Measurements of charm mixing and $C P$ violation parameters from the decaytimedependent ratio of $ D^0 \to K^+ \pi^ $ to $ D^0 \to K^ \pi^+ $ decay rates and the chargeconjugate ratio are reported. The analysis uses $\overline{B}\to D^{*+} \mu^ X$, and chargeconjugate 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 centerofmass 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 $C P$ symmetry, (ii) mixing assuming no direct $C P$ violation in the Cabibbofavored or doubly Cabibbosuppressed decay amplitudes, and (iii) mixing allowing either direct $C P$ violation and/or $C P$ 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 $C P$ symmetry.
The timeintegrated $ D ^0 \pi^+_s$ invariant mass distributions, after samesign subtraction, for (a) RS decays and (b) WS decays. Fit projections are overlaid. Below each plot are the normalized residual distributions. 
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Efficiency corrected and SS background subtracted ratios of WS/RS decays and fit projections for the DT sample. The top plot shows the $ D ^0$ $(R^+(t))$ sample. The middle plot shows the $\overline{ D }{} {}^0$ $(R^(t))$ sample. The bottom plot shows the difference between the top and middle plots. In all cases, the error bars superposed on the data points are those from the $ \chi^2 $ minimization fits with no accounting for additional systematic uncertainties. The projections shown are for fits assuming $ C P$ symmetry (solid blue line), allowing no direct $CPV$ (dasheddotted green line), and allowing all forms of $CPV$ (dashed magenta line). Bins are centered at the average value of $t/\tau$ of the bin. 
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Efficiencycorrected data and fit projections for the DT (red open circles) and prompt (black filled circles) samples. The top plot shows the $ D ^0$ $(R^+(t))$ samples. The middle plot shows the $\overline{ D }{} {}^0$ $(R^(t))$ samples. The bottom plot shows the difference between the top and middle plots. In all cases, the error bars superposed on the data points are those from the $ \chi^2 $ minimization fits without accounting for additional systematic uncertainties. The projections shown are for fits assuming $ C P$ symmetry (solid blue line), allowing no direct $CPV$ (dasheddotted green line), and allowing all forms of $CPV$ (dashed magenta line). Bins are centered at the average $t/\tau$ of the bin. 
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Animated gif made out of all figures. 
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Summary of systematic uncertainties for the DT analysis for each of the three fits described in the text. 
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Fitted parameters of the DT sample. The first uncertainties include the statistical uncertainty, as well as the peaking backgrounds and the $K\pi$ detection efficiency, and the second are systematic. 
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Simultaneous fit result of the DT and prompt samples. The promptonly results from [1] are shown on the right for comparison. Statistical and systematic errors have been added in quadrature. 
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Systematic uncertainties for the simultaneous fits of the DT and prompt datasets. 
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Correlation matrix for the no $CPV$ fit to the DT data. 
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Correlation matrix for the no direct $CPV$ fit to the DT data. 
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Correlation matrix for the all $CPV$ allowed fit to the DT data. 
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Correlation matrix for the no $CPV$ simultaneous fit to the prompt + DT data sets. 
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Correlation matrix for the no direct $CPV$ simultaneous fit to the prompt + DT data sets. 
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Correlation matrix for the all $CPV$ allowed simultaneous fit to the prompt + DT data sets. 
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Created on 20 February 2021.