Different SO(10) Paths to Fermion Masses and Mixings

Recently SO(10) models with type-II see-saw dominance have been proposed as a promising framework for obtaining Grand Unification theories with approximate Tri-bimaximal (TB) mixing in the neutrino sector. We make a general study of SO(10) models with type-II see-saw dominance and show that an excellent fit can be obtained for fermion masses and mixings, also including the neutrino sector. To make this statement more significant we compare the performance of type-II see-saw dominance models in fitting the fermion masses and mixings with more conventional models which have no built-in TB mixing in the neutrino sector. For a fair comparison the same input data and fitting procedure is adopted for all different theories. We find that the type-II dominance models lead to an excellent fit, comparable with the best among the available models, but the tight structure of this framework implies a significantly larger amount of fine tuning with respect to other approaches.


Introduction
In this article we make a quantitative comparison of the performance of different types of SO(10) Grand Unification Theories (GUT's) in reproducing the observed values of fermion masses and mixing, also including the neutrino sector. By now we have a rather precise knowledge of the leptonic mixing angles [1][2][3][4] which, within the experimental accuracy, are consistent with the Tri-Bimaximal (TB) pattern [5], [6] and, as such, are very different from the quark mixing angles. A still open and challenging problem is that of formulating a natural model of Grand Unification based on SO(10), leading not only to a good description of quark masses and mixing but also, in addition, of charged lepton masses and to approximate TB neutrino mixing. In SO(10) the main added difficulty with respect to SU (5) is clearly that all fermions in one generation belong to a single 16dimensional representation, so that one cannot separately play with the properties of the SU(5)-singlet right-handed neutrinos in order to explain the striking difference between quark and neutrino mixing. There are a number of rather complete SO(10) models, with different architectures, that, without having a built-in TB mixing yet are able to reproduce the data on neutrino mixing angles [13,14,16,17,19,38,40]. These models fall in different classes: renormalizable or not, with lopsided or with symmetric mass matrices, with various assumed flavour symmetry, with different types of see-saw and so on. In most of these models some dedicated parameters are available to fit the observed neutrino masses and mixing angles. In these models TB mixing appears as accidental, and if the data would become somewhat different, the new values of the mixing angles could as well be fitted by simply changing the values of the parameters. These models are certainly interesting and, in this article, a number of them will be confronted with the data and their respective performances will be compared and taken as a reference. A more difficult goal would be the construction of SO(10) models where TB mixing is built in and is automatic in a well defined first approximation, due, for example, to an underlying (broken) flavour symmetry. The leading approximation in these models is particularly constrained and the neutrino mixing angles are fixed in this limit. There are a number of GUT models of this type based on SU(5) (see, for example, [7,9,10]), but, as mentioned, the SO(10) case is more difficult and the existing attempts, in our opinion, are still not satisfactory in all respects.
A promising strategy in order to separate charged fermions and neutrinos in SO(10) is to assume the dominance of type-II see-saw [11] (with respect to type-I see-saw [12]) for the light neutrino mass matrix. Grand Unified SO(10) models based on type-II see-saw dominance have been studied in refs. [20-24, 27, 32, 34-36]. If type-II seesaw is responsible for neutrino masses, then the neutrino mass matrix (proportional to) f (see eqs. (1,3,4 )) is separated from the dominant contributions to the charged fermion masses and can therefore show a completely different pattern. This is to be compared with the case of type-I see-saw where the neutrino mass matrix depends on the neutrino Dirac and Majorana matrices and, in SO (10), the relation with the charged fermion mass matrices is tighter. Here we do not consider the problem of formulating a flavour symmetry or another dynamical principle that can lead to approximate TB mixing, but rather study the performance of the type-II see-saw SO(10) models in fitting the data on fermion masses in comparison with other model architectures. Actually, we will show in Sect. 2 that, without loss of generality, we can always go to a basis where the matrix f is of the TB type. In fact, since the TB mixing matrix is independent of the mass eigenvalues, the most general neutrino mass matrix, which is a symmetric complex matrix, can always be transformed into a TB mixing mass matrix by a change of the charged lepton basis. The observed deviations from TB mixing will then be generated by the diagonalisation of charged leptons and, in order to agree with the data, must be small. In turn the charged lepton mixings are related in SO(10) to the quark mixings. Thus, in this class of models TB mixing is exact in the approximation of neglecting charged fermion mixings. When a symmetry guarantees TB mixing in first approximation the corrections from the diagonalisation of charged leptons are automatically small, while in general could be large. The main purpose of our analysis is to see to which extent this particular structure models is supported by the data among different types of SO (10) .
In renormalizable SO(10) models (a non necessary assumption which is only taken here in some cases for simplicity) the fermion masses are generated by Yukawa couplings with Higgs fields transforming as 10, 126 (both symmetric) and 120 (antisymmetric). Alternatively, in non renormalizable SO(10) models the large representations 126 and 120 can be effectively obtained from the tensor products of smaller Higgs representations. The 10 Yukawa couplings contributing to up, down and charged lepton masses in most models have a large 33 term, corresponding to the large third generation masses, while all other entries are smaller and lead by themselves to zero CKM mixing (because the 10 contributes equally to up and down mixing). Quark mixings arise from small corrections due to 126, the same Higgs representation that determines f which, in models with type-II see-saw, is dominant in the neutrino sector, and to 120. Thus, in this approach, in the absence of 120, there is a strict relation between quark masses and mixings and the neutrino mass matrix. The presence of 120 dilutes this connection which however still remains important. In particular the small deviations from TB mixing induced by the diagonalisation of the charged lepton mass matrix, barring cancelations, are of the same order as the quark mixing angles and most of the parameters appear in both.
An interesting question is to see to which extent the data are compatible with the constraints implied by this interconnected structure. A goal of this work is precisely to study how the inclusion of TB mixing along these lines is reflected in the ability of the model in reproducing the data in comparison with different structures. Thus, independent of the problem of determining a flavour symmetry that can fix the parameters to their required values, we study SO(10) models based on the dominance of type-II see-saw with respect to their performance in fitting the fermion masses and mixings in comparison with alternative realistic SO(10) models. Some other analyses have appeared in the literature where SO(10) models with dominance of type-II see-saw have been confronted with the data (see, for example, [32,33,38]). The present analysis is different and, in some respect, more general and, in addition, also includes the comparison with the most established models based on SO(10) that can be considered both realistic (i.e. that have been worked out to a level that a comparison with the data is possible) and complete (i.e. that also include the neutrino sector). Each model will be compared with the same set of data on masses and mixings given at the GUT scale. For this purpose we specify a set of data for reference. We are not much concerned with the uncertainties which for sure exist in evolving the physical quantities measured at the electroweak scale up to the corresponding ones at the GUT scale. We will adopt a reasonable evolution up to the GUT scale, as can be found in the literature [33,[41][42][43], and consider the resulting set of data as the truth and fit all relevant models to that set (to be precise, we actually consider two sets of data, one for models with small tan β and one for those with large tan β). We argue that if a particular model is sizably better than another in fitting these idealized data it will also score better on the real data. For our comparison the first quality factor is the χ 2 or the χ 2 /d.o.f. obtained from the fit for each model. We also introduce a parameter d F T for a quantitative measure of the amount of fine-tuning of parameters which is needed in each model. We find that SO(10) models based on dominance of type-II see-saw can fit the data remarkably well, both in absolute terms and in comparison with other types of models, but at the price of a substantial amount of fine-tuning needed to compensate for the tension introduced by the double role of f in determining both the neutrino mass matrix and the charged fermion masses and mixings. In particular the smallness of the first generation masses requires a precise cancellation of larger terms.
This article is organised as follows. In Sect. 2 we define our reference model, renormalizable with type-II dominance. In Sect. 3 we discuss the relations between parameters of the model and measured quantities. In Sect. 4 we specify the set of data that we adopt at the GUT scale. In Sect. 5 we describe our fitting procedure and the quality indicators that we consider to compare different models. In Sect. 6 we briefly describe a number of alternative SO(10) models and we fit them to the same data as we did for our reference model. Finally in Sect. 7 we present our summary and conclusion. The best fit parameters and observables are listed in Appendix A for all the competing models.

A class of SO(10) models
We consider the class of renormalizable Supersymmetric (SUSY) SO(10) models with dominance of type-II see-saw for neutrino masses [20-24, 27, 32, 34-36]. In the following we indicate the generic model of this class by T-IID (from Type-II Dominance). In renormalizable SO(10) models the Higgs fields that contribute to fermion masses are in 10 (denoted by H), 126 (∆) and 120 (Σ). The Yukawa superpotential of this model is then given by: where the symbol ψ stands for the 16 dimensional representation of SO(10) that includes all the fermion fields in one generation. The coupling matrices h and f are symmetric, while h ′ is anti-symmetric. The representations H and ∆ have two standard model (SM) doublets in each of them whereas Σ has four such doublets. At the GUT scale M GU T , once the GUT and the B − L symmetry are broken, one linear combination of the up-type and one of down-type doublets remain almost massless whereas the remaining combinations acquire GUT scale masses. The electroweak symmetry is broken after the light Minimal Supersymmetric Standard Model (MSSM) doublets (to be called H u,d ) acquire vacuum expectation values (vevs) and they then generate the fermion masses. The resulting mass formulae for different fermion masses are given by (see, for example, [35]): where Y a are mass matrices divided by the electro-weak vev's v u,d and r a (a = 1, 2, 3 and c b (b = e, ν) are the mixing parameters which relate the H u,d to the doublets in the various GUT multiplets.
In generic SO(10) models of this type, the neutrino mass formula has a type-II [11] and a type-I [12] contributions: where v L is the vev of the B − L = 2 triplet in the 126 Higgs field. Note that in general, the two contributions to neutrino mass depend on two different parameters, v L and v R , and it is possible to have a symmetry breaking pattern in SO(10) such that the first contribution (the type-II term) dominates over the type-I term. The possible realisation of this dominance and its consistency with coupling unification has been studied in the literature [24,[27][28][29] and found tricky but not impossible [31]. The neutrino mass formula then becomes Note that f is the same coupling matrix that appears in the charged fermion masses in Eq.
(2), up to factors from the Higgs mixings and the Clebsch-Gordan coefficients. Also note that the neutrino Dirac mass, proportional to Y ν D in eqs. 2, only enters in the neglected type-I see-saw terms and does not play a role in the following analysis. The equations (2) and (4) are the key equations in this approach. For generic eigenvalues m i , the most general matrix that is diagonalised by the TB unitary transformation is given by: where, with a specific phase convention, we can take: In this convention U T B is a real orthogonal matrix and all phases can be included in the eigenvalues m i . Then the matrix f is symmetric with complex entries and, from eq. 5, one obtains: with: An important observation is that, for a generic neutrino mass matrix f ′ , we can always go to a basis where f ′ is diagonalised by the TB unitary transformation in eq. 6 and is of the form in eq. 7. In fact, if we start from a complex symmetric matrix f ′ not of that form, it is sufficient to diagonalise it by a unitary transformation U: As a result the matrices f and f ′ are related by a change of the charged lepton basis induced by the unitary matrix O = UU † T B (in SO(10) the matrix O rotates the whole fermion representations 16 i ). Since TB mixing is a good approximation to the data we argue that this basis is a good starting point. In other words, for the physical neutrino mass matrix, TB mixing is a good approximation so that the unitary transformation O is close to the identity. In this basis the deviations from TB mixing will be generated by the diagonalisation of charged leptons which, in order to agree with the data, must be small. At the same time also the quark mixings must be small in order to correspond to the data.
In the selected basis, the parameterisations adopted for the matrices h (symmetric) and h ′ (antisymmetric) are given by: and The h 33 element of h has been emphasised with the special notation Y because usually it is the dominant term: Y >> h ij . All the matrix elements f ij , Y , h ij and σ ij , as well as r a and c b that appear in eq. 2, will be taken as real. This leads to a crucial economy of parameters (justified by the fact that, as we shall see, the resulting fit is very good) and can be seen to correspond to an underlying "parity" symmetry [34] that implies that all mass matrices obtained from h, h ′ and f are hermitian. Note that, due to the imaginary unit factor i in front of the h ′ matrix, this purely imaginary matrix will in general induce CP violation in the quark and lepton sectors. The charged fermion mass matrices are given by: where tan β = v u /v d , with v u,d being the vacuum expectation values of H u,d , and M u , M d and M e refer to up, down and charged leptons, respectively. Note that all mass matrices are hermitian, hence each of them can be diagonalised by a unitary matrix. We have 6 real parameters in h, 3 in h ′ , 3 in f plus r 1 /tanβ, r 2 , r 3 , c e and the ratio v L /v u , or a total of 17 parameters. The (in principle) measurable quantities are 12 fermion masses, 6 mixing angles, 2 CP violating phases (we do not include Majorana phases) or a total of 20. Only 18 of these observables have been measured (the PNMS phase and one of the neutrino masses are unconstrained). In the following more predictive versions of the model with less free parameters will also be considered (for example, the JK2 model in Sect. 6).

Observables and Parameters
Starting from the neutrino mass matrix in eqs. 4, 7 we obtain: The oscillation frequencies are given by: and their ratio r is: The experimental smallness of r ∼ 1/30 suggests that The deviations of the leptonic mixing angles from the TB values are induced by the diagonalisation of the charged lepton mass matrix given in eq. 13. As M e is hermitian it is diagonalised by a unitary transformation: where M diag e stands for a diagonal matrix with real non-negative elements m e , m µ , m τ and U e is the relevant unitary matrix. Any unitary matrix can in general be written as (see, for example, [8]): where φ i (i=0,...,4) run from 0 to 2π andŨ is the standard parameterization for the CKM mixing matrix, namelỹ where all the mixing angles belong to the first quadrant and δ to [0, 2π]. An approximate form of m e that follows from eq. 17 for m τ >> m µ >> m e , in a linear approximation in the small mixing angles unless non linear terms are rescued by large mass factors, is given by: (20) Going back to eq.13 and similar ones for the heavier generations, in the above approximation, we can write down simple analytic expressions for the mass eigenvalues (except for the first generation mass which is fine tuned in this class of models) and the mixing angles and phases. Note, however, that exact expressions are used in our numerical fits. Starting from the 2-3 family sector we have: We also obtain: The following expression for s e 12 is less accurate but, in general, still sufficiently good, at least for indicative purposes. It is obtained from the 11 entries of the matrices in eqs. 13, 20 by neglecting m e : A remaining phase can be derived from the equation (obtained from the 12 entries of the matrices in eqs. 13,20): From fitting the data one finds that indeed Y is the largest parameter, followed by h 13 , h 22 , h 23 , σ 13 , σ 23 and f 0 while f 1 , f 2 , h 11 , h 12 and σ 12 are still smaller. In a linear approximation in the leptonic mixing angles s e ij the corrections to U T B from the diagonalization of charged leptons are given by (for each matrix element we omit an overall phase): It turns out, however, that the above linear approximations are often not sufficiently accurate because the leptonic mixing angles s e ij are not small enough. This is particularly true for sin θ 12 and sin θ 13 . So the above linearised formulae are only given for indicative purposes and, in our fit, we used the exact expressions for the mixing, obtained from U † e U T B from eqs. 18,19. But the approximate analytic formulae are useful to understand the need of fine tuning to reproduce the observed masses for the light generations of leptons and the neutrino masses and mixing. We see in fact from eqs. (15,16) that neutrino masses impose a strong constraint on the values of f i . But the same f i enter in the charged lepton masses and the leptonic mixing angles eqs. (22,23,24) and they must conspire with the h ij and σ ij parameters in order to reproduce the observed values. And the same is also true for the quark masses that we now discuss.
In the quark sector the approximate formulae for the 2-3 families, analoguous to eqs. 22 for charged leptons, are derived from eqs. 11, 12: Note that, as Y >> |f i |, m b = m τ approximately holds at the GUT scale (bottom-tau unification). Relations analogous to eqs. 23, 24, 25 can be readily written down for the up and down quark sectors. For the CKM mixing matrix we go back to eq. 18 and derive U u and U d that diagonalise m u and m d , given in eqs. 11, 12, respectively. We then construct

A set of idealized data at the GUT scale
For the charged fermion masses and CKM mixings at the GUT scale we used the values given in Tab.1(a)-Tab.1(b) as input for our fitting procedure. These values are based on the analysis of ref. [42], in the MSSM framework, and were obtained from a two-loop Renormalization Group Evolution (RGE) from the SUSY scale at about 1 T eV to the GUT scale ∼ 2 × 10 16 GeV . Ref. [42] can be considered as an upgrade of a previous work [41]. This analysis has been repeated for two different values of the supersymmetric (SUSY) parameter tan β: tan β = 10 and tan β = 55. In fact tan β is the most important SUSY parameter that directly affects the RGE. So these tables have been used to fit two typical classes of models with small or large tan β. Actually for both values of tan β we have used in the fit an updated version of the data which was derived in refs. [33], [43] shown in Tab.1. In particular, refs. [33], [43] use more recent values for m u , m d and m s at low energy with respect to the ones used in ref. [42]. The errors taken on the data points at the GUT scale are those shown in ref. [33] or in ref. [43] and do not include an estimate of the additional ambiguities associated with the chosen procedure. For example, the SUSY spectrum parameters, which mainly enter in the evolution through the threshold corrections, are subject to considerable ambiguities [44]. Moreover at the GUT scale other threshold corrections appear, but they are again affected by large uncertainties and model dependent. As we are only interested in defining a set of idealized data in order to make a meaningful performance test for a number of different models, we decided to ignore the uncertainties on both the SUSY-and GUT-scale threshold corrections. As already discussed we are not much concerned with these uncertainties, rather we prefer to fit all models on the same set of data, arguing that if a particular model is better than another in fitting these idealized data it would also score better on the real data if they were known.  2.2 ± 0.6 Table 1: GUT scale data for charged fermions for tgβ = 10 (ref. [33]) and tan β = 55 (ref. [43]) Concerning the neutrino sector, we have ignored the effects of the evolution from the low energy scale to the GUT scale. In fact for all the models analysed here the mass spectrum is non degenerate, so that the evolution can be considered negligible to a good approximation [46]. The low energy data are taken from [4]

Fitting procedure and quality factors
As already discussed, the class of models T-IID described in Sect. 2 contains a total of 17 independent parameters. We are going to compare these models with the set of 18 "measurements" described in the previous section. As shown in eq. 12, 13 the parameter tan β always enters in the combination r 1 / tan β in all the observables. So in these models it is possible to obtain a large ratio m t /m b without making tan β large (in fact, large tan β can be problematic in the MSSM with a GUT Yukawa t − b − τ unification [45]). Here we are going to fit the model presented in Sect. 2 on the data in Tab.1(a), which refer to the tan β = 10 case. We use the numerical minimization tool Minuit2 developed at CERN.
We introduce a parameter d F T for a quantitative measure of the amount of fine-tuning needed in the models. This adimensional quantity is obtained as the sum of the absolute values of the ratios between each parameter and its "error", defined for this purpose as the shift from the best fit value that changes χ 2 by one unit, with all other parameters fixed at their best fit values (this is not the error given by the fitting procedure because in that case all the parameters are varied at the same time and the correlations are taken into account): It is clear that d F T gives a rough idea of the amount of fine-tuning involved in the fit because if some |err i /par i | are very small it means that it takes a minimal variation of the corresponding parameters to make a large difference on χ 2 . The value of d F T for our best fit output is shown in Tab.3(a). To get a better idea of the significance of this number, one can compare it with a similar number d Data based on the data, i.e. the sum of the absolute values of the ratios between each observable and its error as derived from the input data: In particular for the set of data in Tab

Parameter
Best fit value 1.17 6.06 85.0 6.37 r 1 / tan β 0.0129 r 2 1.66 r 3 0.612 c e 3.85 v L /v u × 10 −9 0.0112 We caution that, due to the non linearity of the problem and the large number of parameters, many local minima are present in all the fits we have performed. Although we have carefully tested the selected minimum for possible improvements, still we cannot be sure that it indeed is the global minimum, within reasonable ranges for the parameters. But, since we obtain an excellent agreement with the data, we are not much concerned with this problem. In fact the resulting χ 2 ∼ 0.13 is very good. We note however that a substantial level of fine tuning is needed. We in fact obtain d F T ∼ 4.7 10 5 from the fit, to be compared with d Data ∼ 3.8 10 3 . As explained in Sect. 3, this is due to the fact that the neutrino oscillation frequency data impose strong constraints on the f i values. Those also enter in the expressions of the mass values and the mixings of charged fermions and in the deviations from TB neutrino mixing. As a result, the strong suppression of the first generation masses and the observed values of mixing angles can only be obtained by a fine tuning among the f i parameters with those of the 10 and 120 matrices.
An important conclusion that we can already give at this stage is that models of the T-IID type formulated in Sect. 2 in the general framework of type-II see-saw dominance can lead at a very good fit of the data but at the price of a pronounced level of fine tuning. In the following section we will compare this class of models with some other "realistic" SO(10) models in the literature.

Comparison with other SO(10) models
In order to appreciate the performance of the above class of models in fitting the fermion masses and mixings we present a comparison with some other realistic SO(10) models present in literature. For a meaningful comparison we applied the same fitting procedure to each model, using the same set of data as described in the previous section. A list of realistic SO(10) theories can be found in [13]. Here we consider a number of models, with different structure and type (renormalizable, non renormalizable, lopsided or symmetric, with type-I and/or type-II see-saw and so on).
The model introduced and discussed in a series of papers by Dermisek and Raby (DR) [14] is an example of non-renormalizable SO(10) theory, with Higgs multiplets in the 10, 45 and 16, based on the flavour symmetry S 3 × U(1) × Z 2 × Z 2 . This model can be considered as a descendant of the model by Barbieri et al, [15], with the two lightest generations in a doublet of U(2) and the third generation in a singlet. In fact, S 3 , the permutation group of 3 objects, is a discrete subgroup of SO(3) (and SU (2) is the covering group of SO(3)) with inequivalent irreducible representations 2, 1 and 1'. In the S 3 symmetry limit only the third generation masses are allowed. Then the second generation masses are generated by a symmetry breaking stage with intensity proportional to a parameter ǫ, and finally the first generation masses only arise when an additional stage with intensity ǫ ′ is switched on. The magnitudes of ǫ and ǫ ′ are determined by a Froggatt-Nielsen mechanism, which is induced by a set of heavy fields that are then integrated away. In the neutrino sector a type-I see-saw mechanism with a hierarchical Majorana mass matrix is adopted. New SO(10)-singlet neutrino and scalar fields are introduced, so that, enough freedom is allowed that is essential to reproduce the observed neutrino properties. In this model the ratio of top to bottom masses is of order tan β, which must be large: m t (m t )/m b (m t ) ∼ tan β ∼ 50.
A good realisation of the lopsided idea is given by the model by Albright, Babu and Barr (ABB) [16] [17], also non-renormalizable and with type-I see-saw. In minimal SU(5) the down quark and the charged lepton mass matrices are connected by a transposition, as the roles of5 and 10 are interchanged in the respective mass matrices. So, if the mass matrices are asymmetric (lopsided), left-handed mixings of charged leptons can be large without implying large left-handed mixings for quarks (in fact, this only implies large right-handed quark mixings which are not observable). In turn large charged lepton lefthanded mixings contribute to the observed large neutrino mixings. In the ABB model the breaking of SO(10) in fact preserves SU(5) in a first stage, which makes this connection with lopsidedness relevant. This model is based on a flavour symmetry U(1) × Z 2 × Z 2 , which, however, plays a different role than in the DR model. In fact, while in the DR model the flavour symmetry and its breaking are mainly used to reproduce the hierarchy of fermion masses and mixings, in the ABB construction, the main goal is to select the Lagrangian terms which are desired and reject those that would not reproduce the data. In particular, in the Higgs sector the symmetry is crucial in order to implement and preserve the mechanism of Dimopoulos and Wilczek for the solution of the doublet-triplet splitting problem [18]. As is often the case in non renormalizable models, the Higgs sector is inspired by a minimality requirement that demands the smallest possible representations compatible with realistic properties. Accordingly the Higgs sector contains 10, 16 + 16, 45 representations and a few SO(10) singlets. The ABB model was originally formulated when the neutrino frequencies and mixing angles were not as precisely measured as now. Later the lepton sector has been revised in ref. [17] with some ad hoc arbitrary ingredients and we have adopted this last description here as it can be fitted to the present data.
We have also considered a variation of this model proposed by Ji, Li, Mohapatra (JLM) [19] which was motivated by a less ad-hoc treatment of the neutrino sector. Precisely, the down quark and the charged lepton mass matrices are the same as in ABB, while the up and Dirac neutrino mass matrices are modified by introducing some new vertices. The structure of neutrino mixing is not attributed to the Majorana matrix, which in this model is diagonal, but rather to the modified Dirac matrix. The model is again based on type-I see-saw and the new added operators introduce a sufficient number of new free parameters to accommodate the neutrino mixing angles.
Turning now to renormalizable models we have analysed the model referred to as BSV in Tab.4. In this version one is introducing the minimal Higgs content in the Yukawa sector (10,126). This minimal model has been discussed in ref. [23] and compared with the data in ref. [33] (where also the cases of type-I and mixed type-I and type-II were considered). A different perspective on this model is presented in ref. [26], also including some comparison with the data (with mixed type-I and type-II see-saw). With this restricted Higgs content one cannot impose the L-R parity that leads to real hermitian h and f matrices, otherwise there is no CP violation and the too restricted number of parameters does not allow a good fit. Thus in ref. [33] complex h and f matrices were taken. As a consequence there are more parameters than observables. We have repeated the fit of that model within our procedure and the results are listed in Tab.4. We see from our fit in Tab.4 that, in spite of this multitude of parameters, with type-II dominance, in the absence of the 120, no good fit of the data can be obtained.
Another particular class of renormalizable models with type-II see-saw dominance has been discussed by Joshipura and Kodrani (JK) [38]. This model comes in two versions, one with type-I dominance (JK1) and one with type-II dominance (JK2). The Higgs coupled to fermions are in 10, 126 and 120, like in the T-IID model. The characteristic feature of the model is the presence of a broken µ − τ symmetry (an explicit breaking is present in the 10 mass terms) in addition to the parity symmetry which leads to hermitian mass matrices. In the case of type-II see-saw dominance this model is a particular case of T-IID with some restrictions on the parameters imposed by the ansatz of broken µ−τ symmetry.
The renormalizable model proposed by Grimus and Kuhbock [40] is of particular interest for us because it closely corresponds to the T-IID model (the fermions masses arise from Higgs in 10, 126 and 120 and the parity symmetry of ref. [34] is assumed) except for the fact that it is based on type-I see-saw dominance. There is one more parameter, v R (see eq.(3)), than in the model T-IID. The only other difference is that this model was fitted in the basis where the matrix h from the 10 is diagonal and real, while for the T-IID we worked in the TB basis for the matrix f of the 126.
The model by Dermisek and Raby is the only one that demands a large value of tan β, so we fit it on the data in Tab  The results in Tab.4 are in some cases somewhat different than those from the fits described by the authors of the various models. One obvious reason is that the set of the input data for the fit is different, as explained in Sect. 4. In some cases (e.g. for JK2 and GK) the difference also comes from the fact that, for hermitian matrices, we fitted the eigenvalues of m i , that is of the mass matrices and not those of m † i m i . Indeed, from fitting the squares of masses one can end up with solutions of negative mass which we discard.
From the results in Tab.4 we see that the most established realistic SO(10) models, DR, ABB, JLM, which are non renormalizable with type-I see-saw, achieve a χ 2 /d.o.f. smaller than 1 with a moderate level of fine tuning, defined by the parameter d F T . In the above list of models DR is special as it has large tan β, so that it was fitted to a different set of data, given in the right panel of Tab.1, which were obtained by evolving up to the GUT scale with large tan β. The model T-IID which we have introduced and described in Sect. 2, realizes an excellent fit, but with a level of fine tuning considerably larger than in the DR, ABB and JLM models. As already mentioned, this large fine tuning arises from the difficulty of fitting the light 1st generation charged fermion masses, together with the neutrino oscillation frequencies and mixing angles. In fact the neutrino oscillation frequencies and mixing angles lead to f matrix elements of comparable size that need cancelations to occur with the parameters in h and h ′ in order to reproduce the light quark and lepton masses. Note that in the DR, ABB, JLM the fine tuning is less pronounced because in all these models new parameters appear in the neutrino sector, so that the neutrino masses and mixings are more independent from the charged fermion sectors. In the JK2 model the constraints from the broken µ − τ symmetry reduce the number of parameters within the general framework of the T-IID model. As a consequence the quality of the fit worsens and the level of fine tuning is the same. Instead the poor result of the BSV model, both in terms of χ 2 and of d F T shows that the presence of the 120 is crucial [27,32]. In ref. [24,25,29,33] it was also shown that the contribution from the 120 is essential for consistency with the existing proton decay bounds. Finally the GK model shows that if one takes the same framework of T-IID, except that the assumption of type-I see-saw dominance is made, an excellent fit is also obtained, still with a substantial level of fine tuning.
We conclude this section by recalling that in this work we have compared the models only on the basis of their ability to fit the fermion masses and mixings. Clearly for a model to be complete and satisfactory many more aspects are important, like the mechanism of the GUT symmetry breaking, the running of the couplings towards the GUT unification, the compatibility of the model with the proton decay bounds and so on (see ref. [29] and refs therein). Also, for the models based on type-II see-saw dominance we assumed here that the dominance is absolute, while in reality one should estimate the corrections to this approximation or include the normal type-I see-saw terms [33]. Additional quality factors could be considered. In models with an underlying flavour symmetry the smallness of some of the parameters is guaranteed by the broken flavour symmetry. Thus an important quality factor is the percentage of small quantities that are predicted to be small. For some of the models there is a broken flavour symmetry that reproduces, at least in part, the observed hierarchies. But this is not the case for T-IID where the problem of an underlying flavour symmetry was not addressd here.

Summary and conclusion
We have made a quantitative comparison of the performance of different types of SO(10) Grand Unification Theories (GUT's) in reproducing the observed values of fermion masses and mixing, also including the neutrino sector. All models, chosen among the most complete and sufficiently realistic, have been confronted with the same set of data (except for DR that requires a large value of tan β), using the same fitting procedure. We have shown that a SO(10) model with type-II see-saw dominance can achieve a very good fit of fermion masses and mixings also including the neutrino sector (provided that the representations 10, 126 and 120 are all included). The quality of the fit in terms of χ 2 and χ 2 /d.o.f. is comparable with the best realistic SO(10) model that we have tested. However, the tight structure of the T-IID model implies a significantly larger amount of fine tuning with respect to more conventional models like the DR or the ABB and JLM models. But those models have no built-in TB mixing and in fact could accommodate a wide range of mixing angle values. A model with type-II see-saw dominance can offer a convenient framework for obtaining a GUT model with approximate TB mixing in the neutrino sector. For this goal a suitable flavour symmetry should be introduced in order to enforce TB mixing as a first approximation. Such a flavour symmetry construction was attempted in ref. [36] but the resulting model is very sketchy and can only be considered as a first step. In fact, when the model of ref. [36] is submitted to our fitting procedure it leads to χ 2 ∼ 342, χ 2 /d.o.f ∼ 38 and d F T ∼ 1.2 10 4 which is quite far from the performance of all models that we considered. The formulation of a natural and elegant model along these lines is not an easy task and more work is demanded.

Appendix A
In this section we show the Yukawa matrices for every model that we have analysed together with the parameter values obtained from our fitting procedure. M u/d/e/ν are the Dirac matrices for up-quarks/down-quarks/charged leptons/neutrinos. M R/L are the Majorana matrices for the right/left neutrinos. As specified by the authors of the model we have taken tan β = 50.34. Some of the parameters are complex and, in these cases, we have set z = |z|e iδz and all the phases are expressed in radiants. The quantity ω is given by ω = 2σ 2σ−1 .

Parameter
Best fit value 0.369 0.110 r 1 / tan β 0.0148 |r 2 | 0.515 δ r 2 1.52 v L /v u × 10 −9 0.0108 Here again the complex parameters are understood as z = |z|e iδz and the phases are in radiants.