The Rare decay K+ ---> pi+ nu anti-nu at the next-to-next-to-leading order in QCD

The authors calculate the complete next-to-next-to-leading order QCD correction of the charm quark contribution to the branching ratio for the rare decay K{sup +} {yields} {pi}{sup +}{nu}{bar {nu}} in the standard model. The inclusion of these {Omicron}({alpha}{sub s}) contributions leads to a significant reduction of the theoretical uncertainty from {+-} 10.1% down to {+-} 2.4% in the relevant parameter P{sub c}, implying the left over scale uncertainties in {Beta}(K{sup +} {yields} {pi}{sup +}{nu}{bar {nu}}) and in the determination of |V{sub td}|, sin 2{beta} and {gamma} from the K {yields} {pi}{nu}{bar {nu}} system to be {+-} 1.3%, {+-} 1.0%, {+-} 0.006 and {+-} 1.2{sup o}, respectively. for the charm quark {ovr MS} mass m{sub c}(m{sub c}) = (1.30 {+-} 0.05) GeV and |V{sub us}| = 0.2248 the next-to-leading order value P{sub c} = 0.37 {+-} 0.06 is modified to P{sub c} = 0.37 {+-} 0.04 at the next-to-next-to-leading order level with the latter error fully dominated by the uncertainty in m{sub c}(m{sub c}). Adding the recently calculated long-distance contributions we find {Beta}(K{sup +} {yields} {pi}{sup +}{nu}{bar {nu}}) = (8.0 {+-} 1.1) x 10{sup -11} with the quoted error almost entirely due to the present uncertainties in m{sub c}(m{sub c}) and themore » Cabibbo-Kobayashi-Maskawa elements.« less