Indirect CP violation in the neutral kaon system beyond leading logarithms.

We have calculated the short distance QCD coefficient 3 of the effective j Sj=2hamiltonian in the next-to-leadingorder of renormalization group improved perturbation theory. Since now all coefficients 1, 2 and 3 are known beyond the leading log approximation, one can achieve a much higher precision in the theoretical analysis of "K , the parameter of indirect CP-violation in K0 K0 -mixing. The measured value for "K yields a lower bound on each of jVcbj, jVub=Vcbj, the top quark mass mt and the non-perturbative parameter BK as a function of the remaining three quantities. E.g. mpole t = 176GeV, jVcbj = 0:040 and BK = 0:75 implies jVub=Vcbj 0:0778, if the measured value for "K is attributed solely to Standard Model physics. We further discuss the implications on the CKM phase , jVtdj and the key quantity for all CP-violating processes, Im t = Im [V tsVtd]. These quantities and the improved Wolfenstein parameters  and  are tabulated and the shape of the unitarity triangle is discussed. We compare the range for jVtdj with the one obtained from the analysis of B0d B0d -mixing. For 0:037 jVcbj 0:043, 0:06 jVub=Vcbj 0:10 and 0:65 BK 0:85we find from a combined analysis of "K and the B0d B0d -mixing paramater xd: 49 146 , 7:4 10 3 jVtdj 12:4 10 3, 0:85 10 4 Im t 1:60 10 4, 0:36  0:28 and 0:21  0:44. We predict the mass difference of theB0 s system to lie in the range 6:5 ps 1 mBs 28 ps 1. Finally ye-mail: [email protected] ze-mail: [email protected] Work supported by BMBF under contract no. 06-TM-743.