Distributions of (\subref*{fig:mass:KKPiPi:D},\subref*{fig:mass:FourPi:D},\subref*{fig:mass:KThreePi:D}) $ m(hhhh)$ and (\subref*{fig:mass:KKPiPi:DELTAM},\subref*{fig:mass:FourPi:DeltaM},\subref*{fig:mass:KThreePi:DELTAM}) $\Delta m$ for (\subref*{fig:mass:KKPiPi:D},\subref*{fig:mass:KKPiPi:DELTAM}) $ D ^0 \rightarrow K ^- K ^+ \pi ^- \pi ^+ $ , (\subref*{fig:mass:FourPi:D},\subref*{fig:mass:FourPi:DeltaM}) $ D ^0 \rightarrow \pi ^- \pi ^+ \pi ^+ \pi ^- $ , and (\subref*{fig:mass:KThreePi:D},\subref*{fig:mass:KThreePi:DELTAM}) $ D ^0 \rightarrow K ^- \pi ^+ \pi ^+ \pi ^- $ candidates for magnet up polarity. Projections of the two-dimensional fits are overlaid, showing the contributions for signal, combinatorial background, and random soft pion background. The contributions from $ D ^0 \rightarrow K ^- \pi ^+ \pi ^- \pi ^+ \pi ^0 $ and $ D _{s}^{+} \rightarrow K ^- K ^+ \pi ^- \pi ^+ \pi ^+ $ contamination are also shown for the $ D ^0 \rightarrow K ^- K ^+ \pi ^- \pi ^+ $ sample.

Invariant mass-squared distributions for $ D ^0$ meson (black, closed circles) and $\overline{ D }{} ^0$ meson (red, open squares) decays to the final state $ K ^- K ^+ \pi ^- \pi ^+ $ . The invariant mass-squared combinations s(1,2), s(2,3), s(1,2,3), s(2,3,4), and s(3,4) correspond to s( $ K ^-$ , $ K ^+$ ), s( $ K ^+$ , $\pi ^-$ ), s( $ K ^-$ , $ K ^+$ , $\pi ^-$ ), s( $ K ^+$ , $\pi ^-$ , $\pi ^+$ ), and s( $\pi ^-$ , $\pi ^+$ ) , respectively for the $ D ^0$ mode. The charge conjugate is taken for the $\overline{ D }{} ^0$ mode. The phase-space distribution of the $ D ^0 \rightarrow K ^- K ^+ \pi ^- \pi ^+ $ decay is expected to be dominated by the quasi-two-body decay $ D ^0 \rightarrow \phi\rho^0 $ with additional contributions from $ D ^0 \rightarrow K _{1}(1270)^{\pm} K ^{\mp} $ and $ D ^0 \rightarrow K ^{*}(1410)^\pm K ^{\mp} $ decays [10].

Invariant mass-squared distributions for $ D ^0$ meson (black, closed circles) and $\overline{ D }{} ^0$ meson (red, open squares) decays to the final state $\pi ^- \pi ^+ \pi ^+ \pi ^- $ . The invariant mass-squared combinations s(1,2), s(2,3), s(1,2,3), s(2,3,4), and s(3,4) correspond to s( $\pi ^-$ , $\pi ^+$ ), s( $\pi ^+$ , $\pi ^+$ ), s( $\pi ^-$ , $\pi ^+$ , $\pi ^+$ ), s( $\pi ^+$ , $\pi ^+$ , $\pi ^-$ ), and s( $\pi ^+$ , $\pi ^-$ ) , respectively for the $ D ^0$ mode. The charge conjugate is taken for the $\overline{ D }{} ^0$ mode. Owing to the randomisation of the order of identical final-state particles the invariant mass-squared distributions s(2,3,4) and s(3,4) are statistically compatible with the invariant mass-squared distributions s(1,2,3) and s(1,2), respectively. As such the invariant mass-squared distributions s(2,3,4) and s(3,4) are not shown. The phase-space distribution of the $ D ^0 \rightarrow \pi ^- \pi ^+ \pi ^+ \pi ^- $ decay is expected to be dominated by contributions from $ D ^0 \rightarrow a_{1}(1260)^{+} \pi^{-} $ and $ D ^0 \rightarrow \rho^{0} \rho^{0} $ decays [21].

Distributions of $ S_{ C P }$ for \subref{fig:Toy:SCP:NoCPV} a typical pseudo-experiment with generated $ D ^0 \rightarrow \pi ^- \pi ^+ \pi ^+ \pi ^- $ decays without CPV and for \subref{fig:Toy:SCP:CPV} a typical pseudo-experiment with a generated 10$^{\circ}$ phase difference between $ D ^0 \rightarrow a_{1}(1260)^{+} \pi^{-}$ and $\overline{ D }{} ^0 \rightarrow a_{1}(1260)^{-} \pi^{+}$ resonant decays. The points show the data distribution and the solid line is a reference Gaussian distribution corresponding to the no CPV hypothesis. The corresponding $p$-values under the hypothesis of no asymmetry for \subref{fig:Toy:SCP:NoCPV} decays without CPV and \subref{fig:Toy:SCP:CPV} decays with a 10$^{\circ}$ phase difference between $ D ^0 \rightarrow a_{1}(1260)^{+} \pi^{-}$ and $\overline{ D }{} ^0 \rightarrow a_{1}(1260)^{-} \pi^{+}$ resonant components are 85.6% and $1.1\times10^{-16}$, respectively.

Distributions of (\subref*{fig:SCP:KKPiPi},\subref*{fig:SCP:FourPi},\subref*{fig:control:SCP}) $ S_{ C P }$ and (\subref*{fig:RAW:KKPiPi},\subref*{fig:RAW:FourPi},\subref*{fig:control:RAW}) local $ C P$ asymmetry per bin for (\subref*{fig:SCP:KKPiPi},\subref*{fig:RAW:KKPiPi}) $ D ^0 \rightarrow K ^- K ^+ \pi ^- \pi ^+ $ decays partitioned with 32 bins, for (\subref*{fig:SCP:FourPi},\subref*{fig:RAW:FourPi}) $ D ^0 \rightarrow \pi ^- \pi ^+ \pi ^+ \pi ^- $ decays partitioned with 128 bins, and for (\subref*{fig:control:SCP},\subref*{fig:control:RAW}) the control channel $ D ^0 \rightarrow K ^- \pi ^+ \pi ^+ \pi ^- $ partitioned with 128 bins. The points show the data distribution and the solid line is a reference Gaussian distribution corresponding to the no CPV hypothesis.

Animated gif made out of all figures.

The $\chi^2/\mathrm{ndf}$ and $p$-values under the hypothesis of no CPV for the control channel $ D ^0 \rightarrow K ^- \pi ^+ \pi ^+ \pi ^- $ . The $p$-values are calculated separately for data samples taken with magnet up polarity, magnet down polarity, and the two polarities combined.

The $\chi^2/\mathrm{ndf}$ and $p$-values under the hypothesis of no CPV with three different partitions for $ D ^0 \rightarrow K ^- K ^+ \pi ^- \pi ^+ $ decays and $ D ^0 \rightarrow \pi ^- \pi ^+ \pi ^+ \pi ^- $ decays. The $p$-values are calculated for a combined data sample with both data taken with magnet up polarity and data taken with magnet down polarity.

Supplementary material full pdf

This ZIP file contains supplemetary material for the publication LHCb-PAPER-2013-041. The files are: Supplementary.pdf : An overview of the extra figures *.pdf, *.png, *.eps, *.C : The figures in variuous formats