Δ(27) family symmetry and neutrino mixing

The observed neutrino mixing, having a near maximal atmospheric neutrino mixing angle and a large solar mixing angle, is close to tri-bi-maximal. This structure may be related to the existence of a discrete non-Abelian family symmetry. In this paper the family symmetry is the non-Abelian discrete group Δ(27), a subgroup of SU(3) with triplet and anti-triplet representations. Different frameworks are constructed in which the mixing follows from combining fermion mass terms with the vacuum structure enforced by the discrete symmetry. Mass terms for the fermions originate from familon triplets, anti-triplets or both. Vacuum alignment for the family symmetry breaking familons follows from simple invariants.


Introduction
The observed neutrino oscillation parameters [1][2][3] are consistent with a nearly tri-bimaximal (TBM) structure (1.1) It has been observed that this simple form might be a hint of an underlying family symmetry. The observation of non-zero θ 13 of comparable magnitude to the Cabibbo angle excludes exact TBM, motivating the search for frameworks where θ 13 is naturally of the observed magnitude and the remaining angles are close to their TBM values. The discrete non-Abelian group ∆ (27) is an interesting candidate family symmetry for such frameworks. It is a subgroup of SU(3) of order 27 with a triplet and anti-triplet representation. After T 7 (order 21) it is the smallest group with this appealing feature, and the smallest in the series of groups ∆(3n 2 ), which contains C 3 , A 4 and ∆ (27) for n = 1, 2, 3 respectively.
In section 2.1 the simplest options to align familon VEVs with F-terms are shown, significantly improving the basic ideas proposed in [44], and providing an alternative to the D-term alignment [8,41,42]. One of the possibilities found allows to obtain in SUSY frameworks the (ω, 1, 1) class of VEV involved in spontaneous geometrical CP violation [5,18]. Section 2.2 features several frameworks that couple the fermions to the familons. To complete these frameworks, a method of safeguarding against terms that would invalidate them is required. An auxiliary symmetry protecting a minimal framework is presented in section 2.3, in order to show a specific example of a complete framework with ∆(27) family symmetry and neutrino mixing. Finally in section 3 the conclusions are presented.

∆(27) family symmetry
A complete family symmetry model is usually constituted by alignment of familon VEVs (section 2.1), the coupling of the familons to fermions (section 2.2), and methods to eliminate terms that would invalidate the model (section 2.3). Appendix A contains the relevant details about ∆ (27), namely product rules for the representations in the notation used throughout the paper to build invariants.

Aligning the familons
An important advantage of building family symmetry models in SUSY frameworks is the holomorphic superpotential. This facilitates the separation of distinct familons as noted in [43,44], through F-term alignment involving alignment superfields (often referred to as driving fields). Although D-term alignment can be more minimal in the sense that it dispenses the introduction of alignment fields [8,41,42], F-term alignment can often proceed through simpler invariants.
Although there are some interesting non-renormalisable alignment terms including the one introduced in [44], here the focus is exclusively on renormalisable F-term options. These are preferred in particular because in standard UV completions of non-renormalisable alignment terms some messengers act as additional alignment fields and usually spoil the desired alignment, as pointed out in [47].
The simplest possibilities for aligning the familon VEVs used throughout the paper are described in this section: triplet alignment fields ϕ and specific singlet familons align anti-triplet familonsφ, similarly anti-triplet alignment fieldsφ align triplet familons φ, and singlet alignment fields ς lead to relative alignment between triplet and anti-triplet familons. Appendix B discusses these and other options in more detail, as well as their applicability to other discrete non-Abelian groups that have similar product rules and representations.

JHEP08(2015)157
Triplet alignment field ϕ 1 combined with 1 i0 singlet familons leads to VEVs with two zeros: Triplet alignment field ϕ 123 combined with 1 0i singlet familons leads instead to VEVs with equal entries: the F-term with respect to the ϕ 123 i components giving conditions on the componentsφ i Although it isn't used further in this paper, it is relevant to note that using singlet familons 1 ij with i, j = 0 in this type of term enables the alignment of directions such as (ω, 1, 1) in a SUSY framework. This class of VEV is relevant in that it leads to spontaneous geometrical CP violation [5,18]. While the VEV has clearly been obtained in non-SUSY frameworks, a way to align this direction in a SUSY framework had not been presented so far.
If aligning a single familon direction, the alignment field and familons can be neutral under additional symmetries and the σ 00 is superfluous. In order to have both alignments simultaneously requires alignment fields and familons separated by some mechanism, usually auxiliary symmetries -see section 2.3 for a specific example.
Analogous terms relying on anti-triplet alignment fields can align triplet familons.

JHEP08(2015)157
Assuming φ 1 , φ 123 triplet VEVs have been aligned, singlet alignment fields lead to the orthogonal anti-triplet VEV ς 02 [φ 123φ23 ] 01 (2.11) of which one solution is where a trivial singlet alignment field ς 00 imposing orthogonality with φ 1 can guarantee φ 1 23 = 0. 1 The representation of ς 02 can be any of the 1 0i due to the equal components of φ 123 . Note that the orthogonality is between triplets and anti-triplet (or vice-versa).
Another relevant VEV direction is which can be obtained as another solution of eq. (2.1) or generalising it to use two nontrivial singlets. If the anti-triplet familon is accompanied by a triplet familon, the direction with two zeros for one of the familons can be obtained by combining 3 singlet alignment fields, as discussed in more detail in appendix B see e. If required for contractions with a triplet into singlets,φ 1 andφ 3 can be used interchangeably if used in conjunction with 1 0i singlets e.g. [Lφ 1 ] 00 σ 00 ∼ [Lφ 3 ] 01 σ 02 as both isolate L 1 , the first component of triplet L. In this sense, having both VEV directions is often not necessary in ∆ (27) frameworks, as demonstrated by the specific example in section 2.3, featuring a minimal alignment superpotential S V .

Building the fermion mass terms
The Standard Model (SM) SU(2) L doublet leptons are contained in L, which is assigned as a triplet of ∆ (27) and e c , µ c , τ c can be singlets. H u , H d are the two SU(2) L doublets required in SUSY frameworks and are trivial singlets of ∆ (27). The charged lepton mass matrix can be made diagonal in the basis where the familon VEVs are presented, referred to henceforth as the familon basis. One option is using an anti-triplet familon φ 3 = (0, 0,ā 3 ) together with trivial singlets e c , µ c , τ c and a non-trivial 1 01 singlet familon σ 01 where coefficients are implicit and M represents the masses of messenger fields which would be specified in a specific UV completion of the family symmetry model [46,47]. In the remainder of the paper M is omitted in the non-renormalizable superpotential terms. If σ 01 in eq. (2.14) is not neutral under an additional auxiliary symmetry (as discussed in more detail in section 2.3), the different powers of σ 01 can match different charges of e c , µ c , τ c under the auxiliary symmetry in a version of the Froggatt-Nielsen mechanism [63].

JHEP08(2015)157
In this way σ 01 M explains the hierarchy of the charged lepton masses, similarly to [45]. Another option is to have non-trivial singlets e c , µ c , τ c similarly to [43], and either add a separate Froggatt-Nielsen symmetry or leave the hierarchy unexplained as in the SM.
With the charged leptons diagonal in the familon basis, all the leptonic mixing is present in the neutrino sector. It will be determined in particular by a familon VEV in the (1, 1, 1) direction, which is an ingredient in all of the frameworks discussed here.
The effective neutrino terms can be obtained through a seesaw mechanism, such as type I seesaw. In such a case one can include Dirac terms featuring explicit geometrical CP violation [32][33][34]: It could also be that there is more than one type of seesaw involved and related with the deviations from TBM [64]. Here the analysis is performed at the level of the effective neutrino terms ]. The frameworks presented here deviate from TBM but often preserve one of the TBM eigenvectors, making an eigenvector based approach particularly useful to study the mixing [65].
In the following, different types of frameworks are illustrated through simple invariants for presentational purposes. These simpler examples can be the starting point to build complete models, but it remains necessary to distinguish the familons, and to allow this to be done through an auxiliary symmetry may require modifying the simpler terms for example with additional familon insertions (this procedure is illustrated for one of the examples in section 2.3).

Invariant frameworks
In terms of neutrino invariants, the simplest type of ∆(27) framework relies exclusively on cubic invariants specific to ∆ (27). They will be referred to as Invariant frameworks. One example relies only on triplet familons φ 1 and φ 123 : where the coefficients are explicitly shown to be associated with the entries in the respective mass matrix

JHEP08(2015)157
In this simple example the two familons couple in exactly the same way to the fermions, which is not compatible with separating their alignments as described above. A complete model requires some modification of the terms e.g. replacing the simpler [φ 1 [LL] I,S ] 00 terms with [φ 1 [LL] I,S ] 00 σ 00 . The matrix structure is determined by the contractions and the VEVs. In this simple framework it leads to TBM mixing in the limit i 1 = s 1 , and in general the corresponding mixing scheme can be denoted as TM3 as it preserves the third TBM eigenvector (0, 1, −1). Like TBM, TM3 is not viable due to θ 13 = 0. An alternative but less predictive Invariant framework introduces a deviation from TM3 by using a non-trivial singlet familon such as σ 01 , with terms where the coefficients corresponding to the 6 invariants were omitted. In order to make this a complete model one Like the previous examples, in order to have a complete model some variation of the terms is required in order to allow the two triplet familons to be distinguished. This simple framework has a few interesting limits. For i 23 = s 23 it preserves the second TBM eigenvector (1, 1, 1), which corresponds to the tri-maximal mixing scheme TM2. For i 23 = −2s 23 it preserves the first TBM eigenvector (−2, 1, 1), which corresponds to the tri-maximal mixing scheme TM1. In general if the TBM rotation is applied: leaving the familon basis shows the particular TM2 and TM1 limits of this framework more clearly.

SU(3) frameworks
Similarly to the strategy described in [66] and implemented in the grand unification ∆(27) model [6], this type of framework relies on invariants with anti-triplet familons e.g.
[  20 requires underlying assumptions about the messenger sector acting as the UV completion of the family symmetry model [46,47]. In this case a possibility would be that the messengers in the neutrino sector are exclusively ∆(27) trivial singlets. An advantage of SU(3) frameworks is that they can be rather predictive, with only one effective invariant for each set of familons.
In order to illustrate the mass terms, one example relies on anti-triplet familonsφ 123 andφ 23 : where the coefficients of two terms were absorbed into the VEVs and a third effective parameter is chosen as the coefficient a controlling the last term. A complete model using the same anti-triplet familons would require additional insertion of familons in some of the terms. With a = 0, this type of model leads to TBM [6] (see also [67]). The last term deviates TBM preserving the (2, −1, −1) eigenvector which makes this an SU(3) framework for the TM1 mixing scheme. This can be confirmed by leaving the familon basis through the TBM rotation to the mass matrix: The consequences in terms of mixing angles are the same for any TM1 models and can be found in [48] and references therein. In this case the effective parameter a is fixed by the observed value of θ 13 , which consequently predicts the deviations of the other angles from the TBM values. With only two other effective parameters, this TM1 framework is particularly predictive. It is clear that the lightest neutrino is massless, and on closer inspection the squared mass differences ∆m 2 a , ∆m 2 s are controlled mostly byb 2 andc 2 respectively, requiring a mild hierarchy in the VEVs.
Alternatives to deviate from TBM in SU (3)

Alignment frameworks
Although the Invariant frameworks are minimal in terms of messengers and the SU(3) frameworks are minimal in terms of effective parameters, as all the frameworks rely on familon VEVs it is also interesting to consider frameworks with minimal requirements in terms of alignment fields and familons. They will be referred to as Alignment frameworks.
An interesting possibility to dispense with some alignments is to use effective familons. The contraction of triplet familons φ 123 and φ 1 into anti-triplets leads to effective antitriplet familons where the anti-symmetric contraction is particularly interesting as it leads toφ 23 . Therefore, an Alignment framework could rely on the effective anti-triplet familons e.g.
Alignment frameworks withφ 123 and φ 23 could appear as

JHEP08(2015)157
which is a very interesting Alignment framework. As [L[φ 123 φ 123 ] I,S ] 00 give the same structure due to eq. (2.26), only the sum of the two contributions is relevant and denoted through a. This framework leads to TM1 models similar to those of eq. (2.23), but with 4 relevant parameters. In comparison with the similar SU(3) framework (with only 3 parameters), it allows non-zero determinant for M ν and therefore a mass for the lightest neutrino. θ 13 is directly related with a which governs the TM1 deviations from TBM. Additionally, within this Alignment framework (and the SU(3) framework of eq. (2.23)) one can naturally explain the hierarchy between neutrino mass eigenstates by having a mild hierarchy in the VEVs of the two familons, which then establishes a relationship between the size of θ 13 and ∆m 2 s ∆m 2 a . 4

Adding auxiliary symmetries
A complete framework matches a set of familon alignments with a set of fermion mass terms, without generating terms that invalidate the framework. Typically this is achieved by adding an auxiliary symmetry which eliminates those terms, possibly in conjunction with specific UV completions [46,47].
The complete framework proposed here results from combining an auxiliary symmetry with a variation of the Alignment framework in eq. (2.31) and a variation of the charged lepton terms in eq. (2.14). The effective familons strategy is employed requiring a minimal set of familons, φ 3 , σ 01 , φ 123 andφ 23 : Although a specific UV completion will not be considered in full detail here, this framework relies on some underlying assumptions regarding the messengers. As mentioned in section 2. 00 e c can be due to specific charged lepton messengers, 4 Relations between θ13 and ∆m 2 s /∆m 2 a are particularly interesting in the context of unified models like [42], where θ13 ∼ ∆m 2 s ∆m 2 a ∼ 0.15 can be further related to the size of the Cabibbo angle and to the hierarchy in quark masses (see also [66]). which could be in this case non-trivial ∆(27) 1 0i singlets. The charged lepton messengers are distinct from neutrino messengers due to SM hypercharge. The presence of the φ 3 familon in S ν or conversely the presence of φ 123 ,φ 23 in S C would invalidate the framework, and likewise for terms in S V . The familons need to be separated to avoid this. Table 1 lists the field content together with symmetries and assignments for the set S C in eq. (2.32), S ν in eq. (2.33) and S V in eq. (2.34), including an auxiliary U(1) a that eliminates terms that would invalidate the framework. The charges of alignment fields and fermions are expressed in terms of the familon charges, which are denoted by curly brackets (e.g. {φ 3 } is the U(1) a charge of φ 3 ). Specific models correspond to a choice of the familon charges, and the existence of choices with U(1) a integer charges was explicitly verified: 2 (equivalent) choices remain for familon triplet charges {φ 3 } = {σ 00 } = −{φ 123 } = ±1, 20 choices for integer charges ranging between −2 and +2 and many more for charges between −3 and +3. For each viable choice of charges it is possible to replace the continuous U(1) a symmetry with a sufficiently large cyclic subgroup C n without invalidating the model. Similarly the R-symmetry can be discrete [68,69]. Table 1 corresponds to a subset of models where H u , H d are neutral under U(1) a , for the sake of simplicity.

Conclusion
In this paper ∆(27) is studied as a promising candidate for a family symmetry. The group has triplet and anti-triplet representations which makes it particularly suitable for grand unification, and has interesting CP properties.
Different options can provide the vacuum alignment of multiple family symmetry breaking familons. In supersymmetric frameworks there is D-term and F-term alignment.
Many frameworks for obtaining neutrino mixing were suggested in section 2.2, including a simple predictive framework with only 3 parameters controlling directly the squared mass differences ∆m 2 a , ∆m 2 s and θ 13 . Viable frameworks can be constructed by combining a set of alignment terms and mass terms with an auxiliary symmetry. A minimal complete framework was presented in section 2.3. The irreducible representations are 9 singlets and 2 triplets. The singlets 1 ij have c 1 ij = ω i and d 1 ij = ω j , where ω ≡ e i2π/3 . The two triplets can be denoted 3 01 and 3 02 . The generator c is represented equally for both and d is represented as a diagonal matrix with entries that are powers of ω related to the subscripts of the triplet representation: i.e. the SU(3) invariant contraction. The non-trivial singlets can be built as

B F-term alignments in ∆(27) and similar groups
To discuss alignment options in more detail, in this appendix triplet alignment fields are referred as A, anti-triplet alignment fields asB, triplet familons are θ, and anti-triplet familons are α (with no bar, but upper indices). Singlets have labels of their representation, ς ij for alignment fields and σ ij for familons. The notation for VEVs is dropped such that e.g.φ 1 23 = 0 implicitly refers to φ 1 23 = 0. Some of the best alignment options were already introduced in section 2.1 and used in eq. (2.34) of section 2.3. Proceeding in a systematic fashion, one can start with the simplest renormalisable superpotentials.

B.1 Triplet alignment field with familon triplet and familon singlets
In terms of alignment fields, the choice is A,B, or one of nine singlets ς ij . The basic invariant for triplet familon θ is then which leads to non-vanishing VEVs only for a special relation between the arbitrary couplings a I and a S . Somewhat similar relations without this issue are obtained by adding one singlet familon σ ij allowing VEV directions that depend on the representation of the singlet σ ij (cf. eq. (2.34) which employed this type of invariants). As discussed in section 2.1, for familons σ i0 the possibilities include θ ∝ (1, 0, 0) and similar VEVs (i.e. those related by action of ∆ (27) group elements, like the cyclic permutations (0, 1, 0) and (0, 0, 1)). For familons σ 0i the possibilities include θ ∝ (1, 1, 1), (1, ω, ω 2 ) and similar VEVs. For familons σ ij with i, j = 0 the possibilities include θ ∝ (ω, 1, 1) and similar VEVs. Although this last class of VEVs was not used in this paper, it is particularly relevant due to spontaneous geometrical CP violation [5,18], and had not been obtained previously in SUSY frameworks.

B.2 Triplet alignment fields with familon triplet and anti-triplet
If an anti-triplet α is present together with the triplet θ, it can contribute to both the A andB terms: the F-terms with respect to the alignment field triplet components A i would then give a I θ 1 θ 1 + 2a S θ 2 θ 3 + aα 1 = 0 (B.9) a I θ 2 θ 2 + 2a S θ 3 θ 1 + aα 2 = 0 (B.10) which can relate the alignment between an anti-triplet familon α and triplet familon θ, but is not sufficient to constrain the direction of either. Nevertheless, if one of the familons is separately aligned in a direction in the class (1, 0, 0) or (1, 1, 1), that special direction is passed into the other familon through this type of term, but this doesn't apply to other directions. Similarly from the F-terms with respect to the alignment field anti-triplet As the directions passed between familons only remain invariant for special directions, combining the triplet alignment field and the anti-triplet alignment field with arbitrary parameters should only allow special solutions. In addition this fixes the absolute magnitude of both VEVs. A simple example of this occurs for the solution where both familons JHEP08(2015)157 mutually align in the (1, 0, 0) direction: and similarly for the (1, 1, 1) direction where the symmetric coefficients are involved e.g.

B.3 Singlet alignment fields with familon triplet and anti-triplet
Without an anti-triplet familon to couple to, there is no renormalisable coupling of θ to any singlet alignment field ς ij . But with an anti-triplet familon α: which enforces a relation between the triplet and anti-triplet components which is a kind of singlet specific orthogonality condition between θ, α (cf. eq. (2.34) which employed this type of invariants).
If there are multiple alignment field singlets it is possible to restrict the possible directions. One example is but one can also sum the 3 while multiplying specific powers of ω to isolate: meaning this set of 3 alignment fields enforces one of the two familons to have two vanishing entries, while the other familon must have the other one vanishing. This is a very interesting option to simultaneously obtain a φ 1 familon with φ 1 2 = φ 1 3 = 0 while guaranteeingφ 1 23 = 0.

JHEP08(2015)157
Similarly, ς 01 , ς 11 , ς 21 (note the second label is the same on all three, as in S i0 ) constituting S i1 would lead to (θ 3 α 1 ) = 0 (B.25) which is particulary interesting for a φ 3 3 = 0,φ 1 23 = 0 solution. The set ς 02 , ς 12 , ς 22 constituting S i2 leads to Combining two of these 3 singlet sets (a total of 6 singlet alignment fields for the same pair of triplet and anti-triplet familons) restricts the directions such that both familons have 2 zero entries: the same non-zero entry for S i1 + S i2 and either the cyclic pairs for S i0 + S i2 (from triplet to anti-triplet, e.g. θ 1 = 0 together with α 2 = 0) or the anti-cyclic pairs for S i0 + S i1 (from triplet to anti-triplet, e.g. θ 1 = 0 together with α 3 = 0). Adding another singlet alignment field to one of these sets of 6 makes one of the familon VEVs vanish.
Summary and applications for other groups. In order to align triplet or anti-triplet familon VEVs with renormalisable superpotential terms in ∆ (27), one must necessarily have another anti-triplet or triplet field and there are three possibilities. The first is the anti-triplet (or triplet) is an alignment field and one can obtain relevant VEVs in conjuction with additional familons singlets. The second is both the alignment fields and the additional familons are triplets. The third option is having singlet alignment fields, and one can obtain relevant VEVs in conjuction with triplet and anti-triplet familons. The main results are summarised in table 2 (where triplet and anti-triplets can be reversed). The D-term alignments found in ∆(27) could be used in other groups, and the same is true for the F-term alignments. Given that the product rules for T 7 triplet, anti-triplet and singlets are so similar to those of ∆(27), many of the F-term alignments discussed in this appendix can be directly applied to T 7 frameworks -namely, the options that do not involve ∆(27) singlets other than the three 1 i0 . This includes some of the options in eq. (B.6) leading to (1, 0, 0), the mutual alignment option which relies only on pairs of triplet and anti-triplet leading to both being aligned as (1, 0, 0) or both being aligned as (1, 1, 1), and the S i0 option which relies on 3 alignment field singlets to align a pair of JHEP08(2015)157 triplet and anti-triplet where one of them has two zeros and the other is orthogonal, e.g.
The S i0 option is particularly versatile as it does not rely on the product of two triplets or on the product of two anti-triplets, and as such can be used for ∆(3n 2 ) and Σ(3n 3 ) groups in general. ∆(3n 2 ) and Σ(3n 3 ) groups with n multiple of 3 (e.g. Σ(81)) have 9 singlets like ∆ (27). For such groups all options in eq. (B.6) and those involving sets of alignment field singlets beyond S i0 are available to align all of the directions discussed here.
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.