EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2020-159 LHCb-PAPER-2020-025 September 1, 2020

Amplitude analysis

of the ${{{B}^{+}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{+}}$ decay

LHCb collaboration†††Authors are listed at the end of this paper.

Results are reported from an amplitude analysis of the
${{{B}^{+}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{+}}$ decay. The analysis is
carried out using LHCb proton-proton collision data taken at $\sqrt{s}=7,8,$
and $13\text{\,Te\kern-1.00006ptV}$, corresponding to a total integrated
luminosity of 9$\text{\,fb}^{-1}$. 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$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$, 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.

Submitted to Phys. Rev. D

© 2024 CERN for the benefit of the LHCb collaboration. CC BY 4.0 licence.

##  1 Introduction

Decays of $B$ mesons to multibody final states involving two open-charm mesons
and a strange meson, henceforth labelled $B\rightarrow D{\kern
1.79993pt\overline{\kern-1.79993ptD}}K$ decays, proceed at quark level through
$\bar{b}\rightarrow c\bar{c}\bar{s}$ transitions and comprise a relatively
large fraction of the total width of the $B$ mesons. Their branching fractions
have been measured previously [1, 2, 3, 4], but few studies of their resonant
structure exist. Such analyses are valuable as a means to study resonant
structure in both $D{\kern 1.79993pt\overline{\kern-1.79993ptD}}$ and charm-
strange systems. Conventional $c\bar{c}$ charmonium states can produce
resonant structures in a neutral $D{\kern
1.79993pt\overline{\kern-1.79993ptD}}$ system, but it is now known that exotic
charmonium-like states, which can decay to both neutral and charged $D{\kern
1.79993pt\overline{\kern-1.79993ptD}}$ combinations, also exist [5, 6, 7].
Conventional resonances can also be observed in charged $DK$ systems,
containing charm and antistrange ($c\bar{s}$) quarks.111The inclusion of
charge-conjugate processes is implied throughout this document. There is no
previous experimental evidence of exotic hadrons containing a charm and a
strange quark ($cs$), and the possible existence of such states has not been
widely discussed in the theoretical literature, although some predictions do
exist [8, 9, 10].

In the ${{{B}^{+}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{+}}$ decay, resonances
in the ${{D}^{-}}{{K}^{+}}$ channel must have minimal quark content
$\bar{c}d\bar{s}u$ and hence would be exotic, as would doubly charged
${{D}^{+}}{{K}^{+}}$ states. Since conventional resonances can only contribute
in the ${{D}^{+}}{{D}^{-}}$ channel, this $B$ decay stands to provide a clean
environment to study charmonium states and to address open questions
concerning $c\bar{c}$ resonant structure, in particular to identify and
determine the properties of spin-0 and spin-2 states [11, 12, 13]. Properties
of the vector charmonium states are better known from studies of their
production in ${e^{+}e^{-}}$ collisions, but improved knowledge of their rates
of production in ${{B}^{+}}$ decays will aid characterisation of the
$c\bar{c}$ contribution in ${{{B}^{+}}}\rightarrow{{K}^{+}}{\mu^{+}\mu^{-}}$
decays [14, 15]. A more detailed discussion of the current knowledge of
charmomium spectroscopy, as relevant to the
${{{B}^{+}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{+}}$ decay, is given in Sec.
7.1.

No prior study of ${{{B}^{+}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{+}}$ resonant
structure has been published, but a few previous amplitude analyses of other
$B\rightarrow D{\kern 1.79993pt\overline{\kern-1.79993ptD}}K$ decays exist.
The Belle collaboration analysed the resonant structure of the
${{{B}^{+}}}\rightarrow{{D}^{0}}{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}{{K}^{+}}$ decay [2], while
Dalitz-plot analyses of both the ${{{B}^{+}}}\rightarrow{{D}^{0}}{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}{{K}^{+}}$ and
${{B}^{0}}\rightarrow{{D}^{0}}{{D}^{-}}{{K}^{+}}$ final states have been
performed by the BaBar collaboration [16]. The signal yields in these previous
measurements ranged from about 400 to just under 2000, with relatively high
background levels giving a maximum signal purity of 40%. Contributions from
the vector $\psi(3770)$ and $\psi(4160)$ charmonium states, and the
$D^{*}_{s2}(2573)^{+}$ and $D^{*}_{s1}(2700)^{+}$ charm-strange resonances,
were determined. A large nonresonant contribution to the
${{B}^{0}}\rightarrow{{D}^{0}}{{D}^{-}}{{K}^{+}}$ decay was also found.

In this paper the first amplitude analysis of the
${{{B}^{+}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{+}}$ decay is described. The
analysis is based on LHCb proton-proton ($pp$) collision data taken at
$\sqrt{s}=7,8,$ and $13\text{\,Te\kern-1.00006ptV}$, corresponding to a total
integrated luminosity of 9$\text{\,fb}^{-1}$. In Secs. 2 and 3, the dataset
and candidate selection are described. The procedure to determine the signal
and background yields, using a fit to the ${B}$-candidate invariant-mass
spectrum, is presented in Sec. 4. The amplitude modelling formalism used is
detailed in Sec. 5, and a description of the selection efficiency and residual
background modelling is given in Sec. 6. The development of the model itself
follows in Sec. 7, with results given in Sec. 8. Sources of systematic
uncertainties that affect the measurements are described in Sec. 9. Studies of
the significance of various features in the model are presented in Sec. 10,
and a summary of the results is provided in Sec. 11.

A key outcome of this amplitude analysis is the observation of structure in
the ${{D}^{-}}{{K}^{+}}$ system. This conclusion is confirmed with a model-
independent analysis that is described in a companion article [17].

##  2 Detector and simulation

The LHCb detector [18, 19] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region [20], a large-area silicon-strip detector located upstream
of a dipole magnet with a bending power of about $4{\mathrm{\,Tm}}$, and three
stations of silicon-strip detectors and straw drift tubes [21, 22] placed
downstream of the magnet. The tracking system provides a measurement of the
momentum, $p$, of charged particles with a relative uncertainty that varies
from 0.5% at low momentum to 1.0% at 200$\text{\,Ge\kern-1.00006ptV\\!/}c$.
The minimum distance of a track to a primary $pp$ collision vertex (PV), the
impact parameter (IP), is measured with a resolution of
$(15+29/p_{\mathrm{T}})\,\upmu\text{m}$, where $p_{\mathrm{T}}$ is the
component of the momentum transverse to the beam, in
$\text{\,Ge\kern-1.00006ptV\\!/}c$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors
[23]. Photons, electrons and hadrons are identified by a calorimeter system
consisting of scintillating-pad and preshower detectors, an electromagnetic
and a hadronic calorimeter. Muons are identified by a system composed of
alternating layers of iron and multiwire proportional chambers [24]. The
online event selection is performed by a trigger [25], which consists of a
hardware stage, based on information from the calorimeter and muon systems,
followed by a software stage, which applies a full event reconstruction.

At the hardware trigger stage, events are required to have a muon with high
$p_{\mathrm{T}}$ or a hadron, photon or electron with high transverse energy
in the calorimeters. For hadrons, the typical transverse energy threshold is
3.5 GeV. The software trigger requires a two-, three- or four-track secondary
vertex with a significant displacement from any primary $pp$ interaction
vertex. At least one charged particle must have a transverse momentum
$p_{\mathrm{T}}>1.6\text{\,Ge\kern-1.00006ptV\\!/}c$ and be inconsistent with
originating from a PV. A multivariate algorithm [26, 27] is used for the
identification of secondary vertices consistent with the decay of a $b$
hadron.

Simulation is required to model the effects of the detector acceptance and the
imposed selection requirements. In the simulation, $pp$ collisions are
generated using Pythia [28, *Sjostrand:2006za] with a specific LHCb
configuration [30]. Decays of unstable particles are described by EvtGen [31],
in which final-state radiation is generated using Photos [32]. The interaction
of the generated particles with the detector, and its response, are
implemented using the Geant4 toolkit [33, *Agostinelli:2002hh] as described in
Ref. [35]. For the samples corresponding to 2017 and 2018 data, the underlying
$pp$ interaction is reused multiple times, with an independently generated
signal decay for each [36].

The particle identification (PID) response in the simulated samples is
corrected by sampling from distributions of
${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}},{{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}$
decays in LHCb data, considering their kinematics and the detector occupancy.
An unbinned method is employed, where the probability density functions are
modelled using kernel density estimation [37]. The event multiplicity is also
corrected in the simulated samples to match more closely that observed in
events containing selected ${{{B}^{+}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{+}}$
candidates. Good agreement is seen between the simulated samples and data for
the variables used in the analysis.

The momentum scale is calibrated using samples of
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\\!\rightarrow{\mu^{+}\mu^{-}}$ and
${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}$ decays
collected concurrently with the data sample used for this analysis [38, 39].
The relative accuracy of this procedure is estimated to be $3\times 10^{-4}$
using samples of other fully reconstructed ${b}$ hadrons, $\mathchar
28935\relax$ and ${{K}^{0}_{\mathrm{S}}}$ mesons.

##  3 Selection

Data samples collected in $pp$ collisions during the Run 1 (2011 and 2012) and
Run 2 (2015–2018) data-taking periods of the Large Hadron Collider are used,
corresponding to integrated luminosities of 3$\text{\,fb}^{-1}$ and
6$\text{\,fb}^{-1}$, respectively. Signal ${{{B}^{+}}}$ candidates are built
from sets of well-reconstructed pions and kaons, where intermediate charm
mesons are reconstructed via the
${{D}^{+}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ decay.

The final-state particles are ensured to be well displaced from the
interaction point by requiring that their $\chi^{2}_{\text{IP}}$ with respect
to any PV be greater than 4, where $\chi^{2}_{\text{IP}}$ is defined as the
difference in the vertex-fit $\chi^{2}$ of a given PV reconstructed with and
without the particle under consideration. The PV that fits best to the flight
direction of the $B$ candidate is taken as the associated PV. All charged
final-state particles are required to have momentum greater than
$1\text{\,Ge\kern-1.00006ptV\\!/}c$ and transverse momentum above
$0.1\text{\,Ge\kern-1.00006ptV\\!/}c$. At least one of them must have momentum
greater than $10\text{\,Ge\kern-1.00006ptV\\!/}c$ and transverse momentum
exceeding $1.7\text{\,Ge\kern-1.00006ptV\\!/}c$, whilst also having an impact
parameter with respect to the $B$-candidate’s associated PV of at least
$0.1\text{\,mm}$. The $D$-candidates’ invariant masses are required to lie
within $20$$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ of the known ${D}^{\pm}$
mass [40] and their decay vertices must be well reconstructed. The
reconstructed momentum and the vector between production and decay vertices
are required to be well aligned for both $B$ and $D$ candidates. The flight
time (distance significance) from the associated PV for the $B$\- ($D$)-meson
candidates is required to exceed $0.2\text{\,ps}$ (6). Finally, PID
information is employed to aid identification of final-state $K$ and $\pi$
mesons.

A boosted decision tree (BDT) [41, 42] algorithm, implemented in the TMVA
toolkit [43, *TMVA4], is employed to separate signal from background. The
boosting algorithm assigns weights during training both to correct for
classification error and to prioritise uniformity in the Dalitz plot
variables. The signal sample for the training consists of correctly
reconstructed simulated ${{{B}^{+}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{+}}$
candidates and the background sample is composed of candidates from the data
samples where the ${B}$-candidate mass exceeds
5.6$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$. No evidence of overtraining is
observed. Candidates are retained if the BDT response exceeds a threshold,
chosen to maximise the product of signal significance and sample purity,
$S^{2}/\left(S+B\right)^{3/2}$, where $S$ and $B$ are the expected signal and
background yields in the range
$5.265\text{\,Ge\kern-1.00006ptV\\!/}c^{2}<m({{D}^{+}}{{D}^{-}}{{K}^{+}})<5.295\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$.
The invariant mass is calculated from a kinematic fit in which the masses of
the charm-meson candidates are fixed to the known ${{D}^{\pm}}$ mass value and
the ${B}$ meson is constrained to originate from its associated PV. Given the
variations in hardware and software trigger criteria, separate BDT classifiers
are developed for Run 1 and Run 2 data. The variables entering the BDT are:
the $\chi^{2}$ of the reconstructed $B$-meson decay vertex; the angle between
the $B$-meson flight direction from the associated PV and its reconstructed
momentum; the $\chi^{2}_{\text{IP}}$ of the $B$\- and $D$-meson candidates and
of the final state pions and kaons; the ratio of the flight distance, parallel
to the beampipe, of each of the ${D}^{\pm}$ candidates to its uncertainty; and
the PID variables of the final-state $K$ and $\pi$ mesons.

Decays of ${{{B}^{+}}}$ mesons to the same set of final-state pions and kaons,
having only one or no intermediate ${{D}^{\pm}}$ mesons or where final-state
particles are associated to the wrong $D$ meson, are a potentially important
source of background since they produce a peak in the reconstructed
${{D}^{+}}{{D}^{-}}{{K}^{+}}$ invariant-mass distribution. To suppress these
backgrounds, vetoes are imposed on narrow invariant-mass structures formed
between specific pairs of final-state pions and kaons where the two particles
originate from different ${{D}^{\pm}}$ mesons or the pair involves the kaon
produced directly in the $B$-meson decay. In addition, the two ${{D}^{\pm}}$
mesons are required to be displaced significantly, parallel to the beampipe,
from their production vertex. These requirements are efficient for the
${{{B}^{+}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{+}}$ signal, and examination of
the sidebands of the reconstructed ${{D}^{\pm}}$ invariant-mass distributions,
illustrated in Fig. 1, confirms that there is negligible residual background
contamination from this source.

The fraction of events containing more than one reconstructed candidate is
measured to be below 1%. All such candidates are retained.

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.

##  4 $B$-candidate invariant-mass fit

An extended maximum-likelihood fit is applied to the
$m({{D}^{+}}{{D}^{-}}{{K}^{+}})$ distribution shown in Fig. 2, for candidates
in the range between 5.22 and 5.60$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$. The
selected candidates in this region are predominantly from signal with a small
amount of combinatorial background. There is no significant contribution from
partially reconstructed $B$ decays, which appear at lower
$m({{D}^{+}}{{D}^{-}}{{K}^{+}})$ values.

The probability density function (PDF) used to model the
${{{B}^{+}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{+}}$ signal component consists
of a double-sided Crystal Ball (DSCB) function [45], having tails on opposite
sides of the peak in order to describe the asymmetric power-law tails of the
distribution due to detector resolution and final-state radiation. An
exponential function accounts for combinatorial background. In the
simultaneous fit to each year of the Run 1 and Run 2 datasets, the mean and
width of the signal component’s Gaussian core are allowed to vary separately
for the two periods, and the parameters of the DSCB tails are fixed to their
values obtained in fits to simulated samples. The sample purities are very
high, so if the background yield falls below 0.01 candidates for one subset of
the data, the background component is removed for that subset and the fit re-
run to ensure stability. The fit projection is shown in Fig. 2, the yields of
the included components are given in Table 1, and the values of the varying
parameters are recorded in Table 2.

Of the 1374 candidates to which the invariant-mass fit is applied, 1260 have a
value of $m({{D}^{+}}{{D}^{-}}{{K}^{+}})$ within
20$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ of the known ${{B}^{+}}$ mass, which
is the window applied for the amplitude analysis. Within this signal window,
the purity is greater than 99.5%.

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.  Table 1: Signal and background component yields obtained from the simultaneous fit to the Run 1 and Run 2 data-taking years.  Year | Signal | Background
---|---|---
2011 | $\phantom{0}84\pm\phantom{0}9$ | —
2012 | $217\pm 15$ | $16\pm 5$
2015 | $\phantom{0}41\pm\phantom{0}6$ | —
2016 | $300\pm 18$ | $19\pm 6$
2017 | $302\pm 18$ | $21\pm 6$
2018 | $359\pm 19$ | $15\pm 5$
Table 2: 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.  Parameter | Result
---|---
Signal |  |  |
$\mu$ ($\text{Me\kern-1.00006ptV\\!/}c^{2}$) | Run 1 |  $5278.90$$~{}\pm~{}$ | $0.39$
| Run 2 |  $5278.70$$~{}\pm~{}$ | $0.27$
$\sigma$ ($\text{Me\kern-1.00006ptV\\!/}c^{2}$) | Run 1 |  $6.22$$~{}\pm~{}$ | $0.33$
| Run 2 |  $7.77$$~{}\pm~{}$ | $0.23$
Background |  |  |
Coefficient (10$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$)-1 | 2012 |  $-38$$~{}\pm~{}$ | $31$
| 2016 |  $-93$$~{}\pm~{}$ | $31$
| 2017 |  $-66$$~{}\pm~{}$ | $28$
| 2018 |  $2$$~{}\pm~{}$ | $36$

##  5 Amplitude analysis formalism

The distribution of ${{{B}^{+}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{+}}$ decays
across the Dalitz plot is fitted using the Laura++ software package [46].
Generic details of the formalism and its implementation in the analysis of
LHCb data can be found in the literature [47, 48, 49]; only aspects
specifically relevant to the current analysis are described here.

The PDF used to fit the Dalitz plot structure of the selected candidates is
composed of signal and background contributions and is a function of position
in the ${B}$-decay phase space, $\vec{x}$. It includes dependence upon model
parameters such as mass, width, or spin of individual components in the signal
model. The fit procedure maximises the likelihood,

$\mathcal{L}=\exp{\left[-\sum_{c}\left(\frac{(p_{c}-\mu_{c})^{2}}{2\sigma_{c}^{2}}\right)\right]}\prod_{j=1}^{N_{c}}\left(N_{\rm
sig}\mathcal{P}_{\rm sig}(\vec{x}_{j})+N_{\rm bg}\mathcal{P}_{\rm
bg}(\vec{x}_{j})\right)\,,$ (1)

where $N_{\rm sig}$ and $N_{\rm bg}$ are the signal and background yields
obtained from the invariant-mass fit, respectively, $N_{c}$ is the total
number of candidates in the data sample, and $\mathcal{P}_{\rm
sig,bg}(\vec{x}_{j})$ are the PDFs for candidate $j$, which differ for Run 1
and Run 2 data since different efficiency and background models are employed.
Gaussian constraints with parameters $\mu_{c}$ and $\sigma_{c}$ are applied to
the values of model parameters, $p_{c}$, such as the masses or widths of
intermediate resonances given in Sec. 7. The background PDF, $\mathcal{P}_{\rm
bg}(\vec{x})$, is an empirical shape used to represent the residual
combinatorial background that enters the selected sample of
${{{B}^{+}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{+}}$ candidates, and is
described further in Sec. 6.

The signal PDF is given by

$\mathcal{P}_{\rm sig}(\vec{x})=\frac{1}{{\cal N}}\times\epsilon_{\rm
total}(\vec{x})\times|\mathcal{A}_{\rm sig}(\vec{x})|^{2}\,,$ (2)

where ${\cal N}$ is a normalisation factor that ensures the integral of
$\mathcal{P}_{\rm sig}(\vec{x})$ over the Dalitz plot ($\vec{x}$) is unity,
and $\epsilon_{\rm total}(\vec{x})$ is the total efficiency for the selected
candidates, described further in Sec. 6. The signal amplitude,
$\mathcal{A}_{\rm sig}(\vec{x})$, is constructed according to the isobar
formalism [50, 51, 52] and contains a coherent sum of resonant and nonresonant
amplitudes,

$\mathcal{A}_{\rm sig}(\vec{x})=\sum_{j=1}^{N}c_{j}F_{j}(\vec{x})\,,$ (3)

where the sum runs over the components in the model, indexed by $j$. The
$c_{j}$ factors are complex coefficients that multiply the complex amplitudes
$F_{j}(\vec{x})$, which contain information about the dynamics of each
component in the amplitude model. For a ${D}^{+}$ ${D}^{-}$ resonance, for
example,

$F(\vec{x})=R\left(m({{D}^{+}}{{D}^{-}})\right)\times T(\vec{p},\vec{q})\times
X(|\vec{p}|)\times X(|\vec{q}|)\,,$ (4)

where $R$ and $T$ describe the invariant-mass and angular dependence of the
amplitude, and the $X$ functions are Blatt–Weisskopf barrier factors. The
invariant-mass dependence, $R\left(m({{D}^{+}}{{D}^{-}})\right)$, is given by
a relativistic Breit–Wigner function for all resonant contributions and the
angular terms, $T(\vec{p},\vec{q})$, are constructed using the non-
relativistic Zemach tensor formalism [53, 54]. Nonresonant contributions are
described with a lineshape that includes an exponential form factor, with
alternative models also considered during the model-building and determination
of systematic uncertainties. The momenta $\vec{p}$ and $\vec{q}$ are those of
the third particle (not involved in the resonance) and one of the particles
produced in the resonance decay, respectively, both evaluated in the rest
frame of the resonance.

The choice of which of the particles produced in the resonance decay is taken
to define $\vec{q}$ corresponds to a convention for the definition of the
helicity angle of the resonance. The helicity angle is defined to be, in the
rest frame of the resonance, the angle between one of the two particles
produced in the resonance decay and the third particle. In this study, the
choice is:

  * •

$\theta({{D}^{+}}{{D}^{-}})$ is the angle between the ${K}^{+}$ and ${D}^{-}$
particles, in the ${D}^{+}$ ${D}^{-}$ rest frame,

  * •

$\theta({{D}^{+}}{{K}^{+}})$ is the angle between the ${D}^{-}$ and ${K}^{+}$
particles, in the ${D}^{+}$ ${K}^{+}$ rest frame, and

  * •

$\theta({{D}^{-}}{{K}^{+}})$ is the angle between the particles ${D}^{+}$ and
${K}^{+}$, in the ${D}^{-}$ ${K}^{+}$ rest frame.

The square Dalitz plot (SDP) provides a useful representation of the phase
space. The large ${{B}^{+}}$ mass means that resonant structure is often found
close to the edge of the regular Dalitz plot, and the SDP provides greater
granularity in exactly these regions. Moreover, the SDP aligns a rectangular
grid with the edges of the phase space, avoiding edge effects associated with
rectangular binning of the regular Dalitz plot.

The two degrees of freedom used to define the SDP are the variables
$m^{\prime}\left({{D}^{+}}{{D}^{-}}\right)$ and
$\theta^{\prime}\left({{D}^{+}}{{D}^{-}}\right)$, which are defined as

$\displaystyle m^{\prime}\left({{D}^{+}}{{D}^{-}}\right)$
$\displaystyle\equiv$
$\displaystyle\frac{1}{\pi}\arccos{\left(2\;\frac{m({{D}^{+}}{{D}^{-}})-m_{{{D}^{+}}{{D}^{-}}}^{\rm
min}}{m_{{{D}^{+}}{{D}^{-}}}^{\rm max}-m_{{{D}^{+}}{{D}^{-}}}^{\rm
min}}-1\right)}\,,$ (5)
$\displaystyle\theta^{\prime}\left({{D}^{+}}{{D}^{-}}\right)$
$\displaystyle\equiv$
$\displaystyle\frac{1}{\pi}\theta({{D}^{+}}{{D}^{-}})\,,$ (6)

where $m_{{{D}^{+}}{{D}^{-}}}^{\rm min,max}$ are the minimum and maximum
kinematically allowed values of $m({{D}^{+}}{{D}^{-}})$ (equal to
$2m_{{{D}^{+}}}$ and $m_{{{{B}^{+}}}}-m_{{{K}^{+}}}$, respectively). With
these definitions both $m^{\prime}$ and $\theta^{\prime}$ are bounded in the
range 0–1.

The complex coefficients, $c_{j}$ in Eq. (3), depend on choices of phase
convention and normalisation. In order to be able to compare results between
different analyses, it is therefore helpful to report the convention-
independent fit fractions, which are defined as the integral of the absolute
value of the amplitude squared for each component, $j$, divided by that of the
coherent matrix-element squared for all intermediate contributions,

$\mathcal{F}_{j}=\frac{\int|c_{j}F_{j}(\vec{x})|^{2}d\vec{x}}{\int|\mathcal{A}_{\rm
sig}(\vec{x})|^{2}d\vec{x}}\,.$ (7)

Interference between amplitudes in the coherent sum within $\mathcal{A}_{\rm
sig}(\vec{x})$ can cause the sum of the fit fractions to depart from unity.
This deviation can be quantified by means of interference fit fractions,

$\mathcal{I}_{ij}=\frac{\int
c_{i}c_{j}^{*}F_{i}(\vec{x})F_{j}^{*}(\vec{x})d\vec{x}}{\int|\mathcal{A}_{\rm
sig}(\vec{x})|^{2}d\vec{x}}\,.$ (8)

Interference effects between different partial waves in the same two-body
combination cancel when integrated over the helicity angle, due to the angular
terms having the form of Legendre polynomials, which form an orthogonal basis.

##  6 Efficiency and background models

The absolute efficiency is not needed for the amplitude analysis but the
variation of the efficiency across the Dalitz plot must be accounted for.
Efficiency variations as a function of position in the Dalitz plot are
evaluated using simulated samples. Four contributing factors are considered:

$\epsilon_{\rm total}(\vec{x})=\epsilon_{\rm
offline}(\vec{x})\times\epsilon_{\rm reco}(\vec{x})\times\epsilon_{\rm
trig}(\vec{x})\times\epsilon_{\rm geom}(\vec{x})\,.$ (9)

The geometrical efficiency, $\epsilon_{\rm geom}$, quantifies the probability
for all final state particles to be within the LHCb detector acceptance. This
efficiency is found not to vary significantly across the phase space. The
efficiencies of the trigger requirements, $\epsilon_{\rm trig}$, and that of
the reconstruction, $\epsilon_{\rm reco}$, all with respect to the preceding
step, do however have significant dependence on Dalitz-plot position. The BDT,
which dominates the offline selection criteria and is designed to minimise
induced efficiency variations across the Dalitz plot, behaves as expected with
$\epsilon_{\rm offline}$ being approximately independent of position in phase
space. The total efficiency, $\epsilon_{\rm total}$, is shown in Fig. 3 as a
function of position in both standard Dalitz plot and SDP. Smooth functions
are obtained by kernel estimation [37] and the model obtained using the SDP is
used in the analysis to avoid edge effects. Given the differences between Run
1 and Run 2 data for every element of Eq. (9), separate efficiency maps are
used for the two data-taking periods.

Figure 3: 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.

The residual combinatorial background contribution, though small, is accounted
for in the fit. A model is derived from candidates in the high $B$-candidate
mass sideband, between 5.35 and 5.69$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$. In
order to increase the sample size available for this modelling, the BDT
requirement is relaxed by an amount that is seen not to influence the
distribution of the background candidates in the Dalitz plot significantly. A
kernel estimation procedure is applied to the selected background candidates
to reduce the impact of statistical fluctuations. Due to the different
selections applied to Run 1 and Run 2 data, both online and offline, separate
background models are obtained for each. The background candidates in the
regular Dalitz plot are shown in Fig. 4, along with the derived background
model as a function of SDP position, obtained using a kernel density
estimation [37].

Figure 4: 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.

##  7 Amplitude model

###  7.1 Model content

The masses of the particles involved in the
${{{B}^{+}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{+}}$ decay give rise to limits
on the allowed masses of on-shell intermediate resonances:
$3.74\text{\,Ge\kern-1.00006ptV\\!/}c^{2}<m\left({{D}^{+}}{{D}^{-}}\right)<4.79\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$
and
$2.36\text{\,Ge\kern-1.00006ptV\\!/}c^{2}<m({{D}^{+}}{{K}^{+}}),m({{D}^{-}}{{K}^{+}})<3.41\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$.
As described in Sec. 1, only charmonium resonances in the ${{D}^{+}}{{D}^{-}}$
channel are anticipated. Moreover, only states with natural spin-parity
($J^{P}=0^{+},1^{-},2^{+},...$) can decay strongly to a pair of pseudoscalar
mesons, and resonances with very high intrinsic spin are unlikely to be
produced in the decay of a pseudoscalar ${{{B}^{+}}}$ meson. Given these
considerations, the resonances initially considered are listed, with their
properties, in Table 3.

Table 3: 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) [40]. For the
$\psi(3770)$ mass and the mass/width of both the $\chi_{c2}(3930)$ and
$X(3842)$, the values in Ref. [55] are used.

Partial wave ($J^{PC}$) |  Resonance | Mass ($\text{Me\kern-1.00006ptV\\!/}c^{2}$) | Width (MeV)
---|---|---|---
S wave ($0^{++}$) |  $\chi_{c0}(3860)$ |  $3862$ $\pm$ | $43$ |  $201$ $\pm$ | $145$
|  $X(3915)$ |  $3918.4$ $\pm$ | $1.9$ |  $20$ $\pm$ | $5$
P wave ($1^{--}$) |  $\psi(3770)$ |  $3778.1$ $\pm$ | $0.9$ |  $27.2$ $\pm$ | $1.0$
|  $\psi(4040)$ |  $4039$ $\pm$ | $1$ |  $80$ $\pm$ | $10$
|  $\psi(4160)$ |  $4191$ $\pm$ | $5$ |  $70$ $\pm$ | $10$
|  $\psi(4260)$ |  $4230$ $\pm$ | $8$ |  $55$ $\pm$ | $19$
|  $\psi(4415)$ |  $4421$ $\pm$ | $4$ |  $62$ $\pm$ | $20$
D wave ($2^{++}$) |  $\chi_{c2}(3930)$ |  $3921.9$ $\pm$ | $0.6$ |  $36.6$ $\pm$ | $2.1$
F wave ($3^{--}$) |  $X(3842)$ |  $3842.71$ $\pm$ | $0.20$ |  $2.79$ $\pm$ | $0.62$

Contributions to the S wave can be expected, but there are few previous
experimental results on scalar $c\bar{c}$ resonances. The Belle collaboration
[56] has reported the observation of a $\chi_{c0}(3860)$ state,222 The PDG
convention, which is followed in this paper, is that the symbol used to denote
a particle depends only on its quantum numbers and does not imply any
interpretation of its substructure.  seen as a $D{\kern
1.79993pt\overline{\kern-1.79993ptD}}$ resonance in the process
${e^{+}e^{-}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}D{\kern
1.79993pt\overline{\kern-1.79993ptD}}$, where the $J^{PC}=0^{++}$ hypothesis
is favoured over the $2^{++}$ hypothesis at the level of $2.5\,\sigma$. This
resonance is yet to be confirmed, and there could be other states or
nonresonant S-wave $D{\kern 1.79993pt\overline{\kern-1.79993ptD}}$
contributions. The PDG listing [40] includes a $X(3915)$ state, with
$J^{PC}=0^{++}$ or $2^{++}$, seen produced in $\gamma\gamma$ collisions by the
Belle [57] and BaBar [58] collaborations (and also possibly in $B\rightarrow
X(3915)K$ decays [59, 60]) and decaying to the
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\omega$ final state — it has not been seen
in the $D{\kern 1.79993pt\overline{\kern-1.79993ptD}}$ final state. It appears
that this structure may be caused by the $\chi_{c2}(3930)$ state [61], which
has also been seen by BaBar to be produced in $\gamma\gamma$ collisions [62]
and has been studied more recently and precisely by LHCb in $pp$ collisions
[55]. However, the existence of both spin-0 and spin-2 states near
$3930\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ [63, 13] is not excluded. At higher
mass, the $\chi_{c0}(4500)$ and $\chi_{c0}(4700)$ states have been seen as
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\phi$ resonances in an LHCb amplitude
analysis of
${{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}\phi{{K}^{+}}$ decays
[64, 65], with masses and widths
$M=4506\,^{+16}_{-19}\text{\,Me\kern-1.00006ptV\\!/}c^{2}$, $\Gamma=92\pm
29\text{\,Me\kern-1.00006ptV}$ and
$M=4704\,^{+17}_{-26}\text{\,Me\kern-1.00006ptV\\!/}c^{2}$, $\Gamma=120\pm
50\text{\,Me\kern-1.00006ptV}$ respectively. Given that their quantum numbers
have been measured as $J^{PC}=0^{++}$, these could in principle be seen in
${{{B}^{+}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{+}}$ decays, but since their
composition is unclear it is difficult to make any prediction as to whether
this is likely or not.

A larger number of vector $c\bar{c}$ states have been observed, since these
can be produced directly in ${e^{+}e^{-}}$ collisions. The $\psi(3770)$,
$\psi(4040)$, $\psi(4160)$ and $\psi(4415)$ states are all well established
and known to decay to $D{\kern 1.79993pt\overline{\kern-1.79993ptD}}$,
therefore all might be expected to appear in
${{{B}^{+}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{+}}$ decays. The $\psi(3770)$
and $\psi(4160)$ resonances were included in the previous BaBar [16] and Belle
[2] amplitude analyses of the ${{{B}^{+}}}\rightarrow{{D}^{0}}{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}{{K}^{+}}$ decay, while
$\psi(4040)$ and $\psi(4415)$ components were additionally included in an LHCb
amplitude analysis of ${{{B}^{+}}}\rightarrow{{K}^{+}}{\mu^{+}\mu^{-}}$ decays
[66] but found not to contribute significantly. The $\psi(4260)$ state,
originally called $Y(4260)$, was observed by the BaBar collaboration through
radiative return in $e^{+}e^{-}$ production to the
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{\pi}^{+}}{{\pi}^{-}}$ final state [67].
Subsequently confirmed by CLEO, Belle and BESIII collaborations [68, 69, 70],
including through direct $e^{+}e^{-}$ production, it has not been observed in
the $D{\kern 1.79993pt\overline{\kern-1.79993ptD}}$ final state, nor is there
convincing evidence for its production in $B$ decays. The only $\psi(4260)$
decays to be observed to date contain a ${J\mskip-3.0mu/\mskip-2.0mu\psi}$
meson in the final state, although a $\psi(4230)$ state with similar mass and
width ($M=4218\,^{+\,5}_{-\,4}\text{\,Me\kern-1.00006ptV\\!/}c^{2}$,
$\Gamma=59\,^{+\,12}_{-\,10}\text{\,Me\kern-1.00006ptV}$) has been seen by
BESIII to be produced in $e^{+}e^{-}$ collisions in the $\chi_{c0}\omega$,
$h_{c}{{\pi}^{+}}{{\pi}^{-}}$ and $\psi(2S){{\pi}^{+}}{{\pi}^{-}}$ final
states [71, 72, 73]. It is sufficient to consider one of the two as a
candidate contribution to the
${{{B}^{+}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{+}}$ Dalitz plot; the
$\psi(4260)$ is used as it is considered to be better established in the PDG
2019 listings.333 In its 2020 edition, the PDG has changed its treatment of
the $\psi(4230)$ and $\psi(4260)$ states, but this does not impact
significantly on the analysis.  Two further vector states, the $\psi(4360)$
and $\psi(4660)$, have been seen in radiative return from $e^{+}e^{-}$
collisions to the $\psi(2S){{\pi}^{+}}{{\pi}^{-}}$ final state by the BaBar
and Belle collaborations [74, 75]. Moreover, a BESIII scan of the energy
dependence of the
${e^{+}e^{-}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{\pi}^{+}}{{\pi}^{-}}$
cross-section [70] suggests that the structure around
$4260\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ is composed of two states: one with
$M=4222.0\pm 3.1\pm 1.4\text{\,Me\kern-1.00006ptV\\!/}c^{2},\Gamma=44.1\pm
4.3\pm 2.0\text{\,Me\kern-1.00006ptV}$ and another with $M=4320.0\pm 10.4\pm
7.0\text{\,Me\kern-1.00006ptV\\!/}c^{2},\Gamma=101.4\,^{+25.3}_{-19.7}\pm
10.2\text{\,Me\kern-1.00006ptV}$. In the PDG 2019 edition, the results for the
first are included in the averages of the properties of the $\psi(4260)$,
while those for the second are included in the $\psi(4360)$ averages. Both the
$\psi(4360)$ and $\psi(4660)$ are considered unlikely to be present in
${{{B}^{+}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{+}}$ decays since they have
never previously been observed to either be produced in $B$ decays or to decay
to $D{\kern 1.79993pt\overline{\kern-1.79993ptD}}$ final states. They are
therefore not included in Table 3.

In the $D$ wave, the $\chi_{c2}(3930)$ state has recently been studied by LHCb
in $pp$ collisions [55], leading to significant improvement in the knowledge
of its properties. However, its quantum numbers are assumed, and while
previous analyses have indicated a preference for a spin-2 particle in this
mass range [76, 62] it is not experimentally excluded that the measured
structure is spin-0 or, at least, has a spin-0 contribution. Therefore, it is
important to determine the spin of the $\chi_{c2}(3930)$ resonance in this
analysis.

Finally, a candidate for the spin-3 $\psi_{3}(1^{3}D_{3})$ charmonium state,
the $X(3842)$, has recently been observed by LHCb decaying to $D{\kern
1.79993pt\overline{\kern-1.79993ptD}}$ [55]. Its quantum numbers have not been
measured, but its properties fit the expectation for that state. Production of
spin-3 states in ${B}$-meson decays is suppressed, especially when there is
little phase space available, and therefore this state is not expected to
contribute at a significant level in
${{{B}^{+}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{+}}$ decays.

###  7.2 Model development

Selected signal candidates entering the invariant-mass fit shown in Fig. 2 are
further filtered by applying a window of width
$40\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ around the known ${{B}^{+}}$ mass.
The 2011 and 2012 data are combined into a single Run 1 dataset, and the
2015–2018 data are combined into a single Run 2 dataset. The Dalitz plot and
its projections are shown in Figs. 5 and 6, for Run 1 and Run 2 respectively.
The Dalitz-plot coordinates are determined after refitting the candidate
decays, imposing the constraints that the reconstructed ${{B}^{+}}$ and
${D}^{\pm}$ masses should match their known values and that the reconstructed
${{B}^{+}}$ meson should originate at its associated primary vertex. This
improves the resolution of the Dalitz-plot coordinates; for example, the
$m({{D}^{+}}{{D}^{-}})$ resolution is reduced from
$10$–$13\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ to
$1.5$–$3.5\text{\,Me\kern-1.00006ptV\\!/}c^{2}$, depending upon position in
the Dalitz plot. As the resolution is much smaller than the width of the
narrowest resonance considered in the analysis, it is neglected in the
amplitude fit. A simultaneous fit of the Run 1 and Run 2 datasets is carried
out with separate efficiency maps, background models, and fixed signal yields
for the two samples. All other model parameters are shared.

Figure 5: 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.

Figure 6: 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.

Models which reproduce the Dalitz plot distribution of the data are developed
by considering resonances listed in Table 3 and additional resonant and
nonresonant components. The $\psi(3770)\rightarrow{{D}^{+}}{{D}^{-}}$ and
$\chi_{c2}(3930)\rightarrow{{D}^{+}}{{D}^{-}}$ resonances, which are both
clearly seen in the data, are taken as a starting point. Further components
are included in the model if they cause a significant reduction in the
negative log-likelihood obtained from the fit to data, while not causing
instabilities in the fit or producing excessively large inference effects and
hence a sum of fit fractions far from 100%. The complex coefficients
associated with all resonant or nonresonant components are allowed to vary
freely, with the exception of that for the
$\psi(3770)\rightarrow{{D}^{+}}{{D}^{-}}$ component, which is fixed to unit
length along the real axis to serve as a reference amplitude. The masses and
widths of contributing resonances are all allowed to vary, though Gaussian
constraints, with parameters corresponding to the central values and
uncertainties in Table 3, are applied to those of the $\psi(3770)$,
$\psi(4040)$, $\psi(4160)$, and $\psi(4415)$ states.

It is observed that significantly better agreement between the model and the
data is obtained when including a spin-0 $D{\kern
1.79993pt\overline{\kern-1.79993ptD}}$ component that overlaps with the
$\chi_{c2}(3930)$ state, labelled $\chi_{c0}(3930)$. The presence of a spin-0
component in this $\chi_{cJ}(3930)$ region may mean that previous measurements
of the mass and width of the $\chi_{c2}(3930)$ state, based on an assumption
of a single resonance, are not reliable. Therefore, the masses and widths of
both the spin-0 and spin-2 components are allowed to vary freely.

It is found that the inclusion of at least one nonresonant component is
essential to obtain a good fit to data. A number of parameterisations are
considered, including the case of completely uniform Dalitz plot density and
modulation of the nonresonant amplitude by either polynomial or exponential
form factors, and the possibility of a spin-1, instead of spin-0, angular
term. A quasi-model-independent partial wave description of the S wave, as
used for example in Refs. [77, 48, 49], is also attempted, but is not viable
with the current sample sizes. In all cases, parameters associated with the
nonresonant model are allowed to vary freely in the fit to data.

For each configuration, the minimisation is repeated 100 times, randomising
the starting parameters at each iteration. The minimisation that is
consistently found to yield the best likelihood value is selected. In order to
assess the fit quality, a $\chi^{2}$ computation is performed, with an
adaptive binning scheme ensuring a minimum of 20 candidates in each bin. The
associated number of degrees of freedom is determined using an ensemble of
pseudoexperiments generated at the fit minimum. The goodness of fit is
assessed using this figure of merit as well as the change in negative log-
likelihood value between different configurations.

##  8 Results

###  8.1 Model excluding ${{D}^{-}}{{K}^{+}}$ resonances

The data in Figs. 5 and 6 exhibit a striking excess at
$m^{2}({{D}^{-}}{{K}^{+}})\approx
8.25\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$, in both Run 1 and Run 2, which
cannot be accounted for by introducing resonances only in the
${{D}^{+}}{{D}^{-}}$ decay channel. To illustrate this, the first model
presented excludes any resonant content from the ${{D}^{-}}{{K}^{+}}$ channel.
The model includes the $\psi(3770)$, $\chi_{c0}(3930)$, $\chi_{c2}(3930)$,
$\psi(4040)$, $\psi(4160)$, and $\psi(4415)$ resonances, which are necessary
to describe structure in the $m({{D}^{+}}{{D}^{-}})$ spectrum. A nonresonant
component is included and described by an exponential S-wave lineshape in the
${{D}^{-}}{{K}^{+}}$ spectrum.

The Dalitz-plot projections from this fit are compared to the data in Fig. 7.
Contributions from individual components are superimposed. The goodness of fit
is quantified in Fig. 8, where the largest deviations are seen in the
$m^{2}({{D}^{-}}{{K}^{+}})\approx
8.25\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$ region of the Dalitz plot. To
illustrate this more clearly, a comparison between data and the result of the
fit is made in Fig. 9 after excluding low-mass charmonium resonances through
the requirement $m({{D}^{+}}{{D}^{-}})>4\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$.

It is concluded that a satisfactory description of the data cannot be obtained
without including one or more components that model structure in
$m({{D}^{-}}{{K}^{+}})$ explicitly. The same conclusion is reached with a
model-independent analysis, as described in Ref. [17].

Figure 7: 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.  Figure 8: 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.  Figure 9: Comparison of the
$m({{D}^{-}}{{K}^{+}})$ distribution and the fit projection for a model
excluding any ${{D}^{-}}{{K}^{+}}$ resonance, after requiring
$m({{D}^{+}}{{D}^{-}})>4\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$ to suppress
reflections from charmonium resonances. The different components are shown as
indicated in the legend of Fig. 7.

###  8.2 Baseline model including ${{D}^{-}}{{K}^{+}}$ resonances

The simplest way to account for the $m({{D}^{-}}{{K}^{+}})$ structure is by
adding resonances to the model. Analysis of the current data sample cannot,
however, exclude the possibility that hadronic effects such as rescattering
may be important, in particular given the observation that the structure
appears near to the $D^{*}K^{*}$ threshold. More detailed investigations of
plausible explanations for the observed structure will require new theoretical
models to be developed and larger data samples to be analysed.

The baseline model includes the same components as in Sec. 8.1, but adds both
spin-1 and spin-0 ${{D}^{-}}{{K}^{+}}$ resonances. An exponential S-wave
lineshape in the ${{D}^{-}}{{K}^{+}}$ channel remains the best description of
the nonresonant contribution. The projections of the Dalitz plot, with fit
results superimposed, are shown in Fig. 10. In Appendix A, the results are
compared to the helicity-angle distributions in eight bins of the invariant-
mass distribution of each pair of particles. A comparison to the distributions
of the angular moments (defined in Ref. [17]) of each pair of particles is
made in Appendix B. The results for the fit parameters and the fit fractions
for each component are shown in Tables 4 and 5, where $X_{1}(2900)$ and
$X_{0}(2900)$ are used to label the new spin-1 and spin-0 ${{D}^{-}}{{K}^{+}}$
states, respectively. These results include systematic uncertainties, the
evaluation of which is described in Sec. 9. The coefficient of the nonresonant
exponential lineshape is found to be
$($0.0809$\pm$0.0496$)~{}(\text{Ge\kern-1.00006ptV}^{2}\\!/c^{4})^{-1}$, where
the uncertainty is statistical only. The interference fit fractions are given
in Table 6, with their statistical and systematic uncertainties.

Figure 10:  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.  Table 4: 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. 9). The first uncertainty is statistical and the second is the sum in quadrature of all systematic uncertainties.  Resonance | Magnitude | Phase (rad) | Fit fraction (%)
---|---|---|---
${{D}^{+}}{{D}^{-}}$ resonances |  |  |  |  |  |  |  |
$\psi(3770)$ | 1 (fixed) | 0 (fixed) |  $14.5$ $\pm$ |  $1.2$ $\pm$ | $0.8$
$\chi_{c0}(3930)$ |  $0.506$ $\pm$ |  $0.064$ $\pm$ | $0.017$ |  $2.162$ $\pm$ |  $0.184$ $\pm$ | $0.034$ |  $3.7$ $\pm$ |  $0.9$ $\pm$ | $0.2$
$\chi_{c2}(3930)$ |  $0.703$ $\pm$ |  $0.064$ $\pm$ | $0.012$ |  $0.827$ $\pm$ |  $0.170$ $\pm$ | $0.133$ |  $7.2$ $\pm$ |  $1.2$ $\pm$ | $0.3$
$\psi(4040)$ |  $0.585$ $\pm$ |  $0.078$ $\pm$ | $0.035$ |  $1.416$ $\pm$ |  $0.176$ $\pm$ | $0.084$ |  $5.0$ $\pm$ |  $1.3$ $\pm$ | $0.4$
$\psi(4160)$ |  $0.668$ $\pm$ |  $0.084$ $\pm$ | $0.052$ |  $0.898$ $\pm$ |  $0.225$ $\pm$ | $0.092$ |  $6.6$ $\pm$ |  $1.5$ $\pm$ | $1.2$
$\psi(4415)$ |  $0.797$ $\pm$ |  $0.080$ $\pm$ | $0.061$ |  $-1.458$ $\pm$ |  $0.197$ $\pm$ | $0.091$ |  $9.2$ $\pm$ |  $1.4$ $\pm$ | $1.5$
${{D}^{-}}{{K}^{+}}$ resonances |  |  |  |  |  |  |  |
$X_{0}(2900)$ |  $0.619$ $\pm$ |  $0.079$ $\pm$ | $0.025$ |  $1.091$ $\pm$ |  $0.193$ $\pm$ | $0.095$ |  $5.6$ $\pm$ |  $1.4$ $\pm$ | $0.5$
$X_{1}(2900)$ |  $1.449$ $\pm$ |  $0.086$ $\pm$ | $0.032$ |  $0.367$ $\pm$ |  $0.102$ $\pm$ | $0.049$ |  $30.6$ $\pm$ |  $2.4$ $\pm$ | $2.1$
Nonresonant |  $1.293$ $\pm$ |  $0.088$ $\pm$ | $0.043$ |  $-2.410$ $\pm$ |  $0.119$ $\pm$ | $0.508$ |  $24.2$ $\pm$ |  $2.2$ $\pm$ | $0.5$
Table 5: 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.  Resonance | Mass ($\text{Ge\kern-1.00006ptV\\!/}c^{2}$) | Width (MeV)
---|---|---
$\chi_{c0}(3930)$ |  $3.9238$ $\pm$ |  $0.0015$ $\pm$ | $0.0004$ |  $17.4$ $\pm$ |  $\phantom{1}5.1$ $\pm$ | $0.8$
$\chi_{c2}(3930)$ |  $3.9268$ $\pm$ |  $0.0024$ $\pm$ | $0.0008$ |  $34.2$ $\pm$ |  $\phantom{1}6.6$ $\pm$ | $1.1$
$X_{0}(2900)$ |  $2.8663$ $\pm$ |  $0.0065$ $\pm$ | $0.0020$ |  $57.2$ $\pm$ |  $12.2$ $\pm$ | $4.1$
$X_{1}(2900)$ |  $2.9041$ $\pm$ |  $0.0048$ $\pm$ | $0.0013$ |  $110.3$ $\pm$ |  $10.7$ $\pm$ | $4.3$

Table 6: 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.

| $\chi_{c0}(3930)$ | $\chi_{c2}(3930)$ | $\psi(4040)$ | $\psi(4160)$ | $\psi(4415)$ | $X_{0}(2900)$ | $X_{1}(2900)$ | Nonresonant
---|---|---|---|---|---|---|---|---
$\psi(3770)$ | — | — |  $-$$3.7$ $\pm$ |  $0.5$ $\pm$ | $0.1831$ |  $-$$1.7$ $\pm$ |  $0.6$ $\pm$ | $0.3$ |  $-3.3$ $\pm$ |  $0.7$ $\pm$ | $0.6$ |  $-0.6$ $\pm$ |  $0.2$ $\pm$ | $0.1$ |  $-4.6$ $\pm$ |  $0.5$ $\pm$ | $0.6$ |  $-0.4$ $\pm$ |  $0.3$ $\pm$ | $0.5$
$\chi_{c0}(3930)$ |  | — | — | — | — |  $-$$0.1$ $\pm$ |  $0.1$ $\pm$ | $0.0$ |  $-$$0.5$ $\pm$ |  $0.1$ $\pm$ | $0.0$ |  $-$$3.2$ $\pm$ |  $0.7$ $\pm$ | $1.5$
$\chi_{c2}(3930)$ |  |  | — | — | — |  $-0.2$ $\pm$ |  $0.0$ $\pm$ | $0.1$ |  $-1.5$ $\pm$ |  $0.2$ $\pm$ | $0.4$ |  $-$$0.0$ $\pm$ |  $0.0$ $\pm$ | $0.0$
$\psi(4040)$ |  |  |  |  $-1.2$ $\pm$ |  $1.3$ $\pm$ | $0.1$ |  $-0.3$ $\pm$ |  $0.7$ $\pm$ | $0.3$ |  $-$$0.1$ $\pm$ |  $0.1$ $\pm$ | $0.0$ |  $-0.6$ $\pm$ |  $0.5$ $\pm$ | $0.2$ |  $-0.6$ $\pm$ |  $0.4$ $\pm$ | $0.5$
$\psi(4160)$ |  |  |  |  |  $-5.1$ $\pm$ |  $1.3$ $\pm$ | $0.9$ |  $-0.2$ $\pm$ |  $0.1$ $\pm$ | $0.0$ |  $-2.8$ $\pm$ |  $0.5$ $\pm$ | $0.4$ |  $-0.2$ $\pm$ |  $0.2$ $\pm$ | $0.4$
$\psi(4415)$ |  |  |  |  |  |  $-$$0.0$ $\pm$ |  $0.1$ $\pm$ | $0.1$ |  $-$$3.1$ $\pm$ |  $0.5$ $\pm$ | $0.2$ |  $-$$0.5$ $\pm$ |  $0.3$ $\pm$ | $0.4$
$X_{0}(2900)$ |  |  |  |  |  |  | — |  $-$$2.3$ $\pm$ |  $1.4$ $\pm$ | $3.0$
$X_{1}(2900)$ |  |  |  |  |  |  |  | —

As described in Sec. 7.1, $D{\kern 1.79993pt\overline{\kern-1.79993ptD}}$
resonant structure has previously been observed in the $\chi_{cJ}(3930)$
region, however it has usually been assumed to arise from the
$\chi_{c2}(3930)$ resonance. The mass and helicity-angle distributions of
candidates in this region, shown in Fig. 11, clearly demonstrate that both
spin-0 and spin-2 contributions are necessary. The masses and widths of these
two components are completely free to vary in the fit; they are found to have
consistent masses while the fit prefers a narrower width for the spin-0 state.
If both spin-0 and spin-2 states are present at the same mass, one would
generically expect the spin-0 state to be broader since its decay to a
${{D}^{+}}{{D}^{-}}$ pair is in S wave, as compared to D wave for the spin-2
state, and therefore is not suppressed by any angular momentum barrier. This
expected pattern is seen in some explicit calculations of the properties of
the $\chi_{cJ}(2P)$ states [11], however the observed pattern is consistent
with other theoretical predictions [13]. Moreover, the fitted
$\chi_{c0}(3930)$ parameters are consistent with those of the $X(3915)$ state.

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.

The $\chi_{c0}(3930)$ state is the only component in the ${{D}^{+}}{{D}^{-}}$
S wave in the baseline model. The broad $\chi_{c0}(3860)$ state, reported by
the Belle collaboration [56], has been included in alternative fit models but
is disfavoured. Fits in which additional S-wave structure is introduced e.g.
through a nonresonant component, have been attempted but tend to destabilise
the fit, which is understood as a consequence of there being too much freedom
in the S wave. In fact the nonresonant component in the ${{D}^{-}}{{K}^{+}}$
projection covers most of the $m({{D}^{+}}{{D}^{-}})$ range, as can be seen in
Fig. 10 top row, but only allows a small contribution at low
$m({{D}^{+}}{{D}^{-}})$ values.

A good description of the intermediate $m({{D}^{+}}{{D}^{-}})$ region is
obtained by including the $\psi(4040)$, $\psi(4160)$, and $\psi(4415)$
contributions, together with reflections from the ${{D}^{-}}{{K}^{+}}$
structures. Inclusion of the $\psi(4260)$ resonance was also considered during
the model-building process, but its inclusion together with the $\psi(4160)$
state leads to fit instabilities, due to the similarity of their masses and
widths. Between the two, a slight preference was visible in negative log-
likelihood value for the $\psi(4160)$ component.

Figure 12:  Comparison of data and the fit projection of the baseline model,
for (a) the ${{D}^{-}}{{K}^{+}}$ invariant-mass distribution requiring
$m({{D}^{+}}{{D}^{-}})>4\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$ to suppress
reflections from charmonium resonances and (b) helicity angle in the region
$2.75\text{\,Ge\kern-1.00006ptV\\!/}c^{2}<m({{D}^{-}}{{K}^{+}})<3.05\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$.
The different components are shown as indicated in the legend of Fig. 10.

The impact of the $X_{1}(2900)$ and $X_{0}(2900)$ states on the agreement
between the data and the model is highlighted in Fig. 12(a) by restricting the
phase space to exclude low mass charmonium resonances in the same way as in
Fig. 9. The need for both spin-1 and spin-0 components is seen in the
helicity-angle distribution shown in Fig. 12(b).

###  8.3 Other models

Numerous variations in the composition of the decay amplitude are considered
in the process of establishing the baseline model. These include consideration
of one or two states with different spins in the $\chi_{cJ}(3930)$ region, and
zero, one or two states in the $X(2900)$ region, as well as the inclusion of a
contribution from the $X(3842)$ state (assumed to be spin 3). The impact of
these different model choices on the negative log-likelihood resulting from
the fit is summarised in Table 7. Models with two components with the same
spin in the same two-body combination, and with freely varying masses and
widths, tend to make the fit unstable and are therefore not included.
Similarly, variations in the description of the nonresonant component that
destabilise the fit are not included as the obtained negative log-likelihood
values are not reliable.

Table 7: Model variations and the associated negative log-likelihood (NLL) and $\chi^{2}$ values. Model | NLL | $\chi^{2}$
---|---|---
Baseline | $-3540$ | 86.1
Variations to $\chi_{cJ}(3930)$ region |  |
$\chi_{c0}(3930)$ only |  $-$3508 | 104.2
$\chi_{c2}(3930)$ only |  $-$3502 | 111.1
$\chi_{c0}(3930)$ \+ $\psi(3930)$ |  $-$3540 | 94.0
Variations in ${{D}^{-}}{{K}^{+}}$ channel |  |
No ${{D}^{-}}{{K}^{+}}$ resonances |  $-$3382 | 288.9
One ${{D}^{-}}{{K}^{+}}$ resonance (spin-0) |  $-$3491 | 175.8
One ${{D}^{-}}{{K}^{+}}$ resonance (spin-1) |  $-$3497 | 107.2
One ${{D}^{-}}{{K}^{+}}$ resonance (spin-2) |  $-$3463 | 152.6
Two ${{D}^{-}}{{K}^{+}}$ resonances (spin-1 + spin-2) | $-3536$ | 91.6
Other |  |
Addition of $X(3842)$ |  $-$3541 | 85.3

Among the models with variations to the description of the $\chi_{cJ}(3930)$
region, those including a spin-1 state (denoted $\psi(3930)$) are considered
unlikely since any vector state in this region would have been seen by
previous experiments, as discussed in Sec. 7.1. Moreover, including such a
state in the model, either by itself or together with a $\chi_{c2}(3930)$
state, has a large impact on other components of the model. The $X_{1}(2900)$
component moves to higher mass and much broader width, with the nonresonant
lineshape also changing significantly. These models are therefore excluded
from Table 7. The model with $\chi_{c0}(3930)$+$\psi(3930)$ states does not
suffer this problem but, like other models including a $\psi(3930)$ component,
has large interference effects due to the overlap between spin-1 states in the
model. This causes a higher sum of fit fractions compared to the baseline
model. All models containing the $\psi(3930)$ are thus disfavoured, leaving
the approach of including $\chi_{c0}(3930)$ and $\chi_{c2}(3930)$ states as
the only candidate to describe the data in the $\chi_{cJ}(3930)$ region.

Among the variations in the ${{D}^{-}}{{K}^{+}}$ channel, the need for two
states is clear from the improvement in the NLL and $\chi^{2}$ values. Noting
the proximity to the $D^{*}K^{*}$ threshold, a model with spin-0 and spin-2
states is theoretically well motivated. However, when the masses and widths of
the states are allowed to vary freely in the fit, the spin-2 component takes
an extremely large ($>500\text{\,Me\kern-1.00006ptV}$) width, effectively
becoming a nonresonant spin-2 component. While this may be due to residual
imperfections in the model (discussed below), this configuration cannot be
considered further in the current analysis and is therefore excluded from
Table 7. Studies of larger data samples may help to shed light on whether it
is possible to describe the structure in $m({{D}^{-}}{{K}^{+}})$ with spin-0
and spin-2 components. A model with spin-1 and spin-2 ${{D}^{-}}{{K}^{+}}$
resonances gives comparable, but less favourable, goodness-of-fit indicators
to the baseline model.

The model with the inclusion of the $X(3842)$ state, assumed to be spin-3,
demonstrates that there is no significant contribution from that component.
This supports the assumption, made in Ref. [17], that only states of spin up
to 2 are present in ${{{B}^{+}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{+}}$
decays. Fits with this model are, for simplicity, made neglecting resolution
effects since this is done for all other fits. If the narrow $X(3842)$ state
were present in the data it would be necessary to account for resolution
effects properly, but the fit neglecting them is sufficient to confirm
qualitatively the absence of this contribution at any significant level.

###  8.4 Residual imperfections in the baseline model

Figure 13: 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.

The goodness of fit is visualised using the binned normalised residual
distribution in Fig. 13. The $\chi^{2}$/ndf is 86.1/38.3 = 2.25, where the
number of degrees of freedom, ndf, is an effective value obtained from
pseudoexperiments and only statistical uncertainties are considered. While an
overall reasonable description of the data is achieved with the baseline
model, there are regions of the Dalitz plot where significant imperfections
remain. The largest contributions to the binned $\chi^{2}$ are at
$\left(m^{2}({{D}^{-}}{{K}^{+}}),m^{2}({{D}^{+}}{{D}^{-}})\right)\sim(10.5\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4},13.5\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4})$
and
$\sim(10.5\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4},18.5\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4})$.
The disagreement in the first of these regions can also be seen in the
${D}^{+}$ ${D}^{-}$ helicity angle distribution at low
$m^{2}({{D}^{+}}{{D}^{-}})$, shown in Fig. 14, which shows a clear asymmetry
most likely originating from interference between the $\psi(3770)$ P-wave
state and S-wave ${{D}^{+}}{{D}^{-}}$ structure. Since the baseline model has
only very limited S wave in this region, the asymmetry observed in data cannot
be reproduced in the model. This disagreement can also be seen in some other
projections, for example at high $m({{D}^{-}}{{K}^{+}})$ in the projection of
the whole Dalitz plot (Fig. 10).

Figure 14: 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. 10.

The second of the aforementioned regions of data-model disagreement
corresponds to low values of $m^{2}({{D}^{+}}{{K}^{+}})$. No particular
disagreement is seen in other projections of this region, and therefore it is
not considered a source of concern. There does seem to be some disagreement at
high $m({{D}^{+}}{{D}^{-}})$ values (Fig. 10), but this does not make a large
contribution to the $\chi^{2}$ value. While the region around the $\psi(4415)$
resonance does not appear to be perfectly modelled in the projection, it is
probable that at least some of this is statistical, since a very sharp
structure at $m({{D}^{+}}{{D}^{-}})\sim
4.47\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$ seems unlikely to be physical.

In summary, while the baseline model does not perfectly reproduce the observed
Dalitz-plot distribution, it gives the best description of the currently
available data, with a stable fit, among a large range of considered models.
Analysis of a larger sample in future will be of great interest to resolve
issues associated with the imperfections of the baseline model, as will
improved knowledge of ${{D}^{+}}{{D}^{-}}$ and ${{D}^{-}}{{K}^{+}}$ structures
that may be obtained by analysis of other systems.

##  9 Systematic uncertainties

Systematic uncertainties arising from a variety of sources are investigated,
and their impact on the model amplitudes, phases, and fit fractions is
quantified. The effects on the masses and widths of resonances that are
determined from the fit are also evaluated. Sources of systematic uncertainty
are separated into those related to experimental effects and those related to
model composition. The various systematic uncertainties on the complex
coefficients and fit fractions are detailed in Table 8, while those on the
masses and widths of resonances are given in Table 9.

The yield of the signal component in the amplitude fit is fixed according to
the results of the invariant-mass fit. Repeats of the amplitude fit to data
are performed where the signal yield is varied, each time being sampled from a
Gaussian PDF centred at the value obtained from the invariant-mass fit, having
a width equal to the statistical uncertainty on that yield. The RMS of the
values of the fit parameters is taken as the systematic uncertainty. The
magnitude of this uncertainty is negligible, and it is therefore omitted from
Table 8.

The PDF used to model the signal component in the invariant-mass fit may be
imperfect. A conservative estimate of the impact of mismodelling the signal
shape is obtained by replacing the DSCB shape by a simple Gaussian function.
The deviation of the fit parameters from their nominal values is taken as an
estimate of the systematic uncertainty.

The size of the sideband sample limits knowledge of the residual background
model in the amplitude fit. An ensemble of bootstrapped sideband data is
prepared, from which an ensemble of background models is extracted. Repeated
fits to data using the different models are performed, and the RMS of the fit
parameters in the resulting ensemble of fit results is taken to represent the
systematic uncertainty. This uncertainty is negligible, and is therefore
omitted from Table 8.

The effect of the limited size of the simulated samples, used to determine the
efficiency model, is quantified. A large ensemble of simulated samples is
prepared by bootstrapping the original sample, such that variations within the
ensemble are representative of statistical fluctuations expected for the size
of that sample. For each variant the efficiency is obtained for Run 1 and Run
2 in the same way as for the nominal efficiency model. The fit to data is then
repeated once per efficiency-model-variant, and the RMS of the values of the
fit parameters is taken to represent the systematic uncertainty.

The PID response in the data is obtained from calibration samples. The
systematic uncertainty incurred through this procedure principally arises from
the kernel width used in the estimation of the PDFs. An alternative PID
response is simulated using an alternative kernel estimation with changed
width, and the efficiency models are regenerated. The fit to data is repeated
with these alternative efficiency models in place and the absolute change in
the fit parameters is taken as the systematic uncertainty. This uncertainty is
omitted from Table 8 since it is negligible.

The hardware-level trigger decision is not expected to be perfectly modelled
in the simulated samples. To estimate the impact of this mismodelling of this
trigger, a correction obtained from data control samples is applied to the
efficiency map. The fit is repeated with this alternative efficiency map and
displacement in each parameter is computed. This procedure overestimates the
effect, since the mismodelling only affects the efficiency for candidates
triggered by hardware-level hadron requirement. Each displacement is therefore
scaled according to the fraction of such candidates (64%) to evaluate the
systematic uncertainty.

The default Blatt–Weisskopf barrier radii for the parent and intermediate
resonances are set to $4.0\text{\,Ge\kern-1.00006ptV}^{-1}$. To evaluate the
systematic uncertainty arising from the fixed radii, the fit to data is
repeated where the radius for each category — parent, charmonia or
${{D}^{-}}{{K}^{+}}$ resonances — is sampled randomly from a Gaussian
distribution centred at $4\text{\,Ge\kern-1.00006ptV}^{-1}$ and with a width
of $1\text{\,Ge\kern-1.00006ptV}^{-1}$, which is the approximate size of the
uncertainty on the Blatt–Weisskopf barrier radii measured in comparable
systems [47]. The RMS of the values of the fit parameters under these
perturbations is taken to represent the systematic uncertainty, where the
largest effect is seen when varying the Blatt–Weisskopf barrier radius of the
charmonium resonances, which dominate the model. This is the largest
systematic uncertainty for several of the parameters determined from the fit.

The baseline model includes contributions that are clearly established, but
the true amplitude may include components that are not significant at the
current level of precision and which are consequently omitted. In addition,
the most appropriate way to model some of the components is not established,
and mismodelling is a source of potential systematic uncertainty. While many
possible model variations could be considered, including too many would lead
to an artificial inflation of the uncertainty. Therefore this procedure is
limited to specific variations in the partial waves where the modelling
uncertainty is largest. With reference to the discussion in Sec. 7.1, these
are

  * •

${{D}^{+}}{{D}^{-}}$ S wave: Inclusion of an additional constant nonresonant
component. Introducing such a component with a freely varying complex
coefficient, alongside the existing nonresonant shape, destabilises the fit so
instead the amplitude and phase are chosen such that the new component
acquires a fit fraction of 5%.

  * •

${{D}^{+}}{{D}^{-}}$ P wave: Inclusion of the $\psi(4320)$ state, with fixed
parameters [40].

These effects related to the composition of the amplitude model constitute the
largest systematic uncertainty for many of the parameters determined from the
fit.

The statistical behaviour of the fit is investigated using pseudoexperiments,
and the outcome of this study is used to correct the results of the fit to
data as summarised in Table 8. The model obtained from the best fit to data is
used to generate an ensemble of datasets. Each dataset includes the efficiency
variation across the Dalitz plot and a background contribution, the yield of
which is sampled for each pseudoexperiment from a Poisson distribution centred
at the observed background yield in data. Separate datasets are generated for
Run 1 and Run 2 data. The standard fit is then applied to each dataset, where
the signal yield is fixed to the generated value. Both the residual,
$\left(P_{\rm fit}-P_{\rm gen}\right)$, and normalised residual or “pull”,
$\left(P_{\rm fit}-P_{\rm gen}\right)/\sigma_{\rm fit}$, are determined for
the value $P$ of each parameter, determined with uncertainty $\sigma_{\rm
fit}$, in the fit to each dataset. The distribution of the residual for each
fit parameter is fitted with a Gaussian function and the mean (“Bias”) is used
to correct the central value. The pull distribution for each fit parameter is
also fitted with a Gaussian function, and the obtained width (“Pull width”) is
used to scale the reported statistical uncertainty for the parameter. For the
fit fractions, which are calculated from the fitted complex coefficients, the
width obtained from the fit of the distribution of the residuals with a
Gaussian function is taken as the statistical uncertainty.

Table 8: 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).

Parameter | Raw | Bias | Pull width | Corrected | (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (total)
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\psi(3770)$ | Magnitude | 1 (fixed) |  |  |  |  |  |  |  |  |  |  |  |
| Phase (rad) | 0 (fixed) |  |  |  |  |  |  |  |  |  |  |  |
| Fit fraction | $0.144$ | $-0.001$ |  |  $-$$0.145$ $\pm$ | $0.012$ | $0.001$ | $0.001$ | $0.002$ | $0.000$ | $0.006$ | $0.000$ | $0.001$ | $0.004$ | $0.008$
$\chi_{c0}(3930)$ | Magnitude |  $-$$0.515$ $\pm$ | $0.068$ | $0.009$ | $0.945$ |  $-$$0.506$ $\pm$ | $0.064$ | $0.001$ | $0.003$ | $0.001$ | $0.003$ | $0.016$ | $0.002$ | $0.003$ | $0.004$ | $0.017$
| Phase (rad) |  $-$$2.168$ $\pm$ | $0.189$ | $0.006$ | $0.970$ |  $-$$2.162$ $\pm$ | $0.184$ | $0.000$ | $0.011$ | $0.003$ | $0.003$ | $0.027$ | $0.009$ | $0.011$ | $0.012$ | $0.034$
| Fit fraction | $0.038$ | $0.001$ |  |  $-$$0.037$ $\pm$ | $0.009$ | $0.000$ | $0.001$ | $0.001$ | $0.000$ | $0.001$ | $0.000$ | $0.001$ | $0.001$ | $0.002$
$\chi_{c2}(3930)$ | Magnitude |  $-$$0.704$ $\pm$ | $0.064$ | $0.001$ | $1.011$ |  $-$$0.703$ $\pm$ | $0.064$ | $0.001$ | $0.003$ | $0.005$ | $0.004$ | $0.006$ | $0.001$ | $0.006$ | $0.003$ | $0.012$
| Phase (rad) |  $-$$0.783$ $\pm$ | $0.170$ | $-0.044$ | $1.003$ |  $-$$0.827$ $\pm$ | $0.170$ | $0.002$ | $0.008$ | $0.002$ | $0.002$ | $0.011$ | $0.013$ | $0.036$ | $0.127$ | $0.133$
| Fit fraction | $0.072$ | $-0.001$ |  |  $-$$0.072$ $\pm$ | $0.012$ | $0.001$ | $0.001$ | $0.000$ | $0.001$ | $0.002$ | $0.000$ | $0.000$ | $0.003$ | $0.003$
$\psi(4040)$ | Magnitude |  $-$$0.608$ $\pm$ | $0.077$ | $0.024$ | $1.002$ |  $-$$0.585$ $\pm$ | $0.078$ | $0.003$ | $0.005$ | $0.003$ | $0.001$ | $0.021$ | $0.007$ | $0.009$ | $0.024$ | $0.035$
| Phase (rad) |  $-$$1.384$ $\pm$ | $0.188$ | $-0.032$ | $0.938$ |  $-$$1.416$ $\pm$ | $0.176$ | $0.005$ | $0.005$ | $0.000$ | $0.001$ | $0.075$ | $0.001$ | $0.035$ | $0.006$ | $0.084$
| Fit fraction | $0.053$ | $0.003$ |  |  $-$$0.050$ $\pm$ | $0.013$ | $0.000$ | $0.001$ | $0.000$ | $0.000$ | $0.002$ | $0.001$ | $0.002$ | $0.003$ | $0.004$
$\psi(4160)$ | Magnitude |  $-$$0.670$ $\pm$ | $0.081$ | $0.002$ | $1.040$ |  $-$$0.668$ $\pm$ | $0.084$ | $0.003$ | $0.005$ | $0.004$ | $0.000$ | $0.014$ | $0.004$ | $0.015$ | $0.047$ | $0.052$
| Phase (rad) |  $-$$0.805$ $\pm$ | $0.232$ | $-0.093$ | $0.970$ |  $-$$0.898$ $\pm$ | $0.225$ | $0.015$ | $0.005$ | $0.003$ | $0.002$ | $0.084$ | $0.011$ | $0.027$ | $0.014$ | $0.092$
| Fit fraction | $0.065$ | $-0.001$ |  |  $-$$0.066$ $\pm$ | $0.015$ | $0.000$ | $0.000$ | $0.000$ | $0.000$ | $0.000$ | $0.001$ | $0.001$ | $0.012$ | $0.012$
$\psi(4415)$ | Magnitude |  $-$$0.770$ $\pm$ | $0.083$ | $-0.027$ | $0.962$ |  $-$$0.797$ $\pm$ | $0.080$ | $0.000$ | $0.005$ | $0.009$ | $0.005$ | $0.011$ | $0.015$ | $0.053$ | $0.022$ | $0.061$
| Phase (rad) |  $-1.606$ $\pm$ | $0.229$ | $-0.148$ | $0.860$ |  $-1.458$ $\pm$ | $0.197$ | $0.001$ | $0.015$ | $0.006$ | $0.013$ | $0.009$ | $0.023$ | $0.083$ | $0.019$ | $0.091$
| Fit fraction | $0.086$ | $-0.006$ |  |  $-$$0.092$ $\pm$ | $0.014$ | $0.000$ | $0.001$ | $0.001$ | $0.001$ | $0.006$ | $0.004$ | $0.011$ | $0.008$ | $0.015$
$X_{0}(2900)$ | Magnitude |  $-$$0.628$ $\pm$ | $0.080$ | $0.009$ | $0.982$ |  $-$$0.619$ $\pm$ | $0.079$ | $0.003$ | $0.006$ | $0.001$ | $0.001$ | $0.001$ | $0.002$ | $0.005$ | $0.024$ | $0.025$
| Phase (rad) |  $-$$1.076$ $\pm$ | $0.201$ | $-0.015$ | $0.961$ |  $-$$1.091$ $\pm$ | $0.193$ | $0.002$ | $0.008$ | $0.005$ | $0.005$ | $0.036$ | $0.001$ | $0.007$ | $0.086$ | $0.095$
| Fit fraction | $0.057$ | $0.001$ |  |  $-$$0.056$ $\pm$ | $0.014$ | $0.000$ | $0.001$ | $0.001$ | $0.000$ | $0.002$ | $0.000$ | $0.003$ | $0.003$ | $0.005$
$X_{1}(2900)$ | Magnitude |  $-$$1.432$ $\pm$ | $0.085$ | $-0.018$ | $1.010$ |  $-$$1.449$ $\pm$ | $0.086$ | $0.005$ | $0.010$ | $0.010$ | $0.002$ | $0.012$ | $0.001$ | $0.011$ | $0.023$ | $0.032$
| Phase (rad) |  $-$$0.370$ $\pm$ | $0.109$ | $0.003$ | $0.939$ |  $-$$0.367$ $\pm$ | $0.102$ | $0.003$ | $0.009$ | $0.003$ | $0.000$ | $0.034$ | $0.006$ | $0.003$ | $0.033$ | $0.049$
| Fit fraction | $0.296$ | $-0.010$ |  |  $-$$0.306$ $\pm$ | $0.024$ | $0.000$ | $0.002$ | $0.000$ | $0.001$ | $0.007$ | $0.000$ | $0.006$ | $0.019$ | $0.021$
Nonres | Magnitude |  $-$$1.301$ $\pm$ | $0.087$ | $0.008$ | $1.007$ |  $-$$1.293$ $\pm$ | $0.088$ | $0.006$ | $0.009$ | $0.010$ | $0.003$ | $0.029$ | $0.003$ | $0.002$ | $0.029$ | $0.043$
| Phase (rad) |  $-2.466$ $\pm$ | $0.125$ | $-0.056$ | $0.953$ |  $-2.410$ $\pm$ | $0.119$ | $0.003$ | $0.011$ | $0.004$ | $0.003$ | $0.025$ | $0.011$ | $0.492$ | $0.123$ | $0.508$
| Fit fraction | $0.244$ | $0.002$ |  |  $-$$0.242$ $\pm$ | $0.022$ | $0.001$ | $0.001$ | $0.001$ | $0.001$ | $0.001$ | $0.000$ | $0.001$ | $0.004$ | $0.005$

Table 9: Systematic uncertainties on the masses
($\text{Ge\kern-1.00006ptV\\!/}c^{2}$) and widths (GeV) 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).

Parameter | Raw | Bias | Pull width | Corrected | (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (total)
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\chi_{c0}(3930)$ | Mass |  $-$$3.9239$ $\pm$ | $0.0015$ | $0.0002$ | $1.0004$ |  $-$$3.9238$ $\pm$ | $0.0015$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0001$ | $0.0004$ | $0.0004$
| Width |  $-$$0.0170$ $\pm$ | $0.0050$ | $-0.0004$ | $1.0255$ |  $-$$0.0174$ $\pm$ | $0.0051$ | $0.0001$ | $0.0002$ | $0.0002$ | $0.0001$ | $0.0005$ | $0.0000$ | $0.0003$ | $0.0003$ | $0.0008$
$\chi_{c2}(3930)$ | Mass |  $-$$3.9262$ $\pm$ | $0.0023$ | $-0.0006$ | $1.0500$ |  $-$$3.9268$ $\pm$ | $0.0024$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0000$ | $0.0002$ | $0.0000$ | $0.0004$ | $0.0006$ | $0.0008$
| Width |  $-$$0.0337$ $\pm$ | $0.0064$ | $-0.0005$ | $1.0287$ |  $-$$0.0342$ $\pm$ | $0.0066$ | $0.0000$ | $0.0002$ | $0.0002$ | $0.0003$ | $0.0005$ | $0.0001$ | $0.0003$ | $0.0009$ | $0.0011$
$X_{0}(2900)$ | Mass |  $-$$2.8670$ $\pm$ | $0.0063$ | $0.0007$ | $1.0262$ |  $-$$2.8663$ $\pm$ | $0.0065$ | $0.0000$ | $0.0002$ | $0.0000$ | $0.0002$ | $0.0003$ | $0.0002$ | $0.0002$ | $0.0019$ | $0.0020$
| Width |  $-$$0.0570$ $\pm$ | $0.0121$ | $-0.0003$ | $1.0123$ |  $-$$0.0572$ $\pm$ | $0.0122$ | $0.0006$ | $0.0004$ | $0.0004$ | $0.0001$ | $0.0007$ | $0.0005$ | $0.0007$ | $0.0038$ | $0.0041$
$X_{1}(2900)$ | Mass |  $-$$2.9053$ $\pm$ | $0.0050$ | $0.0011$ | $0.9629$ |  $-$$2.9041$ $\pm$ | $0.0048$ | $0.0002$ | $0.0001$ | $0.0002$ | $0.0001$ | $0.0007$ | $0.0003$ | $0.0004$ | $0.0009$ | $0.0013$
| Width |  $-$$0.1088$ $\pm$ | $0.0105$ | $-0.0015$ | $1.0154$ |  $-$$0.1103$ $\pm$ | $0.0107$ | $0.0008$ | $0.0003$ | $0.0004$ | $0.0002$ | $0.0012$ | $0.0010$ | $0.0001$ | $0.0038$ | $0.0043$

##  10 Significance of resonant structures

Pseudoexperiments are used to determine the significance of the
${{D}^{-}}{{K}^{+}}$ structure. The pseudoexperiments are generated using an
amplitude model where no ${{D}^{-}}{{K}^{+}}$ resonances are included, with
parameters obtained by fitting the data (see Sec. 8.1). For each dataset, the
yields of the signal and background components are sampled from a Poisson
distribution centred at the yields observed in the data, and the efficiency is
applied to the signal component. Each dataset is fitted with both the model
used for generation ($H_{0}$) and the baseline fit model ($H_{1}$) and the
test statistic $t=-2(\log(\mathcal{L}(H_{1})-\log(\mathcal{L}(H_{0})))$ is
determined. The test statistic observed in data is compared to the
distribution from the pseudoexperiments in Fig. 15(a), where the preference
for the nominal hypothesis is overwhelming. These results confirm those of
Sec. 8.3.

The significance of the $X_{1}(2900)$ and $X_{0}(2900)$ states in this
amplitude analysis is much larger than the significance of exotic
contributions obtained in the model-independent analysis of the same data
sample [17]. This is expected since in the model-independent analysis the
contributions from S, P and D waves in the ${D}^{+}$ ${D}^{-}$ system are
independent in each $m({{D}^{+}}{{D}^{-}})$ bin, while in the amplitude
analysis each partial wave is a continuous function of $m({{D}^{+}}{{D}^{-}})$
that is prescribed by the model. The amplitude analysis consequently has less
freedom to absorb any structure in the $m({{D}^{-}}{{K}^{+}})$ distribution
compared to the model-independent approach, unless explicit components are
included to describe it, and correspondingly a higher significance is
obtained.

A similar approach is taken to determine the significance of the presence of
both spin-0 and spin-2 states in the $\chi_{cJ}(3930)$ region. Three
alternative configurations are considered, where these two components are
replaced by a single resonance, having spin 0, 1, or 2. The results are shown
in Fig. 15. The smallest, though still compelling, significance of the two
state fit occurs when comparing to a single spin-1 resonance in the
$\chi_{cJ}(3930)$ region. Hence the need for two states in this region is
clearly established. These results also confirm those of Sec. 8.3, where
issues with fits including a spin-1 state in the $\chi_{cJ}(3930)$ region are
discussed, leaving the configuration with spins 0 and 2 as the only candidate
to describe the data.

Figure 15: 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.

##  11 Summary

The first amplitude analysis of the
${{{B}^{+}}}\rightarrow{{D}^{+}}{{D}^{-}}{{K}^{+}}$ decay has been carried
out. The analysis is performed using LHCb $pp$ collision data taken at
$\sqrt{s}=7,8,$ and $13\text{\,Te\kern-1.00006ptV}$, corresponding to a total
integrated luminosity of 9$\text{\,fb}^{-1}$, from which a highly pure sample
of 1260 signal candidates are selected.

It is not possible to describe the distribution across the Dalitz plot using
only resonances in the ${{D}^{+}}{{D}^{-}}$ system; this conclusion is
supported by a model-independent analysis of the same data sample [17].
Reasonable agreement with the data is achieved by including new spin-0 and
spin-1 resonances in the ${{D}^{-}}{{K}^{+}}$ channel, described with
Breit–Wigner lineshapes, the parameters of which are determined to be

$\displaystyle X_{0}(2900):\quad
M=$2.8663$\pm$0.0065$\pm$0.0020$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}\,,$
$\displaystyle\Gamma=\phantom{0}$57.2$\pm$12.2$\pm$4.1$\text{\,Me\kern-1.00006ptV}\,,$
$\displaystyle X_{1}(2900):\quad
M=$2.9041$\pm$0.0048$\pm$0.0013$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}\,,$
$\displaystyle\Gamma=$110.3$\pm$10.7$\pm$4.3$\text{\,Me\kern-1.00006ptV}\,,$

where the first uncertainties are statistical and the second systematic. While
the significance of these contributions is overwhelming, and this model gives
a good description of the data in this region, it cannot be ruled out that
alternative models incorporating additional hadronic effects such as
rescattering may also be able to accommodate these ${{D}^{-}}{{K}^{+}}$
structures. Nonetheless, if the ${{D}^{-}}{{K}^{+}}$ structures are
interpreted as resonances, these results constitute the first clear
observation of exotic hadrons with open flavour, and the first that do not
contain a heavy quark-antiquark pair. More detailed investigations will
require larger data samples and studies of additional decay modes. For
example, it will be interesting to see if similar structures can be observed
in ${{{B}^{+}}}\rightarrow{{D}^{-}}{{K}^{+}}{{\pi}^{+}}$ decays, where an
analysis of a subset of the existing LHCb data sample [78] gave an indication
of an excess — though not statistically significant — in the
$m({{D}^{-}}{{K}^{+}})$ region where structure is now observed.

The model also includes contributions from the $\psi(3770)$, $\psi(4040)$,
$\psi(4160)$ and $\psi(4415)$ vector charmonia states. In addition, it is
found necessary to include both spin-0 and spin-2 states in the
$\chi_{cJ}(3930)$ region, the parameters of which are determined from the fit
to be

$\displaystyle\chi_{c0}(3930):\quad M=3.9238\pm 0.0015\pm
0.0004\text{\,Ge\kern-1.00006ptV\\!/}c^{2}\,,$ $\displaystyle\Gamma=17.4\pm
5.1\pm 0.8\text{\,Me\kern-1.00006ptV}\,,$ $\displaystyle\chi_{c2}(3930):\quad
M=3.9268\pm 0.0024\pm 0.0008\text{\,Ge\kern-1.00006ptV\\!/}c^{2}\,,$
$\displaystyle\Gamma=34.2\pm 6.6\pm 1.1\text{\,Me\kern-1.00006ptV}\,.$

Previous measurements of the properties of the $\chi_{c2}(3930)$ state have
assumed a single state in this region and, in the light of these results, may
be unreliable. There is no evidence for the $\chi_{c0}(3860)$ state reported
by the Belle collaboration [56]. Further investigation and independent
confirmation of these results concerning spin-0 and spin-2 charmonium states
may be obtained in future by studies of
${{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}\omega{{K}^{+}}$
decays.

The size and purity of the sample demonstrates the potential impact of further
studies of $B\rightarrow D{\kern 1.79993pt\overline{\kern-1.79993ptD}}K$
decays in the LHCb dataset. In particular, the
${{{B}^{+}}}\rightarrow{{D}^{0}}{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}{{K}^{+}}$ mode is likely to shed
further light on the production of charmonium states in $B$-meson decays,
while analysis of ${{B}^{0}}\rightarrow{{D}^{0}}{{D}^{-}}{{K}^{+}}$ may
provide crucial additional information on the ${{D}^{-}}{{K}^{+}}$ structures.
In both cases, however, contributions from ${{D}^{+}_{s}}$ excitations
decaying to ${{D}^{0}}{{K}^{+}}$ will also need to be considered. The
significantly larger sample anticipated to be collected by LHCb with an
upgraded detector during Run 3 of the Large Hadron Collider also provides
exciting prospects for further discoveries in this area.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST
and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN
(Italy); NWO (Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MSHE
(Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC
(United Kingdom); DOE NP and NSF (USA). We acknowledge the computing resources
that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN
(Italy), SURF (Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and
Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-
GRID (Poland) and OSC (USA). We are indebted to the communities behind the
multiple open-source software packages on which we depend. Individual groups
or members have received support from AvH Foundation (Germany); EPLANET, Marie
Skłodowska-Curie Actions and ERC (European Union); A*MIDEX, ANR, Labex P2IO
and OCEVU, and Région Auvergne-Rhône-Alpes (France); Key Research Program of
Frontier Sciences of CAS, CAS PIFI, and the Thousand Talents Program (China);
RFBR, RSF and Yandex LLC (Russia); GVA, XuntaGal and GENCAT (Spain); the Royal
Society and the Leverhulme Trust (United Kingdom).

## Appendices

##  Appendix A Helicity-angle distributions in slices of Dalitz-plot variables

To allow detailed inspection of the agreement between the result of the fit
and the data, helicity angle distributions are shown in slices of the three
invariant mass-squared combinations. Figure 16 defines the slices for these
projections, with the helicity angle distributions themselves shown in Figs.
17–18.

Figure 16: 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.

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.

##  Appendix B Angular moments

The angular moments of the data, in bins of $m({{D}^{+}}{{D}^{-}})$, are
central to the model-independent analysis presented in Ref. [17]. They also
present a further way of checking the agreement between the result of the fit
and the data. Moments 1–5, for each of $m({{D}^{+}}{{D}^{-}})$,
$m({{D}^{-}}{{K}^{+}})$ and $m({{D}^{+}}{{K}^{+}})$ are presented in Fig. 19,
with moments 6–9 in Fig. 20.

Figure 19: 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. 10.

Figure 20: 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. 10.

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LHCb collaboration

R. Aaij31, C. Abellán Beteta49, T. Ackernley59, B. Adeva45, M. Adinolfi53, H.
Afsharnia9, C.A. Aidala84, S. Aiola25, Z. Ajaltouni9, S. Akar64, J.
Albrecht14, F. Alessio47, M. Alexander58, A. Alfonso Albero44, Z. Aliouche61,
G. Alkhazov37, P. Alvarez Cartelle47, S. Amato2, Y. Amhis11, L. An21, L.
Anderlini21, A. Andreianov37, M. Andreotti20, F. Archilli16, A. Artamonov43,
M. Artuso67, K. Arzymatov41, E. Aslanides10, M. Atzeni49, B. Audurier11, S.
Bachmann16, M. Bachmayer48, J.J. Back55, S. Baker60, P. Baladron Rodriguez45,
V. Balagura11, W. Baldini20, J. Baptista Leite1, R.J. Barlow61, S. Barsuk11,
W. Barter60, M. Bartolini23,h, F. Baryshnikov80, J.M. Basels13, G. Bassi28, B.
Batsukh67, A. Battig14, A. Bay48, M. Becker14, F. Bedeschi28, I. Bediaga1, A.
Beiter67, V. Belavin41, S. Belin26, V. Bellee48, K. Belous43, I. Belov39, I.
Belyaev38, G. Bencivenni22, E. Ben-Haim12, A. Berezhnoy39, R. Bernet49, D.
Berninghoff16, H.C. Bernstein67, C. Bertella47, E. Bertholet12, A. Bertolin27,
C. Betancourt49, F. Betti19,e, M.O. Bettler54, Ia. Bezshyiko49, S. Bhasin53,
J. Bhom33, L. Bian72, M.S. Bieker14, S. Bifani52, P. Billoir12, M. Birch60,
F.C.R. Bishop54, A. Bizzeti21,r, M. Bjørn62, M.P. Blago47, T. Blake55, F.
Blanc48, S. Blusk67, D. Bobulska58, J.A. Boelhauve14, O. Boente Garcia45, T.
Boettcher63, A. Boldyrev81, A. Bondar42,u, N. Bondar37, S. Borghi61, M.
Borisyak41, M. Borsato16, J.T. Borsuk33, S.A. Bouchiba48, T.J.V. Bowcock59, A.
Boyer47, C. Bozzi20, M.J. Bradley60, S. Braun65, A. Brea Rodriguez45, M.
Brodski47, J. Brodzicka33, A. Brossa Gonzalo55, D. Brundu26, A. Buonaura49, C.
Burr47, A. Bursche26, A. Butkevich40, J.S. Butter31, J. Buytaert47, W.
Byczynski47, S. Cadeddu26, H. Cai72, R. Calabrese20,g, L. Calefice14, L.
Calero Diaz22, S. Cali22, R. Calladine52, M. Calvi24,i, M. Calvo Gomez83, P.
Camargo Magalhaes53, A. Camboni44, P. Campana22, D.H. Campora Perez47, A.F.
Campoverde Quezada5, S. Capelli24,i, L. Capriotti19,e, A. Carbone19,e, G.
Carboni29, R. Cardinale23,h, A. Cardini26, I. Carli6, P. Carniti24,i, K.
Carvalho Akiba31, A. Casais Vidal45, G. Casse59, M. Cattaneo47, G.
Cavallero47, S. Celani48, J. Cerasoli10, A.J. Chadwick59, M.G. Chapman53, M.
Charles12, Ph. Charpentier47, G. Chatzikonstantinidis52, C.A. Chavez
Barajas59, M. Chefdeville8, C. Chen3, S. Chen26, A. Chernov33, S.-G. Chitic47,
V. Chobanova45, S. Cholak48, M. Chrzaszcz33, A. Chubykin37, V. Chulikov37, P.
Ciambrone22, M.F. Cicala55, X. Cid Vidal45, G. Ciezarek47, P.E.L. Clarke57, M.
Clemencic47, H.V. Cliff54, J. Closier47, J.L. Cobbledick61, V. Coco47, J.A.B.
Coelho11, J. Cogan10, E. Cogneras9, L. Cojocariu36, P. Collins47, T.
Colombo47, L. Congedo18, A. Contu26, N. Cooke52, G. Coombs58, G. Corti47, C.M.
Costa Sobral55, B. Couturier47, D.C. Craik63, J. Crkovská66, M. Cruz Torres1,
R. Currie57, C.L. Da Silva66, E. Dall’Occo14, J. Dalseno45, C. D’Ambrosio47,
A. Danilina38, P. d’Argent47, A. Davis61, O. De Aguiar Francisco61, K. De
Bruyn77, S. De Capua61, M. De Cian48, J.M. De Miranda1, L. De Paula2, M. De
Serio18,d, D. De Simone49, P. De Simone22, J.A. de Vries78, C.T. Dean66, W.
Dean84, D. Decamp8, L. Del Buono12, B. Delaney54, H.-P. Dembinski14, A.
Dendek34, V. Denysenko49, D. Derkach81, O. Deschamps9, F. Desse11, F.
Dettori26,f, B. Dey72, P. Di Nezza22, S. Didenko80, L. Dieste Maronas45, H.
Dijkstra47, V. Dobishuk51, A.M. Donohoe17, F. Dordei26, A.C. dos Reis1, L.
Douglas58, A. Dovbnya50, A.G. Downes8, K. Dreimanis59, M.W. Dudek33, L.
Dufour47, V. Duk76, P. Durante47, J.M. Durham66, D. Dutta61, M. Dziewiecki16,
A. Dziurda33, A. Dzyuba37, S. Easo56, U. Egede68, V. Egorychev38, S.
Eidelman42,u, S. Eisenhardt57, S. Ek-In48, L. Eklund58, S. Ely67, A. Ene36, E.
Epple66, S. Escher13, J. Eschle49, S. Esen31, T. Evans47, A. Falabella19, J.
Fan3, Y. Fan5, B. Fang72, N. Farley52, S. Farry59, D. Fazzini24,i, P. Fedin38,
M. Féo47, P. Fernandez Declara47, A. Fernandez Prieto45, F. Ferrari19,e, L.
Ferreira Lopes48, F. Ferreira Rodrigues2, S. Ferreres Sole31, M. Ferrillo49,
M. Ferro-Luzzi47, S. Filippov40, R.A. Fini18, M. Fiorini20,g, M. Firlej34,
K.M. Fischer62, C. Fitzpatrick61, T. Fiutowski34, F. Fleuret11,b, M.
Fontana47, F. Fontanelli23,h, R. Forty47, V. Franco Lima59, M. Franco
Sevilla65, M. Frank47, E. Franzoso20, G. Frau16, C. Frei47, D.A. Friday58, J.
Fu25, Q. Fuehring14, W. Funk47, E. Gabriel31, T. Gaintseva41, A. Gallas
Torreira45, D. Galli19,e, S. Gambetta57, Y. Gan3, M. Gandelman2, P. Gandini25,
Y. Gao4, M. Garau26, L.M. Garcia Martin55, P. Garcia Moreno44, J. García
Pardiñas49, B. Garcia Plana45, F.A. Garcia Rosales11, L. Garrido44, D.
Gascon44, C. Gaspar47, R.E. Geertsema31, D. Gerick16, L.L. Gerken14, E.
Gersabeck61, M. Gersabeck61, T. Gershon55, D. Gerstel10, Ph. Ghez8, V.
Gibson54, M. Giovannetti22,j, A. Gioventù45, P. Gironella Gironell44, L.
Giubega36, C. Giugliano20,g, K. Gizdov57, V.V. Gligorov12, C. Göbel69, E.
Golobardes83, D. Golubkov38, A. Golutvin60,80, A. Gomes1,a, S. Gomez
Fernandez44, M. Goncerz33, P. Gorbounov38, I.V. Gorelov39, C. Gotti24, E.
Govorkova31, J.P. Grabowski16, R. Graciani Diaz44, T. Grammatico12, L.A.
Granado Cardoso47, E. Graugés44, E. Graverini48, G. Graziani21, A. Grecu36,
L.M. Greeven31, P. Griffith20, L. Grillo61, S. Gromov80, L. Gruber47, B.R.
Gruberg Cazon62, C. Gu3, M. Guarise20, P. A. Günther16, E. Gushchin40, A.
Guth13, Y. Guz43,47, T. Gys47, T. Hadavizadeh68, G. Haefeli48, C. Haen47, J.
Haimberger47, S.C. Haines54, T. Halewood-leagas59, P.M. Hamilton65, Q. Han7,
X. Han16, T.H. Hancock62, S. Hansmann-Menzemer16, N. Harnew62, T. Harrison59,
C. Hasse47, M. Hatch47, J. He5, M. Hecker60, K. Heijhoff31, K. Heinicke14,
A.M. Hennequin47, K. Hennessy59, L. Henry25,46, J. Heuel13, A. Hicheur2, D.
Hill62, M. Hilton61, S.E. Hollitt14, P.H. Hopchev48, J. Hu16, J. Hu71, W. Hu7,
W. Huang5, X. Huang72, W. Hulsbergen31, R.J. Hunter55, M. Hushchyn81, D.
Hutchcroft59, D. Hynds31, P. Ibis14, M. Idzik34, D. Ilin37, P. Ilten52, A.
Inglessi37, K. Ivshin37, R. Jacobsson47, S. Jakobsen47, E. Jans31, B.K.
Jashal46, A. Jawahery65, V. Jevtic14, M. Jezabek33, F. Jiang3, M. John62, D.
Johnson47, C.R. Jones54, T.P. Jones55, B. Jost47, N. Jurik47, S. Kandybei50,
Y. Kang3, M. Karacson47, J.M. Kariuki53, N. Kazeev81, M. Kecke16, F.
Keizer54,47, M. Kenzie55, T. Ketel32, B. Khanji47, A. Kharisova82, S.
Kholodenko43, K.E. Kim67, T. Kirn13, V.S. Kirsebom48, O. Kitouni63, S.
Klaver31, K. Klimaszewski35, S. Koliiev51, A. Kondybayeva80, A.
Konoplyannikov38, P. Kopciewicz34, R. Kopecna16, P. Koppenburg31, M.
Korolev39, I. Kostiuk31,51, O. Kot51, S. Kotriakhova37,30, P. Kravchenko37, L.
Kravchuk40, R.D. Krawczyk47, M. Kreps55, F. Kress60, S. Kretzschmar13, P.
Krokovny42,u, W. Krupa34, W. Krzemien35, W. Kucewicz85,33,k, M. Kucharczyk33,
V. Kudryavtsev42,u, H.S. Kuindersma31, G.J. Kunde66, T. Kvaratskheliya38, D.
Lacarrere47, G. Lafferty61, A. Lai26, A. Lampis26, D. Lancierini49, J.J.
Lane61, R. Lane53, G. Lanfranchi22, C. Langenbruch13, J. Langer14, O.
Lantwin49,80, T. Latham55, F. Lazzari28,s, R. Le Gac10, S.H. Lee84, R.
Lefèvre9, A. Leflat39, S. Legotin80, O. Leroy10, T. Lesiak33, B.
Leverington16, H. Li71, L. Li62, P. Li16, X. Li66, Y. Li6, Y. Li6, Z. Li67, X.
Liang67, T. Lin60, R. Lindner47, V. Lisovskyi14, R. Litvinov26, G. Liu71, H.
Liu5, S. Liu6, X. Liu3, A. Loi26, J. Lomba Castro45, I. Longstaff58, J.H.
Lopes2, G. Loustau49, G.H. Lovell54, Y. Lu6, D. Lucchesi27,l, S. Luchuk40, M.
Lucio Martinez31, V. Lukashenko31, Y. Luo3, A. Lupato61, E. Luppi20,g, O.
Lupton55, A. Lusiani28,q, X. Lyu5, L. Ma6, S. Maccolini19,e, F. Machefert11,
F. Maciuc36, V. Macko48, P. Mackowiak14, S. Maddrell-Mander53, L.R. Madhan
Mohan53, O. Maev37, A. Maevskiy81, D. Maisuzenko37, M.W. Majewski34, S.
Malde62, B. Malecki47, A. Malinin79, T. Maltsev42,u, H. Malygina16, G.
Manca26,f, G. Mancinelli10, R. Manera Escalero44, D. Manuzzi19,e, D.
Marangotto25,n, J. Maratas9,t, J.F. Marchand8, U. Marconi19, S.
Mariani21,47,21, C. Marin Benito11, M. Marinangeli48, P. Marino48, J. Marks16,
P.J. Marshall59, G. Martellotti30, L. Martinazzoli47, M. Martinelli24,i, D.
Martinez Santos45, F. Martinez Vidal46, A. Massafferri1, M. Materok13, R.
Matev47, A. Mathad49, Z. Mathe47, V. Matiunin38, C. Matteuzzi24, K.R.
Mattioli84, A. Mauri31, E. Maurice11,b, J. Mauricio44, M. Mazurek35, M.
McCann60, L. Mcconnell17, T.H. Mcgrath61, A. McNab61, R. McNulty17, J.V.
Mead59, B. Meadows64, C. Meaux10, G. Meier14, N. Meinert75, D. Melnychuk35, S.
Meloni24,i, M. Merk31,78, A. Merli25, L. Meyer Garcia2, M. Mikhasenko47, D.A.
Milanes73, E. Millard55, M.-N. Minard8, L. Minzoni20,g, S.E. Mitchell57, B.
Mitreska61, D.S. Mitzel47, A. Mödden14, R.A. Mohammed62, R.D. Moise60, T.
Mombächer14, I.A. Monroy73, S. Monteil9, M. Morandin27, G. Morello22, M.J.
Morello28,q, J. Moron34, A.B. Morris74, A.G. Morris55, R. Mountain67, H. Mu3,
F. Muheim57, M. Mukherjee7, M. Mulder47, D. Müller47, K. Müller49, C.H.
Murphy62, D. Murray61, P. Muzzetto26, P. Naik53, T. Nakada48, R. Nandakumar56,
T. Nanut48, I. Nasteva2, M. Needham57, I. Neri20,g, N. Neri25,n, S. Neubert74,
N. Neufeld47, R. Newcombe60, T.D. Nguyen48, C. Nguyen-Mau48, E.M. Niel11, S.
Nieswand13, N. Nikitin39, N.S. Nolte47, C. Nunez84, A. Oblakowska-Mucha34, V.
Obraztsov43, D.P. O’Hanlon53, R. Oldeman26,f, C.J.G. Onderwater77, A.
Ossowska33, J.M. Otalora Goicochea2, T. Ovsiannikova38, P. Owen49, A.
Oyanguren46, B. Pagare55, P.R. Pais47, T. Pajero28,47,q, A. Palano18, M.
Palutan22, Y. Pan61, G. Panshin82, A. Papanestis56, M. Pappagallo18, L.L.
Pappalardo20,g, C. Pappenheimer64, W. Parker65, C. Parkes61, C.J. Parkinson45,
B. Passalacqua20, G. Passaleva21, A. Pastore18, M. Patel60, C. Patrignani19,e,
C.J. Pawley78, A. Pearce47, A. Pellegrino31, M. Pepe Altarelli47, S.
Perazzini19, D. Pereima38, P. Perret9, K. Petridis53, A. Petrolini23,h, A.
Petrov79, S. Petrucci57, M. Petruzzo25, A. Philippov41, L. Pica28, M.
Piccini76, B. Pietrzyk8, G. Pietrzyk48, M. Pili62, D. Pinci30, J. Pinzino47,
F. Pisani47, A. Piucci16, Resmi P.K10, V. Placinta36, S. Playfer57, J.
Plews52, M. Plo Casasus45, F. Polci12, M. Poli Lener22, M. Poliakova67, A.
Poluektov10, N. Polukhina80,c, I. Polyakov67, E. Polycarpo2, G.J. Pomery53, S.
Ponce47, A. Popov43, D. Popov5,47, S. Popov41, S. Poslavskii43, K. Prasanth33,
L. Promberger47, C. Prouve45, V. Pugatch51, A. Puig Navarro49, H. Pullen62, G.
Punzi28,m, W. Qian5, J. Qin5, R. Quagliani12, B. Quintana8, N.V. Raab17, R.I.
Rabadan Trejo10, B. Rachwal34, J.H. Rademacker53, M. Rama28, M. Ramos
Pernas55, M.S. Rangel2, F. Ratnikov41,81, G. Raven32, M. Reboud8, F. Redi48,
F. Reiss12, C. Remon Alepuz46, Z. Ren3, V. Renaudin62, R. Ribatti28, S.
Ricciardi56, D.S. Richards56, K. Rinnert59, P. Robbe11, A. Robert12, G.
Robertson57, A.B. Rodrigues48, E. Rodrigues59, J.A. Rodriguez Lopez73, A.
Rollings62, P. Roloff47, V. Romanovskiy43, M. Romero Lamas45, A. Romero
Vidal45, J.D. Roth84, M. Rotondo22, M.S. Rudolph67, T. Ruf47, J. Ruiz Vidal46,
A. Ryzhikov81, J. Ryzka34, J.J. Saborido Silva45, N. Sagidova37, N. Sahoo55,
B. Saitta26,f, C. Sanchez Gras31, C. Sanchez Mayordomo46, R. Santacesaria30,
C. Santamarina Rios45, M. Santimaria22, E. Santovetti29,j, D. Saranin80, G.
Sarpis61, M. Sarpis74, A. Sarti30, C. Satriano30,p, A. Satta29, M. Saur5, D.
Savrina38,39, H. Sazak9, L.G. Scantlebury Smead62, S. Schael13, M.
Schellenberg14, M. Schiller58, H. Schindler47, M. Schmelling15, T.
Schmelzer14, B. Schmidt47, O. Schneider48, A. Schopper47, M. Schubiger31, S.
Schulte48, M.H. Schune11, R. Schwemmer47, B. Sciascia22, A. Sciubba30, S.
Sellam45, A. Semennikov38, M. Senghi Soares32, A. Sergi52,47, N. Serra49, J.
Serrano10, L. Sestini27, A. Seuthe14, P. Seyfert47, D.M. Shangase84, M.
Shapkin43, I. Shchemerov80, L. Shchutska48, T. Shears59, L. Shekhtman42,u, V.
Shevchenko79, E.B. Shields24,i, E. Shmanin80, J.D. Shupperd67, B.G. Siddi20,
R. Silva Coutinho49, G. Simi27, S. Simone18,d, I. Skiba20,g, N. Skidmore74, T.
Skwarnicki67, M.W. Slater52, J.C. Smallwood62, J.G. Smeaton54, A. Smetkina38,
E. Smith13, M. Smith60, A. Snoch31, M. Soares19, L. Soares Lavra9, M.D.
Sokoloff64, F.J.P. Soler58, A. Solovev37, I. Solovyev37, F.L. Souza De
Almeida2, B. Souza De Paula2, B. Spaan14, E. Spadaro Norella25,n, P.
Spradlin58, F. Stagni47, M. Stahl64, S. Stahl47, P. Stefko48, O.
Steinkamp49,80, S. Stemmle16, O. Stenyakin43, H. Stevens14, S. Stone67, M.E.
Stramaglia48, M. Straticiuc36, D. Strekalina80, S. Strokov82, F. Suljik62, J.
Sun26, L. Sun72, Y. Sun65, P. Svihra61, P.N. Swallow52, K. Swientek34, A.
Szabelski35, T. Szumlak34, M. Szymanski47, S. Taneja61, Z. Tang3, T.
Tekampe14, F. Teubert47, E. Thomas47, K.A. Thomson59, M.J. Tilley60, V.
Tisserand9, S. T’Jampens8, M. Tobin6, S. Tolk47, L. Tomassetti20,g, D. Torres
Machado1, D.Y. Tou12, M. Traill58, M.T. Tran48, E. Trifonova80, C. Trippl48,
A. Tsaregorodtsev10, G. Tuci28,m, A. Tully48, N. Tuning31, A. Ukleja35, D.J.
Unverzagt16, A. Usachov31, A. Ustyuzhanin41,81, U. Uwer16, A. Vagner82, V.
Vagnoni19, A. Valassi47, G. Valenti19, N. Valls Canudas44, M. van Beuzekom31,
H. Van Hecke66, E. van Herwijnen80, C.B. Van Hulse17, M. van Veghel77, R.
Vazquez Gomez45, P. Vazquez Regueiro45, C. Vázquez Sierra31, S. Vecchi20, J.J.
Velthuis53, M. Veltri21,o, A. Venkateswaran67, M. Veronesi31, M. Vesterinen55,
D. Vieira64, M. Vieites Diaz48, H. Viemann75, X. Vilasis-Cardona83, E. Vilella
Figueras59, P. Vincent12, G. Vitali28, A. Vollhardt49, D. Vom Bruch12, A.
Vorobyev37, V. Vorobyev42,u, N. Voropaev37, R. Waldi75, J. Walsh28, C. Wang16,
J. Wang3, J. Wang72, J. Wang4, J. Wang6, M. Wang3, R. Wang53, Y. Wang7, Z.
Wang49, D.R. Ward54, H.M. Wark59, N.K. Watson52, S.G. Weber12, D. Websdale60,
C. Weisser63, B.D.C. Westhenry53, D.J. White61, M. Whitehead53, D. Wiedner14,
G. Wilkinson62, M. Wilkinson67, I. Williams54, M. Williams63,68, M.R.J.
Williams57, F.F. Wilson56, W. Wislicki35, M. Witek33, L. Witola16, G.
Wormser11, S.A. Wotton54, H. Wu67, K. Wyllie47, Z. Xiang5, D. Xiao7, Y. Xie7,
H. Xing71, A. Xu4, J. Xu5, L. Xu3, M. Xu7, Q. Xu5, Z. Xu5, Z. Xu4, D. Yang3,
Y. Yang5, Z. Yang3, Z. Yang65, Y. Yao67, L.E. Yeomans59, H. Yin7, J. Yu70, X.
Yuan67, O. Yushchenko43, K.A. Zarebski52, M. Zavertyaev15,c, M. Zdybal33, O.
Zenaiev47, M. Zeng3, D. Zhang7, L. Zhang3, S. Zhang4, Y. Zhang47, A.
Zhelezov16, Y. Zheng5, X. Zhou5, Y. Zhou5, X. Zhu3, V. Zhukov13,39, J.B.
Zonneveld57, S. Zucchelli19,e, D. Zuliani27, G. Zunica61.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4School of Physics State Key Laboratory of Nuclear Physics and Technology,
Peking University, Beijing, China
5University of Chinese Academy of Sciences, Beijing, China
6Institute Of High Energy Physics (IHEP), Beijing, China
7Institute of Particle Physics, Central China Normal University, Wuhan, Hubei,
China
8Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy,
France
9Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
10Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
11Université Paris-Saclay, CNRS/IN2P3, IJCLab, Orsay, France
12LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3,
Paris, France
13I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
14Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
15Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
16Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
17School of Physics, University College Dublin, Dublin, Ireland
18INFN Sezione di Bari, Bari, Italy
19INFN Sezione di Bologna, Bologna, Italy
20INFN Sezione di Ferrara, Ferrara, Italy
21INFN Sezione di Firenze, Firenze, Italy
22INFN Laboratori Nazionali di Frascati, Frascati, Italy
23INFN Sezione di Genova, Genova, Italy
24INFN Sezione di Milano-Bicocca, Milano, Italy
25INFN Sezione di Milano, Milano, Italy
26INFN Sezione di Cagliari, Monserrato, Italy
27Universita degli Studi di Padova, Universita e INFN, Padova, Padova, Italy
28INFN Sezione di Pisa, Pisa, Italy
29INFN Sezione di Roma Tor Vergata, Roma, Italy
30INFN Sezione di Roma La Sapienza, Roma, Italy
31Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
32Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, Netherlands
33Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
34AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
35National Center for Nuclear Research (NCBJ), Warsaw, Poland
36Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
37Petersburg Nuclear Physics Institute NRC Kurchatov Institute (PNPI NRC KI),
Gatchina, Russia
38Institute of Theoretical and Experimental Physics NRC Kurchatov Institute
(ITEP NRC KI), Moscow, Russia, Moscow, Russia
39Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
40Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
41Yandex School of Data Analysis, Moscow, Russia
42Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
43Institute for High Energy Physics NRC Kurchatov Institute (IHEP NRC KI),
Protvino, Russia, Protvino, Russia
44ICCUB, Universitat de Barcelona, Barcelona, Spain
45Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
46Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain
47European Organization for Nuclear Research (CERN), Geneva, Switzerland
48Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
49Physik-Institut, Universität Zürich, Zürich, Switzerland
50NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
51Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
52University of Birmingham, Birmingham, United Kingdom
53H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
54Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
55Department of Physics, University of Warwick, Coventry, United Kingdom
56STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
57School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
58School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
59Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
60Imperial College London, London, United Kingdom
61Department of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
62Department of Physics, University of Oxford, Oxford, United Kingdom
63Massachusetts Institute of Technology, Cambridge, MA, United States
64University of Cincinnati, Cincinnati, OH, United States
65University of Maryland, College Park, MD, United States
66Los Alamos National Laboratory (LANL), Los Alamos, United States
67Syracuse University, Syracuse, NY, United States
68School of Physics and Astronomy, Monash University, Melbourne, Australia,
associated to 55
69Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
70Physics and Micro Electronic College, Hunan University, Changsha City,
China, associated to 7
71Guangdong Provencial Key Laboratory of Nuclear Science, Institute of Quantum
Matter, South China Normal University, Guangzhou, China, associated to 3
72School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
73Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 12
74Universität Bonn - Helmholtz-Institut für Strahlen und Kernphysik, Bonn,
Germany, associated to 16
75Institut für Physik, Universität Rostock, Rostock, Germany, associated to 16
76INFN Sezione di Perugia, Perugia, Italy, associated to 20
77Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 31
78Universiteit Maastricht, Maastricht, Netherlands, associated to 31
79National Research Centre Kurchatov Institute, Moscow, Russia, associated to
38
80National University of Science and Technology “MISIS”, Moscow, Russia,
associated to 38
81National Research University Higher School of Economics, Moscow, Russia,
associated to 41
82National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
38
83DS4DS, La Salle, Universitat Ramon Llull, Barcelona, Spain, associated to 44
84University of Michigan, Ann Arbor, United States, associated to 67
85AGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
lUniversità di Padova, Padova, Italy
mUniversità di Pisa, Pisa, Italy
nUniversità degli Studi di Milano, Milano, Italy
oUniversità di Urbino, Urbino, Italy
pUniversità della Basilicata, Potenza, Italy
qScuola Normale Superiore, Pisa, Italy
rUniversità di Modena e Reggio Emilia, Modena, Italy
sUniversità di Siena, Siena, Italy
tMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
uNovosibirsk State University, Novosibirsk, Russia