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Deposited in DRO: 04 January 2019 Version of attached le: Published Version Peer-review status of attached le: Peer-reviewed Citation for published item: King, Stephen F. and Zhou, Ye-Ling (2018) 'Spontaneous breaking of SO(3) to nite family symmetries with supersymmetry  an A4 model.', Journal of high energy physics., 2018 (11). p. 173. Further information on publisher's website: https://doi.org/10.1007/JHEP11(2018)173

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Durham University Library, Stockton Road, Durham DH1 3LY, United Kingdom Tel : +44 (0)191 334 3042 | Fax : +44 (0)191 334 2971 https://dro.dur.ac.uk JHEP11(2018)173 , 4 A Springer October 8, 2018 : November 20, 2018 November 28, 2018 U(1), which leads : : model × 4 Received A Accepted Published . We then propose a supersym- 2 Z to finite family Published for SISSA by and https://doi.org/10.1007/JHEP11(2018)173 3 Z SO(3) a,b [email protected] , . and its finite subgroups 3 4 A and Ye-Ling Zhou 1809.10292 a The Authors. Gauge Symmetry, Discrete Symmetries, Neutrino Physics, Supersymmetric c

We discuss the breaking of SO(3) down to finite family symmetries such as , using supersymmetric potentials for the first time. We analyse in detail the case model of leptons along these lines, originating from SO(3) [email protected] 5 4 A A and School of Physics andSouthampton Astronomy, SO17 University 1BJ, of United Southampton, Kingdom Institute for Particle PhysicsSouth Phenomenology, Road, Department Durham of DH1 Physics, 3LE, Durham United University, Kingdom E-mail: 4 b a Open Access Article funded by SCOAP Keywords: Effective Theories ArXiv ePrint: to a phenomenologically acceptablecorrections pattern are of taken lepton into mixing account.having and We a masses also gauged once discuss SO(3), subleading theproblems leading phenomenological are to consequences resolved massive of in gauge this bosons, model. and show that all domain wall Abstract: S of supersymmetric metric Stephen F. King Spontaneous breaking of symmetries with supersymmetry — an JHEP11(2018)173 4 7 8 32 32 33 12 13 13 12 4 S , 4 A 24 21 2 15 Z 26 32 U(1) and × 4 3 33 Z 4 A 29 of – i – 00 1 9 3 to residual with 0 4 4 27 1 A A 2 3 model from SO(3) -invariant vacuum → Z Z 16 2 4 4 5 4 Z 4 A S A A → → 4 4 18 → → → A A 15 non-Abelian discrete symmetries 1 3 2 28 → Z Z with supersymmetry → → 5 4 4 2.2.3 SO(3) A A 2.2.1 SO(3) 2.2.2 SO(3) A B.1 Solutions forB.2 SO(3) Solutions for B.3 Solutions for 4.3 Lepton masses 4.4 Subleading corrections4.5 to the vacuum Subleading (are corrections4.6 negligible) to flavour mixing Phenomenological (are implications4.7 important) of gauged SO(3) Absence of domain walls 3.3 Spontaneously splitting 4.1 The model4.2 Vacuum alignments 2.3 Representation decomposition 3.1 3.2 and 2.1 The SO(3)2.2 group SO(3) C Deviation from the 5 Conclusion A Clebsch-Gordan coefficients of SO(3) B Solutions of the superpotential minimisation 4 A supersymmetric 3 The further breaking of Contents 1 Introduction 2 Spontaneous breaking of SO(3) to finite non-Abelian symmetries JHEP11(2018)173 4 ]. 2 A was 4 ]. It is ]; A ] and/or 21 ]. 40 28 36 ]. However, – – 15 34 23 – , ], when they are 12 13 33 breaking phase tran- 4 A ]. In supersymmetric (SUSY) 20 ]. When supersymmetry (SUSY) is ] not only represents the first labora- ]. 1 5 11 – 9 ]. However, even if such symmetries do arise ]. 32 – 19 – 1 – – ]. However, it is not enough to completely solve 29 17 39 discrete symmetry is anomalous, and hence it is only a sym- breaking terms in the Lagrangian, possibly in the form of ]. Subsequently, non-Abelian discrete symmetries such as 4 4 8 A – A ], perhaps as a subgroup of the modular group [ 6 ]. Since, in general, non-Abelian discrete symmetries do not imply 22 16 ] where three possible solutions were discussed: 38 , or SO(3) [ 37 is never restored after reheating following inflation. 1 4 ] A 4 , 3 Planck scale suppressed higher order operators, arising from gravitational effects; sition happens during inflationthe which effectively dilutes the domain walls, and that by quantum corrections.symmetry For to example the this could quarklevel due be sector to due such the to thatthe QCD extending problem the anomaly since the symmetry [ this discrete is anomaly cannot broken remove at all the the vacuum quantum degeneracy [ metry of the classical action and not a full symmetry of the theory, being broken to include explicit to suppose that, in the thermal history of the Universe, the to suppose that the The problem of domain walls with non-Abelian discrete symmetries such as Although the motivation for non-Abelian discrete symmetries remains strong, there SU(3) has recently been considered in extra dimensions [ 2. 3. 1. 1 conflict with standard cosmology, and appear to “over-close thediscussed Universe” in [ [ possible that the non-Abelian discrete symmetriessuch could as arise string from some theory high [ from energy theory the orbifolding of extra dimensionsfrom [ string theory, and survivespontaneously quantum broken and they gravitational would corrections implyan [ that energy distinct barrier, degenerate leading vacua exist to separated a by network of cosmological domain walls which would be in and most obvious questionModel is (SM) from we where are do familiarsymmetries such with of symmetries the nature, originate? idea of but In(C), gauge parity discrete the theories (P) symmetries Standard and being time-reversal seem fundamental invariancemodels, (T) and only symmetry Abelian robust [ relevant discrete symmetries to are commonly charge used conjugation to ensure proton stability [ tri-bimaximal values [ either a zero reactor anglewidely or used exact in tri-bimaximal current lepton model mixing, these building symmetries [ are still are a few question marks surrounding the use of such symmetries in physics. The first were introduced, for example toof understand (approximate) the tri-bimaximal theoretical lepton originincluded, the of mixing problem the [ of observed vacuum pattern can alignment which be is more crucial readily tocurrent addressed the data success using of involves the such a flat theories, non-zero directions reactor of angle the and potential a [ solar angle which deviate from their The discovery of neutrino masstory and particle lepton physics mixing beyond [ thepuzzles Standard such as Model why (BSM) the but neutrinoEarly also masses family are raises symmetry so additional models small, flavour focussed andSU(3) why on [ lepton continuous mixing non-Abelian is gauge so theories large such [ as 1 Introduction JHEP11(2018)173 ]. ]. 19 48 – , ] have ]. 17 47 [ 44 56 , 55 ]. In order to 43 , 42 -plet of SO(3) with the ], the authors in [ SUSY flavour models 43 , 42 ], or, as in the present paper, in the 41 by introducing 7 . To date, the problem of how to achieve 4 ]. As stated earlier, the main advantage of A masses. The problem of how to achieve tri- ] but the problem of determining the required 19 – → – 2 – τ ] and the phenomenological implications of the 46 , 17 54 – has not been addressed, even though there are many 45 and 49 among parameters had to be considered in order to µ 2 without SUSY − ] has been applied to flavour models based on non-Abelian ]. There is also a strong motivation for considering such 43 , 12 42 realising tri-bimaximal mixing in a non-SUSY flavour model. How- 4 A in a SUSY framework To avoid the domaindiscrete wall flavour problem symmetry of is SUSYing just flavour from an models, the approximate breaking sincesymmetry of effective the is the residual non-Abelian broken continuous symmetry it symmetry. aris- does When not the lead approximate to discrete domain walls. To provide a natural explanationmetries in of SUSY the flavour origin models. of non-Abelian discrete flavour sym- The above literature has been concerned with breaking a continuous gauge theory to The above approach [ An alternative solution to the domain wall problem, which we pursue here, is to suppose • • breaking in a SUSY framework, inIn order to addition, make the contact with usual motivationsa for gauge embedding theory the also non-Abelian apply discrete in symmetry the into SUSY context as well, namely: such a breaking SUSY flavour models insuch the SUSY literature models [ isthe the potential, possibility to which achieve enablesstability vacuum some of alignment technical using the simplifications flat alignment directions and [ of enhances the theoretical Extended discussions including thesymmetries breaking have of been SU(2) discussed andbreaking in SU(3) of [ to SU(3) non-Abelian flavour symmetry discrete in flavour models hasa been non-Abelian discussed discrete in [ symmetry bimaximal mixing at leadingalso order been from discussed by non-Abelian otherflavon continuous authors VEVs flavour [ remains symmetries unclear. has handed One fermions to idea separately is transforming toand under different require realise continuous the maximal flavour symmetries, electroweak atmospheric doublets mixing and from right- the minimisation of the potential [ continuous gauge symmetries.considered For the example, breaking following offurther [ gauged breaking of SO(3) ever, a fine-tuning ofget around the correct 10 hierarchy between achieve this, a scalarvalue potential (VEV) was which constructed breaks the suchkey continuous leading requirement gauge for to symmetry having the to a vaccuum remnantscalar the non-Abelian expectation discrete field discrete symmetry. which symmetry The seems breaks(irrep) to the be of gauge that the the symmetry continuous gauge is group. in some large irreducible representation that the non-Abelian discretethe symmetry spontaneous arises breaking of asplace some a either non-Abelian low within continuous energy the gauge remnant frameworkframework theory. of of symmetry quantum This string after field theory could theory [ take be (QFT). spontaneously For example broken it to has various been non-Abelian shown discrete how symmetries SO(3) [ can JHEP11(2018)173 , 5 4 A we and and A 3 using 3 B Z Z 5 and we list we show A 4 A S C , 4 and we discuss the A 4 2 S will lead to three to residual , 4 4 4 A A A → -plet, and then generalise ]. For example, we discuss 19 – 17 model along these lines, originating 4 A to finite non-Abelian flavour symme- by introducing a 7 4 – 3 – A . 5 A we discuss the further breaking of , with SUSY preserved by the symmetry breaking. We 3 4 and A 4 S -invariant vacuum. 2 may be subsequently broken to smaller residual symmetries → gauged flavour symmetries Z gauge theory to a non-Abelian discrete symmetry using a potential 4 . As stated above, this is the first time that such a symmetry breaking A with supersymmetry SUSY 5 U(1) model of leptons which uses this symmetry breaking pattern and show U(1), and show that it leads to a phenomenologically acceptable pattern of A × × we construct in detail a supersymmetric 4 concludes the paper. The paper has three appendices. In appendix and 5 preserves SUSY 4 The breaking of tries implies new massivephenomenological gauge signatures. bosons For indegenerate instance, the gauge SUSY bosons spectrum, plus SO(3) with their possibly superpartners. observable The layout of the remainder of the paper is then as follows. In section In the present paper, motivated by the above considerations, we discuss the breaking S • symmetries, showing how it may be achieved from a supersymmetric SO(3) potential. , still preserving SUSY, which may be used to govern the mixing patterns in the charged 2 2 The key point toirrep break of SO(3) SO(3) and to requireSO(3), non-Abelian it we gain discrete discuss a symmetries non-trivial how VEV.our is to discussion In break introducing this to SO(3) a section, SO(3) to high after a brief review of the Clebsch-Gordan coefficients ofdisplay SO(3) explicitly which the are solutions used ofthe the in deviation superpotential the from minimisation. paper. the In In appendix appendix 2 Spontaneous breaking of SO(3) to finite non-Abelian symmetries from SO(3) lepton mixing and masses,this model, once we subleading also discuss correctionsleading the to are phenomenological massive taken consequences gauge of intoSection having bosons, account. a and gauged Within show SO(3), that all domain wall problems are resolved. spontaneous breaking of SO(3)with to supersymmetry. finite In section non-AbelianZ symmetries such as In section lepton and neutrino sectors, leading toSUSY a SO(3) predictive framework. We thenthat present it an explicit leadsFinally to we discuss a the phenomenologically phenomenological consequencesmassive acceptable of gauge having bosons, pattern a and gauged of show SO(3), that leading lepton all to domain mixing wall problems and are masses. resolved in such models. the breaking ofsupersymmetric SO(3) potentials down to forbreaking finite the of family first SUSY symmetries time. SO(3)further such show to how the as In particular,Z we focus in detail on the of a continuous which has been discussed inthe the literature, numerous and SUSY the flavour formalism models developed here in may the be applied literature to [ that: Finally, if the continuous symmetry is gauged, there is the phenomenological motivation JHEP11(2018)173 , + ψ (2.5) (2.4) (2.1) (2.2) (2.3) -plet, +3 | q − p is a 9 | . -plet. Each . 4 S -plet flavon + 2 ··· +1 A . p + →    +1 2 | 1 in the 3d space, and q   − a } . − τ α and a 13 , p a α | ρ { α 0 0 0 0 0 0 1 0 = 0 g 3 , , = 0 ]. In this work, we introduce    = 2 2 p , iik ... b = ...j X 43 =1 2 , in the 3d space. This tensor is a j +1 3 , ξ ...i 1 q a j p 42   2 φ . Each element can be expressed by kji ...i 1 2 ), i.e., ...i p 3 ξ 2 , τ j × φ -plet flavon i τ p 1 i + 2.3 i =    -plet [ =1 a O T b +1 i 1 3 τ and p jik a X = − ξ ··· 2 a α i τ 2 3 j = , , 2 2 , i 1 0 0 0 0 0 1 0 0 , tensor – 4 – τ O ikj ... X =1 p    1 a ξ . The minimal irrep for SO(3) a j is using a 7 1 ...i i = 4 = 1 b + 2 O A 1 ...i kij φ ξ ] gives an incomplete list of subgroups which could be → -plet. Applying a 9 → = + 1 dimensions and we denote it as a 2 , τ p = 43 =   p ...i a ...    2 b τ i jki 1 a ξ is a 13 i ...i α 1 0 a φ 5 = 3 , − A ...i 2 , φ 0 1 00 0 0 0 ijk X ξ → 4 =1    a are shown in table A 3 real orthogonal matrix. Here and in the following, doubly repeated   = non-Abelian discrete symmetries to achieve this goal. In the 3d flavour space, it is represented as a rank-3 and the Clebsch-Gordan coefficients are given in appendix × 1 → . It transforms under SO(3) as ξ τ p → )+1 6 = exp q , which satisfies the requirements in eq. ( } is transformation matrix corresponding to the element a + α ijk a, b p { ξ O g -plet can be represented as a rank- Products of two irreps can be reduced as 2 Each irrep of SO(3) has 2 + 2( -plet flavon +1 p The simplest irrep to break SO(3) a 7 tensor obtained after the relevant irrep getby a irreps VEV. up For instance, to somewhile 13 of that those for subgroup SO(3) obtained respectively, we will realise these breakings in a2.2.1 SUSY framework in SO(3) the following. ··· 2.2 SO(3) SO(3) can be spontaneously brokening to other different non-Abelian high discrete irreps. symmetries by introduc- Ref. [ where it is always aindices 3 are summed. for any 2 symmetric and traceless, In the fundamental three dimensional (3d) space, the generators are represented as The rotation group SO(3) ismatics. one It of is the generated most by widely three generators used Lie groups in physics and mathe- 2.1 The SO(3) group JHEP11(2018)173 (2.8) (2.9) (2.6) (2.7) (2.10) 4 5 4 1 A S A 13 . ) to be the 2.9 = 0 = 0 is identical i i 11 333 ξ ]. The main idea is /∂φ h f 44 = – ∂V i . 42 . 233 4 , ,    ξ S 9 i i h 333 , which can be chosen as ijk ijk = ξ ξ ξ , ξ i , h h 0 1 0 0 0 1 1 0 0 233 = = 133    ··· -invariant one in eq. ( ξ i i 4 4 , ξ 3 h 4 0 0 = k k + 1 A Z D SO(2) SO(3) 7 0 0 A t = j j 133 2 . 0 0

i i i ξ ξ i f 1 2 , ξ h h 0 0 , g Z 113 ∂φ ξ ∂w 123 kk kk h

× – 5 – ) )    t s , ξ 2 g 1 = g i ( Z SO(2) SO(3) 5 ( 0 X i 0 − 113 has been given in refs. [ jj jj 1 0 is determined by the minimisation. This idea cannot ) = ) 112 , ξ t 4 s ξ ξ − f g g h A v ( ( V 112 0 1 00 0 0 0 0 ii = ii ) , ξ )    → i t s g SO(3) 3 g ( = 111 ( 111 ξ s ξ ]. h g 43 ), there are 7 free components of , 6 ξ 2.5 v √ i ≡ symmetry, we work in the Ma-Rajasekaran (MR) basis, where the generators is geometrically shown in figure 4 ξ 123 A ξ h represent any scalars in the theory, and the dots are negligible soft breaking = 0 has a solution, the minimisation of the potential i i in the 3d irreducible representation are given by . The not systematical stabiliser subgroups in the low-dimensional irreducible representa- -invariant VEV, satisfying φ 4 t subgroups SO(3) SO(2) irrep 1 A The discussion of SO(3) For the /∂φ f and where terms and D-terms forconstrained the than the fields non-supersymmetric charged version. under If∂w the the minimisation of gauge the group. superpotential This potential is more minimum of the potential,be where directly applied to supersymmetric flavouris models. directly In related the to later the case, the flavon flavon superpotential potential The VEV of constructing flavon potential and clarifying the is given by The s Table 1 tions of the group SO(3) [ Constrained by eq. ( JHEP11(2018)173 . - 4 133 A and ξ 5 (2.11) (2.12) (2.13) − 3. Points ∼ 111 , ξ 2 d 5 , − ξ , = 1 = 1 . Taking the 221 ∼ ξ B ], the driving fields d 1 all in green, etc. As a i, j, k = ξ , take non-zero values. 13 211 321 212 ξ ξ ξ = = = , ) is automatically satisfied 1 121
312 122 ξ 5 ξ charges. Minimisation of the ξ ) 2.13 = = R ξξ , ( d 5 112 213 ξ ξ ξ = 0 2 . = 2 ξ c µ as a tank-3 tensor with can be achieved in SUSY. + 231 ) are listed in appendix = 0 − 4
ξ ijk 2 ξ 1 5 A ξ ) ) = µ 2.13 ξξ ξξ → – 6 – − ( ( 132 1 2 1 ξ -plet ) c c ), we see that eq. ( = ξξ = = ( ) and ( 1 ξ ξ , we introduce two driving fields d d 2.13 123 1 5 c 4 ξ ∂ξ ∂ξ A ∂w ∂w 2.12 d 1 ξ = ) and ( ξ w 2.12 are complex dimensionless coefficients. As required [ 2 c and . A geometrical description of the 7 1 c In order to break SO(3) to -invariant VEV, only those in red, 4 The explicit expressions of eqs.invariant ( VEV to eqs. ( do not gainpotential non-zero is VEVs, identical realised todriving by fields the imposing as minimisation follows, U(1) of the flavon superpotential respecting to the consider the following superpotential terms where These properties leave onlyA 7 independent components, showing in 7 differentto colours. the minimisation For of the SUSY, the it superpotential. is necessary Since to most consider if flavour SO(3) models have been built in Figure 1 in the same colour representtraceless the identical tensor, components, points e.g., in grey are dependent upon the rest, e.g., JHEP11(2018)173 ) 2.13 (2.18) (2.15) (2.16) (2.17) (2.14) and and the t g , /µ s ) and ( g 133 . Constrained . symmetry and , which has an 4 . δξ 0 4 2.12 ijkl 1 i A c 2 . A ξ ρ 3 h = 0 i q is the only symmetry 3333 , − 4 , , , . Eqs. ( , ρ A 2333 . The unconstrained per- = i i i ρ = 0 δξ h 3 133 2333 symmetry is consistent with α ijkl ijkl ijkl + = δξ , ρ ρ ρ 4 233 i h h h , ρ i ξ A δξ /µ h = = = and + 1333 1333 i i i = 112 0 0 0 ρ l l l also preserves the symmetry. The following constraints 0 . 0 0 0 h , ρ δξ i k k k 113 112 0 4 0 0 0 1 ξ i j j j = c    2 h S 0 0 0 δξ ξ i i i 3 i h 1233 , ρ ρ ρ h h h q 0 0 0 , δξ , ρ 112 ll ll ll 1233 0 1 0 1 0 0 0 0 1 ) ) ) , which can be chosen as = ρ t ), uniquely breaks SO(3) to s δξ u , ρ h since it is not clear if g 2 g 1133 g = 0    i ( ( – 7 – ( α 4 0 . Therefore, the 0 0 = 1 − , , ρ 2.13 A kk c i kk 333 kk is represented as a rank-4 tensor 1133 ) ) 6 ) t = /µ s ρ ρ u δξ g h g 1123 √ u g ( 1123 ( ). Therefore, 2 0 / ( g 0 113 -invariant one, = ρ 0 ξ 4 , ρ − h jj jj ) and ( µ jj δξ ) 2.2 ) -plet A t ) 1 s 111 = =  g c u 2 invariant under the g 1113 ( 3 g i i ( i 0 δξ . Directly taking them into account, we get the constraints 2.12 ( 0 = 0 0 ρ ii i , ρ ii q h ) i ii = ) ξ t ) 3333 1113 s h g = u ρ ρ g ( 123 can be rotated away if we consider a SO(3) basis transformation, corresponds to only a basis transformation of SO(3). Such a basis h h 1112 g ( 1 ξ 123 ( 0 h i α the uniqueness of = = δξ , ρ δξ ξ i i h 4 S 1111 to ρ 1111 1112 i ρ ρ ) with → ξ h h h being given in eq. ( ) leads to 2.1 i symmetry, the generators in the 3d irreducible space are given by τ ), there are 9 free components of 4 2.12 S 2.3 in eq. ( . Straightforwardly, we obtain In the 3d flavour space, the 9 We need to check } δξ a α { which are equivalent to are required, by eq. ( In order to require the VEV the superpotential, i.e., eqs. ( 2.2.2 SO(3) For the g generators the shift from transformation has no physical meaning. We conclude that the minimisation equation of on leaving only three unconstrained parameters turbation parameters the vacuum solution obtained from the minimisation of theafter superpotential. SO(3) breaking.infinitesimal We deviation assume from the theremust is also another be satisfied vacuum for solution and eq. ( JHEP11(2018)173 , → /µ ρ . (2.24) (2.19) (2.20) (2.21) (2.22) (2.26) (2.23) (2.25) and . The . Con- 4 1113 5 t S g δρ , ∼ 1 s ρ 5 c g d 5 ijklmn , . , 6 , ρ , i i i ψ q . The uncon- , we succeed to ρ = 0 and = 112333 2333 ijklmn ijklmn ijklmn 1 2 ψ ψ ψ , ψ -invariant VEV, δρ h h h α 1333 -plet 4 ∼ , S δρ = = = d 1 i i i ρ /µ 0 0 0 and + ) is automatically satisfied 111333 . n n n 0 0 0 5 by a 9 ) , ψ m m m 1112 0 0 0 1113 2.21 l l l ξ 0 0 0 ρρ 1113 δρ k k k ( , , 0 0 0 1 d 5 j j j δρ ρ 0 0 0 , which can be chosen to be 5 ρ i i i c ) uniquely break SO(3) to 111233    2 6 , ψ symmetry. The following constraints , δρ ψ ψ ψ -plet ρ = 0 1 h h h 1 2 c . 4 . 0 0 0 , ψ 1 b b q − 2.21 S ) away from the 1112 + nn nn nn = 0 1 ) ) ) 2 1 =
ρ ρρ t = 0 δρ s b b 1 w ) are still satisfied. Then, we will get the ( − g g ) 111133 333333 1 5 1 g ( ( ρ . ) 0 ( 1 0 3333 2 1 α 0 c 1 ρρ b b ) and ( ρ , ψ , ψ − ( 2.21 ρρ δρ c 1 mm – 8 – − ( mm 1). mm ρ ) ) 2    ) 2 ρ t = c ρ s 30 2.20 g − with 1 2 µ c w g We vary ( ( g − 111123 233333 can be rotated away if we consider a SO(3) basis 0 p 5 0 ( − a ll = = / 0 2 ρ ll 1233 . ) and ( ) τ √ ρ ) ll t µ , ψ , ψ ρ ρ 4 a = d 1 d 5 δρ s ) µ g − δρ g α S ( w ( w 1 d (  0 ∂ρ ∂ρ g 1 2 0 ∂w ∂w 2.20 g ρ = invariant under the ( + kk 0 = kk i ) 133333 111113 → = ) = 3 t i kk ψ s g × 2 ) ρ h 1133 g ( 3 , ψ , ψ b 0 ( w w 0 1 g 1133 δρ jj ( jj ρ ) 0 ) t h = = s jj g . Therefore, eqs. ( 123333 111112 ) ( g 5 } 0 ( w in SUSY by introducing two driving fields 0 1) and a ii ), there are 13 free components of /µ in the 3d flavour space is represented as a rank-6 tensor A g , ψ , ψ α ) ii 1123 ( t 4 { ) − 0 , which are straightforwardly expressed as s g ψ g 2.3 S ) to the above equations, we see that eq. ( δρ ii ( 5 2333 g → ) δρ ( √ w = δρ 113333 111111 ( ) leads to g 1 2.18 1 2 ψ ψ ( symmetry, the generators in the 3d irreducible space are given by ρ -plet 5 c 6 5 1111 = and require that eqs. ( 2.20 A 1 q δρ b δρ − The 13 Follwing a similar procedure but replacing the 7 + = i 3 ρ are required, strained by eq. ( In order to require the VEV where α 2.2.3 SO(3) For the leaving only threestrained unconstrained perturbation parameters parameters transformation, and eq. ( The uniqueness of SO(3) h constraints on Taking eq. ( break SO(3) to flavon superpotential is constructed as Minimisation respect to the driving fields gives rise to JHEP11(2018)173 , 9 . of 5 , 3 ∼ i A , and , (2.32) (2.27) (2.28) (2.29) (2.30) (2.31) d 9 ψ 1 ψ /µ 111111 ψ and h . The uncon- 5 1 111333 − , -invariant VEV, ∼ δψ , 5 5 d 1 10 5 123333 21 A √ ψ 7 , of SO(3) with q δψ 111123 − 4 111333 ψ 1 , 3 ψ = . b 1 → and 5 9 i b ) is automatically satisfied 123333 2 ) 5 ψ √ α 3 1 √ , b − ψψ 2.30 ) uniquely break SO(3) to ψ ( , 113333 111333 d = = = 9 ψ /µ ψ h , δψ 2 2.30 away from the ) are still satisfied. Then, we will = 0 , ψ , i c . 1 ) ψ = 0 111233 133333 233333 , we identify 1 , . 123333 ). + 2.30 4 + 5 1
= 0 A ψψ ψ δψ 111123 1 ( ) and ( 111111 c 9 = 0 = 0 ) 1 7 ) + 3 ψ 15 i i ψ δψ 333333 h 21 c ψψ 5 2.29 q ψψ ( ) and ( δψ p – 9 – 4 − ( = 1 1 We vary − 2 ρ (4 2 ψ 3 = 111233 233333 c ψ 5 10 / µ c . 2.29 ψ ψ ψ × √ − h h 5 → − µ 7 = = 3 , δψ 1 2 ψ A = =  , we introducing two driving fields µ α d d 9 1 ψ ψ 113333 = , δψ i i 5 = i d A 1 δψ ∂ψ ∂ψ → ∂w ∂w i , δψ can be rotated away if we consider a SO(3) basis transfor- ψ 111333 with = 111123 133333 123333 = δψ a 111133 δψ 1 ψ ψ τ ψ h h b δψ 111111 ψ . Therefore, eqs. ( , a 111123 h 2 5 w ψ ψ α b 3 = = h 111133 , √ δψ δψ 5 i i i 2 /µ + − b δψ √ 3 × = = = = 3 111113 112333 333333 1 ) to the above equations, we see that eq. ( 111123 ψ ψ ψ and require that eqs. ( h h h = δψ ) leads to } 2.27 111113 112333 111111 111112 7 = = = . The flavon superpotential is constructed as a 15 δψ α i i i δψ δψ δψ δψ { + 2.29 q g 4 i ]. ψ 51 111111 111112 111333 For irreps of SO(3) decomposed to irreps of In order to break SO(3) to ψ ψ ψ → − , respectively and compare the Kronecker products h h h → h 4 3 one [ A α 2.3 Representation decomposition After SO(3) is brokenirrep to of a SO(3) non-Abelian to discrete areduction group, couple of of it irreps is Kronecker of necessary products the to discrete of decompose one. representations each This of task is SO(3) achieved by with comparing those of the discrete leaving also three unconstrainedstrained parameters small parameters mation, Taking eq. ( and eq. ( The uniqueness ofψ SO(3) get the constraints on instead of 5 Minimisation respect to the driving fields gives rise to They are equivalent to JHEP11(2018)173 . , 5 4 in 5 A A χ + (2.37) (2.33) (2.34) (2.35) (2.36) (2.38) 4 and + 0 4 3 , there are S , there are 5 4 , + 4 S A 3 A , 0 . = , 4 3 5    . In . In 5 A 0 5 ) + , × 3 + 0 00 A of 2 3 in the 3d space. It is 3 4 3 3 ωχ + + ij = = of SO(3) is decomposed and 0 0 1 2 χ and + 2 5 + 1 3 χ χ 3 0 4 3 2 2 × + , × 1 1 χ + S ) to irreps of 0 √ √ 3 2 2 are used. Continuing to play 3 + 1 1 ω 00 ), = 0. Independent components 3 ( , 4 1 = = 3 3 -plet is decomposed to two non- 1 5 0 = A 33 √ 2 + 3 + χ + 0 3 ) . It is useful to re-parametrise of × 1 5 3 4 00 × + 0 × 2 χ A + 1 3 + + 2 00 22 4 + 7 . The 5 1 00 ω 1 of 1 3 χ , 1 + 33 , it is also a triplet , χ χ 0 0 . Keeping in mind the Kronecker products , + + 3 + χ T 2 2 5 3 3 5 + 5 0 + 1 1 5 ) 3 and √ √ 0 , 3 + + 11 + + 0 0 1 ωχ 3 + 2 χ 3 3 4 ( 3 = 3 , ϕ and – 10 – and + 3 3 2 = = + + + 1 5 + 4 √ 1 23 1 3 3 3 , ϕ 2 , × 2 χ 0 ) 1 ) = = = = = 00 × 3 3 , (different from × ϕ 3 0 + 0 χ 0 3 4 5 , 0 3 13 3 2 1 1 1 3 1 + × × χ + × χ χ = ( × 0 00 0 0 4 2 2 , = and one triplet 3 3 1 1 1 ϕ 3 χ 0 √ √ 12 ( 00 3 + 3 χ 5 1 1 0 √ , and traceless, , × + , where 1 , can be represented as a rank-2 tensor , 0 ji ( 11 5 3 χ 1 3    χ and χ , + × + 0 = = = 4 5 = 3 1 3 of SO(3), 0 , respectively. . One further compares right hand side of 3 χ + + 1 5 4 ij + 4 4 χ × A A × as follows, which will be useful for our discussion in the next two sections. 1 , + + 0 , 3 0 0 4 5 of 1 = 3 3 5 (the trivial singlet), (the trivial singlet), A and + 3 + + + 4 1 1 4 3 3 3 + S + . -plet of SO(3), + + + 0 00 2 3 1 1 1 1 the form For a triplet 3 A 5 symmetric, can be chosen as trivial singlets , and comparing them with Kronecker products in SO(3), we obtain irrep decompo- = = = = . Since the right hand sides of both equations are identical, 5 + , and 5 4 We summarise decomposition of irreps of SO(3) (up toBefore 13 ending this section, we show more details of how a irrep of SO(3) is decomposed This game is directly applied into irrep decomposition in 0 4 4 4 5 3 • • 1 A S A × × × × 3 4 5 3 sitions in in in table into irreps of in this game, we can get decomposition of as high irrepfive of irreps: SO(3) as we wantfive into irreps: irreps of with that of and obtains 7 in to in SO(3) with JHEP11(2018)173 , (2.41) (2.40) (2.42) (2.43) (2.39) 3 ξ 5 0 1 6 ξ 5 . 1 + µ √ 3 4 . + 5 4 ∂ χ 0 5 0 3 + 0∗ 1 . µ + 5 + + ξ 3 ξ A 0 3 0 4 ∂ µ ξ 4 3 ∗ 3 3 + 10 A ∂ 2 χ . + 3 10 √ µ . Results of decom- 1 + of 4 5 ∂ 1 √ − 3 A A 3 ξ + 2 µ = = 0 ∼ of ∂ χ 3 . The former mentioned ∗ and 3 ) µ 4 3 0 3 ξ 113 333 + ∂ 4 A µ ∗ 2 0 0 0 S , ξ ∼ ∂ 3 3 3 χ 0 2 , 0 ) of µ 4 3 3 + , ξ + + + ∂ 0 , ξ , ξ 0 3 A 3 0 1 2 ξ + 3 3 3 3 , χ 2 2 ξ ξ + µ 4 ( 2 ξ ξ 3 1 µ 3 ∂ + + + + S 6 6 χ ∂ χ ≡ 1 1 , χ 0† 3 ∗ µ 2 µ + 2 3 2 √ √ 1 ξ ξ 0 0 3 ∂ ∂ µ χ µ ∗ 1 † 3 + + + . ( − + ∂ ∂ 0 χ χ 0 3 0 0 2 2 gets a non-zero VEV, SO(3) will be bro- 1 µ µ ξ ≡ + ξ ξ , ξ + ∂ ∂ µ 0 3 3 1 + 3 2 ∂ ξ 10 10 ξ + + ξ 1 1 0∗ 3 µ µ 00 00 ∼ √ √ ξ ∂ ∂ χ χ ]. ) µ – 11 – † ∗ and two triplets 3 1 , χ µ µ 3 = = ∂ ξ ξ 3 1 44 00 ∂ ∂ 1 µ µ , ξ , once 1 + + ∂ ∂ 2 00∗ 00∗ 4 112 233 0 2 χ χ 3 1 3 2 3 ∼ , ξ A , ξ + + µ µ 1 4 µ 00 0 0 ∂ ∂ ξ + + + ∂ ξ ξ A ( + + 3 2 3 3 3 µ µ 0∗ 2 , ξ 0 0 3 1 ∂ ∂ ξ ≡ 1 χ χ + + + + ∗ ∗ , χ 0 0 µ ξ + µ µ 3 0 ξ ξ ∂ 00 00 00 4 3 6 ∂ ∂ µ µ 1 1 1 1 3 3 3 3 1 1 1 1 A ∂ ∂ 0∗ 0∗ + √ + , ξ ∼ + χ χ + + + , ξ 00 = = 0 0 1 . This parametrisation has two advantages. One is the simple 0 0 0 µ µ − 1 1 1 ξ 3 χ ∂ ∂ 1 1 1 1 , 0 1 have been given in [ ) ξ 0 + ∼ ξ 10 = = 4 iπ/ + + 2 ξ 0 2 µ 0 1 A 6 10 √ 1 1 1 e 1 is still preserved. This is consistent with the discussion in the former ξ ) ∂ 1 ∗ √ √ χ − 4 + ξ µ = is a trivial singlet of µ A 1 ∂ = = = ∂ ∗ 0 ω ( ξ χ -plet of SO(3) is a symmetric and traceless rank-3 tensor in the 3d space. It is 123 111 133 µ ξ ξ ξ ∂ . ( 1 3 5 7 9 . Decomposition of some irreps of SO(3) into irreps of can be re-labelled as 11 13 4.6 SO(3) Since ken but subsection. And the kinetic term is also normalised, Here, decomposed to one trivialξ singlet The 7 The other is the normalised kinetic term, where transformation property in Here we ignore the gauge interactions. Consequence of the gauge interactions will be given later in • 2 section Table 2 position to irreps of JHEP11(2018)173 T T in 1) 1) i , -plet ξ (3.2) (3.3) (3.1) − h 1 , leading − 1 , 2 − 1 , Z − , (1 } , and 2 = 0 to determine 3      , and ( Z 2 ϕ } , t, t    2 µ of SO(3). We denote ) 1 1 1 -invariant VEV . { symmetry. Each VEV − 1 4 st − − 1 1 ( 4 ) A =     A 5 2 t 3
ϕϕ , st, 5 Z , Z ( ) 1 1 { f in the framework of supersym-    ϕϕ = 1 1 , ( 2 , 1 and ξ st 3 Z − − 3 Z , and 5 d = 0    = 0 2 3 Z ϕ preserves ϕ 2 ϕ 5 and  ,
µ T 3 5 2 = -invariant VEV, we introduce an 1 are preserved respectively in the charged ) Λ f    3 Z − 1) 2 2 2 and consider the following SO(3)-invariant , 1 1 1 Z . Starting from the ϕ + ϕϕ 1 ) Z 1 becomes very non-trivial. In this section, we → ( − − d 5 preserves ,
breaking. These residual symmetries are not B = ξ 2 4 ϕ ϕ 2 ϕϕ T    4 2 1 – 12 – Z ( A µ and 2 1 ϕ 1) A Λ f groups are conjugate to each and have no physical , f − 3 − to residual , all four VEVs break the 3 , 1 Z and = = ϕ    ) 1 4 Z v 3 , 1 1 1 d 1 d 5 ϕ ϕ 1 A has to be broken to generate flavour mixing. In most Z ϕϕ .    ( − 4 ∂ϕ ∂ϕ 4 ∂w ∂w 1 to      A f ,( A 4 ϕ } d 1 . These 2 v A ) } = 0 to derive ϕ  can be simply realised by using a triplet 3 2 5 ) group. In detail, (1 3 -plet driving field = sts
= ( 5 ts Z 3 ϕ ) (    Z . For non-zero w to the continuous SO(3) symmetry forces strong constraints on cou- to i i i 1 ϕϕ . In order to obtain the 2 3 1 f ( , ts, , sts, 4 4 ϕ 3 ϕ ϕ ϕ ξ 1 1 ]. h h h A and a 5 A { { √ 3 . Here, we directly write out the following complete list of solutions / d 1    = 2 1 58 = ϕ Z , ϕ ϕ µ ts sts 3 3 57 → Z Z = 4 ϕ ), we use lepton flavour models, v A 4 2.9 Minimisation of the superpotential gives rise to Embedding A where preserves a different preserves preserves difference [ the value of whose detailed formula is listedeq. in ( appendix the scale of SO(3) breaking to superpotential Here as appearing in the non-renormalisable term, the scale Λ is assumed to be higher than 3.1 The breaking of such a flavon as driving field plings, and the breakingwill of show, for definiteness, howmetric to SO(3)-invariant theory. realise In of these models, residuallepton symmetries sector and neutrinoprecise sector but after good approximateto symmetries. a The mixing misalignment with between tri-bimaximal mixing pattern at leading order. 3 The further breaking of JHEP11(2018)173 (or and and c = 0. have (3.5) (3.6) (3.4) VEV 4 ξ τ 3 = 0 or , 00 ξ A χ 1 χ 3

5 and 5 , = ) ) c 0 and ]. There is a µ              χξ χ χχ 2 ( , (        d 5 c = 0. We obtain χ 58 ξ e , χ , 2 χ    . As we will prove 1 57 µ 4 χ χ g v 0 0 0 0 − , we will construct a 4.4 +  1 4 ) , respectively. All these 1
}    , 3 χχ st
(        2 = 0 leads to 9 1 ) = 0 g , t 5 , ) 3 1
χχ {        ( 9 χξ ) ξ =    d 3 -plets of SO(3). Minimisation of the st χχ χ , specifically the non-renormalisable 2 ( χ v 0 0 0 0 ξ t 2 and the requirement ξ 4  Z are usually assigned to 3 3 , Λ A g , 3
00    Λ 9 } g and 1 ) 2 = 0. Therefore, two of - and 5 + of        = 0 breaking. The relevant superpotential terms 1 − symmetries. In details, the first, second, and 1 ϕ ) as input, (
χχ 00 . 2 χ 3 χ -, 3 2 2 ( , , tst 3 and
3 µ ξ Z – 13 –
1 Z 5 0 2.9 . These terms may influence the VEV of χ ) 5 {        4 1 − = 0 ) → , = A 1 are 1 5 =    χχ ) 1 ) 4 ( χχ 3 d 5 with 1 2 χ ( ξ χ A χ v 0 ξ 0 0 0 0 χχ χξ 2 tst 2 ( ( in eq. (  2 χ 1 4 Z d 3 Λ g i g g , χ    and ξ = } h = = = d 3 2 2        , s Λ χ g χ χ χ χ d 1 d 3 d 5 1              1 , -invariant one. In general, this shifting effect is small enough { d 1 masses are independent with each other. + 4 χ ∂χ ∂χ ∂χ ∂w ∂w ∂w = χ
A τ = 2 χ s 2        µ to achieve the Z . These VEVs satisfy    − and 1 χ i i i i i 1 g takes the same form as 0 1 2 3 00 µ ) √ χ χ χ χ 3 χ , 2 h h h h / h
e χχ -invariant VEV introduced in the superpotential. 5 χ Z -plet ( 4 ) ) involves interactions between    1 µ χ A g = 0 results in µ →        χχ = 3.2 3 ( d 1 4
χ ξ χ 9 v models, the three singlet irreps A ) = 4 Eq. ( χχ A χ ( w ξ 3.3 Spontaneously splittingIn 1 their permutation), respectively. Thesethe irreps generated are independent with each other in where third pairs preserve VEVs are conjugate withnew each scale other and have no physical differences [ the following complete list of solutions, Given the Then, to be zero. And the rest non-vanishing one is determined by where the driving fields superpotential results in equations therein, the shift effect is suppressed by3.2 the scale Λ and in generalWe very use small. the 5 could be considered as follows term which results inshift the it breaking away of fromdue the to suppression offlavour the model, higher and dimensional baseddue operator. on to non-normalisable the In interactions model, section with the we other will flavons in discuss section in detail the shift of the JHEP11(2018)173 τ with (3.9) (3.7) (3.8) and 0 (3.10) 1 . These µ 2 . This can ξ = 0). These v 2 ζ . ), is determined 4 µ . 00 1 A are obtained from ζ , − ) 00 3 ζζ    1 ) (or ( , 00 ) d 1 0 ζ 00 ˜ ζ ( ζ              3 1 and Λ ωζ h h        0 and achieve to split 2 1 1 + + ζ ζ i 2    0 2 2 1 ω Z 1 1 ζ 0 0 0 0
ζ √ √ of SO(3), we have to face a fine 2 etc), as shown in table 3 v ) ω    = 0 ( and 3 2 ζ ζξ        1 = 0, (or ( 3 √ µ d 3 2 ζ Z , ζ ) − µ 00 . 1 2 = 0. It leaves the third row simplified to to ζ       
− h 2 3 4 5 i ζ ω    ) 3 = 0 3 1 = 0 + A ) ω = 0). The rest one, 0 0 0 0 0 ζ ζ ζ 1 + = 3 ζζ ζ 0 2 2  ) v ) ( 0 1 1 ( ζ 2 ζ 2 ) to the same 5    √ √ ζ – 14 – 1 c ζ µ , Λ ζξ ζζ h ωζ as effective descriptions from higher dimensional τ ( (        ζ ( 2. We will see how this VEV can separate 1 2 3 = − , 2 ζ 3 Λ h              with the following superpotential h h 1 1 1 1 µ √ ζ
, d 1 and = = = = 5 ˜ ζ ) ) = 0 (or c 00 ζ ζ ζ d d d 1 3 1        µ = 0 ζ 00 and ˜ ζζ ζ 3 2 ζ i ( ∂ζ ∂ζ ∂ and    ζ ζ ∂w ∂w ∂w + 2 χ ζ 2 2 i 0 i i i i µ d 1 1 3 0 00 1 2 3 ζ -plet flavon √ √ ζ 1 ζ ( ζ ζ ζ , ζ h Λ h h h h , 3 h 2 ϕ 1 ) and d 1 √ µ 1  ζ    of SO(3) (or higher irreps, e.g., 9 h d 1    ζ        (2 / = = Λ , should be much lower than the scale of SO(3) breaking ζ ζ 2 ζ ζ v w µ 3 = 0, resulting in √ 00 and ζ 3 0 q ζ χ based on SO(3)-invariant superpotential. The scales representing the breaking of v = 4 , ζ = 2 A ϕ v v To summarise, we realise the breaking of We introduce another 5 In the framework of SO(3), the non-trivial singlet irreps 1 ) of , 4 00 ζζ 1 A be satisfied by treating operators. One may noticenot that written there out may but exist cannot some be unnecessary forbidden interactions based which on are current field arrangements. A detailed with masses in the next section. results are summarised as The second row directly determines ( by the first row, which is simplified to and three driving fields Minimisaiton of the superpotential gives to the decomposition of 5 singlets are always correlatedtwo with of the each charged other. leptonstuning (e.g., As of masses a of consequence, thesehow if two to charged we avoid leptons. this directly In problem arrange this from subsection, the we spontaneous are symmetry going breaking to of consider JHEP11(2018)173 00 is ad 1 00 and 1 0 U(1), 1 , using , × 2 from -plets of 1 and 0 0 ,Z Four extra 1 1 3 3 Z . τ in the charged R 3 → Z 4 masses. A and split τ and 2 µ Z R and are arranged as to two different 5 in the charged lepton sector µ c → 4 3 τ 4 A Z A -plets of , and U(1) model, based on SO(3) 3 00 c Z 1 4 µ × , 2 A c → , respectively. These particles should e Z and τ 4 0 R A 1 4 (3) and A µ to two different 5 – 15 – SO c R τ 3 Lepton mixing and Z -plet leads to fine tuning between c and subsequently (at a lower scale) µ model from SO(3) 4 4 models, we embed A . The U(1) symmetry is used to forbid couplings which are 4 A 2 A → into the same 5 in the neutrino sector gives rise to lepton mixing. The model building c τ 2 Z and c µ models, the right-handed charged leptons . A sketch of the symmetry breaking in the model and how flavour mixing is generated. 4 A in the neutrino sector, with supersymmetry preserved throughout. The misalignment of the 2 In discrete symmetries are introduced in this model. Z Imbedding (or their permutation), respectively. In SO(3), the minimal irrep containing 3 , which then breaks, at a lower scale, to the residual symmetry 4 00 . In order to match with 5 SO(3). In our model,right-handed we embed leptons are introduceddecouple for at low energy theory to avoid unnecessary experimental constraints. We achieve strategy is shown inunnecessary figure to generate thehoc required flavon VEVs and flavour mixing.1 Note that no 4.1 The model In this sectionwith we the will breaking construct SO(3) the a vacuum supersymmetric alignments discussed previously,lepton where sector the and misalignment of different driving fields tomodel achieve building, them. which will This not difference be further discussed leads in to this4 paper. the difference of A supersymmetric and residual symmetries gives rise to flavour mixing. discussion on how tothe forbid model the unnecessary building. coupling Besides,showing will above the be are ways given not to the in realise unique the ways. next One section can on introduce different irreps, combined with Figure 2 The flavour symmetry at highA energy is assumed to be supersymmetric SO(3). It is broken first to JHEP11(2018)173 1
after (4.2) (4.1) 5 . We ) 4 0 τ A χξ L ( d 5 , 1 3 d 1 0 χ 1 ˜ ζ µ − 4 L g , . . 1 2 + d 3 4 3 1    ζ , ··· − . A
1 2 3 . Besides SO(3), we τ τ τ + 3    1
3 of L L L    ) ) 9 1 2 ) τ d 1 00 µ ) 1 0 3 c    3 ζ L − ζζ ( = χχ ωτ ωR d 1 ( 2 1 ˜ 2 1 ζ τ 2 3 ξ µ µ d 5 µ + τ τ 3 + 1 5 3 χ R R L h c 0 τ R R d 3 2 2 µ 2 2 1 1 and two singlets R χ + ,L 1 1 2 √ √ 2 √ √ 1 τ ω
0 3 ω 7 3 d 3    ( Λ 3 τ u,d ( g 3 0 + 0 1 1 L 1 2 3 ) 3 1 χ 3 L 1 µ µ µ transform as
1 , + √ ) are the three SM lepton doublets. √ 5 ζξ L L L µ 1 3 ) 1 ( T )
) L ) d    3 , l  00 0 µ c 3 5 6 1 6 3 d 1 ξξ ζ 3 5 µ
1 5 1 τ τ ( R
ν χ = +2 + 5 2 + + d L 5 2 2 5 ) ξ ) U(1) are listed in table ω 3 1 µ h ,R ω 3 1 µ µ 2 τ τ 2 -plets = ( + χχ × + τ 7 3 1 3 4 3 τ d 5 + c R R ϕϕ ( R R breaking in the superpotential involving flavons 5 5 1 – 16 –  ( 0 τ 2 2 c ϕ ,L ξ R 2 2 3 − − − U(1) and decompositions of these fields in + 1 1 η ξ 1 1 ,R 4 ` √ √ √ √
1 1 d 3 A    ωµ × τ
η ωR d 5 ( 1 2 3 χ ( ξ 2 1 5 µ d 1 ϕ R 3 5 3 5 ) 3 ϕ ( 1 N N N A R − − +1 1  2 √ √ Λ g 2 −    ϕχ ) and ≡ ) Λ ) f 1 ( 2 + c 2 3 7 3 1 3 00 µ ) c 3 d ζ 5 = 5 7 5 τ , l τ + ξ  R 3 2 − − + 3 2 2 ξξ 3 2 µ µ R τ τ ζ  ( ν + ¯ η + 3 1 h Λ R R R R . 0 τ c η χ c 2 3 2 3 2 3 4 ,N 2 2 2 2 d Λ 1 g ¯ 1 1 1 1 3 η ξ ϕ χ ζ H 1 1 − µ R ϕ ξ = ( A √ √ √ √ and d Λ 1 − + − ( ( 1 f −    ξ 3 3
2 1 1 T 1 2 3 5 η − √ √ ) ` + ` ` ` ) 1 1 2 3 2 3 3 d
1 ) ) ` N e η µ          2 η η ζζ ), − ( µ 1 χχ = = = ϕϕ ζ ,R ( , l ( − 2 ` τ 0 µ 1 1 1 ¯ µ η 1 Λ g ν f Λ R R η h 1   ,R  d d 1 1 d 1 d 1 U(1) 0 SO(3) 1 U(1) U(1) + Fields SO(3) 1 Fields SO(3) 3 Fields = ( µ ˜ ϕ χ ζ d 1 . Field arrangements in SO(3) U(1) is broken to R 1 η + + + ( ` × ⊃ ≡ Charges for all relevant fields in SO(3) f 3 w µ 4.2 Vacuum alignments Terms leading to SO(3) breakingand and driving fields are given by Here, R introduce additional U(1) symmetry to forbid unnecessary couplings. Table 3 SO(3) this goal by introducingwrite two out explicitly left-handed each 3 components of the fermion multiplets in the 3d space as follows, JHEP11(2018)173 . , i 2 ¯ η h are Λ 4.4 (4.5) (4.4) (4.3) η η/ 1 = 0 ; and
vanishes gets this i i 5 and ¯ . ) 5 η η ) h 1 4 η 333 ξ ϕχ ! h ( ϕϕ ¯ 3 η ζ 3 = v Λ 2 ζ . i ζ 2 1 h . Here η h h χ η v 0 1 233 . g ξ χ g ) takes a very similar ξ ξ ϕ v 1 h A is effectively expressed f f ϕ 4.2 2 v = 1 − Λ and ζ g i / v

3 2 ζ η ¯ η 133 h v Λ ξ χ , eq. ( , respectively. Once h 2 g ¯ η = 3 v √ = ζ i Λ, i / − = 0, we know that both 3 and and , v η d 1 113 = 2 ξ ϕ ¯ η η η 2 ζ h f v v v does not contribute since ( 0 1 /∂η χ = g g f 1 , µ i  ¯ 3 r η 5 ∂w ¯ η v
112 5 v ξ χ ) h Λ – 17 – g ; , charge can be arranged for = = 2 ϕϕ = χ        (        i Z ξ 2 χ ;       111 , v d 5 gains the VEV. In this case, are replaced by 0 0 1 0 0 ξ 0 0 0 1 0    ϕ h η , µ 2 ζ Λ η  v 1 1 1    1    3 η µ ϕ , f v f 3                  6 ϕ ξ Λ ζ ϕ χ f v and after √ s is not explicitly written out, but effectively obtained from the v v v η 2 χ = η 1 v = = = ) 1 are respectively given by µ 2 ϕ is effectively obtained. This treatment is helpful for us to arrange ) i ≡ , = is not explicitly written out, but replaced by ζ µ                  2 ϕ η χχ v ξ 2 ϕ i i i ( χχ v 123 µ 1 2 3 ( d 1       µ ξ ξ Λ. The term d 1 h . Otherwise only a i i i i i i i i ϕ ϕ ϕ χ i i / A 0 0 00 h h h ξ 1 2 3 1 2 3 00 χ 1 , v and η ζ ζ ζ ζ ζ 0 g 1 χ χ χ χ χ v h Λ η h h h h = g h -invariant VEV. h h h χ h    0 1 v 4 1 v g 2 ξ ξ c       A , µ A = ϕ               v 1 s , g ξ : : : : = v as at the The term operator VEV, charges for The constants respectively. These constants are justoperators effective after description the of relevant the flavons get higher VEVs, dimensional The constant SO(3) singlets. From the minimisation cannot be zero, and thus, we denote them as 0 2 4 3 ξ 1 The approach for how the flavons obtained the required VEVs have been discussed in Z Z v A • • • ζ ξ χ ϕ where the former section. We doVEVs not of repeat flavons, the relevant discussion here but just list the achieved Compared with the superpotential termsform in except sections the following differences: Here, the dots represent subleading corrections, which will be discussed in section JHEP11(2018)173 , is ϕ v η , v (4.8) (4.9) (4.6) (4.7) ξ (4.10) v  . ) ϕ χ v all assuming , , v , ¯ 3 η 1 ϕ 1 . v ) ). For the scale v Z
µ ( 5 3 4 )  4.5 2  τ → ζR represents the scale . η ¯ η if the dimension one ( R , respectively. VEVs η ,Z Λ 4 v v ξ 2 τ 0 ¯ η v 3 v A µ  v Z L Z ( η L ,  ξ 1 v d ¯ → µ η 2 and and v Y ∼ H 1 4 τ c
η 3 ζ Y A e 5 + v  Z 1 ) , ¯ η + ξ  N 1 1 , v ` d 1 →

A w 3
¯ 3 η η 5 NN ) H 4 τ ) ) v v ( c + ϕ  µ µ A e R χ 5 τ 5 r ) R R scales of ) R ¯ η    χ µ µ v 1 1 1 w λ L L ζζ ϕϕ ( ( ( (    ∼  + + ξ as ϕ 0 T  1 µ χ η τ ` 2 ) ) 2 2 3 R v ξ µ e L µ Λ 3 v 3 Λ ϕ Y w y 1 – 18 – y ( , v v Λ τ Λ NN + + e 4 η Y + ( + and Λ y v c breakings. For the scale of d 2 A d d e ¯ η ¯ η + H 4 v = r H w η 1 d c → η can be much larger than , Λ ¯ A ηH
λ e v ξ eff e 1 = H ξ 7 1
1 ) v ) A ` w + is given by 3 ∼
µ ) 3 w represent the scales of u c ϕ` ϕ ) µ e ( `R τ χ H 1 ( 1 v ) `R ξ ) , v ( `R ( ) is obtained by requiring Λ ϕ ξ 3 η ϕϕ ϕ `N µ ( v Λ and A . ( 1 Y 1 2 4.6 3 2 can be achieved by either assuming a hierarchy e µ τ ϕ N e . The VEV scale of SO(3) Λ Λ s Λ y y y v y y η + χ η v g v ⊃ ⊃ ⊃ ⊃  + 4 c ∼  τ µ and N e 1 is large enough. With the above treatment (but not the unique treatment), R R ξ e χ 4 w w v y v w ξ w A do not break any non-Abelian symmetries but U(1), their role is to connect , A are determined by Λ, 2 η = 3 → Z ζ e v y → and ¯ The Yukawa coupling of In the following, we simplify our discussion by assuming all dimensionless parameters We briefly discuss the scales involved in the model. The VEV 4 UV scale (Λ) and η A χ where at leading order.couplings for After leptons the and Majorana flavons masses get for their right-handed neutrinos. VEVs, we arrive at the effective Yukawa with The hierarchy in eq. ( 4.3 Lepton masses Lagrangian terms for generating charged lepton masses are given by in the flavon superpotential beingv of order one. In this case, orders of magnitude of a small coefficient parameter we can easily achieve a hierarchy of energy scales of SO(3) of the scales of SO(3)naturally achieved breaking due and to theof suppression of Λ in the dominator in eq. ( JHEP11(2018)173 0 µ L , (4.16) (4.15) (4.11) (4.12) (4.14) (4.13)    3 00 µ c µ decouples µ R R . 00 µ       R decouple. 3    0 τ c τ 3 d τ µ R R . After the heavy 3 H c R ×    3 µ 3 × 1     3 3 and 3 ξ 1 d × v × 3 with 3 ξ 2 1 Λ H µ v µ 1 ) 0 00 µ 2 3 L d Y 3 ξ µ R v ϕ 3 3Λ). This mass term should 3Λ √ v × , Y v , which are much heavier than . These mass are much heavier . 2 1 √ ξ √ √ 3 d ξ . τ 0 d (      v τ 2 v Y √ / 2 H y 2 H 3 c can decouple from the low energy 2 τ 2 ( ϕ ζ    3Λ µ c ϕ 3Λ v 2 v µ ϕ Y v 3Λ √ √ Y τ 0 τ d 1 O ω v 1 √ 3 ω 2 3 √ τ τ τ R ∗    H ω    y τ √ Y    2 d ∗ √ 2 2 ω 2, we arrive at the charged lepton mass y V 1 ω 1 ζ ω ω ωy and it provides a stronger constraint to ¯ V H ω ω η = v √ 2 v ζ 2 ω c 3Λ ¯ η 1    3Λ / 1 v ∗ 2    ϕ ¯ v η 2 ω τ µ V 2 d ¯ √ v η × √ 3Λ 2 v ϕ ¯ η v v Y 3 T 3Λ 2 v 1 ϕ v T 3Λ √ ϕ ϕ 3Λ ` µ v 0 3Λ τ ` √ ϕ v v . After all these heavy particles decouple, we 1 √ = y ,V √ v term obtained via a seesaw-like formula. Y c √ µ µ 2 ¯ µ η – 19 – i τ with y y µ µ 2 v d ϕ y 2    3Λ y µ 0 3Λ τ d ϕ v heavier than the electroweak scale, 2 ωy y ω H d 1 v ω √ R h ω √ H ω ζ τ 3 3 3 H 3 3 3 ϕ ϕ ϕ µ ω v V with y    ∗ v v v Λ Λ Λ obtain a mass 2 ω y ¯ V 1 η e e e v 00 V µ 2 3Λ 0 τ ¯ × η = = y y y 0 0 ϕ = 3 v R √ v R ϕ 3Λ 3Λ ω ϕ 0      at low energy as v eff τ 2 v eff √ µ V √ with the µ . For c w 1 = τ y w ζ are given by τ 2 µ y l and v µ y are given by µ 1     0 M µ    R τ τ /Y Y gets the VEV ) 3 L R ) 3 µ d 3 T τ Y T µ H 2 ,L µ ,L 0 y 0 τ µ − ,L 1 ,L T µ obtain three degenerate heavy masses 2 obtain three degenerate heavy masses 2 than that splitting ` T y 3 3 ` ζ τ µ v = = ( obtain a mass R R = ( µ τ y 00 µ µ eff R eff R R and and After the Higgs Those coupling to Couplings involving w w 3 3 µ τ matrix the electroweak scale. also be heavier than thetheory. electroweak This scale mass such term that the aims scale to split obtain Yukawa coupling for L where than the electroweak scale, andand thus for the low energyfrom theory, the low energy theory.leptons In are integrated this out, way, we we are successfully left split with the following couplings at the low energy theory, L where JHEP11(2018)173 = via l , in is of M N (4.19) (4.20) (4.21) (4.22) (4.23) (4.18) (4.17) M 0 τ T l R M U via and l 0 is diagonalised U τ ν L . M

. are all of order one, d . ) τ v 1 2Λ masses. a y ϕ
, v √ 5 τ − | ) τ b b y and

− and µ NN = a y ( | µ , τ χ , e = = arg ( e y m χ    3 3 , 2 λ ω , we obtain that

ν , T + and Majorana mass matrix for , D d P 2 2 3 v 1 , β ω ω ¯ ) ω η M ×       N ) 1 2Λ 3 v i 2 2 1 11 1 1 a i 1 ϕ − M − √ √ √ v ,M b a u 0 0 a b    Λ : 1. The mass between NN | 2 M v µ ( and / 0 0 3 a 0 0 a 2 ¯ y η | D D √ 1

¯ ν η 2 2 v y √    1 1 0 1 0 η – 20 – M : √ √ = arg ( = . It is straightforward to diagonalise = Λ λ , are respectively given by = = ≡ χ − 2 2 2    µ v T . ω + ) D χ M = d Λ) m 3 = U λ v u / ν M , β M ν 2 ϕ and H

= ,N v ) are totally independent free parameters, there is no need U M 1  . The three mass eigenvalues for light neutrinos are given √ 2 3 ( l } a ) d } τ with ϕ v U 3 ,M y 3 2 ∼ + 2Λ ,N β 3 ϕ | } `N i 1 /v . This matrix is diagonal by a unitary matrix = 2 b v b 3 ( τ √ 2 ζ T , m N , e e v ) N b 2 + m y c 2 y 2 ,M , and β : a 3 2 | i µ , m , τ = y √ µ c 1

, e and ( = = arg ( , ,M N m 1 e m 2 , µ 1 1 1 β T = y c : w { Λ and i ) β e with e e / M 3 M e ¯ 2 η { { } m v , ν m τ ¯ 2 η λ and ( , ν , m = diag 1 ` µ ν ν = diag = diag = 2 Λ. It should be much heavier than the electroweak scale to avoid the constraints U ∗ ν , m / ν ν e 2 ζ a U P v M m M T The realisation of neutrino masses is straightforward. The relevant superpotential If we naturally assume the dimensionless parameters ν { M U † ν Applying the seesaw mechanism, as where where U the bases ( terms at leading order are given by The generated Dirac mass matrix between to introduce a fine tuning of them to fitwe the then hierarchy obtain of order from collider searches, i.e., and In this model, since diag in the basis JHEP11(2018)173 , 3 η 1  (4.27) (4.25) (4.26) (4.24) χ d 5 are non- ξ  =5 d f w . Dimensionless = 0, but , 4 5 -invariant VEV is ) and 4 ··· ξξ A ). The PMNS matrix + | =4 d f a ¯ η . We denote the shifted w η in table − -invariant vacuum is the , 4 5 d 3 5 ) b ξ | Z ξξ ··· . (2 ( / 2 + 2 u 1 and the quintic term ··· Λ ¯ v η - or 2 η ¯ η 2 D + include not just the renormalisable + 1 Z y 1 ) 3 d 5  in detail. =5 ξ = 5 ξξ d f ) ( χη ζ 3 w 2 2 , = 0, respectively. These requirements are 1 1 ξϕ m Λ Λ ξ ( = 0 does not lead to ( + ), a lot of new terms appear here. We will 3 δ d 5
and d 5 + ξ + 5 =4 4.2 ) +  ¯ 1 χ η d f ) 4 2 , ) and /∂ξ – 21 – w | η A χχ f ϕ ( a ξ ϕϕ 1 | Λ + , ξ ( ξ . We are left with the tri-bimaximal mixing. 3 ∂w (2 χ = ν 6 / 1 = d Λ f 2 u P η v 5 ξ w ) + 2 D A = ¯ y η TBM χξ − 5 f U ) = 1 w ) 2 = ξϕ , classified by the driving fields. As mentioned above, we follow ( m ξξ ν ( 4 U 1 Λ = 0, while those to derive a ), = 0 or ( † l | a 5 U 5 =
) 5 U(1). The full flavon superpotential should be given by + 5 = ) ) ξξ b | × , but also the quartic term ξξ ϕϕ 1 ( (2 ( , couplings involving the driving field /  , etc. The minimisation ξ ξ 2 u PMNS 5 ¯ η ) v U η 2 D ξξ 1 y (  d 5 5 ξ = ) represents renormalisable terms in the superpotential, and  1 ξξ First for In the model discussed so far, the crucial point in deriving an 3 ( m is modified into 6 d 5 d f ξ d 1 where the dots represent contributionsVEV of as the rest terms in table where the dots represent contribution of allfree rest parameters terms are involving omitted hereξ and in the following. Couplings involving the driving field  the general arrangement in mostcouple supersymmetric to models that flavon driving fields. fieldsdiscuss Compared always how linearly with they eq. modify ( the VEVs of term w renormalisable quartic andterms quintic are couplings, listed respectively. in table Up to quintic couplings, all 4.4 Subleading corrections toWe the first vacuum list (are termssymmetry negligible) in SO(3) the flavon superpotential which cannot be avoided by the flavour in the superpotential and leadconsequence, to the that relevant the vacuums above dosubsection, requirements not we do will preserve not prove the that hold symmetries afterthe explicitly. including explicitly. former As these In terms, a symmetric the the ones, flavon next the VEVs but subsequent do the deviate subsection size from we ofthat consider the it subleading gives deviations effects important are to corrections. safely the very flavour small. mixing and Then show in is given by the requirement ( requirement obtained via the minimisation of the superpotential. However, extra terms may be involved by JHEP11(2018)173 , 1 1 )
5 ) ϕϕ , ( 2 , , 1 ϕχ ¯ ¯ 1 2 η η , 2 ) (
¯ η
9 1 η 1 5 5 ) 1 ) 1   1 ) )
5 3  5 ) ζζ ,
χχ ζζ 3 ζζ ) ( 1 5 ( (
( , ) 9 , d 5  3 , 1 ϕϕ ) ζζ d 1 1 ) 5 1 ( d ϕ 1 (  η
)  ξ χχ d 5 5 d ξξ 5 9 5 5 , ( χζ
, , ξ χ ) )
¯ ( η (( ϕ 3 2 1 ,  9 1 ) ) ¯ ϕ d 1 η ) 1 ζζ ,
, ξϕ d 3 ) 1 ( χ ( 1 η ) χ d ζζ 1 3 5 χ ζζ 5 ϕχ , ( 5  ) 1 ) ζ ( ζζ ) (  ) 1 5 9 ( 1 5   ξ ) ϕϕ )
, )
5 ξξ 1 5 ( , 7 ϕϕ ¯ η 5
) d
5 ϕϕ d ) 1 ( η ζζ ) 1 (( 1 5 ζζ ( 5 ξ ξ ( 1 ) ) ) ( ) d 1 ) d 5  9 5 , χχ d ϕχ 1 ) ) ϕϕ ϕ ( , d ϕ η 5 ( ζζ χχ η ( χχ 5 ¯ 1 ξ χζ , η ( ξ (  ( d 5 ) ζζ , (
¯ d η 1 η 5  ϕχ , , 1 ϕ η ϕ ξ ) d ξ ( 5 η (( 1 1 , ) ξξ 1 1 7 5 ξ 1 ϕ 1 1
d ) 1 
d 5 )  ϕ ) , , (( d 3 ,  5 ξ 5   3 d χ χ 3 η ) η ¯ η d 1
5 5 , χ
, ξξ ζ 5  1 ϕχ 1 1 5 1

ϕϕ 5 1 ) 1 χ ) ( ξξ ) (( 5 9 ( ) , )  ,(    ( ,( 9 , ) ) ¯ d 1 η 5 3 η 2 5 5 ξξ d ) 5 1 d ¯ 1 d 5 η 1 ) ζζ (
¯
ζζ
˜ η η 1 ) ϕ
ζ ξ ξ η ( ζζ ζζ ( 5 ) 5 5 ϕ 1 1 , , 5 ξϕ ( ( , ) 5 ) ζζ 2 d 1 5 1 ) 5

) ( 2 ¯ 5 9 , ) η ( ¯ ) η χχ
η ) Λ 5 3 9 ) ) 3 η 5 ( η 3 η χζ ζζ ) ) ) , ) ϕχ 1 1 ) η 1 1 ( ( ϕϕ ξξ χχ 2 ( ) ) × ϕχ ζζ ζζ ) 1 ) ( ( ( ξξ ¯ ( ϕ ξ ϕ ( ( η ) ξχ ξχ ζζ ζξ ( ( 2 ζ ξξ ξξ ( ζζ ξξ χ (( d 5 d 5 d 3 =5 ( ( (( d ( – 22 – d 5 d 3 d d d d 3 5 5 5 5 η ( d ( 3 d 5 d f d 1 d 1 d 1 ζ χ ϕ ϕ ξ ξ ξ ξ d d d 1 d d ζ χ χ 1 1 1 1 ξ ˜ ζ  ζ  χ   η ϕ ϕ ξ ξ quintic terms w (     , , 1 , , 1 1 1 1 1 1 1       , 5   , 5 3 3 5 5 ¯ , ) η ) 5 5 ¯ ) ) , η 1

¯ 1 η 1 5 ) ) 5 are free parameters with one mass unit to balance the dimension 5 9 1 1 1
5 9 1 ) )
) 5 9 ) ) ) ) 1 ) ) , 
ξ χ ) ) ξ 5
, 5 5 η 5 7 ) Λ 3 ζζ ¯ ) A ) η ϕϕ ζζ ζζ ) ϕϕ ) 1 χχ χχ ϕϕ ) ζζ ζζ ( ( ( ( ( 1 ) ( ( ( ( ( 1 , × ζζ ) 1 ξ ξ ξ χ χ ) χ ξϕ ξ ξ , ( ¯ ) ϕχ η ( ( ( ϕχ ϕϕ ζϕ ( ( ( ( 3 3 ( ζ 2 χ χχ ( ( ( ξξ χ d 5 d 5 d 5 d 3 d 3 and ¯ =4 ( η ( η d 5 d d d d 5 5 5 5 d 5 ( d η 3 d 5 d f d 1 d 1 d 1 χ χ χ χ ϕ ξ ξ ξ ξ d d d d 1 1 1 d ζ χ 1 1 η χ ξ 0 ζ (     χ χ  η η ϕ ξ η quartic terms w  (    µ 3 U(1). 1 1 1 6


× d f 5 1 3 5 ) 1 ) ) w 1 ) , ) ) η , ξχ ζξ ξξ d d 1 ¯ 1 ( η ϕϕ ξξ ( ( ζζ ξ ( ( η d 5 η ( d d 3 5 ξ d 1 2 η d d 1 d ζ χ ξ 1 1 ˜ ζ 0 0 0 0 µ η ϕ ξ A terms renormalisable . All terms up to quintic couplings in the flavon superpotential allowed by the flavour d 5 d 1 d 5 d 3 1 d d d d d 1 d d 1 3 1 1 5 ˜ ζ ζ ζ χ χ χ ϕ η ϕ ξ ξ Table 4 symmetry SO(3) in the superpotential. JHEP11(2018)173 χ δ 3 are ) , = 0. 9 ϕ ϕ ) 5 (4.31) (4.30) (4.28) (4.29) δ 2 ) Z , 4 without A χ ξ and represents ξ χ v ; ··· 4 δ ξ χ ( ) and A δ 4 , + ξ ··· ξ A ¯ η 4.4 ξ + η ( 5 . We denote the Λ, this correction 3 1 ) ) and ζ ) g Λ 4 0 2 1 A  4.4 ζ ξ , , ξ 0 on the right hand side + 4 and 1 v ¯ η A 3 3 ξ ζ ) v ξ Z ( v ··· χ 9 ( ϕ 5  ) ϕ Λ 2 ) v , 2 + ; 1 3 ϕ Λ ϕ Z ¯ Z η = χ , v 1 ϕ ··· and + 2 ) χ 3 2 ) Z 1 v 3 Z 2 ) Z χ η 0 Z ϕ ( 2 χ 1 ( ··· χ ξ 2 Z ζ , 2 δ 0 , Z 1 χ 0 ( 1 4 Λ χ , 2 3 ζ A ( 1 Λ 2 g ξ ( 3 η Λ ξ , + , ϕ 2 v v ). Since . , 2 2 Z ϕ 5 χ ξ χ η + + 1 , ζ δ δ
5 v v χ Λ Λ 3 δ 5 1
) , ··· ) 1 ) 3 Λ + + 5 (Λ 3 3 + 2 + ϕ ··· 2 Z ξ ) 3 / 2 is used on the left hand side and + Z Z v 0 2 2 v – 23 – + ϕ Z Z ϕ 1 + χ ¯ 5 ϕ ¯ Z ϕ η η 5 Λ v ¯ η ϕ χ ζ ) v 3 3 ) ϕ η ¯ χ ξ η 5 4 ) 4 , Z δ 1 3 v χ ) 2 ( A ) represent leading-order value in eq. ( ¯ A ) η ) is approximately simplified to 2 δ ϕ ξ 0 4 Z ξ 3 ξ ( ( v 0 Z 1 v ξ A 4 4 χ 2 Z 1 ϕ ζ δ χ 4 Λ ξ ) only gives an all overall small correction to 0 Z A A v ζ ϕ 3 4.25 0 1 A ξ ξ χ 2( Z -invariant VEVs 1 ( ( ξ ζ ( 2 3 1 ( ( ϕ ζ Λ 2 f 4.26 Λ ≈ ( ( 2 2 2 Z g 2 Λ and 1 1 1 5 2 1 1 + Λ Λ Λ Λ Λ as
2 max + ¯ ) η Z ζ + ξ ≈ ≈ ≈ ≈ ≈ 5 5 χ . δ 3

5 3 1 - and ) , 5 3 ) ) ) ) has been used. In the above estimation, we include all corrections ξ ξ 2 5 ) 3 + 4 ζ Z δ v and χ ) 3 Z Z δ 4 δ 2 A ϕ 0 Z 4.7 4 ϕ χ ξ A Z 4 1 -, -invariant part with each components given in eq. ( ϕ ζ A ξ , A ζ χ 4 4 3 δ ξ ( 2 ξ symmetry. Eq. ( ϕ ( )( Z A 3 A Z ξ 2 ϕ 4 h δ χ h + ( ( 1 2 Λ ( A ξ 5 ξ + and pick the largest one δ + ) δ ≈ 4 2 ( 3 4 is the ), respectively. In our paper, since we only care about the order of magnitude ) Z A 2 2 5 4 ξ Λ Λ ) ξ g f χ δ 4 ( A are subleading order corrections. Once subleading high dimensional operators are 4.4 ξ 0 ξ A δ 1 ζ ( ξ ζ δ Similarly, we can estimate corrections to the VEVs of ξ ( δ 2 -breaking corrections. Eq. ( 4 h 2( respectively, where and included, the minimisation of the superpotential gives rise to from table is very small andestimation can but be must be safely smaller ignored. than it. The exactshifted correction VEVs may of be different from the where the fourth term in theThe curly relation bracket has in a eq. vanishing contribution ( since ( where replaced by the of eq. ( of corrections, we neglect CGorder coefficients of in magnitude. the Then products we and obtain do a naive estimation of the A breaking the where JHEP11(2018)173 (4.34) (4.35) (4.36) (4.32) (4.33) VEV is and the χ ϕ /v ) are used in ϕ δ 4.7 are stable under , ζ . The second type shift and f , 001Λ . w ¯ η 0 ξ , ϕ , v v χ . ϕ , ∼ ¯ 2 η Λ . v v χ v ζ -invariant VEV ¯ Λ η 03Λ ξ , 3 . v 005 . v ξ . ϕ Z = , v 0 ξ Λ ξ v δ v = 0 ) ∼ ¯ η = = 03Λ χ χ . ···  δ v , 0 , ) , v ¯ ¯ η η ξ ξ , ∼ ··· v v v v ··· , ϕ 1Λ ϕ χ , . Λ Λ 0 v v ξ , 005 , 2 ζ = 0, and the relation in eq. ( v . ¯ η ξ v ∼ χ , v 0 = 0 v – 24 – Λ ¯ 2 1 η v v ) ϕ , v η χ χ 0 Λ . Λ v 0 δ v 1 , , , ζ 01Λ ζ ζ . 0 ξ ξ ξ ξ ξ ξ δ v , v Λ. However, we calculate this correction in detail in 1 δ δ 0 v v δ v / ζ . The upper bound of the correction to the , ¯ η ξ (  ( ∼ 3Λ ϕ ϕ v . δ v η ϕ /v v ξ , δ max max max = 0 √ ξ ξ , v ξ δ v . . . ∼ A ζ 1Λ ζ χ χ ϕ ϕ ) . δ v δ v δ v χ 0 v ∼ (Λ ξ / v = 0, as well as ( ¯ 2 η v 5 ) 4 are as small as . A ζ ξ χ and find that the true correction 4 /v /v A C ζ ξ χ δ δ We numerically give an example of the size of these corrections. By setting and higher dimensional operators in the flavon superpotential VEV ` subleading higher dimensional operators in superpotentialw terms for lepton mass generation the flavon VEVs and furthercorrections modify in the this mixing. model Asdiscuss are discussed corrections less in from than the the 1%, last first safely subsection, origin. negligible. these In the following, we will only subleading corrections. 4.5 Subleading corrections toAt flavour leading mixing order, the (are flavour important) mixingafter appears subleading as corrections the are considered. tri-bimaximal pattern. There are Deviation two arises origins of subleading corrections: and All corrections are less than 1%. Therefore, VEVs of we obtain which is also very small. ζ larger, appendix Again, ( the above. Upper bounds of relevant corrections to the A naive estimation gives the upper bounds of corrections JHEP11(2018)173 is or up l 3 d µ , and M (4.41) (4.38) (4.39) (4.37) (4.40) d H d R τ H 1 ηH ), respec- `R ) 1 and d
ξξ 5 ` or ( ) 4.15 plane, ηH d 1 d 1
ϕχ H
ξ ( eµ 3 µ 7 ηH 5 ) ) 1 ) τ
µ `R 5 . Their contributions ) and ( ϕχ ) c ( . There are five terms `R `R 3 µ ( ( χ ) ζζ . ( µ 4.13 5 , , or +  ) d and .    d `R τ 2 d 2 d d ζ 0 0 ( ), (
` v v H √ H √ ηH Λ `R , 1 1 1 eµ ¯ ( η +
       v  4.10 iφ o 3 7 d 3 µ ) 0 0 0 0 0 d , eµ −

+ y 7 and those to -breaking effects are included. The θ 9 e d 9 2 χ 3 d d ) d d ¯ ) ηH ϕχ v ηH eµ 6 dω Z η ( 1 η − θ ξξ dω ηH − − ξξ cos 1 3 ¯ d
v ηH ( c ( ) +
c c c ) 7 ξ η e ξ + µ ξ ) 2 , d 1 − sin 7 1 leaves , no 7
3 ) χ  ) η `R 0 0 2 0 2 l ) + cω cω µ ϕχ ϕ ` v eµ ( τ τ ( 3 c up to 3 η 00 00 0 7 ) iφ    ( Λ , as in eqs. (
v `R ) d , δM τ e ¯ `R η 5 ) `R ( T eµ – 25 – ϕ d µ ( and ¯ )    3 ( − v ) d θ H v eµ  µ 0 0 1  c `R η ϕ n θ χ ( y `R + 3Λ , ω ηH ϕϕ v v ( , + `e 3 + 3 2 1 ( cos c η χ 1 ξ √ 
ξ d ( sin Λ d v v is a complex rotation matrix on the Λ 5 , , ω η 3 + ) H H − 1  ϕ v + = = arg = Λ eµ d 1 1 4 ζζ d l     U 1  ( = eµ eµ + symmetry are those involving Λ 7 3 5 H ηH θ l φ
= )
d 1 1 3 δM 5 5 τ and (1 +

) ) Z T 5 ω eµ 5 sin d δM ) ηH ) T `R U ξξ U ξξ 1 ) ( H ( (
ζζ 2 3 , where ξ ξ ϕχ ( ξ η , ( 5 7 c eµ 7 5 d 3 ) ) ) e ) ) U µ τ 1 τ . The first two terms only contribute to coupling between H τ µ , ω, ω ω ) on the left hand side of d 1
`R `R U `R `R `R ( ( T 5 ϕ` ω ( ( ( ) ( ηH , (1 ' U are real dimensionless parameters. The unitary matrix to diagonalise   1 2 2 4 T l 1 1 1
ζζ d Λ Λ Λ 7 ( + + + U 1) ) 5 , ) ⊃ ⊃ ⊃ 1 µ , ϕχ c and τ µ ( e 6 include R R 7 `R c w ) ( . Acting w w l µ 6 Subleading terms contributing to . The rest three terms contributing to couplings between symmetry always guarantees that the corrected effective Yukawa couplings take the M d 3 `R τ 3 ( with modified to where to tively. Terms breaking the left, R to the charged lepton mass matrix are characterised by adding a new matrix For terms involving only someZ of forms (1 to JHEP11(2018)173 0 τ = to F ν U (4.45) (4.43) (4.44) (4.42) Λ). For . PMNS ¯ η v U ijk ( with . ξ / . 1 0 µ χ
3 ) v B ξ η eµ 0 1 µ -invariant VEV. v θ g gain VEVs, mass 4 D Q ∗ ∼ A χ ξ µ 13 ] + θ D + sin 2 -invariant mass term for and 5 have only trivial cor- ijl 4 , ξ A eµ ϕ 0 µ 6 kl get the , φ ) B ζ a d ξ 0 1 , τ are gauge couplings of SO(3) g is required. This is not hard to 0 1 Q eµ , respectively. Their interactions 05. In order to generate sizeable 0 + ( . µ φ ) + 0 B + 3) sin /y a ilk eµ ] ) ∼ ξ cos τ θ and g eµ 8 d a jl . 2 θ – 0 and ) µ eµ 0 3 Λ) . Therefore, the unitary matrix 6 + a ) is replaced by ( θ . The degenerate mass spectrum is also 3 F ¯ η } eµ , sin since it is too small compared with the c τ 2 3 2 v θ transformation, an ) ¯ , , η l ( 2 ξ 1 for each field is listed in table 2 − 4 2, an almost maximal CP-violating phase η , / 0 (cos 2 2.43 v eµ + ( 1 i A χ 0 3 θ , X
F Q =1 δM π/ 2 sin 2 sin v gain masses once 5 a − 3(2 η T ljk ) ω eµ 3 0 3 – 26 – v − − ξ , 2 cos U θ → 2 eµ il , 2 2 ) √ φ 1 = (2g in eq. ( a + g NN 0 sin q µ eµ s ( 3 τ ξ µ 0 φ F χ [( 2 ∂ F = = = Later after the rest flavons ) cos a to perform a 0 µ M = 4 + t 13 12 23 eµ F θ θ θ θ µ u ), we have = 3 , 2) entry of D 2 and 2 on the left hand side does not change the third row of the , , sin sin sin 0 ¯ ηH s 2 4.35 F η X → =1 can be eliminated to yield sum rules which have been widely 1 eµ a ) µ M + cos 4 U 0 3 ∂ eµ symmetry. = eµ `N φ 4 ( θ + g has been used. 1 ]. In the limit 0 { A ξ 2 F in the first octant is predicted. The reactor angle 2 ijk 64 Λ. 1 ) Λ – / M ξ 23 d µ θ v (3 cos 2 ⊃ ∂ keeps the same as that in the leading order. 59 ϕ , v , a relatively large value of the ratio ( ν 8 N = ( – 13 w . multiplying ∼ M = +1 for 6 θ 2 is predicted. τ ijk = arg Q ) m π/ δ TBM ξ Specifically, the kinetic term for In the neutrino sector, terms for neutrino masses up to Including the subleading correction, the PMNS matrix is modified into 3 µ U One may use the generators 4 D † ∼ eµ ( where We obtain that consistent with the is obtained only if all masses are degenerated. and U(1), respectively, and the U(1) charge We label the gauge field of SO(3)with and flavons U(1) or as fermions are simply obtained with the replacement in the kinetic terms of the relevant fields. Here, g The unknown phase studied [ δ 4.6 Phenomenological implications of gauged SO(3) the numerical value in eq.value ( of be achieved. The Dirac-type CP-violating phase is predicted to be In this model, U PMNS matrix. Three mixing angles are given by [ mass rections, diagonal Here, we have ignored the (3 JHEP11(2018)173 0 , 4 1 → B ) A ξ ) or 4 van- ξ µ , the TeV, v A 7 i 4 D ( 3 and ∗ ) 0 given by ∼ ξ ξ ∗ µ c F and ξ ξ ∼ O ). After τ D ( ¯ , 2 η 4 h 1 v A . The resulted + and 1 ∼ -plet contraction can be tested at
2 → and domain walls η . This can be sim- c 0 3 0 0 v mass. Interactions ) and ( ) µ 3 B B 0 B χ , Z + , ∗ c B 3 2 ξ ξ e 4.45 ( v -invariant vacuums exit, , be around 100 TeV. For and ( µ 2 ` ∼ 0 2 ζ Z 0 1 F 0 are much smaller than that v F are still nearly degenerate. F µ χ . Here, the 3 = g 3 , and the residual symmetries 1 B , . 4
0 2 and ξ , 3 2 1 B breaking to A from eq. ( and ) 0 . The mixing between ϕ /v 0 0 ξ ). All vacuums are perturbatively 4 v M ϕ ∗ F χ B B A , ξ , v , . Since ( 2.1 χ η ζ µ v ξξ -invariant or , ¯ 0 3 F η and and B Z 0 0 0 µ -invariant vacuum as an example, different get the VEV, where only one of the seven F 3 F F B ξ and Z , the anti-symmetric contraction ξ ξ – 27 – TeV and v 3 1 TeV). Its interaction with = i ∼ 0 . Since VEVs of , degenerate ξ , there are degenerate vacuums which are continuously 3 2 h , , the breaking of a gauge symmetry does not introduce Λ 2 2 4 , / . Without considering gauge interactions, there are not ,Z 1 2 0 contribute small corrections to the ζ A 4 3 χ v , TeV-scale gauge bosons are predicted. Then, interesting ). Taking the v F Z 3 χ → − 3.6 or is suppressed by the ratio → 0 4 invariance, there is no mixing between 4 ϕ and are flavour-dependent, with charges for B v 4 A 0 ϕ breaking, induced by terms such as A B , ) and ( , respectively. 4 ζ and 1 3 )), respectively, if gauge coefficients are of order one. However, they A are anti-symmetric. Once ¯ 0 η 3.3 − ξ F , v η , v and ξ 1 and v , are just phenomenologically effective symmetries at lower scales. The usual are predicted to be around 10 ∗ 2 − ξ η Z , obtains a mass from VEVs of both v 7 3 . We discuss more on why the topological defect of domain walls does not exit in 0 2 (max( − At the second step, At the first step, SO(3) In our model, we actually have a two-step phase transition SO(3) These gauge bosons are supposed to be very heavy, with masses around In the limit of the B , the mass splittings are very small, and masses of and , and ξ -invariant vacuums are randomly generated during ,Z 2 3 3 3 3 ξ as shown in eqs. ( Z separating different vacuums arise.inside These the domain wall walls around store energy with energy density the model. domain walls. As noted inconnected section by SO(3) basisequivalent. transformation as in eq. ( The domain wall problem ispaper, a well-known all problem flavour for symmetries discreteZ symmetry at breaking. high In scale this aredomain wall gauged. problem for the global symmetry breaking doesZ not apply here. mass also around TeVcolliders. scale ( 4.7 Absence of domain walls could be much lighterv if gauge couplings area tiny. gauge For coupling example, aroundsignatures if involving 10 gauge Λ interactions is can be fixedof tested at charged at leptons. colliders 10 or Another precision measurements interesting point is the prediction of a heavy tau lepton with ishes. Therefore, there isis no generated mixing after between mixing between ∼ O ply explained as follows.if The mixing exists, between can beonly SO(3) only invariant generated formed by viabetween these coupling fields is components has a non-zero value, of breaking, VEVs of between leptons and − splitting are generated among JHEP11(2018)173 4 A - and U(1), 3 × Z -, 4 symmetries A 2 Z supersymmetric and . We have proposed 2 3 Z Z , 4 and A U(1), we have shown how 3 × breaking, in accordance with Z -plet, i.e., a rank-3 tensor in 3d 4 after they gain the . The A , and domain walls may directly 2 χ ϕ Z v in SUSY, using high irreps of SO(3) 4 . and A and 0 model, we have demonstrated that such ϕ F 3 , 4 → Z symmetries are preserved in charged lepton ξ M A 2 Z – 28 – potentials for the first time. We have analysed in U(1). This study is important in order to verify and × 3 and its finite subgroups Z 4 A , and then to 4 A . Such 2 Z supersymmetric model of leptons along these lines, originating from SO(3) at a lower scale (below the SO(3) breaking scale) to the residual and 4 4 3 A A using Z 5 by using different higher irreps. A 5 A and or 4 -invariant VEVs, respectively. We have found that tri-bimaximal mixing (with 2 4 S Starting from a supersymmetricSO(3) is flavour broken group first SO(3) to are respectively achieved by theZ flavons zero reactor angle) is realised at leading order. One technical point is that the singlet We have shown that it is possiblebreaking to of also achieve, at thesymmetries level of SO(3), the subsequent sector and neutrino sector,the respectively, semi-direct after model building the strategy. We have achieved the breakingand of flat SO(3) directions. Inspace, this to paper, achieve we the have chosen breaking.S a We 7 have shown that it is possible to break SO(3) to , 4 About a half of the paper is devoted to the study of the realistic supersymmetric The main achievement of the paper is to show for the first time that • • • A phenomenological implications due to the massive gauge bosons. model of leptons, arisingthat from it SO(3) is reallyachievements possible of the to specific construct model a may fully be working summarised model as along follows: these lines. The main SO(3) flavour symmetry can beBy the focussing origin in of detaila finite on non-Abelian strategy a family can symmetry supersymmetric leadSUSY models. to being a preserved in viablescales). the lepton low model Moreover, energy we which spectrum have can (below shown explain the that, all flavour if symmetry oscillation the breaking data SO(3) with is gauged, there may be interesting which leads to a phenomenologicallysubleading acceptable corrections are pattern taken of into account. leptonconsequences We mixing have of also and having discussed masses the a once phenomenological gaugedthat SO(3), all leading domain to wall massive problems gauge are bosons, resolved and in have this shown model. 5 Conclusion In this paper we have discussedas the breaking of SO(3) down todetail finite the family case symmetries of such supersymmetric a supersymmetric fore, domain walls survive.decay to light Once particles gauge mediatedthe interactions by gauge gauge are bosons bosons. included, maydecay For domain into be the gauge walls case light bosons. of should enough, small i.e., gauge couplings, enough energy inputted to force one vacuum jumping across the wall into another. There- JHEP11(2018)173 . δ (A.2) being gain a 0 4 + 1 and with the A B p 3 , 2 , ) (A.1) 1 0 F )+1 q . + ) p ij of dimension 2 φ + (2( models are resolved here. ··· 4 and the CP-violating phase A masses, we have introduced an . Another gauge boson 13 ) + 4 θ τ + (perms of A a +3 -invariant VEVs are stable even after | 2 Ψ q a and Z − φ p µ ij | δ . 1 3 00 - and , 1 3 b discrete symmetries introduced, and − ) + (2 Z Ψ – 29 – , ja a -, a and φ +1 4 Ψ | , Ψ 0 A q ia a iab 1  φ φ ad hoc − + 5 p | ∼ ∼ ∼ i 1 ij
+ 3
3 , giving rise to a non-zero 5 Ψ) always accompany each other after SO(3) breaking. To avoid µ φ -invariant VEV. This welcome correction leads to additional ) = (2 Ψ) = 1 ( 2 Ψ) 4 φ φ 3 ( Z to split the A +1 ( and q × ζ of e (2 U(1), with no 00 U(1) is gauged, the model predicts three gauge bosons , 3 × 1 3 × + 1 are decomposed as follows: × ) q ∼ +1 and Ψ p 0 1 ∼ (2 φ , which gains the For just an effective flavourthat symmetry the below usual the domain SO(3) wall breaking problems scale. encountered in We have shown nearly degenerate masses after SO(3) breakingmass to after U(1) is broken.tions These with gauge leptons bosons will with lead their to flavour-dependent phenomenological interac- We signatures emphasise worthy that of the further flavour study. metry symmetry SO(3) at high scale is the continuous gauge sym- mass matrix is modifiedχ by higher dimensional operators,mixing between due to the coupling with If the SO(3) any fine tuning ofadditional flavon parameters related to We have considered the influence of themodel. higher dimensional We operator have shown corrections that to the subleading the corrections are included. However, we have seen that the charged lepton irreps • • • • Some useful Clebsch-Gordan coefficients offollowing: these products in the 3d space are listed in the A Clebsch-Gordan coefficients ofIn SO(3) SO(3), the product ofΨ two of irreducible dimension representations (irreps) 2 S.F.K. acknowledges theUnion’s STFC Horizon 2020 Consolidated Research and Grantgrant Innovation agreements programme ST/L000296/1 Elusives under and ITN Marielodowska-Curie No. Sk supported the 674896 by and European European InvisiblesPlus RISE ResearchERC-CG No. Council 617143). 690575. under Y.L.Z. Y.L.Z. ERC thanks is Grant Luca Di NuMass Luzio (FP7-IDEAS-ERC for a useful discussion. Acknowledgments JHEP11(2018)173 . (A.7) (A.5) (A.6) (A.3) (A.4) ) ijkl . ) . ) ijkl . . ) , ) ijklm ) , ijkl ijk ) + (perms of ijkl ijk , mab ) , , + (perms of Ψ ) ) , , ) ijk ab ab ) , ijk φ ij , Ψ ) ijkl + (perms of ij kl ) ab δ + (perms of + (perms of ij φ ijk ij + (perms of δ kl klma ka kla δ + (perms of , 2 Ψ Ψ lbc Ψ 21 ij ) , a a a bc δ Ψ φ φ φ ij + Ψ 2 + (perms of ac ij ij ij 35 , δ ac δ φ δ + (perms of + (perms of + 11 + (perms of , 9 4 + (perms of φ 2 5 3 7 lma + kab jk jb δ + 9 jk Ψ jkb ab + (perms of Ψ , la − − , − + (perms of δ – 30 – + 9 Ψ + 7 jklb , Ψ ka iab Ψ a ab Ψ ia a  jk Ψ ija φ iab φ φ + 7 jkl ab a φ ka  2 7 Ψ ijka Ψ ij jklm + 7 + 9 + 11 Ψ ij + 5 φ Ψ a i φ φ 1 5 δ iab δ a i Ψ iab Ψ ij φ φ  + (perms of − ij 2 3 a i φ  φ 2 5 + 5 δ iab δ − , φ  φ + 5 + 9 + 7 1 3 = 3 ∼ ∼ ∼ 4 7 − klb jbc ∼ ∼ ∼ − , kb bc i 5 ∼ ∼ ∼ ij + 3 Ψ Ψ
− , ij − Ψ Ψ ijk
3 = 3 = 7 = 5 ijk
iab × jka
ac ja 5 klm ijkl ab
ja 5 ac ja kl ijk 7
7 9 7 φ φ ijkl Ψ 7 Ψ Ψ Ψ)
φ φ = 1 9 Ψ Ψ
Ψ Ψ) , 3 7 ijklm φ Ψ) ia ij ab 9 × × × Ψ) iab iab 5 5 φ ia ij ab
( Ψ) iab iab φ  φ  φ φ Ψ) ( φ  φ  φ φ ( Ψ) φ ( 11 × Ψ) φ ∼ ( , 5 , 3 , 3 φ ( ∼ ∼ ∼ ∼ ∼ φ 7 9 7 ∼ ∼ ∼ ∼ ∼ ( ( i Ψ) i , 5 ij
φ ij
∼ ∼ ∼ ijk
5 3 ijkl ijkl ( ijk
3
5 ijkl 1


5 7
∼ 9 7 Ψ) and Ψ 9 11 Ψ) Ψ) ,Ψ ,Ψ ,Ψ Ψ) φ Ψ) Ψ) 5 3 Ψ 3 3 φ φ ( Ψ) Ψ) φ φ ( Ψ) ( φ Ψ) ( φ ( φ ( φ ∼ ∼ ∼ ∼ ∼ φ ( ( ( ( φ φ φ φ φ For For For For For • • • • • JHEP11(2018)173 , ) (A.9) (A.8) (A.10) ijkl , ) , ) bcd nabc ijk Ψ . Ψ ) ijkl , , acd abcd ) mabc ) , , φ Ψ mabc ) ) Ψ . lm φ ijk ij nab ) δ + (perms of kl abc ijk abcd ijklmn Ψ δ jk φ ijklm φ , δ ij abc , kl kl δ δ mab + (perms of δ ) iab Ψ 2  φ ij ij 11 ij + 17 + (perms of ijklmn δ δ kl abc 2 mbcd 63 bcdf δ , + φ 2 2 21 35 ij Ψ Ψ lbcd , kl + δ δ ) + 15 + (perms of + (perms of + Ψ , + 2 ij + 15 + (perms of lacd ij 11 acdf + (perms of mnab δ + (perms of mbc φ acd bcd φ Ψ labc abcd 2 + φ Ψ lmab Ψ 35 jk + 13 + 13 jk kabc δ Ψ Ψ abcd δ jk + 13 Ψ bcdf lac klab Ψ δ + (perms of + acd + (perms of Ψ iab φ Ψ φ mna iab φ  kab abcd kabc  iab abc . , ij + 11 + 11 jk lab Ψ abc 4 9 φ  abc φ jk φ 1 5 δ ) abcd φ δ ) + 11 acdf δ , Ψ 2 7 ij Ψ ij ij Ψ φ ij kla 9 ) − δ φ δ δ − 11 iab δ , + 9 + 9 iab φ + (perms of 2 3  − 1 3 4 7 abc kab ijkl abc ijkl  2 5 mn – 31 – lm + 9 ij 4 9 − φ φ δ φ δ 1 5 δ , − − − lmbc kbcd bcdf ij ij − kl + 7 + 7 , jbcd klbc jk − δ δ 9 ijklm δ − mn , Ψ Ψ Ψ δ 11 + 7 Ψ Ψ 1 3 4 7 δ ij abcd lmna klab jabc δ iabc iab klma kl jkab − + 5 + 5 lmb bcd kbc − − Ψ , acd jac  Ψ jkac acdf jacd δ Ψ Ψ + 5 Ψ Ψ Ψ Ψ Ψ Ψ φ φ 2 φ φ φ ij 2 63 231 δ abc lmn kla jab + 3 + 3 abc iab ija ijka abcd iabc ijab iab iab iab iab iab jac jka acd φ  φ  φ Ψ = 3 Ψ Ψ  φ φ  φ  φ Ψ + (perms of + (perms of + − 2 φ φ φ 231 9 ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ = 1 = 1 iab ija ijk abc iab iab iab i i φ  φ  φ φ  × − + (perms of 9 7 ij ij

ijk
ijk
3 3 ijkl ijkl ijkl ∼ ∼ ∼ ∼ ∼ ∼ ∼ 1
5 × ×
5

, 7
7 7 i ijklm 9 9 9 Ψ) ij Ψ)

ijklmn 11 Ψ) Ψ) Ψ) ijk
, 9 , 7 3 φ
φ ijkl Ψ) 1 Ψ) φ
5 φ ∼ ( 11 φ Ψ) 9 7 ( Ψ)
( ( 7 φ ijklm ( Ψ) φ 13 9 φ φ Ψ) ( (
ijklmn Ψ) φ ( ( Ψ) ∼ ∼ Ψ)
φ ( Ψ) φ Ψ) ,Ψ 11 φ ( φ Ψ) ( ( φ 13 ( Ψ 7 Ψ φ φ ( ( ( Ψ) ∼ ∼ ∼ Ψ) φ ( φ φ φ φ ( For For For • • • JHEP11(2018)173 , = in ξ ) are + 25 (B.4) (B.2) (B.3) 133 ξ (A.11) 2.13 = + 23 abcdfg = 0, part of 113 Ψ , , . , , ξ , + 21 ) ) and ( 333 = ξ ij = 0 = 0 = 0 = 0 ;= 0 = 0 (B.1) abcdfg

2.12 φ = 112 + 19 2 333 233 333 233 ξ kl 2 2 ξ ξ ξ ξ 333 333 δ ξ ξ 333 ) is left with 233 = ij 2 ξ ξ 133 113 133 δ + 17 + + 2 ξ ξ ξ − B.1 3 2 113 = 111 35 in eqs. ( ξ ξ 2 2 233 233 + 6 − + 6 ). Taking the VEV of ξ ξ 2 ξ 233 + (perms of + 15 + ξ 133 w + 3 + 2 ξ 3.2 233 333 333 + 2 ξ ξ ξ . , , − 2 = 113 + 13 233 labcdf abcdfg ξ ξ 123 112 123 2 133 ). , Ψ ξ ξ ξ Ψ ξ 333 = 0 = 0 = 0 113 6 ξ 112 + 3

ξ 3.3 1 2 ϕ ξ + 11 2 1 2 3 + 6 + 3 − f + 3 = 0 µ 113 ϕ ϕ = − ξ kabcdf abcdfg 233 3 ξ − φ φ + 9 133 113 133 − − 2 123 2 123 ξ ξ ξ 2 3 ξ ξ ij ij 2 2 2 2 2 112 − 112 112 δ . δ ξ ϕ ξ ). Then, we arrive at the special solution in ϕ ϕ ξ ) 2 113 112 112 4 7 + 7 1 3 1 +3 – 32 – + 3 ξ ξ ξ + 233 c is given in eq. ( = + 3 ξ − , − − 2 2 2 ξ 1 (3 123 123 ϕ ijkl + 5 c µ ϕ / + 9 + 6 − ξ ξ 2 112 112 2 ξ w 111 133 4 4 ξ ξ − ξ + ξ µ 4 − + 3 233 123 123 abcdfg 2 1 ), respectively. By setting klabcd jabcdf − ξ ξ ξ q d 5 111 A ϕ + 2 Ψ Ψ Ψ ξ  3 3 2 111 112 111 ∂ξ 112 = 1 ξ ξ ξ 2 ξ → 133 Z c 3 6 − = ξ ( 13 ijabcd abcdfg iabcdf + − / φ φ φ ) corresponds to it (11), (12), (13), (23) and (33) entries of two + (perms of → ξ 111 × 2 111 ) + 6 ξ 123 ξ 2 111 ) is automatically satisfied. Then, eq. ( 4 ∼ ∼ ∼ ξ ξ 2 ∂w B.2 333 1 A ij − , 13 + 3 ξ B.2 2
≡ ijkl 5 Ψ) 13
+ 2 5 2 φ 9 111 ) ( ξ Ψ) ∼ 113 φ Ψ) ξξ ξ ( Ψ φ + 2 ( (2 ξ 2 ∼ 1 = 0, eq. ( c µ 111 φ ξ − 333 3 ) into these equations, i.e., ). ξ For 2.9 2.9 = • 233 It is straightforward to derive all solutions in eq. ( B.2 Solutions for Equations for theeq. minimasation ( of these equations are automatically satisfied, the left vanishing ones are simplified as from which we obtain eq. ( Five equations in eq.rank-2 ( tensor ( ξ Equations for the minimisation ofrespectively the and superpotential explicitly term written out as B Solutions of theB.1 superpotential minimisation Solutions for SO(3) JHEP11(2018)173 = 23 (B.8) (B.7) (C.2) (C.1) χ 3 ) 12 ). 9 ) χ 2 3.6 Z = , , , . χ χ 13 δ χ ( = 0 ;= 0 (B.5) = 0 = 0 ; (B.6) -invariant VEV, , 4 4 12 ) = 0 ) = 0 2 1 A    . 2 χ ¯ η A χ g 3 3 3 ξ . The minimisation 33 33 3 µ ( g g g 2 ) χ χ χ              3 6 6 6 2 g , Λ − √ √ √ Z 7 7 7        + 2 χ + 2 2 96 96 96 χ 2 23 4 11 v ··· + √ 0 0 0 0 χ get the A 11 χ + + + 3 ξ  χ + . ) ξ 2 2 2 ( + ( (2 9 g g g ¯ 2 η ξ ξ . )        1 2 3 2 3 2 3 1 v v 2 13 2 Λ ··· 0 ) 2 3 2 3 Z χ 2 1 q q q , ) χ Z ≈ ≈ 2 2 2 0 q q 2 + χ 1 3 5 4 4    Z 2        ) ) ), we are left with ζ g g ). After 2 12 0 χ Z 9 χ 2 13 23 23 ( 1 χ ) − χ δ χ ξ v 2 χ χ χ √ ζ 4 3.5 0 0 0 0 ( B.6 δ ( Z + ), we obtain the result in eq. ( A ( ) is simplified to ϕ 12 12 13 2  χ ξ vanishing. The only non-vanishing one is 3 2 Z χ χ χ ( χ Λ 2 33 1 g ξ ξ ξ        χ Λ δ C.1 χ v v v 2.38 23 ( 1 Λ + 4 + + + + + , A 3 , χ – 33 – 2 + ) ξ    ¯ η 5 ( 33 13 ¯ 2 3 2        η ) 3 3 χ g g g 5 ) 2 2 g , χ χ ) Λ 2 3 6 2 3 in eq. ( 2 Z 11 v 2 2 √ 0 0 0 0 Z √ 12 7 χ 3 Z χ , q q χ  χ χ 2 + 72 χ 4 , 2 2 δ + 3 are given in eq. ( 1 3 ( A        -invariant vacuum Z ) 4 − χ − − ξ χ 2 5 2 11 2 ( ϕ A 2              3 3 ) ( w χ 2 ξ Z g Z 2 g g 1 ( = 0. Taking it to eq. ( 2 3 1 Z 6 6 = Λ Λ 2 and χ √ √ 7 7 g Λ → 33 q 00 χ        ≈ ≈ 2 72 72 2 δ χ i i i i i 4 χ 5 (    ). All solutions are listed here, ) , + 4 11 12 13 23 33 A 0 = χ A 3 23 13 33 χ χ χ χ χ χ δ ) ξ h h h h h B.5 4 χ χ χ 11 5 ( ) ) A 2 χ 11 12 by        2 ξ g 33 Λ Z χ χ 2 χ ξ ξ ij χ v v + ( 2 χ + Z 5 ) χ 11 2 ( Z ξ χ ) leads to δ ( χ = 0, and therefore two of ( ξ ξ v 2 δ Λ ( g B.7 23 χ 13 Ignoring all the other subleadingusing operators, the naive we estimation. calculate its In correction this in case, detail eq. instead ( of The naive estimation onlymay gives be the smaller upper than it.of bound the It of superpotential happens the including for correction. subleading the higher correction The dimensional to true operators the correction is VEV given of by Representing C Deviation from the determined by eq. ( Eq. ( χ Equations of minimisation of they are explicitly written out as B.3 Solutions for JHEP11(2018)173 , (C.4) (C.5) (C.3) ] 0. It means , ∼ is approximate ¯ 2 η 2 χ v f ]. family symmetry family symmetry χ /v V 3Λ v 00 ξ χ √ v δ hep-ph/0310204 SPIRE since the first equation    SU(3) SU(3) ∼ [ 1 0 0 IN . 00 χ χ ][    0 δ /v . 0 ≈ ≈ ¯ η χ ξ or δ v ξ . v χ 0 v ϕ ]. v Λ χ Λ v 2 3 ξ δ (2004) 107 v ··· = r i +     67 )  SPIRE 2   

00 0 IN 3 2 χ f d 5 χ χ χ ··· δ ][ hep-ph/0307190 δ δ , , we see that the first term is much smaller ∂χ ωδ ¯ [ ∂w η + f

ξ

3 3 v 0 V v VEV since it is too small as discussed in the − . g g χ ϕ 0 Λ + δ ξ v 00 ( 7 7 to – 34 – 2 , χ 144 144

δ ξ ξ 4 2 2 f d 3 δ v + + . Therefore, the minimisation of g ω 2 2 3  2 2

(2003) 239 (2016) 1 ∂χ 0 ∂w g g Rept. Prog. Phys. f d 5

χ − , δ 2 3 2 3 ∂χ 3 2 ∂w = hep-ph/0108112 max

) and the second one gives g ω ), which permits any use, distribution and reproduction in [ f χ q q 7 72 ∼ 0 0 v B 574 V B 908 −  Fermion masses and mixing angles from Fermion masses and mixing angles from χ χ = 0 cannot hold at the same time. In other word, there is no 00 2 (Λ δ v

    χ / d 5 f d 3 ¯ 2 η ωδ v ∂χ ∂w 0

/∂χ (2001) 243 ∼ χ f CC-BY 4.0 δ χ Phys. Lett. Special Issue on “Neutrino Oscillations: Celebrating the Nobel Prize in Nucl. Phys. 0 0 0 − ∂w , This article is distributed under the terms of the Creative Commons /v , ) 00 00 χ B 520 δ χ Neutrino mass models δ    ]. = 0, and the correction is given by ]. In the model discussed here, the flavon potential is given by + d 5 0 = 0 and χ 65 δ d 3 SPIRE /∂χ f IN [ Phys. Lett. and unification Physics 2015” Without flat direction, one has to calculate the VEV correction via the minimisation /∂χ S.F. King and G.G. Ross, S.F. King and G.G. Ross, T. Ohlsson ed., S.F. King, f ∂w [3] [4] [1] [2] References Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. Taking the superpotential terms in table than the second term, to e.g., ref. [ predict ( that after subleading higher dimensional operators∂w are included in the flavon superpotential, flat direction for the flavon. of the flavon potential. For similar discussion in only non-Abelian discrete symmetry, see This equation cannot give a self-consistent solution for above. The above equation is explicitly written out as Here, we has ignored the correction to the JHEP11(2018)173 , B ]. ]. ] (1999) ]. B 368 SPIRE 521 SPIRE IN SPIRE IN Phys. Lett. ][ (2015) 123001 Non-Abelian , IN ][ Rept. Prog. (2007) 153 , ][ (2010) 1 G 42 Nucl. Phys. , hep-ph/0607045 ]. (2006) 071 B 644 [ ]. 183 ]. 11 and the modular symmetry Lect. Notes Phys. J. Phys. SPIRE 4 (2018) 128 , Neutrino Mass and Mixing: , SPIRE IN arXiv:1402.4271 A ]. SPIRE hep-ph/0206292 Stringy origin of non-Abelian IN [ ][ [ 10 JHEP hep-ph/0611020 IN ]. (2007) 201 ][ , [ Phys. Lett. ][ , family symmetry and quark-lepton SPIRE ]. 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