Observation of $D_{s1}(2536{)}^{+}\to D^{+}{\pi }^{-}{K}^{+}$ and angular decomposition of $D_{s1}(2536{)}^{+}\to D^{*+}{K}_S^0$

Using $462{\mathrm{fb}}^{-1}$ of $e^{+}{e}^{-}$ annihilation data recorded by the Belle detector, we report the first observation of the decay $D_{s1}(2536{)}^{+}\to D^{+}{\pi }^{-}{K}^{+}$. The ratio of branching fractions $\frac{\mathcal{B}(D_{s1}(2536{)}^{+}\to D^{+}{\pi }^{-}{K}^{+})}{\mathcal{B}(D_{s1}(2536{)}^{+}\to D^{*+}{K}^0)}$ is measured to be $(3.27\pm 0.18\pm 0.37)\%$. We also study the angular distributions in the $D_{s1}(2536{)}^{+}\to D^{*+}{K}_S^0$ decay and measure the ratio of $D$- and $S$-wave amplitudes. The $S$-wave dominates, with a partial width of ${\Gamma }_S/{\Gamma }_{\mathrm{total}}=0.72\pm 0.05\pm 0.01$.

Figure 1

Observed invariant mass spectra of selected $D^{+}{\pi }^{-}{K}^{+}$ (top) and $D^{*+}{K}_S^0$ (bottom) combinations. The hatched histogram in the top plot shows the corresponding spectrum of wrong sign $D^{+}{\pi }^{+}{K}^{-}$ combinations. The fit is described in the text. The fit results are listed in Table 1.

Figure 2

$D^{+}$ mass spectrum for the $D_{s1}(2536{)}^{+}$ signal band ($|\Delta M_{D^{+}{\pi }^{-}{K}^{+}}|<5\mathrm{MeV}/c^2$, open histogram in the top plot) and the sidebands ($10<|\Delta M_{D^{+}{\pi }^{-}{K}^{+}}|<20\mathrm{MeV}/c^2$, normalized to the signal interval, hatched histogram). $\Delta M_{D^{+}{\pi }^{-}{K}^{+}}$ is measured relative to the peak position in the top plot of Fig. 1. The bottom plot shows the sideband subtracted distribution. The solid curve shows the results of the fit described in the text.

Figure 3

Normalized mass spectra of $D^{+}{\pi }^{-}$ (top) and $K^{+}{\pi }^{-}$ (bottom) pairs from $D_{s1}(2536{)}^{+}\to D^{+}{\pi }^{-}{K}^{+}$ decay obtained from fits to the $D^{+}{\pi }^{-}{K}^{+}$ mass distributions in different $D^{+}{\pi }^{-}$ or $K^{+}{\pi }^{-}$ mass bins. The dashed histograms show the corresponding MC distributions for $D_{s1}(2536{)}^{+}\to D^{+}{\pi }^{-}{K}^{+}$ decays simulated according to a phase space distribution.

Figure 4

Definitions of the angles $\alpha$, $\beta$, and $\gamma$. The first two are measured in the $D_{s1}(2536{)}^{+}$ rest frame, the third in the $D^{*+}$ frame. “Boost” refers to the direction of the $e^{+}{e}^{-}$ center of mass in the $D_{s1}(2536{)}^{+}$ rest frame.

Figure 5

Background subtracted, efficiency corrected, and normalized one-dimensional $\cos \alpha$, $\beta$, $\cos \gamma$ angular distributions. Projected results of the three-dimensional fit are shown by the solid curves.

Figure 6

$D_{s1}(2536{)}^{+}$ recoil mass spectrum (top), probability ${\rho }_{00}$ that $D_{s1}(2536{)}^{+}$ helicity is zero (bottom). The decrease at low values in the bottom plot can be attributed to the contribution of two-body $e^{+}{e}^{-}\to D_{s1}(2536{)}^{+}X$ reactions.