Amplitude analysis of the $B^{+}\to D^{+}{D}^{-}{K}^{+}$ decay

Results are reported from an amplitude analysis of the $B^{+}\to D^{+}{D}^{-}{K}^{+}$ decay. The analysis is carried out using LHCb proton-proton collision data taken at $\sqrt{s}=7$, 8, and 13 TeV, corresponding to a total integrated luminosity of $9{\mathrm{fb}}^{-1}$. In order to obtain a good description of the data, it is found to be necessary to include new spin-0 and spin-1 resonances in the $D^{-}{K}^{+}$ channel with masses around $2.9\mathrm{GeV}/c^2$, and a new spin-0 charmonium resonance in proximity to the spin-2 ${\chi }_{c2}(3930)$ state.

Figure 1

Invariant-mass distributions for the selected candidates for the $D$ meson having (a) the opposite and (b) the same charge, $Q$, as the $B$ meson, and in the two-dimensional plane showing the two invariant masses in (c) Run 1 and (d) Run 2 data. In (c) and (d) the blue rectangles correspond to regions of charmless background and the green and red where both single-charm and charmless processes contribute. The magenta rectangle indicates the signal region.

Figure 2

Invariant-mass distribution for $B$ candidates with the results of the fit superimposed, where the signal component is indicated in red and background (barely visible) in blue.

Figure 3

Efficiency maps for (a),(b) Run 1 and (c),(d) Run 2, where the variation as a function of position in the (a),(c) standard Dalitz plot and (b),(d) SDP are shown. The $z$-axis scale is arbitrary as the absolute efficiency does not affect the analysis.

Figure 4

Visualization of the sideband candidates in the (a),(c) standard Dalitz plot and (b),(d) derived background models in the SDP for (a),(b) Run 1 and (c),(d) Run 2 data.

Figure 5

Run 1 data entering the amplitude fit shown in the Dalitz plot (a) and its projection onto the invariant-mass squared for each of the three pairs (b),(c),(d) of the final-state particles.

Figure 6

Run 2 data entering the amplitude fit shown in the Dalitz plot (a) and its projection onto the invariant-mass squared for each of the three pairs (b),(c),(d) of the final-state particles.

Figure 7

Comparisons of the invariant-mass distributions (a),(b),(c) of $B^{+}\to D^{+}{D}^{-}{K}^{+}$ candidates to the fit projections without any resonant component in the $D^{-}{K}^{+}$ channel. The total fit function (solid black line) and contributions from individual components (nonsolid colored lines) are shown as detailed in the legend.

Figure 8

Normalized residual between the data and the model excluding any $D^{-}{K}^{+}$ components shown across the Dalitz plot with a minimum of 20 data entries in each bin.

Figure 9

Comparison of the $m(D^{-}{K}^{+})$ distribution and the fit projection for a model excluding any $D^{-}{K}^{+}$ resonance, after requiring $m(D^{+}{D}^{-})>4\mathrm{GeV}/c^2$ to suppress reflections from charmonium resonances. The different components are shown as indicated in the legend of Fig. 7.

Figure 10

Comparisons of the invariant-mass distributions (a),(b),(c) of $B^{+}\to D^{+}{D}^{-}{K}^{+}$ candidates in the data to the fit projection of the baseline model. The total fit function and contributions from individual components are shown as detailed in the legend.

Figure 11

Comparison of the data and fit projection in the ${\chi }_{cJ}(3930)$ region shown for the (a) $D^{+}{D}^{-}$ invariant-mass squared and (b) helicity angle. The different components are shown as indicated in the legend of Fig. 10.

Figure 12

Comparison of the data and the fit projection of the baseline model, for (a) the $D^{-}{K}^{+}$ invariant-mass distribution requiring $m(D^{+}{D}^{-})>4\mathrm{GeV}/c^2$ to suppress reflections from charmonium resonances and (b) helicity angle in the region $2.75\mathrm{GeV}/c^2<m(D^{-}{K}^{+})<3.05\mathrm{GeV}/c^2$. The different components are shown as indicated in the legend of Fig. 10.

Figure 13

Normalized residual between the data and the baseline model including $D^{-}{K}^{+}$ resonances, shown across the Dalitz plot with a minimum of 20 entries in each bin.

Figure 14

Comparison of the data and fit projection in the region of the $\psi (3770)$ states shown for the $D^{+}{D}^{-}$ (a) invariant-mass squared and (b) helicity angle. The different components are shown as indicated in the legend of Fig. 10.

Figure 15

Distributions of the test-statistic $t$ in ensembles of pseudoexperiments generated according to various hypotheses and compared to values found in the data (indicated by dashed vertical lines). In (a), the $H_0$ hypothesis is a model fit to the data without $D^{-}{K}^{+}$ resonances. In (b), (c), and (d) plots, the $H_0$ hypothesis assumes a single ${\chi }_{cJ}(3930)$ state, which has spin-0, spin-1, and spin-2, respectively.

Figure 16

Division of the square Dalitz plot in slices of invariant-mass squared. The binning is used for (top left) the $\cos (\theta (D^{+}{D}^{-}))$ distribution, (top right) the $\cos (\theta (D^{-}{K}^{+}))$ distribution, and (lower) the $\cos (\theta (D^{+}{K}^{+}))$ distribution.

Figure 17

Helicity-angle distributions divided according to the binning scheme shown in Fig. 16 (bins 1–4). The different components are shown as indicated in the legend of Fig. 10.

Figure 18

Helicity-angle distributions divided according to the binning scheme shown in Fig. 16 (bins 5–8). The different components are shown as indicated in the legend of Fig. 10.

Figure 19

Projections of moments 1–5 of each pair of final-state particles in the $B^{+}\to D^{+}{D}^{-}{K}^{+}$ Dalitz plot. As usual, data points are shown in black and the total, and individual components’ PDFs are overlaid. The different components are shown as indicated in the legend of Fig. 10.

Figure 20

Projections of moments 6–9 of each pair of final-state particles in the $B^{+}\to D^{+}{D}^{-}{K}^{+}$ Dalitz plot. As usual, data points are shown in black and the total, and individual components’ PDFs are overlaid. The different components are shown as indicated in the legend of Fig. 10.