Theorems on the Renormalization Group Evolution of Quark Yukawa   Couplings and CKM Matrix
  We analyze the two loop renormalization group equations in the standard model and its extensions for the coupling constants and the quark Yukawa couplings. The key point of our analysis is the observed hierarchy of the quark masses and the CKM matrix. For the one loop evolution we find the explicit solution for the evolution of the Yukawa couplings and show the following: 1. the CKM matrix depends on the energy on only one function of energy; 2. the ratios of the down quark masses depend on the energy through the same function as the CKM matrix; 3. the diagonalizing matrices of the biunitary transformation of the up quarks are energy independent. Next we give the explicit form of the two loop corrections to the evolution of the quark Yukawa couplings and show that the relative corrections are of the order (\lambda ^{4}) for the quark Yukawa couplings and of the order (\lambda ^{5}) for the CKM matrix. Finally we give the equations of the one loop evolution of the squares of the matrix elements of the CKM matrix and their explicit solution. 
P. Kielanowski, S. R. Ju\'arez W. and J. G. Mora H.