Trimaximal-Cabibbo neutrino mixing: A parametrization in terms of deviations from tri-bimaximal mixing
aa r X i v : . [ h e p - ph ] M a r Trimaximal-Cabibbo neutrino mixing: A parametrization interms of deviations from tri-bimaximal mixing
Bo Hu ∗ Department of Physics, Nanchang University, Nanchang 330031, China
Abstract
In this paper we study a parametrized description of neutrino mixing from a phenomenologicalpoint of view. We concentrate on the parametrization in terms of higher order corrections to theleading order mixing matrix. A method to describe subleading contributions and its applicationsto tri-bimaximal mixing are discussed. We show that mixing matrices similar to tri-bimaximal-Cabibbo mixing can be obtained by straightforward choices of parameters. To achieve betteragreement with the experimental data without increasing the number of free parameters, we imposea simple phenomenological relation from which a trimaximal-like mixing matrix, parametrized by U e = sin θ e − iϕ , can be derived straightforwardly without imposing additional requirements. Itcan describe the current global fit to three-neutrino mixing with good accuracy. Its theoreticalexplanation and phenomenological applications are discussed briefly. ∗ [email protected] . INTRODUCTION Neutrino mixing is one of the most extensively studied topics in neutrino physics. Themixing pattern observed in neutrino oscillation experiments provides clear evidence, imply-ing a non-trivial but perhaps simple flavor structure of the lepton sector. Many interestingmixing patterns are proposed to describe the mixing data, including tri-bimaximal mixing(TBM) [1], bimaximal mixing [2], golden-ratio mixings [3], democratic mixing [4] and hexag-onal mixing [5], etc. Some of these often appear as exact mixing matrices in models whereneutrino mixing is determined by underlying discrete flavor symmetries (see, e.g. [8, 9]).This paradigm worked quite well before the recent discovery of a relatively large θ (forrecent global fits, see [10–12]), which signals a deviation from those exact mixing patterns.For example, a recent global fit by M. C. Gonzalez-Garcia et al . [12] givessin θ = 0 . ± . , sin θ = 0 . +0 . − . ⊕ . +0 . − . , (1)sin θ = 0 . ± . , where two 1 σ ranges for sin θ are given because the present data cannot resolve the θ octant degeneracy [12]. In the global fit given above, θ ij are the mixing angles defined inthe standard parametrization. From Eq. (1), the squared mixing matrix elements | ( U ν ) ij | can be calculated and given collectively in a matrix as (cid:13)(cid:13) | ( U ν ) ij | (cid:13)(cid:13) = . +0 . − . . +0 . − . . +0 . − . · · · · · · . +0 . − . ⊕ . +0 . − . · · · · · · . +0 . − . ⊕ . +0 . − . (2)in which | ( U ν ) ij | with i = e and j = 3 are omitted because they are affected by the DiracCP phase whose experimental value has a relatively large error. In our discussion they aredetermined by other parameters and, hence, the CP phase can be extracted by standardformalism.Although most exact mixing patterns including those mentioned above are not in preciseagreement with the experimental data, improvement can be made by introducing small By ”exact mixing matrix”, we mean the mixing matrix that does not depend on other parameters,including lepton masses. The corresponding effective neutrino mass matrix is sometimes called the ”formdiagonalizable matrix” [6, 7]. In this paper, we use a different method where the mixingmatrix is given by the product of a matrix describing deviations and the LO mixing matrix.In some cases it is simpler than the method dealing with mixing angles, and the physicalrelevance is more transparent. Although this method can be used for any LO mixing, weshall concentrate on tri-bimaximal mixing in this paper. We show that mixing matricessimilar to tri-bimaximal-Cabibbo (TBC) mixing [14] can be obtained by this method withstraightforward choices of parameters.However, these TBC-like mixing patterns agree with the data only marginally. To achievebetter agreement, sizable corrections to θ and θ must be taken into account. To do that,one can introduce more parameters. Nevertheless, parametrizations with fewer parameterscan lead to simplified descriptions of neutrino phenomenologies and may provide clues tounderlying physics. Therefore, it is also worthwhile to look for and study simple descriptionsof neutrino mixing suggested by the experimental data. In this respect, phenomenological orempirical relations are very useful, e.g., the relation between θ and the Cabibbo angle θ C : θ ≈ θ C / √ | ( U ν ) e | − | ( U ν ) e | = 2 /
3, which agreeswith the data quite well, as can be seen from Eq. (2). We show that this relation leads toa parametrization that can describe the current global fit with good accuracy. It is derivedfrom the result obtained in Sec. II and parametrized by U e = sin θ e − iϕ . Hence it can beregarded as an improved TBC mixing. Moreover, it is also a trimaximal-like mixing withthe TM trimaximal condition [15] being perturbed by a small correction. Hence, we referto it as trimaximal-Cabibbo mixing. Majorana phases can be ignored since they do not affect neutrino oscillations [18].
3n summary, the paper is organized as follows. In Sec. II, we introduce a method todescribe small deviations from LO mixing matrices. Several cases are discussed in detail.In Secs. III and IV, the results obtained in Sec. II are used to derive TBC-like mixingand trimaximal-Cabibbo mixing. In particular, the latter and its derivation are discussedin greater detail in Sec. IV. In Sec. V, we summarize and discuss briefly the theoreticalexplanation and phenomenological applications of trimaximal-Cabibbo mixing.
II. DEVIATIONS FROM LEADING ORDER MIXING MATRIX
In this section we discuss a method to describe deviations from LO mixing matrices.Note that the lepton mixing matrix U ν can always be written as a product of two unitarymatrices, i.e., U ν = U ν T (3)When T = I , the identity matrix, one has U ν = U ν . Hence, we use U ν to denote theLO mixing matrix and the matrix T to describe the deviations of U ν from U ν . Just forconvenience, in the following the matrix T will be referred to as a perturbation matrix,although it is appropriate only when the deviations are very small. The formalism developedin the following can also be used for the case where the matrix T multiplies U ν from theleft, i.e., U ν = T U ν , which will be discussed briefly at the end of this section.In Eq. (3), the mixing matrix U ν and the LO mixing matrix U ν can be written as U ν = ( U , U , U ) , U ν = ( K , K , K ) . (4)where U i and K i for i = 1, 2, or 3 are column vectors of U ν and U ν , i.e. U i = ( U ν ) ei ( U ν ) µi ( U ν ) τi , K i = ( U ν ) ei ( U ν ) µi ( U ν ) τi . (5)The perturbation matrix T in Eq. (3) is a unitary matrix that can be parametrized by threeangles denoted by ς ij and six phases. As usual, angle ς ij corresponds to the rotation anglein the ( i, j ) plane.We consider first the simplest case where T has two vanishing angles. For example, when4 = ς = 0 and ς = 0, T can be written as T = e iω cos ς ς e − iα − sin ς e iα ς P ω (6)where P ω = diag { e iω , , e iω } . Since P ω and e iω can be adsorbed by Majorana phases orcharged leptons, we will ignore them in the following discussions. Thus, from Eqs. (3), (4),and (6) one has U = ( K − x ∗ K ) /f , U = K , U = ( K + xK ) /f (7)where f = f = p | x | , x = tan ς e − iα As we can see, instead of angles and phases, one can also use x to parametrize U ν .The above procedure can be applied consecutively and iteratively. Below we consider twocases that are relevant to later discussions. In the first case the perturbation matrix T isgiven by a rotation in (1 ,
3) plane followed by a rotation in (1 ,
2) plane. By applying Eq. (7)twice on these rotations, it is straightforward to find that U = ( K − y ∗ f K − x ∗ K ) /f ,U = ( f K + yK − x ∗ yK ) /f , (8) U = ( K + xK ) /f , where f = f = p (1 + | x | )(1 + | y | ) , f = p | x | . (9)Equations (8) and (9) can be expanded in terms of x and y . When | x | and | y | are small,the expansions can be simplified by ignoring higher order terms. Since in our discussions,parameter x and y are, at most, of order O ( λ C ) where λ C = 0 . ± . O ( | x | ), O ( | y | ), or higher can be ignored and, henceone has U ≃ (1 − a ) K − y ∗ K − x ∗ K ,U ≃ (1 − b ) K + yK − x ∗ yK , (10) U ≃ (1 − c ) K + xK , a = (cid:0) | x | + | y | (cid:1) / , b = | y | / , c = | x | / . (11)As another example, we consider the case where the perturbation matrix T is given by a(1 ,
3) rotation followed by a (2 ,
3) rotation. Similarly, one finds that U ≃ (1 − c ) K − x ∗ K ,U ≃ (1 − b ) K − y ∗ K − xy ∗ K , (12) U ≃ (1 − a ) K + yK + xK , where a , b , and c are given in Eq. (11).The perturbation matrix T may depend on two or more different rotations. The corre-sponding mixing matrices can be obtained in the same way. This method can also be usedin the case where the perturbation matrix T multiplies U ν from the left, i.e. U ν = T U ν byapplying this method to its transpose, i.e. ˜ U ν = ˜ U ν ˜ T .Obviously, the method discussed above is different from the one dealing with mixingangles. It can provide a simple way to construct mixing matrices from LO mixing matriceswhen the deviations are small. It can also lead to some interesting results, including thetrimaximal-Cabibbo mixing derived in Sec. IV. This method can be applied to any LO mixingmatrix, but note that even if the LO mixing matrix U ν is in the standard parametrization,the mixing matrix U ν may not be in the standard parametrization. This is not physicallysignificant, although it may require slightly more work to extract mixing parameters usedin the global fits, especially the CP phase. In addition, in some cases, its physical relevanceis more transparent. For example, when U ν is given by T U ν , the perturbation matrix T canbe related to charged lepton corrections. III. TRI-BIMAXIMAL-CABIBBO MIXING
In the rest of this paper we consider deviations from tri-bimaximal mixing, i.e. U ν = U TBM = 1 √ √ − √ −√ − √ √ . (13)It may be instructive to show some simple applications of the method introduced in theprevious section. We begin with the simplest case, where the perturbation matrix T is6escribed by a single rotation. From Eq. (7), one finds that the second column of U ν remains intact. Hence U ν is a trimaximal mixing matrix (see, e.g. [15]). Since this case isvery simple, below we consider the other two cases discussed in the previous section.We consider first the case where U ν is given by Eqs. (12). From the last equation in (12)and Eq. (5) one has U = ( U ν ) e ( U ν ) µ ( U ν ) τ ≃ (1 − a ) K + yK + xK (14)where a is given in Eq. (11). Note that, as discussed in the previous section, in Eq. (14),terms of order O ( | x | ), O ( | y | ) or higher are ignored. Since U ν = U TBM , then from Eq. (5)and Eq. (13), one has K = 1 √ − − , K = 1 √ , K = 1 √ − . (15)Substituting K i given above into Eq. (14) leads to (cid:12)(cid:12)(cid:12)(cid:12) y √ x √ (cid:12)(cid:12)(cid:12)(cid:12) = | ( U ν ) e | = (cid:12)(cid:12)(cid:12)(cid:12) λe − iϕ √ (cid:12)(cid:12)(cid:12)(cid:12) = λ √ . (16)where ( U ν ) e is parametrized as λe − iϕ / √
2. Parameters x and y can be written as x = x λe iϕ √ , y = y λe iϕ r . (17)Then one has | y + x | = 1 (18)Because U ν contains only one physical Dirac CP phase, for simplicity we require that x and y are real. In addition, we require that they do not depend on other parameters. Inprincipal, one may use any x and y as long as | x | and | y | are small. As an example, let y = 1 / x = 2 /
3. Then from Eqs. (12) and (17), one has U ν = q (cid:16) − λ (cid:17) √ (cid:16) − λ (cid:17) λe − iϕ √ − √ (cid:16) − λe iϕ − λ (cid:17) √ (cid:16) λ e iϕ + λ (cid:17) − √ (cid:16) − λ (cid:17) − √ (cid:16) λe iϕ − λ (cid:17) √ (cid:16) − λ e iϕ + λ (cid:17) √ (cid:16) − λ (cid:17) + O ( λ )7hich is similar to the TBC mixing introduced in [14], which is also parametrized by sin θ whose global-fit value is in good agreement with the relation sin θ = λ C / √
2. Note that λ C ≃ .
011 and, hence, terms of order O ( λ ) or higher can be neglected. One may checkexplicitly that U ν is unitary up to O ( λ ) corrections. The mixing angles extracted from U ν are given by sin θ = 13 − λ , sin θ = 12 , sin θ = λ λ C for λ , one finds that the deviation of sin θ from its TBM or TBC valueis λ C / . U ν = U TBM T . Below we consider another example inwhich U ν = T U
TBM . Taking the transpose leads to ˜ U ν = ˜ U TBM ˜ T . Therefore, as discussed inSec. II, one can use the method in the example above to obtain ˜ U ν , the transpose of U ν . Forinstance, one can use Eqs. (10) to construct ˜ U ν . Denote ˜ U TBM by ( K ′ , K ′ , K ′ ) and substitute K ′ i for K i in Eqs. (10), from | ( ˜ U ν ) e | = | ( U ν ) e | = λ/ √ | y ∗ − x ∗ | = λ . Asabove, one may set x = x λe iϕ and y = y λe iϕ and then one has | y − x | = 1. Letting y = − x = 1 / θ = 13 , sin θ = 12 − λ , sin θ = λ . IV. TRIMAXIMAL-CABIBBO MIXING
As we can see, for the TBC-like mixings discussed above, one has sin θ ≃ sin θ TBM12 =1 / θ ≃ sin θ TBM23 = 1 /
2, which fit the data within the range between the 2 σ and 3 σ experimental bounds. To achieve a better agreement with the data, corrections to θ and θ should also be taken into account. As discussed in the first section, to avoidintroducing additional parameters, we will impose a phenomenological relation. We notethat, to some extent, several well-known mixing patterns such as TBM and BM can beregarded as phenomenological mixing patterns. Trimaximal mixing [15] and TBC mixing[14] can also be derived with certain phenomenological relations in mind. The relation usedin our discussion is given by | ( U ν ) e | − | ( U ν ) e | = 23 (19)8hich is consistent with the present data. Like most other phenomenological or empiricalrelations, it can hardly be generated as an exact relation. Nevertheless, note that thisrelation is also satisfied by TBM and, hence, it can be considered as a phenomenologicalconstraint on the deviations from TBM induced by higher order corrections. Its possibletheoretical explanation and phenomenological applications are discussed in the next section.We discuss first its implication on mixing parameters. As in the previous section, wedenote ( U ν ) e by λe − iϕ / √
2. From Eq. (19) we have | ( U ν ) e | = 23 + λ , | ( U ν ) e | = 13 − λ , | ( U ν ) e | = λ θ = λ , sin θ ≃ (cid:18) − λ (cid:19) where higher order terms are neglected. Using the global-fit value of sin θ given in Eq. (1),one finds that the deviation of sin θ from sin θ TBM12 = 1 / θ / . ± . σ experimental error. Therefore, to satisfy Eq. (19), thecorrection to θ TBM12 cannot be neglected. Also note that this relation does not constrain theatmospheric mixing angle θ . Nevertheless, below we will show that even without making aparticular choice of parameters, the mixing matrix given by Eqs. (10) leads straightforwardlyto an improved TBC or trimaximal-like mixing that can fit the data very well, including θ .We consider first the case where the perturbation matrix T multiplies U TBM from the right,i.e. U ν = U TBM T . Since U ν = U TBM , from Eq. (20) one finds that | ( U ν ) e | > | ( U ν ) e | = 2 / U ν ) e = 0, ( U ν ) e must receive a contribution from ( U ν ) e , i.e., the first elementof K , which is the second column of U ν . Therefore we use Eqs. (10) to construct U ν ,which is the only choice made in this case. The mixing matrix U ν can then be derivedstraightforwardly in a similar manner as in the example discussed in the previous section.From the last equation in (10) and | ( U ν ) e | = λ/ √
2, one finds that | x | ≃ √ λ/
2. Then fromthe first equation in (10) and Eq. (20) one has (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − | x | + | y | (cid:19) r − √ y ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≃ (cid:18)
23 + λ (cid:19) For simplicity, we assume that y is real. Since | x | ≃ √ λ/
2, it is straightforward to solvethe above equation for y . Ignoring higher order terms, one has x = √ λe − iϕ , y = − √ λ . (21)9hen from Eqs. (10) it follows that U ν ≃ q (cid:16) λ (cid:17) √ (cid:16) − λ (cid:17) λe − iϕ √ − √ (cid:16) − λe iϕ − λ (cid:17) √ (cid:16) λ (cid:17) − √ (cid:16) λe − iϕ − λ (cid:17) − √ (cid:16) λe iϕ − λ (cid:17) √ (cid:16) λ (cid:17) √ (cid:16) − λe − iϕ − λ (cid:17) + O ( λ ) . (22)which is slightly more complicated than the TBC mixing matrix [14] but in a better agree-ment with the data. Hence it can be considered as an improved TBC mixing. For the reasondiscussed in Sec. III, terms of order O ( λ ) or higher can be neglected. One can also checkexplicitly that U ν is unitary up to O ( λ ) corrections.From Eq. (22) a simple relation between sin θ , sin θ , and the CP phase ϕ can bederived: sin θ ≃
12 (1 + λ cos ϕ + λ ϕ ) . where the λ term can also be neglected. The Jarlskog invariant is given by J ≃ − ( λ sin ϕ ) / O ( λ ) . Note that ϕ is not the CP phase in the standard parametrization, but their difference isvery small, as will be shown in the next section.In the case with vanishing CP phase, substituting p θ for λ and using the global-fitvalue for sin θ given in Eq. (1), one hassin θ ≃ . +0 . − . , sin θ ≃ . +0 . − . which agree with the data within 1 σ range. Note that by switching λ → − λ or ϕ → ϕ + π ,sin θ can be brought into the 1 σ range in the first octant given in Eq. (1). For thenonvanishing phase, one has 0 . . sin θ . . σ experimental range [12].In addition, it is interesting to see that U ν given in Eq. (22) may also be regarded as avariant of the TM trimaximal mixing [15] with the TM condition | ( U ν ) α | = 1 / α = e , µ , τ ) being perturbed by small corrections of order O ( λ ). Hence, we refer to it as trimaximal-Cabibbo mixing since substituting λ C for λ in Eq. (22) leads tosin θ ≃ . , sin θ ≃ . , sin θ ≃ . σ range.Now we consider another case where the perturbation matrix T multiplies U TBM fromthe left, i.e. U ν = T U
TBM . As discussed in previous sections, in this case, one can movethe perturbation matrix to the right by taking transpose. Since the discussion is similar, wejust give the result. From Eqs. (10) we find that x and y should satisfy |− x ∗ + y ∗ | ≃ λ , (cid:12)(cid:12)(cid:12)(cid:12) − | x | + | y | − x ∗ − y ∗ (cid:12)(cid:12)(cid:12)(cid:12) ≃ − λ . One can verify that x = − λe − iϕ λ , y = λe − iϕ λ U ν ≃ q (cid:16) λ (cid:17) √ (cid:16) − λ (cid:17) λe − iϕ √ − √ (cid:16) − λe iϕ − λ (cid:17) √ (cid:16) λe − iϕ + λ (cid:17) − √ (cid:16) − λ (cid:17) − √ (cid:16) λe iϕ − λ (cid:17) √ (cid:16) − λe − iϕ + λ (cid:17) √ (cid:16) − λ (cid:17) + O ( λ ) . (23)which leads to a nearly maximal θ , i.e.sin θ ≃
12 (1 − λ . Although acceptable, it does not fit the data as well as the trimaximal-Cabibbo mixingderived above. One may improve that by adjusting x and y , which is possible for this case,but the choice of parameters is not very straightforward, so we will leave that for futureconsiderations when more experimental data are available. V. SUMMARY AND DISCUSSIONS
In this paper a general method to parametrize the neutrino mixing matrix in terms ofdeviations from leading order mixing is discussed. Using this method, we show that mixingmatrices similar to tri-bimaximal-Cabibbo mixing can be derived by straightforward choicesof parameters. However, these mixing matrices fit the data only marginally. To improve thatwithout increasing the number of free parameters, we introduce a phenomenological rela-tion, i.e., | ( U ν ) e | − | ( U ν ) e | = 2 /
3. Two mixing matrices satisfying this relation are con-structed. The one referred to as trimaximal-Cabibbo mixing provides a good two-parameter11escription of the present mixing data and, hence, can serve as a useful parametrization aslong as future experimental data do not change the current global fit significantly. Belowwe discuss briefly its phenomenological applications and possible theoretical explanations.Since the trimaximal-Cabibbo mixing given in Eq. (22) involves only two free param-eters, when it is used to parametrize the lepton mixing matrix, the expressions for manyphenomenological quantities can be greatly simplified. As an interesting application, weconsider the neutrino mixing probabilities for phase-averaged propagation with oscillationphase (∆ m ) L/ E ≫
1. To simplify our discussion, we use the results given in [16] in whichmore details can be found. Because in [16] the neutrino mixing probabilities are expressed interms of a set of parameters different from those in trimaximal-Cabibbo mixing, one needsto find first the relations between them. Substituting Eq. (22) into the formalism in [16] onefinds that the parameters we need can be written as ǫ ≃ − λ √ , ǫ ≃ λ ϕ, ǫ ≃ λ √ . (24)where λ and ϕ are the two parameters in trimaximal-Cabibbo mixing. In addition, one alsoneeds the Dirac CP phase in the standard parametrization which is denoted by ϕ D . FromEq. (22), one can derive the relation between ϕ D and ϕ , which is given bycos ϕ D ≃ cos ϕ − λ cos ϕ sin ϕ. For phenomenological applications, the second term can be ignored since it is roughly oforder O ( λ C ) and, hence, one may set ϕ D = ϕ . From Eq. (24) and the results given in [16],one finds that flavor mixing probabilities can be expressed in terms of ǫ ≡ λ cos ϕ , i.e. P ν e ↔ ν e = 5 / , P ν µ ↔ ν µ = (cid:0) ǫ + 3 ǫ (cid:1) / , P ν τ ↔ ν τ = (cid:0) − ǫ + 3 ǫ (cid:1) / P ν e ↔ ν µ = (2 − ǫ ) / , P ν e ↔ ν τ = (2 + ǫ ) / , P ν µ ↔ ν τ = (cid:0) − ǫ (cid:1) / . The ratio Φ µ / Φ τ of the ν µ flux to the ν τ flux arriving at earth, which measures the violationof µ − τ symmetry, can be written asΦ µ / Φ τ = 1 + 26 ǫ / . Comparing with [16], one finds that the expressions for mixing probabilities are considerablysimplified. For more details about mixing probabilities, see [16] and references therein.12efore ending this paper, we briefly discuss possible theoretical explanations for trimaximal-Cabibbo mixing. As discussed in Sec. IV, one may begin with a trimaximal-mixing model.For example, consider the one proposed in [26]. It is shown that an A model with a 1 ′ (and/or a 1 ′′ ) flavon can lead to an effective neutrino mass matrix given by M ν = U TBM a + c − d √ d a + 3 b + c + d √ d a − c + d ˜ U TBM (25)= a + b + c + d (26)where a , b , c , and d depend on model parameters. One can show that M ν leads to TM trimaximal mixing with | ( U ν ) e || ( U ν ) µ || ( U ν ) τ | = / √ / √ / √ . Based on this model, one may introduce higher order corrections or additional contributionsto produce the trimaximal-Cabibbo mixing given by Eq. (22).Before we proceed, one can compare M ν with the effective neutrino mass matrix corre-sponding to trimaximal-Cabibbo mixing, which is given by M TMC ν ≃ U TBM m + ∆ λ − √ ∆ λ √ ∆ λ − √ ∆ λ m √ ∆ λ m − ∆ λ ˜ U TBM (27)where m i are neutrino masses and ∆ ij ≡ m i − m j . It can also be written as M TMC ν = U TBM a + c − d + e − √ e √ d − √ e a + 3 b + c + d + e √ d a − c + d − e ˜ U TBM (28)where a , b , c , d , and e can be determined by comparing the two equations above. One canshow that M TMC ν can be decomposed as M TMC ν = M ν + M ν = M ν + e (29)13here M ν is the mass matrix given by Eq. (25) or Eq. (26). Note that a vanishing CP phaseis assumed for simplicity and the second term in Eq. (29), i.e., M ν , can be replaced by acombination of (cid:16) (cid:17) , (cid:16) (cid:17) , (cid:16) (cid:17) and (cid:16) (cid:17) , e.g. (cid:16) (cid:17) or (cid:16) (cid:17) .Now one can see that, in the A model discussed above, if certain additional contributionscan be introduced to account for the difference between M ν and M TMC ν , trimaximal-Cabibbomixing can then be obtained from the latter. For instance, from Eq. (29) it follows that M TMC ν can be produced by introducing higher order corrections that contribute dominantlyto the (2 ,
3) element of the effective neutrino mass matrix. On the other hand, by comparingEq. (28) with Eq. (25), one finds that it can also be generated by an additional nonvanishing(1 ,
2) mass term in the TBM basis, which can be obtained by, e.g. adding Higgs triplets (see,e.g. [27]). Nevertheless, in both cases, to suppress other possible contributions, a certainamount of fine-tuning might be necessary. It would be interesting to see a more concretemodel that can lead to trimaximal-Cabibbo mixing naturally. We leave that for future work.
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