A search for $C P$ violation in the $\Lambda_b^0 \to p \pi^ \pi^+ \pi^$ decay is performed using LHCb data corresponding to an integrated luminosity of 6.6$ fb^{1}$ collected in $pp$ collisions at centreofmass energies of 7, 8 and 13$ Te V$. The analysis uses both triple product asymmetries and the unbinned energy test method. The highest significances of $C P$ asymmetry are 2.9 standard deviations from triple product asymmetries and 3.0 standard deviations for the energy test method. Once the global $p$value is considered, all results are consistent with no $C P$ violation. Parity violation is observed at a significance of $5.5$ standard deviations for the triple product asymmetry method and $5.3$ standard deviations for the energy test method.
Invariantmass distribution for $\Lambda ^0_ b \rightarrow p \pi ^ \pi ^+ \pi ^ $ candidates with the result of the fit overlaid. The solid and dotted lines describe the projections of the fit results for various components as listed in the legend. 
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Measured asymmetries for the binning scheme (left) $A_1$ and $A_2$ and (right) $B_1$ and $B_2$. The error bars represent the sum in quadrature of the statistical and systematic uncertainties. The $\chi^2$ per ndof is calculated with respect to the null hypothesis and includes statistical and systematic uncertainties. 
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Animated gif made out of all figures. 
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The $p$values from the energy test for different distance scales and test configurations. 
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Definition of binning scheme $A$. This binning scheme is based on the helicity angles of the decay topology $\Lambda ^0_ b \rightarrow \left( N^{*+} \rightarrow \left(\Delta ^{++} \rightarrow p \pi ^+ \right)\pi ^ \right)\pi ^ $ where $\varphi$ is the azimuthal angle of the proton in the $\Delta^{++}$ rest frame and $\theta_{\Delta ^{++}}$ ($\theta_ p $) is the polar angle of the $\Delta ^{++}$ ($ p $) in the $N^{*+}$ ($\Delta ^{++}$) rest frame. 
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Supplementary material full pdf 
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This ZIP file contains supplemetary material for the publication LHCbPAPER2019028. The files are: Supplementary.pdf : "Search for CP violation and observation of P violation in Lb > p pi^ pi^+ pi^ decays" Supplementary.tex 
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Created on 06 March 2021.