The Complete |Delta S|=2 Hamiltonian in the Next-To-Leading Order

Abstract

We present the complete next-to-leading order short-distance QCD corrections to the effective \dstwo -hamiltonian in the Standard Model. The calculation of the coefficient $\eta_3$ is described in great detail. It involves the two-loop mixing of bilocal structures composed of two \dsone\ operators into \dstwo\ operators. The next-to-leading order corrections enhance $\eta_3$ by 27\% to $\eta_3=0.47 \errorpm{+0.03}{-0.04}$ thereby affecting the phenomenology of $\epsilon_K$ sizeably. $\eta_3$ depends on the physical input parameters $m_t$, $m_c$ and $\laMSb$ only weakly. The quoted error stems from renormalization scale dependences, which have reduced compared to the old leading log result. The known calculation of $\eta_1$ and $\eta_2$ is repeated in order to compare the structure of the three QCD coefficients. We further discuss some field theoretical aspects of the calculation such as the renormalization group equation for Green's functions with two operator insertions and the renormalization scheme dependence caused by the presence of evanescent operators.