Neutrino and CP-even Higgs boson masses in a nonuniversal \mathrm{U}(1)' extension
NNeutrino and CP-even Higgs boson masses in a nonuniversalU(1) (cid:48) extension
S.F. Mantilla ∗ , R. Martinez † , F. Ochoa ‡ May 11, 2017
Departamento de F´ısica, Universidad Nacional de Colombia, Ciudad Universitaria, K. 45 No. 26-85,Bogot´a D.C., Colombia
Abstract
We propose a new anomaly-free and family nonuniversal U(1) (cid:48) extension of the standard modelwith the addition of two scalar singlets and a new scalar doublet. The quark sector is extendedby adding three exotic quark singlets, while the lepton sector includes two exotic charged leptonsinglets, three right-handed neutrinos and three sterile Majorana leptons to obtain the fermionicmass spectrum of the standard model. The lepton sector also reproduces the elements of thePMNS matrix and the squared-mass differences data from neutrino oscillation experiments. Also,analytical relations of the PMNS matrix are derived via the inverse see-saw mechanism, andnumerical predictions of the parameters in both normal and inverse order scheme for the mass ofthe phenomenological neutrinos are obtained. We employed a simple seesaw-like method to obtainanalytical mass eigenstates of the CP-even 3 × Despite all its success, the Standard Model (SM) of Glashow, Weinberg and Salam [1] has some un-explained features, which has motivated many models and extensions. In particular, the observedfermion mass hierarchies, their mixing and the three family structure are not explained in the SM.From the phenomenological point of view, it is possible to describe some features of the mass hierarchyby assuming zero-texture Yukawa matrices [2]. Models with spontaneously broken flavor symme-tries may also produce hierarchical mass structures. For example, in models with gauge symmetrySU(2) L ⊗ SU(2) R ⊗ U(1) B − L , the electroweak doublets exhibit a discrete symmetry after the sponta-neous symmetry breaking, obtaining Fritzsch zero-texture mass matrices [3] in the basis U = ( u , c , t )of the form: − (cid:104)L Y,U (cid:105) = U L a a ∗ b b ∗ c U R + h . c . (1)The zero-texture of the above matrix can describe the mass spectrum in the quark sector and the CPviolation phase observed in the experiments. This mass structure can also be obtained in the leptonsector, as shown by Fukugita, Tanimoto y Yanagida [4], where very small mass values are predicted ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ h e p - ph ] M a y ormal Ordering (NO) Inverted Ordering (IO)sin θ . +0 . − . . +0 . − . sin θ . +0 . − . . +0 . − . sin θ . +0 . − . . +0 . − . δ CP +38 − +39 − ∆ m − eV . +0 . − . . +0 . − . ∆ m (cid:96) − eV +2 . +0 . − . − . +0 . − . Table 1: Three-flavor oscillation parameter values at 1 σ reported by [24, 22]. (cid:96) = 1 for NO and 2 forIO.through a seesaw mechanism. In addition, these type of models contain Majorana neutrinos whichinduce matter-antimatter asymmetry through leptogenesis [5].Another issue that the SM can not explain is the observation of neutrino oscillations. Theseobservations have been confirmed by many experiments from four different sources: solar neutrinos asin Homestake [6], SAGE [7], GALLEX & GNO [8], SNO [9], Borexino [10] and Super-Kamiokande [11]experiments, atmospheric neutrinos as in IceCube [12], neutrinos from reactors as KamLAND [13],CHOOZ [14], Palo Verde [15], Daya Bay [16], RENO [17] and SBL [18], and from accelerators as inMINOS [19], T2K [20] and NO ν A [21]. The experimental data are compatible with the hypothesisthat at least two species of neutrinos have mass, where the left-handed flavor neutrino fields are linearcombinations of mass eigenstates | ν aL (cid:105) = (cid:88) i =1 , , U ai (cid:12)(cid:12) ν iL (cid:11) , a = e, µ, τ (2)where U is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix, which can be parameterized asfunction of three mixing angles and one CP violating phase [22, 23]. However, the experiments can notdetermine the true nature of the active neutrinos (Majorana or Dirac) nor the absolute values of theirmass. Table 1 shows the parameters from references [22, 24] and available at NuFIT 3.0 [25], wheretwo hierarchies are assumed: normal ordering (NO), where the squared mass difference between thethird and first species accomplish ∆ m >
0, and inverted ordering (IO), where ∆ m < M ),which gives masses to light neutrinos as m ν = v w /M . There are some basic ways to implementthis mechanism: a heavy right-handed Majorana neutrino ν R mixed to the corresponding left-handedneutrino ν L via the SM scalar doublet (type I seesaw), a heavy scalar triplet bosons (type II), or aheavy fermionic triplet (type III). Since the new scale M associated with the new fields is high ( ∼ GeV), this mechanism has the problem that is not accessible to be test in experiments. However, thereis another possibility: the inverse see-saw mechanism (ISS), where a very light Majorana neutrino N R is incorporated, such that in the basis ( ν L , ν Rc , N Rc ) the mass matrix has the form of Fristzschzero-texture: M ν = m T ν m ν m T N m N M N . (3)where the submatrix m N has components of the order of the TeV scale, while M N is of the order ofthe KeV scale, in order to obtain active neutrinos at the sub-eV scale. The inverse see-saw mechanismwas proposed in [26]. This mechanism has also been implemented in SU(3) L ⊗ U(1) X models in orderto study the µ → eγ decay [27].On the other hand, the discovery of the Higgs boson at ATLAS[28] and CMS [29] whose mass is 125GeV opens the window to propose other scalar fields. A new scalar sector is considered as extensionto the SM in order to explain some phenomenological aspects. One of the most studied SM extensionis the two-Higgs-doublet-model (2HDM) which proposes the existence of two scalar doublets whose2calar potential mixes them together obtaining two charged scalar bosons H ± , a CP-odd pseudoscalar A and two CP-even scalar bosons h and H [30]. This model was motivated in order to give massesto up-like and down-like quarks [31] where vacuum expectation values (VEV) v and v are related tothe electroweak VEV by v = v + v .There are also extensions to the 2HDM adding a new scalar singlet χ , as in the Next-to-Minimal2HDM (N2HDM) [33]. In some cases, this additional singlet implement the spontaneous symme-try breaking (SSB) of an additional U (1) (cid:48) gauge symmetry through the acquisition of non-vanishingVEV v χ , and consequently its imaginary part become in the would-be Goldstone boson eaten by thecorresponding gauge boson of U (1) (cid:48) [34]. Furthermore, if this SSB happens at a higher scale thanthe electroweak ( v (cid:28) v χ ), the CP-even mass matrix exhibits an internal hierarchy which allows usto employ a perturbative see-saw-like method in order to obtain analytical expressions for the masseigenvalues and angles of the corresponding mixing matrix.Models with extra U(1) (cid:48) symmetry are one of the most studied extensions of the SM, which impliesmany phenomenological and theoretical advantages including flavor physics [35], neutrino physics [36],dark matter [37], among other effects [38]. A complete review of the above possibilities can be found inreference [39]. In particular, family non-universal U(1) (cid:48) symmetry models have many well-establishedmotivations. For example, they provide hints for solving the SM flavor puzzle, where even thoughall the fermions acquire masses at the same scale, υ = 246 GeV, experimentally they exhibit verydifferent mass values. These models also imply a new Z (cid:48) neutral boson, which contains a large numberof phenomenological consequences at low and high energies [40]. In addition to the new neutral gaugeboson Z (cid:48) , an extended fermion spectrum is necessary in order to obtain an anomaly-free theory. Also,the new symmetry requires an extended scalar sector in order to (i) generate the breaking of the newAbelian symmetry and (ii) obtain heavy masses for the new Z (cid:48) gauge boson and the extra fermioncontent. A non-universal U(1) (cid:48) model in the quark sector was proposed in [34], obtaining zero-texturequark mass matrices with hierarchical structures, where three quarks [up, down and strange] acquiremasses at the MeV scale, and three quarks [charm, bottom and top] exhibit masses at the GeV scale.Additional phenomenological consequences of this model were studied in [41, 42, 43] including effectson scalar DM.The main purpose of this paper is to construct an anomaly-free and family non-universal U(1) (cid:48) symmetry model in both the quark and leptonic sector, with extra lepton and quark singlets, twoscalar doublets, and two scalar singlets. The leptonic sector includes new charged and right-handedneutral leptons, and sterile Majorana neutrinos in order to reproduce the PMNS matrix and theobserved mass structure of the leptons. In Sec. 2, we describe the spectrum and most importantproperties of the model. We also show the scalar and gauge Lagrangians, including rotations into masseigenvectors. In Sec. 3 we show how mass structures in the fermion sector are predicted in the model,first for the quark sector in subsection 3.1, and later for the leptonic sector in subsection 3.2. Sec. 4 isdevoted to obtain some phenomenological parameters from neutrino oscillation data at 1 σ . Finally,the Sec. 5 outlines the main results of the article. U(1) X symmetry The model proposes the existence of a new non-universal gauge group U (1) (cid:48) whose gauge boson andcoupling constant are Z (cid:48) µ and g X , respectively. It brings the following triangle anomaly equations:3uarks X Z Leptons X Z SM Fermionic Isospin Doublets q L = (cid:18) U D (cid:19) L +1 / (cid:96) eL = (cid:18) ν e e e (cid:19) L q L = (cid:18) U D (cid:19) L − (cid:96) µL = (cid:18) ν µ e µ (cid:19) L q L = (cid:18) U D (cid:19) L (cid:96) τL = (cid:18) ν τ e τ (cid:19) L − U , R U R D , , R +2 / / − / −− e e,τR e µR − / − / −− Non-SM Quarks Non-SM Leptons T L T R +1 / / −− ν e,µ,τR N e,µ,τR / −− J , L E L , E R − J , R − / E L , E R − / X quantum number and Z parity for SM and non-SM fermions.[SU(3) C ] U(1) X → A C = (cid:88) Q X Q L − (cid:88) Q X Q R , (4)[SU(2) L ] U(1) X → A L = (cid:88) (cid:96) X (cid:96) L + 3 (cid:88) Q X Q L , (5)[U(1) Y ] U(1) X → A Y = (cid:88) (cid:96),Q (cid:2) Y (cid:96) L X (cid:96) L + 3 Y Q L X Q L (cid:3) − (cid:88) (cid:96),Q (cid:2) Y (cid:96) R X L R + 3 Y Q R X Q R (cid:3) , (6)U(1) Y [U(1) X ] → A Y = (cid:88) (cid:96),Q (cid:2) Y (cid:96) L X (cid:96) L + 3 Y Q L X Q L (cid:3) − (cid:88) (cid:96),Q (cid:2) Y (cid:96) R X (cid:96) R + 3 Y Q R X Q R (cid:3) , (7)[U(1) X ] → A X = (cid:88) (cid:96),Q (cid:2) X (cid:96) L + 3 X Q L (cid:3) − (cid:88) (cid:96),Q (cid:2) X (cid:96) R + 3 X Q R (cid:3) , (8)[Grav] U(1) X → A G = (cid:88) (cid:96),Q [ X (cid:96) L + 3 X Q L ] − (cid:88) (cid:96),Q [ X (cid:96) R + 3 X Q R ] , (9)where the sums in Q run over quarks while (cid:96) runs over leptons with nontrivial U(1) X values. Y is the corresponding weak hypercharge. The fermion content compatible with the above conditionsis composed by ordinary SM particles but also new exotic non-SM particles, as shown in table 2,where column X contains the quantum numbers of the extra U(1) X and the Z column presents theircorresponding Z -parity under a new Z discrete symmetry. Some properties of this spectrum areoutlined below:1. The U(1) X symmetry is only non-universal in the left-handed SM quark sector: the first family1 has X = 1 / , X = 0. Leptons exhibit non-universal charges inboth left- and right-handed sectors: X = 0 for the left-handed components e, µ and X = − τ , while for the right-handed components X = − / e, τ and X = − / µ . We use thefollowing assignation for the phenomenological families: U , , = ( u, c, t ) , D , , = ( d, s, b ) , e e,µ,τ = ( e, µ, τ ) , ν e,µ,τ = ( ν e , ν µ , ν τ ) . (10)2. In order to ensure cancellation of the gauge chiral anomalies, the model includes extra isospinsinglets. The quark sector has an up T and two down J , quarks. For the lepton sector, threeright-handed neutrinos ν e,µ,τR and two charged leptons E and E are added with non-trivial U (1) X charges, as shown in Tab. 2. 4calar bosons X Z Higgs Doublets φ = φ +1 h + v + iη √ / φ = φ +2 h + v + iη √ / − Higgs Singlets χ = ξ χ + v χ + iζ χ √ − / σ − / − Table 3: Non-universal X quantum number for Higgs fields.3. The most natural way to obtain massive neutrinos, according to neutrino oscillations, is througha see-saw mechanism, which requires the introduction of extra Majorana neutrinos. Thus, forobtaining a realistic model compatible with massive neutrinos, three sterile Majorana neutrinos N e,µ,τR are included.The scalar sector of the model is shown in table 3, which exhibits the following properties:1. Two scalar doublets φ , are included with U(1) X charges +2 / / v = (cid:112) v + v . Theinternal Z symmetry is introduced in order to obtain adequate zero texture matrices.2. An extra scalar singlet χ with VEV υ χ is required for the SSB of U(1) X and also to generatemasses to exotic isospin singlets. We assume that it happens at a larger scale υ χ (cid:29) υ thanelectroweak.3. Another scalar singlet σ is introduced. Since it is not essential for the symmetry breakingmechanisms, we may choose υ σ = 0 for its VEV.Finally, in the vector sector, an extra gauge boson Z (cid:48) µ is required to obtain a local U(1) X symmetry.The covariant derivative of the model is D µ = ∂ µ − igW αµ T α − ig (cid:48) Y B µ − ig X XZ (cid:48) µ , (11)where 2 T α corresponds to the Pauli matrices for isospin doublets and T α = 0 for isospin singlets. Theelectric charge is defined by the Gell-Mann-Nishijima relation: Q = T + Y . (12) The scalar potential of the model is V = µ φ † φ + µ φ † φ + µ χ χ ∗ χ + µ σ σ ∗ σ + f √ (cid:16) φ † φ χ ∗ + h . c . (cid:17) + f (cid:48) √ (cid:16) φ † φ σ ∗ + h . c . (cid:17) + λ (cid:16) φ † φ (cid:17) + λ (cid:16) φ † φ (cid:17) + λ ( χ ∗ χ ) + λ ( σ ∗ σ ) + λ (cid:16) φ † φ (cid:17) (cid:16) φ † φ (cid:17) + λ (cid:48) (cid:16) φ † φ (cid:17) (cid:16) φ † φ (cid:17) + (cid:16) φ † φ (cid:17) [ λ ( χ ∗ χ ) + λ (cid:48) ( σ ∗ σ )]+ (cid:16) φ † φ (cid:17) [ λ ( χ ∗ χ ) + λ (cid:48) ( σ ∗ σ )]+ λ ( χ ∗ χ ) ( σ ∗ σ ) + λ (cid:48) [( χ ∗ σ ) ( χ ∗ σ ) + h . c . ] . (13)5fter symmetry breaking the mass matrices for the scalar sector are found. For the charged scalarbosons the mass matrix is obtained in the basis ( φ ± , φ ± ) M = 14 − f v χ v v − λ (cid:48) v f v χ + λ (cid:48) v v f v χ + λ (cid:48) v v − f v χ v v − λ (cid:48) v (14)which is diagonalized by R C = (cid:18) c β s β − s β c β (cid:19) , (15)where tan β = s β /c β = v /v . The mass matrix has the eigenvalues m G ± W = 0 ,m H ± = − f v χ s β c β − λ (cid:48) v , (16)yielding the Goldstone bosons G ± W , which provide the mass to the physical W ± µ gauge bosons, and twophysical charged Higgs bosons H ± .Regarding to the neutral scalar sector, the mass matrix of the CP-odd sector in the basis ( η , η , ζ χ )is: M = − f v v χ v − v χ v − v χ v v χ v − v v − v v v v χ , (17)which can be diagonalized by the following transformation R I = c β s β − s β c β
00 0 1 c γ s γ − s γ c γ , (18)where γ describes the doublet-singlet mixing tan γ = s γ /c γ = v χ /vs β c β . When R I acts on M thefollowing eigenvalues are obtained m G Z = 0 ,m G Z (cid:48) = 0 ,m A = − f v χ s β c β s γ , (19)where the first two are the would-be Goldstone bosons of the neutral vector bosons Z µ and Z (cid:48) µ ,respectively, while the latter is a physical CP-odd pseudoscalar boson A .On the other hand, the CP-even scalar mass matrix is M = λ v − f v χ v v ˆ λ v v + 14 f v χ λ v v χ + 14 f v ˆ λ v v + 14 f v χ λ v − f v χ v v λ v v χ + 14 f v λ v v χ + 14 f v λ v v χ + 14 f v λ v χ − f v v v χ , (20)where ˆ λ = ( λ + λ (cid:48) ) /
2. Since this matrix exhibits a characteristic third order polynomial with non-trivial eigenvalues, it is convenient to use another approximation in order to obtain the eigenvaluesand mixing angles. We propose a seesaw-like mechanism by assuming a hierarchy of VEVs through6he condition | f | υ χ , υ χ (cid:29) υ in the matrix elements. Thus, the matrix (20) can be written in blocksas M = (cid:18) M M T12 M M (cid:19) , (21)where M = λ v − f v χ v v ˆ λ v v + 14 f v χ ˆ λ v v + 14 f v χ λ v − f v χ v v , M T12 = λ v v χ f v λ v v χ f v ≈ λ v v χ λ v v χ , M = λ v χ − f v v v χ ≈ λ v χ . (22)According to the block diagonalization procedure shown in Appendix A, the mass matrix (21) canbe decoupled into two independent blocks through a unitary transformation as: R TS M R S = (cid:18) M hH m H χ (cid:19) . (23)where the transformation matrix can be approximately written as R S = (cid:18) F T R − F R (cid:19) , (24)with F R ≈ M − M ,m H χ ≈ M = λ v χ ,M hH ≈ M − M T12 M − M T12 , (25)and M hH = ˜ λ v s β − f v χ t β ˜ λ v s β c β + 14 f v χ ˜ λ v s β c β + 14 f v χ ˜ λ v c β − f v χ t β . (26)where the new tilde constants are ˜ λ = λ − λ λ − λ λ t β , ˜ λ = λ − λ t β λ − λ λ , ˜ λ = ˆ λ − λ t β λ − λ λ t β . (27)In order to obtain the largest eigenvalue of M hH , we neglect non-dominant terms from the conditionthat f v χ (cid:29) v , v , v v , which leads us to M hH ≈ − f v χ (cid:18) cot β − − β (cid:19) . (28)7ue to this approximation, the new matrix has null determinant and its trace is of the order of thelargest eigenvalue m H ≈ Tr (cid:2) M hH (cid:3) ≈ − f v χ s β c β . (29)The lightest mass eigenvalue can be calculated through the ratio of the determinant and the trace of(26), i.e. Det (cid:2) M hH (cid:3) Tr [ M hH ] = m h m H m h + m H ≈ m h , (30)obtaining m h ≈ (cid:16) ˜ λ s β + 2˜ λ c β s β + ˜ λ c β (cid:17) v , (31)which we associate to the observed 125 GeV Higgs boson. The mixing angle associated to (26) isdefined as t α = tan 2 α , where t α = f v χ + 2˜ λ s β c β v f v χ + 2 t β ( s β ˜ λ − c β ˜ λ ) v t β . (32)Finally, the diagonalization of the CP-even matrix (20) is achieved by R R parametrized by a CKM-like matrix R R = c s − s c c s − s c c α s α − s α c α
00 0 1 , (33)where t α = s α /c α and s = 12 λ vs β λ v χ , s = 12 λ vc β λ v χ , (34)whose corresponding cosines are approximated as c ≈ − s / c ≈ − s /
2. In fact, the R S matrix which block-diagonalizes M is the product of the former two rotation matrices with mixingangles θ and θ .In conclusion, the scalar spectrum of the model is: • Four would-be Goldstone bosons: G ± W , G Z y G Z (cid:48) . • Three scalar CP-even h , H y H χ fields with mass m h ≈ (cid:16) ˜ λ c β + 2˜ λ c β s β + ˜ λ s β (cid:17) v ,m H ≈ − f v χ s β c β ,m H χ ≈ λ v χ . (35) • A pseudoscalar CP-odd A whose mass is m A = − f v χ s β c β s γ . (36) • Two charged scalar bosons H ± with mass m H ± = − f v χ s β c β − λ (cid:48) v . (37)8 .2 Gauge boson masses The kinetic terms of the scalar fields are L kin = (cid:88) i ( D µ S ) † ( D µ S ) . (38)After the symmetry breaking, the charged bosons W ± µ = ( W µ ∓ W µ ) / √ M W = gv/ W µ , B µ , Z (cid:48) µ ): M = 14 g v − gg (cid:48) v − gg X v (1 + c β ) ∗ g (cid:48) v g (cid:48) g X v (1 + c β ) ∗ ∗ g X v χ (cid:104) c β ) (cid:15) (cid:105) , (39)where (cid:15) = υ/υ χ . Taking into account (cid:15) (cid:28)
1, the matrix can be diagonalized with only two angles,obtaining the following mass eigenstates: A µ Z µ Z µ ≈ R W µ B µ Z (cid:48) µ , (40)with: R = s W c W c W c Z − s W c Z s Z − c W s Z s W s Z c Z , (41)where tan θ W = s W /c W = g (cid:48) /g defines the Weinberg angle, and s Z = sin θ Z is a small mixing anglebetween the SM neutral gauge boson Z and the U(1) X gauge boson Z (cid:48) such that in the limit s Z → Z = Z and Z = Z (cid:48) . This mixing angle is approximately s Z ≈ (1 + c β ) 2 g X c W g (cid:18) M Z M Z (cid:48) (cid:19) , (42)where the neutral masses are: M Z ≈ gυ c W , M Z (cid:48) ≈ g X υ χ . (43) We find the Yukawa Lagrangian compatible with the SU(2) L ⊗ U(1) Y ⊗ U(1) X gauge symmetry. Forthe quark sector we obtain −L Q = q L (cid:16) (cid:101) φ h U (cid:17) j U jR + q aL ( (cid:101) φ h U ) aj U jR + q L (cid:0) φ h D (cid:1) j D jR + q aL (cid:0) φ h D (cid:1) aj D jR + q L ( φ h J ) m J mR + q aL (cid:0) φ h J (cid:1) am J mR + q L (cid:16) (cid:101) φ h T (cid:17) T R + q aL ( (cid:101) φ h T ) a T R + T L (cid:0) σh Uσ + χh Uχ (cid:1) j U jR + T L (cid:0) σh Tσ + χh Tχ (cid:1) T R + J nL (cid:0) σ ∗ h Dσ + χ ∗ h Dχ (cid:1) nj D jR + J nL (cid:0) σ ∗ h Jσ + χ ∗ h Jχ (cid:1) nm J mR + h.c., (44)where (cid:101) φ , = iσ φ ∗ , are conjugate fields, a = 2 , n ( m ) = 1 , J n ( m ) quarks. A sum over the indices i, a and n is understood.We can see in the quark Lagrangian that due to the non-universality of the U(1) X symmetry, not all9ouplings between quarks and scalars are allowed by the gauge symmetry, which leads us to specificzero-texture Yukawa matrices. However, these structures are not inherited by the mass matrices of thequarks, due to the interactions of the four scalar fields φ , φ , σ and χ that couple simultaneouslyto all quark flavors. In order to reproduce the observed mass spectrum, we must restrict further thenumber of couplings in the Lagrangian, which can be done by assuming the Z discrete symmetriesshown in tables 2 and 3. Assuming these discrete symmetries, the Lagrangian (44) after the symmetrybreaking leads us to the following mass terms at tree level: −(cid:104)L Q (cid:105) = U iL ( M U ) ij U jR + D iL ( M D ) ij D jR + T L ( M T ) T R + J nL ( M J ) nm J mR + T L ( M T U ) j U jR + U iL ( M UT ) i T R + D iL ( M DJ ) im J mR + h.c., (45)where the mass matrices generate the following zero structures: M U = 1 √ υ a υ a υ a υ a , M D = υ √ B B B ,M J = υ χ √ (cid:18) k k k k (cid:19) , M T = υ χ √ h Tχ ,M T U = υ χ √ , c , , M UT = 1 √ υ y υ y M DJ = 1 √ υ j υ j υ j υ j , M JD = 0 , (46)which leads us to the following extended mass matrices: M (cid:48) U = M U | M UT — — — — — M T U | M T = 1 √ υ a | υ y υ a | υ y υ a υ a |
0— — — — —0 υ χ c | υ χ h Tχ ,M (cid:48) D = M D | M DJ — — — — — M JD | M J = 1 √ | υ j υ j | υ j υ j υ B υ B υ B | | υ χ k υ χ k | υ χ k υ χ k . (47)After diagonalization, the above structures leads us to hierarchies of the phenomenological quarks,as detailed below. Up sector
First, we consider the up-type matrix M (cid:48) U in equation (47). We obtain its symmetrical quadratic formas: M U = M (cid:48) U ( M (cid:48) U ) T = 12 υ (cid:0) a + y (cid:1) υ υ ( a a + y y ) 0 | υ υ χ (cid:0) a c + y h Tχ (cid:1) υ υ ( a a + y y ) υ (cid:0) a + y (cid:1) | υ υ χ (cid:0) a c + y h Tχ (cid:1) υ (cid:0) a + a (cid:1) |
0— — — — — υ υ χ (cid:0) a c + y h Tχ (cid:1) υ υ χ (cid:0) a c + y h Tχ (cid:1) | υ χ (cid:0) c + h T χ (cid:1) . (48)10he above mass matrix can be written as M U = (cid:18) A CC T D (cid:19) , (49)which has the same structure as the general form of equation (129) in the Appendix A, where eachblock is: A = 12 υ (cid:0) a + y (cid:1) υ υ ( a a + y y ) 0 υ υ ( a a + y y ) υ (cid:0) a + y (cid:1)
00 0 υ (cid:0) a + a (cid:1) ,C = 12 υ υ χ (cid:0) a c + y h Tχ (cid:1) υ υ χ (cid:0) a c + y h Tχ (cid:1) ,D = 12 υ χ (cid:0) c + h T χ (cid:1) . (50)We can see that each block are of the order A ∼ υ , , C ∼ υ , υ χ and D ∼ υ χ , respectively, obeyingthe hierarchy from equation (130). Thus, according to Appendix A, the mass matrix (49) can be blockdiagonalized as m U = (cid:16) V ( U ) L (cid:17) T M U V ( U ) L = (cid:18) m U m T (cid:19) , (51)where: m U ≈ A − CD − C T ,m T ≈ D, (52)and the rotation matrix has the approximated form: V ( U ) L ≈ (cid:18) I F U − F TU I (cid:19) , F U ≈ CD − . (53)Since the block D is just a number (see equation (50)), from (52) we obtain directly the mass ofthe heavy T quark: m T ≈ υ χ (cid:0) c + h T χ (cid:1) . (54)On the other hand, from the matrices in (50), and after some algebra, the matrix m U in (52), whichcontains the SM sector, can be put into the form: m U ≈ υ r υ υ r r υ υ r r υ r
00 0 υ (cid:0) a + a (cid:1) , (55)where: r = (cid:0) a h Tχ − y c (cid:1)(cid:113) c + h T χ ,r = (cid:0) a h Tχ − y c (cid:1)(cid:113) c + h T χ . (56)11e see that the 33 component of (55) appears decoupled, which corresponds to one of the eigenvalues.We associate this component to the top quark: m t = 12 υ (cid:0) a + a (cid:1) , (57)which leaves us with the 2 × m uc ≈ (cid:18) υ r υ υ r r υ υ r r υ r (cid:19) . (58)It is evident that the above matrix has null determinant, which leads us to at least one null eigenvalue.In fact, this structure produces one massless quark, which we associate to the lightest quark: the upquark ( u ), while the other eigenvalue, associated to the charm quark, corresponds to the trace of thematrix: m c = Tr[ m uc ] = 12 (cid:0) υ r + υ r (cid:1) ≈ υ r , (59)Since the mass of the top quark in (57) depends only on υ , we take υ (cid:28) υ , which leads us to theapproximation in equation (59).In order to generate mass to the u quark, we consider the one-loop radiative correction shown infigure 1-(a). This contribution add an input into the 11 component in the original 4 × M (cid:48) U in(47), which produces the one-loop quadratic mass matrix M U (1) = M U + ∆ M U , (60)where the small one-loop contribution is:∆ M U = 12 υ Σ υ a Σ |
00 0 0 | υ a Σ |
0— — — — —0 0 0 | , (61)and Σ the value of the diagram in figure 1-(a) which obey the following analytical expression:Σ = − π f (cid:48) (cid:0) h Uσ (cid:1) (cid:0) h T (cid:1) √ M T C (cid:18) M M T , M σ M T (cid:19) , (62)where: C ( x , x ) = 1(1 − x ) (1 − x ) ( x − x ) (cid:20) x x ln (cid:18) x x (cid:19) − x ln x + x ln x (cid:21) , (63)and M is a charateristic mass arised from the internal φ line as linear combinations of mass eigevalues.The new one-loop contribution only has effect on the 3 × m U in (55), which changeinto the one-loop mass matrix m U (1-loop) ≈ υ r + υ Σ υ υ r r υ a Σ υ υ r r υ r υ a Σ m t , (64)where m t is the top mass at tree level obtained in (57). The new 13 component emerged from the 1loop diagram will correct the top mass. However, we will neglect this correction, which leads us againto a 2 × m uc (1-loop) ≈ (cid:18) υ r + υ Σ υ υ r r υ υ r r υ r (cid:19) , (65)12igure 1: Mass one-loop correction for ( a ) up and ( b ) down sector, where k, l, m, n = 1 , j = 1 , , which exhibits determinant different from zero. The trace of the matrix corresponds to the sum of theeigenvalues, i.e.: Tr[ m uc (1-loop) ] = m u + m c = 12 (cid:0) υ r + υ r (cid:1) + 12 υ Σ . (66)If we approximate the mass of the charm quark according to (59), we obtain for the quark u that: m u = 12 υ Σ . (67) Down sector
For the down-type matrix M (cid:48) D in (47), for simplicity we take in the heavy sector, proportional to υ χ ,a diagonal form, i.e., k ij = 0 for i (cid:54) = j . In this scenery, its quadratic form can also be put in the blockform M D = (cid:18) A CC T D (cid:19) , (68)where: A = 12 υ (cid:0) j + j (cid:1) υ υ ( j j + j j ) 0 υ υ ( j j + j j ) υ (cid:0) j + j (cid:1)
00 0 υ (cid:0) B + B + B (cid:1) ,C = 12 υ υ χ j k υ υ χ j k υ υ χ j k υ υ χ j k ,D = υ χ (cid:18) k k (cid:19) . (69)After block diagonalization, the matrix become: m D = (cid:16) V ( D ) L (cid:17) T M D V ( D ) L = (cid:18) m D m J (cid:19) , (70)where: m D ≈ A − CD − C T ,m J ≈ D, (71)13ith: V ( D ) L ≈ (cid:18) I F D − F TD I (cid:19) , F D ≈ CD − . (72)First, since the matrix D appears diagonal, we obtain directly the mass of the heavy down-type quarks: m J = 12 υ χ k , m J = 12 υ χ k . (73)Second, for the SM down sector, the matrix m D in (71) gives: m D = 12 υ (cid:0) B + B + B (cid:1) , (74)which exhibits two massless quarks: the down ( d ) and strange ( s ) quarks, and one massive quarkassociated to the bottom ( b ): m b = 12 υ (cid:0) B + B + B (cid:1) . (75)In order to obtain mass for d and s , we again consider the one-loop contribution shown in figure 1-(b),which produces new entrances different from zero in (74) as follows: m D (1-loop) =12 υ (cid:0) Σ + Σ + Σ (cid:1) υ υ (Σ Σ + Σ Σ + Σ Σ ) υ (Σ B + Σ B + Σ B ) ∗ υ (cid:0) Σ + Σ + Σ (cid:1) υ υ (Σ B + Σ B + Σ B ) ∗ ∗ m b , (76)where the one-loop correction is:Σ lj = − π f (cid:48) (cid:0) h Jl (cid:1) lm (cid:0) h Dσ (cid:1) nj √ M J C (cid:18) M l M J , M σ M J (cid:19) . (77)If the matrix in (76) is grouped as m D (1-loop) = (cid:18) m nn T m b (cid:19) , (78)where the bottom mass is dominant, we can block diagonalize it as: R TL m D (1-loop) R L ≈ (cid:18) m ds
00 2 m b (cid:19) , (79)with: m ds = m − nn T m b = 12 m b (cid:18) s υ s υ υ s υ υ s υ (cid:19) , (80)and s = (Σ B − Σ B ) + (Σ B − Σ B ) + (Σ B − Σ B ) ,s = (Σ B − Σ B ) + (Σ B − Σ B ) + (Σ B − Σ B ) ,s = B (Σ Σ + Σ Σ ) + B (Σ Σ + Σ Σ ) + B (Σ Σ + Σ Σ ) − B B (Σ Σ + Σ Σ ) − B B (Σ Σ + Σ Σ ) − B B (Σ Σ + Σ Σ ) . (81)14he eigenvalues of m ds in (80) will lead us to the down and strange masses. For example, if the mixingcomponent s is null, we obtain: m d ≈ s υ m b ,m s ≈ s υ m b . (82) The non-universal U (1) X also forbids some Yukawa couplings between leptons and scalar bosons. Theallowed couplings are shown below for neutral and charged leptons, respectively: −L Y,N = h νe e (cid:96) eL ˜ φ ν eR + h νµ e (cid:96) eL ˜ φ ν µR + h ντ e (cid:96) eL ˜ φ ν τR + h νe µ (cid:96) µL ˜ φ ν eR + h νµ µ (cid:96) µL ˜ φ ν µR + h ντ µ (cid:96) µL ˜ φ ν τR + h νjχi ν i CR χ ∗ N R + 12 N i CR M ijN N jR + h . c ., (83) −L Y,E = η(cid:96) eL φ e µR + h(cid:96) µL φ e µR + ζ(cid:96) τL φ e eR + H(cid:96) τL φ e τR + q (cid:96) eL φ E R + q (cid:96) µL φ E R + h Eσe E L σe eR + h E σµ E L σ ∗ e µR + h Eστ E L σe τR + H E L χE R + H E L χ ∗ E R + h . c . (84)Since the Higgs doublet φ has the discrete symmetry φ → − φ , all the right-handed leptonsexcept E R and E R also have Z negative parities in order to obtain the adequate zero textures, i.e.: e e,µ,τR → − e e,µ,τR , ν e,µ,τR → − ν e,µ,τR , N e,µ,τR → − N e,µ,τR . (85) Neutral leptons
Evaluating in the VEVs, the terms obtained from (83) can be written in the following mass term usingthe basis N L = (cid:16) ν e,µ,τL , ( ν e,µ,τR ) C , ( N e,µ,τR ) C (cid:17) T for the neutral sector − L Y,N = 12 N CL M ν N L , (86)where the mass matrix is M ν = m T D m D M T D M D M M , (87)with M D = h νχ v χ / √ ν cR and N R , where h Nχ is a 3 × m D = v √ h νe e h νµ e h ντ e h νe µ h νµ µ h ντ µ , (88)is a Dirac mass matrix between ν L and ν R . M M is the mass of the Majorana neutrino N R .Considering that M M (cid:28) m D and M D , the matrix M ν can be diagonalized through the inverseseesaw mechanism [26, 27]. If the following blocks are defined M ν = (cid:18) m D (cid:19) , M N = (cid:18) M T D M D M M (cid:19) , (89)the mass matrix becomes M ν = (cid:18) M T ν M ν M N (cid:19) , (90)15hich has the same form as the block matrix (129) from Appendix A in the limit with A = 0. Thus,we define the rotations W SST M ν W SS = (cid:18) m light m heavy (cid:19) , (91)with W SS ≈ (cid:18) I F N − (cid:0) F N (cid:1) T I (cid:19) , F N ≈ ( M N ) − M ν , (92)and m light ≈ −M T ν M − N M ν , (93) m heavy ≈ M N . (94)Since M − N = (cid:32) − ( M D ) − M M (cid:0) M T D (cid:1) − M − D (cid:0) M T D (cid:1) − (cid:33) , (95)the light mass term is m light = m T D ( M D ) − M M (cid:0) M T D (cid:1) − m D . (96)Now, a unitary matrix V is considered which diagonalizes the 3 × M N [27]: V T M N V = V T (cid:18) M D M T D M N (cid:19) V = (cid:18) V ∗ M diag1 V † V ∗ M diag2 V † (cid:19) , (97)with V and V sub-rotation matrices. V may be formally expressed as [27] V = 1 √ (cid:18) − (cid:19) (cid:32) − SS † S − S † − S † S (cid:33) . (98)Using (97), and assuming that M D = M T D , M M S † = S T M M , M M S = S ∗ M M , M D S † = S T M D and M D S = S ∗ M D , from the off-diagonal elements we find S = S † = − M − D M M , (99)and substituting for the diagonal elements, we get the mass matrices V ∗ M diag1 V † = M M − M D − M M M − D M M ≈ − M D , (100) V ∗ M diag2 V † = M M M D + 18 M M M − D M M ≈ M D . (101)The mass eigenstates n L are constructed as: N L = U N n L , (102)with n L = (cid:16) ν , , L , N , , L , N , , L (cid:17) , and the rotation matrix as U N = W SS W H W B , (103)with W SS the seesaw matrix rotation from (92), W H = (cid:18) V (cid:19) (104)the matrix rotation of the heavy neutrinos, and W B = block diag ( U ν , V , V ) (105)the matrices that diagonalize each 3 × Mass one-loop correction for charged leptons, where n = e, τ and k = e, µ . Charged leptons
For the charged sector in the flavor basis E = ( e e , e µ , e τ , E ), the mass terms obtained from (84) afterthe symmetry breaking are − L Y,E = E L M E E R + H v χ √ E L E R + h . c ., (106)where the lepton mass matrix M E has de following form: M E = v √ η | q t β h | q t β ζ H | − − − − − | H v χ /v , (107)which exhibits one massless lepton (the electron). To obtain a massive electron, we include the one-loopcorrection shown in figure 2, which add a new term M E (1) = M E + ∆ M E , (108)with: ∆ M E = υ Σ | |
00 0 0 |
0— — — — —0 0 0 | . (109)Since M E (1) is not hermitian, there are two rotation matrices V EL and V ER for left- and right-handed electrons. Hence, the left-handed rotation is obtained by diagonalizing M E M † E obtaining thecorresponding eigenvalues m e = h Σ v η + h ) ≈ v ,m µ = v (cid:0) η + h (cid:1) ≈ v h ,m τ = v (cid:0) ζ + H (cid:1) ≈ v H ,m E = H v χ . (110)In addition, the flavor eigenstates are related to mass eigenstates e = ( e, µ, τ, E (cid:48) ) T by:17 L = V EL e L (111)where the corresponding left-handed rotation matrix can be expressed as: V EL = V E SS ,L V E SM ,L , (112)which diagonalizes as: M E M † E = 12 (cid:18) M ee M eE M eE M EE (cid:19) , (113)whose blocks are M ee = v q t β + η + Σ + Σ q q t β + hη + Σ Σ + Σ Σ ζ Σ + H Σ ∗ q t β + h + Σ + Σ ζ Σ + H Σ ∗ ∗ H + ζ , M eE = v v χ H q q , M EE = v χ H . (114)The former matrix V E SS ,L is V E SS ,L = (cid:18) I F E − F E † I (cid:19) , (115)with F E = M eE (cid:0) M EE (cid:1) − . The latter rotation is: V E SM ,L = (cid:18) V E SM ,L
00 1 (cid:19) , (116)where the top-left block diagonalizes the SM charged lepton masses V E SM ,L = c α eµ s α eµ Σ H − s α eµ c α eµ Σ H − Σ H − Σ H . (117)The angle α eµ is defined by t α eµ = tan α eµ ≈ η/h , which is a free parameter of the model as shownbelow. To explore some phenomenological consequences of the above structures, we assume for simplicity that M D is diagonal and M M is proportional to the identity M D = h Nχ h Nχ
00 0 h χN v χ √ M M = µ N I × . (119)Thus, V = V = I × in (97). On the other hand, replacing the Dirac matrix from (88) into the lightmass eigenvalues in (96), we obtain m light = µ N v h Nχ v χ ( h νe e ) + (cid:0) h νe µ (cid:1) ρ h νe e h νµ e + h νe µ h νµ µ ρ h νe e h ντ e + h νe µ h ντ µ ρ h νe e h νµ e + h νe µ h νµ µ ρ ( h νµ e ) + (cid:0) h νµ µ (cid:1) ρ h νµ e h ντ e + h νµ µ h ντ µ ρ h νe e h ντ e + h νe µ h ντ µ ρ h νµ e h ντ e + h νµ µ h ντ µ ρ ( h ντ e ) + (cid:0) h ντ µ (cid:1) ρ , (120)18here ρ = h Nχ /h Nχ . The matrix m light has zero determinant, obtaining at least, one masslessneutrino. The above matrix is diagonalized through U T ν m light U ν = m diaglight , (121)where U ν contains the mixing angles that transform the weak eigenstates ν e,µ,τL into mass eigenstates ν , , L . The PMNS matrix is defined as the product of the above rotation matrix and the rotationmatrix of the charged sector V E SM ,L U PMNS = (cid:0) V E SM ,L (cid:1) † U ν . (122)We use the following parametrization for the PMNS matrix [44]: U PMNS = c c s c s e − iδ − s c − c s s e iδ c c − s s s e iδ s c s s − c c s e iδ − c s − s c s e iδ c c . (123)The mixing angles can be obtained from some matrix componenets as s = | U e | ,s = | U µ | − | U e | ,s = | U e | − | U e | . (124) Parameter values
In order to has a model consistent with neutrino oscillation data [22], the values of the Yukawaparameters h νe e , h νµ e , h ντ e , h νe µ , h νµ µ , h ντ µ and α eµ must be properly adjusted. To achieve this, weimplement a MonteCarlo method to generate random numbers in the parameter space, where onlythe numbers which match up the mass matrix to experimental data are accepted, while the othersare rejected. It is worth mentioning that the other two rotation parameters described by Σ /H andΣ /H were approximated to m e /m τ , while h νµ e was chosen null to simplify the search.On the other hand, the appropriate mass scale and mass ordering can be obtained by adjusting theouter factor of the mass matrix and the ratio ρ . For NO the Yukawa coupling can be set by h Nχ = 0 . ,ρ = 0 . , (125)while for IO h Nχ = 0 . ,ρ = 0 . . (126)In the same way, the mass scale is set by v = 7 GeV ,v χ = 7 TeV ,µ N = 1 keV . (127)The above values fix the outer factor of the mass matrix (120) at 50 meV, which yields to the correctsquared-mass differences. Nevertheless, there exist other possible values for the parameters µ N , h N χ , v χ and tan β that lead us the factor at 50 meV.If the following constraint is assumed µ N v h Nχ v χ = 50 meV , (128)19 v v χ (a) NO: h Nχ = 0 . v v χ (b) NO: h Nχ = 0 . v v χ (c) NO: h Nχ = 1 . Figure 3: Contour plots of v χ vs. v from eq. (128) for different values of h Nχ and µ N . From belowto above there are the corresponding contour plots for the following values of µ N : 500 eV (gray, line),1 keV (black, line), 5 keV (gray, dashed), 10 keV (black, dashed), 50 keV (gray, dot-dashed) and 100keV (black, dot-dashed). α eµ = 0 o α eµ = 15 o α eµ = 30 o h νe e . → .
278 0 . → .
299 0 . → . h νµ e − . → − . − . → − . − . → − . h νµ µ − . → − . − , → − . − . → − . h ντ e . → .
748 0 . → .
677 0 . → . h ντ µ . → .
462 0 . → .
460 0 . → . σ neutrino oscillation data for NO reported by [22]. h νe µ = 0 for simplifying the MonteCarlo search. α eµ = 0 o α eµ = 1 o α eµ = 2 o h νe e . → .
107 1 . → .
105 1 . → . h νµ e − . → − . − . → − . − . → − . h νµ µ . → .
060 0 . → .
070 1 . → . h ντ e . → .
127 0 . → .
138 0 . → . h ντ µ . → .
030 0 . → .
010 0 . → . σ neutrino oscillation data for IO reported by [22]. h νe µ = 0 for simplifying the MonteCarlo search. 20ontour plots can be done for different values of µ N in the v χ vs. v plane, as shown in figure 3.The tables 4 and 5 show regions where the neutrino Yukawa couplings and the angle α eµ makeconsistent this model with neutrino oscillation data reported by [22] at 3 σ .The Yukawa coupling h χN is not fixed by oscillations of the light neutrinos; however, they maycontribute into the total rotation matrix U N in (103). Thus, the neutral spectrum of the model iscomposed by three active light neutrinos ν , , L and six quasidegenerated steril neutrinos N , , L and N , , L at the TeV scale. Abelian nonuniversal gauge extensions of the SM are very well-motivated models which involve a widenumber of theoretical aspects. In this work, by requiring nonuniversality in the left-handed quarksector and in lepton sector, we propose a new G SM × U(1) (cid:48) gauge model. We obtained a free-anomalytheory with invariant Yukawa interactions, predicting hierarchical mass structures in the quark andcharged lepton sector with few free parametersFor the quark sector, we identify three energy scales. First, at the breaking scale of the U (1) X symmetry, we obtain heavy masses to the extra heavy quarks J n and T , with M J n ≈ M T ∼ υ X .Second, at tree level, we obtain masses at the electroweak scale for the c , t and b quarks, with M c,t,u ∼ υ , . Finally, at one-loop level, we obtain light masses for the u , d and s quarks, with M u,d,s ∼ υ , /υ χ .For the leptonic sector, we also obtain the same hierarchical structure, where the extra leptons E and E acquire masses at the υ χ scale, the µ and τ have masses at the electroweak scale, and the electronobtain masses at one-loop, which is suppressed as υ , /υ χ .On the other hand, with the addition of extra Majorana neutrinos, we found that neutrinos mayacquire tiny masses via the inverse seesaw mechanism. The selection of a small Majorana mass term(from eV to KeV scale) and the experimental limits on observables from neutrino oscillations allowsus to perform numerical adjustment for the values of the Yukawa couplings of neutrinos in NO and IOscenarios. In addition, because the non-universal U(1) X charges, the electron remains massless at treelevel but a non-vanishing mass term emerges at one-loop corrections which gives a viable explanationfor its small mass compared to the electroweak scale. Acknowledgment
This work was supported by
El Patrimonio Autonomo Fondo Nacional de Financiamiento para laCiencia, la Tecnolog´ıa y la Innovaci´on Francisco Jos´e de Caldas programme of COLCIENCIAS inColombia. RM thanks to professor Germ´an Valencia for the kindly hospitality at Monash Universityand his useful comments.
A Block Diagonalization
Let us take a generic matrix with arbitrary dimension of the form: M = (cid:18) A CC T D (cid:19) , (129)with A, D and C sub-matrices whose elements obey the hierarchy A (cid:28) C (cid:28) D. (130)The matrix (129), as shown in reference [46], can be block diagonalized approximately by a unitaryrotation of the form: V = (cid:18) I F − F T I (cid:19) , (131)where I is an identity matrix, and F a small sub-rotation with F (cid:28)
1. Keeping only up to linearterms on F , the rotation gives: V T M V = (cid:18) A − CF T − F C T C + AF − F DC T + F T A − DF T D + C T F + F T C (cid:19) , (132)21hich, by definition, must lead us to a diagonal block form m = (cid:18) a d (cid:19) , (133)with a and d non-diagonal matrices, and 0 the null matrix. By matching the upper right non-diagonalblock in (132) and (133), we obtain that C + AF − F D = 0. Taking into account the hierarchy in(130), we may neglect the term with A , finding the following approximate solution: F ≈ CD − . (134)On the other hand, if we match the diagonal blocks in (132) and (133), and using the solution(134), we can obtain the form of the submatrices a and b in terms of the original blocks A , C and D .We obtain at dominant order that: a ≈ A − CD − C T b ≈ D. (135)The above matrices can be diagonalized independently. References [1] S.L. Glashow, Nucl. Phys. 22, 579 (1961); S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967); A.Salam, in
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