Top-Down Anatomy of Flavor Symmetry Breakdown

A top-down approach to the flavor puzzle leads to eclectic flavor groups which include modular and traditional flavor symmetries. Based on examples of semirealistic T2/Z3 orbifold compactifications of heterotic string theory, we discuss the breakdown patterns of the eclectic flavor group via the interplay of vacuum expectation values (vevs) of moduli and flavon fields. This leads to an attractive flavor scheme with various possibilities to obtain"flavor hierarchies"through the alignment of these vevs. Despite the fact that the top-down approach gives strong restrictions for bottom-up flavor model building, it seems to be well suited to provide a realistic flavor pattern for quarks and leptons.


Introduction
In his influential work [1], Feruglio considered finite modular groups for flavor model building. This proposal has triggered intense activity in the construction of bottom-up (BU) models of quark and lepton interactions based on various finite modular groups [2][3][4][5][6][7] (see e.g. the review ref. [8] for a complete set of references and further details on the BU approach). One successful result of these constructions is that they can provide good fits to data by typically requiring a smaller number of free parameters than in models endowed with traditional flavor symmetries. However, the predictability of such BU constructions may be challenged through the appearance of uncontrollable terms in the Kähler potential [9]. In the BU approach there are thus many working models with some degree of predictability but we still lack a baseline theory coming from an underlying fundamental principle.
To address this problem, there has been extensive work devoted toward top-down (TD) derivations of modular flavor symmetries from ultraviolet complete models based on string theory [10,11]. Apart from heterotic orbifold compactifications [11,12], TD models include scenarios based on compactifications on toroidal orientifolds [13] and magnetized tori [14][15][16][17]. TD constructions typically give strong restrictions on the allowed symmetries and the particle spectrum, and may also allow one to constrain (or even compute) the otherwise free terms of the Kähler potential [17][18][19][20]. Furthermore, TD models naturally include unification of flavor with CP-like transformations [11,12,21,22]. This unification becomes even more transparent in models where the modular group is extended to its metaplectic [17,23,24] or symplectic cover [25][26][27][28][29] as recently discussed both in the BU and TD approaches.
In the present paper, we concentrate on the TD approach. This approach to the flavor problem leads to a holistic view that necessarily encompasses all available kinds of discrete flavor symmetries. It thus has to include traditional flavor symmetries and R-symmetries as well as finite modular flavor symmetries and their associated CP transformations [11]. These symmetries reflect the symmetries of the underlying UV complete theory, which we consider here in the framework of heterotic strings with compactifications with elliptic fibrations. The discrete flavor symmetries can be derived in full generality as the outer automorphisms of the Narain space group [11,12]. This leads to the concept of the eclectic flavor group [18,20,30,31] as a multiplicative closure of all flavor symmetries. The eclectic flavor group is the maximal possible flavor group, but it is only partially linearly realized. The linearly realized flavor subgroup is nonuniversal in moduli space, a property that leads to the concept of "local flavor unification" [12]. This holistic picture teaches us some general lessons: • One cannot just consider a specific flavor group (e.g. modular flavor) without the others.
• There is always a traditional flavor group (universal in moduli space) that could give severe restrictions to the Kähler potential and superpotential of the theory.
• Apart from the finite discrete modular flavor group and its specific representations, one has to consider the modular weights of the matter fields as well, as they might lead to further R-symmetries that play the role of "shaping symmetries".
Flavor symmetries have to be spontaneously broken and thus the full eclectic picture requires several sources of breakdown. On the one hand this leads to a serious complication of the picture. On the other hand, it is welcome once we want to obtain the hierarchical structure of masses and mixing angles of quarks and leptons. While the modular group can be broken via the moduli (with hierarchical patterns close to the fixed points of SL(2, Z)), we have to consider additional flavon fields to break the traditional flavor symmetries via nontrivial vacuum expectation values (vevs). These flavon vevs might then lead to a further breakdown of the discrete modular flavor symmetry as well. It is this subtle interplay of breakdown via moduli and flavons that is the main subject of the present paper. We shall see that there are various ways to obtain hierarchical patterns from the breakdown of the eclectic flavor group via moduli and flavons.
Flavor symmetries should, of course, also be discussed from a BU perspective. There one has the free choice of the groups, the representations and modular weights of matter fields to confront a model with existing data. Ideally one might hope to find a specific model as a "best fit" to masses and mixing angles of quarks and leptons. This then might give useful hints towards a fundamental theory of flavor. Unfortunately, no one has yet been able to identify such a distinct model (or even a class of models). Good fits to the data can be achieved for various groups and representations. Still, even in the BU approach, we might try to find some theoretical guidelines. If we, for example, have a traditional flavor group G, we could design an eclectic scheme with a discrete modular flavor group that is included in the outer automorphisms of the group G [30]. This could be a first step to bridge the gap between the TD and BU approaches to flavor. Both approaches enjoy various desirable properties as e.g. the appearance of local flavor unification with enhanced symmetry at certain fixed points or regions in moduli space, eventually broken spontaneously via moduli vevs.
The TD approach is very restrictive and a challenge for realistic model building: • We first have to design models with the desired flavor groups explicitly in string compactifications.
• The explicit representations of the flavor group are then fixed and cannot be chosen by hand.
• Likewise, modular weights are fixed and determined as well.
• There are various restrictions from R-symmetries that appear in the six-dimensional compactification.
Given these restrictions, there remains still a wide gap between TD attempts and the models considered in the BU approach.
There have not been any attempts yet in TD model building, and in this paper, we want to make a first step in this direction. We are particularly interested in string models with elliptic fibrations, and these are classified according to two-dimensional T 2 /Z k orbifold sectors with k = 2, 3, 4, 6. The traditional flavor symmetries of these scenarios were analyzed some time ago [32,33]. In those works, it is found that the T 2 /Z 3 orbifold sector (which in some sense also qualitatively covers the Z 6 case as well) leads to the most promising class of string models that reproduce the matter spectrum of the minimal supersymmetric standard model (MSSM). These models are endowed with traditional flavor symmetry ∆ (54) and twisted states that transform as (irreducible) triplet representations of this group. Therefore, we want to concentrate on this class of models with traditional flavor symmetry ∆(54) and modular flavor symmetry T . Fortunately, there have been explicit semirealistic model constructions of heterotic string theory that exhibit elliptic fibrations of type Z 3 [34]. A full classification of these models with the relevant field content is given in table 3 of the present paper.
All of these models share the same eclectic flavor group but differ in the available representations of candidate flavon and matter fields. A first step toward phenomenological applications would then be the analysis of breakdown patterns of the eclectic flavor group and this is the main goal of the present paper. We have to clarify which fields are needed for an efficient breakdown of the eclectic flavor group which, in addition, allow a hierarchical pattern for a successful description of masses and mixing angles of quarks and leptons (originating partially through the proximity to local gauge group enhancements). Before proceeding to explicit model building, we would therefore like to know the qualitative breakdown pattern of ∆ (54) and T . In future work [35] we shall then use these results for the selection of suitable models (from the classes in table 3) for phenomenological applications.
The paper is structured as follows: In section 2, we describe a T 2 /Z 3 orbifold sector with traditional flavor group ∆(54) and eclectic flavor group 1 Ω(2) ∼ = [1944,3448] (as the multiplicative closure of ∆(54), T and Z R 9 ), derived as the two-dimensional elliptic fibration of a T 6 /(Z 3 × Z 3 ) orbifold. We identify the representations of the groups in the massless sector including the modular weights. At a generic point in moduli space, we have a flavor symmetry ∆(54)∪Z R 9 = ∆ (54, 2, 1) ∼ = [162,44]. We discuss in detail the enhancements at the fixed points of the Kähler modulus T = i, 1, and ω := exp ( 2πi /3) that lead to the groups Ξ(2, 2) ∼ = [324, 111] (for T = i) or [468,125] (for T = 1, ω). In section 3 we discuss the possible breakdown of ∆(54) via flavon vevs. Candidate flavons are identified via the inspection of the massless sector of the T 6 /(Z 3 × Z 3 ) orbifold under consideration. We show that triplets of ∆(54) are very efficient in the breakdown of the traditional flavor group. We extend this analysis to the breakdown of the eclectic flavor group in section 4, with particular attention to the groups [324,111] and [468,125] at the fixed points T = i and T = 1, ω, respectively. The various breakdown patterns are summarized in the figures 1, 2, and 3 with details given in tables 4, 5, and 6. In section 5 we conclude by discussing the relevance of our analysis to flavor model building and give an outlook to future work.
2 Holistic picture of the eclectic flavor symmetry Eclectic flavor groups [30] arise naturally in compactifications of string theory [16,18,20,31]. They consist of a nontrivial combination of the effective traditional flavor symmetries G traditional and finite modular symmetries G modular under which string states are charged. Eclectic flavor groups can also include CP-like transformations and discrete R-symmetries. Let us here discuss the appearance of these symmetries in detail.
We focus on factorizable six-dimensional orbifolds, which contain three two-dimensional T 2 /Z N orbifold sectors. Each of the T 2 is endowed with two modular groups, SL(2, Z) U and SL(2, Z) T , associated respectively with the complex structure modulus U and the (stringy) Kähler modulus T of the torus. For N = 2, the values of both T and U remain unrestricted. For N > 2, due to its geometric nature, U has to be fixed to a value that is compatible with the orbifold twist Z N . In these cases SL(2, Z) U is broken down to a discrete remnant of the Lorentz group of discrete rotations in the compact dimensions, which is an Abelian group compatible with the orbifold. This subgroup appears as a discrete R-symmetry in the effective theory [31]. SL(2, Z) T , however, remains a symmetry of the effective theory. This symmetry is nonlinearly realized, as can be seen from its action on the modulus T and matter fields Φ n .
An element γ ∈ SL(2, Z) T transforms these fields according to [38,39] Here, Φ n denotes a multiplet of string matter states with identical quantum numbers, except for their location at different orbifold singularities. Furthermore, n denotes the modular weight of the matter multiplets Φ n , (cT + d) n is known as automorphy factor, and ρ s (γ) is an sdimensional matrix representation of γ in a (discrete) finite modular group G modular that depends on the Z N twist of the orbifold sector.
On the other hand, the traditional flavor symmetries G traditional can be identified through the geometric features of toroidal orbifolds [32]. First, G traditional includes the permutations among the various equivalent orbifold singularities where matter states comprising the multiplets Φ n are located. Since permutations are non-Abelian, so are these traditional flavor symmetries. Second, G traditional contains the discrete symmetries governing the admissible couplings among the states in Φ n , which are known as string selection rules. The resulting traditional flavor group is obtained by multiplicative closure of these two types of symmetries. It is universal in moduli space, as all its elements only act on Φ n .
In addition to traditional flavor and classical modular transformations, there are CP-like transformations. On the moduli T and U these act with an element γ of determinant −1, thereby enhancing the respective modular groups to SL(2, Z) T,U Z 2 ∼ = GL(2, Z) T,U . Fixing U by orbifolding selects specific CP-like transformations compatible with the twist Z N and fixed value of U [12,40]. The general action on the modulus T is given by [11,12] (see also [40] as well as [21] for the BU approach) while string matter multiplets transform as Here, bars denote complex conjugation. 2 Interestingly, with the help of the Narain formulation of toroidal orbifolds [43,44] (see [45] for further technical details), it is found that the outer automorphisms of the orbifold space group yield all symmetries of the effective theory of these compactifications. In particular, all of G traditional and G modular , as well as the discrete R-symmetries and the CP-like transformations turn out to have their origin in these outer automorphisms, revealing a unified origin of flavor in string compactifications [11].
This rich set of flavor symmetries containing all these elements builds the eclectic flavor group of a toroidal orbifold. Thus, an eclectic flavor group is the multiplicative closure of the various flavor subgroups, i.e.
where G R denotes here the Abelian discrete R-symmetry, remnant of SL(2, Z) U .
One of the phenomenological advantages of models endowed with eclectic flavor symmetries is that they provide control over the structure of the effective superpotential and Kähler potential, preventing in particular the loss of predictability that is found in models based on finite modular symmetries only [9]. The predictive power of the eclectic picture results from the large amount of symmetry of G eclectic . Phenomenological applications require a spontaneous breakdown of this huge symmetry, and this is a challenge for flavor model building. One of the goals of this paper is precisely to address this question in an illustrative example of the eclectic picture based on string theory.

The eclectic flavor symmetry of T 2 /Z 3
We chose the T 2 /Z 3 orbifold sector as an example to illustrate the phenomenological potential of eclectic flavor symmetries. The outer automorphisms of the corresponding Narain space group yield the following symmetries [12,18,31]: • an SL(2, Z) T modular symmetry which acts as a Γ 3 ∼ = T finite modular symmetry on matter fields and their couplings, 2 We note that, in general, not all of the transformations γCP correspond to transformations that map representations of string matter multiplets s to their complex conjugate representationss. Whether or not this is the case crucially depends on the nature of the outer automorphisms of the involved finite groups and the specific representations [41]. However, as we will consider throughout this work only the massless spectrum of T 2 /Z3, for which complex conjugation applies [42], we restrict ourselves to the statement of eq. (3).
• a ∆(54) traditional flavor symmetry, • a Z R 9 discrete R-symmetry as remnant of SL(2, Z) U , and • a Z CP 2 CP-like transformation.
As explained in detail in [31,40] and summarized in table 1, the SL(2, Z) T modular generators S and T arise from rotational outer automorphisms. These generators can be represented by and act on the modulus T and the matter fields Φ n according to eq. (1). Further, there is a reflectional outer automorphism which corresponds to a Z CP 2 CP-like transformation. It can be chosen to be represented by which acts on the modulus and matter fields as in eq. (2).
The traditional flavor symmetry ∆(54) is generated by two translational outer automorphisms of the Narain space group A and B of order 3 together with the Z 2 rotational outer automorphism C := S 2 . The automatic identification of C with S 2 implies that ∆(54) and the modular symmetry have a nontrivial overlap and, furthermore, that ∆(54) is actually a non-Abelian R-symmetry.
Finally, in our example, the complex structure modulus is geometrically stabilized at U = exp( 2πi /3) in order for T 2 to be compatible with the Z 3 point group. If T 2 /Z 3 is embedded in a six-dimensional orbifold, U breaks the original SL(2, Z) U of T 2 to a discrete remnant generated by R that acts as a Z R 9 symmetry on matter fields (normalizing their Rcharges to be integers).
Since ∆(54) and Z R 9 both act trivially on the modulus T , the traditional flavor symmetry 44]. Including the additional Z CP 2 CP-like transformation (3) the full eclectic group according to eq. (4) is a group of order 3888 given by where In semirealistic heterotic orbifold compactifications endowed with a T 2 /Z 3 orbifold sector, the massless spectrum consists of (i) untwisted string states that transform as flavor singlet states Φ 0 or Φ −1 free to move in the bulk, and (ii) twisted string states transforming as flavor triplets, attached to the three orbifold fixed points. The modular weights of twisted states depend on the twisted sector they belong to. As shown in table 2, in the θ sector only n ∈ { −2 /3, T representations of the modular generators [12] which form the representation s = 2 ⊕ 1 of the finite modular group Γ 3 ∼ = T ∼ = [24,3].
Furthermore, the action of the ∆(54) generators on Φ−2 /3 is given by and they generate the representation r = 3 2 of ∆(54). The representations associated with the other twisted triplets are expressed in terms of eqs. (8) and (9), according to the prescription given in table 2. Finally, the integer Z R 9 R-charges of matter fields can be uniquely determined by using their SL(2, Z) U properties and the fixed value U , see [40, sec. 4.2].
It is important to stress that the massless spectrum of the T 2 /Z 3 orbifold sector of heterotic orbifold compactifications does not include all possible representations of ∆(54). In particular, note that there are no massless doublet representations. In fact, doublets arise as winding modes of strings around different singularities and, hence, correspond to massive states [42].
At different points T in moduli space, SL(2, Z) T and T are broken down to the stabilizer subgroup, i.e. to the modular subgroup that leaves T invariant. Since the modulus is no longer transformed at T , the surviving symmetry yields an enhancement of the traditional flavor symmetry ∆(54). This enhancement, associated with the specific location in moduli space, has been called local flavor unification [12]. Next, we study the most relevant scenarios of local flavor unification in the T 2 /Z 3 orbifold sector. sector matter eclectic flavor group Ω(2) fields modular T subgroup traditional ∆(54) subgroup Table 2: T and ∆(54) irreducible representations and Z R 9 R-charges of massless matter fields Φ n with modular weights n in semirealistic heterotic orbifold compactifications with a T 2 /Z 3 sector. T , ∆(54) and Z R 9 combine nontrivially to the eclectic flavor group Ω(2) ∼ = [1944, 3448], as described in table 1. Untwisted matter fields Φ n (with integer modular weights n) form one-dimensional representations, while twisted matter fields Φ n (with fractional modular weights n) build triplet representations.

Flavor enhancement at T = i
First, we discuss the point T = i in moduli space (see section 6.2 in ref. [40] for details of the derivation). In this case, SL(2, Z) T is broken to a Z 4 subgroup generated by S. Twisted matter fields transform as Here we use c = −1, d = 0 for the automorphy factor (c T + d) n = (−i) n of S ∈ SL(2, Z) T at The superpotential also transforms under S as W S − → i W. This implies that also S generates a discrete R-symmetry, which altogether leads to a non-Abelian discrete R-symmetry [46].
At T = i, the explicit representation matrices (which include the associated automorphy factors) of the unified flavor group of twisted matter fields Φ−2 /3 and Φ−5 /3 are given by  The CP-like transformation generated by eq. (6) acts on the modulus as T → −T , which is conserved for T = i. The corresponding automorphy factor and representation matrix in eqs. (2) and (3) are trivial, see [11,40], such that 3 Φ n CP − − →Φ n .
Altogether, the linearly realized unified flavor group at T = i is found to be 2.3 Flavor enhancement at T = ω Next, following section 6.3 in ref. [40], we consider the point T = ω in moduli space. There, SL(2, Z) T is broken to a Z 3 subgroup generated by ST such that our twisted matter fields transform as Here we use c = d = −1 for ST ∈ SL(2, Z) T which yields (c T + d) n = (−ω − 1) n = (ω 2 ) n and, hence, the automorphy factors ( In addition, the superpotential transforms under ST as W ST − − → ω W. Hence, also here the residual flavor symmetry is a non-Abelian discrete R-symmetry [46].
The explicit representation matrices of the unified flavor group of twisted matter fields Φ−2 /3 and Φ−5 /3 are given by A representative of the Z CP 2 transformation is given by K * T, which acts on the modulus and matter fields as such that T = ω is left invariant.
The linearly realized unified flavor group at T = ω is thus 3 For explicit computations, it is useful to combine Φ ⊕Φ and extend also the other generators to this r ⊕r-dimensional representation.

Flavor enhancement at T = 1 and T = i ∞
Since the point T = 1 has not been investigated in the literature for residual symmetries, we give more details here. First, note that the discussion for T = 1 also applies to the point T = i ∞ because these points are dual via conjugation by the element ST −1 , i.e. working in the limit → 0 + , To find the local enhancement of the traditional flavor symmetry by the stabilizer of the modulus, note that T = 1 is left invariant by the γ ∈ SL(2, Z) T transformations satisfying Combining this with the constraint ad − bc = 1, we obtain Recalling that all variables here are integers, we observe that This yields the stabilizer at T = 1. All solutions to eq. (22) then read for b ∈ Z. Therefore, the stabilizer can be generated by the two elements The first generator corresponds to the ∆(54) element C, while the second generator locally Hence, for matter fields in ∆(54) representations r = 3 2 (for Φ−2 /3 ) or r = 3 1 (for Φ−5 /3 ) the explicit representation matrix of the generator that locally enhances the traditional flavor symmetry is given by ρ r, T =1 (ST) = ρ(ST) .
We note that the s = −1 case would give rise to a nontrivial automorphy factor and, hence, different matrix generators for 3 2 and 3 1 , but ultimately lead to exactly the same group.
Hence, altogether, the unified flavor group at T = 1 can be generated by One can show that the groups generated by eqs. (17) and (29), as well as by eqs. (18) and (30) are identical.
To identify the CP-like stabilizer at T = 1, we solve for an element γ CP ∈ GL(2, Z) T with det γ CP = −1. Hence, altogether Once more, recalling that the matrix representation of γ CP has integer entries, we see that This yields the CP-like stabilizer at T = 1. The solutions to eq. (31) can be written as for b ∈ Z. Since all these transformations differ only by an element of the stabilizer group itself, it is clear that they give rise to equivalent resulting groups. For definiteness, we choose for the generator of the CP-like stabilizer the element with s = 1 and b = 2, i.e. ρ( ST −2 −1 S K * ) = ρ(T 2 K * ), such that the resulting Z CP 2 acts on the modulus and matter fields as Combining this with the unified flavor group, the full linearly realized unified flavor group at T = 1 results as We realize that this result at T = 1 coincides with the linearly realized unified flavor group at T = ω, eq. (20).
For completeness, we note that the generator of the CP-like stabilizer at the dual point T = i ∞ can be represented by S 2 K * = −K * , which acts as T → −T and clearly leaves T = i ∞ invariant. The resulting linearized flavor group at T = i ∞ is, as expected, the same as for T = 1.

Heterotic configurations of T 6 /(Z 3 × Z 3 ) with Ω(2) eclectic symmetry
The T 2 /Z 3 orbifold sector appears naturally in six-dimensional T 6 /Z 3 , T 6 /Z 6 -II, T 6 /(Z 3 ×Z 3 ) and T 6 /(Z 3 × Z 6 ) orbifolds. These orbifolds have been explored in the search of models arising from heterotic string compactifications with the MSSM spectrum plus vectorlike exotics [33,47,48]. The details of the matter spectra of these constructions are determined by the choices of the embedding of the six-dimensional orbifold into the gauge degrees of freedom, subject to consistency requirements, such as modular invariance, see e.g. [49][50][51][52].
Inspecting the consistent semirealistic string models classified in [33,47,48] endowed with a T 2 /Z 3 orbifold sector, we observe that the light MSSM matter superfields appear only in a reduced number of field configurations. For example, considering the T 6 /(Z 3 × Z 3 ) (1,1) orbifold geometry (see ref. [53] for details on this geometry) with one and two vanishing Wilson lines, we identify five types of configurations of massless MSSM matter superfields, as summarized in table 3 in terms of the field labels used in table 2. Given these configurations, one can arrive at the flavor phenomenology of the Ω(2) eclectic flavor symmetry by using the effective superpotential and Kähler potential given in ref. [18], including the spontaneous breakdown of the eclectic flavor group triggered by the vevs of the indicated flavon representations and the Kähler modulus T . This shall be done explicitly for models of configuration type A in a companion paper [35].
To conclude this section, let us stress an important empirical TD observation. As we see in table 2 (and use in table 3), the modular weights alone suffice to characterize the transformation behavior of fields under the flavor symmetries. That is, there is here a one-toone relation between the modular weights and all flavor symmetry charges of the fields. This also holds for other known TD constructions, see e.g. [17,20,[54][55][56]. If this feature turned out to be correct for generic TD models, it would suggest that consistent BU constructions should abide to the same rule. Namely, fields of the same modular weight should also transform in the very same representation of all modular flavor symmetries.   3 (see e.g. [57] for a pedagogical introduction): We see that only the representation 3 1 associated with the matter field Φ−5 /3 branches into a trivial singlet of S 3 . Therefore, any nonzero vev of Φ −1 or Φ−2 /3 breaks not only ∆(54) but also the S  Table 4: Overview of subgroups of the ∆(54) traditional flavor group and the corresponding vevs for the breakdown. The second column shows the branching of the relevant representations into the subgroups. The 1 here denotes the trivial singlet, and nontrivial singlets are labeled by their eigenvalue under the Abelian generator. We provide examples for generators of each subgroup (up to conjugation) that are left unbroken by the different choices of vevs specified in the last two columns. The notation "⊕ Φ −1 " means that a vev of a nontrivial ∆(54) singlet field has to be switched on in addition to the vev of a triplet in order to achieve the breaking to the respective subgroup. The provided generators and vevs are not unique since their conjugation can yield equivalent ∆(54) subgroups. All the subgroups are stated up to conjugation, i.e. the groups are distinct and not related by conjugation. We omit here an arbitrary global (normalization) factor for each of the vevs, α is an arbitrary complex number, while "-" denotes the absence of a suitable vev. ρ 3 1 (A) = ρ(A) and ρ 3 1 (C) = −ρ(C) with eigenvalue 1, which is found to be (1, 1, 1) T , up to an arbitrary overall factor. Note that there are physically equivalent subgroups conjugate to the S (1) 3 group generated by A and C. Each choice of conjugate generators naïvely yields a different vev structure, but all of them are related by conjugation with an element of ∆(54). By contrast, any pattern of Φ−5 /3 vevs that cannot be related to (1, 1, 1) T by conjugation will break S Altogether we see that vevs Φ −1 , Φ−2 /3 , and Φ−5 /3 are quite efficient in breaking the traditional flavor symmetry. However, they can leave remnant symmetries unbroken and Ξ(2, 2) this might happen by default, as potentials are known to be often minimized at symmetry enhanced points (i.e. by vevs with remnant subgroups). In order to break the flavor symmetry completely, vevs can be misaligned from their symmetry enhanced directions (thereby still allowing for approximate or softly broken symmetries that can give rise to "flavor hierarchies"), or multiple vevs could be present simultaneously. From the stated patterns of single vevs it is straightforward to work out remnant groups also if multiple vevs are present. The respective remnant groups would be given as the nontrivial intersection of the preserved symmetries of each individual vev. We will further explore the phenomenological consequences of these scenarios in our forthcoming paper [35].
At a generic point in moduli space, the traditional flavor symmetry is ∆(54) and the results of the previous section apply. In this section, we investigate analogous breaking patterns at the symmetry enhanced points in moduli space. In detail, we study the points T = i and T = ω, 1, i∞, where the traditional flavor symmetry ∆(54) is enhanced, respectively, to Ξ(2, 2) ∼ = [324, 111] and H(3, 2, 1) ∼ = [486, 125] as discussed in sections 2.2 to 2.4. As derived above, these flavor symmetries are additionally enhanced by CP-like transformations. However, for practical reasons, we first focus on non-CP-like transformations and comment on the possible enhancements by CP-like transformations in the end.
We focus on the matter fields Φ−2 /3 and Φ−5 /3 from the θ sector of a T 2 /Z 3 orbifold, as well as on the bulk field Φ −1 , because these fields arise as potential flavons in the models H(3, 2, 1) Our results for the breakdown patterns induced by flavon vevs Φ−2 /3 , Φ−5 /3 and/or Φ −1 are depicted in figure 2 for the linearly realized unified flavor symmetry Ξ(2, 2) at T = i, and in figure 3 for the unified flavor symmetry H(3, 2, 1) at T = ω, 1, i∞. Details of our results are summarized in table 5 (for T = i) and table 6 (for T = ω, 1, i∞). Just as in the previous section, we determine the branching of the respective irreducible representations into subgroups of the flavor symmetries. If at least one trivial singlet 1 is present in the branching, the associated flavon vev can break the original unified flavor group to the listed subgroup. We do not list subgroups that cannot be realized by any of the given vevs because they would actually preserve a larger subgroup. We give explicit generators for all subgroups and the associated explicit vevs (up to conjugation). There exist multiple equivalent subgroups which can be obtained from the stated examples by conjugation.
As noted already above, the flavor symmetry enhancement at T = ω, T = 1 and at T = i∞ leads to isomorphic unified flavor groups. We have confirmed explicitly that also the respective matrix groups for the triplets generate identical groups (and irreducible representations). Hence, the analysis of the triplet vevs is exactly the same at all three points. Ξ(2, 2) subgroup branchings subgroup generator(s) Table 5: Maximal subgroups (up to conjugation) of Ξ(2, 2) that can be achieved from its breakdown by vevs of flavon fields at T = i. We provide the branchings of the flavon representations under the resulting subgroups, followed by samples of the generators of such subgroups and the flavon vevs that yield the corresponding breakdown. Note that the Z 2 generator satisfies C = (S 3 ) 2 . We follow the same notation as in table 4.
For definiteness, we chose in table 6 the generator convention at T = ω provided explicitly in eqs. (17) and (18).
Here, we comment on the breakdown induced by the vev of the nontrivial singlet field Φ −1 . The transformation behavior of Φ −1 can be read off from table 2. Generators of the remnant modular group S and T furthermore have to be amended by the correct automorphy factors, stated in (11) and (16) We see that, unlike the triplet winding states, the breaking induced by the vev of the nontrivial singlet bulk field Φ −1 differs between points T = ω and T = 1, i∞. However, the structure of the groups [81,9] and [81,7] coincides in both cases with ∆(27) Z 3 , which is the notation that we adopt in figure 3.
To conclude, let us discuss the stabilizers of the CP-like type, i.e. the possibility of enhanced CP-like transformations. Whether or not CP-like transformations are broken or preserved is often a model dependent statement. As we will see, this is because rephasing transformations of fields are important in this context, and whether or not those rephasings are physical or unphysical depends on the specifics of the model. This is why in our previous discussion we have focused on the non-CP-like flavor symmetries, where model independent statements are possible. However, independently of the model one can answer the question of whether or not the vev of a specific field would automatically break the associated CP-like transformation (i.e. whether or not the respective vev has a CP-like stabilizer). The transformations we investigate here are described by eq. (3), i.e. they transform Φ → UΦ in short. In addition, the transformation might include an element of the initial (non-CP-like) eclectic flavor symmetry that is preserved at the specific location T , i.e. a transformation of the type Φ → ρ(g)UΦ H(3, 2, 1) subgroup branchings subgroup generator(s) (1, 0, 0) T +α(0, 1, 1) T Table 6: Maximal subgroups (up to conjugation) of H(3, 2, 1) that can be achieved from its breakdown by vevs of flavon fields at T = ω (similar results hold for T = 1, i∞). We provide the branchings of the flavon representations under the resulting subgroups, followed by samples of the generators of such subgroups and the flavon vevs that yield the corresponding breakdown. In the first row of the top block, the subindices in the branching representations correspond to the charges with respect to the Z 3 generated by (B 2 A 2 ) 2 R(ST) 4 . We use the definitions a := −1+η−η 4 +η 5 −η 8 and b := −η+η 4 −η 8 with η := e 2πi /18 . Whenever the vev preserves not only the subgroup but a larger symmetry, the resulting symmetry is explicitly given in parentheses. In all other aspects, we follow the same notation as in table 4. with g ∈ G traditional ∪ G modular ∪ G R . Note that there is a crucial difference here with respect to generic bottom-up constructions: From the BU perspective, U has to be a representation matrix of a suitable outer automorphism transformation of the flavor symmetry that maps the representations to their complex conjugates. Such matrices are only defined up to a global phase by construction. Therefore, in solving the equation UΦ ! = Φ, a phase can always be absorbed in U . This is to be contrasted with the TD construction. Here, first of all, not all outer automorphisms of the flavor symmetry are possible, but only those which are part of the eclectic group and preserved by the specific vev T of the modulus. Second, as the matrix U itself is a representation matrix within the eclectic group, arbitrary global phases for U are not admissible.
In order to clarify whether or not the vevs listed in tables 5 and 6 have a CP-like stabilizer, we take the representative U of the respective CP-like transformation, stated in eqs. (35) and (19), and before eq. (14). Furthermore, we allow to amend this generator by any element ρ(g) of the unified flavor symmetries at the respective moduli location T . Then we check whether the flavon vev can solve the equation We find that all of the flavon vevs listed in tables 5 and 6 exhibit CP-like stabilizers, with one exception. The sole exception is the vev (1, 1, 1) T for H(3, 2, 1) (as always irrespective of whether T is ω, 1 or i∞). This vev does not allow for a solution of eq. (39) unless a rephasing by a global phase of i to (i, i, i) T is admitted. Note that also the omission of the global prefactors for all other vevs is, of course, an explicit choice of a global phase, in the sense that eq. (39) would be spoiled also for these vevs upon a "wrong" choice of global phase. Altogether, we see that all of the vevs have CP-like stabilizers (modulo the global choice of phase). In other words, none of the discussed breaking patterns leads itself to CP violation in a model independent way. Nonetheless, CP will certainly be broken (just as the other residual symmetries) once the flavon and/or modulus vevs are deflected away from their symmetry enhanced points, or, alternatively, if multiple vevs with incommensurable stabilizers are switched on at the same time.

Conclusions and Outlook
We have analyzed in detail the breakdown of the eclectic flavor group as it appears in the top-down approach based on string theory. In a realistic setup, the eclectic flavor symmetry has to be broken. This breaking is induced by two mechanisms: First, the vev of the modulus breaks the finite modular symmetry, at least partially. Second, the traditional flavor symmetry is universal in moduli space and, hence, unbroken by the modulus vev. It can be enhanced by elements of the finite modular symmetry at specific, symmetry enhanced points in moduli space. Thus, in the top-down approach, one cannot just consider the finite modular flavor symmetry and ignore the traditional flavor group. The latter enhances the predictive power of the scheme as it gives severe restrictions on the Kähler potential and superpotential of the theory (see [18, section 3] for a detailed discussion). Of course, the (possibly enhanced) traditional flavor symmetry has to be broken as well, and this requires the introduction of flavon fields. This increases the number of parameters, but there is no alternative. The flavon fields that break the traditional flavor symmetry might break the modular symmetries as well. This leads to an attractive flavor structure due to the subtle interplay of the symmetry breakdown via flavons and moduli as it allows the incorporation of various possibilities for "flavor hierarchies" through the alignment of vevs.
We illustrate the scheme in detail for an example based on the T 2 /Z 3 orbifold sector with traditional flavor group ∆(54), modular flavor group T , and eclectic flavor group Ω(2) as displayed in table 1. For the top-down approach, we consider a representation content as it appears in orbifold compactifications of the heterotic string (here, the T 6 /(Z 3 × Z 3 ) orbifold, as discussed in [34]). All possible massless representations are given in table 2. As usual in the top-down approach, the spectrum is very selective, and only a few representations of ∆(54) and T appear as massless modes. In the present example, we also observe that the automorphy factors of the fields are strictly correlated with the corresponding representations of the discrete modular group T . Thus, there is here no freedom to choose modular weights by hand, they are fixed by the underlying string theory construction.
As a warm-up example, we discuss the breakdown of ∆(54) via flavon fields in section 3.
The breakdown pattern is shown in figure 1 and table 4. The main result of the paper concerns the breakdown patterns of the eclectic flavor group Ω(2), which is derived in section 4. We specifically consider the flavor groups Ξ(2, 2) = [324, 111] and H(3, 2, 1) = [486, 125], which appear as unbroken subgroups of Ω(2) at the fixed points T = i and T = 1, ω (as well as at T = i∞ which is dual to T = 1), respectively. The qualitative breakdown patterns via flavon fields are summarized in figures 2 and 3. The specific form of the corresponding flavon vevs is given in tables 5 and 6. This shows that even a simple system like the T 2 /Z 3 orbifold sector exhibits a rich web of breakdown patterns via flavon and modulus vevs that might be suitable to be applied to discuss the flavor structure of quarks and leptons in the standard model of particle physics. In a companion paper [35], we shall show that a successful fit of the masses and mixing angles of quarks and leptons can be achieved.