Warped flavor symmetry predictions for neutrino physics

A realistic five-dimensional warped scenario with all standard model fields propagating in the bulk is proposed. Mass hierarchies would in principle be accounted for by judicious choices of the bulk mass parameters, while fermion mixing angles are restricted by a $\Delta(27)$ flavor symmetry broken on the branes by flavon fields. The latter gives stringent predictions for the neutrino mixing parameters, and the Dirac CP violation phase, all described in terms of only two independent parameters at leading order. The scheme also gives an adequate CKM fit and should be testable within upcoming oscillation experiments.

inherent in the use of non-Abelian flavor symmetries with the presence of warped extra-dimensions. We propose a warped five-dimensional scenario in which all matter fields propagate in the bulk and neutrinos are treated as Dirac particles. Our model can accommodate all the strengths of the standard model Yukawa couplings and resulting fermion mass hierarchies by making adequate choices of fermion bulk mass parameters, while the fermion mixing parameters can be restricted by means of the assumed flavor symmetry. We present a ∆ (27) based flavor symmetry which nicely describes the neutrino oscillation parameters in terms of just two independent parameters, leading to interesting correlations involving the neutrino mass hierarchy and the leptonic Dirac CP phase, not yet reliably determined by current global oscillation fit [31]. Our predictions include a neat leading order relation between the solar and reactor mixing parameters which should be tested at future oscillation experiments.

II. BASIC STRUCTURE OF THE MODEL
In this section we present the basic setup of a warped five-dimensional (5D) model for fermions, constructed under a ∆(27) ⊗ Z 4 ⊗ Z ′ 4 flavor symmetry. The 5D field theory is defined on a slice of AdS 5 , where the bulk geometry is described by the metric with η µν = diag(1, −1, −1, −1) and k as the AdS 5 curvature scale. The fifth dimension y is compactified on S 1 /Z 2 , and two flat 3-branes of opposite tension are attached to the orbifold fixed points, located at y = 0 (UV brane) and y = L (IR brane). The electroweak symmetry of the model is promoted to G bulk = SU (2) L ⊗ SU (2) R ⊗ U (1) B−L in order to avoid excessive contributions to the Peskin-Takeuchi T parameter [33,34]. The gauge group G bulk breaks down to the standard model electroweak (EW) group G SM = SU (2) L ⊗ U (1) Y on the UV brane by the boundary conditions (BCs) of the gauge bosons. Furthermore, a bulk Higgs field with (SU (2) L , SU (2) R ) quantum numbers is responsible for the spontaneous symmetry breaking (SSB) of G SM . The 5D Higgs field H(x µ , y) can be decomposed into Kaluza-Klein (KK) modes as For an adequate choice of BCs, its zero mode profile f H (y) can be written as [35] f H (y) = 2kL(1 − β) 1 − e −2(1−β)kL e kL e (2−β)k(y−L) , where we have introduced the Higgs localization parameter β = 4 + m 2 H /k 2 in terms of the Higgs field bulk mass parameter m H . In the present work, we assume that the vacuum expectation value (VEV) of the Higgs zero mode is of the form and it is peaked toward the IR brane, allowing for a TeV scale EW SSB and inducing the G bulk breakdown to SU (2) D ⊗ U (1) B−L on that brane. Three families of fermion fields are required to describe each generation (labeled by i = 1, 2, 3) of quarks and leptons. All fermion fields propagate into the bulk and transform under the minimal representation of the gauge Notice that we have a separate SU (2) R doublet for every right handed fermion. In the above equations, fields with different sign assignments must be understood as independent. The bracketed signs indicate Neumann (+) or Dirichlet (−) BCs for the left-handed component of the corresponding field, on both UV and IR branes. The right-handed part of the field satisfies opposite BCs. Only fields with [++] BCs have left-handed zero modes, whereas right-handed zero modes exist solely for fields with [−−] BCs. The KK decomposition for such fields has the form where c L and c R are the bulk mass parameters of the 5D fermion fields in units of the AdS 5 curvature k. Thus, the low energy spectrum contains left-handed doublets ℓ iL = (ν iL , e iL ), Q iL = (u iL , d iL ), alongside right-handed singlets ν iR , e iR , u iR , d iR . In the following, we identify all standard model fields with this set of zero modes (i.e. the so called zero mode approximation, ZMA). For future convenience, we denote the flavor components of charged leptons and quarks as e 1,2,3 = e, µ, τ ; Q 1,2,3 = U, C, T ; u 1,2,3 = u, c, t; d 1,2,3 = d, s, b.
In the present work, we choose the flavor symmetry to be ∆ (27), augmented by the auxiliary symmetry Z 4 ⊗ Z ′ 4 . The group ∆ (27) was originally proposed to explain the fermion masses and flavor mixing in Refs. [39,40], and has been used for Dirac neutrinos in [41] by one of us. Here we study its implementation in a warped extra dimensional theory. The flavor symmetry ∆(27) ⊗ Z 4 ⊗ Z ′ 4 is broken by brane localized flavons, transforming as singlets under G bulk . We introduce a set of flavons ξ, σ 1 , σ 2 localized on the IR brane, and a flavon ϕ localized on the UV brane. Both ξ and ϕ are assigned to the three dimensional representation 3 of ∆(27), while σ 1 and σ 2 transform as inequivalent one dimensional representations 1 0,1 and 1 0,0 respectively. A summary of the ∆(27) group properties and its representations can be found in Appendix A. There are two different scenarios for the model, determined by the two possible VEV alignments for ξ, namely: with ω = e 2πi/3 . As indicated above, we will denote the models described by each alignment as cases I and II, respectively. Note that the case II vacuum pattern frequently appears in the context of geometrical CP violation [42,43]. The VEVs for the remaining flavon fields are Further details regarding this vacuum configuration are offered in Appendix B.

III. LEPTON SECTOR
Once the basic framework has been laid out, we are in position to discuss the structure of the lepton sector and its phenomenological implications. As we will show below, charged lepton as well as Dirac neutrino masses are generated at leading order (LO), and non-zero values for the "reactor angle" θ 13 arise naturally. The model is predictive, in the sense that the three mixing angles and the Dirac CP phase will ultimately be determined in terms of only two parameters.
At leading order, the lepton mixing matrix U PMNS = U † l U ν becomes In both cases, the solar, atmospheric and reactor angles can be written in terms of θ ν and ϕ ν as A convenient description for the CP violating phase in this sector is the Jarlskog invariant J CP = [44], which in this parameterization takes the compact form It is worthy of attention the independence of J CP upon ϕ ν , and the simple predicted relation between the solar and reactor angles θ 12 and θ 13 :

B. Phenomenological implications
As shown above, only two parameters are required to generate the three angles and the Dirac CP violating phase characterizing the lepton mixing matrix, making this model highly predictive. In the remaining part of this section we explore in detail the predictions for the lepton mixing parameters and the neutrino mass spectrum.
In Figure 1, the θ ν -ϕ ν parameter region compatible with experimental data is delimited using the global fit of neutrino oscillations given in [31] for each mass ordering, shown as the left and right hand panel. The model can reproduce successfully the best-fit values for the atmospheric and reactor angles, reaching simultaneously the 2σ region for the solar angle. The intersecting points of the "central" or best fit curve in the sin 2 θ 13 contour and the corresponding ones in the sin 2 θ 23 contour are located at where NH 1 denotes the best-fit contour of sin 2 θ 23 , and NH 2 corresponds to its local minimum in the first octant. Notice that the numbers in parenthesis denote the intersection values within the range Once we have determined θ ν and ϕ ν from the central values of the atmospheric and reactor oscillation global fits, the predictions for the solar angle and the Jarlskog invariant can be straightforwardly obtained using Eqs. (34,35). For completeness, in Table II we present the full set of mixing parameters derived from the points defined in Eq. (37).
We conclude this section bringing forth a consistent realization of lepton masses and mixing angles. In the numerical analysis, we assume that the fundamental 5D scale is k ≃ Λ ≃ M Pl , with M Pl ≃ 2.44 × 10 18 GeV as the reduced Planck mass. We also set the scale Λ ′ ≃ k ′ = ke −kL ≃ 1.5 TeV in order to account for the hierarchy between the Planck and the electroweak scales, allowing for the lowest KK gauge boson resonances (with masses m KK = 3 ∼ 4 TeV) to be within reach of the LHC experiments. The Higgs VEV is identified with its standard model value v ≃ 246   [31]. The sign of JCP in the parentheses corresponds to the bracketed prediction for θν in Eq. (37).
GeV, and the ratios  Table III. There, the four BPs are labeled according to their hierarchy scheme and case as NH-I, NH-II, IH-I, IH-II. One sees that, indeed, the large disparity between charged lepton masses is reproduced for Yukawa couplings of the same order of magnitude.  The neutrino masses, splittings and mixing angles associated to each BP are displayed in Table IV. All the obtained neutrino oscillation parameters are consistent with the global fit in Ref. [31]. In particular, the reproduced atmospheric and reactor angles lie comfortably in their respective 1σ region, whereas the solar angle values are contained in the 2σ range, very close to the 1σ boundary.  ∆(27) 10,2 10,1 10,0 10,2 10,0 10,2 10,1 10,0 10,1 The quark transformation properties under the family group ∆(27) ⊗ Z 4 ⊗ Z ′ 4 are given in Table V. At leading order, the most general invariant Yukawa interactions can be written as Again, after spontaneous electroweak and flavor symmetry breaking, the mass matrices for the up and down quark sectors read where y uiuj = y uiuj F (L, c Qi , c uj ) , The up-type quark mass matrix is already block-diagonal. The diagonalization of the down-type mass matrix m d requires a more careful treatment. For the sake of simplicity, in the following analysis we denote the ij element of m u (m d ) as m u ij (m d ij ). The product of the down-type mass matrix and its adjoint can be diagonalized in two steps: in first place, an approximate block diagonalization is accomplished with the aid of the transformation matrix and subsequently the diagonalization is completed through a unitary rotation of the upper block. This approximation is consistent provided |m d 33 | >> |m d 12 |, |m d 13 |, |m d 22 | and |ǫ| << 1. The resulting diagonalization matrices for the up and down sectors can be parameterized as and Correspondingly, the quark mass eigenvalues can be expressed in terms of M ± , defined in Eq. (30), as so that the CKM matrix is given by Hence, the quark sector Dirac CP phase (in PDG convention) and the Jarlskog invariant take the form According to Eq. (40), the size of up and down mass matrix elements is determined by the overlap of the 5D quark field zero mode profiles, i.e., m u ij ∝ f then the elements of m u and m d approximately satisfy justifying the perturbative diagonalization performed on m d m d † . These relations imply that X + u,d ≫ |Y u,d | holds, and therefore, a rough estimate for the mixing parameters and quark mass spectrum is Thus, in order to reproduce plausible quark masses and mixings, namely: the quark zero mode profiles must observe the following hierarchy: The resulting quark masses and mixings are consistent with the current experimental data [1], and the precision of the results can be improved by incorporating high order corrections, addressed in the next section.

V. HIGH ORDER CORRECTIONS
From the particle content and above transformation properties, one finds that nontrivial high order corrections to the charged lepton sector are absent in the present model. The next-to-leading order (NLO) corrections to the neutrino Yukawa interactions are given by However, the contribution of these terms to the neutrino masses and mixing parameters can be absorbed by a proper redefinition of the parameter y 22 after SSB. Hence, in order to estimate the effects of higher order corrections in this sector, we need to investigate the Yukawa terms involving an additional (v IR /Λ ′ ) 2 suppression with respect to the lowest order terms in Eq. (12), where we have introduced v IR to characterize the magnitude of v ξ ∼ v σ1 ∼ v σ2 . The contraction of the field products Ψ l HΨ ν1 , Ψ l HΨ ν3 , transforming as (3, −1, −1) under ∆(27) ⊗ Z 4 ⊗ Z ′ 4 , with the flavon operators 1 (Λ ′ ) 11 2 (ξξ * ) 1a,2 ξσ 1 , 1 as well as the combination of Ψ l HΨ ν2 ∼ (3, i, −1) and provide the desired high order corrections to the neutrino Yukawa interactions. In the above expressions, the indices a, b = 0, 1, 2 label the different singlets of ∆ (27). Additional terms that can be absorbed into y 11 , y 13 , y 22 , y 31 and y 33 have been omitted. Taking into consideration these corrections, the neutrino mass matrix m ν can be roughly written as with x ij = x ij F (L, c l , c νj ), and x ij as dimensionless parameters of order O(1). Working under the same numerical framework established in Section III, one can readily estimate the shift in the neutrino oscillation parameters induced by high order corrections of the Yukawa interaction. Particularly, in Case I, taking x ij as random complex numbers with magnitudes ranging from 2 to 6, and v IR = 0.1, the resulting deviations in the neutrino mixing parameters with respect to their LO values can be estimated as From Eq. (63), it is clear that high order corrections can easily drive s 2 12 into its 1σ region while keeping the remaining parameters optimal.
Turning to the quark sector, every bilinear formed by Ψ Qi and Ψ ui or Ψ di can produce a high order correction to the Yukawa interaction whenever it is contracted with the adequate cubic flavon operator. Beside terms that can be absorbed by a redefinition of y uiuj or y didj in Eq. (38), all the NLO contributions can be classified into three categories: Again, after symmetry breaking, the quark mass matrices m u and m d can be approximately written as Here we have defined x uiuj = x uiuj F (L, c Qi , c uj ) and x didj = x didj F (L, c Qi , c dj ), where the couplings x uiuj and x didj represent dimensionless parameters of order O(1). As a numerical example, taking x uiuj , x didj as random complex numbers with magnitudes ranging from 1 to 4 for x uc , x cu , x bd , x cc , x tu , x dd , and from 2 to 6 for x ut , x sb , x bs , while keeping the values of c Qi , c ui , c di , y uiuj and y didj reported in Section IV, the order of deviation with respect to the LO values of the quark masses is The corresponding correction to the first order CKM matrix is of order and the values for the quark CP violating phase and the Jarlskog invariant are displaced by As for the lepton sector, it is not difficult to find parameter values reproducing the quark mass and mixing parameters required to fit the current experimentally observed values.

VI. CONCLUSIONS
We have proposed a five-dimensional warped model in which all standard model fields propagate into the bulk. Its structure is summarized in the "cartoon" depicted in Figure 2. Mass hierarchies in principle arise from an adequate choice of the bulk shape parameters, while fermion mixing angles are constrained by relations which follow from the postulated ∆(27) flavor symmetry group, broken on the branes by a set of flavon fields. The neutrino mixing parameters and the Dirac CP violation phase are described in terms of just two independent parameters at leading order. This leads to stringent predictions for the lepton mixing matrix which should be tested in future neutrino oscillation experiments. Likewise the scheme also includes the quark sector, providing an adequate description of the quark mixing matrix. The effect of next-to-leading order contributions is estimated to be fully consistent with the experimental requirements. The ∆ (27) group is isomorphic to (Z 3 ⊗ Z 3 ) ⋊ Z 3 . It can be conveniently expressed in terms of three generators a, a ′ and b which satisfy the following relations: All ∆(27) elements can be written into the form b k a m a ′ n , with k, m, n = 0, 1, 2. The group has 11 conjugacy classes, given by The ∆ (27) has nine one dimensional representations, which we denote as 1 k,r (k, r = 0, 1, 2), and two three dimensional irreducible representations 3 and 3. The explicit form of the group generators in each irreducible representation χ1 0,0 χ1 0,1 χ1 0,2 χ1 1,0 χ1 1,1 χ1 1,2 χ1 2,0 χ1 2,1 χ1 2,2 χ3 χ 3 1C1  1  1  1  1  1  1  1  1  1  3  3 1 where ω = e 2πi/3 is the cube root of unity. Notice that 3 and 3 are complex representations dual to each other. From the character table of the group, shown in Table VI, we can straightforwardly obtain the Kronecker products between the various representations where [n] stands for n mod 3, whenever n is an integer. Starting from the representation matrices of the generators in different irreducible representations, we can calculate the Clebsch-Gordan (CG) coefficients for the Kronecker products listed above. All CG coefficients are presented in the form α ⊗ β, where α i stands for the elements of the first representation and β j those of the second one. In the following, we adopt the convention α [3] = α 0 ≡ α 3 .
where the subscripts "S " and "A" denote symmetric and anti-symmetric combinations respectively.

(B8)
The right-handed side of these expressions can be positive in a finite region of parameter space. Analogously, for the Case II vacuum configuration ξ = (1, ω, 1)v ξ , the minimization conditions are ∂V IR ∂ξ *