Amplitude analysis and measurement of the time-dependent $CP$ asymmetry of $B^0\to K_S^0{K}_S^0{K}_S^0$ decays

We present the first results on the Dalitz-plot structure and improved measurements of the time-dependent $CP$-violation parameters of the process $B^0\to K_S^0{K}_S^0{K}_S^0$ obtained using $468\times {10}^{6}B\overline{B}$ decays collected with the BABAR detector at the PEP-II asymmetric-energy $B$ factory at SLAC. The Dalitz-plot structure is probed by a time-integrated amplitude analysis that does not distinguish between $B^0$and ${\overline{B}}^0$ decays. We measure the total inclusive branching fraction $\mathcal{B}(B^0\to K_S^0{K}_S^0{K}_S^0)=(6.19\pm 0.48\pm 0.15\pm 0.12)\times {10}^{-6}$, where the first uncertainty is statistical, the second is systematic, and the third represents the Dalitz-plot signal model dependence. We also observe evidence for the intermediate resonant states $f_0(980)$, $f_0(1710)$, and $f_2(2010)$. Their respective product branching fractions are measured to be $({2.70}_{-1.19}^{+1.25}\pm 0.36\pm 1.17)\times {10}^{-6}$, $({0.50}_{-0.24}^{+0.46}\pm 0.04\pm 0.10)\times {10}^{-6}$, and $({0.54}_{-0.20}^{+0.21}\pm 0.03\pm 0.52)\times {10}^{-6}$. Additionally, we determine the mixing-induced $CP$-violation parameters to be $\mathcal{S}=-{0.94}_{-0.21}^{+0.24}\pm 0.06$ and $\mathcal{C}=-0.17\pm 0.18\pm 0.04$, where the first uncertainty is statistical and the second is systematic. These values are in agreement with the standard model expectation. For the first time, we report evidence of $CP$ violation in $B^0\to K_S^0{K}_S^0{K}_S^0$ decays; $CP$ conservation is excluded at 3.8 standard deviations including systematic uncertainties.

Figure 1

Lines of constant helicity angle in the Dalitz plot of $s_{\min }$ versus $s_{\max }$ (left), and the magnitude of the Jacobian (gray scale on the right) mapping $(s_{\min },s_{\max })$ to $(h_{\min },h_{\max })$. For the latter see Eq. (20).

Figure 2

Two-dimensional scans of $-2\Delta \ln \mathcal{L}$ (gray scale) as a function of the mass and the width of an additional resonance. These scans were performed to look for an additional scalar resonance (left) and an additional tensor resonance (right). The baseline model of the scans for additional scalar resonances contains $f_0(980)$, ${\chi }_{c0}$, and NR intermediate states. The baseline model of the scans for additional tensor resonances contains $f_0(980)$, ${\chi }_{c0}$, NR, and $f_0(1710)$ intermediate states. The ellipses indicate the world average parameters [17] for the $f_0(1710)$ and $f_2(2010)$ resonances that are added to the model.

Figure 3

Symmetrized (left) and square (right) DP for MC simulated signal events using the amplitudes obtained from the fit to data. The low population in bins along the edge of the symmetrized DP is due to the fact that the phase-space boundaries do not coincide with the histogram bin boundaries.

Figure 4

${}_s\mathcal{P}lots$ (points with error bars) and PDFs (histograms) of the discriminating variables: $m_{\mathrm{ES}}$ (top), $\Delta E$ (middle), and NN (bottom), for signal events (left) and continuum events (right). Below each bin are shown the residuals, normalized in error units. The horizontal dotted and full lines mark the one and two standard deviation levels, respectively.

Figure 5

Projections onto $\sqrt{s_{\min }}$ (left) and $\sqrt{s_{\max }}$ (right). On-resonance data are shown as points with error bars while the dashed (dotted) histogram represents the signal (continuum) component. The solid-line histogram is the total PDF. Below each bin are shown the residuals, normalized in error units. The horizontal dotted and full lines mark the one and two standard deviation levels, respectively.

Figure 6

One-dimensional scans of $-2\Delta \ln \mathcal{L}$ as a function of magnitudes (left) and phases (right) of the resonances $f_0(980)$, $f_0(1710)$, $f_2(2010)$, and ${\chi }_{c0}$ (top to bottom). The horizontal dashed lines mark the one and two standard deviation levels.

Figure 7

Signal ${}_s\mathcal{P}lots$ (points with error bars) and PDFs (histograms) of $\Delta t$ (left) and the derived asymmetry (right) for the $B^0\to 3K_S^0({\pi }^{+}{\pi }^{-})$ submode (top), the $B^0\to 2K_S^0({\pi }^{+}{\pi }^{-})K_S^0({\pi }^0{\pi }^0)$ submode (middle), and the combined fit (bottom). In the $\Delta t$ distributions on the left-hand side, points marked with $\times$ and solid lines correspond to decays where $B_{\mathrm{tag}}$ is a $B^0$ meson; points marked with $\circ$ and dashed lines correspond to decays where $B_{\mathrm{tag}}$ is a ${\overline{B}}^0$ meson. Points of the asymmetry ${}_s\mathcal{P}lots$ that are outside the range of a figure are marked by arrows.

Figure 8

Signal ${}_s\mathcal{P}lots$ (points with error bars) and PDFs (histograms) of the discriminating variables: $m_{\mathrm{ES}}$ (top), $\Delta E$ (middle), and the NN output (bottom) for the $B^0\to 3K_S^0({\pi }^{+}{\pi }^{-})$ submode (left) and for the $B^0\to 2K_S^0({\pi }^{+}{\pi }^{-})K_S^0({\pi }^0{\pi }^0)$ submode (right). Below each bin are shown the residuals, normalized in error units. The horizontal dotted and full lines mark the one and two standard deviation levels, respectively.

Figure 9

Continuum ${}_s\mathcal{P}lots$ (points with error bars) and PDFs (histograms) of $m_{\mathrm{ES}}$, $\Delta E$, the NN output, and $\Delta t$ (top to bottom). Plots on the left-hand side correspond to the $B^0\to 3K_S^0({\pi }^{+}{\pi }^{-})$ submode, and on the right-hand side to the $B^0\to 2K_S^0({\pi }^{+}{\pi }^{-})K_S^0({\pi }^0{\pi }^0)$ submode. In the $\Delta t$ distributions, points marked with $\times$ and solid lines correspond to decays where $B_{\mathrm{tag}}$ is a $B^0$ meson; points marked with $\circ$ and dashed lines correspond to decays where $B_{\mathrm{tag}}$ is a ${\overline{B}}^0$ meson. Below each bin are shown the residuals, normalized in error units. The horizontal dotted and full lines mark the one and two standard deviation levels, respectively.

Figure 10

One-dimensional statistical scan of $-2\Delta \ln \mathcal{L}$ as a function of $\mathcal{S}$ (left) and the two-dimensional scan, including systematic uncertainty, as a function of $\mathcal{S}$ and $\mathcal{C}$ (right). In the left-hand plot, red points marked with $\times$ correspond to the $B^0\to 3K_S^0({\pi }^{+}{\pi }^{-})$ submode, blue points marked with $\circ$ to the $B^0\to 2K_S^0({\pi }^{+}{\pi }^{-})K_S^0({\pi }^0{\pi }^0)$ submode, and black points marked with $*$ to the combined fit. In the right-hand plot, the gray scale is given in units of $\sqrt{-2\Delta \ln \mathcal{L}}$. The result of the BABAR analyses of $B^0\to c\overline{c}{K}^{(*)}$ decays [27] is indicated as a white ellipse and the physical boundary (${\mathcal{S}}^2+{\mathcal{C}}^2\le 1$) is marked as a gray line. The scan appears to be trimmed on the lower left since the PDF becomes negative outside the physical region (i.e., the white region does not indicate that the scan flattens out at $5\sigma$).