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

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Abstract

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 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 GeV$/c^2$, and a new spin-0 charmonium resonance in proximity to the spin-2 $\chi_{c2}(3930)$ state. The masses and widths of these resonances are determined, as are the relative contributions of all components in the amplitude model, which additionally include the vector charmonia $\psi(3770)$, $\psi(4040)$, $\psi(4160)$ and $\psi(4415)$ states and a nonresonant component.

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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.

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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.

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

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Visualisation 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.

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

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

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Comparisons of the invariant-mass distributions of $ B ^+ \rightarrow 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 (non-solid coloured lines) are shown as detailed in the legend.

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Normalised 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.

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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\text{ Ge V /}c^2 $ to suppress reflections from charmonium resonances. The different components are shown as indicated in the legend of Fig. ???.

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Comparisons of the invariant-mass distributions of $ B ^+ \rightarrow D ^+ D ^- K ^+ $ candidates in 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.

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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. ???.

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

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Normalised 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.

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

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Distributions of the test-statistic $t$ in ensembles of pseudoexperiments generated according to various hypotheses and compared to values found in data (indicated by dashed vertical lines). In (a), the $H_0$ hypothesis is a model fit to 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.

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

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

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

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Projections of moments 1--5 of each pair of final-state particles in the $ B ^+ \rightarrow 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. ???.

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Projections of moments 6--9 of each pair of final-state particles in the $ B ^+ \rightarrow 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. ???.

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Tables and captions

Signal and background component yields obtained from the simultaneous fit to the Run 1 and Run 2 data-taking years.

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Fitted values of shape parameters of the DSCB and exponential PDFs used to model signal and background, respectively, in the simultaneous fit to Run 1 and Run 2 data.

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Components which may appear in the $ D ^+ D ^- $ spectrum of $ B ^+ \rightarrow D ^+ D ^- K ^+ $ decays, and their properties as given by the Particle Data Group (PDG) \cite{PDG2019}. For the $\psi(3770)$ mass and the mass/width of both the $\chi_{c2}(3930)$ and $X(3842)$, the values in Ref. \cite{LHCb-PAPER-2019-005} are used.

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Magnitude and phase of the complex coefficients in the amplitude model, together with fit fractions for each component. The quantities are reported after correction for fit biases (see Sec. ???). The first uncertainty is statistical and the second is the sum in quadrature of all systematic uncertainties.

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Lineshape parameters for the $\chi_{c0,2}(3930)$ and $X_{0,1}(2900)$ resonances determined from the fit. The first uncertainty is statistical and the second is the sum in quadrature of all systematic uncertainties.

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Interference fit fractions (%) obtained from the results of the amplitude fit with the baseline model. Uncertainties are statistical and systematic, respectively. Absent entries correspond to pairs of resonances that do not interfere, because they either inhabit separate regions of phase space or belong to different partial waves in the same two-body combination.

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Model variations and the associated negative log-likelihood (NLL) and $\chi^2$ values.

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Systematic uncertainties on the complex coefficients and fit fractions of each component of the amplitude model: mass-fit signal shape (1), size of simulated sample for efficiencies (2), hardware trigger modelling (3), modelling: parent Blatt--Weisskopf radius (4), modelling: charmonia Blatt--Weisskopf radius (5), modelling: $ D ^- K ^+ $ resonances' Blatt--Weisskopf radius (6), model composition: S wave (7), model composition: P wave (8).

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Systematic uncertainties on the masses ( $\text{Ge V /}c^2$ ) and widths ( $\text{Ge V}$ ) of the $\chi_{c0,2}(3930)$ and $X_{0,1}(2900)$ resonances: mass-fit signal shape (1), size of simulated sample for efficiencies (2), hardware trigger modelling (3), modelling: parent Blatt--Weisskopf radius (4), modelling: charmonia Blatt--Weisskopf radius (5), modelling: $ D ^- K ^+ $ resonances' Blatt--Weisskopf radius (6), model composition: S wave (7), model composition: P wave (8).

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Supplementary Material [file]

Supplementary material full pdf

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Created on 19 February 2021.