Flavour symmetries in a renormalizable SO(10) model

In the context of a renormalizable supersymmetric SO(10) Grand Unified Theory, we consider the fermion mass matrices generated by the Yukawa couplings to a $\mathbf{10} \oplus \mathbf{120} \oplus \bar{\mathbf{126}}$ representation of scalars. We perform a complete investigation of the possibilities of imposing flavour symmetries in this scenario; the purpose is to reduce the number of Yukawa coupling constants in order to identify potentially predictive models. We have found that there are only 14 inequivalent cases of Yukawa coupling matrices, out of which 13 cases are generated by $Z_n$ symmetries, with suitable $n$, and one case is generated by a $Z_2 \times Z_2$ symmetry. A numerical analysis of the 14 cases reveals that only two of them---dubbed A and B in the present paper---allow good fits to the experimentally known fermion masses and mixings.


Introduction
SO (10) is a popular gauge group for the construction of Grand Unified Theories (GUTs). The reason is that its 16-plet accommodates at once all the chiral fields of one fermion family. Now [1,2], (16 ⊗ 16) (16 ⊗ 16) where the subscripts "S" and "AS" stand for, respectively, the symmetric and the antisymmetric parts of the tensor product. Therefore, in a renormalizable theory the scalars occurring in the Yukawa couplings belong solely to the irreducible representations (irreps) 10, 126, and 120. 1 Previously, in the so-called "minimal supersymmetric SO(10) GUT" (for an incomplete list of references see ref. [3,4,5]) the 120 was absent. However, inconsistencies in the fit of the experimental masses and mixings of the fermions-in particular, a tension between the seesaw and GUT scales [6]-led to the inclusion of the 120-plet; the resulting theory has been called [7] the "new minimal supersymmetric SO (10) GUT" (NMSGUT)-see ref. [8] and the references therein. 2 It has turned out that the NMSGUT, which contains three 16-plets of fermionic fields and one multiplet of scalars for each of the irreps in the right-hand sides of equations (1), is quite a successful theory and is capable of accommodating all the available data on the fermion masses and mixings, including the recent neutrino oscillation data [11,12]; this has been demonstrated by numerical fits [13]. 3 However, adding a 120-plet to the 10-plet and the 126-plet of scalars leads to a proliferation of parameters in the Yukawa couplings; one might want to restrict the number of parameters in order to obtain potentially predictive scenarios. Attempts in this direction have been made: in ref. [15], texture zeros were placed in the mass matrices; in ref. [16], a 2 flavour symmetry has been imposed together with a CP symmetry; in ref. [17], real Yukawa couplings were assumed and CP was broken solely by the imaginary vacuum expectation values (VEVs) of the 120.
In the present paper we pursue the approach of ref. [16] by investigating all the possible flavour symmetries acting on the Yukawa couplings in the NMSGUT. We firstly perform a complete discussion by using only minimal assumptions; we thereby identify all the possible cases and their symmetry groups. Thereafter, all the cases are subjected to a numerical analysis in order to identify the viable ones. Partially anticipating our results, no non-Abelian flavour symmetry groups are permitted and there are 14 inequivalent cases, out of which 13 pertain to one-generator Abelian groups and only one case has a two-generator symmetry group 2 × 2 . However, the numerical analysis rules out almost all the cases, leaving only two viable ones which are compatible with the data on the fermion masses and mixings.
In section 2 we fix the notation, display the basic formulas needed for our investigation, and set forth our assumptions. In section 3 we list all the 14 cases. The results of the 1 The representations 10 and 120 are self-conjugate. 2 A completely different approach is SO (10) GUT models in extra dimensions-see for instance ref. [9] and the references therein-or with a hidden sector [10]. 3 Note that skipping the 126 of scalars does not allow for a good fit of even the charged-fermion sector alone [14]. numerical analysis are presented in section 4. The conclusions of our work are given in section 5. The analysis of two specific problems that arise in family symmetry-furnished GUTs is deferred to appendix A. The discussion of the possibility of one further group generator is left to appendix B. Appendix C focuses on the derivation of some inequalities among the VEVs of the various SO(10) scalar representations.

Notation, framework, and assumptions
The relevant fermion mass matrices are given by (see for instance refs. [2,18]) where M d , M u , and M ℓ are the mass matrices of the down-type quarks, the up-type quarks, and the charged leptons, respectively, while M D is the neutrino Dirac mass matrix. The Yukawa-coupling matrices H, G, and F are associated with the scalar irreps 10, 120, and 126, respectively. Those matrices have the (anti)symmetry properties The coefficients k d , v d , κ d , and κ ℓ are the VEVs of the Higgs doublet components in the respective SO(10) scalar irreps which contribute to the Higgs doublet H d of the Minimal Supersymmetric Standard Model (MSSM). The remaining coefficients-k u , v u , κ u , and κ D -refer to H u . The light-neutrino mass matrix is obtained as with where w L and w R are the VEVs of scalar triplets of the Pati-Salam [19] group SU(4) c × SU(2) L × SU(2) R , which are part of the scalar 126-plet of SO(10). The first term in the right-hand side of equation (4) corresponds to the contribution of the type II seesaw mechanism [20] and the second term to the contribution of the type I seesaw mechanism [21].
where M F d ≡ v d F is the component of the down-type-quark mass matrix arising from the Yukawa coupling to the 126 of scalars. One sees that • a complex factor w L w R /v 2 d parameterizes the strength of the type II seesaw contribution relative to the strength of the type I seesaw contribution; and • the overall magnitude of the neutrino masses relative to the charged-fermion masses is parameterized by a dimensionless factor |w R /v d |.
The mass Lagrangian of the "light" fermions reads with (ν L ) c = Cν T L being the charge-conjugate of ν L . One diagonalizes the "Hermitian mass matrices" as where the matrices U d,u,ℓ,ν are unitary and |m 2 3 − m 2 1 | ≫ m 2 2 − m 2 1 > 0. The fermion mixing matrices are then The neutrino mass spectrum is dubbed "normal" if m 2 3 > m 2 1 and "inverted" otherwise. We make the following assumptions: • All three matrices H, F , and G are nonzero.
The second assumption is necessary for the type I seesaw mechanism. The third assumption is an experimental fact.
If the Lagrangian is invariant under a flavour symmetry S 0 , then, due to the SO(10) structure of the Yukawa couplings we obtain the following relations: where W is the 3 × 3 unitary matrix which acts on the three matter 16-plets under S 0 . Without loss of generality we take W to be diagonal. The scalar multiplets 10, 120, and 126 transform under S 0 with the phase factors e iα , e iβ , and e iγ , respectively. (One of the phase factors may be absorbed into W .)

A single flavour symmetry
A single symmetry transformation S 0 leads to 13 inequivalent cases. We refrain from going through the tedious arguments leading to these cases; we merely list them instead. In the following, generic non-zero entries in the Yukawa coupling matrices are denoted "×". For each case, we also give the Abelian group through which the Yukawa-coupling matrices can be enforced.
In equations (18a), (19a), and (20a) ω ≡ exp (±i2π/3). We note that only case A had been discussed earlier, in ref. [16]. Cases A 1 and A 2 have Yukawa-coupling matrices which are restrictions (i.e. they contain extra zero matrix elements) of those of case A; cases A ′ 1 and A ′′ 1 have Yukawa-coupling matrices which are more restrictive than those of case A 1 .
We demonstrate in appendix A that the scalar potential of the NMSGUT can consistently be modified in order to incorporate the 2 symmetries present in cases A, B and C.

A second flavour symmetry
The list of 13 cases in the previous subsection does not necessarily comprise all the Yukawa-coupling matrices obtainable through flavour symmetries, because in each of those 13 cases either one or more further symmetry transformations might be operative and lead to more restrictive Yukawa-coupling matrices and thus to new cases. Let us denote a generic further symmetry transformation, different from S 0 of equation (10), by S 1 : In principle, the symmetry S 1 might either commute or not commute with S 0 . However, as shown in appendix B, by using our assumptions of section 2 one may demonstrate that X always commutes with W , i.e. that S 1 commutes with S 0 . Even more surprisingly, only one new case ensues, which we denote by the letter E and is a subcase of both case A and case C: 4

Summary
From the assumptions stated in section 2 we have obtained the following results: • There are 14 inequivalent cases.
• All the cases except E can be obtained from a single flavour symmetry transformation.
• The flavour groups with one generator are the cyclic groups 2 (in the cases A, B, and C), 3 (in the cases D k with k = 1, 2, 3), and 4 (in case A 1 ). The remaining cases have a U(1) symmetry. 5 • In case E there are two symmetry transformations which commute with each other; the flavour group is 2 × 2 .
• Our scenario does not admit non-Abelian flavour groups.

Fitting the cases to the data
In this section we report on our numerical study of cases A, B, C, A 1 , and D k (k = 1, 2, 3). We have not studied the cases A ′ 1 and A ′′ 1 because they are restrictions of case A 1 and we have found that that case is unable to fit the data well (details will be given later). Analogously, the cases D ′ k are restrictions of the cases D k ; since we have found that the cases D k do not work well, we did not need to bother with the cases D ′ k . Finally, case E is a restriction of case C (and also of cases A and B); since case C is unable even to correctly fit the charged-fermion masses, case E can be discarded outright.
We did not attempt to fit case A 2 because we knew beforehand that such an attempt would be unsuccessful. Indeed, case A 2 yields M d and M u of the Fritzsch form [22], which has long been known to be unable to simultaneously fit the quark masses and the CKM matrix.

Parameter counting
In order to get a feeling for the ability for fitting the data that each case ought to have, it is instructive to count the number of parameters in each of the cases-see x for x = d, ℓ, u and 10 phases and 7 phases and 6 phases and 5 phases # extra parameters 3 moduli 3 moduli 3 moduli 3 moduli in M ν M † ν and 2 phases and 2 phases and 2 phases and 2 phases Table 1: The number of parameters in the Hermitian mass matrices for each case.
instance, in case A 1 the charged-fermion mass matrices may be written, after adequate rephasings, with five phases θ 1,2,3,4,5 and nine real and non-negative parameters ("moduli") a, b, c, d, f , g, l, t ≡ |v u /v d |, and r ≡ |k u /k d |. Moreover, the neutrino mass matrix is viz. it contains two extra complex parameters C and h, plus the real parameter |v d /w R |, making an extra three moduli and two phases.
One sees in table 1 that M ν M † ν always contains three moduli and two phases beyond the parameters which appear in the charged-fermion Hermitian mass matrices. It is easy to understand the reasons for that: one extra complex parameter originates in κ D of equation (2d); another complex parameter originates in w L w R /v 2 d in the right-hand side of equation (6); and there is an extra modulus |w R /v d | in the left-hand side of equation (6). 6 Case A is the one that has most parameters, hence most degrees of freedom, in the mass matrices. In ref. [16] that case has been numerically studied under some restrictive assumptions; we have repeated that study under the same restrictive assumptions, but using the updated values for the charged-fermion masses given in ref. [23].
The restriction of case A analyzed in ref. [16] contains 13 moduli and 6 phases in the ℓ, u), plus an extra two moduli and one phase in M ν M † ν . The original "minimal supersymmetric SO(10) GUT" [3] has 11 moduli and 8 phases in the M x M † x , plus an extra two moduli and one phase in M ν M † ν . We see that both those models are comparable to our case B in their numbers of parameters.
The M x M † x are supposed to be able to fit 13 observables: the nine charged-fermion masses and the four observables in the CKM matrix. One must take into account that phases usually do not help much in fitting observables; the moduli are most relevant. Additionally, if one also takes into account M ν M † ν , then we have to fit five parameters more-the three lepton mixing angles, the ratio

χ 2 function
In order to test the viability of each case, and to find adequate numerical values for its parameters, we construct a χ 2 function where n is the total number of observables (masses and mixing parameters) to be fitted. In equation (28), H is the Heaviside step function,Ō i is the central value of each observable O i , δ ± O i are the upper and lower errors of that observable, and f i (x) is the value of that observable, in any given case, when the parameters of that case have the values x = {x α }. The data are fitted by minimizing χ 2 (x) with respect to the x α . We have used the mean valuesŌ and the errors δ ± O given in tables 2-4. We   Table 4: The values of the neutrino and lepton-mixing observables used in our fits. "NH"refers to a normal neutrino mass spectrum and "IH" to an inverted one.
have taken the charged-fermion masses in table 2, which are renormalized at M GUT = 2 × 10 16 GeV, from the last column of table V of ref. [23]. 7 These are values computed using the renormalization-group equations of the MSSM with tan β = 10; we leave it for some later, more detailed study the task of fitting the data for other values of tan β. The values of the CKM mixing angles in table 3 are low-energy values and were taken from equation (12.27) of ref. [25]; we have multiplied the error bars given in that equation by a factor of three in order to obtain adequately large intervals. The values in table 4 are the 3σ intervals given for each observable in ref. [11].
In order to assess the fitting ability of each case, we have firstly attempted to fit only the charged-fermion masses (nine observables, given in table 2), secondly the chargedfermion masses together with the CKM matrix (four more observables, given in table 3), and, finally, all that together with the neutrino masses and the PMNS matrix (four observables more, given in table 4). The total χ 2 function is thus the sum of three terms: For the neutrino masses, we have analysed both possibilities of a normal or inverted neutrino mass spectrum; indeed, for each set of values for the parameters x, we have computed the eigenvalues of M ν M * ν and thereby determined the type of neutrino mass spectrum; we have then chosen accordingly the input values in the computation of the function χ 2 ν . In some cases we have not been able to find a reasonably small value of χ 2 masses alone; in those cases, further analysis by considering χ 2 CKM and χ 2 ν made no sense. Similarly, in some other cases a sufficiently low value of χ 2 masses + χ 2 CKM could not be achieved, so we did not have to consider χ 2 ν . Finally, even when χ 2 total could be correctly fitted, we still had to check whether |w R /v d | turned out in the right range. Indeed, since v d must be of order the Fermi scale 100 GeV and w R must be of order the grand-unification scale 10 16 GeV, we must require |w R /v d | to be 10 14 or even larger. We had to check some other inequalities, the exposition of which we defer to section 4.4.

Numerical method
The minimization of χ 2 (x) is a difficult task because the various parameters x α may differ by several orders of magnitude and because there always is a large number of local minima. We have spent much time in the numerical analysis trying to find absolute minima; this has involved various fitting options and restrictions of the parameters for each particular case. Still, we cannot be 100% sure that we have found the absolute minimum for all cases-the possibility remains that a better solution exists somewhere in parameter space.
For the numerical minimization of the χ 2 functions we have employed the Differential Evolution (DE) algorithm. This is a stochastic algorithm that exploits a population of potential solutions in order to effectively probe the parameter space. It was first introduced in ref. [26] and it has been modified several times since then.
The effectiveness of the DE algorithm strongly depends on control parameters. We have performed preliminary tests in order to hand-tune the appropriate ranges for the control parameters in each case. Also, in the χ 2 function of equation (28), we have modified the errors δ ± O i randomly (within the range of magnitude of the true errors) according to the behaviour of the fits; we have thus been able to test, for each case, more local minima-defined as the points where the minimization algorithm converges-and to find the minima closer to the global minimum.
All the numerical calculations were implemented by using the programming language Fortran.

Case B 4.4.1 Theoretical treatment
We choose a weak basis in which the Yukawa-coupling matrix F is diagonal. After an interchange of the first and third generations, Without loss of generality, we assume the parameters a, b, and c to be non-negative real. Then the mass matrices are given by We rewrite the mass matrices (31) as where t ≡ v u /v d . The k 1,2,..., 16 are not all independent. We choose k 1,2,3,4,5,9,10,13 as parameters, while From equations (6), (30), and (32d) it is easy to compute Next, we multiply M u and M ν by phase factors exp (−i arg t), defining This phase change leads to the redefinitions k ′ p ≡ k p exp (−i arg t) for p = 9, . . . , 16.

Inequalities
We fix the scale |w R /v d | in the left-hand side of equation (37b) by requiring the difference of the squared neutrino masses |m 2 3 − m 2 1 | to be equal to the atmospheric mass scale 2.5 × 10 −3 eV 2 . Afterwards, we compute |v d | by identifying |w R | with the unification scale M GUT = 2 × 10 16 GeV. Finally, we calculate |v u | = |tv d | from the value of the parameter |t| of the fit.
In the supersymmetric GUT that we envisage there is only one scalar doublet with hypercharge +1/2, viz. H d , at the Fermi mass scale; there is also only one scalar doublet with hypercharge −1/2, viz. H u , at that scale. Those two doublets have VEVs respectively, where tan β = 10 in our fit. According to the inequalities (C11), Therefore, we have first of all to enforce the inequalities on our fits. The inequality (40a) reads |w R /v d | > 1.155 × 10 15 , which is a quite useful lower bound. Our fit fixes Therefore, from the inequalities (39), We must now face the additional fact that the Yukawa couplings cannot be too large, lest the theory ceases to be perturbative and/or Landau poles arise in the Yukawa couplings. Let y > 0 denote the maximum value that we accept for the absolute value of the Yukawa couplings; we may take y somewhere between 1 and 10. From equations (30), We have directly enforced the inequality (43c) on our fits. The other two inequalities (43) may be put together with the inequalities (42) to derive To summarize, we have enforced on our fits the inequalities (40), (43c), and (44).

Fit
We have found that case B is able to fit perfectly all the observables. This is true irrespective of whether the neutrino mass spectrum is normal or inverted. However, when the neutrino mass spectrum is inverted some of the inequalities in the previous subsection always turn out to be violated; this happens because either |w R /v d | < 10 15 is too small or |t| > 300 is so large that the inequality (44b) ends up being violated. For a normal neutrino mass spectrum, on the other hand, there are fits in which all the inequalities are observed. In table 5 we give the values of the mass-matrix parameters that lead to the best fit which we have been able to achieve.  fit is smaller than 10 −3 , i.e., for all practical purposes, it is zero. The smallest neutrino mass for this fit is m 1 ≈ 0.006 eV, while m 1 + m 2 + m 3 ≈ 0.07 eV. It is interesting to observe in table 5 that the best fit is achieved for a very large value of Ĉ ∼ 10 6 , meaning that the type-II seesaw mechanism dominates over the type-I. For this fit, the matrices U d and U u are almost diagonal, with U d mostly identical with U CKM . On the other hand, U ν is almost completely a rotation between the first two generations, while U ℓ is largely, but not exclusively, a rotation between the second and third generations; both rotations are almost maximal.
Since very perfect fits can be obtained in case B, we suspect that this case has too many degrees of freedom and has little or no predictive power. However, since such a study is very time-consuming, we leave it for later investigation.

Non-viable cases
We have found that all the cases except cases A and B either fail to fit the observables adequately or give a much too low value for |w R /v d |. (For us, an acceptable fit is one in which all the observables simultaneously are within their ranges in tables 2-4.) Indeed, case C even fails to adequately fit the charged-fermion masses alone, while cases A 1 , D 2 , and D 3 are unable to acceptably fit the charged-fermion masses together with the CKM matrix. The best results that we were able to find for all the cases are given in table 6. case χ 2 of best fit pulls larger than one remarks in absolute value Only case D 1 is able to fit all the observables, but all those good fits yield |w R /v d | < 3 × 10 13 . This is unacceptable since, with tan β = 10, |v d | = H 0 d 0 ≈ 17.3 GeV then leads to |w R | 5 × 10 14 GeV, which is almost two orders of magnitude below the unification scale M GUT = 2 × 10 16 GeV. If we enforce a more realistic |w R /v d | > 10 15 on case D 1 , then we are only able to obtain poor fits with χ 2 total 60.

Case A
Case A has much too many degrees of freedom, so it is adequate to try and constrain it somewhat. We follow ref. [16], in which real Yukawa-coupling matrices (due to an additional CP symmetry) F , G, and H were enforced and, moreover, w L = 0 has been assumed, thereby discarding the type-II seesaw mechanism. Under these assumptions, the authors of ref. [16] have parameterized where In this parameterization, there are six phases (ξ ℓ , ξ u , ξ D , ζ u , ξ d , and ζ d ) and 15 moduli (x, y, z, a, b, c, d, f , g, r ℓ , r H , r u , r F , r D , and |w R /v d |).
As usual, we firstly fit the charged-fermion masses, the mixing angles, and r 2 solar . Secondly we adjust the factor |w R /v d | in the left-hand side of equation (45e) in such a way that |m 2 3 − m 2 1 | = 2.5 × 10 −3 eV 2 . Thirdly we compute |v d | = |v d /w R | (2 × 10 16 GeV) and |v u | = r F |v d |. Finally, we check that We also check that max a 2 , b 2 , c 2 , d 2 < y 2 |v d | 2 (48) for some 1 < y < 10; we also require max x 2 , y 2 , z 2 + 1 + r 2 In ref. [16] an explicit fit of case A-under the above restrictions w L = 0 and real Yukawa-coupling matrices-to some data was presented. However, the authors of ref. [16] have used the charged-fermion masses give in ref. [24] and have used the upper bound on sin 2 θ 13 that existed at the time. We have attempted to fit case A both to the updated charged-fermion masses of ref. [23] and to the now extant value of sin 2 θ 13 . We could achieve an excellent fit when the neutrino mass spectrum is normal and a passable one when the mass spectrum is inverted; those fits are presented in tables 7 and 8, respectively.   Table 8: The values of the parameters for out best fit of case A with an inverted neutrino mass spectrum. This fit has χ 2 total ≈ 0.8.

Conclusions
In this paper we have considered a supersymmetric SO(10) GUT in which the fermion masses are generated by renormalizable Yukawa couplings. Consequently, the scalar multiplets under consideration belong to the irreps 10, 126, and 120 of SO (10). We have assumed that there is a single scalar multiplet belonging to each of these three irreps; some further mild assumptions are listed in section 2. We have analysed the prospects of imposing flavour symmetries in this scenario, potentially making it predictive. An exhaustive discussion has revealed 14 cases compatible with our scenario. For the numerical examination of those cases we have used the charged-fermion masses evaluated at the GUT scale through renormalization-group running in the context of the Minimal Supersymmetric Standard Model. Interestingly, the numerical analysis ruled out all 14 cases except case A-see equation (11)-and case B-see equation (12). We have demonstrated that both cases A and B allow excellent fits to the data when the neutrino mass spectrum is normal; when that spectrum is inverted, case A can still fit the data but we were unable to find a fit for case B. Thus, we have come to the conclusion that within the NMSGUT [7], which has renormalizable Yukawa couplings just as the ones considered here, there are at most two pos-sibilities to reduce the number of Yukawa couplings through flavour symmetries, while remaining in agreement with the data.

A The scalar potential with an additional 45
This appendix addresses two problems that may in general arise in a renormalizable supersymmetric SO(10) GUT furnished with additional symmetries: • How to promote the full mixing among the Higgs doublets residing in the 10, 120, and 126 of SO(10).
• How to achieve the full breaking of SO (10) to the SM gauge group by using only renormalizable interactions.
Both these problems can be solved in the NMSGUT, but they are non-trivial in the context of our symmetry-furnished cases, especially when the symmetry is larger than 2 . According to refs. [4,27,28], in the NMSGUT there are five scalar irreps: the 10, the 120, the 126, the 126, and the 210. The 10, the 120, and the 126 have Yukawa couplings; the 126 and the 210 do not. The 210 is needed, together with the 126 and 126, in order to break SO(10) down to the SM gauge group.

B Investigation of a second symmetry
In this appendix we take all 13 cases of subsection 3.1 and consider, for each of them, the possibility of a second flavour symmetry defined in equation (24). Without loss of generality we set e iβ 1 = 1 in that equation.
The conclusion of this appendix is that, beyond those 13 cases, only one new case arises which does not contradict our assumptions-case E in equation (25).

B.1 Cases
In all these four cases, we find In these four cases the matrix F has the form We require det F = 0, hence a = 0 and b = 0. Using with the matrix X of equation (B3), we obtain that X must be diagonal: Now we look for the consequences of With a diagonal X, equation (B7) can only force either one or more matrix elements of H to be zero. In the case A 1 , if one sets one matrix element of H to zero then one simply recovers the cases A ′ 1 and A ′′ 1 . In the cases A ′ 1 and A ′′ 1 , the number of non-vanishing elements of H is already minimal. In the case A 2 we have X T HX = e −iγ 1 H, therefore either α 1 = γ 1 and H is not restricted by S 1 or α 1 = γ 1 and H = 0, which is excluded by our assumptions.
In summary, departing from cases A 1 , A ′ 1 , A ′′ 1 , or A 2 no new cases can ensue from a second symmetry.

B.2 Cases
In these cases equation (B1) is still valid, therefore equation (B3) also holds. In all four cases with a = 0 and b = 0. Using equation (B5) then yields i.e. X is once again diagonal. We next consider equation (B7). In case D 3 we obtain In case D 2 we have Thus, equation (B7) can at most set either r = 0 or s = 0. If s = 0 then one recovers case D ′ 2 from case D 2 and case D ′ 3 from case D 3 . If r = 0 then, through an interchange of the first and third generations, one recovers case A ′ 1 from case D 2 and case A ′′ 1 from case D 3 . Therefore, no new cases arise from the enforcement of the symmetry S 1 on any of these four cases.

B.3 Cases D 1 and D ′
1 Equations (B1) and (B3) once again hold. Now with a = 0 and b = 0. Equation (B5) then yields that either e iγ 1 = +1 and ) or e iγ 1 = −1 and In case D 1 and with equation (B13) one obtains With equation (B14) one arrives instead at Thus, the possibility (B14) implies H = 0, which contradicts our assumptions. With equation (B15) then either s = 0 and one recovers case D ′ 1 or r = 0 and the second generation decouples. We conclude that the enforcement of the symmetry S 1 on cases D 1 and D ′ 1 cannot lead to new cases.

B.4 Cases A and B, step 1: X may be chosen to be diagonal
In cases A and B we may perform a weak-basis transformation such that G acquires the form (B1) while the forms of H and F are kept unchanged: This is achieved through a unitary rotation of the first and second generations, which does not alter the matrix W = diag (+1, +1, −1) for these cases. In the new basis (B17), equation (B3) holds.
Next we consider equation (B5). With F ≡ (f ij ), it reads Let us suppose that X is not diagonal, i.e. that x 13 is nonzero. Then equation (B18) tells us that f 12 = 0, i.e. that F is diagonal. Now we invoke e iα 1 HX = X * H. In case A the matrix H ≡ (h ij ) has the same form as the matrix F , hence we may conclude, from the analogue of equation (B18), that h 12 = 0 just as f 12 = 0, i.e. H is diagonal too. But then the second generation decouples, which runs against our assumptions. For case B the equation e iα 1 HX = X * H reads hence h 23 = 0 and the second generation decouples. We conclude that the hypothesis x 13 = 0 leads to a contradiction with our assumptions. Thus, cases A and B do not admit a non-diagonal X.
B.5 Cases A and B, step 2: the forms of F and X With a diagonal matrix X, the equation X T F Xe iγ 1 = F yields Since det F = 0, f 33 cannot vanish. Therefore, equation (B20d) gives x 33 = εe −iγ 1 /2 , where ε = ±1. Let us firstly suppose that x 11 = x 22 . In this case we must have x 2 11 = e −iγ 1 , else f 11 = f 22 = f 12 = 0 and det F = 0. Consequently, x 11 = ηe −iγ 1 /2 , where η = ±1. In this case the matrix F cannot be restricted any further by S 1 .
Since X = e −iγ 1 /2 diag (η, η, ε), X T HX = e −iγ 1 H in case A and X T HX = εηe −iγ 1 H in case B. This means that the equation e iα 1 X T HX = H either does not restrict H any further, or it enforces H = 0 (depending on the choice for e iα 1 ). Since H = 0 runs against our assumptions, we conclude that, with x 11 = x 22 , the symmetry S 1 does not restrict the Yukawa-coupling matrices any further, i.e. it does not lead to any new cases.
So we are lead to consider x 11 = x 22 . Then, only two possibilities for X remain, which are compatible with det F = 0: either with F 1 given by equation (B4) and e 2iρ = 1. We must remember that X must be of the form (B3), viz. that x 33 = x * 11 . Therefore, B.6 Cases A and B, step 3: the form of H Case A, X = X a , F = F a : In this case the equation e iα 1 X T HX = H gives If e i(α 1 −γ 1 ) = ±1, then H = 0 contradicts our assumptions. If e i(α 1 −γ 1 ) = 1, then h 12 = 0 and the second generation decouples. If e i(α 1 −γ 1 ) = −1, then we get case E-see equations (25), because the choice e iγ 1 = e iγ 1 /2 = η = ε = 1 indeed leads to the symmetry (2)  2 of that equation.
Case B, X = X a , F = F a : In this case the equation e iα 1 X T HX = H gives In order to avoid decoupling of the second generation, we must choose e i(α 1 −γ 1 ) ηε = −1 and h 13 = 0. We then obtain a case which is equivalent to case E after the interchange of the first and third generations.
Since e 2iρ = 1, through a choice of the phases we may achieve either case A 1 or case A ′ In order to avoid decoupling of the second generation we must choose e i(α 1 −γ 1 −ρ) ε = 1 and h 13 = 0; this case is equivalent to D ′ 2 through the interchange of the first and third generations.

B.7 Case C
In case C, it is convenient to choose a weak basis where This weak basis is achieved, starting from the form (13b) of the matrices H, G, and F , through a unitary rotation mixing the first and third generations; such a rotation does not alter the matrix W in equation (13a). With G of equation (B28) we know that X has to obey equation (B3). It is then easy to see that HXe iα 1 = X * H requires X to be diagonal with e iα 1 x 11 = x * 22 . Therefore, X can be parameterized as X = diag e iψ , e −i(α 1 +ψ) , e −iψ . (B29) With this X, the equation X T F Xe iγ 1 = F can only force one or more matrix elements of F to be zero. If e iγ 1 = 1, one obtains f 13 = f 31 = 0 and, therefore, case E. Then, because of det F = 0, all f ii must be non-zero and it is easy to show that this leads to e iγ 1 = −1, e iψ = iη, and e iα 1 = ε with η 2 = ε 2 = 1. Summarizing, we have X = iη (1, −ε, −1) , e iα 1 = ε, e iβ 1 = +1, e iγ 1 = −1.
By choosing ε = −1 and absorbing (iη) 2 = −1 into the phase factors, we arrive at 2 of equation (25). Note that in this subsection the symmetry S 0 is given by (2) 2 and S 1 by (1) 2 , since we started from case C. Thus, in the present subsection, the notation for (1)  2 and (2) 2 is exchanged compared to equation (25). Moving to e iγ 1 = 1 and taking again into account det F = 0, we have f 22 = 0. However, it is neither possible to enforce f 11 = 0 while keeping f 33 = 0 nor to enforce f 33 = 0 while keeping f 11 = 0; with f 11 = f 33 = 0 one recovers case D ′ 1 . One thus concludes that enforcing an extra symmetry on case C can only lead to cases E or D ′ 1 , or else to a violation of our assumptions.

C Precise definition of the matrices F , G, H
The aim of this appendix is to precisely define the matrices F , G, and H through equations (C5) and thereby to extract the useful inequalities (C11), which we employ in subsection 4.4.2.
The inequalities (C2) only involve the amplitudes of the doublets contained in the 10, 126, and 120. Taking into account that the 126 and the 210 have no Yukawa couplings, the Dirac mass matrices are given by M a = v cos β [c a 1ᾱ 1 Y 10 + c a 2ᾱ 2 Y 126 + (c a 5ᾱ 5 + c a 6ᾱ 6 ) Y 120 ] (a = d, ℓ), with Yukawa-coupling matrices Y 10 , Y 126 , and Y 120 and Clebsch-Gordan coefficients c a,b j ; the latter derive from the SO(10)-invariant Yukawa couplings [5,31]. The absolute values of the Clebsch-Gordan coefficients have no physical meaning and some of their phases are convention dependent. With our conventions, 9 the required information reads v d = v cos βᾱ 2 ,