  I describe in detail the calculation of the coefficients eta_3 and eta_1 of
the low-energy Delta S=2 -hamiltonian describing K^0-K^0-bar-mixing in the
next-to-leading order of renormalization group improved (RG) perturbation
theory. First a general introduction into the application of the operator
expansion and of RG methods to weak processes is presented. This includes the
RG formalism for Green's functions with two operator insertions. Second I
discuss the elimination of unphysical operators with emphasis on evanescent
operators. Then the two-loop calculations needed for eta_3 and eta_1 are shown
and finally the implications on the phenomenology of epsilon_K and the K_L-K_S
mass difference are discussed. The results of this thesis have been published
separately, but they are explained in more detail here.
