Classification of three-generation models on magnetized orbifolds

We classify the combinations of parameters which lead three generations of quarks and leptons in the framework of magnetized twisted orbifolds on $T^2/Z_2$, $T^2/Z_3$, $T^2/Z_4$ and $T^2/Z_6$ with allowing nonzero discretized Wilson line phases and Scherk-Schwarz phases. We also analyze two actual examples with nonzero phases leading to one-pair Higgs and five-pair Higgses and discuss the difference from the results without nonzero phases studied previously.


Introduction
One of the greatest achievements in particle physics is the completion of the Standard Model (SM) as the discovery of the SM-like Higgs boson following the works reported in [1,2]. As is widely known, the SM is consistent with almost all the experimental results which have ever been made, while the origin of the SM configurations is still unknown, e.g., the origin of the SU(3) C × SU(2) W × U(1) Y gauge structure and the related quantum numbers in the matter fields, or why the number of the matter generation is three accompanied with the large mass hierarchy.
Some works toward phenomenologically realistic model building have been done through this direction based on U(N) gauge theory in [63,64,13,32,14], where only the Z 2 orbifolding with trivial Wilson line phases and Scherk-Schwarz phases is considered. See also Refs. [27,28] for discussions based on E 6,7,8 groups. Now, thanks to [51], all the technical difficulties for considering the generalized case have been resolved and it would be phenomenologically meaningful that we start to pursue the situation with nontrivial boundary conditions under the Z 2 or higher Z 3,4,6 . As a first step, in this paper, we classify all the possibilities of the models with suitable three-generation matter structure of the quarks and the leptons within U(N) gauge group being the original gauge structure, and show an example in the Z 2 case with nontrivial boundary conditions after discussing the effect via nonzero Wilson line phases and/or Scherk-Schwarz phases.
fluxes, nontrivial Wilson line phases and Scherk-Schwarz phases. In Section 3, we exhaust all the possible configurations of parameters in all the Z 2 , Z 3 , Z 4 and Z 6 cases with nontrivial boundary conditions. In Section 4, after we examine how much mass hierarchy in the quark sector is deformed by introducing nonzero Wilson line phases and/or Scherk-Schwarz phases, we propose an example for realizing (semi-)realistic quark mass patterns and the mixing structure described by the Cabibbo-Kobayashi-Maskawa (CKM) matrix. Section 5 is devoted to summarizing issues and discussing future prospects. In Appendix A, we show the analytical forms describing a state on the magnetized T 2 /Z 2,3,4,6 orbifolds obtained in Ref. [51]. In Appendix B, we show our result for T 2 /Z 2 by writing down all the independent combinations of parameters explicitly.
(Note: the full list of classification including T 2 /Z 3,4,6 cases is available in the arXiv version of this manuscript [65].)

10D SYM on generalized magnetized orbifolds
Here, we review the basics of the 10D super Yang-Mills theory on generalized magnetized orbifolds of T 2 /Z 2 , T 2 /Z 3 , T 2 /Z 4 , T 2 /Z 6 with nonzero Wilson line phases and/or Scherk-Schwarz phases.

U(N) gauge theory on (T 2 ) 3
Firstly, we focus on the 10D magnetized SYM theory without orbifolding [34,66,64], with adopting notations in Refs. [52,51], which is defined on a product of 4D Minkowski space and three factorizable 2-tori, M 4 × (T 2 ) 3 . The capital roman indices M, N run over μ(= 0, 1, 2, 3), z 1 , z 1 , z 2 , z 2 , z 3 , z 3 , where the i-th (i = 1, 2, 3) 2-torus is described by the complex coordinate z i = y 2i+2 + iy 2i+3 and its complex conjugation z i = y 2i+2 − iy 2i+3 made by the two Cartesian coordinates representing the extra directions, y 2i+2 and y 2i+3 . We take each torus modulus parameter τ i (⊂ C) as Imτ i > 0 for convenience. We use the short-hand notation d 6 z meaning 3 i=1 dz i dz i . On T 2 i , the coordinate z i is identified as z i ∼ z i + 1 ∼ z i + τ i . The bulk Lagrangian holds N = 1 supersymmetry in 10D and consists of the 10D vector fields A M , which are found in the covariant derivative D M and the field strength F MN , and the gaugino fields λ described by 10D Majorana-Weyl spinors. The explicit forms of D M λ and F MN are given by

4)
l,M (z 1 , z 1 ) ⊗ φ (2) m,M (z 2 , z 2 ) ⊗ φ (3) n,M (z 3 , z 3 ), (2.5) where l, m, n are KK indices and ψ (i) l is a 2D spinor describing the l-th KK mode on the i-th T 2 , whose exact form is ψ T and the corresponding 2D chirality (+ or −) is denoted by s i . We adopt the gamma matrices ˜ m (identified by the Cartesian coordinates) corresponding to the i-th torus as where σ 1,2 are Pauli matrices. In the following part, we only focus on the zero modes (l = m = n = 0) and omit the KK indices hereafter. We introduce factorizable Abelian magnetic fluxes on the three T 2 through the classical vector potential of A M in the following forms where C (i) j (j = 1, · · · , n) represent the corresponding Wilson line phases on T 2 i , and M (i) j (j = 1, · · · , n) should be integers because of Dirac's quantization condition on T 2 i . On T 2 i , no restriction is there on possible choices of C (i) j . Under this background, the original gauge group U(N) explicitly breaks down as U(N) → n a=1 U(N a ) with N = n a=1 N a . We can derive the following relations, where d j plays as an exterior derivative on T 2 j . Here, the Lagrangian density in Eq. (2.1) should be single-valued under every torus identification z i ∼ z i + 1 ∼ z i + τ i (i = 1, 2, 3), and then in the gaugino fields λ(x, {z i , z i }), the following pseudo-periodic boundary conditions should be arranged, where α (i) j and β (i) j (j = 1, · · · , n) describe Scherk-Schwarz phases and can take any real numbers. Now, we consider the specific case of U(N) → U(N a ) × U(N b ) for our concrete understanding on the system. In this case, the 1-form potential in Eq. (2.7) takes the shape, 16) and the gaugino fields λ and their i-th torus part are decomposed as follows: The fields λ aa and λ bb play as the gaugino fields under the unbroken gauge group U(N a ) × U(N b ), while λ ab and λ ba correspond to the bi-fundamental matter fields as (N a , N b ) and (N a , N b ), respectively. We obtain the zero-mode equations for these gaugino fields on the i-th T 2 with the 2D chirality (+ or −) as The blockdiagonal parts of the gaugino fields, ψ (i)aa and ψ (i)bb , do not feel magnetic flux and Wilson line phase, which is apparent from Eqs. (2.18) and (2.19). On the other hand, in the block-offdiagonal parts, ψ (i)ab and ψ (i)ba , magnetic fluxes and Wilson line phases affect properties of the matter fields. From the information in Eqs. (2.14) and (2.15), the effective boundary conditions of the fields are easily written down,   with the properties where a and b are real numbers, c, m and n are integers, and ν and τ are complex numbers with Imτ > 0. The following orthonormality condition determines the normalization factor N |M (i) ab | , with d 2 z i := dz i dz i . We note that the total number of zero modes of ψ (i)ab  (2.28) where the index I is identified under the condition, mod |M On flux background, zero-mode profiles are not only split, also localized around points different from each other. Then, we can expect that hierarchical values in Yukawa couplings are created via overlap integrals in the Yukawa sector of this model. The concrete form of the Yukawa couplings are as follows: where c is a constant factor via gauge structure and I = (I 1 , , respectively. In a suitable symmetry breaking like the above examples, we can find conditions on the parameters, K and possible choices of integers . 4 At the end of this subsection, we mention an efficient technique discussed in [52,51]. In the Abelian magnetic flux, we can set either the (complex) Wilson line phase C (i) j , or the Scherk-Schwarz phases α (i) j and β (i) j as zero (in each j individually) without loss of generality. This fact could make the following analysis simplified.

U(N) gauge theory on orbifolded (T 2 ) 3
In this subsection, we examine the U(N) SYM theory on magnetized (T 2 ) 3 with orbifolding. In our configuration where the six extra dimensions are factorized as three 2-tori, apparently, we mainly focus on one of the tori T 2 i for a general discussion. On T 2 i , possible twisted orbifolding is to impose the covariance on the fields under the rotation with the angle ω, z i → ωz i , where ω is e 2πi/N with N = 2, 3, 4, 6. In other words, Z 2 , Z 3 , Z 4 and Z 6 (twisted) orbifoldings are realizable on T 2 i . In non-Abelian gauge theories, a nontrivial gauge structure part P appears in the Z N manipulation as, where P should satisfy the conditions, P ∈ U(N) and P N = 1 N×N . Here, to prevent an additional explicit gauge symmetry breaking via the orbifoldings, we should take the following form in P , Now, the gauge structure under the orbifoldings is declared, and thus we can concentrate on an actual case with the Z N parity η (i) , the (2D) positive chirality and the state-discriminating index I on T 2 i as, where we assume that M (i) is a positive integer. Note that the correspondence to the negative chirality case is basically straightforward by the replacements, Constructing the concrete form of can be done straightforwardly just following the general recipe as where, apparently, this state has the eigenvalue η (i) under z i → ωz i . Since the function (z i ) forms a complete set, then in principle, we can express the Z N -rotated one as a linear combination of (z i ). If we derive the coefficients of such a linear combination analytically, we can construct the T 2 i /Z N mode functions in an exact manner. 5 Here, following the discussions in [51], we take the following basis, for simplicity. Within taking the Abelian magnetic flux, we can always select this choice without any loss of generality [52,51]. By use of the results in [51], the Z N -transformed states (in the basis (2.45)) can be expressed as with the expansion coefficients (ω x z i , τ i ), (2.47) where explicit forms of C (ω x ) I i J i are summarized in Appendix A. Using this information, the relation in (2.44) is written, where the number of independent physical states can be calculated as In general after orbifolding, the number of independent physical states should be reduced as procedure is possible since operator formalism tells us the analytical forms of C (ω x ) [51]. The detailed information on the numbers of the survived physical states and the forms of C (ω x ) Also, operator formalism helps us when we calculate the kinetic terms on T 2 i /Z N described as (2.50) After some calculations in operators and states, we can reach the simple correspondence, Here, we should mention that the kinetic terms K are in general non-diagonal, and thereby we should diagonalize them to know the correct forms with suitable normalization. This process itself is, at least in principle, doable for any larger |M (i) | through the Gram-Schmidt process for orthonormalization in linear algebra. After the unitary transformation with the corre- ) means a discriminator for the states on the original geometry T 2 i , whereas the column index identifies the physical eigenstates, where this index is meaningful only up to Rank K (Z N ;η (i) ) − 1 from zero. Therefore in the physical eigenstates after considering the correct normalization in the kinetic terms, T 2 i /Z N mode function should be (2.53) where I i is the index of physical eigenstates. Based on the knowledge, we briefly discuss the way of calculating Yukawa couplings under the Z N symmetry. The T 2 i part is formulated (with the basis fixing in Eq. (2.45)) in the "T 2 i " eigenbasis as where we find the condition on the Z N parities (via the invariance of the system),

Possibilities of three generations in U(N) models
In this section, we classify all the reasonable possibilities of three-generation models under the Z 2 , Z 3 , Z 4 and Z 6 orbifoldings. When we start from 10D U(N) SYM theory, the basic pattern of the gauge symmetry breaking under the magnetic flux is where the corresponding 1-form potential is  6 In such a situation, when the actual chirality of the gaugino is left (negative), λ ab corresponds to the left-handed quarks and leptons, and λ ca accords with (charge-conjugated) right-handed quarks and leptons, respectively. When the magnetic fluxes are suitably assigned, the situation with three-generations is materialized. Besides, λ bc plays as up-type and down-type Higgsinos. After we assume that (4D N = 1) supersymmetry is preserved at least locally at the ab, bc and ca sectors, the corresponding Higgses via extra-dimensional components of the 10D vector fields are still massless under the fluxes and the number of the fields are the same with Higgsino fields. Also, no tachyonic mode is expected at the tree level. Here, in general, multiple Higgs fields appear from the bc sector. Interestingly, when λ ab , λ bc and λ ca have zero modes, λ ba , λ cb and λ ac cannot contain any zero mode and thus no exotic particle arise from these fermionic sectors. In the case of the actual chirality being right (positive), we should flip the roles of the two categories. As a phenomenological point of view, we can consider the following additional breakdowns originating from flux, U(4) PSC → U(3) C × U(1) 1 and U(2) R → U(1) 2 × U(1) 3 (up to U(1) factors), where U(3) C is the color gauge group (up to a U(1) factor). Under the latter breaking, the up-type and down-type Higgsino/Higgs sectors can feel different magnetic fluxes individually. Consequently, the numbers of the two types of the fields become not the same.

Strategy in classification
In the U(N) model, the effective Yukawa couplings after the integration among the six extra dimensions are symbolically written as where c is a constant factor. ψ L i (z), ψ R j (z) and φ H k (z) denote zero mode wave functions of the left-, right-handed matter fields and the Higgs fields, and their generations are indicated by the indices i, j and k, respectively. When the left-and right-handed matters are localized among different tori, the Yukawa couplings are factorized like The rank of this type of matrices (about i and j ) is one and no room would be there for generating realistic flavor structure. We can reach the same conclusion about the Higgs fields. Thereby, realizing the Yukawa matrix holding rank three requires the situation that all the three types of the fields are localized on the same torus like ij k a (2) a (3) . (3.4) In this case, the contributions from a (2) and a (3) only affect the overall normalization, and the structure of the Yukawa matrix is governed only by a (1) . Thus, we can mainly focus on the torus determining the matter generations. Based on the knowledge which we have obtained through the discussions before, we can easily understand the constraints on the parameters describing each sector in the U(N) theory, where the above parameters are defined by the fundamental ones like M (i) ab = M (i) a − M (i) b except the Z N parities. The Z N parities are described as : In the following analysis, like the discussions in the previous section, we adopt the basis where the Scherk-Schwarz phases are nonzero, and correspondingly the Wilson line phases are zero. The interpretation into the basis with nonzero Wilson line phases and Scherk-Schwarz phases being zero is straightforward, where we just obey the relation in Eq. (3.7). In the following part, we search for all the phenomenological possibilities with three generations in the quarks and the leptons. Here, we adopt the strategy that we scan over all the combinations of the effective parameters describing the ab, bc and ca sectors under the constraints in Eq. (3.5). Note that the ba, cb and ac sectors are completely fixed after we determine the configurations of the above three parts. As we discussed before, the number of generations is exactly calculable with the help of operator formalism and actual numbers are summarized in Appendix A. We use this information for the selection cut in realized matter generations. Since our attention is on the Yukawa structure schematically depicted in Eq. (3.4), it is enough to focus on a 2-torus of the geometry. Thus hereafter in this section, we drop the index i for identifying the tori.
Before going to scanning, we focus on how many configurations we should consider to exhaust possibilities under the requirements in Eq. (3.5). Apparently from (3.5), the information on the ca sector is totally determined after we set the parameters of the other two sectors. Here, we look at the condition on the magnetic fluxes. The first line of Eq. Besides, after ignoring the difference coming from the combinatorics, we can introduce the additional condition, Consequently, we only take into account of the magnetic fluxes being possible under the conditions in (3.8) and (3.9).
We also comment on the other parameters. After the orbifolding, the Wilson line phases and the Scherk-Schwarz phases no longer take arbitrary values [52,51]. At this moment in time, combinatorics degrees of freedom are fixed by our preconditions for analysis in Eqs.

Results
Now, we are ready to evaluate the numbers of allowed possibilities in effective parameter configurations of the Z 2 , Z 3 , Z 4 and Z 6 cases. Summaries of our analysis are found in Tables 1 (Z 2 ), 2 (Z 3 ), 3 (Z 4 ) and 4 (Z 6 ), where "General cases" and "Trivial BC's only" means the cases with and without nontrivial Scherk-Schwarz phases, respectively. Here, we only show the numbers of allowed parameter configurations in M ab , M ca < 0 and M ab < 0, M ca > 0 separately. Corresponding numbers of the Higgs pairs (N H ) are also shown. If an allowed configuration says M bc = 0, where one non-localized Higgs pair appears without magnetic background and this situation would not be interesting in the phenomenological point of view, we discriminate this case as "1 trivial " from the one Higgs cases with magnetic flux "1".
We mention about the Z 2 result. The "Trivial BC's only" case in Z 2 was already analyzed in Ref. [64] and our result is totally consistent with them. The complete information on Z 2 configurations is summarized in Appendix B. We skip to put the other concrete information on Z 3 , Z 4 and Z 6 in this paper since their volume is huge, but those concrete results are available in the arXiv version [65]. Brief comments are also given here. After allowing nonzero Wilson line phases, the configurations with seven Higgs pairs arise in the Z 2 case as shown in Table 1. In the  Table 1 Result in Z 2 case. "General cases" and "Trivial BC's only" means the cases with and without nontrivial Scherk-Schwarz phases, respectively. Corresponding numbers of the Higgs pairs (N H ) are also shown. The case indicated by 1 trivial means the one Higgs pair appears under non-magnetized background in bc sector.

General cases
Trivial BC's only 1 (N H = 9) 136 + 81 = 217 in total 10 + 9 = 19 in total Table 2 Result in Z 3 case. The convention is the same with in Table 1.  Table 3 Result in Z 4 case. The convention is the same with in Table 1.  Tables 2, 3 and 4, we find the situations where two or four Higgs pairs are realized, while those with nine pairs never occur.

Examples in Z 2 with nontrivial twisting
Based on the results which we have achieved in the previous section, we can examine how much nontrivial boundary conditions by the Scherk-Schwarz phases (or the Wilson line phases) affect the flavor structure in some specific models with the Z 2 orbifolding. Here, we adopt the basis with the Wilson line phases being zero and only debate with the torus where all the matter fields are assumed to be localized. As we already discussed, the above assumption is justified by the phenomenological requirement for flavors. Here, we rewrite the expressions of the Yukawa couplings λ I,J,K on T 2 , and λ I,J,K on T 2 /Z N , in Eqs. (2.30) and (2.54), respectively,  The reason behind this simplicity is that the structure of orbifolded functions is uninvolved, where we estimate the number of survived physical modes analytically and explicitly as in Appendix A. 8 In the Z 3,4,6 orbifoldings, nontrivial effects via U Z N ;η I might be expected.

Numerical analysis in one-pair Higgs model
From the result in the previous section on the Z 2 general case, we see the number of the possibilities with one-pair Higgs is strikingly enlarged as 65 from 5. Note that the modulus parameter τ of the torus does not receive any restriction from the Z 2 orbifolding, in contrast to the other three cases. As discussed in [64], when the greater value the modulus parameter τ takes, the more significant hierarchy could be realized. However, in the simple five cases without nonzero Scherk-Schwarz phases, we can check that no sizable hierarchy is generated even when we take a large value. For example, even when we adopt τ = 10 i, only up to O(10 2 )-order hierarchy can be generated.
Through the following example, we will see how the nonzero phases improve the magnitude in the hierarchy. Here, we consider the configuration, with nonzero cν and with cν = 0. Table 5 The mass ratios of the quarks and the absolute values of the CKM matrix elements when we adopt the parameters in Eq. (4.6) with one Higgs pair. We use the experimental values in Ref. [67].
where we omit to show the index about the Higgs because only one Higgs pair is there. Now, we show the resultant values in Table 5, 9 where we notice that the desirable amount of hierarchy is suitably realized in the mass values. 10 However, if we try to generate the realistic CKM mixing, a more complicated mixing pattern would be required.

Gaussian Froggatt-Nielsen mechanism with discrete Scherk-Schwarz phases
Here, we would like to search for a configuration with realistic observed values by following the strategy discussed in [14], named Gaussian Froggatt-Nielsen mechanism. As pointed out in [14], especially when M u,d are as large as O(10 2 ) or more, the forms of η (u,d) N are approximately described by the following simple forms with good precision, (4.10) where the real part of τ only contributes to this as an O(1) phase factor. Also, we would remember that the nonzeroness in "cν" of the theta function does not lead to a sizable effect, only up to O(1) as in Eq. (4.8). Under some assumptions in multiple Higgs VEVs, the Yukawa couplings would be symbolically expressed as where c is a common constant factor and a I and b J are determined only by the magnitudes of the magnetic fluxes on the left-and right-handed fermion sectors, respectively. It seems to be similar to another familiar one named Froggatt-Nielsen form [68] as Y I ,J ∼ e −c(a I +b J ) , (4.12) where in this case, a linear form appears and a I and b J correspond to quantum numbers named Froggatt-Nielsen charges. When we realize the following shape, for I ≤ M and J ≤ N , we would regenerate a desirable form seen in the CKM matrix.
In the following part, we analyze the flavor structure through Gaussian Froggatt-Nielsen mechanism with discrete Scherk-Schwarz phases in the parameters, (4.14) where the notation is the same with that in the previous analysis. This configuration is categorized in the case, "M ab , M ca < 0", where multiple Higgs fields tend to be obtained. The correspondence to Eq.   Table 6 The mass ratios of the quarks and the absolute values of the CKM matrix elements when we adopt the parameters in Eq. (4.14) with five Higgs pairs. We use the experimental values in Ref. [67].  Table 6. 11

Summary and discussion
In this paper, based on the analytical formulation discussed in Ref. [51], we have classified all the individual possibilities of flavor models generating suitable three generations in the quarks and the leptons on generalized T 2 -based magnetized orbifolds, namely T 2 /Z 2 , T 2 /Z 3 , T 2 /Z 4 , T 2 /Z 6 , with nontrivial Wilson line phases and/or Scherk-Schwarz phases. Also, we examined the effects caused by the nontrivial phases in mass hierarchy in a configuration with one Higgs pair, and constructed a concrete model with the phases being nonzero values in the case of five Higgs pairs, accompanied by a realization of the SM quark structure, that is the observed mass values and mixing angles, with rather good precision. Now, there are lots of possible suitable candidates with three generations in the choice of fluxes and related parameters, and thus to search for suitable configurations to describe the nature on T 2 /Z 3,4,6 could be an interesting direction. Here, nontrivial contributions via diagonalizing matrices of kinetic terms would be nontrivial and situations might be different from in the Z 2 . Revealing this point is also an important work in the model building on magnetized background. Also, like in [13,32], full model embedding to 10D SYM theory is a meaningful task. range required in the discussion in Section 3. Note that all the following formulas are for the situation where the value of the magnetic flux is positive. The correspondence to the negative case is understandable, e.g., through Eq. (2.28).
A.1. T 2 /Z 2 On the T 2 /Z 2 (ω = e πi = −1), allowed values of Scherk-Schwarz phases are following four patterns, The analytic form of the expansion coefficient is We can find easily the formula of the number of the physical states by analyzing the matrix The numbers of linearly independent zero-mode eigenstates with Z 3 eigenvalue η for M = odd and (α 1 , α 2 ) = ( 1 6 , 1 6 ), ( 5 6 , 5 6 ) on T 2 /Z 3 .
we obtain the results summarized in Tables A.11 and A.12.