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metry other than A h + ϵ . Eq. (2) can therefore be expressed as 

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To correct for the asymmetries other than A$_{CP}$ , we use reconstructed samples of D + s → φπ + and D 0 → K $^{−}$π + decays and assume that A$_{CP}$ in CF decays is negligibly small at the current experimental sensitivity and that A$_{FB}$ is the same for all charmed mesons. We reconstruct φ mesons via their K $^{+}$K − decay channel for D + s → φπ $^{+}$, requiring the K $^{+}$K − invariant mass to be between 1.01 and 1.03 GeV/ c $^{2}$. 

The measured asymmetry for D + s → φπ + is the sum of A D + s FB and A π + ϵ . Hence one can extract the A$_{CP}$ value for the K 0 $_{S}$π + final states by subtracting the measured asymmetry for D + s → φπ + from that for D + ( s ) → K 0 $_{S}$π $^{+}$. The subtraction is performed in bins of π + momentum, p lab π , and polar angle in the laboratory system, cos θ lab π (because A h + ϵ depends on these two variables while it is uniform in azimuthal angle), and the charmed meson’s polar angle in the center-of-mass system, cos θ CMS D + ( s ) (since cos θ CMS D + ( s ) is correlated with cos θ lab π and A D + ( s ) FB depends on it). The choice of the three-dimensional (3-D) binning is selected in order to avoid large statistical fluctuations in 

each bin. F igure 2 shows the A$_{CP}$ map of D + → K 0 $_{S}$π + in bins of ( p la b π , cos θ lab π , cos θ CMS D + ( s ) ). Calculating a weighted average of t he A$_{CP}$ values over the 3-D bins, we obtain A D $^{+}$→ K 0 $_{S}$π + CP = ( − 0 . 71 ± 0 . 26)% where the uncertainty originat es f rom the finite size of the D + → K 0 $_{S}$π + (0.19%) and D + s → φπ + (0.18%) samples. The χ $^{2}$/d.o.f over the 3-D bin s is found to be 31.4/24 which corresponds to 14% probab ilit y. + 0 + 

The s tat istical precision of the D s → K $_{S}$π sample is too lo w to allow for a 3-D correction to A D + s → K 0 $_{S}$π + rec . For this m ode we correct for asymmetries other than A$_{CP}$ w it h an inclusive correction obtained by subtract- ing A D $^{+}$→ K 0 $_{S}$π + re c from A D $^{+}$→ K 0 $_{S}$π + CP after integrating over the ent ir e ( p lab π , cos θ lab π , cos θ CMS D + ) space. The inclusive correct io n is ( − 0 . 34 ± 0 . 18)% where the uncertainty is en- tirely d u e t o the statistical uncertainty of the D + s → φπ + sample. T he value of A D + s → K 0 $_{S}$π + CP is measured to be (+5 . 45 ± 2 . 50)%, where the uncertainty is statistical only. 

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The dominant source of systematic uncertainty in the A D + ( s $_{)}$→ K 0 $_{S}$π + CP measurement is the uncertainty in the A D + s → φπ + rec determination, which originates from the fol- lowing sources: the statistical uncertainty of the selected D + s → φπ + sample (0.18%); the choice of the M ( K $^{+}$K $^{−}$) interval (0.03%); and the choice of binning for the 3- D map of A D + s → φπ + rec (0.03%). Another source is the choice of fitting parameters for the invariant mass dis- 

