Relating quarks and leptons with the T7 flavour group

In this letter we present a model for quarks and leptons based on T7 as flavour symmetry, predicting a canonical mass relation between charged leptons and down-type quarks proposed earlier. Neutrino masses are generated through a Type-I seesaw mechanism, with predicted correlations between the atmospheric mixing angle and neutrino masses. Compatibility with oscillation results lead to lower bounds for the lightest neutrino mass as well as for the neutrinoless double beta decay rates, even for normal neutrino mass hierarchy.

problem. Many extensions of the Standard Model (SM) have been proposed in order to induce nonzero neutrino masses [6] and to predict the oscillation parameters such as the neutrino mass ordering, the octant of the atmospheric mixing angle and the value of the CP-violating phase in the lepton sector.
A popular approach to tackle these issues is the use of discrete non-Abelian flavour symmetries which are known to be far more restrictive than Abelian ones [7]. In the literature there are many models based on, for instance, A 4 (the group of even permutations of a tetrahedron) whose simplest realizations predict zero reactor mixing angle and maximal atmospheric angle [8][9][10]. However, this nice prediction has now been experimentally ruled out [1][2][3][4] so that the corresponding models need to be revamped in order to account for observations [11].
A variety of possible predictions of flavour symmetry based models can be found, for instance [12]: i) neutrino mass sum rules leading to restrictions on the effective mass parameter |m ee | characterizing neutrinoless double beta decay (0νββ) processes [13][14][15]; ii) neutrino mixing sum rules [16].
Here we concentrate on the alternative possibility of having mass relations in the charged fermion sector. For definiteness we focus on the relation in Eq.(1), This relation was suggested in [17][18][19] and can hold at the electroweak scale 1 . First we note that such a relation between down-type quark and charged lepton masses can be understood because of group structure, when there are three vacuum expectation values and only two invariant contractions (Yukawas) in the product, 3 ⊗ 3 ⊗ 3. For example, such relation can be obtained with other groups containing three-dimensional irreducible representations (irreps) such as, for example, T n ∼ = Z n Z 3 (with n = 7, 13, 19, 31, 43, 49; [20]) as well as T .
In what follows we build a flavour model for quarks and leptons based upon the smallest non-Abelian group after A 4 , namely the flavour group T 7 [21][22][23][24][25][26] leading to the mass relation in Eq.(1). Neutrino masses are generated by implementing a Type-I seesaw [27] in contrast to 1 In an early paper Wilczek and Zee found, by using an SU (2) H symmetry, an extended mass relation the dimensional-five Weinberg operator approach used in previous Refs. [17][18][19]. We discuss the resulting phenomenological predictions, namely, a correlation between the lightest neutrino mass and the atmospheric angle, as well as lower bounds for the effective mass parameter |m ee | characterizing 0νββ decay for both neutrino mass orderings.

II. THE MODEL
Here we consider a model with the multiplet content in Table I where the SM electroweak gauge symmetry is crossed with a global flavour symmetry group T 7 . The down-type quarks Here for simplicity we have omitted the flavour indices, and have defined H d ≡ Hϕ d , H u ≡ Hϕ u andH ≡ iσ 2 H * , where ϕ a are T 7 flavon triplets and Λ is the scale at which these fields get their vacuum expectation values (vevs), ϕ a .
On the other hand, let us assume the existence of four RH-neutrinos accommodated as 3 ⊕ 1 0 under T 7 so that the Lagrangian for the neutrino sector becomes, where,H d ≡Hϕ d . Notice that the additional Abelian symmetry Z 7 couples each T 7 flavon triplet with only one fermion sector (down-type, up-type or neutrino sector) , so that, flavons transform non-trivially under the discrete Abelian group and their charges are unrelated to each other by conjugation. Therefore, in some sense, the order of the Z n symmetry is fixed by the Yukawa sector.
In what follows we will study the flavon potential for three distinct triplets under T 7 . The second column of Table II shows the vacuum expectation value alignments allowed in T 7 [21,28], with small deviations from those alignments shown in the third column.
In our case, ignoring the singlet ξ ν , there are three triplets, ϕ u , ϕ d and ϕ ν , with an additional Z 7 charge so that the flavon potential is given as where V α (with α = ν, d, u) are given by Eq.(4). Then, in components, V α contain the triplet elements ϕ α i and the parameters µ 2 α , λ α and κ α . The mixing part of the potential is the following The vev configuration written down in the second column of Table II is a minimum of the potential Eq.(6) when κ ν > 0, κ u < 0 and κ d < 0 and the terms κ 13 and κ 123 , are suppressed. 2 2 The term proportional to κ 13 in the potential could be suppressed by adding a term like −µ 2 13 (ϕ † ν ϕ d + h.c.) which softly breaks Z 7 . The trilinear term can be forbidden by invoking an additional parity transformation over the fields.
Notice that some vevs are orthogonal (namely, ϕ ν ⊥ ϕ u and ϕ u ⊥ ϕ d ). This property of the vevs has been described in [28,29]. In order to ensure a realistic model we assume small deviations from the vev canonical alignments in the middle column in Table II. Such deviations can be induced by adding soft breaking terms in the flavon potential, Eq.(6).

B. Mass relation in down-type sector
As usual, one obtains the fermion mass matrices after electroweak symmetry breaking from the Lagrangian in Eq. (2). Given and the T 7 multiplication rules (see AppendixA), one finds that the down-type quarks and the charged lepton mass matrices turn out to have the form where f = , d and θ f are unremovable phases contributing to CP-violation in the lepton and quark sector. In addition, we have used the following parameterization, It should be noticed that the matrices M f in Eq.(8) have the same structure as those in obtained with A 4 as flavour symmetry [17][18][19]30]. It is useful to rewrite Eq.(8) in the following way, where Following the reasoning in [17][18][19] we consider the invariants of M f M † f and obtain, at leading order in the limit r f >> α f , 1 and From Eq. (15) and the fact that the same flavon is coupled to the down-type quarks and charged leptons we are led to the mass relation in Eq. (1), It is worth mentioning that even when the phases θ f appear in the invariant det|M f M † f | with f = , d, that is in Eq.(13), the mass relation is preserved.

C. Quark mixing
From the Yukawa Lagrangian in Eq.(2) we have that after electroweak symmetry breaking the mass matrices for up-and down-type quarks are, respectively, where the parameters a d , b d and r d are given by Eq.(15), with ω 3 = 1 and the vevs u i (i = 1, 2, 3) defined through the parameterization It is useful to rewrite the vevs as follows, in that way there are 10 free parameters in the quark sector, listed in Table III. These parameters determine the six quark masses, the three CKM mixing angles and the quark CP-violating phase.
10 free parameters a d b d r d y u 1 y u 2 y u 3 α d α 1 α 2 θ d Table III: Parameters characterizing the quark sector.
The mass and CKM mixing parameters describing the quark sector, very similar to those in Eq. (16), were successfully reproduced, as seen in in Table II in [19], assuming trivial phases, namely θ d = 0, π in Eq. (16). However, even in this trivial case there is CP-violation due to the complex phase ω. Here for simplicity we just take advantage of the results given in [19] for the quark sector of our current T 7 model. Therefore we use the following values, given in the aforementioned A 4 model, and where λ = 0.2 the Cabibbo angle. The parameters r d , a d and b d can be computed by carrying out a substitution of (m 1 , m 2 , m 3 ) with the actual values of the down-type quark masses (m d , m s , m b ) in Eq. (15). One can verify with ease that the predictions for the CKM mixing matrix, quark masses and CP-violation are in agreement with the experimental data [31]. Now we proceed to study the lepton sector, for which some of the parameters will be fixed by the fit in the quark sector, namely the parameters α d and r d .

D. Lepton mixing
As we saw above, the spontaneous breaking of the electroweak symmetry yields the following form for the charged lepton mass matrix, where, from the T 7 multiplication rules in the appendix one finds, On the other hand, as mentioned in the introduction, here we adopt a Type-I seesaw approach for generating the neutrino masses. This is in contrast to previous models leading to the mass formula in Eq.(1) from the A 4 group. In those schemes an effective dimension-five operator approach was employed. In the present case the neutrino mass matrix is given by, where, where M i = κ 1 ϕ ν i (for i = 1, 2, 3) and M 4 = κ 2 ξ ν . The real matrix elements M i satisfy In order to implement the vev alignments in Table II we assume that the vevs u i and v i in Eq.(23) satisfy u 3 u 1,2 and v 1 v 2,3 . The former vev hierarchy has to do with the fit in the quark sector and the latter comes from the mass relation r d α d , 1. Then, the vev alignments can be rewritten as follows, where α 1 = 2.14λ 4 , α 2 = 1.03λ 2 , λ = 0.2 and we have defined u 3 = u, v 2 = v and M 3 = M .
Therefore, using Eqs. (23)(24), the light neutrino mass matrix after the seesaw mechanism turns out to be which is symmetric and α 1 = 2.14λ 4 , α 2 = 1.03λ 2 , λ = 0.2 and we have defined, It is important to note that some parameters in the neutrino mass matrix are fixed by the fit in the quark sector. In Table IV we list the parameters in the lepton sector denoting as Parameters in the lepton sector a b r d α d α 1 α 2 1 2 3 θ θ ν Fixed Free Table IV: Parameters in the lepton sector.
"fixed" those determined by the fit in the quark sector. Bear in mind that down-type quarks and charged leptons couple to the same flavon ϕ d and hence, α d = α and r d = r . This is the origin of the mass relation in Eq. (1).
Gathering all we have in the lepton sector we can compute the lepton mixing matrix, where U and U ν are the matrices that diagonalize the charged and neutral mass matrices, Remind that M is the matrix in Eq.(20) with one unremovable phase θ .

III. RESULTS
In our analysis, we have varied for instance i in the range [0, 5] and the phases θ ,ν in the range [0, 2π]. We make use of the neutrino mass matrix invariants trM 2 ν , detM 2 ν and (trM 2 ν ) 2 − tr(M 4 ν ) and choose to rewrite the three neutrino masses in terms of the square mass differences ∆m 2 atm and ∆m 2 sol and the lightest neutrino mass, m 1 for the case of normal hierarchy and m 3 for inverted hierarchy. We now sum up all our results.
The panel on the left in Fig.1 shows the correlation between the atmospheric angle for normal hierarchy (NH, i.e. |m 3 | > |m 2 | > |m 1 |) and the sum of neutrino masses (defined as Σ ≡ |m 1 | + |m 2 | + |m 3 |). We find that there is a lower bound for the lightest neutrino mass and that the first octant is favored by lighter neutrino masses. For reference we also display the constraint coming from the combination of cosmological CMB data from Planck and WMAP, including baryon acoustic oscillations (BAO) data from [32]. If taken at face value such stringent cosmological bound would disfavor not only heavy neutrinos but also the best fit value for the atmospheric angle lying in second octant [5]. values for the atmospheric angle [5] while the horizontal bands are allowed at 1σ. The vertical dot-dashed line is the cosmological bound from the combination of CMB and BAO data [32] On the other hand, a similar correlation between the atmospheric angle and the sum of neutrino masses, Σ, is also found for the inverted hierarchy case (IH, i.e. |m 2 | > |m 1 | > |m 3 |). This is shown on the right panel of Fig.1 where the dot-dashed vertical line is the constraint coming from the same combination of cosmological data [32]. Taking the most stringent cosmological (BAO) bound into account as well as the oscillation results one sees that, at 1σ, this case would be disfavored. Indeed, if this cosmological bound is taken at face value, the second octant would be excluded for inverse hierarchy. However, as seen in Fig. 2, at 3σ the second octant is certainly allowed for inverted hierarchy. The resulting lower bound for the lightest neutrino mass is much tighter than the one that holds for normal hierarchy. For comparison we also display the future sensitivity of the KATRIN experiment on tritium beta decay, Σ 0.6 eV, [33].
In summary, one sees that for both hierarchies our model implies a correlation between the atmospheric angle and the lightest neutrino mass. The current neutrino oscillation experiments lead to a lower bound for m 1 .
Such a lower bounds have implications for the effective mass parameter |m ee | specifying the neutrinoless double beta -0νββ -decay amplitude.
Let us now turn to the implications for 0νββ. In Fig. (2) we plot the effective parameter |m ee | as function of the lightest neutrino mass. The NH case corresponds to the purple/dark region, while the IH case is denoted by the magenta/light region, respectively. The vertical dotdashed line and labeled as "Cosmology" represents the constraint coming from the combination of CMB data [32], as well as the future sensitivity of KATRIN [33] indicated by the vertical dotted line. Figure 2: Effective neutrino mass parameter |m ee | versus the lightest neutrino mass for normal (purple/dark region) and inverted (magenta/light region) hierarchies. The vertical dotdashed line and labeled as "Cosmology" denotes the bound from the combination of CMB and BAO data [32]. The vertical dotted line is the future sensitivity of KATRIN, [33]. Here the oscillation constraints are taken at 3σ [5].

IV. CONCLUSIONS
In this paper we have suggested a model based on the flavour symmetry group T 7 leading to a very successful canonical mass relation between charged leptons and down-type quarks proposed in [17][18][19]. Previous papers predicting this mass relation have adopted the A 4 flavour symmetry and assumed that neutrino masses were generated through higher order operators.
In our T 7 model the neutrino masses are generated through the conventional Type-I seesaw mechanism.
The model leads to a correlation between the lightest neutrino mass and the atmospheric angle. This correlation implies lower bounds for the lightest neutrino mass which come from applying the neutrino oscillation constraints. These bounds on the lightest neutrino mass also translate to lower bounds on the effective amplitude parameter |m ee | characterizing 0νββ decay for both neutrino mass hierarchies. The group T 7 is a subgroup of SU (3) with 21 elements and isomorphic to Z 7 Z 3 . This group has five irreducible representations (e.i., 1 0 , 1 1 , 1 2 , 3 and3) and is known as the smallest group containing a complex triplet. The multiplication rules in T 7 are the following, Let X a = (x a 1 , x a 2 , x a 3 ) T ,X a = (x a 1 ,x a 2 ,x a 3 ) T , and z i (with i = 0, 1, 2), be triplets, anti-triplets and singlets, respectively, under T 7 then these elements are multiplied as follows: • X × X = X +X +X , where X = (x 3 x 3 , x 1 x 1 , x 2 x 2 ),X = (x 2 x 3 , x 3 x 1 , x 1 x 2 ) andX = (x 3 x 2 , x 1 x 3 , x 2 x 1 ), • X ×X = 2 a=0 z a + X +X , where z a = x 1x1 + ω 2a x 2x2 + ω a x 3x3 , X = (x 2x1 , x 3x2 , x 1x3 ), andX = (x 1x2 , x 2x3 , x 3x1 ), • z a × X = X where X = (z a x 1 , ω a z a x 2 , ω 2a z a x 3 ).