Long-distance dimension-eight operators in BK

Besides their appearance at short distances >∼ 1/MW , local dimension-eight operators also contribute to kaon matrix elements at long distances of order ∼ 1/μope, where μope is the scale controlling the Operator Product Expansion in pure QCD, without weak interactions. This comes about in the matching condition between the effective quark Lagrangian and the Chiral Lagrangian of mesons. Working in dimensional regularization and in a framework where these effects can be systematically studied, we calculate the correction from these long-distance dimension-eight operators to the renormalization group invariant B̂K factor of K − K̄ mixing, to next-to-leading order in the 1/Nc expansion and in the chiral limit. The correction is controlled by the matrix element 〈0|s̄LG̃μνγμdL|K0〉, is small, and lowers B̂K .