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

The first flavor-tagged amplitude analysis of the decay $D^0$ to the self-conjugate final state $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ is presented. Approximately 3000 signal decays are selected from data acquired by the CLEO II.V, CLEO III, and CLEO-c detectors. The three most significant amplitudes, which contribute to the model that best fits the data, are $\varphi {\rho }^0$, $K_1(1270{)}^{\pm }{K}^{\mp }$, and nonresonant $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$. Separate amplitude analyses of $D^0$ and ${\overline{D}}^0$ candidates indicate no $CP$ violation among the amplitudes at the level of 5% to 30% depending on the mode. In addition, the sensitivity to the $CP$-violating parameter $\gamma /{\varphi }_3$ from a sample of 2000 $B^{+}\to {\tilde{D}}^0(K^{+}{K}^{-}{\pi }^{+}{\pi }^{-})K^{+}$ decays, where $\tilde{D}$ is a $D^0$ or ${\overline{D}}^0$, collected at LHCb or a future flavor facility, is estimated to be $(11.3\pm 0.3)°$ using the favored model.

Figure 1

(a) $\Delta m$ (after $m_D$ selection) and (b) $m_D$ (after $\Delta m$ selection) distributions of CLEO II.V $D^0$ candidates.

Figure 2

(a) $\Delta m$ (after $m_D$ selection) and (b) $m_D$ (after $\Delta m$ selection) distributions of CLEO III $D^0$ candidates.

Figure 3

(a) $m_{bc}$ and (b) $\Delta E$ distributions of events passing the kaon-tagged $D^0\to K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ selection in the CLEO-c 3770 data set. Each distribution is plotted after applying the selection cut on the other variable.

Figure 4

Distributions of (a) $m_{bc}$ (after $\Delta E-\Delta E_{\mathrm{sig}}$ selection) and (b) $\Delta E-\Delta E_{\mathrm{sig}}$ (after $m_{bc}$ selection) for events passing the kaon-tagged $D^0\to K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ selection in the CLEO-c 4170 data set.

Figure 5

The (a) $s_{12}$, (b) $s_{13}$, (c) $s_{14}$, (d) $s_{23}$, (e) $s_{24}$, and (f) $s_{34}$ projections for all flavor-tagged data (points with error bars) with the best fit (solid line) superimposed. The indices correspond to $K^{+}=1$, $K^{-}=2$, ${\pi }^{+}=3$, and ${\pi }^{-}=4$. The contributions from mistag (filled region) and background plus mistag (dashed line) are also shown.

Figure 6

Distribution of the three-body invariant-mass observables: The (a) $s_{123}$, (b) $s_{124}$, (c) $s_{134}$, and (d) $s_{234}$ projections for all flavor-tagged data (points with error bars) with the best fit (solid line) superimposed. The contributions from mistag (filled region) and background plus mistag (dashed line) are also shown.

Figure 7

Fitted $a_i$ (points) distribution for the ensemble test of 200 simulated data sets. Also shown is the generated value of $a_i$ (asterisk). The labels correspond to (a) $K_1(1270{)}^{-}({\overline{K}}^{*0}{\pi }^{-})K^{+}$, (b) $K_1(1270{)}^{+}({\rho }^0{K}^{+})K^{-}$, (c) $K_1(1270{)}^{-}({\rho }^0{K}^{-})K^{+}$, (d) $K^{*}(1410{)}^{+}(K^{*0}{\pi }^{+})K^{-}$, (e) $K^{*}(1410{)}^{-}({\overline{K}}^{*0}{\pi }^{-})K^{+}$, (f) $K^{*0}{\overline{K}}^{*0}$ $S$ wave, (g) $\varphi {\rho }^0$ $S$ wave, (h) $\varphi {\rho }^0$ $D$ wave, (i) $\varphi \{{\pi }^{+}{\pi }^{-}{\}}_S$, and (j) $\{K^{-}{\pi }^{+}{\}}_P\{K^{+}{\pi }^{-}{\}}_S$. The values of $\mathrm{Re}(a_i)$ and $\mathrm{Im}(a_i)$ are scaled by 0.1 for the $\{K^{-}{\pi }^{+}{\}}_P\{K^{+}{\pi }^{-}{\}}_S$ amplitude.