Solutions to the Renormalization Group Equations for Yukawa Matrices as an Answer to the Quark and Lepton Mass Problem

Abstract

If the scale dependence of a Yukawa matrix is assumed to be determined entirely by the dominant 33-element, then the renormalization group equations can be expressed in terms of two separate equations: a differential equation for the 33-coupling, and, an algebraic equation for the scale-independent 3x3 matrix that is found to have only two non-trivial, hierarchical, solutions with eigenvalues (0,0,1) and (0,1,1). The mass matrices are constructed from these solutions by rotating them first by the experimentally known mixing matrices-the CKM for quarks and charged leptons, and the CKM-analog for the seesaw generated Majorana neutrinos-and then incorporating the appropriate texture zeros. A uniform, hierarchical, description for the mass matrices of quarks and leptons is thus achieved, in terms of the mixing paramters, that give mass eigenvalues consistent with experiments as well as reproduce the input mixing angles. Inverted hierarchy in neutrinos is also discussed. Only a single scale (approx. 10^13 GeV) for the seesaw neutrinos is involved rather than their mass distribution. No new particles are otherwise invoked.