Time-integrated and time-dependent angular analyses of $B\to J/\psi K\pi$: A measurement of $\cos 2\beta$ with no sign ambiguity from strong phases

We present results on $B\to J/\psi K\pi$ decays using $e^{+}{e}^{-}$annihilation data collected with the BABAR detector at the ${\rm Y}(4S)$ resonance. The detector is located at the PEP-II asymmetric-energy storage ring facility at the Stanford Linear Accelerator Center. Using approximately $88\times {10}^6$ $B\overline{B}$ pairs, we measure the decay amplitudes for the flavor eigenmodes and observe strong-phase differences indicative of final-state interactions with a significance of 7.6 standard deviations. We use the interference between the $K\pi$ $S$-wave and $P$-wave amplitudes in the region of the $K^{*}(892)$ to resolve the ambiguity in the determination of these strong phases. We then perform an ambiguity-free measurement of $\cos 2\beta$ using the angular and time-dependent asymmetry in $B\to J/\psi K^{*0}(K_S^0{\pi }^0)$ decays. With $\sin 2\beta$ fixed at its measured value and $\cos 2\beta$ treated as an independent parameter, we find $\cos 2\beta ={2.72}_{-0.79}^{+0.50}(\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t})\pm 0.27(\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t})$, determining the sign of $\cos 2\beta$ to be positive at 86% C.L.

Figure 1

Definition of the transversity angles $({\theta }_{K^{*}},{\theta }_{tr},{\varphi }_{tr})$ and coordinate system $({\widehat{\mathit{x}}}_{tr},{\widehat{\mathit{y}}}_{tr},{\widehat{\mathit{z}}}_{tr})$. The direction opposite to the $B$ meson momentum in the $J/\psi$ rest frame is ${\widehat{\mathit{x}}}_{tr}$; ${\widehat{\mathit{y}}}_{tr}$ is perpendicular to ${\widehat{\mathit{x}}}_{tr}$ in the plane that contains ${\widehat{\mathit{x}}}_{tr}$ and ${\overrightarrow{\mathit{p}}}_K$, chosen so ${\overrightarrow{\mathit{p}}}_K·{\widehat{\mathit{y}}}_{tr}>0$; ${\widehat{\mathit{z}}}_{tr}={\widehat{\mathit{x}}}_{tr}\times {\widehat{\mathit{y}}}_{tr}$. The helicity angle ${\theta }_{K^{*}}$ of the $K^{*}$decay is the angle between the direction opposite to the $B$ meson flight direction and the kaon momentum, in the $K^{*}$ rest frame. Finally, ${\theta }_{tr}$ and ${\varphi }_{tr}$ are the polar and azimuthal angle of the positive lepton defined in the $J/\psi$ rest frame.

Figure 2

(a) The distribution of $\mathcal{P}(\omega ;\mathit{A})/\mathcal{A}(\omega ;\mathit{A})$, (b) $\mathcal{S}(\omega ;\mathit{A})/\mathcal{A}(\omega ;\mathit{A})$ and (c) $\mathcal{C}(\omega ;\mathit{A})/\mathcal{A}(\omega ;\mathit{A})$, where $\mathcal{A}$, $\mathcal{P}$, $\mathcal{S}$ and $\mathcal{C}$ are defined by Eq. (14), for a set of events generated according to the amplitudes $\mathit{A}$ corresponding to the BABAR values in Table 1.

Figure 3

The $m_{\mathrm{E}\mathrm{S}}$ distributions for the $\Delta E$ intervals described in the text with overlaid Gaussian and ARGUS fit functions, for $J/\psi K\pi$ candidates in data.

Figure 4

Angular distributions. Histogram: Inclusive $J/\psi$ MC sample $(p_{J/\psi }^{*}>1.3\mathrm{G}\mathrm{e}\mathrm{V}/c)$. Points: Data. The spectra are acceptance-corrected, background-subtracted, and normalized to the estimated yields (Table 2). The visible forward-backward discrepancy in the $\cos {\theta }_{K^{*}}$ distribution is due to the $K\pi$ $S$-wave amplitude present in the data, and absent in the MC sample. The related systematic uncertainties in the measurements of the decay amplitudes are listed in line seven of Table 5.

Figure 5

The background-subtracted $K\pi$ invariant mass distributions for $J/\psi K\pi$ candidates in data.

Figure 6

Measured values of (a) $⟨f_8⟩·2/{\kappa }_8$, (b) $⟨f_9⟩·2/{\kappa }_9$ and (c) $⟨f_{10}⟩·2/{\kappa }_{10}$, defined in Eq. (32), as a function of $m_{K\pi }$, for the $J/\psi K^{\pm }{\pi }^{\mp }$ candidates in data. The three distributions show a clear variation near the $K^{*}(892)$ region.

Figure 7

(a) The $P$-wave intensity times number of events;, i.e., $n_S{cos}^2\lambda$; (b) the $S$-wave intensity times number of events;, i.e., $n_S{sin}^2\lambda$. These are the numbers of events that would be observed in each interval for the amplitude under consideration, if it were the only amplitude. The fraction of $S$-wave intensity integrated over the range $0.8<m_{K\pi }<1.0\mathrm{G}\mathrm{e}\mathrm{V}/c^2$ is found to be $(7.3\pm 1.8)\%$. (c) The evolution of $\gamma /\pi$ with $m_{K\pi }$, for the two sets of strong phases. The mirror symmetry described by Eq. (10) is clearly visible as $\gamma \leftrightarrow 2\pi -\gamma$. The error bars represent the statistical uncertainty in the fit of ($\lambda ,\gamma$) in the $m_{K\pi }$ interval considered. All distributions are from fits to $J/\psi K^{\pm }{\pi }^{\mp }$ candidates in data.

Figure 8

(a) Comparison of the $K^{\pm }{\pi }^{\mp }$ $P$-wave intensity with a Breit-Wigner lineshape, including a centrifugal barrier factor, with world average parameters for $K^{*}(892)$ [34]. The lineshape is integrated in each mass interval (star markers) and compared with the measured intensity in that interval, after a minimum${\chi }^2$ fit of the overall normalization to the data in the $0.8\mathrm{G}\mathrm{e}\mathrm{V}/c^2$–$1.3\mathrm{G}\mathrm{e}\mathrm{V}/c^2$ mass range. The ${\chi }^2$ per degree of freedom is 0.86. (b) Pull (i.e. difference of measured and expected intensities, normalized to the uncertainty) in each mass interval.

Figure 9

Comparison of the variation of $\gamma ={\delta }_S-{\delta }_0$ with $m_{K\pi }$ for the $J/\psi K^{\pm }{\pi }^{\mp }$ events, for “Solution I” (open points, Eq. (28)) and “Solution II” (full points, Eq. (29)), with that measured by the LASS experiment [22, 39, 40] (diamond markers).

Figure 10

Contour plots in the $\cos 2\beta$, $\sin 2\beta$ plane. The triangle denotes the result of the fit. The error bars show the statistical uncertainty and the quadratic sum of the statistical and systematic uncertainties. The star denotes the result of the fit with $\sin 2\beta$ fixed at $\sin 2\beta =0.731$ [4]. The value of $\sin 2\beta$ of Ref. [4] and its uncertainties are represented as dashed horizontal lines. The $n\sigma (n=1,2,3)$ contour corresponds to a decrease of $0.5n^2$ in the log-likelihood with respect to the maximum value. The unit circle $({cos}^{2}2\beta +{sin}^{2}2\beta =1)$ on which the true values must lie is also shown.

Figure 11

The distribution of $\Delta t$ for events in the signal region, for (a) $B^0$ and (b) ${\overline{B}}^0$ tags with the fit result (full curve) overlaid. In (c) we show the raw asymmetry in the number of $B^0$ and ${\overline{B}}^0$ tags in the signal region, $(N_{B^0}-N_{{\overline{B}}^0})/(N_{B^0}+N_{{\overline{B}}^0})$, for data, with the fit result (full curve) overlaid. Note that above distributions are not sensitive to $\cos 2\beta$ since this dependence vanishes when integrated over the angular variables.

Figure 12

The moment of $\mathcal{C}$ as a function of $\Delta t$. The overlaid curve corresponds to the fit results (Table 11). The dashed curve corresponds to $\cos 2\beta =+\sqrt{1-{sin}^{2}2{\beta }_0}=+0.68$ [4], the dotted one to the nonstandard solution $\cos 2\beta =-0.68$.

Figure 13

The negative logarithm of the likelihood as a function of $\cos 2\beta$. Continuous line: $\sin 2\beta$ is a free parameter. Dashed line: $\sin 2\beta$ is fixed at $\sin 2{\beta }_0=0.731$ [4].

Figure 14

The distribution of the values of $\cos 2\beta$ ((a) and (c)), and of the statistical uncertainties ((b) and (d)), obtained in 2000 simulated experiments, each based on a sample of the same size as the data;, i.e., 104 events. These are taken from the parametrized MC sample mentioned above, with the generated $\cos 2\beta$ value $+0.68$. In (a) and (b) $\sin 2\beta$ is also free in the fit. In (c) and (d) $\sin 2\beta$ is fixed to the world average. The vertical arrows show the positions of the values obtained from the data.

Figure 15

(a) The $\cos 2\beta$ distribution for 2000 simulated experiments of the same size as the data sample, where the generated values for $\sin 2\beta$ and $\cos 2\beta$ are $+0.731$ and $+0.68$, respectively, [same as Fig. 14]. An unbinned likelihood fit is performed to this distribution, using the sum of two Gaussian functions. The fit result is shown on the plot (full line) with the individual Gaussian contributions (dashed lines). The long vertical arrows show the $\cos 2\beta$values $\pm 2.72$. The small arrows indicate the extent of systematic uncertainties. (b) and (c) zoom on the $\cos 2\beta =\pm 2.72$ regions of Fig. 15. The densities of points at $+2.72$ and $-2.72$ are used to discriminate between the $\cos 2\beta =\pm \sqrt{1-{sin}^{2}2{\beta }_0}=\pm 0.68$ hypotheses, as explained in the text.

Figure 16

Representation in the complex plane of the measured transversity amplitudes $(A_0,A_{\parallel },A_{\perp })$ (Eq. (67)) and the equivalent helicity amplitudes $(H_0,H_{-1},H_{+1})$ obtained using Eq. (69). The values for $H_{+}$ and $H_{-}$ are quoted in Table 12.

Figure 17

Root mean square (RMS) of the pull distributions as a function of signal purity (defined in the $5.2\mathrm{G}\mathrm{e}\mathrm{V}/c^2$–$5.3\mathrm{G}\mathrm{e}\mathrm{V}/c^2$$m_{\mathrm{E}\mathrm{S}}$ range), for the fitted parameters ${\theta }_A,{\varphi }_A,{\delta }_{\parallel }$, and ${\delta }_{\perp }$. Open symbols denote the RMS of the pulls computed with the uncertainties taken directly from Minuit. Closed symbols denote the uncertainties computed according to Eq. (a1).