Amplitude analysis of $B^0\to {\overline{D}}^0{D}_s^{+}{\pi }^{-}$ and $B^{+}\to D^{-}{D}_s^{+}{\pi }^{+}$ decays

Resonant contributions in $B^0\to \overline{D}{{}^{0}D}_s^{+}{\pi }^{-}$ and $B^{+}\to D^{-}{D}_s^{+}{\pi }^{+}$ decays are determined with an amplitude analysis, which is performed both separately and simultaneously, where in the latter case isospin symmetry between the decays is assumed. The analysis is based on data collected by the LHCb detector in proton-proton collisions at center-of-mass energies of 7, 8, and 13 TeV. The full data sample corresponds to an integrated luminosity of $9{\mathrm{fb}}^{-1}$. A doubly charged spin-0 open-charm tetraquark candidate together with a neutral partner, both with masses near 2.9 GeV, are observed in the $D_s\pi$ decay channel.

Figure 1

Feynman diagrams for the dominant tree-level amplitudes contributing to (a) $B^0\to {\overline{D}}^0{D}_s^{+}{\pi }^{-}$ and (b) $B^{+}\to D^{-}{D}_s^{+}{\pi }^{+}$ decays with intermediate $D\pi$ resonances; and nonresonant three body decays of (c) $B^0\to {\overline{D}}^0{D}_s^{+}{\pi }^{-}$ and (d) $B^{+}\to D^{-}{D}_s^{+}{\pi }^{+}$ decays.

Figure 2

Invariant mass spectrum of the signal candidates, split by decay mode and run period. The data are overlaid with the results of the fit.

Figure 3

Signal efficiency maps for the $B^0\to {\overline{D}}^0{D}_s^{+}{\pi }^{-}$, ${\overline{D}}^0\to K^{+}{\pi }^{-}$ decays in (a) run 1 and (b) run 2; $B^0\to {\overline{D}}^0{D}_s^{+}{\pi }^{-}$, ${\overline{D}}^0\to K^{+}{\pi }^{-}{\pi }^{-}{\pi }^{+}$ decays in (c) run 1 and (d) run 2; $B^{+}\to D^{-}{D}_s^{+}{\pi }^{+}$ decays in (e) run 1 and (f) run 2. White regions are caused by $B^0\to D^{*-}{D}_s^{+}$ veto.

Figure 4

Distributions of the SPD background shapes for the $B^0\to {\overline{D}}^0{D}_s^{+}{\pi }^{-}$, ${\overline{D}}^0\to K^{+}{\pi }^{-}$ decays in (a) run 1 and (b) run 2; $B^0\to {\overline{D}}^0{D}_s^{+}{\pi }^{-}$, ${\overline{D}}^0\to K^{+}{\pi }^{-}{\pi }^{-}{\pi }^{+}$ decays in (c) run 1 and (d) run 2; $B^{+}\to D^{-}{D}_s^{+}{\pi }^{+}$ decays in (e) run 1 and (f) run 2. White regions are caused by $B^0\to D^{*-}{D}_s^{+}$ veto.

Figure 5

Invariant mass distributions (a) $M({\overline{D}}^0{\pi }^{-})$, (b) $M(D_s^{+}{\pi }^{-})$, and (c) $M({\overline{D}}^0{D}_s^{+})$ for the $B^0\to {\overline{D}}^0{D}_s^{+}{\pi }^{-}$ candidates compared with the fit results with only $D\pi$ resonances.

Figure 6

Invariant mass distributions of the (a) $M(D^{-}{\pi }^{+})$, (b) $M(D_s^{+}{\pi }^{+})$, and (c) $M(D^{-}{D}_s^{+})$ for the $B^{+}\to D^{-}{D}_s^{+}{\pi }^{+}$ candidates compared with the fit results with only $D\pi$ resonances.

Figure 7

Two-dimensional pull plots of the fits to the (a) $B^0\to {\overline{D}}^0{D}_s^{+}{\pi }^{-}$ and (b) $B^{+}\to D^{-}{D}_s^{+}{\pi }^{+}$ samples.

Figure 8

Projection of the fit result on (a) $M(D\pi )$ and (b) $M(D_s^{+}\pi )$ of $B^0\to {\overline{D}}^0{D}_s^{+}{\pi }^{-}$ decays after including the $T_{c\overline{s}0}^a(2900{)}^0$ state, and on (c) $M(D\pi )$ and (d) $M(D_s^{+}\pi )$ of $B^{+}\to D^{-}{D}_s^{+}{\pi }^{+}$ decays after including the $T_{c\overline{s}0}^a(2900{)}^{++}$ state.

Figure 9

Two-dimensional pull plots of the fits to (a) $B^0\to {\overline{D}}^0{D}_s^{+}{\pi }^{-}$ and (b) $B^{+}\to D^{-}{D}_s^{+}{\pi }^{+}$ samples after including the $T_{c\overline{s}0}^a(2900{)}^0$ and $T_{c\overline{s}0}^a(2900{)}^{++}$ states.

Figure 10

Significance tests for the new $D_s\pi$ states in the (a) $B^0\to {\overline{D}}^0{D}_s^{+}{\pi }^{-}$ and (b) $B^{+}\to D^{-}{D}_s^{+}{\pi }^{+}$ samples. The blue histogram is the distribution of $2\mathrm{\Delta }\mathrm{LL}$ and the red solid curve shows the fitted ${\chi }^2$ PDF. The red dashed and dotted lines are the $2\mathrm{\Delta }\mathrm{LL}$ values corresponding to $3\sigma$ and $5\sigma$, and the purple solid line is the $2\mathrm{\Delta }\mathrm{LL}$ measured in the data.

Figure 11

Spin analysis of (a) $B^0\to {\overline{D}}^0{D}_s^{+}{\pi }^{-}$ and (b) $B^{+}\to D^{-}{D}_s^{+}{\pi }^{+}$ decays. The blue solid and black dashed histograms are the distributions of the $2\mathrm{\Delta }\mathrm{LL}$ for the pseudoexperiments generated based on the fit results with $0^{+}$ or $1^{-}$ $D_s^{+}\pi$ exotic state, respectively. The purple vertical line shows the $2\mathrm{\Delta }\mathrm{LL}$ value for the data fitted with the new $D_s^{+}\pi$ exotic state under the $J^P=0^{+}$ and $J^P=1^{-}$ hypotheses. The red curve is the result of a fit to the black dashed histogram with a Gaussian function.

Figure 12

Argand diagrams of the (a) $B^0\to {\overline{D}}^0{D}_s^{+}{\pi }^{-}$ and (b) $B^{+}\to D^{-}{D}_s^{+}{\pi }^{+}$ decays. Black dots show the line shape of the $T_{c\overline{s}0}^a(2900{)}^0$ and $T_{c\overline{s}0}^a(2900{)}^{++}$ Breit-Wigner functions. The red solid line and blue error bars show the line shape and fit results of spline $0^{+}$ $D_s^{+}\pi$ model in $T_{c\overline{s}0}^a(2900{)}^0$ and $T_{c\overline{s}0}^a(2900{)}^{++}$ mass region.

Figure 13

The MD description of the (a) $M({\overline{D}}^0{\pi }^{-})$ and (b) $M(D_s^{+}{\pi }^{-})$ distributions for the $B^0\to {\overline{D}}^0{D}_s^{+}{\pi }^{-}$ decays.

Figure 14

The MD description of the (a) $M(D^{-}{\pi }^{+})$ and (b) $M(D_s^{+}{\pi }^{+})$ distributions for the $B^{+}\to D^{-}{D}_s^{+}{\pi }^{+}$ decays.

Figure 15

Argand diagrams of the MD and QMI $0^{+}D\pi$ components of the (a) $B^0\to {\overline{D}}^0{D}_s^{+}{\pi }^{-}$ and (b) $B^{+}\to D^{-}{D}_s^{+}{\pi }^{+}$ decays. The red solid line and blue error bars show the line shape and fit results of QMI $0^{+}D\pi$ spline model, while the black line is the line shape of the MD $0^{+}D\pi$ component.

Figure 16

Fit result of simultaneous $D\pi$ fit model, the (a) $M(D\pi )$ and (b) $M(D_s^{+}\pi )$ distributions of $B^0\to {\overline{D}}^0{D}_s^{+}{\pi }^{-}$ decays after including the $T_{c\overline{s}0}^a(2900{)}^0$ state; the (c) $M(D\pi )$ and (d) $M(D_s^{+}\pi )$ distributions of $B^{+}\to D^{-}{D}_s^{+}{\pi }^{+}$ decays after including the $T_{c\overline{s}0}^a(2900{)}^{++}$ state.

Figure 17

Moments analysis of $B^0\to {\overline{D}}^0{D}_s^{+}{\pi }^{-}$ on $M^2({\overline{D}}^0{\pi }^{-})$. The black points indicate the data, while the blue and red histogram indicate the fit results without and with the $T_{c\overline{s}0}^a(2900)$ states separately.

Figure 18

Moments analysis of $B^0\to {\overline{D}}^0{D}_s^{+}{\pi }^{-}$ on $M^2(D_s^{+}{\pi }^{-})$. The black points indicate the data, while the blue and red histogram indicate the fit results without and with the $T_{c\overline{s}0}^a(2900)$ states separately.

Figure 19

Moments analysis of $B^{+}\to D^{-}{D}_s^{+}{\pi }^{+}$ on $M^2(D^{-}{\pi }^{+})$. The black points indicate the data, while the blue and red histogram indicate the fit results without and with the $T_{c\overline{s}0}^a(2900)$ states separately.

Figure 20

Moments analysis of $B^{+}\to D^{-}{D}_s^{+}{\pi }^{+}$ on $M^2(D_s^{+}{\pi }^{+})$. The black points indicate the data, while the blue and red histogram indicate the fit results without and with the $T_{c\overline{s}0}^a(2900)$ states separately.