A search for CP violation in the phase space of the decay $D^+\to\pi^\pi^+\pi^+$ is reported using $pp$ collision data, corresponding to an integrated luminosity of 1.0 fb$^{1}$, collected by the LHCb experiment at a centreofmass energy of 7 TeV. The Dalitz plot distributions for $3.1\times 10^6$ $D^+$ and $D^$ candidates are compared with binned and unbinned modelindependent techniques. No evidence for CP violation is found.
Invariantmass distributions for (a) $ D ^+$ and (b) $ D ^+_ s $ candidates in the momentum range $50<p_{ D ^+ _{(s)}} <100$ $ {\mathrm{ Ge V /}c}$ for magnet up data. Data points are shown in black. The solid (blue) line is the fit function, the (green) dashed line is the signal component and the (magenta) dotted line is the background. 
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Dalitz plots for (a) $ D ^+ \rightarrow \pi ^ \pi ^+ \pi ^+ $ and (b) $ D ^+_ s \rightarrow \pi ^ \pi ^+ \pi ^+ $ candidates selected within $\pm2\tilde\sigma$ around the respective $\tilde m$ weighted average mass. 
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(a) Distribution of $\mathcal{S}_{CP}^i$ with 49 $ D ^+_ s $ adaptive bins of equal population in the $ D ^+_ s \rightarrow \pi ^ \pi ^+ \pi ^+ $ Dalitz plot and (b) the corresponding onedimensional distribution (histogram) with a standard normal Gaussian function superimposed (solid line). 
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Dalitz plot for $ D ^+_ s \rightarrow \pi ^ \pi ^+ \pi ^+ $ control sample decays divided into (a) seven regions R1R7 and (b) three regions P1P3. Region R3 is further divided into two regions of $s_{\rm high}$ at masses smaller (R3l) and larger (R3r) than the $\rho^{0}(770)$ resonance. 
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(a) Pull values of $T$ and (b) the corresponding pvalues for $ D ^+_ s \rightarrow \pi ^ \pi ^+ \pi ^+ $ control sample candidates restricted to each region, obtained using the kNN method with $n_k=20$. The horizontal blue lines in (a) represent $3$ and $+3$ pull values. The region R0 corresponds to the full Dalitz plot. Note that the points for the overlapping regions are correlated. 
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(a) Raw asymmetry $A=(N_N_+)/(N_+N_+)$ and (b) the pull values of $\mu_T$ for $ D ^+_ s \rightarrow \pi ^ \pi ^+ \pi ^+ $ control sample candidates restricted to each region. The horizontal lines in (b) represent $+3$ and $+5$ pull values. The region R0 corresponds to the full Dalitz plot. Note that the points for the overlapping regions are correlated. 
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(a) Pull values of $T$ and (b) the corresponding pvalues for the background candidates restricted to each region obtained using the kNN method with $n_k=20$. The horizontal blue lines in (a) represent $3$ and $+3$ pull values. The region R0 corresponds to the full Dalitz plot. Note that the points for the overlapping regions are correlated. 
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(a) Raw asymmetry and (b) pull value of $\mu_T$ as a function of a region for the background candidates. The horizontal lines in (b) represent $+3$ and $+5$ pull values. The region R0 corresponds to the full Dalitz plot. Note that the points for the overlapping regions are correlated. 
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Distributions of $\mathcal{S}_{CP}^i$ across the $ D ^+$ Dalitz plane, with the adaptive binning scheme of uniform population for the total $ D ^+ \rightarrow \pi ^ \pi ^+ \pi ^+ $ data sample with (a) 49 and (c) 100 bins. The corresponding onedimensional $\mathcal{S}_{CP}^i$ distributions (b) and (d) are shown with a standard normal Gaussian function superimposed (solid line). 
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(a) Distribution of $\mathcal{S}_{CP}^i$ with 98 bins in the uniform binning scheme for the total $ D ^+ \rightarrow \pi ^ \pi ^+ \pi ^+ $ data sample and (b) the corresponding onedimensional $\mathcal{S}_{CP}^i$ distribution (b) with a standard normal Gaussian function superimposed (solid line). 
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Dalitz plot for $ D ^+ \rightarrow \pi ^ \pi ^+ \pi ^+ $ candidates divided into (a) seven regions R1R7 and (b) three regions P1P3. 
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(a) Raw asymmetry and (b) the pull values of $\mu_T$ for $ D ^+ \rightarrow \pi ^ \pi ^+ \pi ^+ $ candidates restricted to each region. The horizontal lines in (b) represent pull values $+3$ and $+5$. The region R0 corresponds to the full Dalitz plot. Note that the points for the overlapping regions are correlated. 
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(a) Pull values of $T$ and (b) the corresponding pvalues for $ D ^+ \rightarrow \pi ^ \pi ^+ \pi ^+ $ candidates restricted to each region obtained using the kNN method with $n_k=20$. The horizontal blue lines in (a) represent pull values $3$ and $+3$. The region R0 corresponds to the full Dalitz plot. Note that the points for the overlapping regions are correlated. 
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Animated gif made out of all figures. 
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Results for the $D^+ \rightarrow \pi^\pi^+\pi^+$ decay sample using the adaptive binning scheme with different numbers of bins. The number of degrees of freedom is the number of bins minus 1. 
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Created on 20 February 2021.