Distribution of the invariant mass for the weighted $ D ^{*+} \rightarrow D ^0 (\rightarrow K ^- K ^+ )\pi ^+ $ decay candidates, from the C$_ D ^+ $ calibration procedure. The result of the fit to this distribution is also shown.

Central values and two-dimensional confidence regions in the $(a_{ K ^- K ^+ }^{d},a_{\pi ^- \pi ^+ }^{d})$ plane for the combinations of the LHCb results obtained with the dataset taken between 2010 and 2018 and the one taken between 2010 and 2012, corresponding to an integrated luminosity of 8.7 $\text{ fb} ^{-1}$ and 3.0 $\text{ fb} ^{-1}$ , respectively.

Distributions of the invariant mass for the weighted charm-meson candidates, for the decays (top left) $ D ^{*+}$ $\rightarrow$ $ D ^0$ ( $\rightarrow$ $ K ^-$ $ K ^+$ ) $\pi ^+$ , (top right) $ D ^{*+}$ $\rightarrow$ $ D ^0$ ( $\rightarrow$ $ K ^-$ $\pi ^+$ ) $\pi ^+$ , (bottom left) $ D ^+$ $\rightarrow$ $ K ^-$ $\pi ^+$ $\pi ^+$ , (bottom right) $ D ^+$ $\rightarrow$ $\overline{ K } {}^0$ $\pi ^+$ . The data are from the $\mathrm{C}_ D ^+ $ calibration procedure. The results of the fits to these distributions are also shown. The top left is repeated from the main text.

Distributions of the invariant mass for the weighted charm-meson candidates, for the decays (top left) $ D ^{*+}$ $\rightarrow$ $ D ^0$ ( $\rightarrow$ $ K ^-$ $ K ^+$ ) $\pi ^+$ , (top right) $ D ^{*+}$ $\rightarrow$ $ D ^0$ ( $\rightarrow$ $ K ^-$ $\pi ^+$ ) $\pi ^+$ , (bottom left) $ D ^+_ s $ $\rightarrow$ $ K ^-$ $ K ^+$ $\pi ^+$ , and (bottom right) $ D ^+_ s $ $\rightarrow$ $\overline{ K } {}^0$ $ K ^+$ . The data are from the $\mathrm{C}_ D ^+_ s $ calibration procedure. The results of the fits to these distributions are also shown.

Measurements of $ {\mathcal{A}}^{ C P } ( K ^- K ^+ ) $ in (left) time-ordered data-taking (referred to as run blocks) and (right) year of data-taking and dipole-magnet polarity for the $\mathrm{C}_ D ^+ $ and $\mathrm{C}_ D ^+_ s $ calibration procedures. The uncertainties are statistical only. The horizontal lines are the averaged values for the $\mathrm{C}_ D ^+ $ and $\mathrm{C}_ D ^+_ s $ methods, while the bands represent the one-standard-deviation regions. The labels Mag-Up and Mag-Down refer to the direction of the magnetic field along the positive and negative directions of $y$-axis, respectively.

Comparison of the results for $ {\mathcal{A}}^{ C P } ( K ^- K ^+ ) $ per different intervals of $ D ^0$ candidates (left) pseudorapidity and (right) decay time for the $\mathrm{C}_ D ^+ $ and $\mathrm{C}_ D ^+_ s $ calibration procedures. The horizontal lines are the averaged values for the $\mathrm{C}_ D ^+ $ and $\mathrm{C}_ D ^+_ s $ methods, while the bands represent the one-standard-deviation regions.

Distribution of the $p$-values resulting from the consistency checks on $ {\mathcal{A}}^{ C P } ( K ^- K ^+ )$ measured in different data subsamples. The blue line represents the expected uniform distribution.

Distributions of the neutral-kaon decay time for background-subtracted and weighted candidates considered in the determination of $ {\mathcal{A}}^{ C P } ( K ^- K ^+ ) $ values. The distributions are normalized to have the maximum equal to unity. The control sample is used to estimate $A(\overline{ K } {}^0 )$ and consists of neutral kaons decaying outside of the vertex detector.

Measured asymmetry in $ D ^+$ $\rightarrow$ $\overline{ K } {}^0$ $\pi ^+$ decays as a function of $\overline{ K } {}^0$ decay time in units of $ K ^0_{\mathrm{S}}$ -meson decay time, in data where the neutral kaon meson decays outside of the vertex detector. The predictions for the neutral kaon asymmetry, from the model with and without an additional linear degree of freedom, are shown as well. An overall shift is applied to the model's predictions to account for unrelated nuisance asymmetries, i.e. $ A_{\rm P} ( D ^+ )$ and $ A_{\rm D} (\pi ^+ )$.

Distribution of the $ K ^- K ^+ $ invariant mass, along with the result of the fit describing the expected background components. The vertical lines indicate the interval applied in the selection of $ D ^0$ candidates for the determination of $ {\mathcal{A}}^{ C P } ( K ^- K ^+ ) $. The fit results are used to evaluate the systematic uncertainties related to the peaking background.

Background-subtracted kinematic distributions for the charm-meson pseudorapidity ($\eta$), the kaon momentum ($p$) and the tagging-pion transverse momentum ( $ p_{\mathrm{T}}$ ) (left column) before and (right column) after the per-candidate $\mathrm{C}_ D ^+_ s $ weighting procedure. The distributions are normalized to unit area.

Background-subtracted kinematic distributions for the kaon and pion transverse momentum ($ p_{\mathrm{T}} $) and pseudorapidity ($\eta$) (left column) before and (right column) after the per-candidate $\mathrm{C}_ D ^+ $ weighting procedure. The distributions are normalized to unit area.

Measurements of $ {\mathcal{A}}^{ C P } ( K ^- K ^+ )$ from various experiments [58,63,64,65,66,67,68]. The presented measurement is highlighted in red. The vertical band corresponds to the average of all measurements previous to the presented, computed by HFLAV [59], where it is assumed that $ C P$ is conserved in mixing and in the interference between decay and mixing. This assumption is necessary when results from different experiments are compared. The inset plot shows the five most precise measurements in a reduced horizontal range.

Animated gif made out of all figures.

Signal yields and statistical reduction factors arising from the kinematic weighting of the sample for the various decay modes and both calibration procedures.

Systematic uncertainties on $ {\mathcal{A}}^{ C P } ( K ^- K ^+ )$ for the two calibration procedures C$_ D ^+ $ and C$_ D ^+_ s $. The total uncertainties are obtained as the sums in quadrature of the individual contributions. Correlations between the systematic uncertainties of the two calibration procedures are also reported.

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