Neutrino Mixing and Masses from a Minimum Principle

We analyze the structure of quark and lepton mass matrices under the hypothesis that they are determined from a minimum principle applied to a generic potential invariant under the $\left[SU(3)\right]^5\otimes \mathcal O(3)$ flavor symmetry, acting on Standard Model fermions and right-handed neutrinos. Unlike the quark case, we show that hierarchical masses for charged leptons are naturally accompanied by degenerate Majorana neutrinos with one mixing angle close to maximal, a second potentially large, a third one necessarily small, and one maximal relative Majorana phase. Adding small perturbations the predicted structure for the neutrino mass matrix is in excellent agreement with present observations and could be tested in the near future via neutrino-less double beta decay and cosmological measurements. The generalization of these results to arbitrary sew-saw models is also discussed.


Introduction
The gauge interactions of the Standard Model (SM) admit a large, global, flavor symmetry. Matter fields in the SM are described by quark and lepton doublets, q L and L , and by right-handed singlets corresponding to up and down quarks and to electron-like leptons: U R , D R , E R . With three quark and lepton generations, massless neutrinos, and omitting U (1) factors unessential to the present work, the flavor group is [1]: (1.1) Masses for the observed neutrinos can be generated with the see-saw mechanism [2], by introducing at least two generations of additional Majorana neutrinos, N i . The latter are endowed with a Majorana mass matrix with possibly large eigenvalues, and coupled to the lepton doublets by Yukawa interactions. In analogy with the quark sector, here we assume three Majorana generations. We also assume the maximal flavor symmetry acting on the N i in the limit of vanishing Yukawa couplings but non-vanishing Majorana masses, i.e. O (3). The flavor group for this case is [3]: (1. 2) The large flavor group in Eq. (1.2), of course, does not correspond to observed symmetries. In the SM, global symmetries are explicitly broken by the Yukawa couplings of matter fields to the SU (2) L scalar doublet, whose vacuum expectation value (vev) is responsible for the spontaneous breaking of the SU (2) L ⊗ U (1) gauge symmetry [4].
Explicit breaking avoids the presence of unseen Goldstone or pseudo Goldstone bosons with the disadvantage, however, that values and textures of the Yukawa coupling matrices are out of reach of any theoretical consideration. Also, this picture presents the danger that any new physics will bring in new independent amplitudes to Flavor Changing Neutral Current (FCNC) processes, spoiling the extremely good SM predictions, see e.g. Ref. [5]. This can be avoided by the so called Minimal Flavor Violation (MFV) principle [6], the statement that in any new physics sector the only sources of flavor symmetry breaking accessible at low energies are the same Yukawa couplings. 1 Even the MFV principle, however, does not provide an explanation of the observed pattern of masses and mixings of quarks and leptons.
In this paper we want to discuss a different option, namely that the Yukawa couplings are the vacuum expectation values of Yukawa fields, to be determined by a minimum principle applied to some potential, V (Y ), invariant under the full flavor group G. In this case, one may use group theoretical methods to identify the natural extrema and characterize the texture of the resulting Yukawa matrices.
The simplest realization of the idea of a dynamical character for the Yukawa couplings is to assume that with Λ some high energy scale and Φ a set of scalar fields with transformation properties such as to make invariant the effective Lagrangians and the potential V (Y ) under G. To avoid the problem of unseen Goldstone bosons, G may be in fact a local gauge symmetry broken at the scale Λ, with an appropriate Higgs mechanism, see e.g. Ref. [11]. The idea that quark masses could arise from the minimum of a chiral SU (3) ⊗ SU (3) symmetric potential was considered in the sixties by N. Cabibbo, in the attempt to determine theoretically the value of the Cabibbo angle, and group theoretical methods were established in Refs. [12] and [13] to identify the natural extrema of the potential. Further attempts towards a dynamical origin of the Yukawa couplings, employing either G 0 or other flavor groups and different flavor-breaking fields, have been proposed in Refs. [15][16][17][18][19][20][21][22][23][24][25]. In all cases, the analysis reduces to determining the possible invariants made out of Yukawa fields, and explore their extrema.
In this paper, we apply the theoretical considerations devised in Refs. [12,13] for chiral symmetry, to the flavor group G in Eq. (1.2). Interestingly, we find different textures for the Yukawa matrices of quarks and leptons. For quarks, we find the hierarchical mass pattern of the third vs the first two generations and unity Cabibbo-Kobayashi-Maskawa (CKM) matrix. However, for leptons, hierarchical masses for charged leptons are accompanied by degenerate Majorana neutrinos with one, potentially two, large mixing angles and one maximal relative Majorana phase.
Both textures are close to the real situation. Moreover, if we add small perturbations to the neutrino mass matrix, we obtain a realistic pattern of mass differences and a Pontecorvo-Maki-Nakagawa-Sakata (PNMS) mixing matrix close to the bimaximal or tribimaximal mixing with small θ 13 , without having to resort to symmetries under discrete groups [26]. The prediction that large mixing angles are correlated to Majorana degenerate neutrinos with an average mass that we estimate could be as large as could be tested in a not too distant future in 0νν double beta decay, see e.g. Ref. [27], and possibly in cosmological measurements [28,29]. As anticipated, the idea that the pattern of the Yukawa couplings can be derived by the vev of appropriate fields is not new. In the present context, it was introduced by Froggat and Nielsen [15] for a global flavor U (1) symmetry. It was retaken for quarks in Refs. [16,18,[21][22][23][24], employing the symmetry group G 0 and analyzing the allowed minima of renormalizable potentials. The case of leptons with Majorana neutrinos was first analysed in a particular model [24], and later in general [25], in the context of the type I see-saw model with two heavy degenerate right-handed (RH) neutrinos, corresponding to an O(2) flavor symmetry, for both two and three light generations. In Ref. [24,25] it has been shown that the minimum of the scalar potential does allow a maximal mixing angle -in contrast to the quark case-and a maximal Majorana phase, associated to degenerate neutrinos. For three RH neutrinos, another two sizable angles were obtained within O(2), albeit at the price of adding a supplementary flavor vector component to the Yukawa field. The O(3) flavor symmetry for light neutrinos had been considered from a different perspective in Ref. [30], where a large angle is obtained after the introduction of small perturbations to the symmetric situation and almost degenerate neutrinos are envisaged. Our results provide a generalization of previous findings to the case of three heavy RH neutrinos. Most important, we show that the configuration with degenerate light neutrinos and one maximal mixing angle and Majorana phase holds irrespectively from the renormalizability of the potential and is therefore stable under radiative corrections.

Couplings and definitions
Yukawa couplings can be organised in a set of matrices, Y , which appear in the flavor symmetry breaking, gauge symmetry conserving, lagrangian: Majorana representation of gamma matrices is used throughout, H is the scalar doublet andH its charge conjugate. For quarks and charged leptons, we find: Integrating over the N fields and keeping the light fields only, one finds, to lowest order : which, upon spontaneous breaking of the gauge symmetry, gives the see-saw formula for the light neutrino mass matrix: The G transformations on Y , or Φ in Eq. (1.3), that make the lagrangian invariant are as follows: with the U unitary and O real orthogonal, 3 × 3 matrices. Using this arbitrariness, we can reduce the Yukawa matrices to a standard form: Because the Yukawa matrices are defined in between Fermi fields, see Eq. (2.1), an overall phase can be eliminated by appropriate (flavor-independent) phase redefinitions of the fields U R , D R , E R , and L . Thus we may assume the determinants of Y U , Y D , Y E , and Y ν in Eq. (2.1) to be real. In this case, all the y are diagonal, real and positive, all the U are unitary matrices and ω is a diagonal phase matrix of unit determinant. For simplicity, we shall assume that the generic fields Φ in (1.3) obey the same (G invariant) condition of having a real determinant. 2 In the quark sector we are left with one unitary matrix embodying the CKM mixing. To read neutrino masses we need in addition to diagonalize the matrix in Eq. (2.4). We write: which introduces the PMNS mixing matrix, withm ν diagonal, real and positive, and Ω a diagonal, Majorana phase matrix. It follows from Eq. (2.7) that there are in all 10 independent real parameters (i.e. invariants) in the quark sector. To count the real parameters appearing in Eq. (2.8), we start from neutrinos. We have 4 parameters in U L , as in the CKM matrix, 3 real eigenvalues in y ν and 3 parameters in U R counted as follows: 8 for a general 3 × 3 special, unitary matrix, less 3, corresponding to an orthogonal transformation we may perform on the Majorana fields, less 2 phases we include in ω. Adding the 3 real eigenvalues of Y E , we obtain a total of 15 parameters (and as many corresponding invariants, see below) for the lepton sector, in agreement with Ref. [31].
Note that the low-energy observable m ν , Eq. (2.9), contains 9 parameters only (4 for the U PMNS matrix, 3 mass eigenvalues and 2 Majorana phases). This is because we can factorize from Y ν a complex orthogonal hermitian matrix, hence 3 parameters, which would drop from the expression in Eq. (2.4), see Ref. [32].

Natural extrema of an invariant potential
We summarize here the elements to identify the natural extrema of an invariant potential V (x), that is those extrema that are less or not at all dependent from specific tuning of the coefficients in the potential, compared to the generic extrema. We do not make any assumption about the convergence of the expansion of the potential in powers of higherdimensional invariants, as done e.g. in Ref. [24,25].
The variables x are the field components, transforming as given representations of the invariance group G. In order to be invariant, , where I i are the independent invariants one can construct out of x. The crucial point is that the space of the x has no boundary, while the manifold M, spanned by I i (x), does have boundaries. The situation is exemplified in Fig. 1, with G = SU (3), and x=octet=hermitian, 3 × 3, traceless matrix. Defining the invariants I 1 = Tr(x 2 ) and I 2 = Det(x), the boundary is In general, let N be the number of algebraically independent invariants. One sees easily that [12,13]: • each point of M represents the orbit of x, namely the set of points in octet space given by: x g = gxg −1 , when g runs over G; • points on each boundary admit little (i.e. invariance) groups , which are the same up to a G conjugation.
The boundaries of M are characterized by the rank of the Jacobian matrix being less than maximum [13]: Boundaries are described by R = N − 1 manifolds (e.g. surfaces, for N = 3), each characterized by a different little group. Such manifolds meet along R = N − 2 dimensional manifolds (e.g. lines) which in turn meet along even lower dimensionality manifolds (e.g. singular points), etc.. Each of these boundaries corresponds to a particular little group. The extrema of V (x) are to be found by solving the equations: We may now state the following.
• V has always extrema on boundaries having as little group a maximal subgroup 3 of G [12]; • extrema of V with respect to the points of a given boundary are extrema of V (x) [13].
The latter extrema are more natural than the generic extrema in the interior of M, since they require the vanishing of only N − 1, or N − 2, etc. derivatives of V given that, on the boundary, J has 1 or 2, etc. vanishing eigenvectors (orthogonal to the boundary). Thus, from Fig. 1 we learn that it is more natural to break SU (3) along the direction of the hypercharge [12] (x with two equal eigenvalues, little group SU (2) ⊗ U (1)) than along the direction of T 3 , which corresponds to elements in the interior of M.
The case of chiral SU (3)⊗SU (3) was considered in Ref. [13], with the symmetry broken by quark masses in the (3,3) ⊕ (3, 3). The elements of such representation are complex, 3 × 3, matrices, M , transforming according to: By one such transformation, one can reduce M to the standard diagonal, positive, form ( up to an irrelevant overall phase [13]): There are three invariants, corresponding to the mass eigenvalues in Eq. (3.4): 5) and the natural extrema located on the boundaries correspond to the subgroups:  Turning to our case, we note that there are no common matrices in the transformations of quarks, Eq. (2.5) or leptons, Eq. (2.6). This implies that the invariants 4 divide in two independent sets: the Jacobian factorizes in two determinants and we can discuss the two cases separately.
To classify the invariants, we define two matrices which transform in the same way under SU (3) q and are singlet under the other transformations: There are six unmixed invariants, which we may take as: and the same for Y D Y † D . Next we define four mixed invariants: As anticipated, 10 independent invariants suffice to characterize in generality the physical degrees of freedom in the Yukawa fields. We stress in particular that the 4 invariants in Eq. (4.3) contain enough information to reconstruct the 4 physical parameters of the CKM matrix, including its CP-violating phase (up to discreet choices, see Ref. [14]), despite none of them vanishes in the limit of exact CP invariance. Unmixed invariants produce extrema corresponding to degenerate or hierarchical patterns as in the chiral case illustrated in (3.6). Mixed invariants involve the CKM matrix U , e.g.: The matrix P ij = |U ij | 2 enjoys the properties that: all elements are between zero and 1, the sum of the elements of any row equals the sum of the elements of any column and both sums are equal to unity. Such matrices (by the so-called Birkhoff-Von Neumann theorem [33]) are convex combinations of permutation matrices, i.e. matrices with a 1 and all other null elements in each row, the 1 being in different columns. Thus, permutation matrices provide us the singular points on the boundary of the domain, without having to compute the rank of the determinant. The upshot is that, after a relabeling of the down quark coupled to each up quark, we end up with U CKM = 1.
A more detailed analysis is given in Ref. [34]. It involves the calculation of the 10 × 10 Jacobian, which factorizes in two, 3 × 3 Jacobians (unmixed invariants) and a 4 × 4 one (mixed invariants) and it confirms the conclusion that one natural solution, for the three families quark case, is a hierarchical one, with dominating third family masses, and trivial CKM matrix.
In the limit of vanishing masses for the first two generations, this solution corresponds to the little group SU (2)

Leptons in three families
For leptons, we need 15 invariants. To construct them, we consider first the two combinations: We may construct unmixed and mixed invariants, as in the quark case, the mixed ones involving the matrix U L , Eq. (2.8). We choose the unmixed ones as: Unmixed, E : and three similar ones (I ν 1−3 ) using ρ ν , while the four mixed invariants containing ρ e and ρ ν are taken to be: Mixed, type 1 : For neutrinos we may construct also a matrix which transforms under the orthogonal group only: The symmetric and antisymmetric parts of σ ν transform separately and can be used to construct two different invariants, such as Here the first term in the product gives back the invariant , but the second one gives rise to new contractions which involve the unitary, symmetric matrix We thus define the following three additional invariants: Mixed, type 2 : Finally, we add two invariants which contain both U L and W : Mixed, type 3 : The discussion of the Jacobian leads to the following results, see again Ref. [34] for details.
• Unmixed invariants produce extrema corresponding to degenerate or hierarchical mass patterns.
• Mixed, type 1, invariants contain |(U L ) ij | 2 and lead, like in the quark case, to the conclusion that U L is a permutation matrix (up to an overall phase).
• Mixed, type 2, invariants contain |W ij | 2 and indicate that W = U R U T R is a also permutation matrix (up to an overall phase).
• Once we impose that U L and W are permutation matrices, the sensitivity of Mixed, type 3 invariants to ω vanishes. The latter remains therefore undetermined.
We may absorb the first permutation matrix in a relabeling of the neutrinos coupled to each charged lepton, but the second matrix leads to a non trivial result for the neutrino mass matrix, Eq. (2.9). The reason for the difference is that, for quarks we could eliminate any complex matrix U D by a redefinition of D R , but this is not possible for leptons, because we can redefine the N i only with a real orthogonal matrix. We use the freedom in the neutrino labeling to set U L = 1 in the basis where charged leptons are ordered according to: There are four possible symmetric permutation matrices that can be associated with W = U R U T R , one of them being the unit matrix. The other three imply non trivial mixing in one of the three possible neutrino pairs, e.g.
We introduced the minus sign for W to have a positive determinant, 5 consistently with the condition Det(U R ) = 1.
Using this expression in Eq.(2.9) leads to where y ν = diag(y 1 , y 2 , y 3 ) and ω= diag(e iα , e iβ , e −i(α+β ). The absence of mixing between the first eigenvector of m ν and those associated to the 2-3 sector implies that the phase α is unphysical and can be set to zero by an appropriate phase redefinition of the neutrino fields. From the second identity in Eq. (2.9) we then find: The non-trivial relative Majorana phase in the 2-3 sector is needed to bring all masses in positive form. The one maximal mixing angle and one maximal Majorana phase stem from the O(2) substructure in Eq. (5.10), as found in Ref. [24,25].
With three families we can go closer to the physical reality if we assume complete degeneracy for y ν . In this case, after the 2 − 3 rotation we are left with degenerate 1 and 2 neutrinos and, a priori, a new rotation will be needed to align the neutrino basis with the basis in which the charged lepton mass takes the diagonal form in Eq. (5.8). We may expect, in this case, the PMNS matrix to have an additional rotation in the 1 − 2 plane: We shall see that small perturbations around the solution in Eq. (5.10) allow to determine this angle, that remains non-zero in the limit of vanishing perturbations.

Group theoretical considerations
One may ask what is the little group corresponding to the extremal solution, Eq. (5.10).
In some sense we have to find the appropriate square root of m ν . By explicit calculation, one sees that the answer is given by 6 : In the limit y 1 = y 2 = y 3 , Y ν becomes proportional to a unitary matrix: where O is an orthogonal matrix generated by λ 2,5,7 . The O(3) diag would remain unbroken only in the case of degenerate charged lepton masses. Combining Y ν in Eq. (6.4) with Y E ∝ (0, 0, 1), we recover the little group SU (2) E ⊗ U (1) diag . Summarizing: Both breaking patterns of Y ν feature: i) at least two degenerate neutrinos; ii) θ 23 = π 4 and θ 13 = 0; iii) one maximal Majorana phase. In addition, the degenerate pattern in Eq. (6.4) implies three degenerate neutrinos and a second large (not calculable) mixing angle.
We finally note that all the features of the degenerate pattern can also be obtained without assuming the existence of heavy right-handed neutrinos, but rather starting from the flavor symmetry group G 0 in Eq. (1.1) and assuming that the effective (light) neutrino mass matrix, where s = sin(θ 12 ), c = cos(θ 12 ), and tan(2θ 12 ) = 2 √ 2 /σ. The PMNS matrix features a generically large θ 12 (that we cannot compute in absence of firm predictions for the values of and σ, but that does not goes to zero in the limit of vanishing perturbations), θ 23 close to π/4, and θ 13 generically small. With a suitable choice of the perturbations one can easily achieve the so-called bimaximal or tribimaximal mixing form.
The spectrum is almost degenerate, with normal or inverted hierarchy according to the signs of the perturbations, and mass splittings not correlated to the mixing matrix. Determining the size of the perturbations from | sin θ 13 | or, equivalently, from the deviation of θ 23 from π/4, and assuming a similar size for the perturbation controlling the largest mass splitting, we find A lightest neutrino mass of this size is within reach of the next generation of 0νν double beta decay experiments [27], and possibly of cosmological measurements [28,29]. Note also that the size of the perturbations is not far from what could be deduced from the charged lepton spectrum, treating m µ /m τ ≈ 0.06 as estimate of the sub-leading terms. So far we considered only real perturbations. Assuming complex perturbations does not change qualitatively the results listed above for the mass spectrum and the mixing angles, but leads to non-vanishing CP-violating phases. In particular, we find a generically large Dirac-type phase and a generically large second Majorana phase (that becomes physical), while small corrections to the maximal Majorana phase in the 2-3 sector. In this context, we note that small Majorana phases are enough for successful leptogenesis with almost degenerate right-handed neutrinos masses (see e.g. Ref. [36]).
In this letter we shall not try to identify the origin of the small perturbations. Given the non-perturbative character of the argument that leads to the U (1) diag symmetric boundary, the latter symmetry cannot obviously be lifted by G symmetric interactions in higher perturbative order. Small perturbations may result however from the effect of other fields, transforming differently from the Y 's and acquiring smaller vevs, like e.g. in Refs. [21,30,37], or by the effect of interactions external to the present scheme, e.g. arising from gravity.

Conclusions and outlook
We have assumed that the structure of quark and lepton mass matrices derives from a minimum principle, with the maximal flavor symmetry [SU (3)] 5 ⊗ O(3) and a minimal breaking due to the vevs of fields transforming like the Yukawa couplings. For leptons we find a natural solution correlating large mixing angles and degenerate neutrinos. This solution generalizes to three familes and arbitrary invariant potential the results found in Ref. [24,25]. The generalization of this result to arbitrary see-saw models has also been discussed. Subject to small perturbations, the solution can reproduce the observed pattern of neutrino masses and mixing angles. Our considerations lead to a value of the common neutrino mass that is within reach of the next generation of neutrinoless double beta decay experiments, and possibly within that of cosmological measurements.