Measurement of branching fractions of $B$ decays to $K_1(1270)\pi$ and $K_1(1400)\pi$ and determination of the CKM angle $\alpha$ from $B^0\to a_1(1260{)}^{\pm }{\pi }^{\mp }$

We report measurements of the branching fractions of neutral and charged $B$ meson decays to final states containing a $K_1(1270)$ or $K_1(1400)$ meson and a charged pion. The data, collected with the BABAR detector at the SLAC National Accelerator Laboratory, correspond to $454\times {10}^6$ $B\overline{B}$ pairs produced in $e^{+}{e}^{-}$ annihilation. We measure the branching fractions $\mathcal{B}(B^0\to K_1(1270{)}^{+}{\pi }^{-}+K_1(1400{)}^{+}{\pi }^{-})={3.1}_{-0.7}^{+0.8}\times {10}^{-5}$ and $\mathcal{B}(B^{+}\to K_1(1270{)}^0{\pi }^{+}+K_1(1400{)}^0{\pi }^{+})={2.9}_{-1.7}^{+2.9}\times {10}^{-5}$ ($<8.2\times {10}^{-5}$ at 90% confidence level), where the errors are statistical and systematic combined. The $B^0$ decay mode is observed with a significance of $7.5\sigma$, while a significance of $3.2\sigma$ is obtained for the $B^{+}$ decay mode. Based on these results, we estimate the weak phase $\alpha =(79\pm 7\pm 11)°$ from the time-dependent $CP$ asymmetries in $B^0\to a_1(1260{)}^{\pm }{\pi }^{\mp }$ decays.

Figure 1

Results of the fit to the $0\le |t^{\prime }|\le 0.05{\mathrm{GeV}}^2$ WA3 data. Intensity (left) and phase relative to the $(K^{*}(892)\pi {)}_S$ amplitude (right) for the (a) $(K^{*}(892)\pi {)}_S$, (b, c) $\rho K$, (d, e) $K_0^{*}(1430)\pi$, (f, g) $f_0(1370)K$, and (h, i) $(K^{*}(892)\pi {)}_D$ channels. The points represent the data, the solid lines the total fit function, and the dashed lines the contribution from the background.

Figure 2

Definition of (a) the coordinate axes in the $K_1$ rest frame, (b) the angles $\Theta$ and $\Phi$ in the $K_1$ rest frame, and (c) the angle $\beta$ in the rest frame of the $X_{s,d}$ intermediate resonance.

Figure 3

sPlot projections of signal onto $m_{\mathrm{ES}}$ (left), $\Delta E$ (center), and $m_{K\pi \pi }$ (right) for $B^0$ class 1 (top), $B^0$ class 2 (middle), and $B^{+}$ class 1 (bottom) events: the points show the sums of the signal weights obtained from on-resonance data. For $m_{\mathrm{ES}}$ and $\Delta E$ the solid line is the signal fit function. For $m_{K\pi \pi }$ the solid line is the sum of the fit functions of the decay modes $K_1(1270)\pi +K_1(1400)\pi$ (dashed), $K^{*}(1410)\pi$ (dash-dotted), and $K^{*}(892)\pi \pi$ (dotted), and the points are obtained without using information about resonances in the fit, i.e., we use only the $m_{\mathrm{ES}}$, $\Delta E$, and $\mathcal{F}$ variables.

Figure 4

sPlot projections of combinatorial background onto $m_{\mathrm{ES}}$ (left), $\Delta E$ (center), and $m_{K\pi \pi }$ (right) for $B^0$ class 1 (top), $B^0$ class 2 (middle), and $B^{+}$ class 1 (bottom) events: the points show the sums of the combinatorial background weights obtained from on-resonance data. The solid line is the combinatorial background fit function. For $m_{K\pi \pi }$ the points are obtained without using information about resonances in the fit, i.e., we use only the $m_{\mathrm{ES}}$, $\Delta E$, and $\mathcal{F}$ variables.

Figure 5

(a, b) $-\ln \mathcal{L}$ scan (systematics not included) in the production parameters $\vartheta$ and $\varphi$ for the (a) $B^0$ and (b) $B^{+}$ modes. The cross in (a) indicates the position of the absolute minimum in the $-\ln \mathcal{L}$ scan. A second, local minimum is indicated by a star and corresponds to an increase in $\Delta (-\ln \mathcal{L})$ of 2.7 with respect to the absolute minimum. (c, d) Contours for the $B\to K_1(1270)\pi +K_1(1400)\pi$ branching fraction (in units of ${10}^{-6}$) extracted from the ML fit for the (c) $B^0$ and (d) $B^{+}$ modes. (e, f) Contours for the statistical error (in units of ${10}^{-6}$) on the $B\to K_1(1270)\pi +K_1(1400)\pi$ branching fraction for the (e) $B^0$ and (f) $B^{+}$ modes.

Figure 6

(a, b) 68% (dark shaded zone) and 90% (light shaded zone) probability regions for $\vartheta$ and $\varphi$ for the (a) $B^0$ and (b) $B^{+}$ modes.