Repressing Anarchy in Neutrino Mass Textures

The recent results that $\theta_{13}$ is relatively large, of the order of the previous upper bound, and the indications of a sizable deviation of $\theta_{23}$ from the maximal value are in agreement with the predictions of Anarchy in the lepton sector. The quark and charged lepton hierarchies can then be reproduced in a SU(5) GUT context by attributing non-vanishing $U(1)_{FN}$ charges, different for each family, only to the SU(5) tenplet states. The fact that the observed mass hierarchies are stronger for up quarks than for down quarks and charged leptons supports this idea. As discussed in the past, in the flexible context of $SU(5)\otimes U(1)_{FN}$, different patterns of charges can be adopted going from Anarchy to various types of hierarchy. We revisit this approach by also considering new models and we compare all versions to the present data. As a result we confirm that, by relaxing the extreme ansatz of equal $U(1)_{FN}$ charges for all SU(5) pentaplets and singlets, better agreement with the data than for Anarchy is obtained without increasing the model complexity. We also present the distributions obtained in the different models for the Dirac CP-violating phase. Finally we discuss the relative merits of these simple models.

not add to zero, the vertex is forbidden in the symmetric limit. However, the U (1) F N symmetry (that we can assume to be a gauge symmetry 1 ) is spontaneously broken by the VEVs v f of a number of flavon fields with non-vanishing charges and GUT-scale masses. Then a forbidden coupling is rescued, but is suppressed by powers of the small parameters λ = v f /M , with M a large mass, with the exponents larger for larger charge mismatch. Thus the charges fix the powers of λ, hence the degree of suppression of all elements of mass matrices, while arbitrary coefficients c ij of order 1 in each entry of mass matrices are left unspecified (so that the number of parameters exceeds the number of observable quantities). A random selection of these c ij parameters leads to distributions of resulting values for the measurable quantities. For Anarchy (A) the mass matrices in the leptonic sector are totally random; on the contrary, in the presence of non-vanishing charges different entries carry different powers of the order parameter and thus some hierarchies are enforced. There are many variants of these models: fermion charges can all be non-negative with only negatively charged flavons, or there can be fermion charges of different signs with either flavons of both charges or only flavons of one charge. In models with no SeeSaw, the5 charges completely fix the hierarchies (or Anarchy, if the case) in the neutrino mass matrix. If Right-Handed (RH) neutrinos are added, they transform as SU (5) singlets and can in principle carry U (1) F N charges, which also must be all equal in the Anarchy case. With RH neutrinos the SeeSaw mechanism can take place and the resulting phenomenology is modified.
Anarchy and its variants, all sharing the dominance of ramdomness in the lepton sector, are to be confronted with models based on discrete flavour groups. These more ambitious models are motivated by the fact that the data suggest some special mixing patterns as good first approximations like Tri-Bimaximal (TB) or Golden Ratio (GR) or Bi-Maximal (BM) mixing, for example. The corresponding mixing matrices all have sin 2 θ 23 = 1/2, sin 2 θ 13 = 0, values that are good approximations to the data (although less so since the most recent data), and differ by the value of the solar angle sin 2 θ 12 .
The observed sin 2 θ 12 , the best measured mixing angle, is very close, from below, to the so called Tri-Bimaximal (TB) value [13][14][15][16][17] of sin 2 θ 12 = 1/3. Alternatively, it is also very close, from above, to the Golden Ratio (GR) value [18][19][20][21] (for a different connection to the GR, see Refs. [22,23]). On a different perspective, one has also considered models with Bi-Maximal (BM) mixing [24][25][26][27], where sin 2 θ 12 = 1/2, i.e. also maximal, as the neutrino mixing matrix before diagonalization of charged leptons. One can think of models where a suitable symmetry enforces BM mixing in the neutrino sector at leading order (LO) and the necessary, rather large, corrective terms to θ 12 arise from the diagonalization of the charged lepton mass matrices . Thus, if one or the other of these coincidences is taken seriously, models where TB or GR or BM mixing is naturally predicted provide a good first approximation (but these hints cannot all be relevant and it is well possible that none is). The corresponding mixing matrices have the form of rotations with fixed special angles. Thus one is naturally led to discrete flavour groups.
Models based on discrete flavour symmetries, like A 4 or S 4 , have been proposed in this context and widely studied [56][57][58][59][60][61][62]. In these models the starting Leading Order (LO) approximation is completely fixed (no chance), but the Next to LO (NLO) corrections still introduce a number of undetermined parameters, although in general much less numerous than for U (1) F N models. These models are therefore more predictive and typically, in each model, one obtains relations among the departures of  the three mixing angles from the LO patterns, restrictions on the CP violation phase δ, mass sum rules   among the neutrino mass eigenvalues, definite ranges for the neutrinoless-double-beta decay effective Majorana mass and so on.
The aim of this note on U (1) F N models is to update, on the basis of the present more precise data, our old analysis [63] that shows that, even if one accepts a mainly chaotic approach to lepton mixing, the Anarchy ansatz is perhaps oversimplified and that suitable differences of U (1) F N charges, which must in any case be present for tenplets, if also introduced within pentaplets and singlets, actually lead to distributions that are in much better agreement with the data with the same number of random parameters. In fact Anarchy can be improved by implementing mechanisms that enforce the relative smallness of θ 13 and of r = ∆m 2 solar /∆m 2 atm . The first goal can be achieved by restricting Anarchy only to the µ − τ (or 2-3) sector, by taking the pentaplet charges as5 ∼ (c, 0, 0), c > 0, ("µ − τ Anarchy", A µτ ) [64,65]. Both improvements can be realised by taking5 ∼ (c, d, 0), c > d > 0, (this "hierarchical", H, pattern was not considered in Ref. [63]). In each case the tenplet charges can be readjusted in order to maintain the correct ratios of charged fermion masses. We only consider models with normal hierarchy (NH) here, because, as shown in Ref. [63], in this framework models with inverse hierarchy (IH) tend to favour a solar angle close to maximal. In models with no SeeSaw, the5 charges completely fix the hierarchies (or the Anarchy) in the neutrino mass matrix, through the dimension-5 Weinberg operator, m ν = Ψ T 5 Ψ5 H u H u /M . The distributions arising from the models A, A µτ and H with no SeeSaw can directly be obtained and compared with the data and we shall see to which extent H is better than A µτ which, in turn, is better than A. If RH neutrinos are added, they transform as SU (5) singlets and can in principle carry U (1) F N charges, which also are all equal in the Anarchy case. With RH neutrinos the SeeSaw mechanism can take place and the resulting phenomenology is modified. It is easy to show that models with all non-negative charges and one single flavon have particularly simple factorization properties [66]. In particular, in the SeeSaw expression for the light neutrino mass matrix m ν = m T D M −1 m D , with m D and M denoting the neutrino Dirac and Majorana mass matrices, respectively, the dependence on the RH charges drops out in this case and only that from the5 remains. In these simplest models the only difference between the version with and without SeeSaw, for each model A, A µτ and H is that the extraction procedure for the random numbers is different: in the no SeeSaw version the entries of the neutrino mass matrix m ν are directly generated while in the SeeSaw case the extraction is done for m D and M and then m ν is derived by the SeeSaw formula. For example, in Anarchy models the smallness of r is to some extent reproduced by the spreading of the mass distribution resulting from the product of three mass matrices. Models with naturally large 23 neutrino mass splittings (so that r is small) are obtained if we allow negative charges and, at the same time, either introduce flavons of opposite charges or allow that matrix elements with overall negative charge are vanishing. For example, one can takē 5 ∼ (c, 0, 0) like in "µ − τ " Anarchy and 1 ∼ (e, −e, 0), e > 0, with two flavons of opposite charges, with equal VEV, and SeeSaw (we denote this model as Pseudo µτ -Anarchy,"P A µτ "). The "lopsided" structure of5 ∼ (c, 0, 0) results in naturally small 23 subdeterminant in the neutrino mass matrix after SeeSaw and to r naturally small.
In the following we discuss in more detail the models, the procedure of extraction of the random coefficients and the value of the expansion parameter λ that maximizes the success rate for each model.
We then discuss the mass distributions and mixing angles that we obtain and, finally, we compare these distributions with the data. The conclusion is that the most effective model is H in the no SeeSaw case and P A µτ in the SeeSaw case, while A µτ and A follow in the order and are much less successful.

II. MODELS AND RESULTS
In the following analysis we adopt the results of the fit of Ref. [1] (see also Ref. [2]). The 2(3)σ ranges in the case of NH are: We consider models with different patterns -Anarchy, genuine or Pseudo µτ -Anarchy, and hierarchy -induced by a U (1) flavour symmetry [12]: we present the transformation properties of all the fields in table I, in a notation that corresponds to the SU (5) GUT embedding. In the non-SeeSaw case, the neutrino mass matrix m ν is generated via the effective Weinberg operator Ψ T 5 Ψ5 H u H u /M . In the SeeSaw case the flavour charges determine the Dirac and RH Majorana mass matrices, m D and M , which give the effective neutrino mass m ν = m T D M −1 m D at low energy. If the RH neutrino charges all have the same sign and there is a single flavon, it is known that the structure of m ν in powers of λ is the same as for the non-SeeSaw case [67]. The coefficients in front of λ are randomly generated complex numbers c = |c|e iφc . In the spirit of the U (1) flavour symmetry, |c| = O(1) while the phase φ c is arbitrary.
We report in Fig. 1 the per cent success probability for each model as a function of the expansion parameter λ. We defined P = n ok /n tot , where n tot is the total number of randomly generated models (typically much larger than 10 6 ) and n ok is the number of models consistent with the 2σ ranges of Eq. (1) [1]. The relative error on P is estimated to be O(1/ n ok ). The thickness of the lines indicates this statistical error. Since the success rate depends on the window selected to extract the random coefficients, to estimate the ambiguity from this effect we flatly generated |c| in the interval [0. have also considered flat distributions for real and maginary parts of c, letting c vary in a square centered at the origin of the complex plane. We have studied the dependence on the size of the square. We got slightly different results for the success probabilities, for the value of the parameter λ that maximizes the success rate and for the distributions of the various observables. However, the relative ability of the different models to fit the data, that we consider the most important outcome of the present analysis, is stable and independent from the distributions we used to generate the input coefficients c. In Fig. 2 we display the probability distributions for r, sin θ 13 , tan 2 θ 12 and tan 2 θ 23 , fixing λ at a representative value and selecting |c| ∈ [0.5, 2]. These distributions have been normalized so that the integrated probability is equal to unity. Notice that we use a logarithmic scale. The (green) shaded vertical regions refer to the experimental data at 2σ [1].
We now discuss each model in turn.
• Anarchy (A) [8,10,11]. Neglecting the randomly generated coefficients, the texture of the mass matrices for charged leptons and neutrinos (with or without SeeSaw), expressed in powers of λ, The success rate for neutrino masses and mixing angles is independent of λ. We then choose λ = 0.2 − 0.3, which ensure a reasonable hierarchy for charged fermions according to the charges selected for the Ψ 10 representation. As can be seen from Fig. 1, the success rate is quite small in both the no SeeSaw and SeeSaw cases: for the no SeeSaw case the success rate has been multiplied by a factor of 10 to facilitate its comparison with the other models. The reasons for such a modest performance can be understood by inspecting Fig. 2: the most severe problem is the prediction of a too large θ 13 and, in the no SeeSaw case, also of a too large value of r. In the SeeSaw case the latter problem is cured by the spreading of neutrino mass eigenvalues produced by the product of three random matrix factors. As for the mixing angles θ ij the distributions are all similar and, with a logarithmic scale, appear peaked near π/4.
• µτ -Anarchy (A µτ ) [64,65,69]. In this case only the µτ block of m ν is anarchical and the success rate is maximized for λ ∼ 0.2 and λ ∼ 0.28 for the no SeeSaw and SeeSaw cases respectively. In both cases the performance of A µτ is better than A. The main problem of the A µτ model is the prediction of a too small θ 12 and, in the no SeeSaw case, also a too large value for r. If by accident the 22 matrix element of m ν is numerically of order λ then θ 12 ∼ O(1) and √ r ∼ O(λ): with a single fine tuning one fixes both problems.
• Pseudo µτ -Anarchy (P A µτ ) [63]. This is a SeeSaw model with two flavons of opposite charges and equal VEVs. The 2 and 3 entries of the pentaplets have the same charges, but the 1 and 2 RH neutrinos have opposite charges. The result is that m ν displays an apparently anarchical 23 sector, • Hierarchy (H). Both for the SeeSaw and no SeeSaw cases, the charged lepton and neutrino mass matrices read The success rate is maximized for λ ∼ 0.4 and λ ∼ 0.45 for the non-SeeSaw and SeeSaw cases, respectively. For non-SeeSaw one finds a successful model every 10, 000 trials, that is a factor of 100 better than the Anarchy texture. For the SeeSaw the success rate is slightly lower, but still a factor of 10 better than Anarchy. This is mainly due to the hierarchical structure in both 12 and 23 sectors, that ensures a small r ∼ λ 4 and tan 2 θ 12 ∼ tan 2 θ 23 ∼ sin θ 13 ∼ λ 2 . As can be seen from Fig. 2, with λ = 0.4, the maxima of the distributions of these observables are nicely close to their experimentally allowed range at 2σ. Notice that the distributions of θ 12 and θ 23 are similar. In particular, θ 23 is peaked at a slightly smaller value than its present experimental 2σ range. Despite this, the rate of success of θ 23 equals those of the three previously studied versions of anarchical models, namely A, A µτ and P A µτ .
We can now directly compare the four models. As from Fig. 1, H is the best performing model for the non-SeeSaw case, for values of λ larger than about 0.3, while for smaller values, A µτ has the best success rate. This suggests that a moderate hierarchy could likely be realized in the neutrino sector. For the SeeSaw case, the performances of H and P A µτ are almost equivalent, although P A µτ is slightly better.

III. DIRAC CP PHASE
It is also interesting to study the distribution of the Dirac CP-violating phase δ in the models considered. At present, there is just a very mild 1σ preference for δ ∼ π in the NH case [1].
In order to extract the phase δ, we consider the following combination that is an invariant under phase transformation of the fields. This is only one of the possible invariants that can be considered (notice that the imaginary part of U e3 U * e2 U * µ3 U µ2 corresponds to the Jarlskog invariant [70]). Adopting the usual PDG parameterisation of the PMNS matrix, we get I |U µ2 | = cos θ 23 cos θ 12 e −iδ − sin θ 23 sin θ 13 sin θ 12 .
The distributions of δ are shown in Fig. 3. As one expects, for the Anarchy model the distribution of δ is completely flat. On the other hand, for H and A µτ there is a mild preference for a vanishing value of δ, while for P A µτ this preference is even weaker.

IV. CONCLUSION
Over the years there has been a continuous progress in the measurement of neutrino mixing angles culminating recently with the determination of a relatively large value of θ 13 and with the indication that θ 23 is not maximal (some hints that cos δ 0, with δ being the Dirac CP-violating phase in neutrino oscillations, are also emerging). In spite of this remarkable experimental progress the data can still be reproduced by a wide range of theoretical models. At one extreme we have models where the assumed dynamics is minimal and the dominant ingredient is pure chance (Anarchy and its variants) and, at the other extreme, models with a high level of underlying symmetry, like, for example, those based on discrete non-Abelian symmetries (which start at LO with TB or BM mixing).
The large value of θ 13 and the departure of θ 23 from maximal both go in the direction of Anarchy and move away from the TB or BM limits, where θ 13 = 0 and θ 23 is maximal. In this note we have made a reappraisal of Anarchy, given the new experimental results. To make connection with quark masses and mixing we have adopted the (SUSY) SU (5) ⊗ U (1) FN GUT framework. The Anarchy prototype model has only tenplet charge differences (among the 3 generations) that are non-vanishing, while all pentaplet and singlet charge differences are taken as vanishing. Here we argue on the most recent data that the Anarchy ansatz, in the context of SU (5) ⊗ U (1) FN models, is simple, elegant and viable but does not provide a unique interpretation of the data in that context. In fact, suitable differences of U (1) F N charges, if also introduced within pentaplets and singlets, lead to distributions that are in much better agreement with the data with the same number of random parameters as for Anarchy.
The hierarchy of quark masses and mixing and of charged lepton masses in all cases impose a hierarchy defining parameter of the order of λ C = sin θ C , with θ C being the Cabibbo angle. The weak points of Anarchy (A) are that with this ansatz all mixing angles should be of the same order, so that the subdeterminant is suppressed and thus r is naturally small with unsuppressed θ 23 . In addition also θ 12 is large, while θ 13 is suppressed. We stress again that the number of random parameters is the same in all these models: one coefficient of O(1) for every matrix element. Moreover, with an appropriate choice of charges, it is not only possible to reproduce the charged fermion hierarchies and the quark mixing, but also the order of magnitude of all small observed parameters can be naturally guaranteed.
Thus finally we agree that models based on chance are still perfectly viable, but we consider Anarchy as a simplest possibility that has to be validated on the data in comparison with other similar models and we argue in favor of less chaos than assumed in Anarchy.