Measurements of charged kaon semileptonic decay branching fractions $K^{\pm} \to \pi^{0} \mu^{\pm} \nu$ and $K^{\pm} \to \pi^{0} e^{\pm} \nu$ and their ratio

Measured ratios of decay rates for $\Re_{K_{e3}/K_{2\pi}}, \Re_{K_{\mu3}/K_{2\pi}}$ and $\Re_{K_{\mu3}/K_{e3}}$ are presented. These measurements are based on $K^{\pm}$ decays collected in a dedicated run in 2003 by the NA48/2 experiment at CERN. The results obtained are $\Re_{K_{e3}/K_{2\pi}} = 0.2496 \pm 0.0009_{(stat)} \pm 0.0004_{(syst)}$ and $\Re_{K_{mu3}K_{2\pi}} = 0.1637 \pm 0.0006_{(stat)} \pm 0.0003_{(syst)}$. Using the PDG average for the $K^{\pm} \rightarrow \pi^{\pm}\pi^{0}$ normalisation mode, both values are found to be larger than the current values given by the Particle Data Book and lead to a larger magnitude of the $|V_{us}|$ CKM parameter than previously accepted. In addition, a new measured value of $\Re_{K_{\mu3}/K_{e3}} = 0.656 \pm 0.003_{(stat)} \pm 0.001_{(syst)}$ is compared to the semi-empirical predictions based on the latest form factor measurements.