A search for $CP$ violation using $T$odd correlations is performed using the fourbody $D^0 \to K^+K^\pi^+\pi^$ decay, selected from semileptonic $B$ decays. The data sample corresponds to integrated luminosities of $1.0 \text{fb}^{1}$ and $2.0 \text{fb}^{1}$ recorded at the centreofmass energies of 7 TeV and 8 TeV, respectively. The $CP$violating asymmetry $a_{CP}^{T\text{odd}}$ is measured to be $(0.18\pm 0.29\text{(stat)}\pm 0.04\text{(syst)})\%$. Searches for $CP$ violation in different regions of phase space of the fourbody decay, and as a function of the $D^0$ decay time, are also presented. No significant deviation from the $CP$ conservation hypothesis is found.
Distributions of the $ K ^+ K ^ \pi ^+ \pi ^ $ invariant mass in the four samples defined by $ D ^0$ ( $\overline{ D }{} {}^0$ ) flavour and the sign of $ C_{T}$ ( $\overline{C}_{T}$ ). The results of the fit are overlaid as a solid line, and a dashed line is used for representing the background. The normalised residuals (pulls) of the difference between the fit results and the data points, divided by their uncertainties, are shown on top of each distribution. 
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Sidebandsubtracted distributions of $ D ^0$ ( $\overline{ D }{} {}^0$ ) candidates in variables of (a, b) $m_{\pi ^+ \pi ^ }$ and (c, d) $m_{ K ^+ K ^ }$ for different values of $ C_{T}$ ( $\overline{C}_{T}$ ). The veto for $ D ^0 \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^+ K ^ $ decays is visible in the $m_{\pi ^+ \pi ^ }$ distribution. 
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Sidebandsubtracted distributions of $ D ^0$ ( $\overline{ D }{} {}^0$ ) candidates in variables of (a, b) $\cos(\theta_{ K ^+ })$, (c, d) $\cos(\theta_{\pi ^+ })$, and (e, f) $\Phi$ for different values of $ C_{T}$ ( $\overline{C}_{T}$ ). The asymmetric distributions with respect to 0 for $\cos(\theta_{ K ^+ })$ and $\cos(\theta_{\pi ^+ })$ variables, and with respect to $\pi/2$ for the $\Phi$ variable, are due to the dynamics of the fourbody decay. 
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Distributions of the asymmetry parameters (a) $ A_{T}$ , (b) $\overline{A}_{T}$ and (c) $ a_{ C P }^{\text{ T odd}}$ in 32 regions of the phase space. 
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Distributions of the asymmetry parameters (a) $ A_{T}$ , (b) $\overline{A}_{T}$ and (c) $ a_{ C P }^{\text{ T odd}}$ as a function of the $ D ^0$ decay time. For $ a_{ C P }^{\text{ T odd}}$ , the value of the $\chi^2/\mathrm{ndf}$ for the $ C P$ conservation hypothesis, represented by a dashed line, is also quoted. The scale is broken for the first and last bin. 
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Animated gif made out of all figures. 
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Number of signal decays obtained from the fit to data for each of the four samples defined by the $ D ^0$ / $\overline{ D }{} {}^0$ flavour and the sign of $ C_{T}$ or $\overline{C}_{T}$ . 
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Considered sources of systematic uncertainty and their relative contributions to the total uncertainty. 
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Definition of the 32 regions of the fivedimensional phase space of the fourbody $ D ^0 \rightarrow K ^+ K ^ \pi ^+ \pi ^ $ decay. 
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Measurements of $ a_{ C P }^{\text{ T odd}}$ , $ A_{T}$ and $\overline{A}_{T}$ in each region of phase space. The uncertainties are statistical only. A common systematic uncertainty of 0.13%, 0.12% and 0.04% should be added to the asymmetries $ A_{T}$ , $\overline{A}_{T}$ and $ a_{ C P }^{\text{ T odd}}$ , respectively. This uncertainty is considered fully correlated among the bins. 
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Measurements of $ a_{ C P }^{\text{ T odd}}$ , $ A_{T}$ and $\overline{A}_{T}$ in different intervals of $ D ^0$ decay time, $t$, expressed in $ {\rm ps}$ . The uncertainties are statistical only. A common systematic uncertainty of 0.13%, 0.12% and 0.04% should be added to the asymmetries $ A_{T}$ , $\overline{A}_{T}$ and $ a_{ C P }^{\text{ T odd}}$ , respectively. This uncertainty is considered fully correlated among the bins. 
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Created on 20 February 2021.