Measurement of branching fractions and direct $CP$ asymmetries for $B\to K\ddot{I}€$ and $B\to \ddot{I}€\ddot{I}€$ decays at Belle II

Abstract

We report measurements of the branching fractions and direct $CP$ asymmetries of the decays $B^0\to K^{+}\ddot{I}{€}^{-}$, $B^{+}\to K^{+}\ddot{I}{€}^0$, $B^{+}\to K^0\ddot{I}{€}^{+}$, and $B^0\to K^0\ddot{I}{€}^0$, and use these for testing the standard model through an isospin-based sum rule. In addition, we measure the branching fraction and direct $CP$ asymmetry of the decay $B^{+}\to \ddot{I}{€}^{+}\ddot{I}{€}^0$ and the branching fraction of the decay $B^0\to \ddot{I}{€}^{+}\ddot{I}{€}^{-}$. The data are collected with the Belle II detector from $e^{+}{e}^{-}$ collisions at the $\hat{I}\yen (4S)$ resonance produced by the SuperKEKB asymmetric-energy collider and contain $387\times 10^6$ bottom-antibottom meson pairs. Signal yields are determined in two-dimensional fits to background-discriminating variables, and range from 500 to 3900 decays, depending on the channel. We obtain $-0.03\pm 0.13\pm 0.04$ for the sum rule, in agreement with the standard model expectation of zero and with a precision comparable to the best existing determinations.