Finite grand unified theories and the quark mixing matrix

In N=1 super-Yang–Mills theories, under certain conditions satisfied by the spectrum and the Yukawa couplings, the beta functions will vanish to all orders in perturbation theory. We address the generation of realistic quark mixing angles and masses in such finite grand unified theories. Working in the context of finite SUSY SU(5), we present several examples with realistic quark mixing matrices. Non-Abelian discrete symmetries are found to be important in satisfying the conditions for finiteness. Our realistic examples are based on permutation symmetries and the tetrahedral symmetry A4. These examples enable us to address questions such as the decay rate of the proton in finite GUTs.