A first search for $CP$ violation in the Cabibbosuppressed $\Xi_c^+\rightarrow pK^\pi^+$ decay is performed using both a binned and an unbinned modelindependent technique in the Dalitz plot. The studies are based on a sample of protonproton collision data, corresponding to an integrated luminosity of $3.0 {\rm fb^{1}}$, and collected by the LHCb experiment at centreofmass energies of $7$ and $8 \rm TeV$. The data are consistent with the hypothesis of no $CP$ violation.
Invariantmass, $M( p K ^ \pi ^+ )$, distributions of selected $\Lambda ^+_ c $ candidates are shown in the (left) 2011 and (right) 2012 data samples. Data points are in black. The overlaid fitted model (blue continuous line) is a sum of two Gaussian functions with the same mean and different widths (red dashed line) and a secondorder Chebyshev polynomial function (green dotted line) describing the signal and background components. 
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Invariantmass, $M( p K ^ \pi ^+ )$, distributions of selected $\Xi ^+_ c $ candidates are shown in the (left) 2011 and (right) 2012 data samples. Data points are in black. The overlaid fitted model (blue continuous line) is a sum of two Gaussian functions with the same mean and different widths (red dashed line) and a secondorder Chebyshev polynomial function (green dotted line) describing the signal and background components. 
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Definition of the Dalitz plot regions for (left) $\Lambda ^+_ c \rightarrow p K ^ \pi ^+ $ and (right) $\Xi ^+_ c \rightarrow p K ^ \pi ^+ $ decays. Additional regions are defined by combining regions. For $\Lambda ^+_ c \rightarrow p K ^ \pi ^+ $ R2=R5$\cup$R6 and for $\Xi ^+_ c \rightarrow p K ^ \pi ^+ $ R2=R8$\cup$R9, R10=R4$\cup$R5 and R11=R4$\cup$R5$\cup$R6$\cup$R7. The presented distributions correspond to the 2012 data sample. 
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Measured values of $A_{\rm Raw}$ in regions of $\Lambda ^+_ c \rightarrow p K ^ \pi ^+ $ candidate decays for 2011 (stars) and 2012 (dots) data samples. R0 corresponds to full Dalitz plot and R2 is separated into R5 and R6, and these regions are correlated and separated by dashed lines. 
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Distributions of $S_ C P ^i$ and corresponding onedimensional distributions for $\Lambda ^+_ c \rightarrow p K ^ \pi ^+ $ decays for the data collected in the 2012 data sample: (top row) 28 samesize bins and (bottom row) 106 samesize bins of the Dalitz plot. The number of analysed bins, nbins, and the {\it p}values are given. 
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(Top left) pulls, $(\mu_T  \mu_{TR} )/\Delta(\mu_T  \mu_{TR} )$, and (top right) the corresponding {\it p}values, (bottom left) pull values of the test statistic $T$ and (bottom right) the corresponding {\it p}values in regions for control $\Lambda ^+_ c \rightarrow p K ^ \pi ^+ $ candidate decays obtained using the kNN method with $n_k=50$ for data collected in 2011 (stars) and 2012 (dots). The horizontal lines in the left figures represent 3 and +3 pull values. R0 corresponds to full Dalitz plot and R2 is separated into R5 and R6, and these regions are correlated and separated by dashed lines. 
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Distributions of $S_ C P ^i$ and corresponding onedimensional distributions for $\Xi ^+_ c \rightarrow p K ^ \pi ^+ $ decays for the combined data collected 2011 and 2012: (top row) 29 uniform bins and (bottom row) 111 uniform bins of the Dalitz plot. The number of analysed bins and the {\it p}values are given. 
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(Top left) pulls, $(\mu_T  \mu_{TR} )/\Delta(\mu_T  \mu_{TR} )$, and (top right) the corresponding {\it p}values; (bottom left) pull values of the test statistic $T$ and (bottom right) the corresponding {\it p}values in regions for signal $\Xi ^+_ c \rightarrow p K ^ \pi ^+ $ candidate decays obtained using the kNN method with $n_k=50$ for combined data collected 2011 and 2012. The horizontal lines in the left figures represent $3$ and $+3$ pull values. R0 corresponds to full Dalitz plot and R2 is separated into R8 and R9, R10 is separated into R4 and R5, R11 is separated into R4, R5, R6 and R7, and these regions are correlated and separated by dashed lines. 
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The measured $A_{\rm Raw}$ in regions in signal $\Xi ^+_ c \rightarrow p K ^ \pi ^+ $ candidate decays for the combined data collected in 2011 and 2012. R0 corresponds to full Dalitz plot and R2 is separated into R8 and R9, R10 is separated into R4 and R5, R11 is separated into R4, R5, R6 and R7, and these regions are correlated and separated by dashed lines. 
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(Left) the pull values of the test statistic $T$ and (right) the corresponding {\it p}value dependence on the $n_k$ parameter for the whole Dalitz plot (region R0) for $\Xi ^+_ c \rightarrow p K ^ \pi ^+ $ candidate decays obtained using the kNN method for the combined data collected in 2011 and 2012. The horizontal lines in the left figures represent $3$ and $+3$ pull values. The points are determined with different $n_k$ using same data sample, therefore are correlated. 
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Animated gif made out of all figures. 
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Definitions of the Dalitz plot regions for the control mode, $\Lambda ^+_ c \rightarrow p K ^ \pi ^+ $ . 
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Definitions of the Dalitz plot regions for $\Xi ^+_ c \rightarrow p K ^ \pi ^+ $ decays. 
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Created on 06 March 2021.