Enhancement of the $K_L$ -- $K_S$ Mass Difference by Short Distance QCD Corrections Beyond Leading Logarithms

We calculate the next-to-leading order short distance QCD corrections to the coefficient ${\eta }_1$ of the effective $\mathrm{\Delta }S=2$ hamiltonian in the standard model. This part dominates the short distance contribution $(\Delta m_K)^{\rm SD}$tothe$K_L$--$K_S$ mass difference. The next-to-leading order result enhances ${\eta }_1$ and $(\mathrm{\Delta }{m}_K{)}^{\mathrm{S}\mathrm{D}}$ by 20% compared to the leading order estimate. Taking $0.200\text{gev}\le \text{laMSb}\le 0.350\text{gev}$ and $1.35 \gev \le m_c(m_c) \le 1.45 \gev$weobtain$0.922 \le \eta_1^{\rm NLO} \le 1.419$comparedto$0.834 \le \eta_1^{\rm LO} \le 1.138$\mathrm{.}For$B_K = 0.7$ this corresponds to 48 – 75 % of the experimentally observed mass difference. The inclusion of next-to- leading order corrections to ${\eta }_1$ reduces considerably the theoretical uncertainty related to the choice of renormalization scales.