Lattice Matrix Elements and CP Violation in B and K Physics: Status and Outlook

Status of lattice calculations of hadron matrix elements along with CP violation in B and in K systems is reviewed. Lattice has provided useful input which, in conjunction with experimenatl data, leads to the conclusion that CP-odd phase in the CKM matrix plays the dominant role in the observed asymmetry in $B \to \psi K_s$. It is now quite likely that any beyond the SM, CP-odd, phase will cause only small deviations in B-physics. Search for the effects of the new phase(s) will consequently require very large data samples as well as very precise theoretical predictions. Clean determination of {\it all} the angles of the unitarity triangle therefore becomes essential. In this regard $B \to K D^0$ processes play a unique role. Regarding K-decays, remarkable progress made by theory with regard to maintenance of chiral symmetry on the lattice is briefly discussed. First application already provide quantitaive information on $B_K$ and the $\Delta I=1/2$ rule. The enhancement in $Re A_0$ appears to arise solely from tree operators, esp. $Q_2$; penguin contribution to $Re A_0$ appears to be very small. However, improved calculations are necessary for \epsilon^'/epsilon as there the contributions of QCD penguins and electroweak penguins largely seem to cancel. There are good reasons, though, to believe that these cancellations will not survive improvements that are now underway. Importance of determining the unitarity triangle purely from K-decays is also emphasized.