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

Abstract

We calculate the next-to-leading order short distance QCD corrections to the coefficient ${\eta }_1$ of the effective $\Delta S=2$ hamiltonian in the standard model. This part dominates the short distance contribution $(\Delta m_K{)}^{\mathrm{SD}}$ to the $K_L$ -- $K_S$ mass difference. The next-to-leading order result enhances ${\eta }_1$ and $(\Delta m_K{)}^{\mathrm{SD}}$ by 20\% compared to the leading order estimate. Taking $0.200\text{gev}\le \text{laMSb}\le 0.350\text{gev}$ and $1.35\text{gev}\le m_c(m_c)\le 1.45\text{gev}$ we obtain $0.922\le {\eta }_1^{\mathrm{NLO}}\le 1.419$ compared to $0.834\le {\eta }_1^{\mathrm{LO}}\le 1.138$. 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.

Comments: 30 pages and 9 figures (available as uuencoded tarred postscript files), LaTeX, TUM-T31-40/93