Measurements of charm mixing and $CP$ violation using $D^0 \to K^\pm \pi^\mp$ decays

Measurements of charm mixing and $CP$ violation parameters from the decay-time-dependent ratio of $ D^0 \to K^+ \pi^- $ to $ D^0 \to K^- \pi^+ $ decay rates and the charge-conjugate ratio are reported. The analysis uses $\overline{B}\to D^{*+} \mu^- X$, and charge-conjugate decays, where $D^{*+}\to D^0\pi^+$, and $D^0\to K^{\mp} \pi^{\pm}$. The $pp$ collision data are recorded by the \lhcb experiment at center-of-mass energies $\sqrt{s}$ = 7 and 8 TeV corresponding to an integrated luminosity of 3~fb$^{-1}$ The data are analyzed under three hypotheses: (i) mixing assuming $CP$ symmetry, (ii) mixing assuming no direct $CP$ violation in the Cabibbo-favored or doubly Cabibbo-suppressed decay amplitudes, and (iii) mixing allowing either direct $CP$ violation and/or $CP$ violation in the superpositions of flavor eigenstates defining the mass eigenstates. The data are also combined with those from a previous LHCb study of $D^0\to K \pi$ decays from a disjoint set of $ D^{*+} $ candidates produced directly in $pp$ collisions. In all cases, the data are consistent with the hypothesis of $CP$ symmetry.