Amplitude analysis of the decay $D^0\to K^{-}{K}^{+}{\pi }^0$

Using $385{\mathrm{fb}}^{-1}$ of $e^{+}{e}^{-}$ collisions, we study the amplitudes of the singly Cabibbo-suppressed decay $D^0\to K^{-}{K}^{+}{\pi }^0$. We measure the strong phase difference between the ${\overline{D}}^0$ and $D^0$ decays to $K^{*}(892{)}^{+}{K}^{-}$ to be $-35.5°\pm 1.9°(\mathrm{stat})\pm 2.2°(\mathrm{syst})$, and their amplitude ratio to be $0.599\pm 0.013(\mathrm{stat})\pm 0.011(\mathrm{syst})$. We observe contributions from the $K\pi$ and $K^{-}{K}^{+}$ scalar and vector amplitudes, and analyze their angular moments. We find no evidence for charged $\kappa$, nor for higher spin states. We also perform a partial-wave analysis of the $K^{-}{K}^{+}$ system in a limited mass range.

Figure 1

Dalitz plot for $D^0\to K^{-}{K}^{+}{\pi }^0$ [2] data (a), and the corresponding squared invariant mass projections (b)–(d). The three-body invariant mass of the $D^0$ candidate is constrained to the nominal value. In plots (b)–(d), the dots (with error bars, black) are data points and the solid lines (blue) correspond to the best isobar fit models.

Figure 2

LASS (solid line, blue) and E-791 (dots with error bars) $K\pi$ $S$-wave amplitudes (a), in arbitrary units, and phase (b). The double headed arrow (red) indicates the mass range available in the decay $D^0\to K^{-}{K}^{+}{\pi }^0$.

Figure 3

The phase-space-corrected $K^{-}{K}^{+}$ $S$- and $P$-wave amplitudes, $|S|$ and $|P|$, respectively, in arbitrary units, as functions of the invariant mass. (a) Line shapes for $f_0(980)$ (solid line, blue) and $a_0(980)$ (broken line, blue), derived from Eq. (7). (b) Line shape for $\varphi (1020)$ (solid line, blue). In each plot, solid circles with error bars correspond to values obtained from the model-independent analysis for $|S|$ and $|P|$ using Eq. (8). In (a), the open triangles (red) correspond to values obtained from the decay $D^0\to K^{-}{K}^{+}{\overline{K}}^0$ (see text).

Figure 4

The mass dependence of the spherical harmonic moments of $\cos {\theta }_H$ after efficiency corrections and background subtraction: $K^{+}{\pi }^0$ (columns I, II) and $K^{-}{K}^{+}$ (columns III, IV). The circles with error bars are data points and the curves (red) are derived from the fit functions (see text). For the sake of visibility, we do not show error bars on the curves.

Figure 5

Results of the partial-wave analysis of the $K^{-}{K}^{+}$ system using Eq. (8) described in the text. (a) Cosine of the relative phase ${\theta }_{SP}={\theta }_S-{\theta }_P$, (b) two solutions for ${\theta }_{SP}$, (c) the $P$-wave phase taken from Eqs. (3) and (4) for the $\varphi (1020)$ meson, and (d) the $S$-wave phase derived from the upper solution in (b). Solid bullets are data points, and open circles (blue) and open triangles (red) correspond, respectively, to isobar models I and II. The number of simulated events used for the two models is 10 times larger than data. Errors for quantities from the isobar models arise from Monte Carlo statistical limitations, and differ from errors derived from Eq. (8).