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Construction of a Minimal Higgs SO(10) SUSY GUT Model

A full account is given of the procedure used by the authors to construct an SO(10) supersymmetric grand unified model of the fermion mass matrices. Various features of the model which gives remarkably accurate results for the quark and lepton masses and mixings were presented earlier in separate publications. The construction of the matrices is first discussed in the framework of effective operators, from which one naturally obtains the maximal \nu_\mu - \nu_\tau mixing, while the small angle or maximal mixing solutions for the solar neutrinos depend upon the nature of the Majorana matrix. A set of Higgs and fermion superfields is then introduced from which the Higgs and Yukawa superpotentials uniquely give the structure of the mass matrices previously obtained. The right-handed Majorana matrix arises from one Higgs field coupling to several pairs of superheavy conjugate neutrino singlets. For the simple version considered, 10 input parameters accurately yield the 20 masses and mixings of the quarks and leptons, and the 3 masses of the right-handed neutrinos.


I. INTRODUCTION
In several recent papers [1][2][3][4] we have developed a highly predictive model of quark and lepton masses based on the grand unified group SO (10). This model grew out of our attempt [1] to construct a realistic grand unified theory (GUT) in which SO (10) was broken down to the standard model in the simplest possible, or "minimal" way [5]. In this model there emerged a new mechanism based on certain well-known features of SU (5) for explaining the large mixing between the mu and tau neutrinos that is seen at SuperKamiokande [6]. In [1,2] we gave the structure of the quark and lepton mass matrices for the second and third families, treating the first family as massless. In [3], it was shown how to extend the model to include the first family, which leads to several interesting predictions. In [4], it was observed that the mixing of the electron neutrino very naturally falls either within the range 0.004 ≤ sin 2 2θ eµ ≤ 0.008, corresponding to the small angle MSW solution [7] of the solar neutrino problem, or very near to the value sin 2 θ eµ = 1, corresponding to what is called "bimaximal mixing".
In this paper we present the model in fuller detail, especially in regard to neutrino phenomenology, and to the structure of the Higgs sector, Yukawa interactions, and flavor symmetries that underlie the quark and lepton mass matrices.
The paper is organized as follows. In Section II we discuss in general terms, that is apart from a particular model, our mechanism for explaining the large mixing of ν µ and ν τ . In Section III, we explain what we mean by a "minimal" SO(10) breaking scheme, and show how such minimal breaking and the requirements of simplicity lead one naturally to a certain form for the mass terms of the heavier two families of quarks and leptons. We then observe that this form realizes the general mechanism for large ν µ − ν τ mixing described in the previous Section. It is important to emphasize that this mechanism emerged not from an attempt to explain neutrino phenomenology, but from other considerations entirely, in particular the attempt to simplify the Higgs structure of SO (10). It is most interesting that the same mechanism has also independently been found by other groups attempting to make sense of neutrino phenomenology. In Section IV, it will be explained how this model is best extended to the first family of quarks and leptons, and how this gives rise to several distinctive predictions. Accurate analytic expressions for the predictions at the GUT scale will be presented. In Section V, the neutrino sector will be examined in detail. It will be seen how either the small-angle MSW solution of the solar neutrino problem or bimaximal mixing can result with equal simplicity. Finally, in Section VI, a concrete model, including all the details of flavor symmetries and of the Higgs and Yukawa superpotentials, will be presented, showing that the basic scheme is technically natural.
II. MECHANISM FOR LARGE ν µ − ν τ MIXING Before explaining our mechanism, it will be helpful to explain why the observed large mixing of ν µ , presumably with ν τ , has been a theoretical puzzle. The basic reason is simple: the mixing that is seen between the quarks of the second and third families is described by a small mixing angle, namely V cb ∼ = 0.04, and therefore it was expected that the mixing between the second and third family of leptons would also be small.
The grounds for this expectation were twofold. First, there is the empirical fact that the masses of the quarks and leptons exhibit roughly similar "hierarchical" patterns, and therefore it was natural to assume that their mixing angles would be similar also. Second, the most promising theoretical approaches to understanding the pattern of quark and lepton masses, namely grand unification and flavor symmetry, tend to treat quarks and leptons in similar ways. For instance, small quark mixing angles might suggest an underlying fundamental "family symmetry" that is weakly broken, in which case the lepton mixings might be expected also to be small. And in grand unification based on SO (10) there is a close connection between the quark and lepton mass matrices.
There are actually two puzzles associated with the mixing of the second and third families: First, why is the lepton mixing |U µ3 | ∼ 0.7 so large? And, second, why is the quark mixing V cb ∼ = 0.04 so small? What we mean by saying that these are distinct puzzles is that they are both unexpected within the most commonly assumed framework for explaining quark and lepton masses, the Weinberg-Wilczek-Zee-Fritzsch (WWZF) idea [8].
The WWZF idea was that the Cabibbo angle could be understood if the mass matrices of the first and second families of quarks had the following form: This gives |m d /m s | ∼ = |b ′ /a ′ | 2 , |m u /m c | ∼ = |b/a| 2 , and V us ∼ = b ′ /a ′ − b/a, and thus the famous relation Since |V us | ∼ = 0.22, m d /m s ∼ = 0.22, and m u /m c ∼ = 0.07, this relation is satisfied for α ∼ ±π/2. If we apply the same idea to the leptons of the first two families we get The second term on the right is not known, but if it is assumed to be small one has the rough prediction that U e2 ∼ m e /m µ ∼ = 0.07. This could be consistent with the small angle MSW solution of the solar neutrino problem, which requires that U e2 ∼ 0.04. Thus the WWZF idea appears to work well where it was originally applied, namely to the first and second families. Fritzsch [9] later extended this idea to explain the mixing of the third family. If a WWZF form is assumed to hold for the second and third family, i.e., if one takes (u, c) −→ (c, t) and (d, s) −→ (s, b) in Eq. (1), one obtains and U µ3 ∼ = m µ /m τ − e iδ m ν 2 /m ν 3 .
Since m s /m b ∼ = 0.14, and m c /m t ∼ = 0.04, one sees that the observed value of V cb ∼ = 0.04 is too small by a factor of three or so. Assuming that the neutrino mass ratio in Eq.
(5) is small, and given that m µ /m τ ∼ = 0.24, one sees that the nearly maximal value of U µ3 ∼ 1/ √ 2 ∼ = 0.7 that is observed is too large by a factor of three or so. These Eqs. (2)(3)(4)(5) are based on the assumption of a hierarchical and symmetric form for the mass matrices. A key feature in our mechanism for understanding the large mixing of the tau neutrino is that it involves highly asymmetric mass matrices. As we shall see, the assumption of asymmetric mass matrices naturally explains why U µ3 is larger than the Fritzsch value and V cb is smaller than the Fritzsch value by approximately the same factor.
Consider, a toy model with SU (5) symmetry, which has a set of Yukawa terms of the following form: λ 33 (5 3 10 3 )5 H +λ 23 (5 2 10 3 )5 H +λ 32 (5 3 10 2 )5 H , with λ 32 ≪ λ 23 ∼ λ 33 and the subscript H denoting a Higgs representation. These terms yield the following mass matrices for the second and third families of down quarks and charged leptons: with ǫ ≪ σ ∼ 1. Here we have labelled the fermions with a family index, instead of the names s, b, µ, and τ , since the mass matrices in this case are far from diagonal. A crucial point to notice is that the matrix for the leptons, which we will denote by L, is the transpose of the matrix for the down quarks, which we will denote by D. This is a feature of minimal SU (5). It arises from the fact that the 5 representation of fermions contains the left-handed leptons, l L , and the charge conjugate of the right-handed down-quarks, d R , while the 10 representation of fermions contains the charge conjugate of the right-handed leptons, l R , and the left-handed down-quarks, d L . Thus, SU(5) relates D to L, but only up to a left-right transposition: D = L T . The transposition feature of SU(5) unification appearing in Eq. (6) results in the large element, σ, of L producing an O(1) mixing of l 2L with l 3L for the leptons, while in D for the quarks it produces a large mixing of the right-handed fields d 2R and d 3R . The mismatch between the large l 2L − l 3L mixing and the ν 2L − ν 3L mixing, which is small (as will soon be seen), leads to a large U µ3 mixing element. But the right-handed mixings of the quarks are not observable through standard model physics. What matters is the left-handed mixing of d 2L with d 3L , which contributes to V cb , and is controlled by the small parameter ǫ.
The common statement that grand unification relates quark and lepton mixing angles, and thus V cb to U µ3 , is very misleading. What is really true in general is that grand unification relates the mixing of quarks of one handedness to the mixing of leptons of the other handedness. Thus V cb and U µ3 need not be directly related to each other. Of course, if the mass matrices are symmetric, as has almost always been assumed, the left-handed and right-handed mixings are the same, and hence V cb is directly related to U µ3 . The most natural interpretation, then, of the experimental discovery that |U µ3 | ≫ |V cb | is that the mass matrices are highly asymmetric. This is the essential point first made in [10].
Not only does a highly asymmetric, or, as we will call it, "lopsided," form of the mass matrices explain the difference between the size of U µ3 and V cb , but it also explains the fact, noted above, that U µ3 is larger than the Fritzsch value and V cb is smaller than the Fritzsch value by about the same factor. The point is that the product of the two off-diagonal elements, ǫ and σ, is controlled by the fermion mass ratio. As is evident from Eq. (6), m s /m b ∼ = ǫσ 1+σ 2 ∼ ǫσ. That means that the Fritzsch prediction for the mixing of d L and s L , which is m s /m b , goes approximately as √ ǫσ. That shows that the Fritzsch prediction for the mixing angles is roughly the geometric mean between the true value of U µ3 ∼ σ and the true value of V cb ∼ ǫ. In other words, in our hypothesis of lopsided mass matrices, the surprising largeness of U µ3 and the surprising smallness of V cb are two sides of the same coin. Another important feature of this mechanism should be emphasized. Almost all published explanations of the largeness of the ν µ − ν τ mixing trace it to some special feature or form of the neutrino mass matrix. Perhaps this is due to the purely linguistic fact that we talk about "neutrino mixing angles". But they could just as well be called the "chargedlepton mixing angles". They are really the angles expressing the mismatch between the neutrino mass eigenstates and the charged lepton mass eigenstates, just as the CKM angles are the mismatch between the up and down quark eigenstates. In our mechanism, the large value of U µ3 is traceable to a peculiarity of the charged lepton mass matrix L, namely, having a large off-diagonal entry σ. As we shall see in the next Section, having such a large entry helps to explain several other features of the quark and lepton mass spectrum.
To sum up, the mechanism for explaining large ν µ −ν τ mixing proposed in [1][2][3] has three salient features: (1) the largeness of this mixing is due to the charged lepton mass matrix, which is (2) highly asymmetric, and which is (3) related to the transpose of the down quark mass matrix by SU (5).
In the next Section we will see how a model with precisely these features arises very naturally in SO(10) from very different considerations.

III. FERMION MASS MATRICES IN MINIMAL SCHEMES OF SO(10) BREAKING
The model that we shall examine in this paper emerged originally from our attempt to construct a realistic model based on SO(10) [11] in which SO(10) is broken to the standard model group, G SM = SU(3) c × SU(2) L × U(1) Y in the simplest possible way. We shall therefore start by explaining what we mean by minimal SO(10) breaking.
Since SO(10) is a rank 5 group, it requires for its breakdown to G SM at least two Higgs fields. One Higgs field is needed to break the rank of the group to 4, but this generally leaves an unbroken SU (5). The second Higgs field is needed to break SU(5) down to G SM . The two breakings can occur in either sequence depending upon which Higgs field has the larger VEV and effects the first breaking.
Whatever Higgs field gives superlarge mass to the right-handed neutrinos, as required for the standard seesaw explanation of the lightness of the left-handed neutrinos, will also break SO(10) down to SU (5), and thus the rank to 4. There are two simple choices for this Higgs field: either an antisymmetric five-index tensor 126 or a spinor 16. In either case, one also expects a Higgs field in the conjugate representation, 126 or 16, to go along with it. A nice feature of the 126 is that this tensorial representation leaves unbroken a Z 2 subgroup of the center of SO(10) that acts as an automatic matter parity, whereas if a spinor Higgs is introduced, then matter parity is not automatic. On the other hand, to introduce 126 + 126 is to introduce quite large representations that tend to make the unified gauge coupling go non-perturbative below the Planck scale, and that may be hard to obtain from superstring theory. In any event, it would seem that the use of a spinor-antispinor pair, 16 + 16, is more economical. Thus we assume that the rank of SO (10) is broken at the unification scale and the right-handed neutrinos get mass from one such spinor-antispinor pair of Higgs fields.
To break the group the rest of the way to G SM requires the existence of Higgs fields in the adjoint representation 45 and/or in the symmetric two-index tensor representation 54. Most published realistic SO(10) models have several of both kinds of multiplets. However, it has been shown that it is possible to break SO (10) to G SM with only a single adjoint Higgs and no larger representations [5].
This, then, is what we call the "minimal breaking scheme for SO(10)": The breaking of SO (10) to G SM is accomplished by the expectation values of a set of Higgs fields consisting of 45 H + 16 H + 16 H , with the model containing no multiplets larger than the single 45. There, of course, have to be other Higgs fields to break the SU(2) L × U(1) Y group of the electroweak interactions.
This minimality assumption is restrictive enough that it is possible to say in which direction the expectation values of these fields point. This can be done by considering the problem of doublet-triplet splitting, whereby the colored partners of the weak-doublet Higgs fields of the standard model become superheavy while the weak-scale masses of the doublets themselves are preserved. In SO(10) the only known way of doing this in a technically natural manner is the Dimopoulos-Wilczek or "missing VEV" mechanism [12]. The idea is that if an adjoint Higgs field that has an expectation value proportional to the SO(10) generator B − L couples to Higgs fields in the vector representation, it will make their colortriplet components heavy (since they have B − L = ±2/3) while leaving their weak-doublet components massless (since they have B −L = 0). The needed coupling is simply of the form 10 1H 45 H 10 2H . Of course, the expectation value of the adjoint, by virtue of the definition of the adjoint representation, is necessarily a linear combination of generators of the group. B − L is one of the SO(10) generators that is picked out by simple forms of the Higgs superpotential for the adjoint multiplet. It should be noted that there is another version of the missing VEV mechanism that works in SO (10), in which the VEV of the adjoint is proportional to the generator I 3R of the SU(2) R subgroup of SO(10) [13]. However, that version is significantly more complicated. Therefore, simplicity dictates the choice that Since the assumption of a minimal SO(10) breaking scheme included the supposition that only one adjoint exists in the model, no adjoint exists except the one that points in the B − L direction. As we shall see, this puts an important limitation on the possibilities for constructing realistic mass matrices for the quarks and leptons. The assumption of a minimal SO(10) breaking scheme thus acts as an important guide in searching for good models.
The simplest possible terms that would give mass to quarks and leptons in SO(10) would be λ ij 16 i 16 j 10 H , where the subscripts i and j are family indices. This would lead to four proportional Dirac mass matrices for the up-quarks (U), down quarks (D), charged leptons (L), and neutrinos (N). In fact one would have D = L ∝ U = N. Moreover, all these matrices would be symmetric, which is why one can write D = L instead of D = L T as in minimal SU (5). Some of the predictions that follow from these relations are good, notably the famous prediction m 0 b = m 0 τ , where the superscript zero stands for quantities evaluated at the unification scale M G . However, D = L also predicts that m 0 s = m 0 µ and m 0 d = m 0 e . Empirically, one finds instead that m 0 s ≃ 1 3 m 0 µ and m 0 d ≃ 3m 0 e . These factors of three are called the Georgi-Jarlskog factors [14]. The simplest possible SO(10) Yukawa terms also predict that all the CKM angles vanish, since U ∝ D. While not exactly true, this is at least a good zeroth order relation, since the CKM angles are all small compared to unity. By contrast, in SU(5) the matrices D and U are not related by the unified symmetry and so the CKM angles are unconstrained. The smallness of the CKM angles can be regarded, therefore, as evidence for SO (10). On the other hand, the proportionality of D and U in SO(10) also predicts that m 0 c /m 0 t = m 0 s /m 0 b , which fails badly by over an order of magnitude. What one can conclude is that a way of going beyond the simplest possible SO(10) Yukawa scheme must be found which preserves some of its predictions while breaking others. One way to do this involves using larger representations to break the electroweak interactions. For instance, in the original Georgi-Jarlskog model, a 45 multiplet of SU(5) (not to be confused with the adjoint of SO(10)) participates in breaking SU(2) L × U(1) Y . In the context of SO (10), this 45 is contained in a 126, which is inconsistent with our minimality assumptions. More economical is to assume that the Higgs fields that break SO(10) at the unification scale, i.e., the 45 H + 16 H + 16 H , couple to quarks and leptons and thus introduce the effects of that SO(10) breaking into the quark and lepton mass relations. This is the assumption we make.
To describe the third family it is simplest to assume the minimal Yukawa term 16 3 16 3 10 H as pictured in Fig. 1(a). By itself, this would make all the mass matrices have the form That would give the following predictions, all of which are at least good zeroth approximations to reality: m 0 b = m 0 τ , V cb = 0, and m 1 /m 3 = m 2 /m 3 = 0, where m i is a mass of a fermion of the i th family. Note that these are just the "good" SO(10) predictions mentioned above.
The second family presents more of a challenge. The main issue is how to get the Georgi-Jarlskog factor of 3 between m 0 µ and m 0 s . Breaking of SU(5) must be involved, since the bad relation m 0 s = m 0 µ arises already at the SU(5) level. The only field that breaks SU(5) in the framework of minimal SO(10) breaking is the adjoint, 45 H . Since 45 H ∝ B − L, and the B − L of leptons is −3 times that of quarks, this field has the possibility of giving the needed Georgi-Jarlskog factor. Thus one must seek an effective Yukawa term that involves the 45 H . The simplest such term [15], in the sense of the term of lowest dimension, is of the form (16 i 16 j )10 H 45 H /M G . Moreover, this term can arise in a simple way by the integration out of a 16 + 16 family plus antifamily at the unification scale, as shown in Fig. 1 There are actually two ways to contract the SO(10) indices of such a term: the product (16 i 16 j ) can be contracted symmetrically or antisymmetrically. It is easy to show that if 45 ∝ B − L, only the antisymmetric contraction contributes to the quark and lepton mass matrices. (The reason is simple. If the VEV of the adjoint is proportional to a generator Q, then the symmetric/antisymmetric contractions give contributions to fermion masses that go as Q(f ) ± Q(f ). Since B − L of an antifermion is minus that of the fermion, the contribution cancels for the symmetric contraction.) Thus, one need only consider the flavorantisymmetric term, which means only ij = 23 and not ij = 22, or 33. Consequently, the only operator of interest is (16 2 16 3 )10 H 45 H which, together with the operator 16 3 16 3 10 H , gives The desired factor of 3 has been achieved between leptons and quarks, due to the generator B − L to which the adjoint VEV is proportional. One also can see that the ǫ entries are flavor antisymmetric for reasons already explained. As they stand, these forms of the matrices are inadequate to explain even the features of the second and third families of fermions. There are three inadequacies. (1) The factor of 3 comes in squared between the mass of the leptons and quarks of the second family. The reason is that, for ǫ small due to the mass hierarchy between families, the second eigenvalue of L is given by the seesaw formula m 0 (2) The matrices D and U are still exactly proportional. This is a consequence of the fact that the generator B − L does not distinguish up and down quarks. Therefore, the CKM angle V cb still exactly vanishes. (3) Because D and U are exactly proportional, one still has the bad prediction m 0 c /m 0 t = m 0 s /m 0 b . It is clear that the breaking of SO(10) due to the adjoint cannot cure all of these problems, since B −L does not distinguish D from U. Thus the breaking of SO(10) done by 16 H +16 H must come into play. As we shall now show, a single simple operator exists, which involves one of these spinor Higgs and cures at one stroke all three of the problems we have identified.
The lowest dimension effective Yukawa operators that involve the spinor Higgs fields are quartic in spinors. Consider, therefore, operators of the form 16 i 16 j 16 ′ H 16 H /M G . The 16 H is the spinor Higgs field that breaks SO(10) at M G down to SU(5). The 16 ′ H is a spinor Higgs that has a weak-scale VEV that breaks SU(2) L × U(1) Y . In principle, these two spinors could be the same field. However, if they were, it would mean that they had to be contracted symmetrically by Bose statistics, which in turn would mean that 16 i and 16 j would also have to be contracted symmetrically. A careful examination shows that the resulting flavor-symmetric contributions to the mass matrices do not lead to realistic forms, though it is possible to achieve realistic mass matrices by adding yet another Yukawa operator, as in the interesting model of Babu,Pati,and Wilczek [16]. Therefore 16 ′ H must be a distinct field. As will be seen later, introducing this 16 ′ H involves no loss of economy, since it allows a very elegant explanation of the largeness of the ratio m t /m b without making tan β large.
There are still several operators of this type to be considered: the family indices can take the values ij = 33, 22, 23, or 32, and there are three ways to contract the four spinors to make an SO(10) singlet. Here again, one must examine the various cases to see which gives the most realistic mass matrices. As it turns out, there is one operator that is much superior to the others, in the sense that it much more cleanly and simply fits the data. It is of the form [16 2 16 H ][16 3 16 ′ H ], where [...] means that the spinors inside are contracted into a 10. This can arise very simply by integrating out a 10 of fermions, as shown in Fig. 1

(c).
Let us write the resulting mass operator in SU (5) language. Denote by p(q) a p multiplet of SU(5) that is contained in a q multiplet of SO (10). The VEV of 16 H lies, of course, in the 1(16) direction, while the VEV of 16 ′ H that breaks the weak interactions lies in the 5(16) direction. Therefore, the resulting mass term is of the form [5(16 2 (5) terms gives effectively the operator (5 2 10 3 )5 H . Note that this has the same form as the SU (5) operator discussed in the last Section, which gave the σ entries in Eq. (6). The result, then, of including this operator [2] is to make the mass matrices take the form: The new term has given the entries we call σ. Note that these lopsided entries appear only in D and L. The reason is simply that the 16 ′ H contains a 5 of SU(5) but no 5. It is easy to see that the new term with σ ≫ ǫ cures at once all three of the problems we identified with the forms given in Eq.
More exact expressions will be given later. Thus the desired Georgi-Jarlskog factor of 1/3 is obtained, instead of 1/9. The σ entry has dominated over one of the factors of ǫ/3 and thus prevented the factor of 1/3 from coming in squared.
(2) The σ entry comes into D but not U, and thus breaks the proportionality of the two matrices. As a result, V cb no longer vanishes, but is given approximately by (ǫ/3)( σ 2 σ 2 +1 ). Note that this is of the same order in ǫ as m s /m b ∼ = (ǫ/3)( σ σ 2 +1 ), rather than m s /m b as is the case with Fritzsch forms, and accords much better with the actual experimental values.
(3) The fact that σ breaks the proportionality of U and D also means that the bad still of order ǫ 2 and therefore much smaller. This also accords well with the experimental numbers. In fact, as we shall see, if one uses V cb and m µ /m τ to fix the parameters σ and ǫ, one finds that m c (1 GeV) is predicted to be in agreement with the experimentally determined value of 1.27 ± 0.1 GeV. It should also be noted that the prediction m 0 b = m 0 τ is only very slightly affected by the addition of the σ term, both m 0 b and m 0 τ being given to leading order in ǫ by √ σ 2 + 1M D . The economy of the above mass matrix forms is seen in the fact that five quantities (V cb , m µ /m τ , m s /m b , m c /m t , and m τ /m b ) are successfully fit with only the two parameters σ and ǫ. No other published form succeeds in accurately reproducing the masses and mixing of the the heavier two families with so few parameters. The predictions and fits will be discussed in detail in Section IV.
We see that the matrices in Eq. (10) were arrived at by a process of reasoning that had nothing to do with the question of neutrino mass but rather with an attempt to get realistic masses and mixings for the quarks and charged leptons using as simple a Higgs sector as possible in SO (10). But what has emerged is a structure with precisely the three critical features identified in the last Section as giving a simple explanation of the large mixing of ν µ and ν τ . In fact, a fit of V cb and m µ /m τ gives σ ∼ = 1.8 and ǫ ∼ = 0.14. Consequently, as can be seen directly from Eq. (10), the angle . This is quite consistent with what is observed. We will look more carefully at these predictions later.
To summarize, with two parameters, ǫ and σ, four mass ratios and two mixing angles are satisfactorily accounted for, if we include U µ3 . No greater economy could be hoped for in explaining the spectrum of the heavy two families. Moreover, as we shall see in the next Section, the forms in Eq. (10) can be extended to include the first family with equal economy: the introduction of two new parameters (one of which is complex) will nicely account for seven quantities pertaining to the first family.
Before explaining how the model is extended to the first family we will expand on a couple of points made earlier. First, we said that the introduction of the 16 ′ H allows a simple explanation [1,16] of why m t ≫ m b that does not require a tan β ≫ 1. The point is that the Higgs doublet of the MSSM that is often called H U is purely contained in the 10 H that couples to 16 3 16 3 and gives rise to the "1" entry in the mass matrices of Eq. (10). However, the Higgs doublet of the MSSM that is called H D does not come purely from 10 H . Rather it is a mixture of doublets in 10 H and 16 ′ H , since they both contain 5's of SU (5). Thus we may write where γ is some mixing angle that depends on the parameters of the Higgs sector. Since the 33 elements of the mass matrices all arise purely from the coupling of the 10 H , the parameters we called M U and M D in Eq. (10) are given by leading to the ratio M U /M D = H U /( H D cos γ) = tan β/ cos γ, where tan β is defined to be the ratio of the Higgs VEV H U giving mass to the top quark to the Higgs VEV H D giving mass to the bottom quark. Hence The point is simply that the large ratio of the top to bottom masses could be the result of cos γ being small rather than tan β being large. In fact, since we do not know anything a priori about the angle γ, we cannot say whether tan β is large or small. It should be noted that if one assumes that H D lies mostly in the 16 ′ H (so that cos γ ≪ 1), it would explain why the parameter σ is large since it comes from a coupling to 16 ′ H , and also explain why tan β might be small.
A second point we wish to underline here has to do with the reasonableness of asymmetric mass matrices. In many models it is assumed that all the mass matrices are symmetric. However, this is not something that is called for by the group theory of grand unification. It is true that with the minimal Yukawa terms SU (5) gives a symmetric U. But even with minimal Yukawa terms SU (5) does not predict any symmetry of the D and L matrices. And in SO (10), as we have seen, once one introduces the effects of SO(10) breaking into the Yukawa sector, as one virtually must, one easily obtains effective Yukawa terms that are asymmetric. Fig. 1(c) shows that very simple diagrams can give terms that are lopsided, in the sense that they contribute only above or below the diagonal. From the point of view of the fundamental grand unified theory, then, lopsided terms are as natural as symmetric ones. The preference for symmetric terms has been the result not of examining what kinds of terms are obtained in a simple way in unification, but rather from the desire to reduce the number of parameters at the level of mass matrices with the aim of making models which are highly predictive. However, putting oneself in the straightjacket of symmetric matrices makes it hard to get a good fit to all the quark and lepton masses and mixings. It turns out, as we have seen, and will see further below, that allowing asymmetric matrices makes possible a model which gives both a very good fit to the data and is actually much more predictive than most models which assume symmetric matrices.

IV. EXTENSION TO THE FIRST FAMILY
In arriving at the form of the mass matrices for the heavy two families we were limited in the choices that were possible by the assumption we made about the simplicity of the SO(10)breaking sector. In extending to the first family we are not quite so limited. Nevertheless, the number of simple possibilities is not very large. There are several discrete choices: Should the contributions to the first row and column of the mass matrices be flavor symmetric like the 1's in Eq. 10, antisymmetric like the ǫ's, or lopsided like the σ's? Should they contribute to all the matrices equally like the 1's, to all the matrices but with non-trivial Clebsch factors like the ǫ's or only to D and L like the σ's? It is fairly easy to run through the various cases and see what kinds of relations among masses and mixings result. As it turns out, one of the simplest possibilities gives a remarkably good fit to the data. This uniquely simple choice [3] is the following: We have already mentioned that fits give σ ∼ = 1.8 and ǫ ∼ = 0.14. The new parameters δ and δ ′ both have magnitude of about 0.008. The parameter η is by far the smallest, being about 8 × 10 −6 . The only role that η plays in the sector of quarks and charged leptons is in giving the up quark a mass, for it makes negligible contributions to the down quark and electron masses as determined from D and L, respectively. In Fig. 2(a) we have illustrated a higher-order diagram that can contribute to the parameter η. Since it is not excluded that the up quark is exactly massless, it is possible to set η to zero. In any event, one can see that η ∼ = m 0 u /m 0 t , which is by orders of magnitude smaller than any other interfamily ratio of masses in the standard model. It will, however, be of some significance for neutrino masses. If η vanishes, this model gives only the small-angle MSW solution to the solar neutrino problem. But even if η is as small as 8 × 10 −6 , it allows either the small-angle MSW solution or bimaximal neutrino mixing to arise in a simple way.
Turning to the parameters δ and δ ′ , we see that they appear symmetrically and only in D and L. Such terms are easily obtained in SO (10) from simple diagrams such as that shown in Figs. 2(b) and 2(c). The effective operators arising from these diagrams are of the form [16 1 16 j where again the spinors in brackets are contracted symmetrically into a 10 of SO (10) which is integrated out. Note, however, that the symmetric contributions δ and δ ′ from the two Higgs contraction [1(16 H H )] contributes only to D and L by virtue of their SU(5) structure. Contrast these effective operators for j = 2, 3 with that occurring previously for the term σ arising from the diagram shown in Fig. 1(c).
The three new parameters we have introduced are, as we shall see, sufficient to account for everything about the first family. Before proceeding, however, we must be careful about complex phases. It is easy to show that if we allow all parameters of the model to be complex all but two phase angles can be rotated away from the mass matrices U, D, L and N, provided we now neglect the negligible η contributions to D and L. We will call these physical phases α and φ which appear as follows, where in these matrices and henceforth ǫ, σ, δ, δ ′ and η denote the magnitudes of these parameters and the phases are written explicitly. The phase α only comes into the fits of masses at higher order in the small quantity ǫ/σ ∼ = 0.08. Numerically, its effect is only a few percent; moreover, the fits (especially to m c ) prefer a value near zero. Therefore, we can ignore α and will do so from now on. That leaves only the phase φ. Its only significant effect, but a very important one, is to give the CP-violating phase angle δ CP .
Instead of using the parameters δ, δ ′ and e iφ , it will be somewhat more convenient to use the parameters t L , t R , and e iθ , which are defined in terms of them as follows: and The significance of these parameters is that they are essentially the left-handed and righthanded Cabbibo angles. This can be seen by taking the forms for D and L given in Eq. (15) and diagonalizing the 2-3 block. When this is done the 1-2 blocks of these matrices take the form In terms of the five dimensionless parameters ǫ, σ, t L , t R , and e iθ with η set equal to zero, we now write down expressions for fourteen observable quantities: seven ratios of quark and lepton masses, three CKM angles and one phase, and three lepton mixing angles.
These expansions have been carried to sufficiently high order in small quantities to be accurate to within 0.2% and are useful in doing the fits to the data. However, the leading terms in these expansions have much simpler forms and thus allow one to see more readily the relationships among various quantities in this model. We therefore write these simpler expressions for purposes of discussion.
It might at first seem surprising that without any information about the Majorana mass matrix M R of the right-handed neutrinos we are able to write down predictions for the three neutrino mixing angles. However, if η = 0, as we are assuming at present, then the Dirac mass matrix of the neutrinos (N) has vanishing first row and column, and therefore, obviously, the same will be true of the mass matrix of the light neutrinos, which is given by the well-known "see-saw" formula M ν = −N T M −1 R N. This means that the two mixing elements of the electron neutrino, U e2 and U e3 , get no contribution from diagonalizing M ν , but come entirely from diagonalizing L. Since L is a known matrix in our model, these two mixing elements are predicted. In the case of the mixing of the mu and tau neutrinos, U µ3 does receive a contribution from diagonalizing M ν . However, as can be seen from the form of N this is an effect of O(ǫ). The contribution to U µ3 coming from diagonalizing L, on the other hand, is of order unity, since it arises from the large parameter σ. Thus U µ3 is predicted, although not precisely.
Since we have written fourteen quantities in terms of five parameters, there are altogether nine predictions of the model. Which quantities one takes as "predicted" depends on which quantities are used to determine the values of the parameters. We will use the lepton mass ratios and the angles V cb and V us for this purpose as they are the best measured. As one can see from the third and eighth of Eqs. (20), one can get the value σ from the ratio 3V 0 cb /(m 0 µ /m 0 τ ). One finds (of course, taking the renormalization effects into account as was done in [2]) that numerically σ ≃ √ 3. Substituting this into the expression for m 0 µ /m 0 τ , one obtains that ǫ ≃ 0.14. This is the small parameter of the model that is responsible for the hierarchy between the second and third families, and is small enough that the expressions in Eqs. (20) are fairly accurate. One can use m e /m µ and V us to determine t L and t R . A careful fit, described later, gives t L = 0.236 and t R = 0.205. That t L ≃ t R is easily understood from Eq. (18) and the well-known Weinberg-Wilczek-Zee-Fritzsch result [8] that the Cabbibo angle is well acounted for by symmetric mass matrices for the first two families; cf. Eq. (1). The near equality of t L and t R is also apparant from the seventh and ninth relations of Eqs. (20) and the fact that numerically V us ∼ = m d /m s . The phase factor e iθ will be determined from the CP-violating phase δ CP .
The nine predictions, then, are the following. To begin with, there are the three famous predictions, (1) The first is the "good" prediction of minimal SU(5) unification, and the latter two are the Georgi-Jarlskog relations. These predictions are manifest from the first, third, fourth, sixth, and seventh of Eqs. (20). It is hardly surprising that the model gives these relations, since we were guided by them in constructing the model. The fourth prediction is (4) m 0 u /m 0 t = 0. Even if the u quark is not exactly massless this relation is a very good approximation to reality. If one takes the favored value of m u ≈ 4 MeV, then, with reasonable assumptions about thresholds in doing the running up to the GUT scale, one obtains η ≃ m 0 u /m 0 t ≈ 8 × 10 −6 . This is far smaller than any other interfamily ratio of masses. For instance, the comparable ratio for down quarks is m 0 d /m 0 b ∼ = 10 −3 , and for leptons is m 0 e /m 0 τ ∼ = 3 × 10 −4 . Like the previous three relations, m u ≈ 0 is a reflection of basic group-theoretical aspects of the model. It comes from the fact, explained above, that the δ and δ ′ entries only appear in D and L.
The prediction for m c is particularly interesting. We see immediately that, for reasons having to do with the group-theoretic structure of the model, the ratio m 0 c /m 0 t is much less than the corresponding ratio m 0 s /m 0 b for the down quarks because it is of higher order in the small parameter ǫ. This is a highly significant success, because the minimal Yukawa terms of SO(10) notoriously give these ratios to be equal. Moreover, the success is not merely a qualitative one. When ǫ and σ are fit (using V cb and m µ /m τ ) and the renormalization effects are later taken into account, it is found that m c comes out within about 5% of the experimentally preferred value, which is quite remarkable given the various experimental and theoretical uncertainties. This success is non-trivial, because the reasoning that led to the forms of the mass matrices did not depend upon the value of m c , and hence it could have been expected that m c would come out wrong by a large factor.
Another non-trivial quantitative hurdle for the model is the prediction for V ub . The eighth, ninth, and tenth relations of Eqs. (20) give In other words the model predicts that V ub should lie on a certain circle in the complex plane. As can be seen from Fig. 3, the circle for V ud V * ub slices neatly through the middle of the presently allowed region. Again, this is a very significant success, since the reasoning that led to the forms in Eq. (15) was not based on the value of V ub .
The prediction for the mixing of ν µ and ν τ has already been discussed. It is one of the key successes of this model that this mixing turns out to be nearly maximal. The fact that σ ≃ √ 3 tells us that the first term in the expression for U 0 µ3 in Eq. (20) corresponds to an angle near π/3. As we shall see in the next Section, the O(ǫ) corrections easily bring this down close to the maximal mixing value of π/4.
The prediction of this model for the mixing of ν e and ν µ with η = 0 is quite interesting. From the sixth relation of Eqs. (20) and the fact that t L ≃ t R , one sees that 1 3 t R ≃ m e /m µ . Thus the model predicts that U e2 ∼ = cos θ µτ m e /m µ . The factor of cos θ µτ is crucial [17] since without it one would have sin 2 2θ solar = 4|U e1 | 2 |U e2 | 2 ≈ 4(m e /m µ ) ∼ = 2 × 10 −2 , which is about twice the value needed for the small-angle MSW solution to the solar neutrino problem. Since atmospheric neutrino data tells us that cos θ µτ ≃ 1/ √ 2, the model gives just the correct value for the small-angle MSW solution.
In the future both V ub and sin 2 2θ solar will be known better and will provide a sharp test of the model. The theoretical uncertainties in the predictions for V ub and U e2 are estimated to be only a few percent.
In discussing the ν e −ν µ mixing above, we have assumed that η = 0. If η does not vanish, but is around 8 × 10 −6 , corresponding to m u ≈ 4.5 MeV, then it turns out that both the small-angle MSW solution we just discussed and bimaximal mixing are possible. This will be discussed in detail in the next Section.
Finally, there is the prediction of ν e − ν τ mixing. One sees from Eq. (20) that there is a prediction that U e3 ∼ = tan θ µτ U e2 ∼ = 0.05. It is interesting that even for the bimaximal mixing case that will be discussed in the next Section, the numerical value of U e3 is virtually unaffected. Thus this prediction is a "robust" one of this model.

V. NEUTRINO MIXING
A. Mixing of ν µ − ν τ In analyzing the predictions of this model for ν µ − ν τ mixing, we may make the approximation that η = 0. This means that N has vanishing first row and column. Therefore, in computing M ν = −N T M −1 R N the first row and column of M −1 R are irrelevant. Thus we may write M −1 R as where X, Y , and Z are in general complex. There are consequently five real parameters (the over all phase does not matter) that come into the masses and mixing of ν µ and ν τ from M R . As observed earlier, this does not prevent us from making a qualitative prediction for the mixing parameter U µ3 , since the contribution to it from diagonalizing the mass matrix M ν is only of order ǫ and U µ3 comes predominantly from diagonalizing the known matrix L. However, in order to see if more precise predictions can be obtained, we shall look at two simple special cases: and (II) In these cases only three parameters in M R , namely Λ R , B and e iβ , contribute to the neutrino observables of the second and third families, since ǫ has appeared previously and is used as a natural scaling parameter. In the first case, The neutrino mixing matrix U, now known as the MNS mixing matrix [18], is given by U = U † L U ν , where U L is the unitary matrix that diagonalizes L † L, and U ν is the unitary matrix that diagonalizes M † ν M ν . For case I, U ν is given by where tan 2θ ν 23 = 2ǫ/K, and K ≡ 2+B −1 e iβ . The ratio of eigenvalues of M ν gives m ν 2 /m ν 3 ∼ = (ǫ 2 / |K| 2 )(1 − ǫ 2 / |K| 2 + ...). One can choose cos θ ν 23 to be real, and one can write where e iξ is the phase of K. One readily sees from the form of the charged-lepton mass matrix L in Eq. (15) that sin θ L 23 ≡ (U L ) 23 is given by tan 2θ L 23 = − 2(σ+ǫ) Using the best fit values of σ and ǫ one finds, more precisely, that θ L 23 ∼ = 63 • . Altogether, then, the mixing parameter of ν µ and ν τ is given by If neutrino masses are hierarchical, and atmospheric neutrino oscillations are ν µ − ν τ oscillations, then m ν 3 ≃ 0.06 eV. And if one further assumes the small-angle MSW solution to the solar neutrino problem, then m ν 2 ≃ 0.003 eV. Thus, m ν 2 /m ν 3 ≃ 0.05, within a factor of two or so. Taking it to have the value 0.05, and the phase ξ to vanish, Eq. (28) gives U µ3 ∼ = −0.756, and sin 2 2θ µτ ∼ = 0.984. With the same value of the neutrino mass ratio and ξ taken to be π/4, sin 2 2θ µτ ∼ = 0.943. We see that there is excellent agreement with the experimental limits from SuperKamiokande if the complex phase is not too large. But if ξ = π/2, with the same mass ratio, sin 2 2θ µτ ∼ = 0.77. The value of m ν 2 /m ν 3 = 0.05 corresponds to |K| = 0.63. Since B −1 e iβ = |K|e iξ − 2, for ξ = 0 this gives B = 0.73, or Bǫ = 0.1. In other words, no very great hierarchy is required in M R .
Turning now to case II, we have that Consequently, for this case tan 2θ ν where The ratio of eigenvalues of M ν gives m ν 2 /m ν 3 ∼ = If we take, m ν 2 /m ν 3 = 0.05, as before, and assume that K ′ is real, we have that K ′ ∼ = 0.6. This gives θ ν 23 ∼ = 12.3 • , θ µτ ∼ = 50.5 • , and sin 2 2θ µτ ∼ = 0.96. Again, there is good agreement with the SuperKamiokande results. Moreover, since this value of K ′ corresponds to B = −2.5, or Bǫ 2 ∼ = −0.05, we see that in this case also no great hierarchy is needed in M R .
The foregoing discussion is all based on the assumption that the mixing with the first family is small, so that one has the small-angle MSW solution to the solar neutrino problem. This will certainly be the case if η = 0. As we will now see, if instead η ∼ = 8 × 10 −6 , as needed to have m u ∼ = 4.5 MeV, either the small mixing of ν e that we have been considering or large mixing of ν e is possible, depending on the form of M R .

B. Mixing of the First Family
In the previous discussion, we set η = 0 in which case, no matter what the form of M R , the matrix M ν = −N T M −1 R N has vanishing first row and column, and the matrix U ν that diagonalizes M † ν M ν has the form of Eq. (26). It is easy to show that the matrix U L which diagonalizes L † L has the form where sin θ L 12 ∼ = 1 3 t R , t R is defined in Eq. (17), and θ L 23 is given after Eq. (27). Putting these together, one has that the total mixing matrix of the neutrinos, where θ µτ = θ L 23 − θ ν 23 . This yields the results, already given in Eq. (20), for U e2 and U e3 . Now we will consider what happens under what is presumably the more realistic assumption that η ∼ = 8 × 10 −6 .
With η = 0, there are two basic possibilities to consider. One possibility is that M R has a form in which its 12,21,13, and 31 elements all vanish or are negligibly small. If such is the case, then the previous analysis applies, and the mixing of ν e is due entirely to the matrix L. The only effect of the parameter η in the lepton sector is then to give ν 1 a mass of about 4 × 10 −7 eV. The second possibility is that M R does have significant 12, 21 and/or 13, 31 elements. If this is the case then a strikingly different situation can arise [4], namely "bimaximal" mixing [19], [20].
We will first illustrate what happens with a simple example. Consider the following form for M R : We normalize A and B by powers of ǫ simply for later convenience. The mass matrix of light neutrinos resulting from this form is One sees that the 2-3 block has vanishing determinant, so that a rotation in the 2-3 plane by an angle θ ν This can be put in a more transparent form by a rotation in the 1-3 plane by an angle θ ν 13 ∼ = −η/(ǫ 2 A). This angle is less than or of order 10 −4 and thus negligible, practically speaking, so It is clear that the 11 element, being higher order in η, is likely to be much smaller than the 12 and 21 elements. The condition for this to be the case is that A/(B −1) > η/ǫ 3 ∼ = 2×10 −3 . If this very weak condition is satisfied, then the form of the matrix manifestly corresponds to the situation in which the ν e and ν µ together form a pseudo-Dirac pair. That in turn would mean that the mixing of these two neutrinos is very close to maximal. One sees from Eq. (36) that m ν 3 = M 2 U /Λ R , and that the splitting between m ν 1 and m ν 2 is given by ∆m 2 For the vacuum solution to the solar neutrino problem, one has ∆m 2 21 ≃ 10 −10 eV 2 , so that ∆m 2 21 /m 2 ν 3 ≃ 3 × 10 −8 ∼ = 2(η 3 /ǫ 5 )(B − 1)/A 3 . This gives A(B − 1) 1/3 ≈ 0.06. Thus no great hierarchy is required in M R to get the vacuum oscillation solution. The reason for this is that in this scheme the smallness of ∆m 2 21 is due to the extreme smallness of the parameter η, which is equal to the ratio m u /m t .
It is easy to see from what has already been said that the matrix U ν needed to diagonalize M † ν M ν is of the form where we have neglected the tiny rotation θ ν 13 . The matrix U L is already given in Eq. (31), so that the full neutrino mixing matrix can be written where U SM A is given in Eq. (32), and is just the form that results in the small mixing angle (SMA) MSW case of this model. In other words, the net result of the large mxing of the first family produced by the A entries in Eq. (34) is simply to multiply the (SMA) MSW form of U on the right by a rotation of π/4 in the 1-2 plane. Consequently, the predictions for U µ3 and U e3 are essentially unaffected. However, U e2 becomes 1/ √ 2 instead of the value given in Eq. (20).
The interesting lesson is that "bimaximal mixing" is easy to achieve if the large mixing of ν µ and ν τ comes from the charged lepton sector, i.e., from diagonalizing L, while the large mixing of ν e comes from diagonalizing M ν .

VI. DETAILS OF A SPECIFIC MODEL
In the previous Sections we have presented the construction of our SO(10) minimal Higgs model in the framework of effective SO(10) and SU(5) operators. We now show that one can construct a technically-natural realization of this scheme by introducing sets of Higgs and matter superfields with a well-defined family symmetry. We first address the Higgs sector.
A. Higgs Sector with U (1) × Z 2 × Z 2 Family Symmetry The doublet-triplet splitting problem in SU (5), and therefore SO (10), arises because the colored Higgs in the 5 − 5 pairs of each 10 H must be made superheavy at the GUT scale, while just one pair of Higgs doublets should remain massless there and be free to develop VEV's at the electroweak scale. This problem has been addressed and solved in [5] by the introduction of just one 45 adjoint Higgs field with its VEV pointing in the B − L direction, together with two pairs of 16 + 16 spinor Higgs fields, two 10 Higgs in the vector representation plus several Higgs singlets. We shall briefly summarize the solution, but first we note that it is necessary to introduce several more Higgs fields in the vector and singlet representations in order to generate the Yukawa structure for the fermion mass matrices presented earlier.
The authors of [5] found that the Higgs superpotential required to solve the doublettriplet splitting problem could be neatly obtained from their list of Higgs fields by introducing a global family symmetry group of the type U(1) × Z 2 × Z 2 , which can arise in a natural fashion from string theory. With this in mind, we now list in Table I all the Higgs fields to be considered together with their family charge assignments.
Higgs Fields Needed to Solve the 2-3 Problem: Additional Higgs Fields for the Mass Matrices: 10: Table I. Higgs superfields in the proposed model.
As noted in the table, in order to complete the construction of the Dirac mass matrices, four more vector Higgs fields and four additional Higgs singlets are needed, while one Higgs singlet is introduced to generate the right-handed Majorana neutrino mass matrix. It is then possible to write down explicitly the full Higgs superpotential from the Higgs SO(10) and family assignments, where we have written it as the sum of five terms: The Higgs singlets are all assumed to develop VEV's at the GUT scale. We can then determine the fate of the other Higgs fields from the F-flat and D-flat conditions. In particular, W A fixes A through the F A = 0 condition where one solution is A ∝ B − L, the Dimopoulos-Wilczek solution [12]. W CA gives a GUT-scale VEV to C and C by the F X = 0 condition and also couples the adjoint A to the spinors C, C, C ′ and C ′ without destabilizing the Dimopoulos-Wilczek solution or giving Goldstone modes, as shown in [5]. W 2/3 gives the doublet-triplet splitting by the Dimopoulos-Wilczek mechanism [12], [5]. W H D mixes the (1, 2, −1/2) doublet in T 1 with those in C ′ (by F C = 0), and in T 0 and T ′ 0 (by F T 0 = 0).

B. Yukawa Sector
We now turn to the Yukawa sector and specify the matter fields and their U(1) × Z 2 × Z 2 charge assignments which will complete the realization of the specific model in question.
For this purpose, we require three spinor fields 16 i , one for each light family, two vector-like pairs of 16 − 16 spinors which can get superheavy, a pair of superheavy 10 fields in the vector representation, and three pairs of superheavy 1 − 1 c fermion singlets. The complete listing is given in Table II. Table II. Matter superfields in the proposed model.
In terms of these fermion fields and the Higgs fields previously introduced, one can then spell out all the terms in the Yukawa superpotential which follow from their SO(10) and U(1) × Z 2 × Z 2 assignments: where the coupling parameters have been suppressed. To obtain the GUT scale structure for the fermion mass matrix elements, all but the three chiral spinor fields in the first line of Table II. will be integrated out to yield Froggatt-Nielsen diagrams [21] of the type pictured earlier. Note that the right-handed Majorana matrix elements will all be generated through the Majorana couplings of the V M Higgs field with conjugate singlet fermions given in the last two terms of Eq. (42). In order to present a clearer description of how the GUT scale mass matrices are determined from the Yukawa and Higgs superpotentials, we shall illustrate the procedure for the up quark mass matrix U. The three massless color-triplet quark states each with charge 2/3 are obtained as linear combinations of all such color and charge states within the fermion supermultiplets given in (41). In particular, the basis for the left-handed states u L and left-handed conjugate states u c L can be ordered as follows: where the states are labeled by their representations and hypercharge according to |SU(3) c , SU(2) L , Y SU (5)(SO (10)) . We then form the Yukawa contribution u cT L C −1 D u u L by using the above bases and the superpotentials to obtain the matrix where we have introduced the following shorthand notation: We can then determine from this matrix the three pairs of zero-mass eigenstates at the GUT scale where the electroweak VEV of T 1 vanishes: and where the states are now simply labeled by their SU (5) and SO (10) representations. Finally, the Dirac matrix U for the three light quark states u, c, t is obtained by bracketing the electroweak contributions by the appropriate u c iL state on the left and the u jL state on the right. The result obtained for U has exactly the form found earlier in Eq. (15) from the previous effective operator approach, with the identifications: Here the subscript on a q signifies a factor of 1/3 arising from the B − L VEV of the A in the adjoint representation, while the subscripts on the t terms specify the appropriate doublet VEV in the 10 for T 1 . We have neglected the state normalization factors in (47) but will later argue that they can all be taken to be approximately unity.
The Dirac matrices, D, N, L are constructed in a similar fashion. In the case of D and L, the bases corresponding to Eqs. (43) and (44) are enlarged by two states lying in the 10 1 and 10 2 representations of SO (10). For N, on the other hand, in addition to the two above states, one must add the singlet fermion contributions from the representations 1 k , k = 1, 2, 3 for B ν L and 1 c k , k = 1, 2, 3 for B ν c L . We list the zero-mass state vectors for D, N and L in analogy to Eqs. (46): In the above we have introduced, in analogy with Eqs. (45), the additional shorthand notation: The Dirac matrices D, N and L are found by forming matrix elements of the electroweak symmetry breaking VEV's with the appropriate basis vectors. Again these matrices have exactly the structures given in Eqs. (15), provided the state normalization factors are approximated by unity, i.e., we assume that the zero-mass states have their large components in the chiral representations 16 1 , 16 2 and 16 3 and that all the other components are small. We shall return to this point in the next Section. In the meantime we make the identifications M D = (t 3 )5 (10) ǫM D = |3(a q /p)(t 2 )5 (10) | σM D = −(c/y)(c ′ )5 (16) in terms of the notation given in Eqs. (45) and (51) and the following: The phase φ appearing in the δ ′ term can be understood to arise from a phase in the VEV for S ′ . The structures of the Dirac matrix elements given in Eqs. (15), (47) and (52) can be understood in terms of the simple Froggatt-Nielsen diagrams of Fig. 1 and 2, with the Higgs fields labeled as in Table I.
Turning to the right-handed Majorana mass matrix, we use the zero mass left-handed conjugate states that were obtained implicitly above for the Dirac matrix N to form the basis for M R . The right-handed Majorana matrix is then obtained by bracketing the Majorana Higgs V M with the appropriate zero mass conjugate neutrino states in (49). We obtain where The lighter two right-handed Majorana masses are degenerate and have opposite CP-parity. Note that the whole right-handed Majorana matrix has been generated in this simple model by one Majorana VEV coupling superheavy conjugate fermion singlets as shown in Fig. 4.
We conclude this Section with a summary of the GUT scale predictions derived from the Dirac and Majorana mass matrices with the particular parameters appropriate for the model in question. For convenience we give the whole set equations which are the counterpart of Eqs. (19).
To the quark equations we have added the ratio m 0 t /m 0 b which involves the ratio of 5(T 1 ) to 5(T 1 ) , i.e., M U /M D , as well as giving the leptonic mass ratios and mixings specific to the model in question.

C. Numerical Evaluation of Matrix Parameters
We have elaborated above how the simple explicit model proposed gives precisely the structure for the Dirac mass matrices that was obtained from the effective operator approach. We now show that the entries are also numerically in the range to fit the quark and lepton mass and mixing data.
In order to compare the GUT scale predictions in Eq. (56) with the low scale data, the GUT scale values are first evolved from Λ G = 2 × 10 16 GeV down to the SUSY scale which is taken to be Λ SUSY = m t (m t ) = 165 GeV by use of the 2-loop MSSM beta functions. For this purpose, the mass ratios at the two scales are related by the η i/j running factors, while the quark mixing elements are scaled by the η ij factor according to The remaining evolutions to the bottom and charm quark or tau lepton running mass scales, or to the 1 GeV scale in the case of the light quarks and leptons, is carried out with the 3-loop QCD and 1-loop QED renormalization group equations. Here the running factors are η i with the mass ratios scaled according to or similarly, with the running mass scale m i replaced by 1 GeV. With tan β = 5 used for the numerical evaluations for reasons that will become apparently shortly, α s (M Z ) = 0.118, α(M Z ) = 1/127.9 and sin 2 θ W = 0.2315, the running factors are given by The upper vertex of the triangle appears to be circled precisely in the allowed experimental region.
Additional predictions follow for the neutrino sector. The effective light neutrino mass matrix of Eq. (34) or (36) with B = 0 leads to bimaximal mixing with a large angle solution for atmospheric neutrino oscillations [6] and the "just-so" vacuum solution [19] involving two pseudo-Dirac neutrinos, if we set Λ R = 2.4 × 10 14 GeV and A = 0.05. We then find m 3 = 54.3 meV, m 2 = 59.6 µeV, m 1 = 56.5 µeV The effective scale of the right-handed Majorana mass contribution occurs two orders of magnitude lower than the SUSY GUT scale of Λ G = 1.2 × 10 16 GeV. The effective twocomponent reactor mixing angle given above should be observable at a future neutrino factory, whereas the present limit from the CHOOZ experiment [23] is approximately 0.1 for the above ∆m 2 23 . In principle, the parameter A appearing in M R can also be complex, but we find that in no case does the leptonic CP-violating phase, δ ′ CP exceed 10 o in magnitude. Hence the model predicts leptonic CP-violation will be unobservable.
The vacuum solar solution depends critically on the appearance of the parameter η in the matrix N, corresponding to the non-zero η entry in U which gives the up quark a mass at the GUT scale. Should we set η = 0, only the small-angle MSW solution [7] would be obtained for the solar neutrino oscillations. The large angle MSW solution is disfavored by the larger hierarchy, i.e., very small A value, required in M R .
Finally we address the issue that the state normalization factors were all replaced by unity in Eqs. (47) and (52) for the various matrix parameters. This is a good approximation provided the three fermion spinor states |16 1 , |16 2 , |16 3 provide the dominant contributions to the zero-mass quark and lepton states at the GUT scale. In particular, the following ratios must be much less than unity: Let us assume for simplicity that the electroweak couplings of T 1 in t 3 , t 2 and t ′ in Eq. (45) and of C ′ in c ′ of (53) are identical. Then with ǫ = |3(a q /p)| = |a ℓ /p| = 0.14, we find the condition (a/p) 2 ≃ 0.02 ≪ 1 holds. To obtain an up quark mass m u (1GeV ) ≃ 4.5 MeV, we need η ∼ = (y ′ /s ′′ ) 2 ≃ 8 × 10 −6 at the GUT scale, which easily satisfies (64).
Requiring that (c/y) 2 ≪ 1 and with the result from Eqs. (52) that leads us to the results that in terms of the T 1 −C ′ mixing angle, γ, in Eq. (11). With c/y ∼ = 0.1, for example, tan γ ≃ 18 which implies tan β ≃ 6, a very reasonable mid-range value allowed by experiment. For this reason, we have chosen to illustrate the numerical results above with tan β = 5. The remaining ratios in Eq. (64) can also be satisfied. For comparablec i 's, Aǫ 3 ∼ 1.4×10 −4 obtained from Eq. (55) requires that Z / X P ∼ 0.01. This ratio is consistent with the VEV's needed in the Higgs superpotential of Eq. (41) in order to solve the doublettriplet splitting problem.
Thus we have found that not only are the desired forms of the Dirac (and Majorana) matrices generated by the model of this Section, but that the numerical values required for the matrix parameters are also quite reasonable.

VII. SUMMARY
Both the largeness of the atmospheric neutrino mixing U µ3 and the smallness of the quark mixing V cb can be elegantly accounted for by the idea that the charged lepton mass matrix L is highly asymmetric or "lopsided" and that the down-quark mass matrix D is related to the transpose of L by an SU (5) symmetry. This idea was discovered independently by several groups and has since been used in numerous models of fermion masses. Remarkably, exactly such mass matrices emerged in our work from quite other considerations than neutrino masses and mixings, specifically from an attempt to construct the simplest possible realistic SO(10) model.
Advances have been made in recent years in simplifying the Higgs structure of SUSY SO(10) models. If one assumes the minimal set of Higgs fields that can break SO (10) down to the standard model group, the possibilities for Yukawa terms for the quarks and leptons become significantly restricted. It turns out that there is what seems to be a uniquely simple set of SO(10) Yukawa terms that gives realistic masses and mixings. This set consists of only six effective Yukawa terms (five if m u = 0) which satisfactorily fits all nine masses of the quarks and charged leptons as well as the four CKM parameters. In addition, large ν µ −ν τ mixing emerges automatically. Moreover, in this uniquely simple model, the simplest possibilities for the Majorana mass matrix M R of the right-handed neutrinos lead either to small angle MSW values for the solar neutrino mixing or to vacuum oscillation values. In this paper we have studied in detail the consequences of different forms of M R for the neutrino mixing angles and mass ratios.
In the published literature no more predictive and economical a model for quark and lepton masses than the one discussed here exists that is also consistent with present knowledge. It is striking that in this model a single term and a single parameter (which we call σ) accounts for no less than four puzzling aspects of the light fermion spectrum: the largeness of U µ3 , the smallness of V cb , the smallness of m c /m t compared to m s /m b , and the Georgi-Jarlskog factor of three between m µ and m s at the GUT scale. It should be emphasized that, while many satisfactory neutrino mixing ideas and also many interesting ideas for explaining the pattern of quark and charged lepton masses have been proposed, very few models exist which not only give a satisfactory account of neutrino phenomenology but are at the same time highly predictive.
We have shown that the model defined by the existence of these five (or six) effective Yukawa terms can be realized in a complete and specific renormalizable SUSY SO(10) model that is technically natural. We have presented the details of such a model, including all the Higgs and quark and lepton superfields, the abelian flavor symmetries, and the transformation properties of the fields under these symmetries. Finally, we have done a quantitative comparison of the predictions of the model to experiment.
In the future this model will be rigorously testable in several ways. The most important are (1) a relation between the real and imaginary parts of V ub including a precise test of the angles of the unitarity triangle; (2) a prediction for U e2 , which in the small angle MSW case gives a sharp relation between the solar and atmospheric angles; and (3) Table I of Section VI. (a) The 33 elements denoted "1". (b) The 23 and 32 elements denoted "ǫ". Note that because of the appearance of the VEV of the adjoint Higgs field 45 H ≡ A, they are proportional to the SO(10) generator B − L. (c) The asymmetric elements denoted "σ" arise from this diagram. That they do not contribute to the up quark masses, and contribute asymmetrically to the down quark and charged lepton mass matrices are consequences of the fact that the SO(10) 10's, i.e., 10 1 and 10 2 , contain 5 but not 10 of SU (5).  (14). (a) The 11 element called "η". (b) The 12 and 21 elements called "δ". Because the 10 H couples to the symmetric product of 16 1 16 2 , δ appears symmetrically in the mass matrices. (c) The 13 and 31 elements called "δ ′ ", which also appear symmetrically.  3. The unitarity triangle for V ud V * ub + V cd V * cb + V td V * tb = 0 is displayed along with the experimental constraints on V ud V * ub , which is the upper vertex in the triangle. The constraints following from |V ub |, B-mixing and ǫ extractions from experimental data are shown in the lightly shaded regions. The experimentally allowed region is indicated by the darkly shaded overlap. The model predicts that V ud V * ub will lie on the dashed circle; cf. Eq. (21). The particular point on this circle used to draw the triangle shown is obtained from the numbers given in Section VI; cf. Eq. (62).