Yukawa couplings in 6D gauge-Higgs unification on $T^2/Z_N$ with magnetic fluxes

We discuss the Yukawa couplings in 6D gauge-Higgs unification models on $T^2/Z_N$ in the presence of magnetic fluxes. We provide general formulae for them, and numerically evaluate their magnitude in a specific model on $T^2/Z_3$. Thanks to the nontrivial profiles of the zero-mode wave functions, the top quark Yukawa coupling can be reproduced without introducing a large representation of the gauge group for matter fields. However, it is difficult to realize small Yukawa couplings only by the magnetic fluxes and the Wilson-line phases because of the complicated structure of the mode functions on $T^2/Z_N$ ($N=3,4,6$).


Introduction
The gauge-Higgs unification (GHU) [1]- [5] is an interesting candidate for the new physics beyond the standard model. The Higgs fields are identified with extra-components of higher-dimensional gauge fields, and we do not need to introduce elementary scalar fields. The higher-dimensional gauge symmetry governs the Higgs and the Yukawa sectors. Namely, the gauge invariance prohibits the Higgs masses at tree-level, 1 and the Yukawa couplings originate from the (higher-dimensional) gauge couplings. In particular, five-dimensional (5D) models have been extensively investigated [7]- [18] because they have the simplest extra-dimensional structure and the 5D gauge invariance protects the Higgs mass against large quantum corrections.
Six-dimensional (6D) GHU models are also phenomenologically attractive because the existence of Higgs quartic couplings at tree-level makes a realization of the observed Higgs mass easier [6]. In our previous work [19], we investigated 6D GHU models on T 2 /Z N orbifolds, and searched for possible gauge groups, orbifolds, and representations of the matter fermions by requiring the theory to have the custodial symmetry and realize the top quark mass. By employing the group theoretical analysis, we found that the minimal candidate is an U(4) gauge theory on T 2 /Z 3 and the third-generation quarks are embedded into 20 ′ of SU(4).
6D models have another important feature. We can introduce magnetic fluxes that penetrate the compact space as a background. Such a background is phenomenologically interesting because it induces gauge symmetry breaking, chiral fermions in four-dimensional (4D) effective theories, and multiple zero-modes from a single bulk field [20]- [24]. Besides, since the magnetic flux deforms the flat profile of zero-mode wave functions in the extra dimensions, it can control 4D effective Yukawa couplings [25].
In this paper, we discuss the Yukawa couplings in 6D GHU models on T 2 /Z N in the presence of background magnetic fluxes. As mentioned above, the Yukawa couplings originate from higher-dimensional gauge couplings. Hence, they become flavor-universal in a simple setup. In 5D models on S 1 /Z 2 , we can vary them by means of the bulk fermion masses that have kink profiles. Unfortunately, they cannot be extended to 6D models because we only have codimension 2 singularities on two-dimensional orbifolds. Instead, we can control them by the magnetic fluxes and the Wilson-line phases. Furthermore, 1 Six-dimensional (6D) models generically allow tadpole terms proportional to the field strength F 45 at the orbifold fixed points. Such terms induce tree-level Higgs masses unless they are cancelled [6]. the magnetic fluxes, which are quantized, can realize the generational structure of quarks and leptons. In Refs. [26]- [29], possibilities of reproducing the realistic Yukawa structure by magnetic fluxes are investigated in the context of ten-dimensional super Yang-Mills or superstring theories, and it is shown to be reproduced in some cases. Their success of the realization of the Yukawa hierarchy is supported by the following two points. One is that the gauge groups they considered are large and contain a lot of U(1) subgroups that have the magnetic fluxes, which means that there exist sufficient number of independent magnetic fluxes to control the Yukawa couplings. The other is that their models are compactified on T 2 or T 2 /Z 2 . 2 Hence the mode functions have simpler structures than those on T 2 /Z N (N = 3, 4, 6), and easier to control. However, these properties are not necessary conditions for the GHU models. In this paper, we discuss realization of the Yukawa hierarchy in smaller gauge groups, and especially focus on a U(3) model on T 2 /Z 3 as a specific example.
The paper is organized as follows. In the next section, we explain our setup and introduce the magnetic fluxes. In Sec. 3, we show explicit forms of the mode functions on T 2 and T 2 /Z N . In Sec. 4, we provide a formula for the Yukawa coupling constants, and evaluate their numerical values in a specific model. Sec. 5 is devoted to the summary.

Setup
We consider a 6D gauge theory compactified on an orbifold T 2 /Z N (N = 2, 3, 4, 6). The gauge group is G × U(1) X , where G is a simple group that includes SU(2) L × U(1) Z . 3 The field content consists of the G gauge field A M , the U(1) X gauge field B M , where M = 0, 1, · · · , 5 is the 6D Lorentz index, and 6D Weyl fermions Ψ f χ 6 (f = 1, 2, · · · ), where χ 6 = ± denotes the 6D chirality. The 6D Lagrangian is where L gf denotes the gauge-fixing terms, Γ M are 6D gamma matrices, and g A and g B are the 6D gauge coupling constants for G and U(1) X , respectively. The field strengths and 2 In Ref. [28], the cases in which three generations are realized are discussed on T 2 /Z N (N = 2, 3, 4, 6).
However, the numerical evaluations of the Yukawa couplings are performed only on T 2 /Z 2 . 3 We do not consider the color group SU(3) C since it is irrelevant to the discussion, and U(1) X is introduced in order to adjust the Weinberg angle to the realistic value.
the covariant derivatives are defined as where q f is the U(1) X charge of Ψ f χ 6 .

Orbifold and boundary conditions
For the coordinates of the extra dimensions, it is convenient to use a complex (dimension- Correspondingly, the extra dimensional components of the gauge fields are written as 3) The orbifold T 2 /Z N is defined by identifying points in the extra space as where ω ≡ e 2πi/N and τ is a complex constant that satisfies Im τ > 0. An arbitrary value of τ is allowed when N = 2 while it must be equal to ω when N = 3, 4, 6. The orbifold T 2 /Z N has the following fixed points in the fundamental domain [30]: We can introduce 4D fields or interactions at these fixed points. Fields at equivalent points on T 2 /Z N do not have to be equal as long as the Lagrangian is single-valued. The torus boundary conditions are expressed as where s = 1, τ . Matrices U s (z) ∈ G and real functions Λ s (z) may depend on z. The orbifold boundary conditions are where χ 4 denotes the 4D chirality, ϕ f and P ∈ G are a real constant and a constant matrix, respectively.
The G gauge field is decomposed as where p·H ≡ i p i H i (p i : real constants). Since (2.7) is a Z N -transformation, the following relations must hold: where n α , n χ 4 χ 6 µf ∈ Z.

Magnetic fluxes
We introduce the magnetic fluxes that penetrate T 2 /Z N as a background. For simplicity, we assume that W α M do not have nonvanishing background and the background values of the field strengths are constants. Then nonvanishing constant fluxes are where C i zz = ∂ z C ī z − ∂zC i z , B zz = ∂ z Bz − ∂zB z , and A ≡ (2πR 1 ) 2 Im τ /N is the area of the fundamental domain of T 2 /Z N . This indicates that the vector potentials C i z and B z have the following background values: where c i and b are complex constants, which correspond to the Wilson-line phases [25,32].
From (2.12), we identify U s (z) and Λ s (z) (s = 1, τ ) in (2.6) as where α i s and β s are real constants, which correspond to the Scherk-Schwarz (SS) phases [25,32]. The magnetic fluxes C i and B are quantized as (2.14) where α and µ are a root and a weight of G, and k α , k µf ∈ Z. The first and the second conditions originate from the requirement for the single-valuedness of W α z and Ψ f on T 2 /Z N , respectively. Using (2.14), the background gauge fields are expressed as We assume that the magnetic fluxes break G to SU(2) L × U(1) X × U(1) r−2 (r: rank of G), and that U(1) Z × U(1) X is broken down to the hypercharge group U(1) Y at one of the orbifold fixed points by some dynamics. The generators of the unbroken SU(2) L and U(1) Z are expressed as where α L is a root of SU(2) L ⊂ G, and a constant real vector η satisfies η · α L = 0. Then the hypercharge Y is expressed in terms of Q Z and the U(1) X generator Q X as

Mode functions
In this section, we provide a brief review of the results in Refs. [25,28,31,32,33] in our notations, and show explicit forms of the mode functions on T 2 and T 2 /Z N .

Kaluza-Klein mode expansion
The 6D fields are expanded into the Kaluza-Klein (KK) modes as where |µ is a vector in the G representation space that corresponds to the weight µ. The fermion fields ψ f ± andλ f ± are the right-and the left-handed 2-component spinors defined as All the mode functions are defined to be dimensionless, and normalized as and from (2.7), they also satisfy The SS phases φ s = φ α s , φ µf We should also note that the SS phases can be converted into the Wilson-line phases by a large gauge transformation, and vice versa [32]. The correspondence is or equivalently, In the following, we choose a gauge where all the SS phases are zero. As mentioned in Refs. [32,34,35,36], the Wilson-line phases can only take finite numbers (which are equal to the numbers of the orbifold fixed points) of values when the theory is compactified on

Mode equations
We choose the following gauge-fixing terms: Then, the mode equations are read off as wherem n ≡ πR 1 m n (m n is the KK mass eigenvalues), 5 and

Mode functions on T 2
Let us first find the mode functions defined on T 2 , which are denoted by letters with tilde.

Gauge fields
Since C i M and B M do not feel the background gauge fields, their mode equations are easily solved, and the solutions are 6 where N ci n,l , N si n,l , N cB n,l and N sB n,l are real constants, and the corresponding mass eigenvalues arem (3.14) Note that the zero-mode functions are constant.
For W α M with k α = 0, the mode functions are affected only by the Wilson-line phases. 7 where N α n,l are normalization constants, and the mass eigenvalues arẽ The other fields feel the magnetic fluxes, 8 and there are degenerate mass eigenstates at each KK level. For W α µ with k α = 0, there are no zero-modes, i.e., As for W α z , only components with k α > 0 have zero-modes. The corresponding mode functions areg where j = 1, 2, · · · , k α , and 6 For these modes, we label the KK level by a pair of integers. 7 Note that k α ζ α = N c · α/2π is independent of the flux C i . It can take nonvanishing values even in the case of k α = 0. 8 For simplicity, we do not consider the case of k µf = 0.
Here, ϑ a b is the Jacobi theta function defined by The function F (j) satisfies the relation: and is normalized as The mode functions for the KK excitation modes arẽ and the mass eigenvalues arem The components of W α z with k α < 0 do not have zero-modes, and
The mode functions for the KK excitation modes are obtained by operating D (for k µf < 0) on the above functions, and their mass eigenvalues arẽ

Mode functions on T 2 /Z N
As we have seen in the previous subsection, {f i 0 (z),f B 0 (z)} and {g i 0 (z),g B 0 (z)} are constants. The former satisfies the orbifold boundary conditions in (3.6), but the latter does not. Thus, C i µ and B µ have zero-modes on T 2 /Z N while C i z and B z do not. As for W α M with k α = 0, zero-modes exist on T 2 only when ζ α = 0 (see (3.16)). 9 Since the corresponding mode functions are constants, they satisfy (3.6) only when p · α = 0 for f α 0 (z), and p · α = 2π/N for g α 0 (z). These are the conditions for W α µ and W α z have zero-modes on T 2 /Z N .
The other modes feel the magnetic fluxes. Thus, they have degenerate modes at each KK level. The orbifold boundary conditions in (3.6) have the form where η is an N-th root of unity, and j = 1, 2, · · · , |K| discriminates the degenerate modes.
Note thatF is a solution of (3.11) that satisfies (3.4), it can be expressed as a linear combination ofF Although j runs from 1 to |K|, not all ofF (j) 0 (z) are independent mode functions [32]. In fact, the matrix M (η) generically has zero eigenvalues. The number of zero-modes is equal to the rank of M (η) . Here, note that the matrix M (η) is hermitian because where λ j (j = 1, 2, · · · , r) are the non-zero (real) eigenvalues, and r ≡ Rank M (η) . Then Therefore, it is convenient to choose independent mode functions on T 2 /Z N as where j = 1, 2, · · · , r. We can easily show that these satisfy the orthonormal condition: which follows from the orthonormal condition ofF In Ref. [33], analytic forms of the matrix M (η) are derived implying the operator formalism.
It is obtained from (3.34) with analytic forms of D (ω l ) jk , which are collected in Appendix A. The mode functions for the KK modes are obtained by operating jk becomes ηω −1 (for K > 0) or ηω (for K < 0). Therefore, the expression corresponding to (3.38) for the KK modes is .
The number of mass eigenstates at each KK level is given by the rank of M (ηω −n ) (for Note that the constants D (ω l ) jk in Appendix A, which are functions of K and ζ, satisfy where l ′ ≡ −l. This indicates that the number of zero-modes for a field that feels a magnetic flux K < 0 and the orbifold twist phase η is equal to that for a field with |K| andη.

Yukawa coupling constants 4.1 General expression
In the gauge-Higgs unification, the Yukawa couplings originate from the 6D gauge interactions: where d 2 z ≡ dzdz. In the 4D effective theory, we have the following Yukawa couplings: where the indices i, j, k run over the degenerate zero-modes, and  From the gauge invariance of the Lagrangian, the following conditions hold: and from the condition that the zero-modes exist, it follows that Then we find that which follows from the formula ((5.8) in Ref. [25]): with (3.21) and (4.4). Therefore, using the orthonormal condition (3.22), we obtain

(other cases)
. (4.9) As a result, we obtain the following expression for the Yukawa coupling constant: From the gauge invariance, K a and ζ a (a = 1, 2, 3) satisfy 11) and the zero-mode conditions are Following the same procedure in the previous case, we obtain
We choose the direction of the G flux in (2.12) as so that G is broken to SU(2) L × U(1) Z . Then, α L and η in (2.17) are identified as The normalization of η is chosen in such a manner that the hypercharge of the Higgs doublet becomes ±1/2 (see (4.20)). The fluxes C 1 and B are determined so that the 11 We do not consider the custodial symmetry, for simplicity. quantization condition (2.14) is satisfied for all the roots and the weights. In this model, (2.14) becomes 0 = 2k ±α 1 π, ±NC 1 = 2k ±α 2 π = 2k ±α 3 π, These can be solved as where κ and κ ′ are integers.
Under the unbroken SU(2) L , the SU(3) adjoint representation is decomposed as where |0 T and |0 S are the states that correspond to the Cartan generators, and Y is the hypercharge. Since the above states do not have the U(1) X charges, Y in (4.20) is equal to the U(1) Z charge. Thus, the Higgs doublets are identified as (ϕ and those of Ψ 2,4 + are .

(4.22)
Thus (λ ) are identified as the left-handed doublets (the right-handed singlets) in the standard model. They are denoted by where L and R denote the 4D chiralities.

Model parameters
We choose the matrix P in (2.7) in such a way that it does not affect the symmetry breaking caused by the magnetic fluxes. Then the possible choices are where n p = 0, 1, 2.
In order for the components in (4.23) to have zero-modes, the integers κ and κ ′ in (4.19) should satisfy which are summarized as Hence, the (ϕ ) are identified as the Higgs doublets H k because k α 2 = k α 3 = 3κ > 0.
The values of the orbifold twist phase η in (3.30) for the relevant components are expressed as where n f (f = 1, 2, 3, 4) are integers (see (2.10)).
In summary, the Yukawa sector of this model is specified by nine integers: κ, κ ′ , l, l ′ , n p and n f (f = 1, 2, 3, 4). The numbers of zero-modes and mode functions are determined by the magnetic flux the field feels K, the orbifold twist phase η, and the Wilson-line phase ζ = 2 K φ(τ − 1), which are summarized in Table I.

Realization of three generations
Here we consider the possibility that the three generations of quarks and leptons are realized by the magnetic fluxes. This occurs when κ = 6, κ ′ = 0, n p = 0, n 1,3 = 0, n 2,4 = 2 and l = l ′ = 0. 12 In this case, we obtain the following terms in the 4D effective Lagrangian 12 If we allow extra zero-modes in addition to (4.23), other parameter choices are also possible.
from the bulk: where g =ḡ A ≃ 0.652 is the 4D SU(2) L gauge coupling, and we have used that 13 Extra SU(2) L -doublets in (4.32) can be made heavy by introducing the following branelocalized terms: whereQ i R andL i R are brane-localized 4D fields, and Q L , Q ′ L , L L and L ′ L are SU(2) Ldoublet components of Ψ 1 − , Ψ 2 + , Ψ 3 − and Ψ 4 + , respectively. The parameters c i Q , c ′i Q , c i L and c ′i L are dimensionless constants. Focusing on the zero-modes, (4.35) is rewritten as 36) 13 We can always redefine the phases of the fields so that the matrix elements in (4.34) are real.
where the ellipsis denotes terms involving non-zero KK modes, and are effective mass parameters. If these mass parameters are large enough, only the following linear combinations remain in the 4D effective theory: 14 where i = 1, 2, 3, and V Q and V L are 6 × 6 unitary matrices that satisfy In order to avoid large flavor-changing processes, we assume that only one Higgs doublet H k 0 acquires a nonvanishing vacuum expectation value (VEV). Then, the fermion masses are obtained as eigenvalues of the mass matrices given by where v ≡ H k 0 . We can control the mass spectrum by tuning the parameters c i Q , c ′i Q , c i L and c ′i L through the unitary matrices V Q and V L . For example, if we choose those parameters in a manner such that V Q ≃ 1 6 , we can realize the hierarchy m t ≫ m b . In such a case, the eigenvalues of the Yukawa matrixỹ (k 0 )U ij are approximately given by those of y (k 0 )U ij , whose absolute values λ (k 0 )U i (i = 1, 2, 3) are shown in Appendix C.1. From (C.1), we find that the top quark Yukawa coupling, which is close to one, can be obtained when k 0 = 2, 5.
However, large hierarchies among the Yukawa couplings cannot be realized.
Besides the Yukawa hierarchy, the existence of the five Higgs doublets may be problematic because it seems difficult to hide so many extra Higgs bosons from the collider experiment. Therefore, in the next subsection we focus on the case that only one Higgs doublet appears.

One-Higgs-doublet case
Here we evaluate the magnitude of the Yukawa coupling constants in the case where only one Higgs doublet appears. This occurs when (κ, n p ) = (1, 2), (2, 0). As an example, we focus on the case (κ, n p ) = (2, 0). The Yukawa couplings are more restricted in the other case. From (4.26), possible values of κ ′ are −1 and 0. In these cases, each component of (4.23) has at most one zero-mode. Hence we will omit the "flavor indices" i and j in the following. The Yukawa coupling constants are expressed as follows: where l is an integer, l ′ is an even number, and where φ a (a = 1, 2) are defined by ζ a = 2φa Ka (τ − 1), and here we choose them as the second argument of V (η) ij instead of ζ a . The possible values of n, φ 1 and φ 2 in (4.44) are n = 0, 1, 2, (mod 3) where u = 0, 1, 2, 3. Numerical values of Y (±) are listed in Table II Numerical values of these are summarized in Tables III and IV  ij in (4.10) and (4.13). In order to see this, let us define the quantity:   We should also note that the top quark Yukawa coupling, which is close to 1, can be reproduced in our model, which only has the small representations 3 and3. This is in contrast to a model without the magnetic fluxes. In the absence of the magnetic fluxes, the zero-mode wave functions are constants unless the brane-localized terms exist. In such a case, the Yukawa couplings are equal to 1/ √ 2. Thus, we need an enhancement factor, which is roughly background. This will be discussed in a subsequent paper.
only take discrete values on T 2 /Z N from the consistency conditions [32]. This is equivalent to only discrete values of the Wilson-line phases being allowed [34,35,36].
Note that D The allowed values of the SS phases are The explicit form of D The allowed values of the SS phases are The explicit forms of D The allowed values of the SS phases are The explicit forms of D The allowed values of the SS phases are The explicit forms of D The sign function sgn(K) comes from the formula: where β is an integer (half-integer) when K is even (odd).

B Normalizations of KK modes
In this appendix, we identify the coefficients in (3.1). Here we focus on those for W α µ and W α z . The other normalization factors are obtained similarly. The 6D Lagrangian (2.1) includes the following terms: The KK expansion is expressed as where N W and N ϕ are positive constants, and the mode functions are normalized as Substituting (B.3) into (B.1), we obtain the 4D effective Lagrangian: where α 1 and {α 2 , α 3 } are the roots such that W ±α 1 µ and W α 2,3 z have zero-modes that are identified with the W boson and the Higgs doublet fields respectively, and We have used that Comparing (B.5) with the standard model, where A a µ (a = 1, 2, 3) are the SU(2) L gauge fields, F a µν are their field strengths, and W ± µ ≡ 1 √ 2 (A 1 µ ∓ iA 2 µ ), the constants N W and N ϕ should be chosen as and the 4D gauge coupling constantḡ A is identified from (B.6) as Solving these, we obtain We have used that after appropriate phase redefinitions of the fields.

C.2 One-Higgs-doublet case
Here we collect numerical values of the Yukawa coupling constants in Sec. 4.2.4.
The absolute values of Y (+) (n, φ 1 , φ 2 ) are listed in Table II   The absolute values of y D,E can be read off from Table. II. Those of y U (n 2 , l, l ′ ) (l = 0, 1) are shown in Table. III. When n 2 = 0, Q ′ L does not have a zero-mode. The coupling constants for the other values of l are related to those in Table. III by y U (n 2 , 2u, l ′ ) = y U (n 2 , 0, l ′ + 8u) , y U (n 2 , 2u + 1, l ′ ) = y U (n 2 , 1, l ′ + 8u) .