Gauge extension of non-Abelian discrete flavor symmetry

We investigate a gauge theory realization of non-Abelian discrete flavor symmetries and apply the gauge enhancement mechanism in heterotic orbifold models to field-theoretical model building. Several phenomenologically interesting non-Abelian discrete symmetries are realized effectively from a $U(1)$ gauge theory with a permutation symmetry. We also construct a concrete model for the lepton sector based on a $U(1)^2 \rtimes S_3$ symmetry.


Introduction
The flavor structure of quarks and leptons in the standard model is mysterious. Why are there three generations? Why are their masses hierarchically different from each other? Why do they show the specific mixing angles? It is challenging to try to solve this flavor mystery. A flavor symmetry could play an important role in particle physics models in order to understand the flavor structure of quarks and leptons. Since the Yukawa matrices of the standard model include many parameters, flavor symmetries are useful to effectively reduce the number of parameters and to obtain some predictions for experiments. In particular, non-Abelian discrete flavor symmetries can be key ingredients to make models with a suitable flavor structure. Indeed, there are many works of flavor models utilizing various non-Abelian discrete flavor symmetries (see [1,2,3] for reviews).
It is known that some non-Abelian discrete flavor symmetries have a stringy origin. In particular, in orbifold compactification of heterotic string theory [4,5,6,7,8,9,10,11,12] (also see a review [13]), non-Abelian discrete symmetries D 4 and ∆(54) respectively arise from one-and two-dimensional orbifolds, S 1 /Z 2 and T 2 /Z 3 , as discussed in [14] 1 . The non-Abelian discrete symmetries originate from a geometrical property of extra-dimensional orbifolds, the permutation symmetry of orbifold fixed points, and a string selection rule between closed strings. Phenomenological applications of string derived non-Abelian discrete symmetries to flavor models are analyzed, e.g. in [19].
Furthermore, in [20], it is argued that the non-Abelian discrete symmetries D 4 and ∆(54) have a gauge origin within the heterotic string theory. Namely, these symmetries are respectively enhanced to continuous gauge symmetries U(1) ⋊ Z 2 and U(1) 2 ⋊ S 3 at a symmetry enhancement point in the moduli space of orbifolds. After certain scalar fields which are associated with the Kähler moduli fields get vacuum expectation values, the U(1) symmetries break down to Abelian discrete subgroups, and there remains a Z 4 ⋊ Z 2 ∼ = D 4 or (Z 3 × Z 3 ) ⋊ S 3 ∼ = ∆(54) symmetry group, respectively. This result suggests that a non-Abelian discrete symmetry can be regarded as a remnant of a continuous gauge symmetry. Also, this result could provide us with a new insight on model building for flavor physics.
Various non-Abelian discrete symmetries other than D 4 and ∆(54) have been used in field-theoretical model building, e.g. S 3 , S 4 , A 4 , ∆(3N 2 ), ∆(6N 2 ) (see [1,2,3]). Thus, it is important to extend the stringy derivation of D 4 and ∆(54) from U(1) ⋊ Z 2 and U(1) 2 ⋊ S 3 , by studying a field-theoretical derivation of other non-Abelian discrete flavor symmetries from U(1) m ⋊ S n or U(1) m ⋊ Z n (See also [21]). That is the purpose of this paper. Some of them may be reproduced from other types of string compactifications.
In this paper we consider an extension of the argument of the gauge origin in [20] to field-theoretical model building. We show that phenomenologically interesting non-Abelian discrete symmetries can be embedded into U(1) m ⋊S n or U(1) m ⋊Z n continuous gauge theory. Spontaneous symmetry breaking of U(1) m to Abelian discrete symmetries leads to non-Abelian discrete flavor symmetries. In the next section we discuss a gauge theory realization of non-Abelian discrete symmetries. In section 3, we show a concrete lepton flavor model based on a U(1) flavor symmetry. Section 4 is devoted to conclusions.

Gauge extension of non-Abelian discrete symmetry
In this section we investigate a field theoretical model building technique in which non-Abelian discrete symmetries have a continuous gauge symmetry origin. We start with a gauge theory with group structure of the form U(1) n ⋊ S m or U(1) n ⋊ Z m . Then, by giving a suitable VEV to a scalar field, a non-Abelian discrete symmetry is realized effectively.

S 3 group
We consider a U(1) ⋊ Z 2 model with the field contents as in Table 1. The action of the Z 2 symmetry on the U(1) charge q is given by By this we mean that the U(1) gauge field A µ transforms as A µ → −A µ , and that the oppositely charged fields in this model transform into each other, e.g.
This implies that the kinetic (and gauge interaction) terms are invariant under the Z 2 . Now, we consider VEVs for fields U i obeying the relation This VEV relation maintains the original Z 2 permutation symmetry, but breaks the U(1) group to a discrete Z 3 subgroup since the field M i has U(1) charge ±1/3. The Z 3 charges are 1 for the field M 1 and 2 for the field M 2 , so the Z 3 action is expressed by with the cubic root ω = e 2πi/3 . The combination of the two actions (3) and (4) gives rise to a non-Abelian discrete symmetry, which is nothing but S 3 ∼ = Z 3 ⋊ Z 2 . It turns out that (M 1 , M 2 ) forms a doublet of this S 3 group.
Next, we read off the S 3 representation of the other matter fields. First, the field M can be regarded as the trivial singlet 1 of the S 3 group. In the case of (M ′ 1 , M ′ 2 ), we see that these fields have trivial Z 3 charges. Then we can perform a change of basis asM ′ In this basis, the Z 2 action is given byM forms a 1 ⊕ 1 ′ of the S 3 group. As a result, we can reproduce all irreducible representations of the S 3 group.

D 4 group
Now, we consider a U(1) ⋊ Z 2 model with the field contents as in Table 2. This model is based on a U(1) symmetry and possesses an additional Z 2 symmetry which acts on the U(1) charge as in the previous case (1), so the fields transform as U 1 ↔ U 2 and M 1 ↔ M 2 etc. We consider the following VEV relation This VEV relation maintains the original Z 2 permutation symmetry, but breaks the U(1) group to its discrete Z 4 subgroup. The Z 4 charges for M 1 and M 2 are 1 and 3 respectively, hence the Z 4 action is written as The combination of actions (6) and (7) leads to the non-Abelian discrete symmetry D 4 ∼ = Z 4 ⋊ Z 2 . It turns out that (M 1 , M 2 ) forms the doublet of the D 4 group.
Next, we read off the D 4 representation of the other matter fields. First, the field M can be regarded as the trivial singlet 1 ++ of D 4 . In the case of a set of fields (M ′ 1 , M ′ 2 ), we make redefinitions asM ′ In this basis, the Z 2 action acts asM For the fields (N 1 , N 2 ), both fields have Z 4 charge 2. Then we can take a linear combination as N 1 ≡ N 1 + N 2 andÑ 2 ≡ N 1 − N 2 , and observe that the Z 2 action acts asÑ 1 →Ñ 1 and N 2 → −Ñ 2 . Then (Ñ 1 ,Ñ 2 ) forms 1 +− ⊕ 1 −+ of the D 4 group. As a result, we can reproduce all irreducible representations of the D 4 group by a suitable field setup.

S 4 group
We consider a U(1) 2 ⋊ S 3 model with the field contents as in Table 3. This model has a gauge U(1) 2 symmetry and fields are characterized by two U(1) charges q 1 and q 2 . We define the two dimensional U(1) 2 charges e 1 , e 2 and e 3 used in the table as The additional non-Abelian discrete S 3 symmetry is generated by a 120 degree rotation and a reflection on the two-dimensional U(1) 2 charge plane (q 1 , q 2 ) as Rotation : Reflection : The S 3 action permutes e 1 , e 2 and e 3 , which corresponds to a permutation of the fields as We consider the VEV relation as This VEV relation maintains S 3 , but breaks the U(1) 2 group down to a discrete Z 2 2 subgroup. The Z 2 charges z 1 and z 2 in Table 3 are determined from the U(1) 2 charges as . Then, the Z 2 2 action is given by The combination of (12) and (13) gives rise to the non-Abelian discrete symmetry Table 3: Field contents of the U (1) 2 ⋊ S 3 model for the S 4 group. Besides the U (1) 2 charges, the charges under the unbroken discrete Z 2 2 subgroup of U (1) 2 are shown. Representations under the resulting S 4 group are also shown.
Next, we read off the S 4 representation of the other matter fields. First, the field M can be regarded as the trivial singlet 1 of S 4 . In the case of the fields (N 1 , N 2 , N 3 ), we make redefinitions asÑ 1 In this basis, the three fields transform as the 1 ⊕ 2 of the S 3 group. After the VEV, these fields have the trivial Z 2 2 charge (0, 0), so they correspond to 1 ⊕ 2 of S 4 . Note, that fields with opposite U(1) 2 charges −e i /2 have, after U(1) 2 breaking, the same Z 2 2 charges as the fields M i . Hence, such fields also lead to the 3 of S 4 . As a result, we can realize the 1, 1 ⊕ 2, 3 representations of the S 4 group in this setup.
We have introduced the specific combination of U(1) 2 charges, e 1 , e 2 , and e 3 which can be interpreted as weights of the fundamental SU(3) triplet (or anti-triplet) representation. Then, the action of the S 3 group on the e i corresponds to the action of the Weyl group of SU(3) on the triplet weights. Thus, one might wonder about a SU(3) origin of this setup. In fact, U(1) 2 ⋊ S 3 is a subgroup of SU(3) where U(1) 2 furnishes maximal torus and S 3 is a lift of the Weyl group into SU(3). Also, note that the representation matrices (12) of S 3 do not actually belong to SU(3), so they give rise to genuine U(1) 2 ⋊ S 3 representations. The fundamental triplet and anti-triplet of SU(3) also give rise to U(1) 2 ⋊ S 3 representations which we did not cover here (in these cases, the representation matrices are given by those in (12) amended by a minus sign). For a short remark on these kinds of representations please refer to the conclusion section.
Furthermore, in a stringy realization of ∆(54), the SU(3) gauge symmetry appears in toroidal compactification at a symmetry enhanced point. Then, by a Z 3 orbifolding the charged root vectors are projected out [20], leaving a symmetry group U(1) 2 ⋊ S 3 .
To realize ∆(54), A 4 and ∆ (27) in the next subsections, we also use the vectors e 1 , e 2 and e 3 , as well as the Weyl reflections and the Coxeter elements.

∆(54) group
We consider a U(1) 2 ⋊ S 3 model for the ∆(54) group, with field contents given as in Table  4. The difference from the previous subsection is that the matter fields now have relative U(1) charges of 1/3 when compared to the fields U i . Then, by the VEV relation (11) for the field U i , the S 3 symmetry remains but U(1) 2 is broken down to its Abelian subgroup Z 2 3 .
Field The two Z 3 charges z 1 , z 2 in Table 4 are determined as , and the Z 2 3 action is described by The actions (12) and (14) together generate the non-Abelian discrete symmetry ∆(54 Next, we read off the representation of the other matter fields under the ∆(54) group. First, the fields (M ′ 1 , M ′ 2 , M ′ 3 ), which have opposite U(1) 2 charges and Z 2 3 charges when compared to the M i field, lead to the 3 1(2) of ∆(54). The field M can be regarded as the trivial singlet 1 + of ∆(54). In the case of the fields (N 1 , N 2 , N 3 ), we use the linear combinationsÑ 1 ≡ In this basis, one sees that they transform as a 1⊕2 of the S 3 group. After the VEV, these fields have trivial Z 2 3 charges, so they correspond to 1 + ⊕ 2 1 of the ∆(54) group. Note, that instead of the M i which have U(1) 2 charges e i /3, we can also introduce fields with charges −2e i /3. Since the Z 2 3 charges of such fields are identical to the M i , they also lead to the 3 1(1) representation. As the result, we can realize 1 + , 1 + ⊕ 2 1 , 3 1(1) , 3 1(2) representations of the ∆(54) group in our setup.
Next, we read off the A 4 representation of the other fields. First the field M can be regarded as the trivial singlet 1 of A 4 . The fields (A 1 , A 2 , A 3 ) have a similar structure to the fields M i , and they also lead to a 3 of A 4 . In the case of the fields (N 1 , N 2 , N 3 ), we use the linear combinationsÑ 1 ≡ ( In this basis, the three fields transform as 1 ⊕ 1 ′ ⊕ 1 ′′ of Z 3 . After the VEV, these fields have trivial Z 2 2 charges, so they correspond to 1 ⊕ 1 ′ ⊕ 1 ′′ of the A 4 group. Note that other fields (M ′ 1 , M ′ 2 , M ′ 3 ) with U(1) 2 charges (2n + 1)e i /2, where n is an integer, also lead to 3 representation since they have same Z 2 2 charges as M i . As a result, we can realize 1, 1 ⊕ 1 ′ ⊕ 1 ′′ , 3 representations of A 4 in this setup.

U (1) 2 ⋊ S lepton flavor model
In this section we present a concrete model for the lepton sector based on the U(1) 2 ⋊ S 3 symmetry, which is related to the ∆(54) discrete symmetry discussed in Section 2.4. Several interesting flavor models based on the ∆(54) symmetry have been investigated in [22,23,24,25,26].
Here we consider a supersymmetric model with U(1) 2 ⋊ S 3 × Z 2 symmetry, and with the field content as in Table 7. There, in addition to the MSSM fields (the lepton doublets (L e , L µ , L τ ), the right-handed lepton fields (e c , µ c , τ c ) and Higgs doublet pairs (H u , H d )) we introduce flavon fields A i , B i , C i and D i . The VEV of the flavon fields breaks the U(1) 2 ⋊ S 3 symmetry completely. Corresponding representations under ∆(54) are also shown in Table 7. It is also possible to add other flavon fields, e.g. fields U i in Table 4, and consider the situation where the VEV of the fields, U 1 = U 2 = U 3 , breaks the symmetry as U(1) 2 ⋊ S 3 → ∆(54) at an intermediate scale. In this paper we do not consider this possibility.

Yukawa mass matrices
First, we consider the Yukawa sector of the model. By invariance under U(1) 2 ⋊ S 3 × Z 2 , the superpotentials of the neutrino sector and the charged lepton sector are given by and respectively. Here, we assume a UV cutoff scale Λ. Then the mass matrices are given by where we used the following definition for the VEVs of the flavon fields: Note that the charged lepton mass matrix is diagonal. Thus, the mixing angles are determined only by the neutrino mass matrix.

Flavon potential and vacuum alignment
Next we consider the flavon sector. The superpotential up to three-point level including only flavon fields is given by The F-flatness condition for the flavon superpotential leads to (for i = j = k = i) There are two branches of solutions: (a) Let us first assume A i = 0 and B i = 0. Then we can solve (29) for B k and insert the solution in to (30). Then, we obtain the condition 4λ 3 4 = −λ 2 λ 2 1 , so we can choose the VEVs as: (b) If not all A i = 0 or B i = 0 then there exist solutions, and they can be brought into the following form by an S 3 transformation: with the condition λ 2 b 1 b 2 + λ 4 a 2 3 = 0.
Furthermore, the VEVs of any two components C i must be zero. In the following we assume The D i are not constrained from F-flatness.

Neutrino mass/mixing properties
In the following we consider only the case (a). By inserting the VEVs the mass matrix becomes For the later convenience we define the following parameters (A, B and C not to be confused with the flavon fields A i , B i and C i ) and rewrite the mass matrix (36) as It turns out that this mass matrix has the following relations, Note that the three equations are dependent. Actually, the third equation is a consequence of the first and the second equations. The first equation (39) can be solved by AC as thus if the mass matrix M ν is fixed, the parameter AC can be derived. Hence, (40) is a prediction for ratios of elements of the neutrino mass matrix M ν . Now, we investigate whether this model can explain the experimental values of mass hierarchies and mixings. In our model, the charged lepton mass matrix (23) already takes a diagonal form, so the PMNS mixing matrix U PMNS is given by a unitary matrix U ν which diagonalizes the neutrino mass matrix (38) as Here, the rotation matrices are defined by three mixing angles (θ 12 , θ 23 , θ 13 ) and three CP phases (δ, β 1 , β 2 ) as For simplicity, here we consider only the case where We also set the mixing angle θ 12 to fit the experimental value as Then the mixing matrix (43) is a real matrix. As for the neutrino mass differences, we wish to reproduce the case of the inverted hierarchy: and regard the third family neutrino mass m 3 as a parameter. These values are consistent with the global analysis in [27] within 2σ range. The neutrino mass matrix is then obtained as where M = diag(m 1 , m 2 , m 3 ). In Figure 1, we show a prediction for various values of (m 3 , θ 13 , θ 23 ) from the ratio condition of this mass matrix (40). In the figure we show solutions of the mixing angle θ 23 against the third generation neutrino mass m 3 for (40) with fixed θ 13 angles, θ 13 = 8.2 • , 8.7 • , 9.1 • , which is in 2σ range.

Charged lepton masses
Next, we consider the charged lepton mass matrix (23). We want to fix the charged lepton masses as can be regarded as UV completions of discrete flavor models. The main difference between a discrete flavor model and a U(1) flavor model as shown in this paper can be seen in the field interactions. Namely, some fields in a discrete flavor model can be distinguished in a U(1) flavor model. For example, the 3 1(1) representation field of the ∆(54) symmetry can be described by several U(1) 2 charges, (e 1 /3, e 2 /3, e 3 /3), (−2e 1 /3, −2e 2 /3, −2e 3 /3) etc. Thus a superpotential in a U(1) flavor model can be different from the one of the corresponding discrete flavor model. In general, U(1) n ⋊ S m and U(1) n ⋊ Z m flavor models are constrained more than flavor models with non-Abelian discrete flavor symmetries, which are subgroups of U(1) n ⋊ S m and U(1) n ⋊ Z m , because symmetries are larger. Our results would provide a new insight on flavor models.
We have introduced the specific combination of U(1) 2 charges, e 1 , e 2 , and e 3 , to realize S 4 , ∆(54), A 4 and ∆ (27). They correspond to weights of the triplet (or anti-triplet) representation of SU(3). In fact, U(1) 2 ⋊ S 3 is a subgroup of SU (3), where S 3 is associated with the Weyl group. We also obtained genuine U(1) 2 ⋊ S 3 representations which are not obtained from SU(3) triplets by spontaneous symmetry breaking. Also, in a stringy realization of ∆(54), the SU(3) gauge symmetry appears in toroidal compactification, and the non-zero roots can be projected out by an orbifold projection [20]. This may also suggest that a similar situation can be realized field-theoretically in a higher-dimensional SU(3) gauge theory with a suitable orbifold boundary condition.
Anomalies of non-Abelian discrete symmetries are important [29]. Anomalous discrete symmetries would be violated by non-perturbative effects, but its breaking effects might be small depending on dynamical scales of non-perturbative effects. By our construction, discrete Abelian symmetries originating from U(1) n of U(1) n ⋊S m and U(1) n ⋊Z m are always anomalyfree and exact symmetries, but S m and Z m of U(1) n ⋊S m and U(1) n ⋊Z m can include anomalous discrete symmetries depending on the model. We have constructed a concrete flavor model for the lepton sector based on the U(1) 2 ⋊ S 3 continuous gauge theory. We have shown that it is possible obtain a realistic flavor structure from this model. Since the model is based on an extended symmetry the number of the parameters is relatively few. In particular, we could show a relation between the angle θ 23 and third generation neutrino mass m 3 .
We have shown six types of gauge realizations of non-Abelian discrete symmetries. However, further extensions are possible. For example, extensions to higher N, ∆(6N 2 ), is possible if we consider models with U(1) charges q = e i /N. It is also possible to include further representations of e.g. U(1) 2 ⋊S 3 which we did not cover here for the sake of simplicity. The general representation theory of these semidirect groups is obtained from the little group method of Wigner, which is familiar from the representation theory of the Poincaré group. Then, e.g. in the case of U(1) 2 ⋊ S 3 one obtains an uncharged singlet representation which transforms as 1 ′ under S 3 while being uncharged under the U(1) 2 .
A phenomenological implication of our U(1) flavor models is that there should be Z ′ boson(s) which originate from U(1) gauge groups in the effective theory. In this framework Z ′ bosons and flavor structures are related. Since we assigned different U(1) charges to the three-generation leptons, the Z ′ bosons have flavor dependent interactions. Thus, if Z ′ bosons are light as e.g. the TeV scale, they can be a probe of the flavor structure. It will be interesting to investigate Z ′ phenomenology by extending well-known discrete flavor models.