Natural fermion mass hierarchy and mixings in family unification

We present an SU(9) model of family unification with three light chiral families, and a natural hierarchy of charged fermion masses and mixings. The existence of singlet right handed neutrions with masses about two orders of magnitude smaller than the GUT scale, as needed to understand the light neutrinos masses via the see-saw mechanism, is compelling in our model.


Introduction
The ideas of family symmetry and family unification has been with us for a while. Grand unified theories lend themselves to construction of such models , but most of the early models did not go as far as considering fermion masses and mixings. We know that there is a five orders of magnitude hierarchy among the charged fermions masses. There is also a two orders of magnitude hiearchy amongst the quark mixing angles. In addition, there are strong suppressions for the flavor changing neutral current processes. With the fairly accurate data on charged fermion masses and quark mixings, we are now in a position to attempt the construction of family symmetry models that include these parameters. The new data from the Tevatron and upcoming LHC will provide further constraints on such family unified models.
We have studied a class of SU(N) family unification models, i.e., models where the families are not due to simple replication of the representation of the first fermion family. Grand unification requires at least an SU(5) gauge group, but the only reasonable choice (that avoids exotic fermions) of representations for the families are (10 +5) F , and family unification is impossible. In SU(N) models, if the fermions all reside in totally antisymmetric irreducible representations (irrep), then there are guaranteed to be no exotic fermions. Such an idea was first proposed by H. Georgi [2], and subsequently used by many authors to build models with three chiral families [3] [4] [5] [6]. We write the h th totally antisymmetric irrep as [1] k . Also in SU(N) there are two invariant tensors from which we can construct group singlets from the [1] k s. They are the Kroneker δ α β and the Levi-Civita tensor ε α 1 α 2 ...α N or it's dual with all upper indices. The indices α, β etc. runs from 1 through N of SU(N).
The number of totally antisymmetric irreps in the groups SU(6) and SU(7) are too small to arrange the realistic mass and mixing relations of the type to follow. An SU(8) model of family unification has been proposed by S. Barr [7][8], but we find it possible to arrange a more detailed phenomenology in SU (9), and this justifies our choice of gauge group.
Note that this assignment is anomaly free.

Fermion Masses and Mixings
The assignment of the three light chiral families in the SU(9) multiplets in our model are as follows.
3rd family: In addition, the Higgs represetations that we shall use are various 9 H s, 36 H s, and 126 H s. The charged fermions will receive masses from the Yukawa interactions with the above Higgs multiplets which has electroweak VEVs. Since the top quark has a mass at the EW scale, its Yukawa coupling is of order one. So it is very reasonable to assume that only the top quark has dimension four Yukawa interaction, while the allowed interactions of the lighter quarks and charged leptons are of higher dimensions suppressed by a parameter ε. We will identify this parameter ε with the ratio of the SU(5) singlet Higgs VEV, < 1 > and the unification scale, M.

Yukawa interaction for the up sector
Consistent with the SU(9) symmetry, and the Z 2 symmetry, the allowed Yukawa interactions for the up sectors are as follows.
Note that the Yukawa interactions involving c L t R has the same structure with h u 32 replaced by h u 23 above, and similarly for theū L c R term. Also no lower dimensional Yukawa interactions are allowed for each terms. This is enforced by the SU(9) invariance, as well as imposing the following Z 2 symmetry: In each of the Higgs multiplets, there are 5 H ,5 H , and 1 H under SU (5). From each of the Yukawa interactions, we use one EW VEV arising from either 5 H , or5 H , and the rest from singlets. Thus a Yukawa interaction of dimension 4 + n above will give rise to the mass matrix elements of the form where < 1 > is the VEV of the SU(5) singlet field contained in the above SU(9) Higgs representations, and M is the SU(9) unification scale. Collecting terms from the above Yukawa interactions, we obtain the following up quark mass matrix:

Yukawa interaction for the down sector
Since the bottom quark mass is very small compared to the EW scale, we do not allow any dimension 4 Yukawa coupling in the bottom sector. Again, consistent with the SU(9) gauge symmetry and the Z 2 symmetry, the allowed Yukawa interactions in the down quark sector hierarchies has its origin in a gauge family symmetry, SU(9 Neutrino Masses and Mixings: Because we need9 F of SU(9) to obtain5 F of SU(5) for the chiral fermion families, SU(5) singlet fermions are unavoidable. Thus, the existence of singlet right handed (RH) neutrinos are required in our model, similar to a SO(10) GUT, and contrary to a SU(5) GUT. These RH neutrinos get Majorana masses from the SU(5) singlet Higgs whose VEVs are about 50 times smaller than the GUT scale, as needed to explain the hierarchy of quark masses and mixings. Thus our model naturally explain why the mass scale of the RH neutrinos are smaller than the SU(9) GUT scale as needed to obtain the light neurino masses at the observed level via the see-saw mechanism. Furthermore, the Dirac mass terms between the light neutrinos and the heavy RH neutrinos occur via dimension 4 operators at the tree level. Hence, in agreement with observation, there will not be large hierarchies among the light neutrino masses or among the neutrino mixing angles.