A search for timeintegrated $CP$ violation in the Cabibbosuppressed decay $ D^0\rightarrow\pi^+\pi^\pi^+\pi^$ is performed using an unbinned, modelindependent technique known as the energy test. This is the first application of the energy test in fourbody decays. The search is performed for $P$even $CP$ asymmetries and, for the first time, is extended to probe the $P$odd case. Using protonproton collision data corresponding to an integrated luminosity of 3.0 fb$^{1}$ collected by the LHCb detector at centreofmass energies of $\sqrt{s}=$7 TeV and 8 TeV, the world's best sensitivity to $CP$ violation in this decay is obtained. The data are found to be consistent with the hypothesis of $CP$ symmetry with a $p$value of $(4.6\pm0.5)\%$ in the $P$even case, and marginally consistent with a $p$value of $(0.6\pm0.2)\%$ in the $P$odd case, corresponding to a significance for $CP$ nonconservation of 2.7 standard deviations.
Distribution of $\Delta m$ with fit overlaid for the selected $ D ^{*+}$ candidates in the 2012 data. The data points and the contributions from signal, background, and their total obtained from the fit are shown. 
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(a,b) Distribution of permutation $T$values fitted with a GEV function for the simulated sample and showing the measured $T$value as a vertical line, and (c,d,e,f) local asymmetry significances. Left column plots are for a $P$even $ C P$ violation test with a $3^\circ$ phase $ C P$ violation introduced in the $a_1(1260)^+$ resonance (see text), projected onto the (c) $m(\pi_1 \pi_2 \pi_3)$ and (e) $m(\pi_1\pi_2)$ axes. Right column plots are for a $P$odd $ C P$ violation test with $3^\circ$ phase $ C P$ violation introduced in the Pwave $\rho ^0 (770) \rho ^0 (770)$ resonance projected onto the same axes. In plots (c,d,e,f) the grey area corresponds to candidates with a contribution to the $T$value of less than one standard deviation. The pink (blue) area corresponds to candidates with a positive (negative) contribution to the $T$value. Light, medium or dark shades of pink and blue correspond to between one and two, two and three, and more than three standarddeviation contributions, respectively. 
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(a,b) Distribution of permutation $T$values fitted with a GEV function and showing the $T$value of the data tests as a vertical line, and (c,d,e,f) local asymmetry significances. Left column plots are for the $P$even $ C P$ violation test, projected onto the (c) $m(\pi_1 \pi_2 \pi_3)$ and (e) $m(\pi_1\pi_2)$ axes. Right column plots are for the $P$odd $ C P$ violation test projected onto the same axes. In plots (c,d,e,f) the grey area correspond to candidates with a contribution to the $T$value of less than one standard deviation. In the $P$even $ C P$ violation test the positive (negative) asymmetry significance is set for the $ D ^0$ candidates having positive (negative) contribution to the measured $T$ value. In the $P$odd $ C P$ violation test the positive (negative) asymmetry significance is set for sample $I+IV$ having positive (negative) contribution to the measured $T$ value (see Sect. 5). The pink (blue) area corresponds to candidates with a positive (negative) contribution to the $T$value. Light, medium or dark shades of pink and blue correspond to between one and two, two and three, and more than three standard deviation contributions, respectively. 
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Animated gif made out of all figures. 
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The yields of signal events in the four samples that obtained from fits to the $\Delta m$ distribution. 
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Overview of sensitivities to various $ C P$ violation scenarios in simulation. $\Delta A$ and $\Delta \phi$ denote, respectively, the relative change in magnitude and the change in phase of the amplitude of the resonance $R$. The Pwave $\rho ^0 (770)\rho ^0 (770)$ is a $P$odd component. The phase change in this resonance is tested with the $P$odd $ C P$ violation test. Results for all other scenarios are given with the standard $P$even test. 
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Results for the $P$even and $P$odd $ C P$ violation tests for three different values of the effective radius $\delta$. The $p$values obtained with both the counting and GEV fitting methods are given (see text). The counting method is the default method. 
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Created on 08 August 2020.