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 $\Delta_{LR}$ 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 $\Delta_{LR}$. Accordingly this will also allow to obtain better constraints on the Wolfenstein parameter $\varrho$ from future measurements of $\Delta_{LR}$. For $-0.25\leq \varrho \leq 0.25$, $V_{cb}=0.040\pm 0.004$ and $m_t=(170\pm 20)GeV$ we find $3.0\cdot 10^{-3}\leq |\Delta_{LR}|\leq 9.6\cdot 10^{-3}$.