Predictions from warped flavordynamics based on the $T'$ family group

We propose a realistic five-dimensional warped scenario where all standard model fields propagate in the bulk. The assumed $T'$ flavor symmetry is broken on the branes by flavon fields, providing a consistent scenario where fermion mass hierarchies are accounted for by adequate choices of the bulk mass parameters, while quark and lepton mixing angles are restricted by the flavor symmetry. Neutrino mixing parameters and the Dirac CP violation phase are all described in terms of just two independent parameters. This leads to predictions for the neutrino mixing angles and the Dirac CP phase, as well as a $\rm 0\nu\beta\beta$ decay rate within reach of upcoming experiments. The scheme provides a good global description of flavor observables also in the quark sector.


I. INTRODUCTION
Understanding flavor from first principles is one of the greatest challenges in particle physics. The coin has two sides. On the one hand there is the problem of understanding the observed hierarchies of quark and lepton masses, explaining why is the muon about 200 times heavier than the electron, or why does the top quark seem to play such a special role in being the heaviest.
On the other hand comes the problem of finding a rationale for the observed pattern of mixing parameters. This problem has only become trickier after the discovery of neutrino oscillations [1, 2] which implies not only the need for neutrino masses -and understanding their smallness with respect to the charged fermion masses -but also the need to understand why the pattern of neutrino mixing is so special when compared to that of quarks [3].
The Standard Model (SM) lacks an organizing principle to account for the observed flavor properties. The existence of flat extra dimensions has been suggested as a way to shed light on the possible nature of the family symmetry [4]. In particular, six-dimensional theories compactified on a torus have been suggested [5,6] and a successful model has recently been proposed [7] in which fermions are nicely arranged within the framework of an A 4 family symmetry, with good predictions for fermion masses and mixings, including the "golden" quark-lepton unification formula [8][9][10][11]. Although intriguingly successful, this theory remains far from giving a complete description of mass hierarchies.
As a possible alternative to the flat-extra-dimensions approach here we turn to the possibility of warped extra dimensions. These have been proposed by Randall & Sundrum [12] in order to address the hierarchy problem without the need to invoke supersymmetry. The fundamental scale of gravity gets exponentially reduced with respect to the Planck scale by having the Higgs sector localized near the boundary of the extra dimensions. Here we assume the standard model fermions to propagate in the bulk, though peaked towards either brane. This allows us to address at once both aspects of the flavor problem: the fermion mass hierarchy problem, as well as their mixing pattern, due to the imposition of a family symmetry group. This follows the approach suggested in Ref. [13]. In such scenario fermion mass hierarchies are accounted for by adequate choices of the bulk mass parameters, while quark and lepton mixing angles are restricted by the assumed family symmetry, broken on the branes by flavon fields. Our present scenario employs the T -based family group and predicts the neutrino mixing parameters and the Dirac CP violation phase in terms of only two independent parameters at leading order. In contrast to Ref. [13] where neutrinos were Dirac particles, here a viable description of neutrino oscillations requires neutrinos to be Majorana particles. Moreover, given the predicted regions for the oscillation parameters it follows that there must be a lower bound on the neutrinoless double beta decay rate even if the spectrum is normal-ordered. We show that the model also provides a successful global description of flavor, consistent with the observed CKM quark mixing matrix, in which the successful Gatto-Sartori relation emerges in leading order.
The paper is organised as follows. In Sec. II we present the theoretical framework for the lepton sector, while in Sec. III we sketch the quarks sector, its field content and quantum numbers. In Sec. IV we give a numerical analysis of the resulting predictions and conclude in Sec. V.

II. LEPTON SECTOR
Here we study the implementation of a flavor symmetry within a warped extra dimensional theory context. For the flavor symmetry we choose the T group. The T flavor symmetry has been studied in the literature [14][15][16][17][18][19][20][21][22]. We introduce four flavon fields ϕ ν and σ ν localized on the IR brane, and flavons ϕ l and σ l localized on the UV brane. The fermion fields and Higgs field live in the bulk, and the profiles of their zero modes in the fifth dimension are displayed in where ω = e 2iπ 3 , v ϕ l , v σ l , v ϕν and v σν are arbitrary complex numbers. As shown in Appendix C, the alignment in Eq. (1) is the minimum of the scalar potential.
The leading order charged lepton Yukawa interactions respecting both gauge and flavor symmetries are of the By performing the seesaw diagonalization procedure [23,24], one finds that the effective light neutrino mass matrix is given as . It is remarkable that, apart from an overall mass scale m 0 , the mass matrix m ν only depends on two complex input parameters y 4 , y 5 . These will describe the three neutrino masses as well as the full lepton mixing matrix. We first perform a tri-bimaximal transformation on the neutrino fields. The resulting light neutrino mass matrix becomes where U T BM is the well-known tri-bimaximal mixing matrix, From the lepton mixing matrix in Eq. (16), one can easily extract the following results for the neutrino mixing angles as well as the Jarlskog invariant, One sees that the three neutrino mixing angles as well as the Dirac CP violation phase are all expressed in terms of just two parameters, θ ν and δ ν . Therefore there are two relations between these mixing angles and the Dirac CP violation phase, that can be expressed analytically as cos 2 θ 12 cos 2 θ 13 = 2 3 , cos δ CP = (3 cos 2θ 12 − 2) cos 2θ 23 3 sin 2θ 23 sin 2θ 12 sin θ 13 .
In Fig. 2 we display the contour plots of sin 2 θ 12 , sin 2 θ 13 , sin 2 θ 23 and Dirac CP violation phase δ CP in the θ ν − δ ν plane. The shaded regions are the ones allowed by individual measurements of the three mixing angles, according to the global oscillation analysis in Ref.
[3]. One sees that the parameter θ ν is constrained to lie within quite narrow regions around θ ν 0.082π and θ ν 0.918π. The left panel in Fig. 3 shows the contour plots of δ CP in the θ ν − δ ν plane. The black bands denote the regions in which all the three lepton mixing angles vary within the experimentally allowed 3σ ranges [3]. In our model, the sign of δ CP can not be fixed uniquely, with the predicted correlation between |δ CP | and sin 2 θ 23 shown in the right panel of Fig. 3.

B. Neutrinoless double beta decay
We start this section by noticing that, in the absence of the Majorana terms in Eq. (7), in this model neutrinos would be unmixed, since both charged lepton and Dirac mass terms are diagonal. Neutrinos would also be degenerate in mass. Hence neutrino mass differences, as well as mixing and CP violation, all result from the seesaw mechanism. This is in sharp contrast with the warped standard model extension proposed in Ref. [13]. This also implies that, in contrast to Ref. [13], in the present model neutrinos must be Majorana particles, implying the existence of neutrinoless double beta decay, or 0νββ for short.
One can determine the expected ranges for the 0νββ decay amplitude, taking into account the allowed neutrino oscillation parameters obtained from experiment [3]. In Fig. 4 we plot the expected values for the mass parameter |m ee | characterizing the 0νββ amplitude. In a generic model the regions expected for inverted-ordered and normal-ordered neutrino masses are indicated by the broad shaded regions indicated in Fig. 4.
The current experimental bound from KamLAND-Zen [26] as well as the estimated experimental sensitivities are indicated by the horizontal lines [27][28][29][30][31][32]. We now show how, within our model, the predictions for the oscillation parameters imply important restrictions for the effective Majorana mass |m ee |. In fact, the allowed ranges are quite narrow. If the neutrino mass spectrum is inverted-ordered (IO), the effective Majorana masss has a lower limit |m ee | ≥ 0.0162 eV, while the lightest neutrino mass satisfies m lightest ≥ 0.0133 eV. In contrast, in the case of normalordering (NO), the effective mass |m ee | lies in the narrow interval [5.2meV, 9.6meV], and the allowed region of m lightest is [4.8meV, 7.2meV] 2 . As indicated in the figure, we expect that these predictions will be tested by the next generation 0νββ decay experiments.  In fact, as indicated in table II, the predicted neutrino mass parameter in β decay and cosmology are also interesting. These should be compared with the recent limits from the KATRIN experiment [33], and the 95% confidence limit for the sum of neutrino masses set by the Planck collaboration [25].

III. QUARK SECTOR
We now extend our model to the quark sector. The classification of the quark fields under the flavor symmetry T × Z 3 × Z 4 is given in table III, and no new flavon fields are required. We show the profiles of the zero models of  the quark fields in Fig. 5. It is straightforward to read off the down-type quark Yukawa interactions where dots stand for higher dimensional operators. Similarly the up-type quark Yukawa interactions take the form In the zero mode approximation, we integrate over the fifth dimension and then obtain the up-and down-type quark effective mass matrices as follows withỹ with The down-type quark masses are determined to be The product of the up-type quark mass matrix with its hermitian conjugate is of the following form The resulting up-type diagonalization matrix can be parameterized as where We find the up-type quark mass eigenvalues are with X ± = |m u 33 | 2 + |m u 32 | 2 + |m u 31 | 2 ± |m u 23 | 2 . As a result, the quark mixing matrix is given by from which we can extract the expressions of CP violation phase and Jarlskog invariant in the quark sector as follows,

IV. GLOBAL FIT OF FLAVOR OBSERVABLES
We have already discussed the predictions for the oscillation parameters in Eq. (21), shown in Figs. 2 and 3, as well as those for neutrinoless double beta decay and the quark sector prediction for the Cabibbo angle in Eq. (38). We now provide a global description of all flavor observables, including the quark and lepton mass parameters.
The resulting predictions for neutrino and charged lepton masses as well as lepton mixing parameters are given in table IV, and they reproduce very well current experimental data. For the quark sector we take The fitted value for the Jarlskog invariant is

V. SUMMARY AND CONCLUSIONS
We have proposed a realistic five-dimensional warped extension of the standard model where all leptons and quarks propagate in the bulk, see Figs. 1 and 5. We have assumed a T ⊗ Z 3 ⊗ Z 4 family symmetry broken on the branes by flavon fields. We have shown that it provides a consistent scenario where fermion mass hierarchies are accounted for by adequate choices of the bulk mass parameters, while quark and lepton mixing angles are restricted by the flavor symmetry. Neutrino masses are generated by the type-I seesaw mechanism, with the seesaw scale determined by the cut-off scale at the IR brane, generally suppressed by the exponential warp factor e −kL with respect to Λ, the fundamental UV scale. This naturally accounts for the large values required for the seesaw mechanism. Neutrino mixing parameters and the Dirac CP violation phase are all described in terms of just two independent parameters. The resulting predictions for the neutrino oscillation parameters are summarized in Figs. 2 and 3. In addition to these oscillation results we predict a 0νββ decay rate within reach of the upcoming generation of experiments, as seen in Fig. 4. Our scheme also provides a good description of the quark sector, as seen in   The T group is the double covering of the tetrahedral group A 4 . It has 24 elements which can be generated by three generators S and T and R obeying the relations 3 , The T group has seven inequivalent irreducible representations: three singlets 1, 1 and 1 , three doublets 2, 2 and 2 , and one triplet 3. The representations 1 , 1 and 2, 2 are complex conjugated to each other respectively. The two-dimensional representations 2, 2 and 2 are faithful representations of T group while the odd dimensional representations 1, 1 , 1 and 3 coincide with those of A 4 . In the present work we shall adopt the basis of [36,37]. with ω = e i2π/3 . In the doublet representations, the generators S and T are given by For the triplet representation 3, the generators are Notice that due to the choice of complex representation matrices for the real representation 3 the conjugate a * of a ∼ 3 does not transform as 3, but rather (a * 1 , a * 3 , a * 2 ) transforms as triplet under T . The reason for this is that T * = U T 3 T U 3 and S * = U T 3 SU 3 = S where U 3 is the permutation matrix which exchanges the 2nd and 3rd row and column. Similarly, notice that the irreducible representations 2 and 2 are complex conjugated to each other by a unitary transformation U 2 with i.e, T * 2 = U † 2 T 2 U 2 and S * 2 = U † 2 S 2 U 2 . Besides, the real doublet representation 2 and its complex conjugation are also related by the unitary transformation U 2 , i.e, T * 2 = U † 2 T 2 U 2 and S * 2 = U † 2 S 2 U 2 . Thus we have The T group can also be equivalently expressed in terms of three generators S, T and R with S 2 = R, RT = T R and (ST ) 3 = T 3 = In the following, we collect the Clebsch-Gordan coefficients for the decomposition of product representations in our basis, all the results are taken from [36,37]. We use α i to indicate the elements of the first representation of the product, β i to indicate those of the second representation. For convenience, we shall denote 1 ≡ 1 0 , 1 ≡ 1 1 , 1 ≡ 1 2 for singlet representations and 2 ≡ 2 0 , 2 ≡ 2 1 , 2 ≡ 2 2 for the doublet representations.
The contraction rules involving singlets representations in the product are as follows, where a, b = 0, 1, 2. The contraction rules for the products of two doublet representations are The products of doublet and triplet representations are decomposed as follows, Finally the contractions of two triplets are given by

Appendix B: 5-D profiles of Higgs and fermion fields
We formulate our model in the framework of Randall-Sundrum model [12], assuming the bulk of our model to be a slice of AdS 5 with curvature radius 1/k. The extra dimension y is compactified, and the two 3-branes with opposite tension are located at y = 0, the UV brane, and y = L, the IR brane. The bulk metric is non-factorizable, We have assumed the Higgs field to be in the bulk, so it has the standard Kaluza-Klein decomposition as where f H (y) is the zero mode profile. In this paper we adopt the zero mode approximation (ZMA) which identify standard model fields with zero modes of corresponding 5-D fields. As in Ref. [38] we take the profile f H (y) to be of the form with β = 4 + m 2 H /k 2 where m H is the bulk mass parameter of the Higgs field. For 5-D fermion fields, the three families of leptons and quarks are given as leptons : where the two signs in the bracket indicate Neumann (+) or Dirichlet (−) BCs for the left-handed component of the corresponding field on UV and IR branes respectively. The Kaluza-Klein decomposition for the two different BCs are where c L and c R are the bulk mass parameters of the 5-D fermion fields in units of k.