Yukawa Coupling Evolution in Susy Guts

The scaling behavior and fixed points in the evolution of fermion Yukawa couplings and mixing angles are discussed. The relevance of fixed points in determining the top quark mass is described. The unification of gauge couplings is a well-known feature of unified theories. To understand the masses of the observed fermions requires ideas beyond the standard model. With certain assumptions about the interactions in the GUT theory, relationships arise between various fermion masses. A mild assumption of minimality yields the well-known relation between the bottom and tau Yukawa couplings at the GUT scale λb = λτ . A more ambitious program is the attempt to formulate ansatze for the Yukawa coupling matrices, and obtain relationships between fermion masses and CKM relations as well. Unlike the gauge couplings or the CKM matrix elements, infrared fixed point solutions exist for the Yukawa coupling. Since the top quark is the only known fermion with a mass of order of the electroweak scale, it is usually the only particle for which the fixed point solution is relevant. The bottom and tau Yukawa couplings can be large in a multi-Higgs doublet model, where the smallness of the masses can be attributed to the properties of the vacuum. The fixed point of the top quark Yukawa coupling in the minimal supersymmetric standard model (MSSM) can be obtained approximately by setting