A modelindependent search for direct CP violation in the Cabibbo suppressed decay $D^+ \to K^ K^+\pi^+$ in a sample of approximately 370,000 decays is carried out. The data were collected by the LHCb experiment in 2010 and correspond to an integrated luminosity of 35 pb$^{1}$. The normalized Dalitz plot distributions for $D^+$ and $D^$ are compared using four different binning schemes that are sensitive to different manifestations of CP violation. No evidence for CP asymmetry is found.
Fitted mass spectra of (a) $ K ^ \pi ^+ \pi ^+ $ and (b) $ K ^ K ^+ \pi ^+ $ candidates from samples 1 and 3, $ D ^+ $ and $ D ^ $ combined. The signal mass windows and sidebands defined in the text are labelled. 
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Dalitz plot of the $ D ^+ \rightarrow K ^ K ^+ \pi ^+ $ decay for selected candidates in the signal window. The vertical $\overline{ K }{} ^{*}(892)^{0}$ and horizontal $\phi(1020)$ contributions are clearly visible in the data. 
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$\mathcal{S}_{CP}$ across the Dalitz plot in a Monte Carlo pseudoexperiment with a large number of events with (a) no CPV and (b) a 4$^{\circ}$ CPV in the $\phi\pi$ phase. Note the difference in colour scale between (a) and (b). 
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Layout of the (a) "Adaptive I" and (b) "Adaptive II" binnings on the Dalitz plot of data. 
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(a) Distribution of $\mathcal{S}_{CP}$ values from $ D ^+ \rightarrow K ^ \pi ^+ \pi ^+ $ from a test with 900 uniform bins. The mean of the fitted Gaussian distribution is $0.015\pm0.034$ and the width is $0.996\pm0.023$. (b) Distribution of $\mathcal{S}_{CP}$ values from $ D ^+_ s \rightarrow K ^ K ^+ \pi ^+ $ with 129 bins. The fitted mean is $0.011\pm0.084$ and the width is $0.958\pm0.060$. 
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Dalitz plots of (a) $ D ^+ \rightarrow K ^ \pi ^+ \pi ^+ $, showing the 25bin adaptive scheme with the $\mathcal{S}_{CP}$ values, and (b) $ D ^+_ s \rightarrow K ^ K ^+ \pi ^+ $, showing the three regions referred to in the text. The higher and lower $ K ^ \pi ^+ $ invariant mass combinations are plotted in (a) as there are identical pions in the final state. 
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Distribution of ${\mathcal S}^i_{\it CP}$ in the Dalitz plot for (a) "Adaptive I", (b) "Adaptive II", (c) "Uniform I" and (d) "Uniform II". In (c) and (d) bins at the edges are not shown if the number of entries is not above a threshold of 50 (see Sect. III). 
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Distribution of ${\mathcal S}^i_{\it CP}$ fitted to Gaussian functions, for (a) "Adaptive I", (b) "Adaptive II", (c) "Uniform I" and (d) "Uniform II". The fit results are given in Table IX. 
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Animated gif made out of all figures. 
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Yield ($S$) and purity for samples 1 and 3 after the final selection. The purity is estimated in the 2$\sigma$ mass window. 
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Number of candidates $(S+B)$ in the signal windows shown in Fig. 1 after the final selection, for use in the subsequent analysis. 
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The CLEOc amplitude model "B" [7] used in the simulation studies. The uncertainties are statistical, experimental systematic and model systematic respectively. 
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Results from sets of 100 pseudoexperiments with different $ C P$ asymmetries and Adaptive I and II binnings. $p(3\sigma)$ is the probability of a 3$\sigma$ observation of CPV. $\langle S \rangle$ is the mean significance with which CPV is observed. 
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Results from sets of 100 pseudoexperiments with $4^{\circ}$ CPV in the $\phi(1020)$ phase and different Dalitz plot models. $p(3\sigma)$ is the probability of a 3$\sigma$ observation of CPV. $\langle S \rangle$ is the mean significance with which CPV is observed. The sample size is comparable to that seen in data. 
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Results ($p$values, in %) from tests with the $ D ^+ \rightarrow K ^ \pi ^+ \pi ^+ $ control channel using the uniform and adaptive binning schemes. The values correspond to tests performed on the whole dataset in the mass windows defined in Sect. II. 
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Results ($p$values, in %) from tests with the $ D ^+_ s \rightarrow K ^ K ^+ \pi ^+ $ control channel using the uniform binning scheme. The values correspond to tests performed separately on Zones AC, with samples 13 and both magnet polarities combined. 
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Results ($p$values, in %) from tests with the $ D ^+ \rightarrow K ^ \pi ^+ \pi ^+ $ and $ D ^+_ s \rightarrow K ^ K ^+ \pi ^+ $ samples divided according to the impact parameter with respect to the primary vertex. The tests are performed using the adaptive binning scheme with 25 bins. 
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Fitted means and widths, $\chi^2/{\rm ndf}$ and $p$values for consistency with no CPV for the $ D ^+ \rightarrow K ^ K ^+ \pi ^+ $ decay mode with four different binnings. 
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Created on 11 July 2020.