = 2( η ′ + η ′′ ). Therefore, the vacuumexpectation values u, v and w are u = { (Λ b Λ c − κ Λ a ) m H + ( η Λ a − λ c ) m χ + (2 κ Λ c − η Λ b ) m ϕ } /Dv = { ( λ Λ a − b ) m H + (2Λ b Λ c − κ Λ a ) m χ + (2 κ Λ b − λ Λ c ) m ϕ } /D (A4) w = { (2 κ Λ b − λ Λ c ) m H + ( κ Λ c − η Λ b ) m χ + (2 ηλ − κ ) m ϕ } /DD = ( κ − ηλ )Λ a + 2 η Λ b − κ Λ b Λ c + λ Λ c .
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