Some model-independent properties of quark mixing

We discuss some new invariants of quark mixing and show their usefulness with a simple example. We also present some other new tools for analyzing quark mixing. ∗ talk presented at the Yukawa couplings and the origin of mass workshop, February 1994, University of Florida, Gainesville, Florida ∗∗address after September 1, 1994: Department of Physics, University of Pennsylvania, Philadelphia, PA 19104 1 Invariants of quark mixing and their applications In recent years the increasingly accurate data on quark mixing has stimulated interest in possible predictions concerning quark mass matrices. The parameters of the CabibboKobayashi-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. In this paper we introduce some new invariants of the quark mass matrices (with respect to the transformation (1)) which can be expressed in terms of the measurable quantities only. These new invariants help one simplify the algebraic expressions which relate the elements of the quark mass matrices to the data. This important advantage allows for some new uses of the invariants which we will illustrate with some examples. 1 We introduce the following new invariants of the transformation (1): Kpq(α, β) = det(αH p u + βH q d) (3) where p, q, α, β 6= 0. The hermitian matrices Hu and Hd can be diagonalized by a unitary similarity transformation: { UuHuU † u = Du UdHdU † d = Dd (4) where Dq = diag((m (q) 1 ) , (m (q) 2 ) , (m (q) 3 ) ) are the diagonal matrices of the quark masses squared. The mixing matrix V can be written then as: V = UuU † d (5) In order to find an expression for Kpq in terms of Uij we will need the following Theorem If A and B are two 3×3 matrices such that det(A) 6= 0 and det(B) 6= 0 then the following relation holds: det(A +B) = det(A) + det(B) + det(A) Tr(AB) + det(B) Tr(AB) (6) Proof: We denote the elements of matrices A and B by Aij and Bij correspondingly. Their co-factors (which are equal to the corresponding minors, up to sign) will be written as Âij and B̂ij. Then each determinant may be decomposed in a sum (Laplace expansion): det(A) = ∑