A search for $CP$ violation in the Cabibbosuppressed $D^0 \rightarrow K^+ K^ \pi^+ \pi^$ decay mode is performed using an amplitude analysis. The measurement uses a sample of $pp$ collisions recorded by the LHCb experiment during 2011 and 2012, corresponding to an integrated luminosity of 3.0 fb$^{1}$. The $D^0$ mesons are reconstructed from semileptonic $b$hadron decays into $D^0\mu^ X$ final states. The selected sample contains more than 160000 signal decays, allowing the most precise amplitude modelling of this $D^0$ decay to date. The obtained amplitude model is used to perform the search for $CP$ violation. The result is compatible with $CP$ symmetry, with a sensitivity ranging from 1% to 15% depending on the amplitude considered.
Mass distribution of the $ D ^0 \!\rightarrow K ^+ K ^ \pi ^+ \pi ^ $ candidates after the final selection, with fit result superimposed. The top plot shows the normalised residuals. 
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Definition of the helicity angles $\theta_{K}$ and $\theta_{\pi}$, and the decayplane angle $\phi$. 
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Distributions of the five CM variables for the selected $ D ^0$ and $ C\!P$ transformed $\overline{ D }{} {}^0$ candidates (black points with error bars). The results of the fivedimensional amplitude fit is superimposed with the signal model (dashed blue), the background model (dotted green) and the total fit function (plain red). The plot on top of each distribution shows the normalised residuals, where the error is defined as the quadratic sum of the statistical uncertainties of the data and simulated samples. 
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Animated gif made out of all figures. 
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Resonances considered in the analysis, classified according to their spinparity $J^P$ and decay products. 
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Modulus and phase of the fit parameters along with the fit fractions of the amplitudes included in the model. The substructures of the threebody resonances are listed in Table 3. The first uncertainty is statistical and the second is systematic. 
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Parameters of the amplitudes contributing to the threebody decays of the $a_1(1260)^+$, $K_1(1270)^+$ and $K_1(1400)^+$. The first uncertainty is statistical and the second is systematic. 
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Parameters of the $\rho\omega$ interference for all relevant amplitudes. The first uncertainty is statistical and the second is systematic. 
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$ C\!P$ violation parameters fitted simultaneously to the $ D ^0$ and ( $ C\!P$ transformed) $\overline{ D }{} {}^0$ samples. The first uncertainty is statistical and the second is systematic. 
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Modulus and phase of the fit parameters along with the fit fractions of the amplitudes included in the alternative model using the $K^*(1410)^0$ instead of the $K^*(1680)^0$ resonance. 
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Modulus and phase of the fit parameters along with the fit fractions of the amplitudes included in the alternative model using five additional amplitudes. 
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Modulus and phase of the fit parameters along with the fit fractions of the amplitudes of the alternative model that includes the amplitude $ D ^0 \!\rightarrow \rho(1450)^0\rho(770)^0$ in $D$wave. 
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Statistical and systematic uncertainties (in %) on the fit fractions. Values smaller than 0.0005% are displayed as "0.000". The sources of systematic uncertainty are described in the text in the same order as shown in this table. 
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Statistical and systematic uncertainties on the fit parameters shown for all floating components. Values smaller than 0.0005 are displayed as "0.000". The sources of systematic uncertainty are described in the text in the same order as shown in this table. For each amplitude, the first value quoted is the modulus and the second is the phase of the complex fit parameter. 
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Statistical and systematic uncertainties (in %) on $A_{\mathcal{F}_k}$. Values smaller than 0.0005% are displayed as "0.000". The sources of systematic uncertainty are described in the text in the same order as shown in this table. 
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Statistical and systematic uncertainties (in %) on the $ C\!P$ violation parameters shown for all floating components. Values smaller than 0.0005% are displayed as "0.000". The sources of systematic uncertainty are described in the text in the same order as shown in this table. For each amplitude, the first value quoted is the modulus asymmetry and the second is the phase difference. 
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Created on 16 February 2019.Citation count from INSPIRE on 22 February 2019.