Spontaneous breaking of $SO(3)$ to finite family symmetries with supersymmetry - an $A_4$ model

We discuss the breaking of $SO(3)$ down to finite family symmetries such as $A_4$, $S_4$ and $A_5$ using supersymmetric potentials for the first time. We analyse in detail the case of supersymmetric $A_4$ and its finite subgroups $Z_3$ and $Z_2$. We then propose a supersymmetric $A_4$ model of leptons along these lines, originating from $SO(3)\times U(1)$, which leads to a phenomenologically acceptable pattern of lepton mixing and masses once subleading corrections are taken into account. We also discuss the phenomenological consequences of having a gauged $SO(3)$, leading to massive gauge bosons, and show that all domain wall problems are resolved in this model.


Introduction
The discovery of neutrino mass and lepton mixing [1] not only represents the first laboratory particle physics beyond the Standard Model (BSM) but also raises additional flavour puzzles such as why the neutrino masses are so small, and why lepton mixing is so large [2]. Early family symmetry models focussed on continuous non-Abelian gauge theories such as SU (3) [3] 3 or SO(3) [5]. Subsequently, non-Abelian discrete symmetries such as A 4 were introduced, for example to understand the theoretical origin of the observed pattern of (approximate) tri-bimaximal lepton mixing [6,7]. When supersymmetry (SUSY) is included, the problem of vacuum alignment which is crucial to the success of such theories, can be more readily addressed using the flat directions of the potential [8][9][10]. However, current data involves a non-zero reactor angle and a solar angle which deviate from their tri-bimaximal values [11]. Since, in general, non-Abelian discrete symmetries do not imply either a zero reactor angle or exact tri-bimaximal lepton mixing, these symmetries are still widely used in current model building [12].
Although the motivation for non-Abelian discrete symmetries remains strong, there are a few question marks surrounding the use of such symmetries in physics. The first and most obvious question is from where do such symmetries originate? In the Standard Model (SM) we are familiar with the idea of gauge theories being fundamental and robust symmetries of nature, but discrete symmetries seem only relevant to charge conjugation (C), parity (P) and time-reversal invariance (T) symmetry [13]. In supersymmetric (SUSY) models, Abelian discrete symmetries are commonly used to ensure proton stability [14]. It is possible that the non-Abelian discrete symmetries could arise from some high energy theory such as string theory [15], perhaps as a subgroup of the modular group [9,16] and/or from the orbifolding of extra dimensions [17]. However, even if such symmetries do arise from string theory, and survive quantum and gravitational corrections [18], when they are spontaneously broken they would imply that distinct degenerate vacua exist separated by an energy barrier, leading to a network of cosmological domain walls which would be in conflict with standard cosmology, and appear to "over-close the Universe" [19][20][21].
The problem of domain walls with non-Abelian discrete symmetries such as A 4 was discussed in [22,23] where three possible solutions were discussed: transition happens during inflation which effectively dilutes the domain walls, and that the A 4 is never restored after reheating following inflation.
An alternative solution to the domain wall problem, which we pursue here, is to suppose that the non-Abelian discrete symmetry arises as a low energy remnant symmetry after the spontaneous breaking of some non-Abelian continuous gauge theory. This could take place either within the framework of string theory [25], or, as in the present paper, in the framework of quantum field theory (QFT). For example it has been shown how SO (3) can be spontaneously broken to various non-Abelian discrete symmetries [26,27]. In order to achieve this, a scalar potential was constructed such leading to the vaccuum expectation value (VEV) which breaks the continuous gauge symmetry to the discrete symmetry. The key requirement for having a remnant non-Abelian discrete symmetry seems to be that the scalar field which breaks the gauge symmetry is in some large irreducible representation (irrep) of the continuous gauge group.
The above approach [26,27] has been applied to flavour models based on non-Abelian continuous gauge symmetries. For example, following [26,27], the authors in [28] have considered the breaking of gauged SO(3) → A 4 by introducing 7-plet of SO(3) with the further breaking of A 4 realising tri-bimaximal mixing in a non-SUSY flavour model. However, a fine-tuning of around 10 −2 among parameters had to be considered in order to get the correct hierarchy between µ and τ masses. The problem of how to achieve tri-bimaximal mixing at leading order from non-Abelian continuous flavour symmetries has also been discussed by other authors [29,30] but the problem of determining the required flavon VEVs remains unclear. One idea is to require the electroweak doublets and right-handed fermions to separately transforming under different continuous flavour symmetries, and realise maximal atmospheric mixing from the minimisation of the potential [31,32]. Extended discussions including the breaking of SU (2) and SU (3) to non-Abelian discrete symmetries have been discussed in [33][34][35][36][37][38] and the phenomenological implications of the breaking of SU (3) flavour symmetry in flavour models has been discussed in [39,40].
The above literature has been concerned with breaking a continuous gauge theory to a non-Abelian discrete symmetry without SUSY. To date, the problem of how to achieve such a breaking in a SUSY framework has not been addressed, even though there are many SUSY flavour models in the literature [12]. As stated earlier, the main advantage of such SUSY models is the possibility to achieve vacuum alignment using flat directions of the potential, which enables some technical simplifications and enhances the theoretical stability of the alignment [8]. There is also a strong motivation for considering such breaking in a SUSY framework, in order to make contact with SUSY flavour models [12]. In addition, the usual motivations for embedding the non-Abelian discrete symmetry into a gauge theory also apply in the SUSY context as well, namely: • To provide a natural explanation of the origin of non-Abelian discrete flavour symmetries in SUSY flavour models.
• To avoid the domain wall problem of SUSY flavour models, since the non-Abelian discrete flavour symmetry is just an approximate effective residual symmetry arising from the breaking of the continuous symmetry. When the approximate discrete symmetry is broken it does not lead to domain walls.
Finally, if the continuous symmetry is gauged, there is the phenomenological motivation that: • The breaking of gauged flavour symmetries to finite non-Abelian flavour symmetries implies new massive gauge bosons in the spectrum, with possibly observable phenomenological signatures. For instance, SUSY SO(3) → A 4 will lead to three degenerate gauge bosons plus their superpartners.
In the present paper, motivated by the above considerations, we discuss the breaking of a continuous SUSY gauge theory to a non-Abelian discrete symmetry using a potential which preserves SUSY. As stated above, this is the first time that such a symmetry breaking has been discussed in the literature, and the formalism developed here may be applied to the numerous SUSY flavour models in the literature [12]. For example, we discuss the breaking of SO(3) down to finite family symmetries such as A 4 , S 4 and A 5 using supersymmetric potentials for the first time. In particular, we focus in detail on the breaking of SUSY SO(3) to A 4 , with SUSY preserved by the symmetry breaking. We further show how the A 4 may be subsequently broken to smaller residual symmetries Z 3 and Z 2 , still preserving SUSY, which may be used to govern the mixing patterns in the charged lepton and neutrino sectors, leading to a predictive framework. We then present an explicit SUSY SO(3) × U (1) model of leptons which uses this symmetry breaking pattern and show that it leads to a phenomenologically acceptable pattern of lepton mixing and masses. Finally we discuss the phenomenological consequences of having a gauged SO(3), leading to massive gauge bosons, and show that all domain wall problems are resolved in such models.
The layout of the remainder of the paper is then as follows. In section 2 we discuss the spontaneous breaking of SO(3) to finite non-Abelian symmetries such as A 4 , S 4 and A 5 with supersymmetry. In section 3 we discuss the further breaking of A 4 to residual Z 3 and Z 2 symmetries, showing how it may be achieved from a supersymmetric SO(3) potential. In section 4 we construct in detail a supersymmetric A 4 model along these lines, originating from SO(3) × U (1), and show that it leads to a phenomenologically acceptable pattern of lepton mixing and masses, once subleading corrections are taken into account. Within this model, we also discuss the phenomenological consequences of having a gauged SO(3), leading to massive gauge bosons, and show that all domain wall problems are resolved. Section 5 concludes the paper. The paper has three appendices. In Appendix A we list the Clebsch-Gordan coefficients of SO(3) which are used in the paper. In Appendix B we display explicitly the solutions of the superpotential minimisation. In Appendix C we show the deviation from the Z 2 -invariant vacuum.
2 Spontaneous breaking of SO(3) to finite non-Abelian symmetries A 4 , S 4 and A 5 with supersymmetry The key point to break SO(3) to non-Abelian discrete symmetries is introducing a high irrep of SO(3) and require it gain a non-trivial VEV. In this section, after a brief review of SO(3), we discuss how to break SO(3) to A 4 by introducing a 7-plet, and then generalise our discussion to SO(3) → S 4 and A 5 .

The SO(3) group
The rotation group SO(3) is one of the most widely used Lie groups in physics and mathematics. It is generated by three generators τ 1 , τ 2 and τ 3 . Each element can be expressed by In the fundamental three dimensional (3d) space, the generators are represented as Each irrep of SO(3) has 2p + 1 dimensions and we denote it as a 2p+1-plet. Each 2p+1-plet can be represented as a rank-p tensor T i 1 i 2 ...ip in the 3d space. This tensor is symmetric and traceless, for any a, b p. It transforms under SO(3) as where O is transformation matrix corresponding to the element g {α α } in the 3d space, and it is always a 3 × 3 real orthogonal matrix. Here and in the following, doubly repeated indices are summed.

SO(3) → non-Abelian discrete symmetries
SO (3) can be spontaneously broken to other non-Abelian discrete symmetries by introducing different high irreps. Ref. [27] gives an incomplete list of subgroups which could be obtained after the relevant irrep get a VEV. For instance, some of those subgroup obtained by irreps up to 13 are shown in Table 1. The minimal irrep for SO(3) → S 4 is a 9-plet, while that for SO(3) → A 5 is a 13-plet. Applying a 9-plet flavon ρ and a 13-plet flavon ψ, respectively, we will realise these breakings in a SUSY framework in the following.
For the A 4 symmetry, we work in the Ma-Rajasekaran (MR) basis, where the generators s and t in the 3d irreducible representation are given by The A 4 -invariant VEV, satisfying is given by The VEV of ξ is geometrically shown in Fig. 1. The discussion of SO(3) → A 4 has been given in Refs. [26][27][28]. The main idea is constructing flavon potential and clarifying the A 4 -invariant one in Eq. (9) to be the minimum of the potential, where v ξ is determined by the minimisation. This idea cannot be directly applied to supersymmetric flavour models. In the later case, the flavon potential is directly related to the flavon superpotential where φ i represent any scalars in the theory, and the dots are negligible soft breaking terms and D-terms for the fields charged under the gauge group. This potential is more constrained than the non-supersymmetric version. If the minimisation of the superpotential ∂w f /∂φ i = 0 has a solution, the minimisation of the potential ∂V f /∂φ i = 0 is identical to the minimisation of the superpotential. Since most flavour models have been built in SUSY, it is necessary to consider if SO(3) → A 4 can be achieved in SUSY.
In order to break SO(3) to A 4 , we introduce two driving fields ξ d 1 ∼ 1, ξ d 5 ∼ 5 and consider the following superpotential terms where c 1 and c 2 are complex dimensionless coefficients. As required [9], the driving fields do not gain non-zero VEVs, realised by imposing U (1) R charges. Minimisation of the potential is identical to the minimisation of the flavon superpotential respecting to the driving fields as follows, The explicit expressions of Eqs. (12) and (13) are listed in Appendix B. Taking the A 4invariant VEV to Eqs. (12) and (13), we see that Eq. (13) is automatically satisfied and Eq. (12) leads to ξ 123 = ±µ ξ / √ 6c 1 . Therefore, the A 4 symmetry is consistent with the vacuum solution obtained from the minimisation of the superpotential.
We need to check the uniqueness of A 4 since it is not clear if A 4 is the only symmetry after SO(3) breaking. We assume there is another vacuum solution ξ , which has an infinitesimal deviation from the A 4 -invariant one, ξ = ξ + δξ. Eqs. (12) and (13) must also be satisfied for ξ . Directly taking them into account, we get the constraints on δξ. Straightforwardly, we obtain leaving only three unconstrained parameters δξ 112 , δξ 113 and δξ 133 . The unconstrained perturbation parameters δξ can be rotated away if we consider a SO(3) basis transformation, with α 1 = 3c 1 2 δξ 113 /µ, α 2 = 3c 1 2 δξ 112 /µ, α 3 = − 3c 1 2 δξ 133 /µ and the generators τ i being given in Eq. (2). Therefore, ξ also preserves the A 4 symmetry and the shift from ξ to ξ corresponds to only a basis transformation of SO(3). Such a basis transformation has no physical meaning. We conclude that the minimisation equation of the superpotential, i.e., Eqs. (12) and (13), uniquely breaks SO(3) to A 4 .

SO(3) → S 4
For the S 4 symmetry, the generators in the 3d irreducible space are given by g s , g t and In the 3d flavour space, the 9-plet ρ is represented as a rank-4 tensor ρ ijkl .
In order to require the VEV ρ invariant under the S 4 symmetry. The following constraints are required, which are equivalent to Follwing a similar procedure but replacing the 7-plet ξ by a 9-plet ρ, we succeed to break SO(3) to S 4 in SUSY by introducing two driving fields ρ d 1 ∼ 1 and ρ d 5 ∼ 5. The flavon superpotential is constructed as Minimisation respect to the driving fields gives rise to Taking Eq. (19) to the above equations, we see that Eq. (22) is automatically satisfied and Eq. (21) leads to ρ 1133 = ±µ ρ / 30c ρ1 .

SO(3) → A 5
For the A 5 symmetry, the generators in the 3d irreducible space are given by g s , g t and The 13-plet ψ in the 3d flavour space is represented as a rank-6 tensor ψ ijklmn . Constrained by Eq. (3), there are 13 free components of ψ, which can be chosen to be In order to require the VEV ψ invariant under the S 4 symmetry. The following constraints are required, They are equivalent to In order to break SO(3) to A 5 , we introducing two driving fields ψ d 1 ∼ 1 and ψ d 9 ∼ 9, instead of 5. The flavon superpotential is constructed as Minimisation respect to the driving fields gives rise to ∂w ψ ∂ψ d Taking Eq. (28) to the above equations, we see that Eq. (31) is automatically satisfied and Eq. (30) leads to ψ 111111 = ±µ ψ /(4 21c ψ1 ).

Representation decomposition
After SO(3) is broken to a non-Abelian discrete group, it is necessary to decompose each irrep of SO(3) to a couple of irreps of the discrete one. This task is achieved by comparing reduction of Kronecker products of representations of SO(3) with those of the discrete one [35].
For irreps of SO(3) decomposed to irreps of A 4 , we identify 1, 3 of SO(3) with 1, 3 of A 4 , respectively and compare the Kronecker products in A 4 . Since the right hand sides of both equations are identical, 5 of SO(3) is decomposed to 1 + 1 + 3 of A 4 . One further compares right hand side of 3 × 5 = 3 + 5 + 7 (35) with that of and obtains 7 = 1 + 3 + 3, where 1 × 3 = 3 and 1 × 3 = 3 are used. Continuing to play this game, we can get decomposition of as high irrep of SO(3) as we want into irreps of This game is directly applied into irrep decomposition in S 4 and A 5 . In S 4 , there are five irreps: 1 (the trivial singlet), 1 (different from 1 of A 4 ), 2, 3 and 3 . In A 5 , there are five irreps: 1 (the trivial singlet), 3, 3 , 4 and 5. Keeping in mind the Kronecker products in S 4 , and in A 5 , and comparing them with Kronecker products in SO (3), we obtain irrep decompositions in S 4 and A 5 , respectively.
We summarise decomposition of irreps of SO(3) (up to 13) to irreps of A 4 , S 4 and A 5 in Table 2.
Before ending this section, we show more details of how a irrep of SO (3) is decomposed into irreps of A 4 as follows, which will be useful for our discussion in the next two sections. • For a triplet 3 of SO (3)

SO(3)
• A 5-plet of SO (3), χ, can be represented as a rank-2 tensor χ ij in the 3d space. It is symmetric, χ ij = χ ji , and traceless, χ 11 + χ 22 + χ 33 = 0. Independent components can be chosen as χ 11 , χ 12 , χ 13 , χ 23 and χ 33 . The 5-plet is decomposed to two nontrivial singlets 1 and 1 and one triplet 3 of A 4 . It is useful to re-parametrise χ in the form where ω = e 2iπ/3 . This parametrisation has two advantages. One is the simple transformation property in A 4 , The other is the normalised kinetic term, • The 7-plet of SO (3) is a symmetric and traceless rank-3 tensor in the 3d space. It is decomposed to one trivial singlet 1 and two triplets 3 of A 4 . The former mentioned ξ can be re-labelled as Here, And the kinetic term is also normalised 4 , Since ξ 0 is a trivial singlet of A 4 , once ξ 0 gets a non-zero VEV, SO(3) will be broken but A 4 is still preserved. This is consistent with the discussion in the former subsection.
3 The further breaking of A 4 to residual Z 3 and Z 2 In A 4 lepton flavour models, A 4 has to be broken to generate flavour mixing. In most of these models, residual symmetries Z 3 and Z 2 are preserved respectively in the charged lepton sector and neutrino sector after A 4 breaking. These residual symmetries are not precise but good approximate symmetries. The misalignment between Z 3 and Z 2 leading to a mixing with tri-bimaximal mixing pattern at leading order.
Embedding A 4 to the continuous SO(3) symmetry forces strong constraints on couplings, and the breaking of A 4 to Z 3 and Z 2 becomes very non-trivial. In this section, we will show, for definiteness, how to realise A 4 → Z 3 and Z 2 in the framework of supersymmetric SO(3)-invariant theory.

A 4 → Z 3
The breaking of A 4 to Z 3 can be simply realised by using a triplet 3 of SO (3). We denote such a flavon as ϕ. In order to obtain the Z 3 -invariant VEV, we introduce an 1-plet driving field ϕ d 1 and a 5-plet driving field ϕ d 5 and consider the following SO(3)-invariant superpotential Minimisation of the superpotential gives rise to whose detailed formula is listed in Appendix B. Starting from the A 4 -invariant VEV ξ in Eq. (9), we use (ξ(ϕϕ) 5 ) 5 = 0 to derive ϕ 2 1 = ϕ 2 2 = ϕ 2 3 , and f 1 (ϕϕ) 1 − µ 2 ϕ = 0 to determine the value of ϕ 2 1 . Here, we directly write out the following complete list of solutions where These Z 3 groups are conjugate to each each and have no physical difference [41,42].
Eq. (46) involves the interaction between ϕ and ξ. This interaction contributes to the potential of ξ and shifts the ξ VEV and makes it deviate from the A 4 -invariant one. This shifting effect is small enough due to suppression of the higher dimensional operator. Once the ξ VEV is shifted, the A 4 symmetry is not precise any more, but should be treated as an approximate and effective symmetry.

Spontaneously splitting 1 with 1 of A 4
In A 4 models, the three singlet irreps 1, 1 and 1 are usually assigned to e c , µ c and τ c (or their permutation), respectively. These irreps are independent with each other in A 4 and the generated e, µ and τ masses are independent with each other.
In the framework of SO (3), the non-trivial singlet irreps 1 and 1 are obtained from the decomposition of 5 of SO(3) (or higher irreps, e.g., 9 etc), as shown in Table 2. These singlets are always correlated with each other. As a consequence, if we directly arrange two of the charged leptons (e.g., µ c and τ c ) to the same 5 of SO (3), we have to face a fine tuning of masses of these two charged leptons. In this subsection, we are going to consider how to avoid this problem from the spontaneous symmetry breaking of A 4 .
To summarise, we realise the breaking of A 4 to Z 3 and Z 2 and achieve to split 1 with 1 of A 4 based on SO(3)-invariant superpotential. The scales representing the breaking of A 4 , v ϕ , v χ and v ζ , should be much lower than the scale of SO (3) breaking v ξ . This can be satisfied by treating µ 2 ϕ , µ 2 χ and µ 2 ζ as effective descriptions from higher dimensional operators. One may notice that there may exist some unnecessary interactions which are not written out but cannot be forbidden based on current field arrangements. A detailed discussion on how to forbid the unnecessary coupling will be given in the next section on the model building. Besides, the ways to realise A 4 → Z 3 , A 4 → Z 2 and split 1 from 1 showing above are not the unique ways. One can introduce different irreps, combined with different driving fields to achieve them. This difference further leads to the difference of model building, which will not be discussed in this paper.

The model
In this section we will construct a supersymmetric A 4 model, based on SO(3) × U (1), with the breaking SO(3) → A 4 and subsequently (at a lower scale) A 4 → Z 3 , Z 2 , using the vacuum alignments discussed previously, where the misalignment of Z 3 in the charged lepton sector and Z 2 in the neutrino sector gives rise to lepton mixing. The model building strategy is shown in Fig. 2. The U (1) symmetry is used to forbid couplings which are unnecessary to generate the required flavon VEVs and flavour mixing. Note that no ad hoc discrete symmetries are introduced in this model.
In A 4 models, the right-handed charged leptons e c , µ c and τ c are arranged as 1, 1 and 1 (or their permutation), respectively. In SO(3), the minimal irrep containing 1 and 1 is 5. In order to match with A 4 models, we embed 1 and 1 of A 4 to two different 5-plets of SO(3). In our model, we embed µ c and τ c to two different 5-plets R µ and R τ 5 . Four extra right-handed leptons are introduced for R µ and R τ , respectively. These particles should decouple at low energy theory to avoid unnecessary experimental constraints. We achieve this goal by introducing two left-handed 3-plets L µ , L τ and two singlets L µ0 , L τ 0 . We write out explicitly each components of the fermion multiplets in the 3d space as follows, 5 Imbedding µ c and τ c into the same 5-plet leads to fine tuning between µ and τ masses The flavour symmetry at high energy is assumed to be supersymmetric SO(3). It is broken first to A 4 , which then breaks, at a lower scale, to the residual symmetry Z 3 in the charged lepton sector and Z 2 in the neutrino sector, with supersymmetry preserved throughout. The misalignment of the residual symmetries gives rise to flavour mixing.
Charges for all relevant fields in SO(3) × U (1) are listed in Table 3. Besides SO(3), we introduce additional U (1) symmetry to forbid unnecessary couplings.

Vacuum alignments
Terms leading to SO(3) breaking and A 4 breaking in the superpotential involving flavons and driving fields are given by Here, the dots represent subleading corrections, which will be discussed in section 4.4. Compared with the superpotential terms in sections 2 and 3, Eq. (56) takes a very similar form except the following differences: • The constant µ 2 ξ is not explicitly written out, but replaced by A ξ η. Here η andη are SO(3) singlets. From the minimisation ∂w f /∂η d 1 = 0, we know that both η and η cannot be zero, and thus, we denote them as v η and vη, respectively. Once η gets this VEV, µ 2 ξ = A ξ v η is effectively obtained. This treatment is helpful for us to arrange charges for ξ. Otherwise only a Z 2 charge can be arranged for ξ.
The approach for how the flavons obtained the required VEVs have been discussed in the former section. We do not repeat the relevant discussion here but just list the achieved VEVs of flavons, where v ξ , v ϕ , v χ and v ζ are respectively given by We briefly discuss the scales involved in the model. The VEV v ξ represents the scale of SO(3) → A 4 and v ϕ and v χ represent the scales of A 4 → Z 3 and Z 2 , respectively. VEVs of η andη do not break any non-Abelian symmetries but U (1), their role is to connect the scales of SO(3) breaking and A 4 breakings. For the scale of A 4 → Z 3 , v ϕ v η is naturally achieved due to the suppression of Λ in the dominator in Eq. (59). For the scale of A 4 → Z 2 , v χ v η can be achieved by either assuming a hierarchy v η vη all assuming a small coefficient g χ . The VEV v ξ can be much larger than v η and vη if the dimension one parameter A ξ is large enough. With the above treatment (but not the unique treatment), we can easily achieve a hierarchy of energy scales In the following, we simplify our discussion by assuming all dimensionless parameters in the flavon superpotential being of order one. In this case, orders of magnitude of v ξ , v ϕ , v χ and v ζ are determined by Λ, A ξ , vη and v η as The hierarchy in Eq. (60) is obtained by requiring Λ A ξ vη v η .

Lepton masses
Lagrangian terms for generating charged lepton masses are given by at leading order. After the flavons get their VEVs, we arrive at the effective Yukawa couplings for leptons and Majorana masses for right-handed neutrinos.
The Yukawa coupling of e c is given by where y e = 3y e1 + 4y e2 .
Couplings involving R µ are given by L µ3 and R µ3 obtain three degenerate heavy masses 2 √ 3Y µ2 v ξ . These mass are much heavier than the electroweak scale, and thus for the low energy theory, L µ3 and R µ3 decouple. L µ0 and R µ obtain a mass Y µ1 v ζ . For v ζ heavier than the electroweak scale, R µ decouples from the low energy theory. In this way, we successfully split R µ with µ c . After the heavy leptons are integrated out, we are left with the following couplings at the low energy theory, where y µ = y µ1 − y µ2 Y µ3 /Y µ2 with the y µ2 term obtained via a seesaw-like formula.
Those coupling to R τ are given by L τ 3 and R τ 3 obtain three degenerate heavy masses 2 √ 3Y τ 2 v ξ , which are much heavier than the electroweak scale. L τ 0 and R τ obtain a mass 2Y ). This mass term should also be heavier than the electroweak scale such that R τ can decouple from the low energy theory. This mass term aims to split R τ with τ c and it provides a stronger constraint to the scale v ζ than that splitting R µ with µ c . After all these heavy particles decouple, we obtain Yukawa coupling for τ c at low energy as After the Higgs H d gets the VEV H d = v d / √ 2, we arrive at the charged lepton mass matrix in the basis and (e c , µ c , τ c ) T . This matrix is diagonal by a unitary matrix U l via U T l M l = diag{m e , m µ , m τ } with and In this model, since y e , y µ , and y τ are totally independent free parameters, there is no need to introduce a fine tuning of them to fit the hierarchy of e, µ and τ masses.
If we naturally assume the dimensionless parameters y e , y µ and y τ are all of order one, we then obtain m e : m µ : m τ ∼ (v ϕ /Λ) 2 : vη/Λ : 1. The mass between L τ 0 and R τ is of order v 2 ζ /Λ. It should be much heavier than the electroweak scale to avoid the constraints from collider searches, i.e., v 2 The realisation of neutrino masses is straightforward. The relevant superpotential terms at leading order are given by The generated Dirac mass matrix between ν and N and Majorana mass matrix for N , in the bases (ν 1 , ν 2 , ν 3 ) T and (N 1 , N 2 , N 3 ) T , are respectively given by where a = 2ληv 2 η /Λ and b = 2 where P ν = diag { e i β 1 2 , e i β 2 2 , e i β 3 2 } and Applying the seesaw mechanism, The three mass eigenvalues for light neutrinos are given by We are left with the tri-bimaximal mixing.
In the model discussed so far, the crucial point in deriving an A 4 -invariant VEV is the requirement (ξξ) 5 = 0, while those to derive a Z 2 -or Z 3 -invariant vacuum is the requirement (ξ(ϕϕ) 5 ) 5 = 0 or (χξ) 5 = (ξ(χχ) 5 ) 3 = 0, respectively. These requirements are obtained via the minimisation of the superpotential. However, extra terms may be involved in the superpotential and lead to that the above requirements do not hold explicitly. As a consequence, the relevant vacuums do not preserve the symmetries explicitly.
In the next subsection, we will prove that after including these terms, the flavon VEVs do deviate from the former symmetric ones, but the size of the deviations are safely very small. Then in the subsequent subsection we consider subleading effects to the flavour mixing and show that it gives important corrections.

Subleading corrections to the vacuum (are negligible)
We first list terms in the flavon superpotential which cannot be avoided by the flavour symmetry SO(3) × U (1). The full flavon superpotential should be given by w d 3 f represents renormalisable terms in the superpotential, and w d=4 f and w d=5 f are nonrenormalisable quartic and quintic couplings, respectively. Up to quintic couplings, all Table 4: All terms up to quintic couplings in the flavon superpotential allowed by the flavour symmetry SO(3) × U (1). µ η and A ξ are free parameters with one mass unit to balance the dimension in the superpotential terms are listed in Table 4, classified by the driving fields. As mentioned above, we follow the general arrangement in most supersymmetric models that driving fields always linearly couple to flavon fields. Compared with Eq. (56), a lot of new terms appear here.
We will discuss how they modify the VEVs of ξ, ϕ, χ and ζ in detail.
First for ξ, couplings involving the driving field ξ d 5 include not just the renormalisable term ξ d 5 (ξξ) 5 1 , but also the quartic term ξ d 5 (ξϕ) 5 1η and the quintic term ξ d 5 χ 1 η 3 , ξ d 5 (ξξ) 5 1 ηη, etc. The minimisation ∂w f /∂ξ d 5 = 0 does not lead to (ξξ) 5 = 0, but where the dots represent contribution of all rest terms involving ξ d 5 in Table 4. Dimensionless free parameters are omitted here and in the following. Couplings involving the driving field ξ d 1 is modified into where the dots represent contributions of the rest terms in Table 4. We denote the shifted VEV as where ξ A 4 is the A 4 -invariant part with each components given in Eq. (58) and δ ξ represents A 4 -breaking corrections. Eq. (80) only gives an all overall small correction to v ξ without breaking the A 4 symmetry. Eq. (79) is approximately simplified to where (ξ A 4 + δ ξ )(ξ A 4 + δ ξ ) 5 ≈ 2(δ ξ ξ A 4 ) 5 is used on the left hand side and ξ, χ and ϕ are replaced by the A 4 -, Z 2 -and Z 3 -invariant VEVs ξ A 4 , χ Z 2 and ϕ Z 3 on the right hand side of Eq. (58), respectively. In our paper, since we only care about the order of magnitude of corrections, we neglect CG coefficients in the products and do a naive estimation of the order of magnitude. Then we obtain where the fourth term in the curly bracket has a vanishing contribution since (ξ A 4 ξ A 4 ) 5 = 0. The relation in Eq. (61) has been used. In the above estimation, we include all corrections from Table 4 and pick the largest one v ξ v ϕ /(Λv ξ ). Since v χ , v ϕ v ξ Λ, this correction is very small and can be safely ignored. The exact correction may be different from the estimation but must be smaller than it.
Similarly, we can estimate corrections to the VEVs of ϕ, χ and ζ. We denote the shifted VEVs of ϕ, χ and ζ as respectively, where ϕ Z 3 , χ Z 2 and ζ 1 represent leading-order value in Eq. (58) and δ ϕ , δ χ and δ ζ are subleading order corrections. Once subleading high dimensional operators are included, the minimisation of the superpotential gives rise to A naive estimation gives the upper bounds of corrections Again, (ξ A 4 ξ A 4 ) 5 = 0, as well as (ζ 1 ζ 1 ) 1 = 0, and the relation in Eq. (61) are used in the above. Upper bounds of relevant corrections to the Z 3 -invariant VEV δ ϕ /v ϕ and the ζ VEV δ ζ /v ζ are as small as δ ξ /v ξ . The upper bound of the correction to the χ VEV is larger, δ χ /v χ v 2 η /(Λv χ ) ∼ √ v η vη/Λ. However, we calculate this correction in detail in Appendix C and find that the true correction which is also very small.
We numerically give an example of the size of these corrections. By setting and All corrections are less than 1%. Therefore, VEVs of ξ, χ, ϕ and ζ are stable under subleading corrections.

Subleading corrections to flavour mixing (are important)
At leading order, the flavour mixing appears as the tri-bimaximal pattern. Deviation arises after subleading corrections are considered. There are two origins of subleading corrections: subleading higher dimensional operators in superpotential terms for lepton mass generation w and higher dimensional operators in the flavon superpotential w f . The second type shift the flavon VEVs and further modify the mixing. As discussed in the last subsection, these corrections in this model are less than 1%, safely negligible. In the following, we will only discuss corrections from the first origin.
Subleading terms contributing to e c H d up to d 7 and those to R µ H d or R τ H d up to d 6 include For terms involving only some of ϕ, ξ, η andη, no Z 3 -breaking effects are included. The Z 3 symmetry always guarantees that the corrected effective Yukawa couplings take the forms (1, 1, 1) T , (1, ω, ω 2 ) T and (1, ω 2 , ω) T , as in Eqs. (64), (67) and (69), respectively. Terms breaking the Z 3 symmetry are those involving ζ or χ. There are five terms left, ( R µ ) 5 (ζζ) 5 1 H d , ( R τ ) 5 (ζζ) 5 1 ηH d , ( R µ ) 3 (ϕχ) 3 1 ηH d , ( R µ ) 5 (ϕχ) 5 1 ηH d , and ( R µ ) 7 (ϕχ) 7 1 ηH d . The first two terms only contribute to coupling between and R µ3 or R τ 3 . The rest three terms contributing to couplings between and µ c . Their contributions to the charged lepton mass matrix are characterised by adding a new matrix to M l . Acting U T ω on the left hand side of δM l leaves where c and d are real dimensionless parameters. The unitary matrix to diagonalise M l is modified to U l U ω U eµ , where U eµ is a complex rotation matrix on the eµ plane, Here, we have ignored the (3, 2) entry of U T ω δM l since it is too small compared with the τ mass m τ ∼ v ϕ v d /Λ.
In the neutrino sector, terms for neutrino masses up to d 5 have only trivial corrections, Therefore, the unitary matrix U ν to diagonal M ν keeps the same as that in the leading order.
Including the subleading correction, the PMNS matrix is modified into U PMNS = U † eµ U TBM . multiplying U eµ on the left hand side does not change the third row of the PMNS matrix. Three mixing angles are given by [5] sin θ 13 = sin θ eµ √ 2 , In this model, θ 23 in the first octant is predicted. The reactor angle θ 13 ∼ v η v χ /(vηΛ).

Phenomenological implications of gauged SO(3)
We label the gauge field of SO(3) and U (1) as F 1,2,3 and B , respectively. Their interactions with flavons or fermions are simply obtained with the replacement in the kinetic terms of the relevant fields. Here, g 3 and g 1 are gauge couplings of SO (3) and U (1), respectively, and the U (1) charge Q for each field is listed in Table 3.
Specifically, the kinetic term for ξ in Eq. (44) is replaced by (D µ ξ * D µ ξ) 1 with where Q = +1 for ξ has been used. F 1,2,3 µ gain masses once ξ get the A 4 -invariant VEV. We obtain that M 2 The degenerate mass spectrum is also consistent with the A 4 symmetry 6 . Later after the rest flavons ζ, ϕ and χ gain VEVs, mass splitting are generated among F 1,2,3 . Since VEVs of ζ, ϕ and χ are much smaller than that of ξ, the mass splittings are very small, and masses of F 1,2,3 are still nearly degenerate.
B obtains a mass from VEVs of both ξ and η,η, M 2 . After A 4 breaking, VEVs of ζ, ϕ and χ contribute small corrections to the B mass. Interactions between leptons and B are flavour-dependent, with charges for , e c , µ c and τ c given by − 2 3 , − 7 3 , −1 and − 1 3 , respectively.
In the limit of the A 4 invariance, there is no mixing between F and B . This can be simply explained as follows. The mixing between F and B from Eq. (99) and (D µ ξ * D µ ξ) 1 , if exists, can be only generated via coupling F B ξξ. Since F ∼ 3, B ∼ 1, ξ ∼ 7, the only SO(3) invariant formed by these fields is B µ F µ (ξ * ξ) 3 1 . Here, the 3-plet contraction between ξ * and ξ are anti-symmetric. Once ξ get the VEV, where only one of the seven components has a non-zero value, ξ 0 = v ξ , the anti-symmetric contraction (ξ * ξ) 3 vanishes. Therefore, there is no mixing between F and B . The mixing between F and B is generated after A 4 breaking, induced by terms such as B µ F µ (ξ * χ) 3 1 . The resulted mixing between F and B is suppressed by the ratio v χ /v ξ .
These gauge bosons are supposed to be very heavy, with masses around ∼ O(v ξ ) or ∼ O(max(v ξ , v η , vη)), respectively, if gauge coefficients are of order one. However, they could be much lighter if gauge couplings are tiny. For example, if Λ is fixed at 10 4 TeV, v ξ and v η are predicted to be around 10 3 TeV and v χ , v ϕ and v ζ be around 100 TeV. For a gauge coupling around 10 −3 , TeV-scale gauge bosons are predicted. Then, interesting signatures involving gauge interactions can be tested at colliders or precision measurements of charged leptons. Another interesting point is the prediction of a heavy tau lepton with mass also around TeV scale (v 2 ζ /Λ ∼ 1 TeV). Its interaction with B can be tested at colliders.

Absence of domain walls
The domain wall problem is a well-known problem for discrete symmetry breaking. In this paper, all flavour symmetries at high scale are gauged. A 4 , and the residual symmetries Z 3 and Z 2 , are just phenomenologically effective symmetries at lower scales. The usual domain wall problem for the global symmetry breaking does not apply here.
In our model, we actually have a two-step phase transition SO(3) → A 4 and A 4 → Z 3 , Z 2 . We discuss more on why the topological defect of domain walls does not exit in the model.
At the first step, SO(3) → A 4 , the breaking of a gauge symmetry does not introduce domain walls. As noted in section 2, there are degenerate vacuums which are continuously connected by SO(3) basis transformation as in Eq. (15). All vacuums are perturbatively equivalent.
At the second step, A 4 → Z 3 , Z 2 , degenerate Z 3 -invariant or Z 2 -invariant vacuums exit, as shown in Eqs. (47) and (50). Taking the Z 3 -invariant vacuum as an example, different Z 3 -invariant vacuums are randomly generated during A 4 breaking to Z 3 and domain walls separating different vacuums arise. These domain walls store energy with energy density inside the wall around v 4 ϕ or v 4 χ . Without considering gauge interactions, there are not enough energy inputted to force one vacuum jumping across the wall into another. Therefore, domain walls survive. Once gauge interactions are included, domain walls should decay to light particles mediated by gauge bosons. For the case of small gauge couplings, the gauge bosons may be light enough, i.e., M F v ϕ , and domain walls may directly decay into gauge bosons.

Conclusion
In this paper we have discussed the breaking of SO(3) down to finite family symmetries such as A 4 , S 4 and A 5 using supersymmetric potentials for the first time. We have analysed in detail the case of supersymmetric A 4 and its finite subgroups Z 3 and Z 2 . We have proposed a supersymmetric A 4 model of leptons along these lines, originating from SO(3) × U (1), which leads to a phenomenologically acceptable pattern of lepton mixing and masses once subleading corrections are taken into account. We have also discussed the phenomenological consequences of having a gauged SO(3), leading to massive gauge bosons, and have shown that all domain wall problems are resolved in this model.
The main achievement of the paper is to show for the first time that supersymmetric SO(3) flavour symmetry can be the origin of finite non-Abelian family symmetry models. By focussing in detail on a supersymmetric A 4 model, we have demonstrated that such a strategy can lead to a viable lepton model which can explain all oscillation data with SUSY being preserved in the low energy spectrum (below the flavour symmetry breaking scales). Moreover, we have shown that, if the SO(3) is gauged, there may be interesting phenomenological implications due to the massive gauge bosons.
About a half of the paper is devoted to the study of the realistic supersymmetric A 4 model of leptons, arising from SO(3) × U (1). This study is important in order to verify that it is really possible to construct a fully working model along these lines. The main achievements of the specific model may be summarised as follows: • We have achieved the breaking of SO(3) → A 4 in SUSY, using high irreps of SO (3) and flat directions. In this paper, we have chosen a 7-plet, i.e., a rank-3 tensor in 3d space, to achieve the breaking. We have shown that it is possible to break SO(3) to S 4 or A 5 by using different higher irreps.
• We have shown that it is possible to also achieve, at the level of SO(3), the subsequent breaking of A 4 at a lower scale (below the SO(3) breaking scale) to the residual symmetries Z 3 and Z 2 . Such Z 3 and Z 2 symmetries are preserved in charged lepton sector and neutrino sector, respectively, after the A 4 breaking, in accordance with the semi-direct model building strategy.
• Starting from a supersymmetric flavour group SO(3) × U (1), we have shown how SO (3) is broken first to A 4 , and then to Z 3 and Z 2 . The A 4 , Z 3 and Z 2 symmetries are respectively achieved by the flavons ξ, ϕ and χ after they gain the A 4 -, Z 3 -and Z 2 -invariant VEVs, respectively. We have found that tri-bimaximal mixing (with zero reactor angle) is realised at leading order. One technical point is that the singlet irreps 1 and 1 of A 4 always accompany each other after SO (3) breaking. To avoid any fine tuning of parameters related to µ and τ masses, we have introduced an additional flavon ζ to split the 1 and 1 .
• We have considered the influence of the higher dimensional operator corrections to the model. We have shown that the A 4 -, Z 3 -and Z 2 -invariant VEVs are stable even after subleading corrections are included. However, we have seen that the charged lepton mass matrix is modified by higher dimensional operators, due to the coupling with χ, which gains the Z 2 -invariant VEV. This welcome correction leads to additional mixing between e and µ, giving rise to a non-zero θ 13 and the CP-violating phase δ.
• If the SO(3) × U (1) is gauged, the model predicts three gauge bosons F 1,2,3 with the nearly degenerate masses after SO(3) breaking to A 4 . Another gauge boson B gain a mass after U (1) is broken. These gauge bosons with their flavour-dependent interactions with leptons will lead to phenomenological signatures worthy of further study.
• We emphasise that the flavour symmetry at high scale is the continuous gauge symmetry SO(3) × U (1), with no ad hoc discrete symmetries introduced, and A 4 being just an effective flavour symmetry below the SO(3) breaking scale. We have shown that the usual domain wall problems encountered in A 4 models are resolved here.