Hierarchical structure of physical Yukawa couplings from matter field K\"ahler metric

We study the impacts of matter field K\"ahler metric on physical Yukawa couplings in string compactifications. Since the K\"ahler metric is non-trivial in general, the kinetic mixing of matter fields opens a new avenue for realizing a hierarchical structure of physical Yukawa couplings, even when holomorphic Yukawa couplings have the trivial structure. The hierarchical Yukawa couplings are demonstrated by couplings of pure untwisted modes on toroidal orbifolds and their resolutions in the context of heterotic string theory with standard embedding. Also, we study the hierarchical couplings among untwisted and twisted modes on resolved orbifolds.


Introduction
The origin of flavor structure of quarks and leptons, in particular the hierarchical structure of their masses is one of unsolved issues in the Standard Model (SM) of particle physics. The Yukawa couplings of quarks and leptons are free parameters such that their masses and mixing angles are fitted with experimental data. There is no guiding principle to choose the Yukawa couplings in the quark sector, which have hierarchical ratios of O(10 5 ). Several mechanisms are proposed to explain the structure of Yukawa couplings, as represented by the localization of matter wavefunctions in the extra-dimensional space [1,2] and a proper charge assignment of quarks/leptons under the continuous or discrete flavor symmetries [3,4]. These mechanisms can be engineered in the ultra-violet completion of the SM, such as its supersymmetric extension and the string theory.
In the four-dimensional (4D) N = 1 supersymmetric theory, physical Yukawa couplings Yâbĉ are obtained from holomorphic Yukawa couplings κ abc in the canonical basis, where a kinetic term of matter fields is canonical, i.e., the diagonal Kähler metric. Note that the Kähler metric is in a non-diagonal, but non-trivial form in a generic supergravity theory derived from the superstring theory. There are two possibilities to realize the hierarchical structure of physical Yukawa couplings: (i) the flavor structure of holomorphic Yukawa couplings controlled by the continuous/discrete flavor symmetries, (ii) a non-trivial structure of matter field Kähler metric which would be connected with the geometry of extra-dimensional space.
In the traditional approach (i), holomorphic Yukawa couplings are well controlled by stringy selection rules, gauge and extra-dimensional geometrical symmetries. It was known that the holomorphic Yukawa couplings of twisted modes on heterotic orbifold models are exponentially suppressed with respect to moduli fields [5][6][7]. Hence, one can realize the hierarchical structure of physical Yukawa couplings of these states due to the fact that they are localized at fixed points. However, the coupling selection rules among twisted modes are very tight [8,9]. It is not straightforward to realize realistic mass matrices for three generations of quarks and leptons. (See for attempts to realize quark and lepton mass matrices by twisted modes, Refs. [10,11].) At any rate, it would be difficult to have the hierarchical structure for other states such as untwisted modes propagating the bulk. It indicates that smooth Calabi-Yau (CY) compactifications will not lead to the hierarchical structure of holomorphic Yukawa couplings for matter zero-modes. For instance, in the context of heterotic string theory with standard embedding, the holomorphic Yukawa couplings are of O(1) in the large volume/complex structure regime of CY threefolds, determined by the third derivative of the prepotential. It results in the trivial structure of the holomorphic Yukawa couplings which are also observed in the untwisted modes on a singular limit of CY threefolds such as toroidal orbifolds. Hence, the understanding of matter field Kähler metric would shed new light on the structure of physical Yukawa couplings.
In this paper, we study whether the matter field Kähler metric provides the hierarchical structure of physical Yukawa couplings. In contrast to the holomorphic Yukawa couplings, the matter field Kähler metric would not be restricted by symmetry arguments. Hence, in the approach (ii), we have to calculate zero-mode wavefunctions themselves living on the extradimensional space to understand the structure of matter field Kähler metric. 1 To overcome this problem, we focus on the heterotic string theory with standard embedding, where matter fields are in one-to-one correspondence with moduli fields on CY threefolds including toroidal orbifolds in the singular limit. In this context, the matter field Kähler metric as well as the holomorphic Yukawa couplings are explicitly calculated by means of conformal field theory as well as the special geometry [17]. To study the impacts of matter field Kähler metric on physical Yukawa couplings, we deal with a simplified CY threefold and toroidal orbifolds preserving the supersymmetry. On toroidal backgrounds, the moduli fields are categorized by untwisted and twisted sectors, where twisted modes are analyzed through resolutions of toroidal orbifolds, capturing the feature of CY threefolds. We find that the kinetic mixings on the moduli space of untwisted and twisted modes are inevitable for a realization of hierarchical Yukawa couplings. Remarkably, O(1) values of the moduli fields provide the hierarchical structure of physical Yukawa couplings, demonstrated on specific toroidal orbifolds with and without resolutions having the non-diagonal matter field Kähler metric.
The remaining part of this paper is organized as follows. In Section 2, we briefly review an effective action of matter fields, focusing on Yukawa couplings for the fundamental and anti-fundamental representations of E 6 gauge group. The off-diagonal entries in the matter field Kähler metric play an important role of distinguishing the holomorphic Yukawa couplings and the physical one as illustrated on toroidal orbifolds in detail. In Section 3, the hierarchical physical Yukawa couplings are demonstrated on T 6 /Z 3 orbifold, blown-up T 6 /(Z 2 ×Z 2 ) orbifold and a mirror dual of T 6 /(Z 3 × Z 3 ) orbifold. Finally, Section 4 is devoted to the conclusion and discussion.

Heterotic string theory with standard embedding
In Section 2.1, we briefly review the effective action of matter fields in the heterotic string theory on CY compactification with standard embedding. In particular, we focus on Yukawa couplings of matter zero-modes, determined by triple intersection numbers of CY threefolds. In Section 2.2, the matter kinetic mixings on a simplified CY threefold and toroidal orbifolds are examined in the light of physical Yukawa couplings. on which the coefficient F I = ∂ I F corresponds to the first derivative of the prepotential F with respect to X I . In what follows, we adopt the gauge X 0 = 1, yielding the complex structure moduli in the flat coordinate: X i = U i , (i = 1, 2, · · · , h 2,1 ) with h 2,1 being the hodge number of CY threefolds. There the complex structure moduli space is governed by the structure of the prepotential whose functional form is known in the large complex structure regime, We now pick up the cubic term determined by the triple intersection numbers κ ijk , but other next-leading terms are irrelevant to the following discussions. Furthermore, it is possible to consider the other corner of complex structure moduli space, but the prepotential has a modeldependent structure as shown in an example of Section 3.3. Then, the Kähler potential (2) is simplified as follows: where it is notable that loop and quantum corrections are assumed to be enough suppressed in comparison with the classical terms. We next deal with the effective action of charged matter fields 27 and 27 under the E 6 gauge group, which are denoted by A a and A i , respectively. The matter Kähler potential is only known in an expansion of A a and A i , that is, A a , A i 1 [17], and the corresponding Kähler metrics become The matter field Kähler metric is in turn in one-to-one correspondence with the moduli Kähler metric. In addition, Yukawa interactions of charged matter fields are described by the following superpotential: stating that the holomorphic 27 3 and 27 3 Yukawa couplings are determined by the triple intersection numbers of the Kähler and complex structure moduli, respectively [20,21]. To derive the physical Yukawa couplings, we perform the unitary transformation L diagonalizing the kinetic terms for multiple 27 matter fields, 3 3 Here, we focus on the 27 sector, but it is straightforward to extend the following calculation to 27 sector. Furthermore, we note that the hat index is introduced to canonically normalized fields in subsequent analyses.
where Λ stands for an eigenvalue matrix satisfying LKL † = Λ. The canonically normalized matter fields Aâ are defined as and the physical Yukawa couplings result in Note that not all the 27 and 27 representations behave as the chiral zero-modes in the lowenergy effective action. The index theorem tells us that the net number of chiral zero-modes is determined by the index corresponding to half the Euler characteristics of CY threefolds χ(M). Hence, the chiral zeromodes in our interest are the remaining zero-modes originating from 27 or 27. Remarkably, the holomorphic Yukawa couplings in both 27 and 27 sectors are integers determined by the triple intersection numbers, indicating that it would be difficult to realize the hierarchical structure of holomorphic Yukawa couplings. Indeed, the difficulties of hierarchical structure of holomorphic Yukawa couplings on CY threefolds and toroidal orbifolds are discussed in detail in the following sections.

Role of Kähler metric on physical Yukawa couplings
In this section, we will focus on kinetic mixings to exemplify the difference between the physical Yukawa couplings and the holomorphic one on the basis of a simplified CY threefold in Section 2.2.1 and toroidal orbifolds in Section 2.2.2.

Calabi-Yau threefolds
We start with CY threefolds to show the impacts of Kähler mixing on physical Yukawa couplings. It was known that typical CY threefolds contain hodge numbers with |h 1,1 − h 2,1 | = O(100) as shown in Kreuzer-Skarke dataset of CY threefolds in toric ambient spaces [22,23], leading to O(100) number of chiral zero-modes. (See for several attempts to construct CY threefolds with a small hodge number, e.g., Ref. [24].) For the illustrative purpose, we assume that CY threefolds have the hodge number h 1,1 − h 2,1 = 3 and the moduli space of remaining three-generation zero-modes is described by the special Kähler manifold SU (1,1) . The Kähler potential of this geometry is described by where T a denotes the Kähler moduli protected from becoming massive. The triple intersection number of these moduli is found as and 0 otherwise, and these correspond to the holomorphic 27 3 Yukawa couplings. For concreteness, we identify the Higgs field with an element of A 1 and suppose that three generations of quarks and leptons appear from A a (a = 1, 2, 3), associated with the massless Kähler moduli. Then, the holomorphic Yukawa couplings of quarks and leptons on the (A 1 , A 2 , A 3 ) basis are of the form: This rank-two matrix does not have non-trivial flavor and hierarchical structures. On these backgrounds, the Kähler metric is of the diagonal form due to the factorizable Kähler metric: yielding the non-vanishing element of physical Yukawa matrix with the eigenvalue matrix Λ and the diagonalizing matrix L of moduli Kähler metric. Since the physical Yukawa couplings have the same structure as the holomorphic Yukawa couplings, it is still difficult to realize the non-trivial flavor and hierarchical structure of Yukawa couplings on such a special Kähler manifold with SU (1,1) When the Kähler moduli space differs from the SU (1,1) structure, a generic Kähler potential for the remaining 3 massless modes would be given by where the ellipsis represents contributions from other massive moduli. The matter kinetic mixings are thus induced by the intersection number κ abc and/or vacuum expectation values of heavy moduli, leading to the non-trivial flavor structure of 27 zero-modes. The structure of physical Yukawa couplings (12) would be different from the holomorphic one due to the matter Kähler mixings. So far, we have focused on the Kähler moduli sector accompanying the 27 zero-modes, but these statements hold for the opposite case with h 2,1 > h 1,1 , where 27 zero-modes are in one-to-one correspondence with the complex structure moduli. Note that the holomorphic 27 3 Yukawa couplings are also determined by triple intersection numbers κ ijk in the moduli space with the large complex structure. However, there is no definite way to analyze the CY moduli space with a large Euler number, although there exist the early three-generation models on a specific CY threefold taking into account freely acting discrete symmetries [25,26]. In this respect, we move on to toroidal orbifolds with and without resolutions, on which physical Yukawa couplings of untwisted and twisted modes are analyzed in detail below. The blown-up toroidal orbifolds would capture the structure of physical Yukawa couplings on smooth CY threefolds.
where the number of untwisted complex structure moduli is listed in Table 1.
The number of the Kähler moduli is larger than that of the complex structure moduli in all the toroidal orbifolds, and consequently the chiral zero-modes are originating from the Kähler moduli. 5 In this respect, we examine the holomorphic 27 3 Yukawa couplings. Let us study the holomorphic 27 3 Yukawa coupling associated with the untwisted Kähler moduli on T 6 /Z 7,8,12 and T 6 /(Z N ×Z M ) orbifolds, on which the non-vanishing triple intersection number is evaluated as Hence, the untwisted Kähler moduli space on these toroidal orbifolds is described by the special . When we identify the Higgs field and three generations of quarks/leptons with, for instance, the elements of A 1 and A 1,2,3 associated with the untwisted Kähler moduli, the holomorphic Yukawa couplings of quarks and leptons become the ranktwo matrix. There is no significant difference between the physical Yukawa couplings and the holomorphic one as analyzed before. In this way, it is difficult to realize the hierarchical structure of Yukawa couplings from the untwisted Kähler moduli. Similar phenomena occur in the 27 3 Yukawa couplings associated with the untwisted complex structure moduli on all the toroidal orbifolds whose Kähler potential is enumerated in Eqs. (20)- (22). One of the possibilities to realize the difference between the holomorphic Yukawa couplings and the physical one is to incorporate the off-diagonal Kähler metric of matter fields. The matter kinetic mixings are sourced by (i) the unfactorizable toroidal orbifolds such as Z 3,4,6 orbifolds and/or (ii) the twisted modes in the effective action. In the next section, we study both possibilities to clarify the crucial role of off-diagonal entries in matter field Kähler metric. Before concluding this section, we exhibit a constraint on the moduli Kähler metric in the analysis of 27 3 Yukawa couplings.
In the context of heterotic string theory, the value of 4D gauge coupling is directly related to the size of internal volume. We recall that the ten-dimensional (10D) heterotic supergravity action in string frame for our interest is of the form: where F denotes the gauge field strength of E 8 × E 8 or SO(32) gauge group with the adjoint representation. The 10D gravitational and gauge couplings are related as with the string length l s = 2π √ α . When we compactify the theory on a 6D internal manifold M with the volume Vol(M) = Vl 6 s measured in units of the string length, the 4D gauge coupling is determined by the internal volume V and the string coupling g s = e φ , It indicates that the internal volume should obey where we impose the small string coupling g s ≤ 1 and the 4D gauge coupling α −1 = 4πg −2 4 25 required in supersymmetric SU (5) grand unified theory. Supersymmetry breaking at a high energy scale may allow larger α −1 . In the following analysis, we allow for V ≤ 30 to be applicable to a broad class of 4D low-energy models. 6 A strong correlation between the 4D gauge coupling and the size of internal volume is well known in a global model such as the heterotic string theory, but we will argue that the limited range of internal volume also constrain the structure of physical 27 3 Yukawa couplings as discussed in detail later.

Hierarchical structure of physical Yukawa couplings
In this section, we explicitly demonstrate the hierarchical structure of physical Yukawa couplings on three backgrounds such as (i) T 6 /Z 3 orbifold without twisted modes in Section 3.1, orbifold with twisted modes in Section 3.3, respectively.

T 6 /Z 3 orbifold
For the first illustrative example, we deal with the T 6 /Z 3 orbifold, where the moduli fields on T 6 /Z 3 geometry consist of 9 untwisted and 27 twisted Kähler moduli. It is known that the Kähler potential of the untwisted Kähler moduli described by the with and the corresponding triple intersection numbers are given by and 0 otherwise. The matter Käher potential and corresponding metric are also known as [36] where α denotes the indices of SU (3) ⊂ E 8 . Note that the indices of α and a are not identified with each other. The superpotential takes the form 7 where κ abc and αβγ correspond to the anti-symmetric tensors for SU (3) isometry and SU (3) ⊂ E 8 , respectively. In contrary to the previous analysis, there exist the non-vanishing matter kinetic mixing. To examine the effects of off-diagonal entries in matter Kähler metric to the physical Yukawa couplings, we focus on 27 3 Yukawa couplings of untwisted modes {A 1 1 , A 2 2 , A 3 3 } by assuming that they are protected from becoming massive. Note that the following analysis is applicable to other untwisted modes and we rewrite {A 1 1 , A 2 2 , A 3 3 } as {A 1 , A 2 , A 3 } to shorten the notation. Furthermore, we restrict ourselves on the locus T 4 = T 5 = T 6 = T 7 = T 8 = T 9 to simplify our analysis. On this locus, the moduli Kähler potential and matter Kähler metric are evaluated as follows: with t a = Re(T a ), which reduces to the diagonal matter Kähler metric in the t 4 → 0 limit, namely the value of t 4 controls whether the Kähler metric is diagonal.
When the vacuum expectation values of the untwisted Kähler moduli are restricted along the isotropic locus, with x > 1 and p ≥ 0, 8 eigenvalues of the moduli Kähler metric are simply obtained as by employing the following diagonalizing matrix of moduli Kähler metric: The physical Yukawa couplings are then computed by substituting these into the formula (12). It turns out that the physical Yukawa couplings Y1âb and Y2âb become the rank-two matrices due to the presence of degenerate eigenvalues in the Kähler metric (35). For that reason, we assume that the Higgs field is originating from A3, and quarks/leptons are the elements of (A1, A2, A3), respectively. Indeed, from an analytical expression of physical Yukawa couplings Y3âb, the determinant turns out to be non-zero unless p = 0. One can realize the rank-three physical Yukawa matrix unless p = 0, although the holomorphic Yukawa coupling (16) is rank-two matrix. Hence, off-diagonal entries in the matter field Kähler metric bring us a rich structure of the physical Yukawa couplings. In the current parametrization of the moduli fields (34), eigenvalues of Y3âb are calculated as with the order |m 1 | > |m 2 | > |m 3 | and these ratios become satisfying |m 3 |/|m 2 | = 1/3. Taking into account the constraint for the internal volume it turns out that the typical ratio |m 3 |/|m 1 | is of O(0.01) to be consistent with the realistic value of 4D gauge coupling.
To exemplify the realization of the hierarchical physical Yukawa couplings, we search the anisotropic untwisted Kähler moduli by changing the moduli locus (34) to with x, y, z > 1. When we adopt the above parametrization of the moduli fields, the moduli Kähler metric (33) is diagonalized as follows: with Λ ± = x + y + z ± x 2 + y 2 + 2xz + 9z 2 − 2y(x + z).
Here, the diagonalization of the Kähler metric is performed by with We arrive at the physical Yukawa couplings by substituting these quantities into Eq. (12).
Since the analytical expression of the physical Yukawa couplings is complicated with respect to the moduli fields, we rely on the numerical search to find out desirable hierarchical structure of physical Yukawa couplings. In particular, we focus on physical Yukawa coupling Y3âb, setting one of moduli fields at specific values. Note that Y1âb becomes the rank-two matrix and Y2âb has the same structure as Y3âb due to the structure of diagonalizing matrix (44).  Remarkably, there exists an overlap region between the region with the hierarchical Yukawa couplings and the desired value of the toroidal volume V ≤ 30. Note that the volume of the toroidal orbifold V = 8(x − z) (xy + (y − 2z)z) goes to zero when y and z approach to x. In Table 2, we show three benchmark values for the ratio (1, r 1 , r 2 ) at several values of the Kähler moduli.  Table 2. Benchmark ratios for the ratio (1, r 1 , r 2 ) and the toroidal volume V at several moduli values.

Blown-up T 6 /(Z 2 × Z 2 ) orbifold
In this section, we discuss the T 6 /(Z 2 × Z 2 ) orbifold including twisted modes. On the T 6 /(Z 2 × Z 2 ) geometry, the Kähler metric of matter fields associated with the untwisted Kähler moduli is factorizable and their physical Yukawa couplings have the same structure of the holomorphic Yukawa couplings. Hence, we study in detail the impacts of twisted Kähler moduli on the physical Yukawa coupling.
The Kähler form on the blown-up T 6 /(Z 2 × Z 2 ) geometry is expanded as where t a denotes the untwisted Kähler moduli associated with divisors R a represented as dz i ∧dz i using the complex coordinates of tori z i , whereas s r denote the twisted Kähler moduli associated with exceptional divisors E r . Note that t a and s r are chosen to be positive to give a geometrical interpretation, and furthermore t a should be larger than s r to ensure the positivity of curves, divisors and entire volume.
In this paper, we restrict ourselves to the simplified case, where all the twisted Kähler moduli are identified with t 4 ≡ s r , (r = 1, 2, · · · , 48). (47) Here and in what follows, we follow the result in Eq. (6.8) of Ref. [37], although there exists a huge possibility of resolutions. (For more details, see, Refs. [37,38].) The Kähler potential on resolutions of the blown-up T 6 /(Z 2 × Z 2 ) orbifold is given by with t a = Re(T a ), approaching to the volume of T 6 /(Z 2 × Z 2 ) orbifold in the limit t 1,2,3 t 4 . Here the effective triple intersection numbers are with a = 1, 2, 3, and 0 otherwise. The inclusion of the blow-up modes gives rise to the matter kinetic mixing. Indeed, the Kähler potential has a similar structure to T 6 /Z 3 orbifold with the identification T 4 = T 5 = T 6 = T 7 = T 8 = T 9 as shown in Eq. (33), where only the difference is the coefficient of (T 4 + T 4 ). In the following subsections, we assume the specific modes protected becoming massive to reveal the structure of physical Yukawa couplings. In particular, we analyze two types of Yukawa couplings: (i) three-point couplings of untwisted modes themselves in Section 3.2.1 and (ii) three-point couplings of (untwisted)-(untwisted)-(twisted) modes in Section 3.2.2.

Yukawa couplings of untwisted modes
To analyze the structure of physical Yukawa couplings of untwisted modes themselves, we search the anisotropic untwisted Kähler moduli: with x, y, z > 1. When we adopt the above parametrization of the moduli fields, the moduli Kähler metric is calculated as Here, the diagonalization of the Kähler metric is performed by with We arrive at the physical Yukawa couplings by substituting these quantities into Eq. (12). In a similar way with analysis on T 6 /Z 3 orbifold, we perform the numerical search to clarify the structure of physical Yukawa couplings by setting x = 4. Since the value of blow-up mode has to be smaller than the other moduli values to justify our effective action, we choose a relatively large value for x, that is, x = 4. We find that physical Yukawa coupling Y1âb becomes the ranktwo matrix due to the structure of diagonalizing matrix (52). In this respect, the Higgs field is assumed to either elements of A 2 or A 3 , and quarks/leptons are the elements of (A 1 , A 2 , A 3 ).
In Figures 3 and 4, we plot two ratios of eigenvalues of the physical Yukawa couplings Y2âb and Y3âb to the maximum one, with the order |m 1 | > |m 2 | > |m 3 |, respectively. The left and right panels in Figures 3 and  4 correspond to the r 1 and r 2 with respect to y and z, respectively. In the same way as the results on T 6 /Z   Figure 4. In the left and right panels, we plot two ratios r 1 and r 2 of eigenvalues of the physical Yukawa couplings (Y3âb) to the maximum one with respect to y and z, respectively. The dotted, dashed, and solid curves correspond to the values for 0.5 (10 −1 ), 10 −1 (10 −2 ), and 10 −2 (10 −3 ), in the left (right) panel, respectively. In both panels, the blue shaded regions represent the allowed volume 1 < V ≤ 30.

Yukawa couplings of the twisted mode
In this section, we analyze the Yukawa couplings among two untwisted modes and the twisted mode by assuming that the matter fields in the SM are originating from the untwisted modes (A 1 , A 2 , A 3 ) and the twisted mode A 4 . Since the holomorphic Yukawa coupling as well as the Kähler metric of the untwisted modes (A 1 , A 2 , A 3 ) is totally symmetric, we focus on the Yukawa coupling of (A 1 , A 2 , A 4 ) without loss of generality. The holomorphic Yukawa couplings are of the form which are rank 1 and 2 matrices, respectively. In the following, we examine the structure of physical Yukawa couplings with an emphasis on three classes of holomorphic Yukawa couplings, namely (i) κ 1ab , (ii) κ 2ab , (iii) κ 4ab , where the Higgs field is identified with the element of A 1 , A 2 and A 4 , respectively. In all cases, quarks/leptons are assumed to the elements of (A 1 , A 2 , A 4 ). The moduli Kähler metric is calculated as with t a = Re(T a ) and hereafter, we adopt the parametrization of the moduli vacuum expectation values (50) in the same way as the previous analysis. We omit the complicated expressions for the eigenvalues of the Kähler metric as well as its diagonalization matrix. The physical Yukawa couplings are calculated from the formula From our numerical search along the moduli locus (50), we find that the rank of physical Yukawa couplings in all three cases (i)-(iii) has two and there is no significant differences among them. Hence, we show the non-vanishing ratio of physical Yukawa coupling in the case (iii) as shown in Figure 5 with x = 4. The difficulty of realizing the full rank of physical Yukawa couplings would be attributable to the trivial structure of holomorphic Yukawa couplings.

A mirror dual of
As a final example, we deal with a mirror dual of rigid T 6 /(Z 3 × Z 3 ) orbifold as developed in Ref. [39]. In contrary to the previous backgrounds, there exist h 2,1 = 30 number of complex structure deformation and no Kähler deformation. Hence, there will be no constraint on the internal volume that determines the value of 4D gauge coupling. The chiral zero-modes for our interest are originating from the complex structure moduli whose physical Yukawa couplings are analyzed in detail below.
In this section, we show the hierarchical structure about the physical Yukawa couplings for chiral zero-modes associated with the complex structure moduli. In particular, we focus on one of the blow-up modes (χ) among totally 27 blow-up modes, which is expanded in the small complex structure regime. On the other hand, the bulk complex structure moduli U i are considered the large complex structure regime, that is, with i = 1, 2, 3. Note that we still call "blow-up modes" on a mirror dual of T 6 /(Z 3 × Z 3 ) orbifold, since they correspond to Kähler moduli associated with exceptional divisor on a rigid T 6 /(Z 3 × Z 3 ) orbifold. The moduli Kähler potential on a mirror dual of rigid T 6 /(Z 3 × Z 3 ) is known as taking into account the regime (58). In addition, the holomorphic Yukawa couplings are also found by calculating the integrals of the holomorphic three-form over three-cycles. The nonvanishing holomorphic Yukawa couplings under the regime (58) are given by where we omit an expression of exponentially-suppressed holomorphic Yukawa couplings of the blow-up mode. Since we are interested in assessing to what extent the Kähler mixing induces the hierarchical structure of physical Yukawa couplings, we focus on the Yukawa couplings of untwisted modes. It is remarkable that the blow-up mode is now expanded in the small complex structure regime and in the blow-down limit χ → 0, other holomorphic Yukawa couplings such as κ χχU i and κ χU i U j vanish. We refer Ref. [39] for the derivation of the moduli effective action, which can be achieved by calculating the holomorphic three-form of CY threefold embedded into the complex projective space CP 8 .
We are ready to analyze the physical Yukawa couplings (12) with the Kähler metric given by (59) and the holomorphic Yukawa couplings (60), in particular for the untwisted modes. The explicit form of the Kähler metric is given by with τ i ≡ Im(U i ), where the value of blow-up mode |χ| encodes the amount of kinetic mixing associated to a non-diagonal Kähler metric. In the blow-down limit |χ| → 0, the off-diagonal components of the Kähler metric go to zero, and these play an important role of realizing the rank-three physical Yukawa matrix with the hierarchical structure as shown below. We numerically analyze the structure of physical Yukawa couplings Y1ˆiĵ, Y2ˆiĵ and Y3ˆiĵ by setting and the other moduli values (τ 2 ,τ 3 ,χ) are chosen as free parameters. It turns out that the rank of physical Yukawa couplings {Y1ˆiĵ, Y2ˆiĵ, Y3ˆiĵ} is three in the moduli space except for τ i = τ j with i = j. When at least two moduli values are coincides with each other, two eigenvalues of the Kähler metric are degenerate due to the symmetric structure of Kähler metric with respect to τ i . The phenomenon is observed in the following numerical results. In a general moduli space, eigenvalues of physical Yukawa couplings (m 1 , m 2 , m 3 ) with |m 1 | ≥ |m 2 | ≥ |m 3 | satisfy the relation for all {Y1ˆiĵ, Y2ˆiĵ, Y3ˆiĵ}. Since we focus on the moduli space within (58), the off-diagonal entries in Kähler metric (61) should be smaller than the diagonal one. It results in two degenerate eigenvalues of physical Yukawa couplings and those are determined by the rank-two holomorphic Yukawa matrix (16). The smallest eigenvalue of physical Yukawa coupling has an origin in the smallness of the off-diagonal entries in the Kähler metric. Indeed, the ratio r 2 = |m 3 |/|m 1 | is sensitive to the value of the blow-up modulus χ as indicated in Figure 6. We plot the ratio r 2 of the physical Yukawa coupling Y3ˆiĵ, but the same structure arises in the other physical Yukawa couplings. It is remarkable that there exist a huge hierarchy between m 3 and others, due to the suppressed off-diagonal entries in Kähler metric (61). Indeed, the ratio of off-diagonal components to diagonal one in |χ| → 0 limit is typically of O(3.5 × 10 −2 |χ| 2 (τ i ) −3 ), when the moduli values τ i are assumed to be the same order. From the formula of physical Yukawa couplings (12), the smallest ratio of eigenvalues of the physical Yukawa couplings, r 2 , is then further suppressed by the value of blow-up mode through the matter Kähler metric. That phenomenon is consistent with our numerical results, indicating that the ratio r 2 becomes smaller and smaller as |χ| decreases.  Figure 6. We plot the smallest ratio r 2 of eigenvalues of the physical Yukawa couplings Y3âb to the maximum one with respect to τ 2 and τ 3 by setting |χ| = 0.1 and |χ| = 0.01 in the left and right panels, respectively and other moduli values are set as in Eq. (62). In both panels, red lines correspond to r 2 = 0, namely rank-two Yukawa matrix. In the left (right) panel, the dotted, dashed and solid curves correspond to the values for 4 × 10 −5 (3 × 10 −7 ), 2 × 10 −5 (2 × 10 −7 ), and 10 −5 (10 −7 ), respectively.

Conclusions
The non-trivial structure of physical Yukawa couplings has an origin in the matter field Kähler metric and the holomorphic Yukawa couplings in the context of 4D N = 1 supersymmetric theory. In the context of heterotic orbifold models, it was known that hierarchical Yukawa couplings are realized in the twisted sector due to the exponentially-suppressed holomorphic Yukawa couplings, whereas the Yukawa couplings among the untwisted modes are typically of O(1) in the heterotic string theory with standard embedding, where matter fields are assigned to the fundamental or anti-fundamental representations of E 6 gauge group. Hence, we have mainly focused on the Yukawa couplings of untwisted modes whether the matter field Kähler metric provide their hierarchical structure. Since the matter fields 27 and 27 are respectively in one-to-one correspondence with the Kähler moduli and complex structure moduli, one can argue definite properties of the matter field Kähler metric in the physical Yukawa couplings.
We have analyzed 27 3 and 27 3 Yukawa couplings of untwisted modes on a simplified CY threefold in Section 2.2.1 and the untwisted and twisted modes on toroidal orbifolds preserving the supersymmetry in Section 2.2.2. It turned out that off-diagonal entries in matter field Kähler metric are necessary to generate the rank-full physical Yukawa couplings with hierarchical structure. There exists the non-vanishing matter Kähler mixing on unfactorizable tori such as T 6 /Z 3,4,6 orbifolds and blown-up toroidal orbifolds. Indeed, we performed the numerical search on T 6 /Z 3 orbifold, blown-up T 6 /(Z 2 × Z 2 ) orbifold, and a mirror dual of T 6 /(Z 3 × Z 3 ) orbifold. In the former two examples, we exhibited that the hierarchical structure of physical Yukawa couplings of matter zero-modes associated with Kähler moduli is provided by O(1) values of moduli fields, taking into account the realistic value of 4D gauge couplings. In the latter example, the complex structure moduli space accompanying chiral matter 27 was analyzed in detail. We found that the small but non-zero value of the blow-up mode, which is required to justify the effective action, leads to the suppressed off-diagonal components of the matter field Kähler metric, thereby providing the hierarchical structure of physical 27 3 Yukawa couplings of untwisted modes.
So far, we have focused on the realization of hierarchical structure of physical Yukawa couplings of untwisted modes by means of the structure of matter Kähler metric. These scenarios would be widely applicable to more realistic models such as SU (5) or SO(10) grand unified theories achieved by introducing Wilson-lines on the Cartan directions of E 6 and theories with smaller gauge groups. For example, many realistic models have been constructed in T 6 /Z 6−II orbifold models [40][41][42]. In these models, the third generation of left-handed (right-handed) quarks and leptons corresponds to the untwisted sector A 1 (A 2 ) and the Higgs field corresponds to the untwisted mode A 3 , while the other two generations of quarks and leptons are originated from twisted sectors. At the orbifold limit, the Yukawa matrices are rank-one, and only the third generation can gain masses [43]. Thus, it would be interesting to extend our analysis to above models on blown-up T 6 /Z 6−II orbifold 9 . Including these topics, we leave a pursue of more phenomenologically interesting three-generation models for future work.
The moduli vacuum expectation values are assumed to be free parameters in our analysis to find the structure of physical Yukawa couplings. However, a realization of the hierarchical structure of physical Yukawa couplings is highly dependent on the stabilization mechanism of the moduli fields. The moduli stabilization in the context of heterotic string theory is still open challenging issue, since it generically induces the backreaction on the background geometry. We comment on the possible mechanism of the moduli stabilization which we leave for future work. When the matter zero-modes have an origin in the Kähler moduli sector, we require some nonperturbative effects to stabilize the Kähler moduli. We often employ the gaugino condensation on the hidden sector with one-loop threshold corrections depending on the Kähler moduli and/or world-sheet instanton effects on the two-cycles of CY threefolds. When the matter zero-modes have an origin in the complex structure moduli sector, a flux compactification is a powerful tool to determine the vacuum expectation value of these moduli fields. Indeed, the flux compactification on a mirror dual of rigid CY threefolds in the example of Section 3.3 is suffice to stabilize all the moduli fields. We hope to come back to these problems in the future.