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

Information

LHCb-PAPER-2020-025
CERN-EP-2020-159
arXiv:2009.00026 [PDF]

(Submitted on 31 Aug 2020)
Phys. Rev. D102 (2020) 112003
Inspire 1814324

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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.

Figures and captions

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.	fig1a.pdf [24 KiB] HiDef png [170 KiB] Thumbnail [115 KiB] *.C file
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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.	fig2.pdf [39 KiB] HiDef png [183 KiB] Thumbnail [226 KiB] *.C file
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.	fig3a.pdf [89 KiB] HiDef png [484 KiB] Thumbnail [337 KiB] *.C file
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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.	fig4a.pdf [111 KiB] HiDef png [445 KiB] Thumbnail [488 KiB] *.C file
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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.	fig5a.pdf [90 KiB] HiDef png [306 KiB] Thumbnail [361 KiB] *.C file
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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.	fig6a.pdf [113 KiB] HiDef png [413 KiB] Thumbnail [443 KiB] *.C file
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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.	fig7a.pdf [117 KiB] HiDef png [420 KiB] Thumbnail [362 KiB] *.C file
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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.	fig8.pdf [105 KiB] HiDef png [559 KiB] Thumbnail [453 KiB] *.C file
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. ???.	fig9.pdf [127 KiB] HiDef png [247 KiB] Thumbnail [277 KiB] *.C file
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.	fig10a.pdf [131 KiB] HiDef png [410 KiB] Thumbnail [355 KiB] *.C file
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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. ???.	fig11a.pdf [97 KiB] HiDef png [434 KiB] Thumbnail [384 KiB] *.C file
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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. ???.	fig12a.pdf [139 KiB] HiDef png [408 KiB] Thumbnail [379 KiB] *.C file
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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.	fig13.pdf [105 KiB] HiDef png [569 KiB] Thumbnail [460 KiB] *.C file
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. ???.	fig14a.pdf [89 KiB] HiDef png [394 KiB] Thumbnail [393 KiB] *.C file
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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.	fig15a.pdf [24 KiB] HiDef png [215 KiB] Thumbnail [265 KiB] *.C file
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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.	fig16a.pdf [59 KiB] HiDef png [533 KiB] Thumbnail [591 KiB] *.C file
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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. ???.	fig17a.pdf [105 KiB] HiDef png [481 KiB] Thumbnail [411 KiB] *.C file
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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. ???.	fig18a.pdf [89 KiB] HiDef png [564 KiB] Thumbnail [454 KiB] *.C file
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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. ???.	fig19a.pdf [66 KiB] HiDef png [504 KiB] Thumbnail [421 KiB] *.C file
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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. ???.	fig20a.pdf [75 KiB] HiDef png [474 KiB] Thumbnail [412 KiB] *.C file
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Animated gif made out of all figures.	PAPER-2020-025.gif Thumbnail

Tables and captions

Signal and background component yields obtained from the simultaneous fit to the Run 1 and Run 2 data-taking years.	Table_1.pdf [38 KiB] HiDef png [96 KiB] Thumbnail [45 KiB] tex code
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.	Table_2.pdf [62 KiB] HiDef png [75 KiB] Thumbnail [37 KiB] tex code
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.	Table_3.pdf [70 KiB] HiDef png [67 KiB] Thumbnail [31 KiB] tex code
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.	Table_4.pdf [71 KiB] HiDef png [115 KiB] Thumbnail [54 KiB] tex code
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.	Table_5.pdf [61 KiB] HiDef png [65 KiB] Thumbnail [30 KiB] tex code
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.	Table_6.pdf [60 KiB] HiDef png [43 KiB] Thumbnail [16 KiB] tex code
Model variations and the associated negative log-likelihood (NLL) and $\chi^2$ values.	Table_7.pdf [71 KiB] HiDef png [131 KiB] Thumbnail [57 KiB] tex code
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).	Table_8.pdf [65 KiB] HiDef png [190 KiB] Thumbnail [61 KiB] tex code
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).	Table_9.pdf [62 KiB] HiDef png [71 KiB] Thumbnail [22 KiB] tex code