Scale dependence of the quark masses and mixings: leading order

Abstract

We consider the Renormalization Group Equations (RGE) for the couplings of the Standard Model and its extensions. Using the hierarchy of the quark masses and of the Cabibbo-Kobayashi-Maskawa (CKM) matrix our argument is that a consistent approximation for the RGE should be based on the parameter $\lambda= |\hat{V}_{ud}| \approx0.22$. We consider the RGE in the approximation where we neglect all the relative terms of the order $\sim\lambda^{4}$ and higher. Within this approximation we find the exact solution of the evolution equations of the quark Yukawa couplings and of the vacuum expectation value of the Higgs field. Then we derive the evolution of the observables: quark masses, CKM matrix, Jarlskog invariant, Wolfenstein parameters of the CKM matrix and the unitarity triangle. We show that the angles of the unitarity triangle remain constant. This property may restrict the possibility of new symmetries or textures at the grand unification scale.