Measurements of branching fractions and $CP$ asymmetries and studies of angular distributions for $B\to \varphi \varphi K$ decays

We present branching fraction and $CP$ asymmetry measurements as well as angular studies of $B\to \varphi \varphi K$ decays using $464\times {10}^6$ $B\overline{B}$ events collected by the BABAR experiment. The branching fractions are measured in the $\varphi \varphi$ invariant mass range below the ${\eta }_c$ resonance ($m_{\varphi \varphi }<2.85\mathrm{GeV}$). We find $\mathcal{B}(B^{+}\to \varphi \varphi K^{+})=(5.6\pm 0.5\pm 0.3)\times {10}^{-6}$ and $\mathcal{B}(B^0\to \varphi \varphi K^0)=(4.5\pm 0.8\pm 0.3)\times {10}^{-6}$, where the first uncertainty is statistical and the second systematic. The measured direct $CP$ asymmetries for the $B^{\pm }$ decays are $A_{CP}=-0.10\pm 0.08\pm 0.02$ below the ${\eta }_c$ threshold ($m_{\varphi \varphi }<2.85\mathrm{GeV}$) and $A_{CP}=0.09\pm 0.10\pm 0.02$ in the ${\eta }_c$ resonance region ($m_{\varphi \varphi }$ in [2.94, 3.02] GeV). Angular distributions are consistent with $J^P=0^{-}$ in the ${\eta }_c$ resonance region and favor $J^P=0^{+}$ below the ${\eta }_c$ resonance.

Figure 1

Distributions of the Fisher discriminant $\mathcal{F}$ for $B\to \varphi \varphi K$ signal (solid black) and $e^{+}{e}^{-}\to q\overline{q}$ continuum (dashed red) Monte Carlo simulation.

Figure 2

Data distribution of $B^{+}\to 5K$ events in the $m_{\varphi 2}$ vs $m_{\varphi 1}$ plane for events with $m_{\varphi \varphi }<2.85\mathrm{GeV}$. To enhance the $B\to 5K$ signal for the figure, we have required $m_{\mathrm{ES}}>5.27\mathrm{GeV}$, $|\Delta E|<0.040\mathrm{GeV}$, and $\mathcal{F}<0.0$. The efficiency of these additional requirements, relative to the nominal selection, is about 70% for the signal. The inset shows the definition of the five zones.

Figure 3

Results of fitting the $B^{+}\to \varphi \varphi K^{+}$ sample for $m_{\varphi \varphi }<2.85\mathrm{GeV}$. The dashed red curve is the sum of the combinatoric and peaking background components. The solid blue curve is for all components. A requirement on a likelihood ratio based on all fit variables except the one plotted is made to reject most of the background. The likelihood ratio requirements are about 84% efficient for the signal.

Figure 4

Results of fitting the $B^0\to \varphi \varphi K_S^0$ sample for $m_{\varphi \varphi }<2.85\mathrm{GeV}$. The dashed red curve is the sum of the combinatoric and peaking background components. The solid blue curve is for all components. A requirement on a likelihood ratio based on all fit variables except the one plotted is made to reject most of the background. The likelihood ratio requirements are about 84% efficient for the signal.

Figure 5

Fitted $B^{+}\to \varphi \varphi K^{+}$ yield as a function of $m_{\varphi \varphi }$. Each point shows the results of a maximum likelihood fit of the events in that bin. The inset is the same data with an expanded vertical range to show the shape of the nonresonant component more clearly. The yield has been divided by the bin width and scaled by 0.027 GeV, which is the bin width of the three bins in the ${\eta }_c$ resonance region ([2.94, 3.02] GeV and dashed vertical lines in the inset). The two narrow bins above the ${\eta }_c$ are centered on the ${\chi }_{c0}$ (bin range [3.400, 3.430] GeV) and the ${\chi }_{c2}$ (bin range [3.552, 3.560] GeV).

Figure 6

Histograms (top) and weighted distributions (bottom) of $m_{\varphi \varphi }$ for the signal region (solid points) and data sideband selection (dashed with open diamonds) defined in the text. The sideband distributions have been normalized to the expected level of combinatoric background remaining after the signal region selection. Events in the bottom distribution were weighted by $P_2(\cos {\theta }_1)\mathrm{Re}[Y_2^2({\theta }_2,\chi )]$ which projects out the $J^P=0^{-}$ component. The yield has been divided by the bin width and scaled by 0.211 GeV, which is the width of the bin covering the ${\eta }_c$ resonance ([2.875, 3.086] GeV).

Figure 7

Background-subtracted angular distributions in the ${\eta }_c$ resonance region ($m_{\varphi \varphi }$ in [2.94, 3.02] GeV for the top row) and below the ${\eta }_c$ resonance ($m_{\varphi \varphi }<2.85\mathrm{GeV}$ for the bottom row). The reconstruction and selection efficiency is uniform in $\chi$ (left) and $\cos \theta$ (center), but dependent on $|\cos {\theta }_{\varphi \varphi }|$ (right), so the right column has been efficiency corrected. The red dashed line shows a least-${\chi }^2$ fit of the points to a uniform distribution while the solid black curve shows a fit to the expectation for a $J^P=0^{-}$ state decaying to $\varphi \varphi$. The fit ${\chi }^2/N_{\mathrm{dof}}$ and ${\chi }^2$ probabilities are given in Table .