The role of flavon cross couplings in leptonic flavour mixing
IIPPP/16/19
The role of flavon cross couplings in leptonic flavour mixing
Silvia Pascoli ∗ and Ye-Ling Zhou , † Institute for Particle Physics Phenomenology, Department of Physics,Durham University, Durham DH1 3LE, United Kingdom Center for High Energy Physics, Peking University, Beijing 100080, China
Abstract
In models with discrete flavour symmetries, flavons are critical to realise specific flavour struc-tures. Leptonic flavour mixing originates from the misalignment of flavon vacuum expectation valueswhich respect different residual symmetries in the charged lepton and neutrino sectors. Flavon crosscouplings are usually forbidden, in order to protect these symmetries. Contrary to this approach,we show that cross couplings can play a key role and give raise to necessary corrections to flavour-mixing patterns, including a non-zero value for the reactor angle and CP violation. For definiteness,we present two models based on A . In the first model, all flavons are assumed to be real or pseudo-real, with 7 real degrees of freedom in the flavon sector in total. A sizable reactor angle associatedwith nearly maximal CP violation is achieved, and, as both originate from the same cross coupling, asum rule results with a precise prediction for the value of the Dirac CP-violating phase. In the secondmodel, the flavons are taken to be complex scalars, which can be connected with supersymmetricmodels and multi-Higgs models. The complexity properties of flavons provide new sources for gener-ating the reactor angle. Models in this new approach introduce very few degrees of freedom beyondthe Standard Model and can be more economical than those in the framework of extra dimension orsupersymmetry. PACS number(s): 14.60.Pq, 11.30.Hv, 12.60.FrKeywords: Lepton flavour mixing, cross couplings, flavour symmetry ∗ E-mail: [email protected] † E-mail: [email protected] a r X i v : . [ h e p - ph ] J un Introduction
Thanks to the discovery of neutrino oscillations [1, 2], leptonic flavour mixing has been observed by aseries of atmospheric [1], solar [2], accelerator [3] and reactor [4] neutrino experiments. The three mixingangles have been measured to a very good accuracy. Both the atmospheric angle θ and the solarangle θ are rather large, and an order 0.1 reactor angle θ has been measured by reactor neutrinoexperiments [5]. The 3 σ ranges of mixing angles from current global analysis of solar, atmospheric,accelerator and reactor neutrino oscillation measurements [6] are given bysin θ ∈ (0 . , . , sin θ ∈ (0 . , . , sin θ ∈ (0 . , . . (1)There is also a preliminary hint [6, 7] for a maximally CP-violating value of the Dirac phase with abest-fit value δ ∼ ◦ by combining the latest T2K [8] and Daya Bay [9] data, but the statisticalsignificance of this result is still low and at 3 σ all possible values of δ are allowed.Motivated by these values of the mixing angles, specific mixing patterns, realised at leading order,have been proposed in the last two decades. Among them, the tri-bimaximal (TBM) mixing predictssin θ = 1 / √ θ = 1 / √
2, which fit well current oscillation data and has therefore attracted alot of attention [10]. However, exact TBM mixing is ruled out due to the prediction of a vanishing θ .To be compatible with current data, corrections to TBM must be introducedsin θ = √ r (cid:48) , sin θ = 1 √ s ) , sin θ = 1 √ a ) (2)with r (cid:48) ∼ . | s | , | a | (cid:46) . Z , and in the neutrino sector, the only choice is Z or Z × Z (cid:48) , if neutrinos are Majorana particles. Torealise TBM, the generators for Z , Z and Z (cid:48) are respectively given by T = ω
00 0 ω , S = 13 − − − , U = , (3)where ω = e i π/ . The most popular and simplest group to realise TBM is the tetrahedral group A ,which is generated by T and S . The residual symmetries Z generated by T and Z generated by S are The explicit expressions of generators are basis-dependent. Here, we show them in the Altarelli-Feruglio basis, whichcan be found in [17]. A breaking, respectively, while the other Z (cid:48) generated by U arises in the neutrino sector accidentally. Many studies have been conducted on howto realise TBM and gain suitable corrections compatible with current oscillation data, for instance, see[19, 20].The flavon fields play a key role in the flavour model construction. To realise different residualsymmetries in the charged lepton and neutrino sectors, we need different flavons, ϕ and φ , and requiretheir VEVs be invariant under the action of T , S , respectively, i.e., T (cid:104) ϕ (cid:105) = (cid:104) ϕ (cid:105) , S (cid:104) φ (cid:105) = (cid:104) φ (cid:105) . (4)The only solution of the above equation takes the following form (cid:104) ϕ (cid:105) = v ϕ , (cid:104) φ (cid:105) = v φ √ . (5)The flavon VEVs are dictated by the minimisation of the potential. The latter will generally containcross couplings between ϕ and φ , which, although allowed by the full symmetry, would violate theresidual ones. This vacuum alignment problem is a general problem of most flavour symmetry groups,not just limited to A models. Extra dimension or supersymmetry is invoked to forbid these crosscouplings. In models with extra dimensions, ϕ and φ can be localised on different branes such that theydo not significantly couple with each other [16]. In models with supersymmetry, a continuous U (1) R symmetry and neutral scalars called driving fields are introduced [17]. The latter take a nontrivialcharge under U (1) R and appear linearly in the superpotential. The minimisation of the flavon potentialis finally simplified to vanishing F -terms of the driving fields. These approaches can solve the flavonVEV alignment problem effectively, but the price is that many degrees of freedom have to be introducedinto the model. There is another solution by extending the flavour group, H , to a larger group N (cid:111) H [21]. Here, N (cid:111) H should admit irreducible representations of H such that the Standard Model leptonsand one flavon can still transform in H , while the other flavon transforms as a different representationthat belongs to N (cid:111) H but not to H .Thanks to the flavon VEVs and the preservation of the two discrete symmetries in the chargedlepton and neutrino sectors, TBM can be generated. However, due to the measurement of the reactorangle, the residual symmetries should be broken and corrections to TBM are needed in order to ren-der models compatible with experimental data. In most models in the literature, this is realised byintroducing higher dimensional operators, which may appear in both the flavon potential (especially thesuperpotential of flavons and driving fields in supersymmetric models) and couplings between leptonsand flavons [15]. These operators involve a certain scale Λ higher than the scale of flavour symmetrycharacterised by v ϕ and v φ . They give rise to corrections to the mixing angles, e.g., r (cid:48) ∼ v ϕ / Λ, implyingthat the new physics scale Λ should not be far above the scale of flavour symmetries.In this paper, we will develop a new approach. Differently from above where the cross couplingsare forbidden, we will allow their existence and investigate how they break the Z and Z residualsymmetries. Similar ideas have been mentioned in Ref. [20], but a detailed discussion of how thevacuum is corrected by these terms and how the flavour mixing is affected is lacking. For definiteness,our flavour symmetry is assumed to be A . To correct TBM in agreement with the experimentallyallowed region, cross couplings should be small, of order O (0 . ϕ /v φ will be derived. As a consequence, corrections of mixing angles to those in TBM are characterisedby v ϕ /v φ and the cross-coupling coefficients.The rest of this paper is organised as follows. In section 2, we discuss the relation between the flavonpotential and flavon VEVs in A . We first point out how the size of coefficients in the flavon potentialdetermines the Z - and Z -symmetric VEVs, and then derive the corrections to these VEVs from crosscouplings. Schematically, we present two flavour models in section 3. In Model I, we introduce only two A pseudo-real triplet flavons ϕ, φ , one A singlet flavon η and an A triplet right-handed neutrino N .This model is very economical since only 7 real degrees of freedom in the flavon sector are introduced.In Model II, we extend the flavons to complex fields and see how the mixing structure deviates fromthe pseudo-real flavon case. This extension is interesting since, if we wanted to draw a connection withsupersymmetric and multi-Higgs models, flavons must be complex. We summarise the results in section4. In appendix A, we list the full solutions of VEVs for a single A triplet flavon. We assume the flavour symmetry to be the tetrahedral group A [22], the group of even permutationsof four objects. It is generated by S and T with the requirement S = T = ( ST ) = 1, and contains12 elements: T , ST , T S , ST S , T , ST , T S , T ST , S , T ST , T ST and the identity element 1.It is the smallest discrete group which has a 3-dimensional (3d) irreducible representation and inthis representation, the generators S and T can be given as in Eq. (3), respectively. Besides, it hasthree 1-dimensional irreducible representations: the trivial singlet and non-trivial singlets (cid:48) , (cid:48)(cid:48) . TheKronecker product of two 3d irreducible representations can be reduced: × = + (cid:48) + (cid:48)(cid:48) + S + A ,where the subscripts S and A stand for the symmetric and anti-symmetric parts, respectively.We introduce a flavon field ϕ = ( ϕ , ϕ , ϕ ) T . It contains three gauge-singlet scalars and transformsas a pseudo-real triplet representation of A which requires ϕ ∗ = ϕ and ϕ ∗ = ϕ . The renormalisableflavon potential invariant under A is generically written as V ( ϕ ) = 12 µ ϕ ( ϕϕ ) + 14 (cid:104) f (cid:0) ( ϕϕ ) (cid:1) + f ( ϕϕ ) (cid:48) ( ϕϕ ) (cid:48)(cid:48) + f (cid:0) ( ϕϕ ) S ( ϕϕ ) S (cid:1) (cid:105) , (6)where all the coefficients µ ϕ and f , , are real. The conditions µ ϕ < f + f > f + f > V ( ϕ ), please see Eq. (55) in appendix A. To simplify our discussion, tri-linear terms suchas (cid:0) ( ϕϕ ) S ϕ (cid:1) are not considered here. These terms can be forbidden by an additional Z symmetry( ϕ → − ϕ ). There are two classes of configurations that are candidates for the vacuum of ϕ . They arecharacterised by (cid:104) ϕ (cid:105) = v ϕ , (cid:104) ϕ (cid:105) = v ϕ √ , (7)where v ϕ = − µ ϕ f + f , v ϕ = − µ ϕ f + f . (8)The potential V ( ϕ ) takes extremal values V ( (cid:104) ϕ (cid:105) ) = − µ ϕ f + f ) , V ( (cid:104) ϕ (cid:105) ) = − µ ϕ f + f ) , (9)4t (cid:104) ϕ (cid:105) and (cid:104) ϕ (cid:105) , respectively, which depend on the relative size of f and f . In the case f > f , V ( ϕ )has the global minimum value at (cid:104) ϕ (cid:105) , and (cid:104) ϕ (cid:105) is just an unstable saddle point. Thus, the vacuum of ϕ is (cid:104) ϕ (cid:105) . In the opposite case f < f , (cid:104) ϕ (cid:105) is the vacuum. For a detailed discussion of the determinationof the vacuum of the potential V ( ϕ ), see appendix A. From here onward, we assume f > f such thatthe VEV of ϕ is fixed at (cid:104) ϕ (cid:105) at leading order.Then, we consider another A pseudo-real triplet scalar φ = ( φ , φ , φ ) T . Its potential V ( φ ) takesthe same form as V ( ϕ ) with coefficients µ ϕ and f i replaced by µ φ and g i , respectively. All the resultsabove for ϕ will apply with the substitutions ϕ → φ and f i → g i . In order to select the other vacuumalignment, we assume g < g , so that V ( φ ) takes the global minimum value at (cid:104) φ (cid:105) .With the assumptions f > f and g < g , we obtain the VEVs of ϕ and φ at (cid:104) ϕ (cid:105) and (cid:104) φ (cid:105) ,respectively, as in Eq. (5). They respect the Z and Z residual symmetries, respectively, and reproducethe TBM mixing pattern.Cross couplings between ϕ and φ would modify the VEVs of ϕ and φ . The most general Lagrangiandescribing flavon cross couplings is given by V ( ϕ, φ ) = 12 (cid:15) ( ϕϕ ) ( φφ ) + 14 (cid:2) (cid:15) ( ϕϕ ) (cid:48)(cid:48) ( φφ ) (cid:48) + h.c. (cid:3) + 12 (cid:15) (cid:0) ( ϕϕ ) S ( φφ ) S (cid:1) , (10)where (cid:15) , are real and (cid:15) is the only complex parameter in the flavon potential. By including thesecouplings, we get the full renormalisable flavon potential invariant under the flavour symmetry A × Z ,i.e., V = V ( ϕ ) + V ( φ ) + V ( ϕ, φ ). The cross couplings will break the residual symmetries. To achieveorder O (0 .
1) corrections to the TBM mixing, we assume the coefficients (cid:15) i to be of the same order.In this case, modifications of the flavon VEVs are small and the residual Z and Z symmetries arepreserved at leading order.As the cross couplings are assumed to be small, we can proceed to compute analytically the correc-tions to the leading terms. Expanding the VEVs as (cid:104) ϕ (cid:105) = v ϕ + δv ϕ , (cid:104) ϕ (cid:105) = δv ϕ , (cid:104) ϕ (cid:105) = δv ϕ , (cid:104) φ (cid:105) = v φ √ δv φ , (cid:104) φ (cid:105) = v φ √ δv φ , (cid:104) φ (cid:105) = v φ √ δv φ , (11)we retain the quadratic terms of V ( ϕ ) and V ( φ ) and the linear terms of V ( ϕ, φ ), which are the onlyones relevant to the vacuum shifts at first order: V (2) ( ϕ ) = 12 m ϕ δv ϕ + m ϕ | δv ϕ | ,V (2) ( φ ) = 16 ( m φ + 2 m φ ) (cid:2) δv φ + 2 | δv φ | (cid:3) + 16 ( m φ − m φ ) (cid:0) δv φ + 2 δv φ δv ∗ φ + h.c. (cid:1) ,V (1) ( ϕ, φ ) = (cid:15) v φ v ϕ δv ϕ + (cid:15) v ϕ v φ √ δv φ + δv φ + δv ∗ φ )+ 12 v φ v ϕ ( (cid:15) δv ∗ ϕ + (cid:15) ∗ δv ϕ ) + 12 (cid:15) v ϕ v φ √ δv φ − δv φ − δv ∗ φ ) , (12)where m ϕ = 2( f + f ) v ϕ , m ϕ = ( f − f ) v ϕ ,m φ = 2( g + g ) v φ , m φ = 32 ( g − g ) v φ . (13)One can check that ϕ and ϕ are actually the mass eigenstates of ϕ after A breaking to Z , withmass eigenvalues m ϕ and m ϕ , respectively. However, φ and φ are not mass eigenstates of φ . By5iagonalising the mass matrix of φ , we derive the mass eigenvalues m φ and m φ . We minimise thepotential in Eq. (12) and derive the modified VEVs of ϕ and φ to be (cid:104) ϕ (cid:105) ≈ (cid:15) ϕ (cid:15) ∗ ϕ v ϕ , (cid:104) φ (cid:105) ≈ − (cid:15) φ (cid:15) φ (cid:15) φ v φ √ , (14)where v ϕ = − µ ϕ f + f , v φ = − µ φ g + g ,(cid:15) ϕ = − (cid:15) v φ m ϕ , (cid:15) φ = (cid:15) v ϕ m φ . (15)Here, we have redefined the effective µ ϕ and µ φ via µ ϕ + (cid:15) v φ → µ ϕ and µ φ + (cid:15) v ϕ → µ φ to absorb (cid:15) ,since the (cid:15) term is a trivial correction keeping the residual symmetries unchanged no matter what kindof VEVs ϕ and φ achieve. The (cid:15) term is the main source of the breaking of the Z symmetry in (cid:104) ϕ (cid:105) ,since (cid:104) ( φφ ) (cid:48)(cid:48) (cid:105) in this term approximates to v φ , which is not invariant under the action of T . (cid:104) ( φφ ) S (cid:105) in the (cid:15) term vanishes and will not break the Z symmetry at leading order. The (cid:15) term is the mainsource of breaking Z in (cid:104) φ (cid:105) . Similarly, the reason is that (cid:104) ( ϕϕ ) S (cid:105) ≈ (1 , , T v ϕ / S , while (cid:104) ( ϕϕ ) (cid:48)(cid:48) (cid:105) in the (cid:15) term vanishes at leading order.The effective complex parameter (cid:15) ϕ , which measures the amount of Z breaking, is a crucial param-eter in our discussion. We parameterize it as (cid:15) ϕ = | (cid:15) ϕ | e iθ ϕ , in which − ◦ < θ ϕ (cid:54) ◦ . In the nextsection, we will construct two lepton flavour models with lepton mixing parameters corrected by (cid:15) ϕ . In this section, we will construct two leptonic flavour models, introducing gauge singlets N and imple-menting the seesaw mechanism. In both models, we assume the flavour symmetry to be A × Z × Z .The additional Z is imposed to forbid unnecessary couplings between flavons and leptons. We intro-duce only two A triplet flavons ϕ and φ and one singlet flavon η . The singlet η is used to give suitableneutrino mass spectra. We will exploit the presence of cross couplings between ϕ and φ to realise leptonflavour mixing compatible with the constraints of neutrino oscillation experiments. The main differenceof these two models is that ϕ and φ in Model I transform as pseudo-real triplets of A , and those inModel II are complex triplets of A . Although the trivial singlet flavon η is real in Model I and complexin Model II, it does not result in any different mixing structure in the two models. In model I, both ϕ and φ are pseudo-real, and η is a real singlet. This model is very economical as only7 real degrees of freedom in the flavon sector are introduced. Transformation properties of ϕ , φ and η under A × Z × Z , together with those of the Higgs H , leptons in the Standard Model (cid:96) L , e R , µ R , τ R and an extra right-handed neutrino N are shown in Table 1.As discussed in the last section, the flavon fields ϕ and φ get VEVs as shown in Eq. (14). Crosscouplings of ϕ , φ with H and the new flavon singlet η will modify the VEVs of ϕ and φ . Since H and η are flavour singlets , their cross couplings with ϕ and φ can only modify the overall sizes of (cid:104) ϕ (cid:105) and6ields (cid:96) L e R , µ R , τ R N H ϕ φ ηA , (cid:48) , (cid:48)(cid:48) Z − − Z i i i − − A × Z × Z . (cid:104) φ (cid:105) , i.e., v ϕ and v φ , but have no influence on the direction (1 , (cid:15) ϕ , (cid:15) ∗ ϕ ) T and (1 − (cid:15) φ , (cid:15) φ , (cid:15) φ ) T .The effect of H and η can be reabsorbed in the redefinition of µ ϕ and µ φ and, having done so, theexpressions of v ϕ , v φ , and (cid:15) ϕ , (cid:15) φ in Eq. (15) remain valid.The Lagrangian terms for generating lepton masses are given by − L l = y e Λ ( (cid:96) L ϕ ) e R H + y µ Λ ( (cid:96) L ϕ ) (cid:48)(cid:48) µ R H + y τ Λ ( (cid:96) L ϕ ) (cid:48) τ R H + h.c. + · · · , −L ν = y D ( (cid:96) L N ) ˜ H + y (cid:0) ( N c N ) S φ (cid:1) + y N c N ) η + h.c. + · · · , (16)where the dots stand for higher dimensional operators. Note that in many models, higher dimensionaloperators are important because they are responsible for corrections to leading order mixing structure.For example, the dimension-6 operator (cid:0) ( N c N ) S (( ϕϕ ) S φ ) S (cid:1) / Λ will modify flavour mixing from theleading order structure since the vacuum direction of the combined term (cid:104) (( ϕϕ ) S φ ) S (cid:105) ∼ (2 , − , − T does not preserve the Z symmetry. Compared with these models, our model assumes that higherdimensional operators are negligible such that they cannot lead to significant modifications.Leptons gain masses with specific mass matrix structures, after breakings of the flavour symmetryand the electroweak symmetry. The resulting charged lepton mass matrix can be written as M l ≈ y e y µ (cid:15) ∗ ϕ y τ (cid:15) ϕ y e (cid:15) ϕ y µ y τ (cid:15) ∗ ϕ y e (cid:15) ∗ ϕ y µ (cid:15) ϕ y τ vv ϕ √ , (17)where (cid:104) H (cid:105) = v/ √ v = 246 GeV being the VEV of the Higgs boson. The Dirac and Majoranamass matrices for neutrinos are given by M D = y D v √ ,M N ≈ √ √ y v η + y v φ (1 − (cid:15) φ ) − y v φ (1 + (cid:15) φ ) − y v φ (1 + (cid:15) φ ) − y v φ (1 + (cid:15) φ ) y v φ (1 + (cid:15) φ ) √ y v η − y v φ (1 − (cid:15) φ ) − y v φ (1 + (cid:15) φ ) √ y v η − y v φ (1 − (cid:15) φ ) y v φ (1 + (cid:15) φ ) , (18)where v η the VEV of the singlet η , and the active neutrinos obtain masses through the seesaw mechanism M ν = − M D M − N M TD . (19)After diagonalising M l and M ν , we compute the masses of the charged leptons and neutrinos m e ≈ | y e v ϕ | v √ , m µ ≈ | y µ v ϕ | v √ , m τ ≈ | y τ v ϕ | v √ ,m ≈ | y D | v |√ y v φ + 2 y v η | , m ≈ | y D | v | y v η | , m ≈ | y D | v |√ y v φ − y v η | . (20)7he PMNS matrix is obtained by the product of the unitary matrices which diagonalise the chargedlepton and neutrino mass matrices and is given by U PMNS ≈ − (cid:15) ∗ ϕ − (cid:15) ϕ (cid:15) ϕ − (cid:15) ∗ ϕ (cid:15) ∗ ϕ (cid:15) ϕ √ √ − √ √ − √ − √ √ √ √ (cid:15) φ −√ (cid:15) φ P ν . (21)The LHS of the PMNS matrix is the correction from the charged lepton sector, the middle is the TBMmixing, and the RHS is the correction from the neutrino sector. The mixing angles can be expressed interms of the model parameters θ ϕ , | (cid:15) ϕ | and (cid:15) φ assin θ ≈ √ | (cid:15) ϕ sin θ ϕ | , sin θ ≈ √ (cid:0) − | (cid:15) ϕ | cos θ ϕ + 2 (cid:15) φ (cid:1) , sin θ ≈ √ (cid:0) | (cid:15) ϕ | cos θ ϕ (cid:1) , (22)and the CP-violating phases are approximately given by δ ≈ (cid:40) ◦ − | (cid:15) ϕ | sin θ ϕ , θ ϕ > , ◦ − | (cid:15) ϕ | sin θ ϕ , θ ϕ < ,α ≈ Arg (cid:40) √ y v φ y v η (cid:41) ,α ≈ Arg (cid:40)(cid:104) √ y v φ + 2 y v η √ y v φ − y v η (cid:105)(cid:104) − i | (cid:15) ϕ | sin θ ϕ (cid:105)(cid:41) , (23)where the Majorana phases α and α are defined in Ref. [5]. If there are no cross couplings between ϕ and φ , we obtain an explicit TBM mixing, which predicts sin θ = 0, sin θ = 1 / √ θ = 1 / √ v ϕ / Λ, the corrections here are functions of the ratio ofthe two flavon VEVs v φ /v ϕ , as well as the coefficients (cid:15) and (cid:15) .We can also express the corrections in terms of the parameters r (cid:48) , s and a introduced earlier as r (cid:48) = | (cid:15) ϕ sin θ ϕ | , s = − | (cid:15) ϕ | cos θ ϕ + 2 (cid:15) φ , a = | (cid:15) ϕ | cos θ ϕ . (24)A sum rule for the corrections to θ and θ r (cid:48) + a = | (cid:15) ϕ | (25)is obtained. From this relation in Eq. (25), we can deduce that the value of | (cid:15) ϕ | is around 0.1-0.2,consistently with the initial assumptions made on this parameter. The non-zero reactor angle θ arisesfrom the imaginary part of (cid:15) ϕ , i.e., the cross coupling Im( (cid:15) )( ϕϕ ) (cid:48)(cid:48) ( φφ ) (cid:48) . To be compatible withthe measured value of sin θ ≈ .
02 [6], the effective parameter | (cid:15) ϕ | sin θ ϕ should be around ± . θ ϕ cannot be too small. The other two parameters | (cid:15) ϕ | cos θ ϕ and (cid:15) φ can take a value (cid:46) .
1, depending on the precise measured values of θ and θ . We show the numerical results for thecorrelation between θ and θ and the allowed parameter space of | (cid:15) ϕ | vs (cid:15) φ of Model I in Fig. 1. Here, | (cid:15) ϕ | , ϑ ϕ , and (cid:15) φ are randomly generated in the range [0 , . , ◦ ) and [ − . , . σ range data of the mixing angles in Eq. (1) are shown in the figure. We seethat 0 . (cid:46) | (cid:15) ϕ | (cid:46) .
17 and − . (cid:46) (cid:15) φ (cid:46) .
17 are required. In the region where (cid:15) φ has a large deviationfrom 0, the correlation | (cid:15) φ | ∝ | (cid:15) ϕ | roughly holds, which is required by the constraint of θ , as shown inEq. (22). The phase of (cid:15) ϕ is given by | tan θ ϕ | = r (cid:48) | a | = sin θ | θ − √ | . (26)Taking the 3 σ ranges of θ and θ in Eq. (1) into account, we get 38 ◦ < | θ ϕ | < ◦ .The Dirac CP-violating phase has a small deviation from the value relative to maximal CP violation: δ ≈ (cid:40) ◦ − √ θ , θ ϕ > , ◦ + √ θ , θ ϕ < . (27)The connection between δ and θ can be expected since they both result from the same cross couplingIm( (cid:15) )( ϕϕ ) (cid:48)(cid:48) ( φφ ) (cid:48) . Taking the best-fit value of θ , we predict δ = 258 ◦ or 102 ◦ , which is very close tothe current best-fit value in the inverted mass ordering. The Majorana CP-violating phases take valuestotally independent of the other mixing parameters. They can be arbitrary, depending on the relativephase of y /y .Figure 1: Theoretical prediction of mixing angles (left panel) and the allowed parameter space of | (cid:15) ϕ | and (cid:15) φ (right panel) in Model I. The 3 σ range data of mixing angles in Ref. [6] have been used as thecut. The straight line in the left panel stands for the correlation of θ vs θ in the limit (cid:15) φ = 0 withthe expression given in Eq. (28).This model can be further simplified in the limit (cid:15) φ →
0. Due to the fact that the ratio (cid:15) ϕ /(cid:15) φ isproportional to v φ /v ϕ , a small hierarchy between v φ and v ϕ would result in a large hierarchy between (cid:15) ϕ and (cid:15) φ . Therefore, if v φ is significantly larger than v ϕ , (cid:15) φ will be much smaller than (cid:15) ϕ , and itscontribution to the mixing angles can be neglected. In this case, all deviations from the tri-bimaximalmixing pattern come from the corrections from the charged lepton sector. A simple relation between θ and θ is derived 3 sin θ + 4 sin θ = 3 , (28)or equivalently, s + 2 a = 0. Taking the 3 σ range of θ to the sum rule in Eq. (28) predicts 0 . < sin θ < . θ being preferred. Finally, let us discuss the constraint on (cid:15) ϕ
9n this simplified case. | (cid:15) ϕ | must be in general in the range from 0.1 to 0.12, as shown in Fig. 1. Thephase of (cid:15) ϕ can be further constrained by the mixing angle θ : | tan θ ϕ | = 2 r (cid:48) | s | = √ θ |√ θ − | . (29)Using the 3 σ ranges of θ and θ , θ ϕ is constrained to be 63 ◦ < | θ ϕ | < ◦ . This leads to | sin θ ϕ | ≈ θ ≈ √ | (cid:15) ϕ | (30)with | (cid:15) ϕ | ≈ . θ =0 .
304 and sin θ = 0 . θ = 0 .
52 in the second octant, with θ ϕ (cid:39) ± ◦ and | (cid:15) ϕ | (cid:39) . In Model I, we have used flavon cross couplings to obtain flavour mixing deviations from the TBM mixing.Connections between corrections of mixing angles and the ratio of flavon VEVs, i.e., δθ ij = f ( v ϕ /v φ ),have been derived. Here, we would like to extend the flavons ϕ and φ from pseudo-real scalars tocomplex scalars. In other words, ϕ is not real and ϕ is not conjugate with ϕ , and the same holdsfor φ . The case of complex flavons is widely used in flavour model building. One example is that invarious supersymmetric flavour models, flavons have to be complex to be consistent with supersymmetry.Another example is that, in some multi-Higgs models [23, 24], the flavon coupling to gauge doublet (cid:96) L is formed by three Higgs fields, which are gauge doublets and thus have to be complex.In the following, we assume ϕ and φ to be neutral complex scalars and transform as triplets of A ,taking the same charges as in Table 1. Note that, in the basis in Eq. (3), as ϕ = ( ϕ , ϕ , ϕ ) is a triplet,˜ ϕ = ( ϕ ∗ , ϕ ∗ , ϕ ∗ ) is also a triplet. To simplify our discussion, we require ϕ and ˜ ϕ to appear in pairs ofthe combination ˜ ϕϕ , and the same applies for φ and ˜ φ . This is easy to be achieved by imposing anadditional U (1) symmetry with ϕ and φ taking different charges q ϕ and q φ , respectively. On the otherhand, if ϕ is a gauge doublet, ϕ and ˜ ϕ always appears in pairs, and no additional symmetry is needed.Also in this case we allow cross couplings in the potential and we study how the VEVs are modifiedcompared to the case of no cross couplings. The most general potential of ϕ is altered to [23] V ( ϕ ) = 12 µ ϕ ( ˜ ϕϕ ) + 14 (cid:104) f (cid:0) ( ˜ ϕϕ ) (cid:1) + f ( ˜ ϕϕ ) (cid:48) ( ˜ ϕϕ ) (cid:48)(cid:48) + f (cid:0) ( ˜ ϕϕ ) S ( ˜ ϕϕ ) S (cid:1) + f (cid:0) ( ˜ ϕϕ ) A ( ˜ ϕϕ ) A (cid:1) + f (cid:0) ( ˜ ϕϕ ) S ( ˜ ϕϕ ) A (cid:1) (cid:105) , (31)where f i are all real couplings. The additional f and f terms originate from the complex propertyof ϕ . Replacing ϕ → φ and the coefficients f i → g i , we obtain the potential for φ , V ( φ ). The crosscouplings between ϕ and φ are modified to V ( ϕ, φ ) = 12 (cid:15) ( ˜ ϕϕ ) ( ˜ φφ ) + 14 (cid:104) (cid:15) ( ˜ ϕϕ ) (cid:48)(cid:48) ( ˜ φφ ) (cid:48) + h.c. (cid:105) + 12 (cid:104) (cid:15) (cid:0) ( ˜ ϕϕ ) S ( ˜ φφ ) S (cid:1) + (cid:15) (cid:0) ( ˜ ϕϕ ) A ( ˜ φφ ) A (cid:1) + (cid:15) (cid:0) ( ˜ ϕϕ ) S ( ˜ φφ ) A (cid:1) + (cid:15) (cid:0) ( ˜ ϕϕ ) A ( ˜ φφ ) S (cid:1) (cid:105) , (32)in which (cid:15) is complex and (cid:15) , (cid:15) , (cid:15) , (cid:15) and (cid:15) are real. If ϕ is a gauge doublet, the following discussion applies to its neutral components. ϕ and φ perturbatively. Wefirst consider the VEV of ϕ without cross couplings. (cid:104) ϕ (cid:105) = (1 , , T v ϕ and (cid:104) φ (cid:105) = (1 , , T v φ / √ V ( ϕ ) at (cid:104) ϕ (cid:105) takes the same value as V ( (cid:104) ϕ (cid:105) ) in Eq. (8). We expandthe VEVs of ϕ and φ as in Eq. (11) and write out the quadratic terms of V ( ϕ ), V ( φ ) and the linearterms of V ( ϕ, φ ) as V (2) ( ϕ ) = 12 m ϕ [Re( δv ϕ )] + 12 (cid:16) δv ∗ ϕ δv ϕ (cid:17) O ϕ (cid:32) m ϕ m ϕ (cid:33) O Tϕ (cid:32) δv ϕ δv ∗ ϕ (cid:33) ,V (2) ( φ ) = 12 (cid:16) Re( δv φ ) Re( δv φ ) Re( δv φ ) (cid:17) O φ m φ m φ
00 0 m φ O Tφ Re( δv φ )Re( δv φ )Re( δv φ ) + 12 (cid:16) Im( δv φ ) Im( δv φ ) Im( δv φ ) (cid:17) O φ m φ
00 0 m φ O Tφ Im( δv φ )Im( δv φ )Im( δv φ ) ,V (1) ( ϕ, φ ) = ( (cid:15) terms) + (cid:0) (cid:15) v φ v ϕ ( δv ∗ ϕ + δv ϕ ) + h.c. (cid:1) + (cid:15) v ϕ v ϕ √ (cid:0) δv φ ) − Re( δv φ ) − Re( δv φ ) (cid:1) + (cid:15) v ϕ v ϕ √ (cid:0) Re( δv φ ) − Re( δv φ ) (cid:1) , (33)where O ϕ = (cid:32) cos ϑ sin ϑ − sin ϑ cos ϑ (cid:33) , O φ = U TBM θ sin θ − sin θ cos θ ,m ϕ = 2( f + f ) v ϕ , m ϕ , m ϕ = (cid:18) f − f + f ∓ (cid:113) (2 f + f − f ) + 4 f (cid:19) v ϕ ,m φ = 2( g + g ) v φ , m φ , m φ = (cid:18) − g + 3 g + g ∓ (cid:113) (3 g − g ) + 3 g (cid:19) v φ , (34)and the flavon mixing parameters ϑ and θ are respectively given bytan 2 ϑ = 2 f + f − f f , tan 2 θ = √ g g − g , (35)with m ϕ (cid:54) m ϕ , m φ (cid:54) m φ and − ◦ < ϑ, θ (cid:54) ◦ required. Here, m ϕ , m ϕ , m ϕ and m φ , m φ , m φ are mass eigenvalues of ϕ and φ after A is broken to Z and Z , respectively. Note that ϕ and ϕ are not mass eigenstates any more since there is a mixing between ϕ and ϕ ∗ , with the mixing anglecharacterised by ϑ . In most cases, the mixing between ϕ and ϕ ∗ is sizable and cannot be neglected,except in the limit 2 f + f − f = 0. It becomes maximal in the limit f = 0. When both conditions2 f + f − f = 0 and f = 0 are satisfied, the two mass eigenvalues are degenerate, being equal to m ϕ inModel I. As for φ , its real components and imaginary components mix separately after A breaks to Z .The real components gain masses with eigenvalues m φ , m φ and m φ , while the imaginary componentsgain masses with eigenvalues 0, m φ and m φ . m φ and m φ can be degenerate only if both conditions3 g = g and g = 0 hold. One necessary condition for the stable vacuum is that mass squares of all11assive scalars are positive, which leads to f + f > , f − f + f > , f − f )( f − f ) − f > ,g + g > , g − g + g > , g − g )(3 g − g ) − g > . (36)The terms in the cross couplings which can shift the direction of the VEV at first order are only thoserelated to (cid:15) , (cid:15) and (cid:15) , and in Eq. (33) we list the linear terms for them. The (cid:15) terms can only giveoverall corrections to v ϕ and v φ and can be absorbed in a redefinition of µ ϕ and µ φ by performing thesame procedure as described in section 2.We minimise the potential in Eq. (33) and directly obtain the following corrections: (cid:32) δv ϕ δv ∗ ϕ (cid:33) ≈ − v φ v ϕ O ϕ (cid:32) m − ϕ m − ϕ (cid:33) O Tϕ (cid:32) (cid:15) (cid:15) (cid:33) , Re( δv ϕ )Re( δv ϕ )Re( δv ϕ ) ≈ − v ϕ v ϕ √ O φ m − φ m − φ
00 0 m − φ O Tφ (cid:15) − (cid:15) − (cid:15) − (cid:15) + (cid:15) . (37)Then, the vacuum shifts for ϕ and φ can be expressed as (cid:104) ϕ (cid:105) ≈ − κ ϕ ) (cid:15) ϕ (1 + κ ϕ ) (cid:15) ∗ ϕ v ϕ , (cid:104) φ (cid:105) ≈ − (cid:15) φ − κ φ ) (cid:15) φ κ φ ) (cid:15) φ v φ √ , (38)respectively with (cid:15) ϕ = − (cid:15) v φ (cid:2) ( m ϕ + m ϕ ) − ( m ϕ − m ϕ ) sin 2 ϑ (cid:3) m ϕ m ϕ ,κ ϕ = ( m ϕ − m ϕ ) cos 2 ϑ ( m ϕ + m ϕ ) − ( m ϕ − m ϕ ) sin 2 ϑ ,(cid:15) φ = v ϕ (cid:2) (cid:15) ( m φ + m φ ) + ( m φ − m φ )( (cid:15) cos 2 θ − (cid:15) √ sin 2 θ ) (cid:3) m φ m φ ,κ φ = ( m φ − m φ )( √ (cid:15) sin 2 θ − (cid:15) cos 2 θ ) − (cid:15) ( m φ + m φ ) (cid:15) ( m φ + m φ ) + ( m φ − m φ )( (cid:15) cos 2 θ − (cid:15) √ sin 2 θ ) . (39)These expressions reflect the complex properties of ϕ and φ . When the mixing between ϕ and ϕ ∗ ismaximal (sin 2 ϑ = ±
1, corresponding to f = 0), κ ϕ vanishes, and we get the (cid:104) ϕ (cid:105) shift similar to thatin Model I. Furthermore, (cid:15) ϕ takes the value − (cid:15) v φ / (2 m ϕ ) and − (cid:15) v φ / (2 m ϕ ) for ϑ = 45 ◦ and − ◦ ,respectively. In the limit sin 2 θ = 0 (corresponding to g = 0), κ φ vanishes, and we also get the (cid:104) φ (cid:105) shiftsimilar to that in Model I, with (cid:15) φ = (cid:15) v ϕ / (2 m φ ) for θ = 0 and (cid:15) φ = (cid:15) v ϕ / (2 m φ ) for θ = 90 ◦ .Let us consider the corrections inducing to the TBM pattern. In A × Z × Z , all fields are assumedto take the same charges as in Model I. With a suitable arrangement of the U (1) charge for these fields,couplings such as ( (cid:96) L ˜ ϕ ) e R H and (cid:0) ( N c N ) S ˜ φ (cid:1) can be forbidden. Eventually, we can obtain the same If ϕ is formed by three Higgses, the squares of the charged Higgs masses are given by m ϕ +2 , m ϕ +3 = ( − f ∓ f ) v ϕ / f < f < f < − f should be imposed. . After the scalars get VEVs, the charged lepton and right-handed neutrinomass matrices are given by M l ≈ y e y µ (1 + κ ϕ ) (cid:15) ∗ ϕ y τ (1 − κ ϕ ) (cid:15) ϕ y e (1 − κ ϕ ) (cid:15) ϕ y µ y τ (1 + κ ϕ ) (cid:15) ∗ ϕ y e (1 + κ ϕ ) (cid:15) ∗ ϕ y µ (1 − κ ϕ ) (cid:15) ϕ y τ vv ϕ √ ,M N ≈ √ √ y v η + y v φ [1 − (cid:15) φ ] − y v φ [1 + (1 + κ φ ) (cid:15) φ ] − y v φ [1 + (1 − κ φ ) (cid:15) φ ] − y v φ [1 + (1 + κ φ ) (cid:15) φ ] y v φ [1 + (1 − κ φ ) (cid:15) φ ] √ y v η − y v φ [1 − (cid:15) φ ] − y v φ [1 + (1 − κ φ ) (cid:15) φ ] √ y v η − y v φ [1 − (cid:15) φ ] y v φ [1 + (1 + κ φ ) (cid:15) φ ] , (40)and the Dirac neutrino mass matrix takes the same form as in Eq. (18). The lepton mass eigenvaluesare the same as those in Model I. The PMNS matrix is given by U PMNS ≈ − (1 + κ ϕ ) (cid:15) ∗ ϕ − (1 − κ ϕ ) (cid:15) ϕ (1 + κ ϕ ) (cid:15) ϕ − (1 + κ ϕ ) (cid:15) ∗ ϕ (1 − κ ϕ ) (cid:15) ∗ ϕ (1 + κ ϕ ) (cid:15) ϕ √ √ − √ √ − √ − √ √ √ √ (cid:15) φ −√ (cid:15) φ √ κ (cid:48) φ (cid:15) φ −√ κ (cid:48) φ (cid:15) φ P ν , (41)where κ (cid:48) φ = κ φ y v φ y v φ − √ y v η . (42)Eventually, we obtain approximate expressions for the mixing anglessin θ ≈ (cid:113) | (cid:15) ϕ | sin θ ϕ + 2( κ ϕ | (cid:15) ϕ | cos θ ϕ + κ (cid:48) φ (cid:15) φ ) , sin θ ≈ √ (cid:2) − | (cid:15) ϕ | cos θ ϕ + 2 (cid:15) φ (cid:3) , sin θ ≈ √ (cid:2) κ ϕ ) | (cid:15) ϕ | cos θ ϕ − κ (cid:48) φ (cid:15) φ (cid:3) , (43)and the CP-violating phases δ ≈ Arg (cid:110)(cid:104) i | (cid:15) ϕ | sin θ ϕ + κ ϕ | (cid:15) ϕ | cos θ ϕ + κ (cid:48) φ (cid:15) φ (cid:105)(cid:104) − i (2 + κ ϕ ) | (cid:15) ϕ | sin θ ϕ (cid:105)(cid:111) ,α ≈ Arg (cid:40)(cid:104) √ y v φ y v η (cid:105)(cid:104) − iκ ϕ | (cid:15) ϕ | sin θ ϕ (cid:105)(cid:41) ,α ≈ Arg (cid:40)(cid:104) √ y v φ + 2 y v η √ y v φ − y v η (cid:105)(cid:104) − i | (cid:15) ϕ | sin θ ϕ (cid:105)(cid:41) . (44)Compared with Model I, two additional parameters κ ϕ and κ φ , which stand for the asymmetric correc-tions between the second and third components of the flavon VEVs, come into the game. They resultin some different features compared to Model I. Here we discuss two such possibilities: • Case A: All coefficients in the potential V ( ϕ, φ ) are real, i.e., (cid:15) ϕ being real. In this case, there willbe no Dirac-type CP violation, and the corrections to the mixing angles are simplified to r (cid:48) = | κ ϕ (cid:15) ϕ + κ (cid:48) φ (cid:15) φ | , s = 2 (cid:15) φ − (cid:15) ϕ , a = (1 + κ ϕ ) (cid:15) ϕ − κ (cid:48) φ (cid:15) φ . (45) If ϕ is a gauge doublet, we can construct a renormalisable model with terms generating charged lepton masses such as −L l = y e ( (cid:96) L ϕ ) e R + y µ ( (cid:96) L ϕ ) (cid:48)(cid:48) µ R + y τ ( (cid:96) L ϕ ) (cid:48) τ R + h.c. . (cid:15) ϕ , (cid:15) φ and κ ϕ , κ φ (top right and bottom panels) in Case A, Model II. The 3 σ ranges ofmixing angles in Ref. [6] are used as inputs. (cid:15) ϕ is assumed to be real. (cid:15) ϕ , (cid:15) φ and κ ϕ , κ φ are samplesrandomly generated in the bounds [ − . , .
2] and [ − , r (cid:48) , the reactor mixing angle depends on two additional parameters,the asymmetric shifts of the VEVs (cid:104) ϕ (cid:105) and (cid:104) φ (cid:105) , characterised by κ ϕ and κ (cid:48) φ , respectively. Each canbe related to the asymmetric coupling f (cid:0) ( ˜ ϕϕ ) S ( ˜ ϕϕ ) A (cid:1) or g (cid:0) ( ˜ φφ ) S ( ˜ φφ ) A (cid:1) , see Eqs. (35)and (39). These couplings also contribute to the correction to θ , but not to θ . In Fig. 2, weshow the prediction of θ vs θ and the allowed parameter spaces of (cid:15) ϕ vs (cid:15) φ and κ ϕ vs κ φ . Wetake the 3 σ allowed ranges of mixing angles in Ref. [6] as inputs and treat (cid:15) ϕ and (cid:15) φ as randomnumbers in the range [ − . , .
2] and κ ϕ and κ φ in the range [ − , κ ϕ and κ φ can take any valuesfrom −∞ to ∞ in principle. Numerically, we have checked that given random values of m ϕ , m ϕ and ϑ and assuming that there is no large hierarchy between (cid:15) and (cid:15) ( | (cid:15) /(cid:15) | ∈ [1 / , κ ϕ and κ φ are located in the range [ − , σ ranges, except for a very weak preference for the second octant of θ as shown in Fig. 2. Sizable (cid:15) φ , (cid:15) ϕ (cid:38) . (cid:15) φ ≈ (cid:15) ϕ are allowed by constraints. They can give rise to a sizable θ and avoida large correction to θ . If (cid:15) ϕ and (cid:15) φ are small, a relatively large κ ϕ or κ φ will be preferred togive a sufficiently large correction to θ . Finally, we note that in this model, one cannot assumethat all corrections come from the neutrino sector. Otherwise we will arrive at | a | = 2 r (cid:48) = 2 | κ (cid:48) φ (cid:15) φ | from Eq. (45), and this is not compatible with current constraints on a and r (cid:48) , e.g., | a | (cid:46) . (cid:15) ϕ , (cid:15) φ and κ ϕ , κ φ (bottom panels) in Case B, Model II, where (cid:15) φ = 0, and | (cid:15) ϕ | , κ ϕ and θ ϕ are samplesrandomly generated in the ranges [0 , . − ,
2] and [0 ◦ , ◦ ), respectively, and the points compatiblewith data are shown in the figure. The same inputs of mixing angles have been employed as in Fig. 2. r (cid:48) ≈ . • Case B: The correction in the neutrino sector is much smaller than the correction in the chargedlepton sector. As discussed in Model I, this happens when v φ is significantly larger than v ϕ . After (cid:15) φ is neglected, the corrections to the mixing angles are simplified to r (cid:48) = | (cid:15) ϕ | (cid:113) sin θ ϕ + κ ϕ cos θ ϕ , s = 2 | (cid:15) ϕ | cos θ ϕ , a = (1 + κ ϕ ) | (cid:15) ϕ | cos θ ϕ . (46)The correlations of mixing parameters and the allowed parameter space of | (cid:15) ϕ | , θ ϕ and κ ϕ areshown in Fig. 3. The Dirac phase δ may deviate from 90 ◦ or 270 ◦ greatly and take a value in therange [0 , ◦ ). However, in most cases, it takes a value in the range (50 ◦ , ◦ ) or (210 ◦ , ◦ ).The sum rule | (cid:15) ϕ | = r (cid:48) + a ( s − a ) is satisfied. To be compatible with data, the value of | (cid:15) ϕ | isin general around 0.1. A small cos θ ϕ is also preferred, similar to that in Model I, which allows | (cid:15) ϕ | sin θ ϕ to give a sizable correction to θ and not significantly modify θ and θ . In detail, θ ϕ is mostly constrained in the ranges (50 ◦ , ◦ ) and (270 ◦ , ◦ ) in this model. Flavons play a key role in leptonic flavour models with discrete flavour symmetries. They gain vacuumexpectation values (VEVs), breaking the high energy flavour symmetry and leaving residual symmetries15ifferent in the charged lepton ( ϕ ) and in the neutrino ( φ ) sectors. This misalignment leads to specificflavour mixing structures. In most models, θ vanishes and the CP invariance is conserved at leadingorder. In order to be compatible with observations, higher dimensional operators are typically introducedto modify θ , θ , θ from their leading order values and induce δ (cid:54) = 0 , π . In this paper, we exploita different approach in which we emphasise the importance of flavon cross couplings to flavour mixing.We find that cross couplings between different flavons can break the residual symmetries, shifting theVEVs of flavons and modifying flavour mixing. These couplings provide new origins for the non-zero θ and CP violation.For definiteness, we present two models based on A . Depending on the coefficients in the flavonpotential, different vacua preserving different residual symmetries can be identified. By appropriatelychoosing them in the charged lepton and neutrino sectors, we can realise the tri-bimaximal (TBM)mixing at leading order. The cross couplings between different flavons result in the breaking of theresidual symmetries and corrections to TBM.In Model I, both flavons ϕ and φ are assumed to be pseudo-real triplets of A . The cross couplingIm( (cid:15) )( ϕϕ ) (cid:48)(cid:48) ( φφ ) (cid:48) leads to the vacuum shift of ϕ and the breaking of the Z residual symmetry in thecharged lepton sector, where the relative size of the breaking is characterised by a complex parameter (cid:15) ϕ . Both δ and θ arise from this term and consequently are connected by a sum rule δ = 270 ◦ − √ θ .Taking account of current oscillation data, we predict δ ≈ ◦ , very close to the current best-fit value.The flavon cross couplings also lead to the breaking of the Z residual symmetry in the neutrino sector,characterised by a real parameter (cid:15) φ . The solar angle θ and the atmospheric angle θ gain correctionsfrom both (cid:15) ϕ and (cid:15) φ . In the interesting case in which the VEV v φ is significantly greater than v ϕ ,the correction (cid:15) φ is negligible compared with (cid:15) ϕ since (cid:15) φ /(cid:15) ϕ is suppressed by v ϕ /v φ . All modificationsto TBM arise from one single parameter (cid:15) ϕ , and an additional sum rule 3 sin θ + 4 sin θ = 3 isobtained.In Model II, flavons are assumed to be complex scalars. This extension is natural due to theconsistency with supersymmetric models and multi-Higgs models and brings some new features whichare absent in Model I. One is that it provides new sources for non-zero θ . Due to the complexproperty of the flavons, some asymmetric couplings are included in the flavon potential and they lead toasymmetric modifications between the second and third components of the flavon VEVs, parametrisedby κ ϕ and κ φ . The latter can induce sizable θ while not affecting CP conservation in some specificregion of the parameter space. If the correction in the neutrino sector is negligibly small compared withthat in the charged lepton sector, the Dirac phase prefers to take a value not far from maximal CPviolation.The flavon couplings and in particular the cross couplings can have other phenomenological conse-quences, which, depending on the flavon mass scales, can be tested directly. They may be at the originof other types of lepton flavour violation, namely lepton-flavour-violating decays of charged leptons suchas µ → eγ and τ → eee . Another type of cross couplings which we have not considered here are thosebetween flavons and Higgs. Although not relevant for the leptonic flavour structure, such couplings pro-vide ways to detect flavons directly and indirectly at colliders, through, e.g., the associated productionwith the Higgs and precision measurement of the couplings of the Higgs, respectively. Detailed studiesof these aspects will be carried out in the future.In conclusion, we have shown that cross couplings between different flavons may be the origin of thereactor mixing angle and CP violation. This is a new way, different from higher dimensional operators,to modify flavour mixing from its leading order result. Very few degrees of freedom are introduced in16odels based on cross couplings, which makes them much simpler than those built in the framework ofextra dimension or supersymmetry. The approach proposed in this paper is not limited in A models,and can be easily applied into other models with different discrete flavour symmetries. Acknowledgement
We would like to thank Dr. Peter Ballett for his useful advice. YLZ is also grateful to Prof. Zhi-zhongXing for useful discussions and to Prof. Wei Wang for warm hospitality at Sun Yat-Sen University.This work was supported by the European Research Council under ERC Grant NuMass (FP7-IDEAS-ERC ERC-CG 617143), and by the European Union FP7 ITN-INVISIBLES (Marie Curie Actions,PITN-GA-2011-289442).
A The vacuum alignment for a pseudo-real triplet flavon of A We calculate the vacuum of the A -invariant potential V ( ϕ ) in Eq. (6) in the Ma-Rajasekaran basis [22].This basis is easier for us to find out all vacuums of ϕ than the Altarelli-Feruglio basis [17], althoughphysics is equivalent in different bases. After we find out the solutions, we will rotate them to theAltarelli-Feruglio basis, in which charged lepton mass matrix is nearly diagonal.In the Ma-Rajasekaran basis, generators of A in the 3d irreducible represention are written as T = , S = − − . (47)The Kronecker product for two triplets a = ( a , a , a ) T and b = ( b , b , b ) T is divided into the followingirreducible representations:( ab ) = a b + a b + a b , ( ab ) (cid:48) = a b + ωa b + ω a b , ( ab ) (cid:48)(cid:48) = a b + ω a b + ωa b , ( ab ) S = √
32 ( a b + a b , a b + a b , a b + a b ) T , ( ab ) A = i a b − a b , a b − a b , a b − a b ) T . (48)In this basis, the flavon triplet ϕ which is pseudo-real in the Altarelli-Feruglio basis becomes real, ϕ ∗ i = ϕ i (for i = 1 , ,
3) and the flavon potential V ( ϕ ) take a simple form V ( ϕ ) = 12 µ ϕ ( ϕ + ϕ + ϕ ) + 14 ( f + f )( ϕ + ϕ + ϕ ) + 34 ( f − f )( ϕ ϕ + ϕ ϕ + ϕ ϕ ) . (49)In order to achieve a nontrivial and stable vacuum, we require a negative-definite quadratic term and apositive-definite quartic term in V ( ϕ ), and this leads to µ ϕ <
0, and f + f , f + f >
0, respectively.A necessary condition for the vacuum of ϕ is ∂V ( ϕ ) /∂ϕ i = 0, which is expressed as ∂V ( ϕ ) ∂ϕ i = ϕ i (cid:20) µ ϕ + ( f + f ) ϕ i + 12 (2 f − f + 3 f )( ϕ j + ϕ k ) (cid:21) = 0 (50)17or i, j, k = 1 , , i (cid:54) = j (cid:54) = k (cid:54) = i . One can obtain all solutions from the above equations directly.These solutions are divided into three classes, according to the corresponding values of V ( ϕ ):(1) (cid:104) ϕ (cid:105) = , − , − , − v ϕ √ ,v ϕ = − µ ϕ f + f , V ( (cid:104) ϕ (cid:105) ) = − µ ϕ f + f ) , (2) (cid:104) ϕ (cid:105) = , , v ϕ ,v ϕ = − µ ϕ f + f , V ( (cid:104) ϕ (cid:105) ) = − µ ϕ f + f ) , (3) (cid:104) ϕ (cid:105) = , , , − , − , − v ϕ √ ,v ϕ = − µ ϕ ( f + f ) + 3( f + f ) , V ( (cid:104) ϕ (cid:105) ) = − µ ϕ ( f + f ) + 3( f + f ) . (51)Here, each solution in the first class of solutions preserves a different Z symmetry, e.g., (1 , , T invariant in a Z generated by T and ( − , , T invariant in another Z generated by ST S . Similarly,each solution in the second class of solutions preserves a different Z symmetry, e.g., (1 , , T invariantin a Z generated by S and (0 , , T invariant in another Z generated by T ST .In order to get a vacuum at (cid:104) ϕ (cid:105) a (for a = 1 , , V ( ϕ ) take a local minimum at (cid:104) ϕ (cid:105) a .And this corresponds to the requirement of the positive-definite second derivative of V ( ϕ ). In detail,the matrix M ϕ defined in the following should be positive-definite at (cid:104) ϕ (cid:105) a :( M ϕ ) ij = ∂ V ( ϕ ) ∂ϕ i ∂ϕ j . (52)In general, M ϕ is a 3 × W T M ϕ W =diag { m ϕ , m ϕ , m ϕ } , with m ϕi the eigenvalues of M ϕ . In the above three classes of solutions, we get(1) m ϕ = 2( f + f ) v ϕ , m ϕ = m ϕ = 2( f − f ) v ϕ (2) m ϕ = 2( f + f ) v ϕ , m ϕ = m ϕ = 32 ( f − f ) v ϕ (3) m ϕ = 12 [( f + f ) + 3( f + f )] v ϕ , m ϕ = − m ϕ = 32 ( f − f ) v ϕ . (53)at (cid:104) ϕ (cid:105) , (cid:104) ϕ (cid:105) and (cid:104) ϕ (cid:105) , respectively. Although m ϕ is always positive in all solutions, m ϕ or m ϕ maybe positive or negative, depending on the sign of the coefficient f − f . The third class of solutions areless interesting since m ϕ and m ϕ always take opposite signs at (cid:104) ϕ (cid:105) . Thus, (cid:104) ϕ (cid:105) is always an unstablesaddle point of V ( ϕ ) and cannot be a vacuum. For the first two classes of solutions, if f − f > m ϕ and m ϕ are positive at (cid:104) ϕ (cid:105) and negative at (cid:104) ϕ (cid:105) . (cid:104) ϕ (cid:105) is an unstable saddle point. V ( ϕ )can only take a local minimum value (thus, also the global minimum value) at (cid:104) ϕ (cid:105) . Therefore, (cid:104) ϕ (cid:105) isthe only choice of the ϕ VEV. On the contrary, if f − f < (cid:104) ϕ (cid:105) is the only choice of the ϕ VEV.18ow we turn to the Altarelli-Feruglio basis in which the 3d irreducible generators are given in Eq.(3). They are obtained through a basis transformation of Eq.(47): U ω T U † ω → T , U ω SU † ω → S , where U ω = 1 √ ω ω ω ω . (54)This basis is widely implied in the literature since the charged lepton mass matrix invariant under T is diagonal in this basis. A real 3d irreducible representation in the Ma-Rajasekaran basis becomespseudo-real in this basis: ϕ ∗ = ϕ , ϕ ∗ = ϕ . The products of two 3d irreducible representations a and b can be expressed as( ab ) = a b + a b + a b , ( ab ) (cid:48) = a b + a b + a b , ( ab ) (cid:48)(cid:48) = a b + a b + a b , ( ab ) S = 12 (2 a b − a b − a b , a b − a b − a b , a b − a b − a b ) T , ( ab ) A = 12 ( a b − a b , a b − a b , a b − a b ) T . (55)Solutions in Eq. (51) transform to(1) (cid:104) ϕ (cid:105) = , − − , − ω − ω , − ω − ω v ϕ , (2) (cid:104) ϕ (cid:105) = , ω ω , ωω v ϕ √ , (3) (cid:104) ϕ (cid:105) = − − , − ω − ω , − ω − ω , −√ i √ i , −√ iω √ iω , −√ iω √ iω v ϕ √ . 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