Explaining Why the u and d Quark Masses are Similar

An approach is suggested for modeling quark and lepton masses and mixing in the context of grand unified theories that explains the curious fact that m_u ~ m_d even though m_t>>m_b. The structure of the quark mass matrices is such as to allow a non-Peccei-Quinn solution of the Strong CP Problem.

It is well known that the grand unification group SU(5) relates the mass matrices of the down quarks and charged leptons. There is some empirical support for the existence of such a relationship in the fact that when the fermion masses are extrapolated to the GUT scale in the MSSM one finds m b ∼ = m τ [1,2], m s ∼ = m µ /3, and m d ∼ = 3m e [3]. However, the pattern of masses of the up quarks is very different. One difference is that the t mass is much greater than the b and τ masses, which is usually explained by saying that the ratio of VEVs v u /v d ≡ tan β is large compared to one. Another difference is that the interfamily mass hierarchies are much stronger for the up quarks than for the down quarks and charged leptons (e.g. m c /m t ≪ m s /m b and m u /m c ≪ m d /m s ). It is tempting to say that the up quark mass matrix (M U ) is more distantly related to the down quark and charged lepton mass matrices (M D and M L ) than the latter are to each other. On the other hand, there is the tantalizing fact that m u /m d ∼ 1.
In this paper we suggest a somewhat new approach which qualitatively explains why m t ≫ m b but m u ∼ m d . The idea is that there are "underlying" mass matrices (denoted by the superscript zero) whose structure is controlled by SO (10) and which satisfy M 0 , but that a strong mixing of the third family with vectorlike fermions at the GUT scale distorts these underlying mass matrices in such a way that m b and m τ are highly suppressed relative to m t . This distortion does not affect the first family much, so the masses m u , m d , and m e remain of the same order.
The approach we will describe has several other virtues: (a) It can be realized in models with very few parameters. (b) It dovetails with the ideas of Ref. [4] for solving the Strong CP Problem. And (c) it implements the "lopsided" mass matrix approach to explaining large neutrino mixing angles [5,6].
To understand the idea, consider an SU(5) model (which will later be assumed to descend from SO(10)) that has three families of fermions in 10 i + 5 i + 1 i , with mass terms of the There is then mixing between the ordinary three families and the vectorlike fermions, more specifically the mixing is between the 5 i and the 5 ′ i . (A similar idea, but with mixing among fermions in 10's of SU(5) was used in [5]. However, the models proposed there were very different in character from the present models.) With the mass terms specified above, we may write where F = L or D, and M 0 F is either M 0 L or (M 0 D ) T , depending on whether the fermions in 5 are ℓ − L or d c L . In order to find the light fermion mass matrices in the effective low-energy theory, we must do a unitary transformation M F = UM 0 F that eliminates the off-diagonal block B in the full mass matrix given in Eq. (1). Such a transformation is where x ≡ BA −1 , Λ ≡ (I + xx † ) −1/2 , and Λ = (I + x † x) −1/2 . (To check the unitarity of U it is useful to note that x † Λ = Λx † and xΛ = Λx.) This gives the result for the low energy mass matrices Basically, the hermitian matrix Λ describes the mixing of 5 i with 5 ′ i . It appears on the left in the equation for M L since (M L ) ij couples to 5 i 10 j . It appears on the right in the equation for M D since (M D ) ij couples to 10 i 5 j . For the Dirac neutrino masses we have giving Since the masses of the up-type quarks come from a 10 i 10 j coupling of the fermions, they are not affected by the mixing of the 5 i with the 5 ′ i . Consequently, Before we discuss how the structure we have described can help us explain the magnitudes of quark and lepton masses and mixings, we note that it is exactly the kind of structure that is used in the solution of the Strong CP Problem proposed in [4]. The idea there was the following. Suppose that CP is a symmetry of the lagrangian that is spontaneously broken, and that the VEV that breaks CP appears in the off-diagonal matrix B in Eq. (1), but not elsewhere in the quark mass matrices. Then M 0 D and A are real, and it is easily shown that the determinant of the full mass matrix M D is therefore real. Also real, of course, is the determinant of M U . Thus, at tree level, the phase θ is zero. At higher order, these matrices can receive complex corrections that induce a non-vanishing θ, but these may be made small. (In SUSY, there can be contributions to the θ parameter that are harder to make small, for example, one-loop corrections to the gluino mass [7]. How large these are depends upon how SUSY is broken. These contributions are not a problem in theories with gauge-mediated SUSY breaking, for example. We imagine that whatever mechanism resolves the usual SUSY flavor and SUSY CP problems will also suppress these extra contributions to θ.) On the other hand, since B is a complex matrix, so is the matrix x = BA −1 and the matrix Λ = (I + xx † ) −1/2 . Consequently, the mass matrix of the light three families of down-type quarks in the effective low-energy theory, given by M D = M 0 D Λ T , is also complex, which means that in general there is a non-vanishing Kobayashi-Maskawa phase.
In short, the structure in Eq. (1) allows a spontaneously generated phase in the matrix B to contribute to δ KM but not at tree level to θ. This can also enhance the predictivity of models by reducing the number of parameters, since one can assume that all parameters in and the right-handed Majorana matrix M R are real, and that the only phase (and only one is needed) comes from Λ. This is the assumption we shall make in the illustrative model we present below.
Returning to the issue of mass and mixing hierarchies, let us assume that the matrix Λ that characterizes the mixing of where the real parameter λ ≪ 1 and the complex parameter t has magnitude of order one.
As we shall see shortly, it is the smallness of λ that gives rise to m b , m τ ≪ m t , while the |t| ∼ 1 explains the large atmospheric neutrino mixing. The phase of t, the only phase in the model, is what produces the KM phase. We shall see later that the form in Eq. (7) is easy to obtain.
To illustrate our basic approach we now present a toy model in which the underlying mass matrices have the following simple "textures": where δ, δ ′ ≪ ǫ u , ǫ d ≪ 1. The similarity of these four matrices is assumed to come from SO(10). In SO (10) and Of course, from Eq. (6) one sees that M U is already given in Eq. (8).
Simply by inspecting these matrices one can observe several significant facts. First, the masses of m b and m τ are suppressed by the small parameter λ, whereas m t is not, so that m b , m τ ≪ m t can be explained without requiring that m U /m D be extremely large. Second, the masses of the first family will be almost unaffected by the parameter λ, so that m d and m e will not be similarly suppressed compared to m u . Indeed for m U /m D of order one, m u ∼ m d , as observed. Third, there emerges naturally the "lopsided" structure discussed in many recent papers [6]. That is, we see that the 23 element of M L is much larger than One can read off from the simple forms in Eqs. (9) and (10) the following approximate relations that hold at the GUT scale: and From the form of the M L (Eq. (10)) and M N (Eqs. (5) and (8)) it can be seen that the tangent of the atmospheric neutrino mixing angle is controlled by t, which therefore must be of order one. That in turn implies, through the equation This succesful qualitative relation between the atmospheric neutrino angle, V cb and m s /m b is characteristic of lopsided models.
In this model there are eight parameters (m U /m D , ǫ u , ǫ d , δ, δ ′ , λ, |t|, and θ t ) to fit twelve quantities (eight mass ratios of charged leptons and quarks, three CKM angles, and the KM phase). There are therefore four quantitative predictions, which can be taken to be m b ∼ = m τ , m s ∼ = m µ /3, m d ∼ = 3m e , and the value of the Cabibbo angle. In addition, the atmospheric angle is predicted to be of order one, though it cannot be predicted more precisely than that without knowing M R .
We can easily determine the approximate values of most of the parameters of the model from Eqs. (11) and (12). We take the values of the quark and lepton masses and CKM  [10].) Finally, assuming all SUSY particle masses to be degenerate at m t we run the masses and mixings to the GUT scale M GU T ≈ 2 × 10 16 GeV using the 2-loop MSSM β functions summarized in Ref. [11].
In the final running we set tan β = 3.
The equation for V cb tells us immediately that |t| ∼ = V cb m b /m s ≈ 2. From Eq. (11) one 1 3 (m µ /m τ )λ(1 + |t| 2 ) gives |ǫ d | ≈ 10 −3 . It is gratifying that ǫ u and ǫ d come out to be of the same order. If we choose the relative sign of ǫ u and ǫ d to be negative, then we get a good fit to the Cabibbo angle: The value of δ is determined from δ 2 ∼ = m u m c /m 2 t to be 10 −4 . Finally, the parameter δ ′ and the phase of t can be determined from the real and imaginary parts of V ub . Specifically, one has One gets a fairly reasonable fit from the following values of the parameters of the model: that is somewhat large compared to the values favored by recent lattice calculations [12].
While this model does not give a perfect fit, it is simple enough to illustrate the basic idea we are proposing in a transparent way. It seems likely that models based on these ideas can be found that give better fits. Another possibility that might be realized in this approach is a "doubly lopsided" model [13]. One could imagine, for example, that the matrix Λ had the form with λ ≪ 1 and |t i | ∼ 1. If the underlying matrix M 0 L has a hierarchical form, with the 33 element being the largest, then the effective low-energy mass matrix M L = ΛM 0 L would have the doubly lopsided form, with the 13, 23, and 33 elements all being of the same order. This is known to be able to give in a simple way the correct "bi-large" pattern of neutrino mixing angles, with U e3 being small [13].
We now turn to the question of whether the form of Λ given in Eq. (7) can arise naturally.
Consider the special case where B is diagonal and where the only non-zero elements of A are the diagonal elements and the 23 element: Then it is easily found that for b 1 < a 1 , b 2 < a 2 , and b 3 > a 3 , the matrix Λ has the form given in Eq. (7) with λ ∼ = |a 3 | 2 /(|a 3 | 2 + |b 3 | 2 ), and t = −(b * 3 b 2 a 4 )/(a 2 |a 3 | 2 ). Of course, there are other forms of A and B that also give Eq. (7). Another simple example is that A is diagonal and B has nonzero diagonal elements and 23 element.
In conclusion, we have found a framework that differs from most "texture" models of quark and lepton masses in several respects. First, it can partially explain the fact, usually treated as an accident, that m u ∼ m d , m e , while also giving m t ≫ m b , m µ . This it does, not by requiring tan β to be large, which might be somewhat unnatural, but by mixing the b and τ strongly with vectorlike fermions at the GUT scale. Second, it combines predictive textures with a structure that realizes a non-axion solution to the strong CP problem proposed many years ago [4]. By allowing most of the parameters to be real, even though CP is violated, it has the potential of giving very predictive models. And it gives rise naturally to the "lopsided" kind of structure that has been proposed to explain the largeness of U µ3 relative to V cb [6].
The toy model we have described illustrates the essential ideas in a transparent way.
However, it would be good to find a model which is more predictive and which does a better job fitting certain quantities, especially m s . It would also be interesting to investigate further models of this type that are "doubly lopsided" [5,13].