{\bf Dominant Feynman diagrams for { $\Lambda ^0_ b \rightarrow p \pi ^- \pi ^+ \pi ^- $ } and { $\Lambda ^0_ b \rightarrow p \pi ^- K ^+ K ^- $ } transitions.} The two diagrams show the transitions that contribute most strongly to $\Lambda ^0_ b \rightarrow p \pi ^- \pi ^+ \pi ^- $ and $\Lambda ^0_ b \rightarrow p \pi ^- K ^+ K ^- $ decays. In both cases, a pair of $\pi ^+ \pi ^- $ ($ K ^+ K ^- $) is produced by gluon emission from the light quarks ( $ u $ , $ d $ ). The difference is in the $ b $ quark decay that happens on the left through a virtual $W^-$ boson emission ("tree diagram") and on the right as a virtual $W^-$ boson emission and absorption together with a gluon emission ("loop diagram"). The magnitudes of the two amplitudes are expected to be comparable, and each is proportional to the product of the CKM matrix elements involved, which are shown in the figure.

{\bf Reconstructed invariant mass fits used to extract the signal yields.} The invariant mass distributions for (a) $\Lambda ^0_ b \rightarrow p \pi ^- \pi ^+ \pi ^- $ and (b) $\Lambda ^0_ b \rightarrow p \pi ^- K ^+ K ^- $ decays are shown. A fit is overlaid on top of the data points, with solid and dotted lines describing the projections of the fit results for each of the components described in the text and listed in the legend. Uncertainties on the data points are statistical only and are calculated assuming Poisson distributed entries.

{\bf Definition of the $\Phi$ angle.} The decay planes formed by the $ p \pi^-_\textrm{fast}$ and the $\pi^-_\textrm{slow}\pi ^+ $ systems in the $\Lambda ^0_ b $ rest frame. The momenta of the particles, represented by vectors, determine the two decay planes and the angle $\Phi \in [-\pi,\pi]$ [19] measures their relative orientation.

{\bf Distributions of the asymmetries}. The results of the fit in each region of binning schemes A and B are shown. The asymmetries $ a_P^{\widehat{T}\text{-odd}}$ and $ a_ C P ^{\widehat{T}\text{-odd}}$ for $\Lambda ^0_ b \rightarrow p \pi ^- \pi ^+ \pi ^- $ decays are represented by open boxes and filled circles, respectively. The error bars indicate the total uncertainties, calculated as the sum in quadrature of the statistical uncertainty resulting from the fit to the invariant mass distribution and the systematic uncertainties estimated as described in the main text. The values of the $\chi^2$ /ndf are quoted for the $P$- and $ C P$ -conserving hypotheses for each binning scheme.

Animated gif made out of all figures.

{\bf Definition of binning scheme A for the decay mode { $\Lambda ^0_ b \rightarrow p \pi ^- \pi ^+ \pi ^- $ }.} Binning scheme A is defined to exploit interference patterns arising from the resonant structure of the decay. Bins 1-4 focus on the region dominated by the $\Delta (1232)^{++}\rightarrow p \pi ^+ $ resonance. The other eight bins are defined to study regions where $ p \pi ^- $ resonances are present (5--8) on either side of the $\rho(770)^{0}\rightarrow \pi ^+ \pi ^- $ resonances (5--12). Further splitting for $|\Phi|$ lower or greater than $\pi/2$ is done to reduce potential dilution of asymmetries, as suggested in Ref. [19]. Masses are in units of $ {\mathrm{ Ge V /}c^2}$ .

{\bf Measurements of $ C P$ - and $P$-violating observables.} The $ C P$ - and $P$-violating observables, $ a_ C P ^{\widehat{T}\text{-odd}}$ and $ a_P^{\widehat{T}\text{-odd}}$ , resulting from the fit to the data are listed with their statistical and systematic uncertainties. Each value is obtained through an independent fit to a region of the phase space as described in the text and Table 1. Results for schemes A and B are outlined for $\Lambda ^0_ b \rightarrow p \pi ^- \pi ^+ \pi ^- $ decays, and in two bins of phase space for $\Lambda ^0_ b \rightarrow p \pi ^- K ^+ K ^- $ decays, as defined in the text. The first column lists the bin number. For both decay modes the measurement integrated over the phase space, performed independently, is also shown.

Supplementary material full pdf

This ZIP file contains supplemetary material for the publication LHCb-PAPER-2016-030. The files are: supplementary.pdf : An overview of the extra figures *.pdf, *.png, *.eps : The figures in various formats