Invariants Describing Quark Mixing and Mass Matrices

We introduce two new sets of invariant functions of quark mass matrices, which express the constraints on these mass matrices due to knowledge of the quark mixing matrix. These invariants provide a very simple method to test candidate forms for mass matrices. Supported in part by NSF contract PHY-89-08495 email: [email protected] In recent years the increasingly accurate data on quark mixing has stimulated one’s interest in possible predictions concerning quark mass matrices. The parameters of the Cabibbo-Kobayashi-Maskawa mixing matrix V are now known reasonably well [1]. This determination has been made possible partly by the finding that there are only three generations of usual standard-model fermions (with corresponding light or massless neutrinos). Since the diagonalization of the quark matrices in the up and down sectors determines V , one can work back from the knowledge of V to put constraints on the possible forms of (original, nondiagonal) quark mass matrices. However, the data on quark mixing determines these mass matrices only up to an arbitrary unitary similarity transformation. This is a result of the fact that if the up and down quark mass matrices, Mu and Md, are both acted on by the same unitary operator U0 according to Mu,d → U0 Mu,d U † 0 (1) then the mixing matrix V remains unchanged. There have been many attempts to study specific assumed forms for quark matrices. While this is worthwhile, it is desirable to express the constraints from data on V on the quark mass matrices in an invariant form. In Refs. [2, 3] certain invariant functions of the quark mass matrices Ipq were introduced, which are expressed in terms of the quark masses squared and the |Vij|: Ipq = Tr(H p u H q d) = ∑ ij (m (u) i ) (m (d) j ) 2q |Vij| 2 (2) where Hq = MqM † q , and m (u) i and m (u) i are the masses of quarks in the “up” and “down” charge sectors. However, in practice, these are somewhat awkward to use because of the high powers of elements of the actual quark mass matrices which are involved. In this paper we introduce two new sets of invariants of the quark mass matrices (with respect to the transformation (1)) which can be expressed in terms of the measurable quantities only (up to the ± sign ambiguity in fermion masses, which, as we will show, can be dealt with by considering all choices of these signs). These invariants have an important 1 advantage relative to (2) that they involve lower powers of the elements of the quark mass matrices and hence yield much less complicated analytic expressions in practice. In the standard model with only left-handed charged weak currents one may choose Mu and Md both to be hermitian without any loss of generality, by re-phasing the right-handed components of quarks. We introduce the following new invariants of the transformation (1): Kpq = Tr(M p u M q d ) (3) Lpq(α, β) = det(αM p u + βM q d ) (4) where p, q, α, β 6= 0. The hermitian matrices Mu and Md can be diagonalized by a unitary similarity transformation: