Parity Violating Longitudinal Muon Polarization in $K^{+}\to {\pi }^{+}{\mu }^{+}{\mu }^{-}$ Beyond Leading Logarithms

We generalize the existing analyses of the parity violating muon polarization asymmetry ${\mathrm{\Delta }}_{LR}$ in $K^{+}\to {\pi }^{+}{\mu }^{+}{\mu }^{-}$ beyond the leading logarithmic approximation. The inclusion of next–to–leading QCD corrections reduces the residual dependence on the renormalization scales, which is quite pronounced in the leading order. This leads to a considerably improved accuracy in the perturbative calculation of the short distance dominated quantity ${\mathrm{\Delta }}_{LR}$. Accordingly this will also allow to obtain better constraints on the Wolfenstein parameter $\varrho$ from future measurements of ${\mathrm{\Delta }}_{LR}$. For $-0.25\le \varrho \le 0.25$, $V_{cb}=0.040\pm 0.004$ and $m_t=(170\pm 20)GeV$wefind$3.0\cdot 10^{-3}\leq |\Delta_{LR}|\leq 9.6\cdot 10^{-3}$.