Variation of the overall phase of the $ D ^+$ decay amplitude in the $\phi$ mass region of the Dalitz plot, from a simulation study based on the CLEO-c amplitude model in which the phase is defined relative to that of the $ K ^* (892)^{0}$ resonance [14]. To calculate $A_{CP}|_{S}$, the region is divided into rectangular zones as shown, corresponding to $1.00 < m( K ^- K ^+ )< 1.02$ $ {\mathrm{ Ge V /}c^2}$ and $1.02 < m( K ^- K ^+ )< 1.04$ $ {\mathrm{ Ge V /}c^2}$ along the $y$-axis, and to $m^{2}( K ^- \pi ^+ ) < 1.48$ $ {\mathrm{ Ge V^2 /}c^4}$ and $m^{2}( K ^- \pi ^+ ) > 1.48$ $ {\mathrm{ Ge V^2 /}c^4}$ along the $x$-axis.

Invariant mass distribution of selected (a) $ D ^+ \rightarrow \phi\pi ^+ $, (b) $ D ^- \rightarrow \phi\pi ^- $, (c) $ D ^+ \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi ^+ $ and (d) $ D ^- \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi ^- $ candidates. The data are represented by symbols with error bars. The red dashed curves indicate the signal lineshapes, the green solid lines represent the combinatorial background shape, and the green dotted lines represent background from mis-reconstructed $ D ^+_ s \rightarrow \phi\pi ^+ \pi ^0 $ decays in (a) and (b), and $ D ^+_ s \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi ^+ \pi ^0 $ or $ D ^+_ s \rightarrow K ^0_{\rm\scriptscriptstyle S} K ^+ $ decays in (c) and (d). The blue solid lines show the sum of all fit components.

Observed density of decays in the $ D ^+ \rightarrow K ^- K ^+ \pi ^+ $ Dalitz plot, with the regions A-D labelled as described in the text.

Distributions of transverse momentum $p_{\rm T}$ and pseudorapidity $\eta$ for (a) $ D ^+ \rightarrow K ^0_{\rm\scriptscriptstyle S} \pi ^+ $ and (b) $ D ^+ \rightarrow \phi\pi ^+ $ candidates with invariant masses $m$ in the range $1845 < m < 1895$ $ {\mathrm{ Me V /}c^2}$ . Candidates that do not fall into the 12 rectangular bins are not used in the analysis.

Invariant mass distribution of selected $ D ^{\pm}_{( s )} \rightarrow \pi ^\mp \pi ^\pm \pi ^\pm $ decays. The data are represented by symbols with error bars. The red dashed peaks indicate the signal decays, the green solid lines represent the combinatorial background shape, and the green dotted lines represent backgrounds from mis-reconstructed $ D ^+_ s \rightarrow \pi ^- \pi ^+ \pi ^+ \pi ^0 $ decays. The blue solid line shows the sum of all fit components.

Animated gif made out of all figures.

Expected mean values of $A_{CP}$ and $A_{CP}|_{S}$ for different types of $ C P$ violation introduced into the simulated Dalitz plots, together with the significance with which a signal could be observed given estimated overall uncertainties in $A_{CP}$ and $A_{CP}|_{S}$ of 0.2%.

Numbers of signal candidates in the four decay modes from the mass fits, with statistical uncertainties only.

Raw charge asymmetries, in %, in samples of the $ D ^+ \rightarrow K ^- \pi ^+ \pi ^+ $ control decay in which a particle not from the signal decay activated various hardware triggers.

Changes to the final result observed with various alternative kinematic binning schemes. The default scheme uses four bins of $ D ^+_{( s )}$ $p_{\rm T}$ and three bins of $ D ^+_{( s )}$ $\eta$. The variable $\phi$ is the azimuthal angle around the proton beams. The statistical uncertainties are determined by subtracting the uncertainties on the alternative result and the default result in quadrature.

Systematic uncertainties on the three measurements. The abbreviation n/a is used where the systematic effect does not apply. The row labelled `Backgrounds' represents the uncertainty in modelling the cross-feed in $A_{CP}$ and the uncertainty from ignoring the background in $A_{CP}|_{S}$.