Lattice matrix elements and CP violation in<Emphasis Type="Italic">B</Emphasis> and<Emphasis Type="Italic">K</Emphasis> physics: Status and outlook

Status of lattice calculations of hadron matrix elements along with CP violation inB and inK systems is reviewed. Lattice has provided useful input which, in conjunction with experimental data, leads to the conclusion that CP-odd phase in the CKM matrix plays the dominant role in the observed asymmetry inB → ψK _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 ofall the angles of the unitarity triangle therefore becomes essential. In this regardB → KD⁰ processes play a unique role. RegardingK-decays, remarkable progress made by theory with regard to maintenance of chiral symmetry on the lattice is briefly discussed. First application already provide quantitative information onB _K and the ΔI = 1/2 rule. In the lattice calculation, the enhancement in Re A₀ appears to arise solely from tree operators, esp. Q₂; penguin contribution toRe A₀ appears to be very small. However, improved calculations are necessary for ε’/ε as 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 fromK-decays is also emphasized.