Gaussian Froggatt-Nielsen mechanism on magnetized orbifolds
Hiroyuki Abe, Tatsuo Kobayashi, Keigo Sumita, Yoshiyuki Tatsuta
aa r X i v : . [ h e p - ph ] M a y WU-HEP-14-06EPHOU-14-012
Gaussian Froggatt-Nielsen mechanismon magnetized orbifolds
Hiroyuki Abe , ∗ , Tatsuo Kobayashi † , Keigo Sumita , ‡ and Yoshiyuki Tatsuta , § Department of Physics, Waseda University, Tokyo 169-8555, Japan Department of Physics, Hokkaido University, Sapporo 060-0810, Japan
Abstract
We study the structure of Yukawa matrices derived from the supersymmetric Yang-Millstheory on magnetized orbifolds, which can realize the observed quark and charged leptonmass ratios as well as the CKM mixing angles, even with a small number of tunable parame-ters and without any critical fine-tuning. As a reason behind this, we find that the obtainedYukawa matrices possess a Froggatt-Nielsen like structure with Gaussian hierarchies, whichprovides a suitable texture for them favored by the experimental data. ∗ E-mail address: [email protected] † E-mail address: [email protected] ‡ E-mail address: [email protected] § E-mail address: y [email protected] ontents
Introduction
The standard model (SM) is a successful theory describing all the elementary particles discov-ered so far. However, it is still mysterious why it has such a complicated structure, e.g., theproduct gauge group, the three generations and the hierarchical Yukawa couplings required byobservational results. Extra dimensional space would give an origin of the flavor structures, andthat has been studied with a great variety. In particular, we focus on toroidal compactificationsof higher-dimensional supersymmetric Yang-Mills (SYM) theories with magnetic fluxes on thetorus, which induce the four-dimensional (4D) chiral spectra, like the SM [1, 2].So far, some phenomenological studies of the higher-dimensional SYM theories with mag-netized extra dimensions have been done. In the works, we constructed a phenomenologicalmodel [3] starting from the ten-dimensional (10D) SYM theory with a certain flux configu-ration, in which the magnetic fluxes on three factorizable two-dimensional (2D) tori give thesemi-realistic flavor structures of the SM. Magnetic fluxes in extra dimensional space breakthe supersymmetry (SUSY) in general. The flux configuration studied in Ref. [3] preserves a N = 1 SUSY as a part of the N = 4 SUSY possessed by the 10D SYM theory with respect tothe 4D supercharges, and we constructed the model within a framework of the low-scale SUSYbreaking scenario. We have also searched the other flux configurations [4] and found that anyother choice of factorizable magnetic fluxes would not induce plausible three generation struc-tures even if we abandon preserving the N = 1 SUSY. Therefore the realistic three generationspectrum is uniquely determined to the one given in Ref. [3] from the 10D magnetized SYMon factorizable tori.The situation changes if the chirality projections due to Yang-Mills fluxes are combinedwith those of the background geometry. The most simple but illustrating example to realizesuch a situation is magnetized orbifold models [5, 6, 7, 8]. The orbifold projections alter therelation between the magnitude of magnetic fluxes and the number of degenerate zero-modesidentified with the number of generations. That is attractive because it provides a rich varietyof phenomenological models obtained from the magnetized SYM theories. However, we foundthat any flux configurations on orbifolds yielding three generations of quarks and leptons do notpreserve 4D N = 1 SUSY [4]. That is, the magnetic fluxes on orbifold backgrounds break theSUSY corresponding to that of the MSSM at the compactification scale as far as those triggerthe SM flavor structure. We think these facts are strong phenomenological predictions derivedfrom the toroidal compactifications of 10D SYM theory. Therefore it would be important tostudy such a non-SUSY orbifold background, which can generate Yukawa structures differentfrom those shown in Ref. [3].In this paper, we construct a magnetized orbifold model with a high-scale SUSY breakingcaused by the magnetic fluxes on orbifolds. First, we focus on the structure of 2D orbifoldbecause all the three generation structures of the SM should arise on a single 2D tori to getfull rank Yukawa matrices [6] and that is adequate to discuss the Yukawa structures. We willidentify a flux configuration whose predicted values of the quark and charged lepton massesand the CKM mixing angles [9] are consistent with the experimental data. The resultantmodel is quite interesting and attractive because it has a smaller number of continuous free We call fluxes factorizable if they do not cross over two tori.
We give a brief review of the 10D SYM theory with the magnetized and orbifold backgrounds.We start from the 10D U ( N ) SYM theory, S = Z d x √− G (cid:26) − g tr (cid:0) F MN F MN (cid:1) + i g tr (cid:0) ¯ λ Γ M D M λ (cid:1)(cid:27) , compactified on a product of 4D Minkowski space and three factorizable tori, R , × ( T ) .The covariant derivative D M and the field strength F MN contain the 10D vector field A M with M, N = 0 , , · · · , λ is the 10D Majorana-Weyl spinor field. The 10D metric G MN whosedeterminant is denoted by G contains three torus metrics, g ( i ) = (cid:0) πR ( i ) (cid:1) (cid:18) τ ( i ) Re τ ( i ) (cid:12)(cid:12) τ ( i ) (cid:12)(cid:12) (cid:19) , for i = 1 , ,
3, where real parameters R ( i ) and complex parameters τ ( i ) correspond to the sizesand shapes of the i -th torus T , respectively. The 10D fields are decomposed as λ (cid:0) x µ , y k (cid:1) = X l,m,n χ lmn ( x µ ) ⊗ ψ (1) l (cid:0) y , y (cid:1) ⊗ ψ (2) m (cid:0) y , y (cid:1) ⊗ ψ (3) n (cid:0) y , y (cid:1) ,A M (cid:0) x µ , y k (cid:1) = X l,m,n ϕ lmn,M ( x µ ) ⊗ φ (1) l,M (cid:0) y , y (cid:1) ⊗ φ (2) m,M (cid:0) y , y (cid:1) ⊗ φ (3) n,M (cid:0) y , y (cid:1) , where ψ ( i ) l ( y i , y i ) is a 2D spinor corresponding to the l -th mode on the i -th T . Weconcentrate on the zero modes l = 0 in the following and denote them as ψ ( i )0 = ψ ( i )+ ψ ( i ) − ! . We introduce factorizable (Abelian) magnetic fluxes F , , F , and F , on the tori, whichhave typically the following forms, F i, i = π M N . . . M n N n , M k ∈ Z because of the Dirac’s quantization condition and N k denotes the ( N k × N k )unit matrix. These fluxes break the U ( N ) symmetry down to U ( N ) × · · · × U ( N n ). Then, wedenote matter fields with the bifundamental representation ( N a , ¯ N b ) under the unbroken gaugegroup U ( N a ) × U ( N b ) as ψ ( i ) ± ab . Using the gamma matricesΓ i = (cid:18) (cid:19) , Γ i = (cid:18) − ii (cid:19) , the Dirac equations for ψ ( i ) ± ab on the i -th T are given by " ¯ ∂ i + πM ( i ) ab τ ( i ) z i ψ ( i )+ ab = 0 , (1) " ∂ i − πM ( i ) ba τ ( i ) ¯ z i ψ ( i ) − ab = 0 , (2)where M ( i ) ab = M ( i ) a − M ( i ) b , z i = y i + τ ( i ) y i and ∂ i = ∂/∂z i .In Eq. (1), degenerate zero-modes can be obtained with the number of degeneracy M ( i ) ab if M ( i ) ab > ψ ( i )+ ab,I = Θ I,M ( i ) ab (cid:0) z i , τ ( i ) (cid:1) , (3)where Θ I,M ( z, τ ) = N I · e iπMz Im z/ Im τ · ϑ " I/M ( M z, M τ ) ,ϑ " ab ( ν, τ ) = X l ∈ Z e πi ( a + l ) τ e πi ( a + l )( ν + b ) , and normalization factors N I are determined as Z T d z Θ I,M (cid:0) Θ J,M (cid:1) ∗ = δ IJ . (4)The index I in Eq. (3) labels degenerate zero-modes, I = 0 , , · · · , M ( i ) ab −
1. In this case,the conjugate field, ψ ( i )+ ba , has no zero-modes because of the negative fluxes, M ( i ) ba <
0. Asfor the other equation (2), ψ ( i ) − ab has | M ( i ) ab | degenerate zero-modes and their wavefunctions aregiven by (cid:0) Θ I,M (cid:1) ∗ if M ( i ) ab <
0, and then ψ ( i ) − ba has no zero-modes. Note that, in the case with M ( i ) ab = 0, each of ψ ( i )+ ab , ψ ( i )+ ba , ψ ( i ) − ab and ψ ( i ) − ba has a single zero-mode and their wavefunctionsare constant on the torus. The total degeneracy is given by Y k =1 (cid:12)(cid:12)(cid:12) M ( k ) ab (cid:12)(cid:12)(cid:12) , M ( k ) ab = 0 and such degenerate zero-modes can be identified as thegenerations of the SM particles. The magnetic fluxes can give a three generation structure.Furthermore, each wavefunction of degenerate zero-modes is localized at a different pointfrom each other by magnetic fluxes and overlap integrals of them on the torus give their Yukawacoupling constants. The Yukawa matrices can be hierarchical depending on the flux configura-tions.We show them up to complex phases and global factors including the 10D gauge coupling g as Y IJ K = λ (1) I J K λ (2) I J K λ (3) I J K , (5) λ ( i ) I i J i K i = Z T d z i Θ I i ,M ( i ) ab ( z i , τ ( i ) )Θ J i ,M ( i ) bc ( z i , τ ( i ) ) (cid:16) Θ K i ,M ( i ) ac ( z i , τ ( i ) ) (cid:17) ∗ , (6)where I = ( I , I , I ), J = ( J , J , J ), and K = ( K , K , K ). Thus the Yukawa couplings aregiven by the overlaps of three Jacobi-theta functions ϑ . Hereafter we often omit the index i forthe i -th torus, when it is clear that we concentrate on one T .This overlap integral (6) can be performed analytically by using the decomposition,Θ I,M ab ( z, τ )Θ J,M bc ( z, τ )= N I N J N K X m ∈ Z Mac Θ I + J + M ab m, M ac ( z, τ ) × ϑ " M bc I − M ab J + M ab M bc mM ab M bc M ac (0 , τ M ab M bc M ac ) , with the normalization (4), and we obtain λ IJK = N I N J N K X m ∈ Z Mac ϑ " M bc I − M ab J + M ab M bc mM ab M bc M ac (0 , τ M ab M bc M ac ) × δ I + J + M ab m,K , (7)where K is defined up to mod M ac .Next, we consider Z projections on the magnetized tori. For instance, we take the orbifold T /Z with coordinates ( y , y ) identified as ( y , y ) → ( − y , − y ) under the Z transformation,and the fields contained in the higher-dimensional SYM theory are projected as A m ( − y , − y ) = − P A m ( y , y ) P for m = 4 , ,A m ( − y , − y ) = + P A m ( y , y ) P for m = 4 , ,ψ ± ( − y , − y ) = ± P ψ ± ( y , y ) P, (8)where P is a projection operator ( P = ). On such an orbifold background, either even or oddzero-modes remain and the number of zero-modes generated by magnetic fluxes are reduced(or vanish). Even and odd wavefunctions on the magnetized background are found as follows[5] Θ I,M even ( z, τ ) = 1 √ (cid:0) Θ I,M ( z, τ ) + Θ M − I,M ( z, τ ) (cid:1) , Θ I,M odd ( z, τ ) = 1 √ (cid:0) Θ I,M ( z, τ ) − Θ M − I,M ( z, τ ) (cid:1) , I,M ( − z ) = Θ M − I,M ( z ). The number of surviving zero-modes canbe counted and that is shown in Table 1. From the table, we find that several magnitudes of M M = 3 could give three generations without orbifoldings.
10D SYM theories have the N = 4 SUSY counted by 4D supercharges and that can be brokenby magnetic fluxes. However, flux configurations satisfying a certain condition can preserve apart of that, the 4D N = 1 SUSY, and then we can derive the MSSM-like models from the10D SYM theory. One of the relevant SUSY preserving conditions is satisfied in the following D -flat direction of the fluxed U (1), A (1) F + 1 A (2) F + 1 A (3) F = 0 , (9)where A ( i ) is an area of the i -th T . This condition guarantees the N = 1 SUSY at thecompactification scale (e.g. GUT scale), and at the same time restricts flux model buildingsstrongly. The D -term contributions to the soft SUSY breaking masses arise to all the scalarscharged under the fluxed U (1) if the magnetic fluxes do not satisfy the condition (9) .We have been trying to construct a model along the D -flat direction (9) on the toroidalbackground [4]. As a result, we find that the model proposed in Ref. [3] is the unique solutionthat has (semi-)realistic spectra compatible with a low scale SUSY breaking scenario. Notethat, in the model, the three generation structure is obtained without orbifold projections andthree-block magnetic fluxes break the U (8) group down to U (4) C × U (2) L × U (2) R , i.e., tothe Pati-Salam gauge group, which is further broken by Wilson-lines. Therefore it has beenconfirmed that no more three-generation models are possible with factorizable fluxes satisfyingEq. (9).Here we remark that the continuous Wilson-lines are forbidden on orbifolds because thecorresponding components of vector fields have Z parity odd. Without Wilson-lines, the fluxesby themselves must further break U (4) C × U (2) L × U (2) R to make a difference between thequarks and the leptons. Then it requires more-than-three blocks structure of the flux matrices,with which the D -flat condition (9) is hardly satisfied with just three torus parameters A ( i ) [4]. See Ref. [10], which gives the manifest description of the N = 1 SUSY in the 10D magnetized SYM theory. Magnetic fluxes other than F , F and F are encoded in the F -term SUSY conditions, and those areassumed to be absent in this paper. N = 1 SUSY.On the other hand, if we abandon the low-scale SUSY in the model building and permitnon-SUSY fluxes out of the D -flat direction (9), then various orbifold models are expected toarise at first glance. However, it is not easy to make such models phenomenologically viable,because Wilson-lines are not allowed on orbifolds as mentioned above, which played a role inRef. [3] to fit the quark/lepton masses/mixings to their experimental values as well as to breakthe Pati-Salam gauge symmetry. Since Wilson-lines shift the localization points of the zero-mode wavefunctions, we can control the size of their overlaps and then order of the magnitudeof their Yukawa couplings. That is, Wilson-lines are significant degrees of freedom to realizeexperimental data in the previous model without orbifoldings.Based on the above discussions, in the following, we try to construct non-SUSY orbifoldmodels with magnetic fluxes yielding realistic flavor structures out of the D -flat direction (9).We will also discuss the effects of D -term contributions to scalar masses of the order of thecompactification scale, that leads to the high-scale SUSY breaking scenario. Here we study model building and derive realistic quark and lepton mass matrices.
In order to obtain full rank Yukawa matrices, three generations of both left- and right-handedmatters must be originated from the same torus. Configurations of magnetic fluxes on theother two tori have to be determined to leave the SM contents unchanged and to eliminateextra fields. As we declared in the previous section, they are no longer restricted by the SUSYcondition (9), that is, we consider the high-scale SUSY breaking scenarios. Since we requirethat the structure of the other two tori must not change or spoil the three-generation structureat all, λ (2) and λ (3) in Eq. (5) cannot carry the flavor index when we consider the case that thethree generations are induced on the first T . Therefore 4D Yukawa couplings are representedby Y IJK = λ (1) IJK λ (2) λ (3) , where λ (2) and λ (3) are O (1) global factors.We restrict ourselves to the single torus for a while, where three degenerate zero-modesarise, and we discuss about the whole extra dimensional space later. As shown in Table 1,three degenerate zero-modes are realized by four types of magnetic fluxes in orbifold models; M = 4 , Z even modes and M = 7 , Z odd modes. Then, magnetic fluxes of theHiggs sector are automatically determined as a consequence of the gauge symmetry and theother consistency if the fluxes of the left- and right-handed sectors are fixed. Twenty fluxconfigurations, which induce the three-generation left- and right-handed matters and certainnumbers of Higgs fields on a single torus, were listed in Ref. [6]. In the following, we constructa model on the orbifold background that contains the three-generation quarks and leptons6imultaneously realizing a (semi-) realistic patterns of the mass ratios and the CKM mixingangles [9], even without the Wilson-line parameters.We start from 10D U(8) SYM theory with the following flux configuration on the first T ( y , y -directions), F = 2 π × × − × . (10)We also have other plausible configurations of magnetic fluxes (summarized in Appendix A).Among them, the configuration (10) is simple and instructive to demonstrate a mechanismwhich is a main proposal of this paper. We concentrate on this one in this paper and the othermodels would be studied elsewhere.The magnetic flux (10) breaks the U (8) group down to SU (3) C × SU (2) L × SU (2) R up to U (1)s and leads to the model shown in Table 2. This table shows magnetic fluxes appearing inthe quark and lepton sectors as well as the Higgs sector and also their Z parities. We knowLeft-handed Right-handed HiggsQuark − − − − T .from Table 1 and 2 that the flux configuration gives the three generations of quarks and leptonsand five generations of Higgs multiplets. The last column in Table 2 shows the magnetic fluxesand the Z parity of the Higgs sector.It is important that the data of Higgs sector is determined by the left and right sectorsto obtain the nonvanishing Yukawa couplings. Such couplings should be Z -even quantities(otherwise vanish) and the sums of magnetic fluxes vanish because of the gauge invariance, M ab + M bc + M ca = 0. We require a difference between the flux configurations of quarks andleptons [11] because their flavor structures are somewhat different from each other. The twodifferent configurations, however, have to lead to the same consistent Higgs sector obtainingthe nonvanishing Yukawa couplings. SU (2) R remains unbroken in this model and the breaking sector (or mechanism) otherthan the magnetic fluxes and orbifoldings (and Wilson-lines of course) is required. We canconsider such an additional sector as extensions of our model. In this paper, we just assumethat the SU (2) R is broken somehow and it gives the different VEVs to the up- and down-typeHiggs fields, because we are focusing on the structures of Yukawa matrices here. Therefore,in this model, the four Yukawa matrices (for U, D, N, E ) have different structures due to theconfigurations of the magnetic fluxes, orbifold projections and VEVs of Higgs fields.We show all the wavefunctions given by this flux configuration in Table 3. Based on thistable, we can evaluate all the Yukawa couplings analytically. We define the following η -function7 u,d L e, ν H0 Θ , √ (Θ , − Θ , ) Θ , √ (Θ , − Θ , ) √ (Θ , − Θ , )1 √ (Θ , + Θ , ) √ (Θ , − Θ , ) √ (Θ , + Θ , ) √ (Θ , − Θ , ) √ (Θ , − Θ , )2 √ (Θ , + Θ , ) √ (Θ , − Θ , ) Θ , √ (Θ , − Θ , ) √ (Θ , − Θ , )3 √ (Θ , − Θ , )4 √ (Θ , − Θ , )Table 3: Wavefunctions of the quarks, leptons and Higgs fields.for the simple description of ϑ -function, which appears in Eq. (7), η N = ϑ " N/M (0 , τ M ) , (11)where M is a product of three fluxes, 5 × ×
12 = 420 for quarks and 4 × ×
12 = 384 forleptons and τ = τ (1) in this section.We have the five Higgs fields in up- and down-sectors respectively, and the Yukawa couplingterms are written by Y IJK H K Q L I Q R J = ( Y IJ H + Y IJ H + Y IJ H + Y IJ H + Y IJ H ) Q L I Q R J . One linear combination of the five can be identified as the SM Higgs fields, and usual Yukawamatrices of the SM are given by a linear combination of the five Yukawa matrices. For thequark sector, the five Yukawa matrices are given with η N as follows [6] Y = 1 √ √ η − η ) √ η − η ) √ η + η ) η − η − η + η η − η − η + η η − η − η + η η − η − η + η η − η − η + η η − η − η + η ,Y = 1 √ √ η − η ) √ η − η ) √ η + η ) η − η − η + η η − η − η + η η − η − η + η η − η − η + η η − η − η + η η − η − η + η ,Y = 1 √ √ η − η ) √ η − η ) √ η − η ) η − η − η + η η − η − η + η η − η − η + η η − η − η + η η − η − η + η η − η − η + η ,Y = 1 √ √ η − η ) √ η − η ) √ η − η ) η − η − η + η η − η − η + η η − η − η + η η − η − η + η η − η − η + η η − η − η + η ,Y = 1 √ √ η − η ) √ η − η ) √ η − η ) η − η − η + η η − η − η + η η − η − η + η η − η − η + η η − η − η + η η − η − η + η . Y = y b − y l √ ( y e − y i ) 0 − y f y h , Y = y c − y k √ ( y b − y h ) 0 √ ( y f − y l )0 0 0 ,Y = − y j y d √ ( y a − y m ) 0 y d − y j , Y = √ ( y f − y l ) 0 √ ( y b − y h )0 y c − y k ,Y = y h − y f √ ( y e − y i ) 0 − y l y b , where y a = η + η + η + η , y b = η + η + η + η ,y c = η + η + η + η , y d = η + η + η + η ,y e = η + η + η + η , y f = η + η + η + η ,y g = η + η + η + η , y h = η + η + η + η ,y i = η + η + η + η , y j = η + η + η + η ,y k = η + η + η + η , y l = η + η + η + η ,y m = η + η + η + η . Note again that the different VEVs of up- and down-type Higgs yield the difference betweenup- and down-type Yukawa matrices.
The tunable continuous parameters relevant to the generation structure in the 6D model withthe flux configuration (10) are the complex structure of the torus τ (1) and the VEVs v ui ≡ h H ui i and v di ≡ h H di i of the five pairs of up- and down-type Higgs fields H ui and H di ( i = 0 , , . . . , to fit the resultant quark and charged leptonmass ratios as well as the CKM mixing angles to the observed values. If we assume somenonperturbative and/or higher-order effects to generate Majorana mass terms for the right-handed neutrinos, the neutrino masses as well as the lepton mixing matrix can be also analyzed,which is beyond the scope of this paper. We do not take renormalization group effects onYukawa couplings into account, those are negligible and dependent to the details of the SU (2) R (as well as of the SUSY) breaking sector. We do not treat complex phases of Yukawa couplings by setting Re τ (1) = 0 and also the (generation-independent) global factors of them are assumed to be of O (1). See e.g. [13].
9e first study the structures of the Yukawa matrices analytically, and utilize η N withIm τ (1) = 1 .
5, which is defined in Eq. (11), η N = ϑ " N/M (0 , . M i ) = X l ∈ Z e − . π ( N /M +2 Nl + Ml ) , where M = 420 for quarks and M = 384 for leptons. For a large M such as M = O (100),contributions from the higher modes l = 0 are strongly suppressed and we can use the followingapproximation, η N ∼ e − . πM N . (12)It is obvious that η N is larger than η N ′ for N < N ′ , which allows us to rewrite the five Yukawamatrices for the quark sector as Y ∼ η − η η − η η − η − η − η η , Y ∼ − η η η η − η η − η − η − η ,Y ∼ η − η η − η η − η η − η η , Y ∼ η η − η η − η η − η η − η ,Y ∼ η − η η η η − η η η η , (13)up to O (1) factors.Note that the ( Y ) ij matrix has the hierarchy( Y ) ij ≪ ( Y ) kℓ , (14)for i ≤ k and j ≤ ℓ , and the ( Y ) ij matrix has a similar hierarchy. (There are a few exceptionalentries with the smaller values.) That is quite useful to realize both the hierarchical quarkmasses, m u ≪ m c ≪ m t and m d ≪ m s ≪ m b , and small mixing angles observed by theexperiments. On the other hand, the ( Y ) ij and ( Y ) ij matrices have the hierarchy opposite tothe above. In the ( Y ) ij matrix, the (2 ,
2) entry as well as the (1 ,
3) and (3 ,
1) entries are largecompared with those in ( Y ) ij and ( Y ) ij , and that would be useful to make the correspondingentries in the mass matrices larger. Thus, the matrices, ( Y ) ij and ( Y ) ij , as well as ( Y ) ij areinteresting. Indeed, we find that Y ≡ Y + Y is estimated as Y ∼ η − η − η η − η η η η η . (15)Interestingly, this matrix can be approximated by Y (G) ij = e − c ( a i + b j ) , a i and b j , where a i ( b j ) depends on only the left-handed (right-handed)flavors. It looks like the Froggatt-Nielsen (FN) form [14] , i.e., Y (FN) ij = e − c ( a ′ i + b ′ j ) , (16)up to O (1) factors, but Y (G) ij is a Gaussian form for a i + b j . In the FN form, a ′ i and b ′ j correspondto some kind of quantum numbers of left-handed and right-handed fermions, respectively. TheFN form has been extensively studied, because it is very useful to obtain realistic values of quarkmasses and mixing angles.For simplicity, let us concentrate on the (2 ×
2) low right matrix, Y (FM)2 − = Y (FN)22 Y (FN)23 Y (FN)32 Y (FN)33 ! , (17)where Y (FN)22 ≤ Y (FN)23 , ≤ Y (FN)33 . In this case, the mass ratio m /m is estimated by m /m ∼ Y (FN)23 Y (FN)32 / ( Y (FN)33 ) , and the (2,3) entry of the diagonalizing matrix V ij for the left-handedsector is estimated as V ∼ Y (FN)23 /Y (FN)33 . Then, we can realize the quark mass ratios andmixing angles by choosing proper values of a ′ i and b ′ j for the up and down-sectors such that Y (FN)23 /Y (FN)33 ∼ V cb = 0 .
04 for both the up and down-sectors and Y (FN)23 Y (FN)32 / ( Y (FN)33 ) ∼ V cb Y (FN)32 /Y (FN)33 ∼ m c /m t ∼ .
007 for the up-sector and Y (FN)23 Y (FN)32 / ( Y (FN)33 ) ∼ V cb Y (FN)32 /Y (FN)33 ∼ m s /m b ∼ .
03, that is Y (FN)32 /Y (FN)33 ∼ . Y (FN)32 /Y (FN)33 ∼ λ = 0 .
22. Then, the realistic values can be realized by the following FN form: Y (FN)2 − ∼ (cid:18) λ λ λ (cid:19) , (18)for the up-sector and Y (FN)2 − ∼ λ ′ (cid:18) λ λ (cid:19) , (19)with a proper value of λ ′ for the down-sector, up to O (1) factors. Similarly, we can obtain massratios and mixing angles including the first generations.One of the differences between Y (FN) and Y (G) is that Y (G) ii Y (G)33 Y (G) i Y (G)3 i ≪ , (20) The FN form of Yukawa matrices can be derived, e.g. in flavor models with extra U(1) [15], five-dimensionalmodels with the warped background [16], and four-dimensional models with strong dynamics such as conformaldynamics [17]. Note that Y (FN)22 Y (FN)33 ∼ Y (FN)23 Y (FN)32 . i = 1 ,
2, but this ratio is of O (1) in the FN form. However, the (1,1) and (2,2) elementscan be almost irrelevant to obtain the realistic values in the above explanation of the FN form.Thus, the form Y (G) could lead to a result similar to the FN form.As mentioned before, our model has the SU (2) R gauge symmetry. That is, the up-sectorand down-sector have the same Yukawa couplings. If the patterns of VEVs h H i i for i = 0 , · · · , SU (2) R breaking happens.As mentioned above, the Yukawa couplings Y and Y would be useful to realize the realisticmass hierarchy, and also Y would be useful to enhance the (2 ,
2) entry. Hence, we assumethat H and H in both the up and down-sectors develop their VEVs and also H in only thedown-sector develops its VEV. We assume that v u ∼ v u , v d ∼ v d and v d ≪ v d . Then, theup-sector quark mass matrix is written by m ( u ) ≈ η v u − η v u − η v u η v u − η v u η v u η v u η v u η v u = v u η ρ u − η − η ρ u η − η ρ u η ρ u η η ρ u η , (21)where ρ u = v u /v u . Similarly, the down-sector quark mass matrix is written by m ( d ) ≈ η v d − η v d η v d − η v d η v d η v d − η v d η v d η v d + η v d η v d η v d (22)= v d η ρ d − η η ρ ′ d − η ρ d η η ρ ′ d − η ρ d η ρ d η ρ ′ d + η η ρ d η , (23)where ρ d = v d /v d and ρ ′ d = v d /v d .First, let us concentrate on the (2 ×
2) low right matrices, again. When ρ u ∼ ρ d , we wouldobtain the unrealistic relation, m c /m t ∼ m s /m b for a negligible value ρ ′ d . That is, the termwith ρ ′ d must be dominant in the (2,2) entry of the down-sector. Then, the mass ratios andmixing angle are written by m c m t ∼ η η ( ρ ( u ) ) , m s m b ∼ η ρ ′ d , V cb ∼ η ( ρ u − ρ d ) . (24)Note that η ≈ η = 1 . × − for τ = 1 . i . In addition, we obtain η ∼ . η ∼ . η ∼ .
5. Then, we can realize the realistic values of the mass ratios and mixing angles by ρ ′ d = O (0 . − O (0 . ρ u = O (0 .
1) and ρ u − ρ d = O (0 . − O (0 . η ρ ′ d is larger than η ρ d . Then,the mixing angle V us is estimated by V us ∼ η η ρ ′ d ρ d . (25)12ample values Observed( m u , m c , m t ) /m t (1 . × − , . × − ,
1) (1 . × − , . × − , m d , m s , m b ) /m b (2 . × − , . × − ,
1) (1 . × − , . × − , m e , m µ , m τ ) /m τ (2 . × − , . × − ,
1) (2 . × − , . × − , | V CKM | .
96 0 .
29 0 . .
29 0 .
96 0 . .
01 0 .
07 1 . .
97 0 .
23 0 . .
23 0 .
97 0 . . .
040 1 . Table 4: The mass ratios of the quarks and the charged leptons, ( m u , m c , m t ) /m t ,( m d , m s , m b ) /m b and ( m e , m µ , m τ ) /m τ , and the absolute values of the CKM matrix V CKM elements. The experimental data are quoted from Ref. [12].Since η ∼ . η ∼ .
08, we can realize V us = O (0 .
1) for ρ ′ d /ρ d = O (0 . − O (1).Furthermore, because det( m ( u ) ) ∼ ( v u ) η η and det( m ( d ) ) ∼ ( v d ) η η η ( ρ ′ d ) , we find m u m t ∼ η η η η ( ρ u ) , m d m b ∼ η η ρ ′ d . (26)Since η ∼ . η ∼ . η ∼ .
3, and η ∼ .
08, we can realize the realistic massratios by ( ρ u ) = O (0 .
1) and ρ ′ d = O (0 . ρ u and ρ ′ d , still.We show an example. We set ρ u = 0 . , ρ d = 0 . , ρ ′ d = 0 . . (27)Then, we obtain the mass ratios and mixing angles shown in Table 4. To obtain these results,we have used the full formula without the approximation (12) and (13). Thus our model canlead to a (semi-)realistic pattern. The specific values of input parameters are not so importantand such (semi-)realistic patterns can be realized in wide regions of the parameter space as faras the VEVs are similar values. That means the experimentally observed hierarchies in thequark sector are realized without any critical fine-tuning.After the relevant parameters (VEVs) are fixed to fit the quark sector so far, we can evaluatethe Yukawa matrices in the charged lepton sector as Y ∼ η − η η − η η , Y ∼ η η η ,Y ∼ − η η η η − η , Y ∼ η η η ,Y ∼ η − η η − η η . m ( ℓ ) ∼ v d η η ρ ′ d − η η ρ d ρ ′ d + η η ρ d η ρ ′ d − η η ρ d η . The numerical results of the charged lepton mass ratios are also shown in Table 4.We have shown analytically that the precise values of the complex structure and the ratiosof Higgs VEVs specified above are not so critical to produce the observed hierarchies thanksto the nice texture of the Yukawa matrices obtained in the magnetized orbifold. Next, wenumerically confirm that the suitable spectra can be given in wide parameter regions in termsof the Higgs VEVs and the complex structure Im τ (1) .We show the stability of the input parameters numerically in Figs. 1 and 2. The two axesof each panel in Fig. 1 represent ρ u and ρ d , which are chosen randomly from 0 to 0.5. Theinput values indicated by each (colored) dot in the parameter space ( ρ u , ρ d ) yield the ratios ofthe theoretical values to the experimental center values (th/ex ratio) within a range 0.2 to 5.0inclusively , with respect to the six mass ratios and the nine elements of the CKM matrix si-multaneously. Three panels correspond to the three cases with ρ ′ d = 0 . , . , . τ (1) = 1 . , . , . , . , . , . ρ ′ d , Im τ (1) ), we try to dot 10 times. When Im τ (1) = 1 . , . O (0 .
1) range of widths. In the parameterspace with respect to VEVs (GeV) of the five Higgs fields, the allowed regions exist over O (1)or wider range of widths with tan β = v MSSMu /v MSSMd = p ( v u + v u ) / ( v d + v d + v d ) = 1,and even in the case with tan β = 50, the values of the most restricted VEVs can vary in O (0 .
1) range. That is, the suitable hierarchies, like the one shown in Table 4, are realized inwide regions of the parameter space. We also show explicitly that the allowed values of ρ ′ d existover a wide range. In Fig. 2, the two axes of each panel represent ρ ′ d and ρ d , which are chosenrandomly from 0 to 0.5 with 10 trials, and we draw a dot when the values of th/ex ratiosare in the range 0 . ≤ th / ex ≤ . ρ u , Im τ (1) ) = (0 . , .
45) in the upper panel and ( ρ u , Im τ (1) ) = (0 . , .
95) in the lower panel,and the other Higgs VEVs are vanishing. From these figures, the obtained nice texture of theYukawa matrices give the suitable hierarchies in wide regions of the parameter space withouta critical fine-tuning.
We show an example of embedding our model into the 10D SYM theory with two independent Z twists. We consider the magnetic fluxes on the second and the third tori as F = 2 π (cid:18) × − (cid:19) , F = 2 π (cid:18) × (cid:19) . (28)14 .0 0.1 0.2 0.3 0.4 0.5 Ρ u Ρ d Ρ u Ρ d Ρ u Ρ d Figure 1: The dots represent the input values of ( ρ u , ρ d ) which yield the theoretical values ofthe mass ratios and the mixing angles within the range 0 . ≤ th / ex ≤ .
0. We choose values of ρ u and ρ d from 0 to 0.5 randomly with N = 10 trials for each combination ( ρ ′ d , Im τ (1) ) with ρ ′ d = 0 . , . , . τ (1) = 1 . , . , . , . , . , . .0 0.1 0.2 0.3 0.4 0.5 Ρ ' d Ρ d Ρ ' d Ρ d Figure 2: The dots represent the input values of ( ρ ′ d , ρ d ) which yield the theoretical valuesof the mass ratios and the mixing angles within the range 0 . ≤ th / ex ≤ .
0. We choosevalues of ρ d and ρ ′ d from 0 to 0.5 randomly with N = 10 trials for each combination of( ρ u , Im τ (1) ) = (0 . , . ρ u , Im τ (1) ) = (0 . , . R , × T / ( Z × Z ′ ). Under the Z twist, the six toroidal coordinates transformas follows, ( y , y , y , y , y , y ) → ( − y , − y , − y , − y , y , y ) , and then, the transformations of the fields are already mentioned in Eq. (8) with the projectionoperator P = (cid:18) − (cid:19) . (29)The other, Z ′ is ( y , y , y , y , y , y ) → ( y , y , − y , − y , − y , − y ) (30)with the same projection operator as that of Z .This background is consistent with the parity assignment on the first torus shown in Table 2and does not change the flavor structure caused by the magnetic fluxes (10). We summarizeall the contents induced by this background in Table 5.Fermions Bosons assignmentSM gauge fields massless massless A µ , λ +++ Q A , A , λ − + − u, d A , A , λ −− + L A , A , λ − + − ν, e A , A , λ −− + H u , H d massless 5 generations A , A , λ + −− Others none none -Table 5: All the contents of the model. The four types of λ ±±± are the 4D Weyl spinorscontained in the 10D Majorana-Weyl spinor, and each sign represents chiralities on the threetori. Others(i.e, λ −−− ) would never appear because of the 10D Majorana condition.We obtain all the SM contents. Amazingly, there are no extra fields at all, such as exoticmodes, vector-like matters and diagonal adjoint fields, so-called “open string moduli”. Theexistence of such extra massless fields is known as a notorious open problem in string phe-nomenology. Indeed, there necessarily exist some extra massless fields in our past works. Inthis model, the two orbifold twists and the magnetic fluxes free from the SUSY conditionseliminate them completely.Finally, we estimate the spectrum of scalar particles caused by the non-SUSY background .Masses of scalar fields on a magnetized torus are given by [2] M = 2 π | M ab |A , As for the massless gauginos and higgsinos, we expect other contributions from moduli, anomaly and gaugemediated SUSY breaking as well as some SUSY mass terms. M ± = 2 π | M ab |A ± π M ab A , where ± represent mass eigenstates instead of the original 2D real vector (e.g., ( A , A )). Eachof A m behaves as a vector field on the corresponding torus as well as a scalar field on the othertwo tori. The masses in the 4D effective field theory are given by the sums of their eigenvalueson the three tori. Accordingly, we can estimate the heavy scalar spectrum, i.e. squarks andsleptons, as follows, m Q = 2 π (cid:18) A (1) + 1 A (2) ± A (2) + 1 A (3) (cid:19) ,m L = 2 π (cid:18) A (1) + 1 A (2) ± A (2) + 1 A (3) (cid:19) ,m u, ˜ d = 2 π (cid:18) A (1) + 1 A (2) + 1 A (3) ± A (3) (cid:19) ,m e, ˜ ν = 2 π (cid:18) A (1) + 1 A (2) + 1 A (3) ± A (3) (cid:19) , and the contribution to the SM Higgs fields is m H = 2 π (cid:18) A (1) ± A (1) (cid:19) . Therefore, the lightest scalar mode in the Higgs sector has a tachyonic mass of the orderof the compactification scale. We find that, in any models with various 10D embeddingseliminating all the extra massless fields, tachyonic modes arise inevitably. We have to assumeadditional perturbative contributions to the soft scalar masses, and even some nonperturbativeeffects or higher dimensional operators to give them enough masses to cancel these tachyoniccontributions. For example, other SUSY breaking contributions mediated by moduli, anomalyand gauge interactions are able to cancel the tachyonic D -term contributions. Another exampleis Wilson lines. It is known that Wilson lines without magnetic fluxes arise massive modes [18].In the above example of 10D embedding, tachyonic Higgs fields have no fluxes on the two toriand a certain mass can be given by the Wilson lines. However, we have to remove the Z ′ projection (30) to introduce the Wilson lines. Even in such a case, the (MS)SM contents shownin Table 5 are unchanged, but there also remain some massless Wilson line moduli. We wouldstudy more about the stability of our 10D embeddings elsewhere. We have realized a new-type of Yukawa structures in magnetized models with Z orbifoldtwists. It is known that magnetic fluxes on a torus induce the chiral spectra and give them thedegenerate (generation) structures. The magnitude of magnetic fluxes determine the number18f generations, then, the orbifold projections give a variety of the model building with the threegenerations. We found such magnetized orbifold backgrounds do not preserve the 4D N = 1SUSY , so in this paper, we have constructed a magnetized orbifold model with a non-SUSYbackground.In Sec. 3, we have constructed the model, which can realize the experimental data of thequarks and the charged leptons simultaneously. For the purpose, we have chosen the differentmagnetic fluxes between the quark sector and the lepton sector under the condition that thedown-sector quarks and the charged leptons couple with the same Higgs sector. Although thenumber of tunable parameters (VEVs) is extremely small in this model, we have been ableto obtain the theoretical values of the masses and mixing angles, which is consistent with theexperimental data, without fine-tunings. There is a reliable reason behind that, which has beenrevealed by our analysis of the Yukawa matrices.The Yukawa matrix has the FN-like form with Gaussian hierarchies. The magnetic fluxeson the orbifold determine the “flavor quantum numbers” and give the new type of texture tothe Yukawa matrices. We can get the theoretical values favored by the experiments in thewide parameter regions thanks to this new texture obtained from the wavefunction localizationon the magnetized orbifold. We have also confirmed these analytic results numerically in thefigures. We would expect such a FN-like Yukawa structure with Gaussian hierarchies inspiresvarious phenomenological studies.Finally, we have given a sample of embedding the above generation structure into theproduct of three tori in 10D spacetime, and studied the spectrum caused by the SUSY breakingmagnetic fluxes. The magnetic fluxes and the orbifold twists can eliminate all the extra masslessfields in this model, although some scalar fields will become tachyonic. Such a model free fromextra massless fields has never realized on SUSY magnetized backgrounds. Acknowledgement
H.A. and K.S. thank H. Kunoh for stimulating discussions in the early stage of this work. Y.T.would like to thank H. Ohki and Y. Yamada for useful discussions. H.A. was supported inpart by the Grant-in-Aid for Scientific Research No. 25800158 from the Ministry of Education,Culture, Sports, Science and Technology (MEXT) in Japan. T.K. was supported in part bythe Grant-in-Aid for Scientific Research No. 25400252 from the MEXT in Japan. K.S. wassupported in part by a Grant-in-Aid for JSPS Fellows No. 25 · A Other flux configurations
We require that the magnetic fluxes in the quark sector are different from ones in the leptonsector, but the quarks and leptons couple with the same Higgs sector. The configuration in It is as far as we try to realize differences of Yukawa matrices between the quark and lepton sector at thetree-level. M = −
5, 8, and − M = −
4, 7, and − SU (2) R symmetry. That is shown inTable 6 or with the quark/lepton exchange. This configuration breaks the SU (2) R symmetry. Q L U D N E H u H d − − − − − − SU (2) R breaking.However, the values of up- and down-type Higgs VEVs could not be predicted practically, be-cause they are depending on other parts of this model and the relevant Higgs phenomenologies,which are not the purpose and we will not mention it in this paper. Since the neutrino Yukawamatrix y ν is the transpose of up-type quark Yukawa matrix y u and the (Dirac) mass hierarchiesamong the three generations are the same in the two sectors, we cannot exchange the magneticfluxes between up and down sectors.In the model shown in Table 2, the three generations are obtained by the magnetic fluxes,( M ab = − M bc = − M ca = 12) or ( M ab = − M bc = − M ca = 12). Y , which is definedabove Eq. (15), with the two types of fluxes have the Gaussian Froggatt-Nielsen texture. Thematrix Y can be approximated by y ij ∼ e − π (Im τ )( a i + b j ) /M ab M bc M ca (31)with ( a i = 20 , ,
0) and ( b i = 10 , ,
0) for ( M ab = − M bc = − M ca = 12), andwith ( a i = 20 , ,
0) and ( b i = 15 , ,
0) for ( M ab = − M bc − M ca = 12). The five-block model shown in Table 6 contains a new type of that, that is, ( M ab = − M bc = − M ca = 9). These fluxes also induce the five generations of Higgs multiplets, and then, Y hasthe Gaussian Froggatt-Nielsen texture and can be written by Eq. (31) with ( a i = 10 , ,
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