EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2018-336 LHCb-PAPER-2018-039

Dalitz plot analysis of the
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{K}^{+}}$ decay

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

The resonant structure of the doubly Cabibbo-suppressed decay
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{K}^{+}}$ is studied for the first
time. The measurement is based on a sample of $pp$-collision data, collected
at a centre-of-mass energy of 8 TeV with the LHCb detector and corresponding
to an integrated luminosity of 2$\mbox{\,fb}^{-1}$. The amplitude analysis of
this decay is performed with the isobar model and a phenomenological model
based on an effective chiral Lagrangian. In both models the S-wave component
in the $K^{-}K^{+}$ system is dominant, with a small contribution of the
$\phi(1020)$ meson and a negligible contribution from tensor resonances. The
$K^{+}K^{-}$ scattering amplitudes for the considered combinations of spin
(0,1) and isospin (0,1) of the two-body system are obtained from the Dalitz
plot fit with the phenomenological decay amplitude.

Published in JHEP 04 (2019) 063.

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

##  1 Introduction

The theoretical treatment of weak decays of charm mesons is very challenging.
The charm quark is not light enough for the reliable application of chiral
perturbation theory, which is successfully applied in predictions of kaon
decays. The charm quark is also not heavy enough for the reliable application
of the factorisation approach and heavy-quark expansion tools, as used in
predictions of properties of b hadrons. The description of charm meson decays
relies on approximate symmetries and phenomenological models. For such models,
the knowledge of branching fractions and the resonant structures, in the case
of multi-body decays, are key inputs. In this paper, the first determination
of the resonant structure of the doubly Cabibbo-suppressed decay
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{K}^{+}}$ is presented.111Charge
conjugation is implied throughout the paper. The analysis is based on a data
sample of $pp$ collisions collected with the LHCb detector, corresponding to
an integrated luminosity of 2 $\mbox{\,fb}^{-1}$ at a centre-of-mass energy of
8 TeV. The determination of the resonant structure of this decay is
complementary to the recent LHCb measurement of its branching fraction [1],
based on the same data set.

The amplitude analysis of the
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{K}^{+}}$ decay is performed using
two methods. The Dalitz plot is fitted with the isobar model, in which the
decay amplitude is a coherent sum of resonant and nonresonant amplitudes [2].
The Dalitz plot is also fitted with a phenomenological model derived from an
effective chiral Lagrangian with resonances [3]. This phenomenological model,
referred to as the multi-meson model, or Triple-M, includes the effects of
coupled channels — $\pi\pi$, $K^{+}K^{-}$, $\pi\eta$, $\eta\eta$ and $\rho\pi$
— in the final state interactions (FSI), in four considered combinations of
spin J and isospin I ($J\\!=\\!0,1$; $I\\!=\\!0,1$). Given the small phase
space of the ${{D}^{+}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{K}^{+}}$ decay and
the lack of tensor resonances with significant coupling to $K^{+}K^{-}$, the
contribution from D-wave is expected to be suppressed.

An additional motivation for the Dalitz plot analysis of the
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{K}^{+}}$ decay is to obtain the
$K^{+}K^{-}$ scattering amplitudes. Most information currently available on
$\pi\pi$ and $K\pi$ scattering is obtained indirectly from meson-nucleon
interactions [4, 5, 6]. In the regime where the momentum transferred to the
nucleon is small enough, the interaction is assumed to be dominated by the
one-pion-exchange amplitude. The asymptotically free incoming meson interacts
with a virtual pion, resulting in what is generally referred as $\pi\pi$ and
$K\pi$ scattering data. The resulting $\pi\pi\rightarrow\pi\pi$ and
$K\pi\rightarrow K\pi$ phase shifts are affected by ambiguities and large
systematic uncertainties. The $\pi\pi\rightarrow K{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}$ scattering was studied both in $\pi
p$ and $\pi n$ reactions [7, 8], and in $p\bar{p}$ annihilation at rest [9].
For the $K{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}\rightarrow K{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}$ scattering, no meson-nucleon data
exists.

Three-body decays of $D$ mesons into kaons and pions are an interesting
alternative for light-meson spectroscopy, as they are complementary to the
meson-nucleon reactions. Large data sets from the $B$-factories and LHCb exist
for these decays. However, it is necessary to isolate the physics of two-body
systems from the rich dynamics of three-body decays, which involve the weak
decay of the $c$ quark, the formation of the mesons and their FSI. This is
achieved with the Triple-M decay amplitude, in which these three stages are
included. The FSI are described in terms of the $K^{+}K^{-}$ scattering
amplitudes for the considered spin-isospin combinations, allowing the
determination of these amplitudes from a fit to the
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{K}^{+}}$ Dalitz plot.

This paper is organised as follows. A brief description of the LHCb detector
is presented in Sec. 2. The signal selection is presented in Sec. 3. In Sec.
4, the efficiency determination and background model are discussed. The
formalism for the Dalitz plot fit is presented in Sec. 5. In Sec. 6, the
results of the fit with the isobar model are presented, whilst the results of
the Dalitz plot fit with the Triple-M amplitude are presented in Sec. 7.
Systematic uncertainties are discussed in Sec. 8. A summary and the
conclusions are presented in Sec. 9.

##  2 Detector and simulation

The LHCb detector [10, 11] 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, 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 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$\mathrm{\,Ge\kern-1.00006ptV}$.222Natural units with
$\hbar=c=1$ are used in this paper. The minimum distance of a track to a
primary vertex (PV), the impact parameter (IP), is measured with a resolution
of $(15+29/\mbox{$p_{\rm T}$}){\,\upmu\rm m}$, where $p_{\rm T}$ is the
component of the momentum transverse to the beam, in GeV. Different types of
charged hadrons are distinguished using information from two ring-imaging
Cherenkov detectors [12]. Photons, electrons and hadrons are identified by a
calorimeter system consisting of scintillating-pad and pre-shower detectors,
an electromagnetic and a hadronic calorimeter. Muons are identified by a
system composed of alternating layers of iron and multi-wire proportional
chambers.

The online event selection is performed by a trigger, 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
$p_{\rm T}$ or a hadron, photon or electron with high transverse energy in the
calorimeters. The software trigger is divided into two parts. The first
employs a partial reconstruction of the candidates from the hardware trigger
and a cut-based selection. In the second stage, a full event reconstruction is
applied and various dedicated algorithms are used in the selection of specific
decays. In this analysis, a dedicated algorithm is used to select
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{K}^{+}}$ decay candidates.

In the simulation, $pp$ collisions are generated using Pythia [13,
*Sjostrand:2007gs] with a specific LHCb configuration [15]. Decays of hadronic
particles are described by EvtGen [16], in which final-state radiation is
generated using Photos [17]. The interaction of the generated particles with
the detector, and its response, are implemented using the Geant4 toolkit [18,
*Agostinelli:2002hh] as described in Ref. [20].

##  3 Candidate selection

The ${{D}^{+}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{K}^{+}}$ decay candidates are
selected offline with requirements that exploit the decay topology by
combining three charged particles identified as kaons according to particle-
identification (PID) criteria. These particles must form a good-quality decay
vertex, detached from the PV. The PV is chosen as that with the smallest value
of $\chi^{2}_{\rm IP}$, where $\chi^{2}_{\rm IP}$ is defined as the difference
in the vertex-fit $\chi^{2}$ of the PV reconstructed with and without the
particle under consideration, in this case the ${D}^{+}$ candidate. The
selection of candidates is based on the distance between the PV and the
${D}^{+}$ decay vertex (the flight distance); the IP of the ${D}^{+}$
candidate; the angle between the reconstructed ${D}^{+}$ momentum vector and
the vector connecting the PV to the decay vertex; the $\chi^{2}$ of the
${D}^{+}$ decay vertex fit; the distance of closest approach between any two
final-state tracks; and the momentum, the transverse momentum and the
$\chi^{2}_{\rm IP}$ of the ${D}^{+}$ candidate and of its decay products. The
invariant mass of the ${D}^{+}$ candidate is required to be within the
interval 1820–1920$\mathrm{\,Me\kern-1.00006ptV}$. In order to suppress the
contamination from $D^{+}_{s}\rightarrow K^{-}K^{+}\pi^{+}\pi^{0}$ decays,
where the neutral pion is not reconstructed and the charged pion is
misidentified as a kaon, more stringent PID requirements are applied to the
kaon candidates with the same charge.

A boosted decision tree (BDT) multivariate classifier [21, 22] is used to
further reduce the combinatorial background. In order to keep the selection
efficiency uniform over the Dalitz plot, the BDT uses only the quantities
related to the $D^{+}$ candidate described above. The BDT is trained using
simulated ${{D}^{+}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{K}^{+}}$ decays for the
signal, and data from the invariant-mass intervals
1820–1840$\mathrm{\,Me\kern-1.00006ptV}$ and
1900–1920$\mathrm{\,Me\kern-1.00006ptV}$ for the background. After the
application of all selection requirements, approximately 0.5% of the events
include more than one signal candidate. All candidates are retained for
further analysis.

The invariant-mass spectrum of the selected
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{K}^{+}}$ sample is shown in Fig.
1. To fit the invariant-mass distribution, the signal probability density
function (PDF) is modeled by a sum of two Gaussian functions with a common
mean and independent widths that are free parameters. The signal model is
validated with simulation. The background PDF is parameterised by an
exponential function. The fitted PDF is overlaid with the mass distribution in
Fig. 1. For the Dalitz plot analysis, only candidates within the range
1861.4–1879.5 $\mathrm{\,Me\kern-1.00006ptV}$ are considered. This interval
corresponds to four times the effective mass resolution, and contains 111
thousand candidates, of which $(90.45\pm 0.07)\%$ correspond to signal.

Figure 1: Invariant-mass spectrum of the $K^{-}K^{+}K^{+}$ candidates with the
fit result overlaid (solid blue line). The orange and green dashed lines
indicate the two Gaussian functions representing the signal and the red dashed
line is the background.

The Dalitz plot of the candidates in the signal region is shown in the left
side of Fig 2. The particle ordering is such that the kaon with charge
opposite to that of the $D^{+}$ meson is always particle 1, and the same-sign
kaons are randomly assigned particles 2 and 3, i.e. $D^{+}\rightarrow
K^{-}(p_{1})K^{+}(p_{2})K^{+}(p_{3})$, where $p_{i}$ are the four-momenta. The
Dalitz plot is represented in terms of the square of the invariant masses of
the two $K^{-}K^{+}$ combinations, $s_{12}\equiv(p_{1}+p_{2})^{2}$ and
$s_{13}\equiv(p_{1}+p_{3})^{2}$. Throughout this paper, the symbol
$s_{K^{-}K^{+}}$ is used to represent the invariant mass squared of both
$K^{-}K^{+}$ combinations. These Lorentz-invariant quantities are computed
constraining the invariant mass of the candidate to the known $D^{+}$ mass
[23]. An accumulation of candidates is visible at $s_{K^{-}K^{+}}$
$\sim$1.04$\mathrm{\,Ge\kern-1.00006ptV^{2}}$ which corresponds to the
$\phi(1020)K^{+}$ component. The difference in the number of candidates in the
regions of the Dalitz plot above and below
1.55$\mathrm{\,Ge\kern-1.00006ptV^{2}}$ (regions I and II in the left side of
Fig. 2, respectively) is caused by interference between the $\phi(1020)K^{+}$
and S-wave amplitudes. This interference also shifts the position of the peaks
of the $s_{K^{-}K^{+}}$ distributions in the two regions. These two effects
are better illustrated in the projections of the Dalitz plot shown in the
right side of Fig. 2.

Figure 2: (left) Dalitz plot of the selected sample, including background.
(right) Dalitz plot projections for candidates from regions I (blue) and II
(red), above and below
$s_{K^{-}K^{+}}\\!=\\!1.5$$\mathrm{\,Ge\kern-1.00006ptV^{2}}$. The
interference between the S- and P-wave amplitudes cause the asymmetry in the
number of candidates in the two regions, as well as the shift in the peak
position. Both figures include all candidates in the selected mass range.

##  4 Efficiency and background model

###  4.1 Efficiency variation over the Dalitz plot

In the fit to the Dalitz plot distribution, the variation of the total
efficiency across the phase space must be taken into account. The total
efficiency is determined from a combination of simulation and methods based on
data, and includes the geometrical acceptance of the detector and the
reconstruction, selection, PID and trigger efficiencies.

The geometrical acceptance, reconstruction and selection efficiencies are
obtained from simulation. The PID efficiency of each $D^{+}$ candidate is
determined by multiplying the efficiencies for each of the final-state kaons.
The PID efficiencies for the kaons are evaluated from calibration samples of
$D^{*+}\rightarrow D^{0}(\rightarrow K^{-}\pi^{+})\pi^{+}$ decays [24] and
depend on the particle momentum, pseudorapidity and event charged-particle
multiplicity. The trigger efficiency is obtained from simulation, with a
correction factor determined from data to account for the small mismatch
between the performance of the trigger in data and simulation.

The total efficiency distribution is a two-dimensional histogram with
$14\times 14$ uniform bins. A two-dimensional cubic spline is used to smooth
this distribution to avoid binning discontinuities, yielding the high-
resolution histogram ($300\times 300$ uniform bins), shown in Fig. 3. This
histograms is used to weight the signal PDF in the Dalitz plot fit. The
binning scheme of the efficiency histogram is a source of systematic
uncertainty.

###  4.2 Background model

The background model is built from the inspection of the mass sidebands of the
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{K}^{+}}$ signal. The Dalitz plots
of candidates from both sidebands, 1820–1840$\mathrm{\,Me\kern-1.00006ptV}$
and 1900–1920$\mathrm{\,Me\kern-1.00006ptV}$, are very similar, with a clear
peaking structure, corresponding to random $\phi(1020)K^{+}$ combinations over
a smooth distribution.

The Dalitz plot variables are computed from the four-momenta determined by a
$D^{+}$ mass constrained fit. This constraint implies an unique boundary of
the Dalitz plot, regardless of the value of the invariant mass of the three-
kaon system. It also improves the mass resolution of signal candidates, but
has the effect to distort and shift any structure present in the Dalitz plot
of the background candidates in the sidebands. This effect depends strongly on
the invariant mass of the three-kaon system and prevents the determination of
the background model from a two-dimensional parameterisation of the Dalitz
plots from the sidebands. An alternative method is used instead. Each
$m(K^{-}K^{+}K^{+})$ sideband is divided into slices of
5$\mathrm{\,Me\kern-1.00006ptV}$. For each slice, the projections onto the
$s_{K^{+}K^{-}}$ axis are fitted using a relativistic Breit–Wigner for the
$\phi(1020)$ component (with floated mass and width) and a phase-space
distribution. The latter serves as a proxy for both the smooth component
spread across the Dalitz plot and the projection of the $\phi$ candidates
appearing in the other $s_{K^{+}K^{-}}$ combination. The fraction of the
$\phi(1020)$ component is nearly constant in both sidebands, indicating that
the background composition is independent of $m(K^{-}K^{+}K^{+})$. A linear
interpolation is used to obtain the fraction of peaking background in the
signal region and is found to be (20.67$\pm$0.28)%.

The fit to the $s_{K^{+}K^{-}}$ projection has the limitation of being less
sensitive to the distribution near the $K^{+}K^{-}$ threshold. The inspection
of the Dalitz plot sidebands shows that the smooth background component has
more candidates at low values of $s_{K^{+}K^{-}}$ and fewer at low values of
$s_{K^{+}K^{+}}\equiv(p_{2}+p_{3})^{2}$, indicating that this smooth
distribution is not uniform over the phase space. A model for the smooth
component of the background is built assuming a sum of two contributions,
random $f_{0}(980)K^{+}$ candidates and a constant amplitude, with equal
proportions. The relative fractions of these two terms in the smooth component
is treated as a source of systematic uncertainty.

Figure 3: Total efficiency, normalised to unity, for the
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{K}^{+}}$ signal over the Dalitz
plot, including the geometrical acceptance and the reconstruction, selection,
PID and trigger efficiencies. Figure 4:  Projection onto $s_{K^{+}K^{-}}$ of
$K^{-}K^{+}K^{+}$ candidates with invariant mass in the range 1820–1830 MeV.

A high-resolution normalised histogram ($300\times 300$ uniform bins) is used
in the Dalitz plot fit to represent the background PDF, and is shown in Fig.
5. This histogram is produced from a large simulated sample, using a PDF in
which the peaking and smooth components are added incoherently with the
estimated relative fractions and weighted by the efficiency function.

Figure 5: High-resolution histogram representing the background model used in
the Dalitz plot fits.

##  5 The Dalitz plot fit procedure

The ${{D}^{+}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{K}^{+}}$ decays are studied
through an unbinned maximum-likelihood fit to the observed Dalitz plot
distribution. The total PDF is constructed as a sum of signal and background
components, and the likelihood function is given by

$\mathcal{L}=\prod^{N_{\rm cand}}f_{\rm S}\times S_{\rm
PDF}(s_{12},s_{13})+(1-f_{\rm S})\times B_{\rm PDF}(s_{12},s_{13}),$ (1)

where $N_{\rm cand}$ is the total number of candidates and $f_{S}$ is the
fraction of signal candidates in the sample, as obtained from the
$m(K^{-}K^{+}K^{+})$ fit described in Sec. 3. The background PDF, $B_{\rm
PDF}(s_{12},s_{13})$, is described in Sec. 4.2.

The normalised signal PDF is written in terms of the total decay amplitude
${\mathcal{T}}(s_{12},s_{13})$,

$S_{\rm PDF}(s_{12},s_{13})=\frac{1}{N_{\rm
S}}{\left|\mathcal{T}(s_{12},s_{13})\right|}^{2}\varepsilon(s_{12},s_{13}),$
(2)

where $\varepsilon(s_{12},s_{13})$ is the detection efficiency, described in
Sec. 4.1. The normalisation factor, $N_{\rm S}$, is given by

$N_{\rm S}=\int{\rm d}s_{12}\ {\rm
d}s_{13}\left|\mathcal{T}(s_{12},s_{13})\right|^{2}\varepsilon(s_{13},s_{13}).$
(3)

For any given model, the amplitude ${\mathcal{T}}(s_{12},s_{13})$ depends on a
set of parameters that are floated in the fit. The optimum values for these
parameters are determined by minimizing the quantity $-2\ln\mathcal{L}$ using
the MINUIT package [25].

In order to compare the fit results of a given model to the Dalitz plot
distribution in data, a large simulated sample is generated according to the
model, including background and efficiency, normalised to the total number of
data candidates. Since there are two identical kaons, the folded Dalitz plot
is used, represented as $s_{K^{+}K^{-}}^{\rm high}$ versus
$s_{K^{+}K^{-}}^{\rm low}$, which are respectively the higher and the lower
values among $s_{12}$ and $s_{13}$. The Dalitz plot distribution is divided
into 1024 bins with approximately 110 candidates each and the normalised
residuals are computed as

$\Delta_{i}=\frac{(N_{\rm pred}^{i}-N_{\rm obs}^{i})}{\sigma_{i}},$ (4)

where, for each bin $i$, $N_{\rm pred}^{i}$ is the predicted number of
candidates from the model, $N_{\rm obs}^{i}$ is the number of candidates in
the data sample, and $\sigma_{i}$ is the statistical uncertainty from data and
simulation added in quadrature. The sum of the square values of $\Delta_{i}$
over all bins is the total $\chi^{2}$ and is used as a metric to compare fit
results with different models.

##  6 Dalitz plot analysis with the isobar model

In the isobar model, the decay amplitude is written as a coherent sum of a
constant nonresonant (NR) component and intermediate resonant amplitudes,

${\cal T}(s_{12},s_{13})=c_{\rm NR}+\sum_{k}c_{k}T_{k}(s_{12},s_{13}).$ (5)

Each resonant amplitude, $T_{k}$, is given by a product of Blatt–Weisskopf
penetration factors [26], $F_{D}^{L}$ and $F_{R}^{L}$, accounting for the
finite size of the $D^{+}$ meson and the resonance, respectively, the spin
amplitude, $\mathcal{S}$, accounting for the conservation of angular momentum,
and a function, $M_{R}$, describing the resonance lineshape, which is either a
relativistic Breit–Wigner (Eq. A.2) or a Flatté lineshape (Eq. A.4). The
Zemach formalism [27] is used for the spin amplitude $\mathcal{S}$. Details of
each of these factors are given in Appendix A. Since there are two identical
kaons in the final state, the resonant amplitudes are Bose-symmetrised,

$T_{k}(s_{12},s_{13})=F_{D}^{L}(s_{12})F_{R}^{L}(s_{12})\,\times\,\mathcal{S}(s_{12},s_{13})\,\times\,M_{R}(s_{12})+(2\leftrightarrow
3).$ (6)

The fit parameters are the complex coefficients $c_{\rm NR}=a_{\rm
NR}e^{i\delta_{\rm NR}}$ and $c_{k}=a_{k}e^{i\delta_{k}}$. The results are
expressed in terms of the magnitude and phase of the complex coefficient for
each component, and the corresponding fit fractions. The fit fractions are
computed by integrating the squared modulus of the corresponding amplitude
over the phase space, and dividing by the integral of the total amplitude
squared,

${\rm FF}_{k}=\frac{\int{\rm d}s_{12}\ {\rm d}s_{13}\ |c_{k}\
T_{k}(s_{12},s_{13})|^{2}}{\int{\rm d}s_{12}\ {\rm d}s_{13}\
\left|\sum_{i}c_{i}\ T_{i}(s_{12},s_{13})\right|^{2}}.$ (7)

The sum of fit fractions for all components is, in general, different from
100% due to the presence of interference; it is less than 100% in the case of
net constructive interference or higher than 100% otherwise.

###  6.1 Signal models

For the ${{D}^{+}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{K}^{+}}$ decay amplitude,
contributions from following resonances are possible: the isoscalars
$f_{0}(980)$, $f_{0}(1370)$ and $f_{0}(1500)$; the isovectors $a_{0}(980)$ and
$a_{0}(1450)$; the vector $\phi(1020)$; the tensor $f_{2}(1270)$.
Contributions from resonances with spin greater than one are suppressed due to
the small phase space of the
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{K}^{+}}$ decay. In the case of the
$f_{2}(1270)$ state, a further suppression is expected due to its small
branching fraction to $K^{-}K^{+}$, $(4.6\pm 0.4)$% [23]. The relatively
narrow $f^{\prime}_{2}(1525)$ state is neglected since it is well beyond the
phase space.

Various combinations of the nonresonant and the possible resonant amplitudes
are considered. All models studied contain the $\phi(1020)K^{+}$, which is
chosen as the reference amplitude, fixing the phase convention and setting the
scale for the magnitudes. The models tested differ by the composition of the
S-wave. Near the $K^{+}K^{-}$ threshold, both the $a_{0}(980)$ and
$f_{0}(980)$ resonances can contribute. Similarly, at higher $K^{+}K^{-}$
invariant mass, contributions from several scalar resonances are possible.

The $\phi(1020)$ mass and width are fixed to the known values [23]; for the
$f_{0}(980)$ state, a Flatté lineshape is used, with parameters from the BESII
collaboration [28].

###  6.2 Results

The simplest model that describes the data, referred to as model A, consists
of three intermediate components: $\phi(1020)K^{+}$, $f_{0}(980)K^{+}$, and
$f_{0}(1370)K^{+}$. As the $f_{0}(1370)$ state has large uncertainties on its
mass and width [23], these parameters are allowed to float in the fit. Its
contribution can also be interpreted, within the isobar formalism, as an
effective representation for the overlap of two or more broad structures at
high $K^{-}K^{+}$ invariant mass.

Further addition of scalar states does not improve the fit quality
significantly, creates more complex interference effects, and provides a very
similar description of the lineshape and phase behaviour of the total S-wave.
For example, in model B, a constant nonresonant contribution is added to the
resonant amplitudes of model A. The resulting fit quality is essentially
unchanged, with the total $\chi^{2}/{\rm ndof}$ being 1.15 and 1.14 for models
A and B, respectively. A similar situation occurs in model C, which has the
same amplitudes as in model B plus the $a_{0}(980)K^{+}$ component. In this
model, the contribution of the $f_{0}(1370)$ is found to be negligible and the
value of $\chi^{2}/{\rm ndof}$ is 1.16. Table 1 summarizes the fit results for
these three models. The total S-wave fit fraction includes the interference
terms between the various S-wave components. In all cases, the total S-wave in
the $K^{+}K^{-}$ system is dominant, a notable feature also observed in other
three-body $D$ decays with a pair of identical particles in the final state,
such as the $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ and
$D^{+}_{(s)}\rightarrow\pi^{-}\pi^{+}\pi^{+}$ decays[23]. The contribution
from the $f_{2}(1270)K^{+}$ component is also tested and found to be
consistent with zero in all models.

Since model A is the simplest model describing all the general features of the
observed Dalitz plot distribution, it is chosen as the baseline result for the
fit with the isobar model. The projections of the Dalitz plot, with the model
A fit result overlaid, are shown in Fig. 6. The green dashed line represents
the phase-space distribution, weighted by the efficiency, evidencing the
presence of at least one broad, scalar contribution not consistent with a
uniform distribution.

Table 1: Results from the ${{D}^{+}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{K}^{+}}$ Dalitz plot fit with the isobar models A, B and C. Magnitudes, $|c_{k}|$ , phases, $\arg(c_{k})$ (in degrees), and fit fractions (in %) are given with statistical uncertainties only. |  | Model A | Model B | Model C
---|---|---|---|---
$\phi(1020)K^{+}$ | Magnitude | 1 [fixed] | 1 [fixed] | 1 [fixed]
| Phase | 0 [fixed] | 0 [fixed] | 0 [fixed]
| Fraction | 6.17 $\pm$ | 0.47 | 6.40 $\pm$ | 0.47 | 6.40 $\pm$ | 0.48
$f_{0}(980)K^{+}$ | Magnitude | 3.12 $\pm$ | 0.10 | 2.64 $\pm$ | 0.08 | 2.84 $\pm$ | 0.13
| Phase |  $-58.9$ $\pm$ | 4.9 |  $-36.5$ $\pm$ | 7.6 |  $-25.9$ $\pm$ | 8.4
| Fraction | 23.7 $\pm$ | 3.0 | 17.7 $\pm$ | 2.1 | 20.4 $\pm$ | 1.5
$f_{0}(1370)K^{+}$ | Magnitude | 3.46 $\pm$ | 0.46 | 2.33 $\pm$ | 0.35 | –
| Phase | 13.1 $\pm$ | 7.7 | 42 $\pm$ | 10 | –
| Fraction | 25.4 $\pm$ | 5.0 | 18.7 $\pm$ | 1.5 | –
|  $f_{0}(1370)$ mass [$\mathrm{\,Ge\kern-1.00006ptV}$ ] | 1.422 $\pm$ | 0.015 | 1.401 $\pm$ | 0.009 | –
|  $f_{0}(1370)$ width [$\mathrm{\,Ge\kern-1.00006ptV}$ ] | 0.324 $\pm$ | 0.038 | 0.178 $\pm$ | 0.031 | –
NR | Magnitude | – | 8.8 $\pm$ | 1.3 | 11.7 $\pm$ | 1.8
| Phase | – |  $-5.5$ $\pm$ | 6.5 |  $-39.0$ $\pm$ | 4.4
| Fraction | – | 18.4 $\pm$ | 5.9 | 32.7 $\pm$ | 8.2
$a_{0}(980)K^{+}$ | Magnitude | – | – | 5.9 $\pm$ | 0.4
| Phase | – | – | 48.5 $\pm$ | 3.0
| Fraction | – | – | 53.5 $\pm$ | 7.4
| S-wave fraction | 92 $\pm$ | 11 | 91 $\pm$ | 13 | 93 $\pm$ | 12
| Fractions sum | 55.4 $\pm$ | 5.9 | 61.2 $\pm$ | 6.4 | 113 $\pm$ | 11
Table 2: Fit results with model A, given in terms of the magnitudes $|c_{k}|$, phases, $\arg(c_{k})$ (in degrees), and fit fractions (in %). For each measurement, the first uncertainty is statistical, the second systematic and the third is a systematic uncertainty due to model.  Component | Magnitude | Phase [deg.] | Fraction (%)
---|---|---|---
$\phi(1020)K^{+}$ | 1.0 [fixed] | 0.0 [fixed] | 6.17 $\pm$ | 0.47 $\pm$ | 0.19 $\pm$ | 0.41
$f_{0}(980)K^{+}$ | 3.12 $\pm$ | 0.10 $\pm$ | 0.13 $\pm$ | 0.33 |  $-58.9$ $\pm$ | 4.9 $\pm$ | 2.3 $\pm$ | 2.0 | 23.7 $\pm$ | 3.0 $\pm$ | 2.1 $\pm$ | 3.3
$f_{0}(1370)K^{+}$ | 3.46 $\pm$ | 0.46 $\pm$ | 0.32 $\pm$ | 0.73 | 13.1 $\pm$ | 7.7 $\pm$ | 1.6 $\pm$ | 3.2 | 25.4 $\pm$ | 5.0 $\pm$ | 3.4 $\pm$ | 3.8
sum |  |  | 55.4 $\pm$ | 5.9 $\pm$ | 0.4 $\pm$ | 0.6

The distribution of the normalised residuals $\Delta_{i}$ over the Dalitz plot
is shown in the left plot of Fig. 7, and their distribution is consistent with
a normal Gaussian, as shown in the right plot of Fig. 7. In Table 2 the
results including the systematic and model uncertainties, as discussed in Sec.
8, are presented.

The squared modulus and phase of the S-wave amplitude from model A are shown
in Fig. 8 as a function of the $K^{+}K^{-}$ mass, with total uncertainties
represented as bands. For comparison, the corresponding central results for
models B and C are overlaid. Although the S-wave composition is different for
these models, the total S-wave description is essentially the same, evidencing
that the isobar model fails to disentangle the individual contributions. The
$f_{0}(1370)$ parameters are found to be $m_{0}=1.422\pm 0.015\pm 0.009\pm
0.028$$\mathrm{\,Ge\kern-1.00006ptV}$ and $\Gamma_{0}=0.324\pm 0.038\pm
0.018\pm 0.038$$\mathrm{\,Ge\kern-1.00006ptV}$, where the first uncertainties
are statistical and the second systematic.

Figure 6: Projections of the Dalitz plot onto (top left) $s_{K^{+}K^{-}}$,
(top right) $s_{K^{+}K^{+}}$, (bottom left) $s_{K^{+}K^{-}}^{\rm high}$ and
(bottom right) $s_{K^{+}K^{-}}^{\rm low}$ axes, with the fit result with model
A overlaid (red histogram). The histogram in magenta represents the
contribution from the background, whereas the dashed green line is the phase-
space distribution weighted by the efficiency.

Figure 7: (left) Normalised residuals $\Delta_{i}$ across the Dalitz plot,
from the result of isobar fit. (right) Distribution of the normalised
residuals with the fit result overlaid. The distribution is fitted with a
Gaussian function and the fit result is consistent with the standard normal
distribution.

Figure 8: (left) Magnitude and (right) phase of the total S-wave from the
result of the Dalitz plot fit with the isobar model. The black line
corresponds to model A and the green band represents the statistical and
systematic uncertainties added in quadrature. For comparison, the results of
models B and C are shown as the blue solid and dashed thick red lines.
Uncertainties on the S-wave magnitude and phase for models B and C are similar
to those from model A and are not shown.

##  7 Dalitz plot analysis with the Triple-M amplitude

Figure 9: Diagrams representing the two quark-level topologies for the
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{K}^{+}}$ decay. In the Triple-M
[3], diagram ($a$) is assumed to be the dominant mechanism of the decay,
whereas diagram ($b$) is suppressed since the production of a $K^{+}K^{-}$
pair from a $d\bar{d}$ pair requires rescattering.

The basic hypothesis of the Triple-M is the dominance of the annihilation
diagram shown in Fig. 9(a). The
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{K}^{+}}$ decay can also proceed
via the diagram in Fig. 9(b), but in this case a $K^{+}K^{-}$ pair could only
be produced from the $d\bar{d}$ pair through rescattering, since charged kaons
have no $d$-valence quark. The same holds for the production of the
$\phi(1020)$ meson which is essentially an $s\bar{s}$ state [23].

Assuming the annihilation diagram is the dominant mechanism for the
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{K}^{+}}$ decay, the Triple-M
amplitude is a product of two axial-vector currents,

$\langle
K^{-}K^{+}K^{+}|\mathcal{T}|D^{+}\rangle=-\left[\frac{G_{F}}{\sqrt{2}}\,\sin^{2}\theta_{C}\right]\langle
K^{-}K^{+}K^{+}|A_{\mu}|0\rangle\langle 0|A_{\mu}|D^{+}\rangle\ ,$ (8)

where $G_{F}$ is the Fermi decay constant, $\theta_{C}$ is the Cabibbo angle
and $A^{\mu}$ are the axial currents. The weak vertex is
$\langle\,0\,|A_{\mu}|\,D^{+}(P)\rangle=-i\,\sqrt{2}\,f_{D}\,P_{\mu}$, where
$P=p_{1}+p_{2}+p_{3}\,$ is the $D^{+}$ four-momentum and $f_{D}$ is the
$D^{+}$ decay constant.

In the Triple-M, the three-kaon system can be produced in two ways, as
illustrated in the diagrams in Fig. 10. Diagram (a) represents the production
of the three kaons directly from the weak vertex, whereas in diagram (b) two
of the three kaons result from the decay of a bare intermediate resonance.
Final state interactions are introduced in diagrams (c) and (d). The full
black circles indicate the unitarised scattering amplitudes,
$A_{K^{+}K^{-}}^{JI}$, representing the scattering $ab\rightarrow K^{+}K^{-}$
with the coupled channels $ab=K^{+}K^{-},\ \pi\pi,\ \eta\pi$ and $\eta\eta$ in
a well-defined spin (J ) and isospin (I ) state. The nonresonant component
corresponds to diagram (a). Due to the existence of two identical kaons,
diagrams (b), (c) and (d) are symmetrised. As in the isobar analysis,
contributions of D-wave are expected to be very small and are not included.

The Triple-M decay amplitude therefore has five components,

$\mathcal{T}=T_{\rm NR}+\sum_{J,I}T^{JI},\hskip 17.07182ptJ,I=0,1.$ (9)

The free parameters in the Triple-M amplitude are the couplings and masses of
the chiral Lagrangian. There are four couplings, $c_{d},\ c_{m},\
\tilde{c}_{d},\ \tilde{c}_{m}$ in the scalar part, contributing to $T^{00}$
and $T^{01}$ terms; two masses, $m_{So},\ m_{S1}$, for the scalar-isoscalar,
$T^{00}$ contribution and one, $m_{a_{0}}$, in the scalar-isovector $T^{01}$
components; one coupling, $G_{V}$, for the vector components, $T^{10}$ and
$T^{11}$, and one mass, $m_{\phi}$, in the vector-isoscalar component. In the
fit to the data, the combination $G_{\phi}\equiv
G_{V}\sin\theta_{\omega-\phi}/F$ is used as free parameter, where
$\theta_{\omega-\phi}$ is the $\omega-\phi$ mixing angle. The parameter $F$ is
the $SU(3)$ pseudoscalar decay constant, common to all components. For
convenience, the formulae of the various components of the Triple-M amplitude
are reproduced from Ref. [3] in Appendix B.

Figure 10: Diagrams contributing to the amplitude $\mathcal{T}$ for the decay
$D^{+}\rightarrow K^{-}\,K^{+}\,K^{+}$: (a) the final state kaons are produced
directly from the weak vertex; (b) a bare resonance is produced directly from
the weak vertex; (c) particles produced at the weak vertex undergo final state
interactions; (d) final state interactions endow finite widths to the
resonances. The full circle represents the unitary $ab\rightarrow K^{+}K^{-}$
scattering amplitude with angular momentum $J$ and isospin $I$, and
$ab=K\overline{K},\ \pi\pi,\ \eta\pi$ and $\eta\eta$.

Equation 9 resembles that of the isobar model, but there are several
significant differences. The free parameters in the Triple-M amplitude are
real quantities from the chiral Lagrangian. Some of these parameters appear in
different spin-isospin components of the model. In the isobar model the free
parameters are the complex coefficients $c_{k}$, from which the individual
contributions of the resonances are determined. In the Triple-M amplitude, the
relative contributions of the various components are fixed by theory. The
nonresonant component is usually represented by an empirical constant in fits
with the isobar model. In the Triple-M amplitude, it is a function of the
Dalitz plot coordinates and is fully determined by chiral symmetry.

###  7.1 Fit results

The optimum values of the Triple-M parameters are determined by an unbinned
maximum-likelihood fit, as described in Sec. 5. The fitted values of the
Triple-M parameters are listed in Table 3, with statistical and systematic
uncertainties.

The quality of the fit with the Triple-M amplitude is tested with the metric
defined in Eq. 4. The value of $\chi^{2}/{\rm ndof}$ is 1.12. The projections
of the Dalitz plot onto the $s_{K^{+}K^{-}}$ and the $s_{K^{+}K^{+}}$ axes, as
well as the projections onto the highest and lowest invariant masses squared
of the two $K^{+}K^{-}$ combinations, $s_{K^{+}K^{-}}^{\rm high}$ and
$s_{K^{+}K^{-}}^{\rm low}$, are shown in Fig. 11, with the fit result
superimposed. The projections indicate that the model is in good agreement
with the data. The distribution of the normalised residuals over the Dalitz
plot is shown in the right panel of Fig. 12. The distribution of normalised
residuals, shown in the left panel of Fig. 12, is consistent with a normal
Gaussian.

Table 3: Results of the $D^{+}\rightarrow K^{-}K^{+}K^{+}$ Dalitz plot fit with the Triple-M amplitude. parameter | value
---|---
$F$ |  $94.3^{+2.8}_{-1.7}$ $\pm$ | 1.5$\mathrm{\,Me\kern-1.00006ptV}$
$m_{a_{0}}$ |  $947.7^{+5.5}_{-5.0}$ $\pm$ | 6.6$\mathrm{\,Me\kern-1.00006ptV}$
$m_{S_{o}}$ |  $992.0^{+8.5}_{-7.5}$ $\pm$ | 8.6$\mathrm{\,Me\kern-1.00006ptV}$
$m_{S_{1}}$ |  $1330.2^{+5.9}_{-6.5}$ $\pm$ | 5.1$\mathrm{\,Me\kern-1.00006ptV}$
$m_{\phi}$ |  $1019.54^{+0.10}_{-0.10}$ $\pm$ | 0.51$\mathrm{\,Me\kern-1.00006ptV}$
$G_{\phi}$ |  $0.464^{+0.013}_{-0.009}$ $\pm$ | 0.007 0
$c_{d}$ |  $-78.9^{+4.2}_{-2.7}$ $\pm$ | 1.9$\mathrm{\,Me\kern-1.00006ptV}$
$c_{m}$ |  $106.0^{+7.7}_{-4.6}$ $\pm$ | 3.3$\mathrm{\,Me\kern-1.00006ptV}$
$\tilde{c}_{d}$ |  $-6.15^{+0.55}_{-0.54}$ $\pm$ | 0.19$\mathrm{\,Me\kern-1.00006ptV}$
$\tilde{c}_{m}$ |  $-10.8^{+2.0}_{-1.5}$ $\pm$ | 0.4$\mathrm{\,Me\kern-1.00006ptV}$

Figure 11:  Projections of the Dalitz plot onto (top left) $s_{K^{+}K^{-}}$,
(top right) $s_{K^{+}K^{+}}$, (bottom left) $s_{K^{+}K^{-}}^{\rm high}$ and
(bottom right) $s_{K^{+}K^{-}}^{\rm low}$ axes, with the fit result with the
Triple-M amplitude superimposed, whereas the dashed green line is the phase
space distribution weighted by the efficiency. The magenta histogram
represents the contribution from the background.

Figure 12: (left) Two-dimensional distribution of the normalised residuals for
the Triple-M fit. (right) Distribution of normalised residuals of each bin.

###  7.2 Interpretation

The resonance masses in the Triple-M are introduced in the denominators, D, of
Eqs. B.21–B.24, where the functions M are imaginary and proportional to
interaction kernels which contain the bare masses of the effective chiral
Lagrangian, $m_{a_{0}}$, $m_{S_{o}}$, $m_{S_{1}}$ and $m_{\phi}$. The Triple-M
amplitude is derived assuming that only the imaginary part of the two-body
propagators in Eqs. B.25–B.28 is relevant. In this approximation, the bare
masses coincide with the masses of the physical states and the association
$m_{S_{o}}=m_{f_{0}(980)}$ and $m_{S_{1}}=m_{f_{0}(1370)}$ can be made. As in
the case of the isobar model, the masses in the Triple-M correspond to the
values of $s_{K^{+}K^{-}}$ for which the real part of the denominator D of
Eqs. B.21–B.24 vanishes. At these values of $s_{K^{+}K^{-}}$, only the
imaginary parts of the denominators remain, corresponding to the model
prediction for the widths. The denominators D would be very similar to those
from the isobar model if no coupled channel was considered. The inclusion of
coupled channels is, therefore, the main difference between the Triple-M and
Breit–Wigner denominators, resulting in widths with different dynamical
content.

####  7.2.1 Resonant structure

The nonresonant contribution in the Triple-M is a three-body amplitude
predicted by chiral symmetry. It can be projected into the S- and P-waves
rewriting Eq. B.3 as

$\displaystyle T_{\rm NR}$ $\displaystyle=$
$\displaystyle\frac{C}{4}\left[\,(m_{D}^{2}-m_{K}^{2}+s_{12})+\,(s_{13}-s_{23})+(m_{D}^{2}-m_{K}^{2}+s_{13})+\,(s_{12}-s_{23})\right]$
(10) $\displaystyle=$ $\displaystyle T_{\rm NR}^{S}+T_{\rm NR}^{P}\;,$

where $C$ is a constant common to all components of the Triple-M amplitude,
and defined in Eq. B.2. The decay amplitude can then be written as the sum of
scalar and vector components

$\mathcal{T}=\left[T^{S}+T^{P}+(2\leftrightarrow 3)\right],$ (11)

with

$T^{S}=T_{NR}^{S}+T^{00}+T^{01}\;$ (12)

and

$T^{P}=T_{NR}^{P}+T^{11}+T^{10}\;.$ (13)

The relative contribution of each individual component of the Triple-M
amplitude is determined by integrating the modulus squared of each term in the
right-hand side of Eq. 9 over the phase space of the
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{K}^{+}}$ decay,

${\rm FF}_{\rm NR}=\frac{\int{\rm d}s_{12}\ {\rm d}s_{13}\ |T_{\rm
NR}(s_{12},s_{13})|^{2}}{\int{\rm d}s_{12}\ {\rm d}s_{13}\
\left|\mathcal{T}(s_{12},s_{13})\right|^{2}},\hskip 22.76228pt{\rm
FF}^{JI}=\frac{\int{\rm d}s_{12}\ {\rm d}s_{13}\
|T^{JI}(s_{12},s_{13})|^{2}}{\int{\rm d}s_{12}\ {\rm d}s_{13}\
\left|\mathcal{T}(s_{12},s_{13})\right|^{2}}.$ (14)

Similarly, the S-wave contribution can be determined by the integral over the
phase space of the modulus squared of the $T^{S}$ component, defined in Eq.
12, and divided by the integral of the modulus squared of the decay amplitude
$\mathcal{T}$. The results are summarised in Table 4. There is a large
destructive interference between the two scalar below-threshold states,
$a_{0}(980)$ and $f_{0}(980)$, yielding an S-wave contribution of $(94\pm
1)$%. The large $a_{0}(980)/f_{0}(980)$ interference may be, in part, due to
the fact that in the $K^{+}K^{-}$ mass spectrum these two states have very
similar lineshapes, since only the tails are visible. This large interference
is also observed in the fit with the isobar model C, yielding similar fit
fractions for the S-wave component. A more accurate determination of the
relative contribution of the $a_{0}(980)$ and $f_{0}(980)$ resonances could be
obtained from a simultaneous analysis of the
$D^{+}\rightarrow\pi^{+}\pi^{-}\pi^{+}$ and
$D^{+}\rightarrow\eta\pi^{+}\pi^{0}$. The contribution of the $\phi(1020)$
resonance, $(7.1\pm 0.5)\%$, is consistent to that observed in the fit with
the isobar model.

Table 4: Relative fractions (%) of the various components of the Triple-M amplitude. The uncertainties correspond to the combined statistical and systematic uncertainties. ${\rm FF_{NR}}$ | ${\rm FF}^{00}$ | ${\rm FF}^{01}$ | ${\rm FF}^{10}$ | ${\rm FF}^{11}$ | ${\rm FF_{S-wave}}$
---|---|---|---|---|---
14 $\pm$ 1 | 29 $\pm$ 1 | 131 $\pm$ 2 | 7.1 $\pm$ 0.9 | 0.26 $\pm$ 0.01 | 94 $\pm$ 1

####  7.2.2 Decay and scattering amplitudes

The phases of the S-wave amplitude, $T^{S}$, and the $K^{+}K^{-}\rightarrow
K^{+}K^{-}$ scattering amplitudes, $A^{0I}_{K^{+}K^{-}}$, for the two allowed
isospin states, are shown in Fig. 13 as a function of the $K^{+}K^{-}$
invariant mass. The bands correspond to the statistical and systematic
uncertainties added in quadrature. The kink in the phase of $T^{S}$ at
$m(K^{+}K^{-})\sim 1.25$ GeV is due to the opening of the $\eta\eta$ channel.
The curves of Fig. 13 illustrate the difference between decay and scattering
amplitudes. The latter, which depends on spin and isospin, is a substructure
of the former, which depends only on spin. The expressions of the various
scattering amplitudes, derived in Ref. [3], are reproduced in Appendix C.

The physics of two-body scattering is encompassed by the phase shifts and
inelasticities. These quantities are obtained from the scattering amplitudes,
following the procedure described in Ref. [3]. The phase shifts,
$\delta^{JI}_{K^{+}K^{-}}$, and inelasticities $\eta^{JI}_{K^{+}K^{-}}$, are
displayed in Fig. 14 for $J\\!=\\!0$ and $I\\!=\\!0,1$.

The interpretation of the phase shifts for $K^{+}K^{-}$ scattering is not as
straightforward as in the case of elastic scattering, since for both isospin
states, the $\pi\pi\rightarrow K^{+}K^{-}$ and $\pi\eta\rightarrow K^{+}K^{-}$
channels are already open at the $K^{+}K^{-}$ threshold. An interesting
feature of the results displayed is that the phase variation of
$\delta^{00}_{K^{+}K^{-}}$ is monotonic and spans over more than
$180^{\circ}$, with a fast variation starting at $m(K^{+}K^{-})\sim 1.4$ GeV,
close to the value of $m_{S_{1}}$ and typical of a resonance at high
$K^{+}K^{-}$ mass. A fast variation of the phases is observed near threshold
for both $\delta^{00}_{K^{+}K^{-}}$ and $\delta^{01}_{K^{+}K^{-}}$, indicating
the contribution from the resonances below threshold.

The $\eta\eta$ channel contributes to $T^{00}$ but not to $T^{01}$ and its
effect is visible in the bottom left plot of Fig. 14 as a kink at
$m(K^{+}K^{-})\sim 1.1$$\mathrm{\,Ge\kern-1.00006ptV}$. As elastic scattering
corresponds to $\eta^{JI}=1$, one sees that the isoscalar component becomes
significantly more inelastic after the mass of the second scalar resonance.

Figure 13: Phase of the $J\\!=\\!0$ component of the decay amplitude (blue)
$T^{S}=T^{00}+T^{01}+T_{\rm NR}^{S}$, compared to the phases of the scattering
amplitudes, (red) $A_{K^{+}K^{-}}^{00}$ and (magenta) $A_{K^{+}K^{-}}^{01}$ as
a function of the $K^{+}K^{-}$ invariant mass.

Figure 14: (top) Phase-shifts $\delta^{0I}_{K^{+}K^{-}}$ and (bottom)
inelasticities $\eta^{0I}$ as a function of the $K^{+}K^{-}$ invariant mass,
for both isospin states.

##  8 Systematic uncertainties

Sources of systematic uncertainties associated to the background model, to the
efficiency correction and to possible biases in the fitting procedure are
common to the fits with the isobar model and the Triple-M. They are summarized
in Tables 5 and 6, respectively. There is an additional source of systematic
uncertainties on the results of the fit with the isobar model due to the
uncertainties on the parameters defining the $f_{0}(980)$ lineshape, which are
fixed in the fit. This additional uncertainty, quoted separately from the
experimental uncertainties, is estimated by repeating the fit varying the
parameters $g_{\pi}$, $g_{K}$ and $m_{0}$ of Eq. A.4 by one standard
deviation, one at a time, and taking the largest deviation as the systematic
uncertainty. The radii of the Blatt–Weisskopf form factors are also fixed in
the fit. However, they impacts only the $\phi(1020)K^{+}$ amplitude. Fits with
alternative values of these parameters are performed. The tested values of the
radii are 4 and 6 GeV-1, for $F_{D}^{L}$, and 1 and 3 GeV-1, for $F_{R}^{L}$.
Since no significant deviation from the baseline fit is observed, no
systematic uncertainty is assigned.

Two types of systematic uncertainties due to the background are investigated.
First, the background level is varied according to the uncertainty from the
fit to the $K^{-}K^{+}K^{+}$ invariant mass. The data is fitted changing the
fraction of the background by $\pm 1\sigma$. No significant change in the fit
parameters is found and no systematic uncertainty is assigned. Uncertainties
due to the background modelling are also investigated. The background model is
built from inspection of the sidebands of the
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{K}^{+}}$ signal. It is a
combination of a peaking structure and a smooth component. The smooth
component corresponds to 80% of the background and is modelled by a sum of a
constant term and an $f_{0}(980)K^{+}$ contribution, in equal proportions. A
systematic uncertainty due to the modelling of the background is assigned by
varying the relative fractions of these two components, fitting the data with
these alternative background models and taking the largest variation as
systematic uncertainty.

Table 5: Systematic uncertainties (%) on the results of the isobar model fit. For comparison, the statistical uncertainties are listed in the last column. parameter | binning | sim. stat. | bkg. | total | model | stat.
---|---|---|---|---|---|---
$|c_{f_{0}(980)}|$ | 1.0 | 1.4 | 3.8 | 4.2 | 11 | 3.0
$\delta_{f_{0}(980)}$ | 3.1 | 1.9 | 1.4 | 3.9 | 3.4 | 8.2
$|c_{f_{0}(1370)}|$ | 3.5 | 3.5 | 7.8 | 9.2 | 21 | 13
$\delta_{f_{0}(1370)}$ | 9.3 | 5.2 | 4.4 | 12 | 24 | 59
$M_{f_{0}(1370)}$ | 0.1 | 0.3 | 0.6 | 0.7 | 2.0 | 1.0
$\Gamma_{f_{0}(1370)}$ | 3.7 | 3.0 | 3.1 | 5.7 | 12 | 12
Table 6: Systematic uncertainties (%) on the results of the Triple-M fit. For comparison, the statistcal uncertainties are listed in the last column. parameter | binning | sim. stat. | PID | bkg. | fit bias | total | stat.
---|---|---|---|---|---|---|---
$F$ | 0.53 | 0.07 | 0.09 | 1.5 | 0.11 | 1.6 | 1.8
$m_{a_{0}}$ | 0.54 | 0.14 | - | 0.40 | 0.16 | 0.70 | 0.54
$m_{So}$ | 0.60 | 0.21 | - | 0.56 | 0.21 | 0.87 | 0.82
$m_{S1}$ | 0.16 | 0.15 | - | 0.13 | 0.04 | 0.26 | 0.41
$m_{\phi}$ | 0.002 | 0.001 | - | - | 0.002 | 0.003 | 0.005
$G_{\phi}$ | 0.86 | 0.25 | 0.02 | 1.2 | 0.15 | 1.5 | 1.9
$c_{d}$ | 0.18 | 0.08 | 0.09 | 2.4 | 0.13 | 2.4 | 3.3
$c_{m}$ | 0.16 | 0.11 | - | 2.7 | 0.10 | 2.7 | 4.7
$\tilde{c}_{d}$ | 0.13 | 0.15 | - | 2.6 | 1.1 | 3.1 | 8.8
$\tilde{c}_{m}$ | 0.19 | 0.11 | 0.08 | 2.8 | 1.9 | 3.4 | 13

Systematic uncertainties are assigned to small biases in the fit using
ensembles of 500 simulated samples. Two sets of samples are generated using
the Triple-M amplitude and the isobar model, both with the fitted values of
the parameters. In the simulations the signal PDFs are weighted by the
efficiency function and the background component is included. Each simulated
sample is fitted independently, resulting in distributions of fitted values of
the parameters and their respective uncertainties. For each parameter, the
mean of the distribution of fitted values is compared to the input. The
difference is assigned as the systematic uncertainty due to the fit bias. A
small bias is observed in the fit with the Triple-M amplitude, whilst no bias
is observed in the fit with the isobar model.

The systematic uncertainty associated to the efficiency variation across the
Dalitz plot includes the effect of the uncertainties on the PID efficiency and
the hardware trigger correction factors, the effect of the finite size of the
simulated sample, and the effect of the binning scheme of the efficiency
histogram prior to the two-dimensional spline smoothing. The uncertainties on
the PID efficiency are due to the finite size of the calibration samples and
imply small systematic uncertainties compared to the other sources of
systematics, in the fit with the Triple-M amplitude, and negligible
uncertainties in the fit with the isobar model. The uncertainty due to the
hardware trigger correction factors is found to be negligible. The effect of
the finite size of the simulated sample is assessed by generating a set of
alternative histograms from the selection efficiency histogram, prior to the
hardware trigger correction and the PID efficiency weighting. The content of
each bin of the selection efficiency histogram is varied according to a
Poisson distribution. For each of these alternative histograms, an efficiency
map is produced and used to fit the data. For each parameter, the root mean
square of the distribution of fitted values is assigned as a systematic
uncertainty. The systematic uncertainty due to the binning scheme of the
efficiency map is accessed by varying the number of bins of the final
efficiency histogram. The histograms with alternative binnings are fitted by
the two-dimensional cubic spline. The data is fitted with these alternative
efficiency maps and the largest variation of each parameter is assigned as
systematic uncertainty.

##  9 Summary and conclusions

In this paper, the first Dalitz plot analysis of the doubly Cabibbo-suppressed
decay ${{D}^{+}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{K}^{+}}$ is performed. The
two goals of the analysis are the determinations of the resonant structure of
the decay and the $K^{+}K^{-}$ scattering amplitudes. The resonant structure
is studied with two different approaches. In the fit with the isobar model,
several variations of the decay amplitude are tested. The Dalitz plot analysis
is also performed with the Triple-M [3], which is a model derived from a
chiral effective Lagrangian. The Triple-M amplitude has a nonresonant
component plus the minimal $SU(3)$ content corresponding to four states, the
$\phi(1020)$, the $a_{0}(980)$ and two isoscalar states, identified with the
$f_{0}(980)$ and $f_{0}(1370)$ resonances. A good description of the data is
achieved with both approaches.

The resonant structure of the
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{K}^{+}}$ is largely dominated by
the S-wave, with a approximately 7% contribution from the $\phi(1020)K^{+}$
component. The dominance of the S-wave contribution is also observed in other
three-body $D^{+}_{(s)}$ decays with a pair of identical particles in the
final state, such as the $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ and
$D^{+}_{(s)}\rightarrow\pi^{-}\pi^{+}\pi^{+}$ decays[23]. The possibility of
determining the individual components of the S-wave, however, is limited by
the lack of structures in the Dalitz plot, other than that from the
$\phi(1020)$ resonance, and by the fact that the $f_{0}(980)$ and $a_{0}(980)$
mesons poles lie below the $K^{+}K^{-}$ threshold. In all the models tested,
large interference between the various S-wave components is observed. In the
fit with isobar model, different combinations of scalar resonances and
nonresonant amplitudes yield fits of same quality and a very similar S-wave
amplitude. In the fit with the Triple-M, a large $a_{0}(980)$ contribution is
observed, with a large destructive interference with the $f_{0}(980)$
component that yields an S-wave fraction of about 94%. The separation between
the $f_{0}(980)$ and $a_{0}(980)$ contributions could better achieved with a
simultaneous analyses of the
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{K}^{+}}$,
$D^{+}\rightarrow\pi^{-}\pi^{+}\pi^{+}$ and
$D^{+}\rightarrow\eta\pi^{+}\pi^{0}$ decays.

Predicitions for the $K^{+}K^{-}\rightarrow K^{+}K^{-}$ scattering amplitudes
are obtained from the Dalitz plot fit using the Triple-M amplitude. This is
possible because the model incorporates explicitely coupled channels and
isospin degrees of freedom. In this respect, the chiral Lagrangian approach
represents an advance towards the description of the hadronic part of weak
decays of $D$ mesons in a more fundamental basis.

## 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); 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
Sklodowska-Curie Actions and ERC (European Union); 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); Laboratory Directed Research and
Development program of LANL (USA).

Appendix

##  Appendix A Decay amplitudes in the isobar model

Intermediate decay amplitudes within the Isobar model are given by Eq. 6. Each
factor appearing in that equation is presented here.

The form factors $F_{D}^{L}$ and $F_{R}^{L}$, for the $D^{+}$ and the
resonance decay, respectively, are parameterized by the Blatt–Weisskopf
penetration factors [26], and depend on $L$, the orbital angular momentum
involved in the transition. Since both the initial state (the $D^{+}$ meson)
and the final state (three kaons) have spin 0, $L$ is equal to the spin of the
resonance. In the rest frame of a resonance formed by particles 1 and 2 , R12,
$q$ is the modulus of the momentum of particle 1 or 2 (the decay momentum),
$q_{0}$ is the decay momentum when $s_{12}=m_{R}^{2}$ ($m_{R}$ being the
nominal resonance mass), and $d$ is a measure of the effective radius of the
decaying meson, fixed in this work to $5.0\mathrm{\,Me\kern-1.00006ptV}^{-1}$
for the $D$ meson and $1.5\mathrm{\,Me\kern-1.00006ptV}^{-1}$ for the
resonance. Defining $z=(qd)^{2}$ and $z_{0}=(q_{0}d)^{2}$, the Blatt–Weisskopf
barrier factors are usually written with two different formulations, $B_{L}$
and $B_{L}^{\prime}$ [23], given in Table 7. The $B_{L}^{\prime}$ formulation
is used in this analysis, consistent with the energy dependent width given
below in Eq. A.3, with the momenta in $F_{D}^{L}$ and $F_{R}^{L}$ computed in
the rest frame of the respective decaying particle.

Table 7: Blatt–Weisskopf form factors for angular momentum $L=0,1,2$ with two distinct formulations.  L | $B_{L}$ | $B_{L}^{\prime}$
---|---|---
0 | 1 | 1
1 | $\sqrt{\frac{2z}{1+z}}$ | $\sqrt{\frac{1+z_{0}}{1+z}}$
2 | $\sqrt{\frac{13z^{2}}{1+z}}$ | $\sqrt{\frac{(z_{0}-3)^{2}+9z_{0}}{(z-3)^{2}+9z}}$

The function $\mathcal{S}(\theta^{R12}_{13})$ describes the angular
distribution of the decay particles, with
$\theta^{R_{12}}_{13}=\theta^{R_{12}}_{13}(s_{12},s_{13})$ being the angle
between particles 1 and 3 momenta measured in the rest frame R12. The Zemach
formalism [27] is used for the angular distribution

${\cal S}=(-2|p_{1}||p_{3}|)^{L}P_{L}(\cos{\theta}^{R_{12}}_{13}),$ (A.1)

where $P_{L}$ is the Legendre polynomial of order $L$. For vector and tensor
resonances, this term introduces nodes in the Dalitz plot in regions where the
helicity angle is either 90∘ or 270∘.

The relativistic Breit–Wigner function [29] is used as the dynamical function,

$M_{R}(s_{12})=\frac{1}{s_{12}-m_{R}^{2}+im_{R}\Gamma(s_{12})},$ (A.2)

where $m_{R}$ is the mass of the resonance and $\Gamma(s_{12})$ is the mass-
dependent width,

$\Gamma(s_{12})=\Gamma_{R}\left(\frac{q}{q_{0}}\right)^{2L+1}\frac{m_{R}}{\sqrt{s_{12}}}\left(\frac{F_{R}^{L}(z)}{F_{R}^{L}(z_{0})}\right)^{2}$
(A.3)

with $\Gamma_{R}$ being the nominal resonance width.

In the case of the $f_{0}(980)$ resonance, the relativistic Breit–Wigner is
replaced by the Flatté formula[30]

$M_{R}(s_{12})=\frac{1}{s_{12}-m_{R}^{2}+im_{R}(\rho_{\pi\pi}\
g_{\pi}^{2}+\rho_{KK}\ g_{K}^{2})},$ (A.4)

where $g_{\pi}$ and $g_{K}$ are dimensionless coupling constants to the
$K{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}$ and $\pi\pi$ channels,
respectively, and $\rho_{\pi\pi}$ and $\rho_{KK}$ are the corresponding phase-
space factors,

$\begin{split}\rho_{\pi\pi}&=\sqrt{\left(\frac{s_{12}}{4}-m_{\pi}^{2}\right)}+\sqrt{\left(\frac{s_{12}}{4}-m_{\pi^{0}}^{2}\right)},\\\
\rho_{KK}&=\sqrt{\left(\frac{s_{12}}{4}-m_{K}^{2}\right)}+\sqrt{\left(\frac{s_{12}}{4}-m_{K^{0}}^{2}\right)}.\end{split}$
(A.5)

All the above formulation holds equally for the resonances in the system
composed by particles 1 and 3, with $s_{12}\rightarrow s_{13}$ and
$\theta^{R_{12}}_{13}\rightarrow\theta^{R_{31}}_{12}$ (angular functions
convention with cyclic permutation $(12)3\rightarrow(31)2$).

##  Appendix B The Triple-M Decay amplitude

All formulae presented in this appendix are reproduced from Ref. [3] for
convenience. The Triple-M decay amplitude for the $D^{+}\rightarrow
K^{-}\,K^{+}\,K^{+}$ decay is given by

$\displaystyle\mathcal{T}$ $\displaystyle\\!=\\!$ $\displaystyle T_{\rm
NR}+\left[T^{(1,1)}+T^{(1,0)}+T^{(0,1)}+T^{(0,0)}+(2\leftrightarrow
3)\right]\;,$ (B.1)

where $T_{\rm NR}$ and the $T^{(J,I)}$ are the nonresonant and resonant
contributions, respectively. All components are proportional to the kaon mass
squared, $m_{K}^{2}$, included in the common factor

$\displaystyle
C=\left\\{\left[\frac{G_{F}}{\sqrt{2}}\,\sin^{2}\theta_{C}\right]\;\frac{2F_{D}}{F}\,\frac{m_{K}^{2}}{(m_{D}^{2}-m_{K}^{2})}\right\\}\,,$
(B.2)

where $F_{D}$ is the $D^{+}$ decay constant, $F$ is the $SU(3)$ pseudoscalar
decay constant, $G_{F}$ is the Fermi decay constant and $\theta_{C}$ is the
Cabibbo angle. The nonresonant contribution is a three-body amplitude, and
therefore is not Bose-symmetrised. It is written as a real polynomial,

$\displaystyle T_{\rm
NR}=C\,\left[(s_{12}-m_{K}^{2})+(s_{13}-m_{K}^{2})\right]\;.$ (B.3)

The amplitudes $T^{(J,I)}$ are

$\displaystyle
T^{(1,1)}=-\,\frac{1}{4}\,\left[\bar{\Gamma}_{KK}^{(1,1)}-\Gamma_{c|KK}^{(1,1)}\right]\,(s_{13}\\!-\\!s_{23})\,,$
(B.4)
$\displaystyle\bar{\Gamma}_{KK}^{(1,1)}=\frac{1}{D_{\rho}(s_{12})}\left[M_{21}^{(1,1)}\,\Gamma_{(0)\,\pi\pi}^{(1,1)}+\left(1\\!-\\!M_{11}^{(1,1)}\right)\,\Gamma_{(0)\,KK}^{(1,1)}\right]\,,$
(B.5) $\displaystyle
T^{(1,0)}=-\,\frac{1}{4}\,\left[\bar{\Gamma}_{KK}^{(1,0)}-\Gamma_{c|KK}^{(1,0)}\right]\,(s_{13}\\!-\\!s_{23})\,,$
(B.6)
$\displaystyle\bar{\Gamma}_{KK}^{(1,0)}=\frac{1}{D_{\phi}(s_{12})}\;\Gamma_{(0)\,KK}^{(1,0)}\,,$
(B.7) $\displaystyle
T^{(0,1)}=-\,\frac{1}{2}\,\left[\bar{\Gamma}_{KK}^{(0,1)}-\Gamma_{c|KK}^{(0,1)}\right]\;,$
(B.8)
$\displaystyle\bar{\Gamma}_{KK}^{(0,1)}=\frac{1}{D_{a_{0}}(s_{12})}\left[M_{21}^{(0,1)}\,\Gamma_{(0)\,\pi
8}^{(0,1)}+\left(1\\!-\\!M_{11}^{(0,1)}\right)\,\Gamma_{(0)\,KK}^{(0,1)}\right]\,,$
(B.9) $\displaystyle
T^{(0,0)}=-\,\frac{1}{2}\,\left[\bar{\Gamma}_{KK}^{(0,0)}-\Gamma_{c|KK}^{(0,0)}\right]\,,$
(B.10)
$\displaystyle\bar{\Gamma}_{KK}^{(0,0)}=\frac{1}{D_{S}(s_{12})}\left\\{\left[M_{21}^{(0,0)}\left(1\\!-\\!M_{33}^{(0,0)}\right)\\!+\\!M_{23}^{(0,0)}M_{31}^{(0,0)}\right]\,\Gamma_{(0)\,\pi\pi}^{(0,0)}\right.$
$\displaystyle\left.+\left[\left(1\\!-\\!M_{11}^{(0,0)}\right)\left(1\\!-\\!M_{33}^{(0,0)}\right)\\!-\\!M_{13}^{(0,0)}M_{31}^{(0,0)}\right]\,\Gamma_{(0)\,KK}^{(0,0)}\right.$
$\displaystyle\left.+\,\left[M_{23}^{(0,0)}\left(1\\!-\\!M_{11}^{(0,0)}\right)\\!+\\!M_{13}^{(0,0)}M_{21}^{(0,0)}\right]\,\Gamma_{(0)\,88}^{(0,0)}\right\\}\;.$
(B.11)

The various functions $\Gamma_{(0)ab}^{(J,I)}$ correspond to diagrams ($a$)
and ($b$) of Fig. 10, and represent the tree-level production of particles
$abK^{+}$ from the weak vertex. The functions $\bar{\Gamma}_{KK}^{(J,I)}$
represent the full decay vertex, from which the decay amplitude is obtained
after subtracting the contribution of the contact terms
$\Gamma_{c|KK}^{(J,I)}$ to avoid double counting. Their explicit form of the
$\Gamma_{(0)ab}^{(J,I)}$ functions are

$\Gamma_{(0)\,\pi\pi}^{(1,1)}=C\left\\{\left[\frac{\sqrt{2}\,G_{V}^{2}}{F^{2}}\right]\,\frac{s_{12}^{2}}{s_{12}^{2}-m_{\rho}^{2}}+\left[-\,\frac{1}{\sqrt{2}}\right]_{c}\right\\},$
(B.12)
$\Gamma_{(0)\,KK}^{(1,1)}=C\left\\{\left[\frac{G_{V}^{2}}{F^{2}}\right]\,\frac{s_{12}^{2}}{s_{12}^{2}-m_{\rho}^{2}}+\left[-\,\frac{1}{2}\right]_{c}\right\\}.$
(B.13)
$\Gamma_{(0)\,KK}^{(1,0)}=C\left\\{\left[\frac{3\,G_{V}^{2}}{F^{2}}\,\sin^{2}\\!\theta\right]\,\frac{s_{12}^{2}}{D_{\phi}^{\pi\rho}(s_{12}^{2})}+\left[-\,\frac{3}{2}\right]_{c}\right\\},$
(B.14) $\displaystyle\Gamma_{(0)\,\pi
8}^{(0,1)}=C\left\\{\left[\frac{2\,\sqrt{2}}{\sqrt{3}\,F^{2}}\right]\;\frac{\left[-c_{d}\,P\\!\cdot\\!p_{3}+c_{m}\,m_{D}^{2}\right]}{s_{12}^{2}-m_{a_{0}}^{2}}\left[c_{d}\,\left(s_{12}^{2}-m_{\pi}^{2}-m_{8}^{2}\right)+2\,c_{m}\,m_{\pi}^{2}\right]\right.$
$\displaystyle\left.+\left[-\,\frac{\sqrt{3}}{\sqrt{2}}\,\left[\,m_{D}^{2}/3-P\\!\cdot\\!p_{3}\right]\right]_{c}\right\\},$
(B.15)
$\displaystyle\Gamma_{(0)\,KK}^{(0,1)}=C\left\\{\left[\frac{2}{F^{2}}\right]\;\frac{\left[-c_{d}\,P\\!\cdot\\!p_{3}+c_{m}\,m_{D}^{2}\right]}{s_{12}^{2}-m_{a_{0}}^{2}}\left[c_{d}\,\left(s_{12}^{2}-2m_{K}^{2}\right)+2\,c_{m}\,m_{K}^{2}\right]\right.$
$\displaystyle\left.+\left[-\,\frac{1}{2}\left[m_{D}^{2}-P\\!\cdot\\!p_{3}\right]\right]_{c}\right\\},$
(B.16)
$\displaystyle\Gamma_{(0)\,\pi\pi}^{(0,0)}=C\left\\{\left[\frac{8\sqrt{3}}{F^{2}}\right]\;\frac{\left[-\tilde{c}_{d}\,P\\!\cdot\\!p_{3}+\tilde{c}_{m}\,m_{D}^{2}\right]}{s_{12}^{2}-m_{S1}^{2}}\;\left[\tilde{c}_{d}\,\left(s_{12}^{2}-2m_{\pi}^{2}\right)+2\,\tilde{c}_{m}\,m_{\pi}^{2}\right]\right.$
$\displaystyle\left.-\;\left[\frac{2}{\sqrt{3}\,F^{2}}\right]\;\frac{\left[-c_{d}\,P\\!\cdot\\!p_{3}+c_{m}\,m_{D}^{2}\right]}{s_{12}^{2}-m_{So}^{2}}\;\left[c_{d}\,\left(s_{12}^{2}-2m_{\pi}^{2}\right)+2\,c_{m}\,m_{\pi}^{2}\right]\right.$
$\displaystyle\left.+\left[-\,\frac{\sqrt{3}}{2}\,\left[m_{D}^{2}-P\\!\cdot\\!p_{3}\right]\right]_{c}\right\\},$
(B.17)
$\displaystyle\Gamma_{(0)\,KK}^{(0,0)}=C\left\\{\left[\frac{16}{F^{2}}\right]\;\frac{\left[-\tilde{c}_{d}\,P\\!\cdot\\!p_{3}+\tilde{c}_{m}\,m_{D}^{2}\right]}{s_{12}^{2}-m_{S1}^{2}}\;\left[\tilde{c}_{d}\,\left(s_{12}^{2}-2m_{K}^{2}\right)+2\,\tilde{c}_{m}\,m_{K}^{2}\right]\right.$
$\displaystyle\left.+\;\left[\frac{2}{3\,F^{2}}\right]\;\frac{\left[-c_{d}\,P\\!\cdot\\!p_{3}+c_{m}\,m_{D}^{2}\right]}{s_{12}^{2}-m_{So}^{2}}\left[c_{d}\,\left(s_{12}^{2}-2m_{K}^{2}\right)+2\,c_{m}\,m_{K}^{2}\right]\right.$
$\displaystyle\left.+\left[-\,\frac{3}{2}\,\left[m_{D}^{2}-P\\!\cdot\\!p_{3}\right]\right]_{c}\right\\},$
(B.18)
$\displaystyle\Gamma_{(0)\,88}^{(0,0)}=C\left\\{\left[\frac{8}{F^{2}}\right]\;\frac{\left[-\tilde{c}_{d}\,P\\!\cdot\\!p_{3}+\tilde{c}_{m}\,m_{D}^{2}\right]}{s_{12}^{2}-m_{S1}^{2}}\;\left[\tilde{c}_{d}\,\left(s_{12}^{2}-2m_{8}^{2}\right)+2\,\tilde{c}_{m}\,m_{8}^{2}\right]\right.$
$\displaystyle\left.+\;\left[\frac{2}{3\,F^{2}}\right]\;\frac{\left[-c_{d}\,P\\!\cdot\\!p_{3}+c_{m}\,m_{D}^{2}\right]}{s_{12}^{2}-m_{So}^{2}}\left[c_{d}\,\left(s_{12}^{2}-2m_{8}^{2}\right)+c_{m}\,\left(-10m_{\pi}^{2}+16m_{K}^{2}\right)/3\right]\right.$
$\displaystyle\left.+\left[-\,\frac{1}{2}\,\left[5\,m_{D}^{2}/3-3\,P\\!\cdot\\!p_{3}\right]\right]_{c}\right\\},$
(B.19)

with

$\displaystyle
P\\!\cdot\\!p_{3}=\frac{1}{2}\left[m_{D}^{2}+m_{K}^{2}-s_{12}^{2}\right]\,.$
(B.20)

In the above equations, $m_{\pi}$ and $m_{D}$ are the $\pi^{+}$ and the
$D^{+}$ masses, respectively, and $\theta$ is the $\omega-\phi$ mixing angle.
The subscripts $8$ refer to the member of the $SU(3)$ octet with the quantum
numbers of the $\eta$. The denominators in Eqs. B.5, B.7, B.9 and B.11 are the
model prediction for the resonance line shapes:

$\displaystyle
D_{\rho}=D^{(1,1)}=\left[\left(1\\!-\\!M_{11}^{(1,1)}\right)\,\left(1\\!-\\!M_{22}^{(1,1)}\right)-M_{12}^{(1,1)}\,M_{21}^{(1,1)}\right]\;,$
(B.21) $\displaystyle
D_{\phi}=D^{(1,0)}=\left\\{1\\!-\\!M^{(1,0)}\right\\}\,,$ (B.22)
$\displaystyle
D_{a_{0}}=D^{(0,1)}=\left[\left(1\\!-\\!M_{11}^{(0,1)}\right)\,\left(1\\!-\\!M_{22}^{(0,1)}\right)-M_{12}^{(0,1)}M_{21}^{(0,1)}\right]\,,$
(B.23) $\displaystyle
D_{S}=D^{(0,0)}=[1\\!-\\!M_{11}^{(0,0)}][1\\!-\\!M_{22}^{(0,0)}][1\\!-\\!M_{33}^{(0,0)}]-[1\\!-\\!M_{11}^{(0,0)}]M_{23}^{(0,0)}M_{32}^{(0,0)}$
$\displaystyle-[1\\!-\\!M_{22}^{(0,0)}]M_{13}^{(0,0)}M_{31}^{(0,0)}-[1\\!-\\!M_{33}^{(0,0)}]M_{12}^{(0,0)}M_{21}^{(0,0)}$
$\displaystyle-
M_{12}^{(0,0)}M_{23}^{(0,0)}M_{31}^{(0,0)}-M_{21}^{(0,0)}M_{32}^{(0,0)}M_{13}^{(0,0)}\;.$
(B.24)

The functions $M_{ij}^{(J,I)}$ read

$\displaystyle
M_{11}^{(1,1)}=-{\cal{K}}_{\pi\pi|\pi\pi}^{(1,1)}\,[\bar{\Omega}_{\pi\pi}^{P}/2]\;,\hskip
28.45274ptM_{12}^{(1,1)}=-{\cal{K}}_{\pi\pi|KK}^{(1,1)}\,[\bar{\Omega}_{KK}^{P}/2]\;,$
$\displaystyle
M_{21}^{(1,1)}=-{\cal{K}}_{\pi\pi|KK}^{(1,1)}\,[\bar{\Omega}_{\pi\pi}^{P}/2]\;,\hskip
28.45274ptM_{22}^{(1,1)}=-{\cal{K}}_{KK|KK}^{(1,1)}\,[\bar{\Omega}_{KK}^{P}/2]\;,$
(B.25) $\displaystyle
M^{(1,0)}=-{\cal{K}}_{KK|KK}^{(1,0)}\,[\bar{\Omega}_{KK}^{P}/2]\;,$ (B.26)
$\displaystyle M_{11}^{(0,1)}=-{\cal{K}}_{\pi 8|\pi
8}^{(0,1)}\,[\bar{\Omega}_{\pi 8}^{S}]\;,\hskip
28.45274ptM_{12}^{(0,1)}=-{\cal{K}}_{\pi
8|KK}^{(0,1)}\,[\bar{\Omega}_{KK}^{S}/2]\;,$ $\displaystyle
M_{21}^{(0,1)}=-{\cal{K}}_{\pi 8|KK}^{(0,1)}\,[\bar{\Omega}_{\pi
8}^{S}]\;,\hskip
28.45274ptM_{22}^{(0,1)}=-{\cal{K}}_{KK|KK}^{(0,1)}\,[\bar{\Omega}_{KK}^{S}/2]\;.$
(B.27) $\displaystyle
M_{11}^{(0,0)}=-{\cal{K}}_{\pi\pi|\pi\pi}^{(0,0)}\,[\bar{\Omega}_{\pi\pi}^{S}/2]\;,\hskip
28.45274ptM_{12}^{(0,0)}=-{\cal{K}}_{\pi\pi|KK}^{(0,0)}\,[\bar{\Omega}_{KK}^{S}/2]\;,$
$\displaystyle
M_{13}^{(0,0)}=-{\cal{K}}_{\pi\pi|88}^{(0,0)}\,[\bar{\Omega}_{88}^{S}/2]\;,\hskip
28.45274ptM_{21}^{(0,0)}=-{\cal{K}}_{\pi\pi|KK}^{(0,0)}\,[\bar{\Omega}_{\pi\pi}^{S}/2]\;,$
$\displaystyle
M_{22}^{(0,0)}=-{\cal{K}}_{KK|KK}^{(0,0)}\,[\bar{\Omega}_{KK}^{S}/2]\;,\hskip
28.45274ptM_{23}^{(0,0)}=-{\cal{K}}_{KK|88}^{(0,0)}\,[\bar{\Omega}_{88}^{S}/2]\;,$
$\displaystyle
M_{31}^{(0,0)}=-{\cal{K}}_{\pi\pi|88}^{(0,0)}\,[\bar{\Omega}_{\pi\pi}^{S}/2]\;,\hskip
28.45274ptM_{32}^{(0,0)}=-{\cal{K}}_{KK|88}^{(0,0)}\,[\bar{\Omega}_{KK}^{S}/2]\;,$
$\displaystyle
M_{33}^{(0,0)}=-{\cal{K}}_{88|88}^{(0,0)}\,[\bar{\Omega}_{88}^{S}/2]\;.$
(B.28)

The imaginary propagators $\bar{\Omega}$ are given by

$\displaystyle\bar{\Omega}_{ab}^{S}=-\,\frac{i}{8\pi}\;\frac{Q_{ab}}{\sqrt{s}}\;\theta(s\\!-\\!(M_{a}\\!+\\!M_{b})^{2})\;,$
(B.29)
$\displaystyle\bar{\Omega}_{aa}^{P}=-\,\frac{i}{6\pi}\;\frac{Q_{aa}^{3}}{\sqrt{s}}\;\theta(s\\!-\\!4\,M_{a}^{2})\;,$
(B.30) $\displaystyle
Q_{ab}=\frac{1}{2}\,\sqrt{s-2\,(M_{a}^{2}+M_{b}^{2})+(M_{a}^{2}-M_{b}^{2})^{2}/s}\;.$
(B.31)

The functions ${\cal{K}}_{ab|cd}^{(J,I)}$ are the scattering kernels,

${\cal{K}}_{(\pi\pi|\pi\pi)}^{(1,1)}=-2\,\left[\frac{G_{V}^{2}}{F^{4}}\right]\;\frac{s}{s-m_{\rho}^{2}}+\left[\frac{1}{F^{2}}\right]_{c},$
(B.32)

${\cal{K}}_{(\pi\pi|KK)}^{(1,1)}=-\sqrt{2}\,\left[\frac{G_{V}^{2}}{F^{4}}\right]\;\frac{s}{s-m_{\rho}^{2}}+\left[\frac{\sqrt{2}}{2\,F^{2}}\right]_{c},$
(B.33)

${\cal{K}}_{(KK|KK)}^{(1,1)}=-\left[\frac{G_{V}^{2}}{F^{4}}\right]\;\frac{s}{s-m_{\rho}^{2}}+\left[\frac{1}{2\,F^{2}}\right]_{c},$
(B.34)

${\cal{K}}_{(KK|KK)}^{(1,0)}=-3\,\left[\frac{G_{V}^{2}\;\sin^{2}\\!\theta}{F^{4}}\right]\;\frac{s}{D_{\phi}^{\pi\rho}}+\left[\frac{3}{2\,F^{2}}\right]_{c},$
(B.35) ${\cal{K}}_{(\pi 8|\pi
8)}^{(0,1)}=-\,\frac{1}{s-m_{a_{0}}^{2}}\,\left[\frac{4}{3\,F^{4}}\right]\;\left[c_{d}\,(s\\!-\\!m_{\pi}^{2}\\!-\\!m_{8}^{2})+c_{m}\,2m_{\pi}^{2}\right]^{2}+\left[\frac{2m_{\pi}^{2}}{3F^{2}}\right]_{c},$
(B.36) $\displaystyle{\cal{K}}_{(\pi
8|KK)}^{(0,1)}=-\,\frac{1}{s-m_{a_{0}}^{2}}\,\left[\frac{2\sqrt{2}}{\sqrt{3}\,F^{4}}\right]\left[c_{d}\,(s\\!-\\!m_{\pi}^{2}\\!-\\!m_{8}^{2})+c_{m}\,2m_{\pi}^{2}\right]\,\left[c_{d}\,s-(c_{d}\\!-\\!c_{m})\,2m_{K}^{2}\right]$
$\displaystyle+\left[\frac{(3s-4m_{K}^{2})}{\sqrt{6}\,F^{2}}\right]_{c},$
(B.37)
${\cal{K}}_{(KK|KK)}^{(0,1)}=-\frac{1}{s-m_{a_{0}}^{2}}\,\left[\frac{2}{F^{4}}\right]\;\left[c_{d}\,s-(c_{d}\\!-\\!c_{m})\,2m_{K}^{2}\right]^{2}+\left[\frac{s}{2F^{2}}\right]_{c},$
(B.38)

$\displaystyle{\cal{K}}_{(\pi\pi|\pi\pi)}^{(0,0)}=-\,\frac{1}{s-m_{S1}^{2}}\,\left[\frac{12}{F^{4}}\right]\;\left[\tilde{c}_{d}\,s-(\tilde{c}_{d}\\!-\\!\tilde{c}_{m})\,2m_{\pi}^{2}\right]^{2}$
$\displaystyle-\,\frac{1}{s-m_{So}^{2}}\,\left[\frac{2}{F^{4}}\right]\;\left[c_{d}\,s-(c_{d}\\!-\\!c_{m})\,2m_{\pi}^{2}\right]^{2}+\left[\frac{2s-m_{\pi}^{2}}{F^{2}}\right]_{c},$
(B.39)
$\displaystyle{\cal{K}}_{(\pi\pi|KK)}^{(0,0)}=-\,\frac{1}{s-m_{S1}^{2}}\,\left[\frac{8\sqrt{3}}{F^{4}}\right]\;\left[\tilde{c}_{d}\,s-(\tilde{c}_{d}\\!-\\!\tilde{c}_{m})\,2m_{\pi}^{2}\right]\,\left[\tilde{c}_{d}\,s-(\tilde{c}_{d}\\!-\\!\tilde{c}_{m})\,2m_{K}^{2}\right]$
$\displaystyle+\frac{1}{s-m_{So}^{2}}\,\left[\frac{2}{\sqrt{3}\,F^{4}}\right]\;\left[c_{d}\,s-(c_{d}\\!-\\!c_{m})\,2m_{\pi}^{2}\right]\left[c_{d}\,s-(c_{d}\\!-\\!c_{m})\,2m_{K}^{2}\right]$
$\displaystyle+\left[\frac{\sqrt{3}\,s}{2F^{2}}\right]_{c},$ (B.40)
$\displaystyle{\cal{K}}_{(\pi\pi|88)}^{(0,0)}=-\,\frac{1}{s-m_{S1}^{2}}\,\left[\frac{4\sqrt{3}}{F^{4}}\right]\;\left[\tilde{c}_{d}\,s-(\tilde{c}_{d}\\!-\\!\tilde{c}_{m})\,2m_{\pi}^{2}\right]\,\left[\tilde{c}_{d}\,s-(\tilde{c}_{d}\\!-\\!\tilde{c}_{m})\,2m_{8}^{2}\right]$
$\displaystyle+\frac{1}{s-m_{So}^{2}}\,\left[\frac{2}{\sqrt{3}\,F^{4}}\right]\;\left[c_{d}\,s-(c_{d}\\!-\\!c_{m})\,2m_{\pi}^{2}\right]\left[c_{d}\,(s\\!-\\!2m_{8}^{2})+c_{m}\,(16m_{K}^{2}\\!-\\!10m_{\pi}^{2})/3\right]$
$\displaystyle+\,\left[\frac{\sqrt{3}\,m_{\pi}^{2}}{3F^{2}}\right]_{c},$
(B.41)
$\displaystyle{\cal{K}}_{(KK|KK)}^{(0,0)}=-\,\frac{1}{s-m_{S1}^{2}}\,\left[\frac{16}{F^{4}}\right]\;\left[\tilde{c}_{d}\,s-(\tilde{c}_{d}\\!-\\!\tilde{c}_{m})\,2m_{K}^{2}\right]^{2}$
$\displaystyle-\frac{1}{s-m_{So}^{2}}\,\left[\frac{2}{3\,F^{4}}\right]\;\left[c_{d}\,s-(c_{d}\\!-\\!c_{m})\,2m_{K}^{2}\right]^{2}+\left[\frac{3s}{2F^{2}}\right]_{c},$
(B.42)
$\displaystyle{\cal{K}}_{(KK|88)}^{(0,0)}=-\,\frac{1}{s-m_{S1}^{2}}\,\left[\frac{8}{F^{4}}\right]\;\left[\tilde{c}_{d}\,s-(\tilde{c}_{d}\\!-\\!\tilde{c}_{m})\,2m_{K}^{2}\right]\,\left[\tilde{c}_{d}\,s-(\tilde{c}_{d}\\!-\\!\tilde{c}_{m})\,2m_{8}^{2}\right]$
$\displaystyle-\,\frac{1}{s-m_{So}^{2}}\,\left[\frac{2}{3\,F^{4}}\right]\;\left[c_{d}\,s-(c_{d}\\!-\\!c_{m})\,2m_{K}^{2}\right]\left[c_{d}\,(s\\!-\\!2m_{8}^{2})+c_{m}\,(16m_{K}^{2}\\!-\\!10m_{\pi}^{2})/3\right]$
$\displaystyle+\,\left[\frac{9s-8m_{K}^{2}}{6F^{2}}\right]_{c},$ (B.43)
$\displaystyle{\cal{K}}_{(88|88)}^{(0,0)}=-\,\frac{1}{s-m_{S1}^{2}}\,\left[\frac{4}{F^{4}}\right]\;\left[\tilde{c}_{d}\,s-(\tilde{c}_{d}\\!-\\!\tilde{c}_{m})\,2m_{8}^{2}\right]^{2}$
$\displaystyle-\,\frac{1}{s-m_{So}^{2}}\,\left[\frac{2}{3\,F^{4}}\right]\;\left[c_{d}\,(s\\!-\\!2m_{8}^{2})+c_{m}\,(16m_{K}^{2}\\!-\\!10m_{\pi}^{2})/3\right]^{2}$
$\displaystyle+\left[\frac{-7m_{\pi}^{2}+16m_{K}^{2}}{9F^{2}}\right]_{c}.$
(B.44)

##  Appendix C Scattering amplitudes

The $K^{+}K^{-}$ scattering amplitudes are written in terms of the
denominators $D^{(J,I)}$ as

$\displaystyle
A_{KK|KK}^{(1,1)}=\frac{1}{D_{\rho}(s_{12})}\left[M_{21}^{(1,1)}\,{\cal{K}}_{\pi\pi|KK}^{(1,1)}+\left(1\\!-\\!M_{11}^{(1,1)}\right)\,{\cal{K}}_{KK|KK}^{(1,1)}\right]\;,$
(C.1) $\displaystyle
A_{KK|KK}^{(1,0)}=\frac{1}{D_{\phi}(m_{12}^{2})}\;{\cal{K}}_{KK|KK}^{(1,0)}\;,$
(C.2) $\displaystyle
A_{KK|KK}^{(0,1)}=\frac{1}{D_{a_{0}}(s_{12})}\left[M_{21}^{(0,1)}\,{\cal{K}}_{\pi
8|KK}^{(0,1)}+\left(1\\!-\\!M_{11}^{(0,1)}\right)\,{\cal{K}}_{KK|KK}^{(0,1)}\right],$
(C.3) $\displaystyle
A_{KK|KK}^{(0,0)}=\frac{1}{D_{S}(s_{12})}\left\\{\left[M_{21}^{(0,0)}\left(1\\!-\\!M_{33}^{(0,0)}\right)\\!+\\!M_{23}^{(0,0)}M_{31}^{(0,0)}\right]\,{\cal{K}}_{\pi\pi|KK}^{(0,0)}\right.$
$\displaystyle\left.\,+\left[\left(1\\!-\\!M_{11}^{(0,0)}\right)\left(1\\!-\\!M_{33}^{(0,0)}\right)\\!-\\!M_{13}^{(0,0)}M_{31}^{(0,0)}\right]\,{\cal{K}}_{KK|KK}^{(0,0)}\right.$
$\displaystyle\left.\,+\left[M_{23}^{(0,0)}\left(1\\!-\\!M_{11}^{(0,0)}\right)\\!+\\!M_{13}^{(0,0)}M_{21}^{(0,0)}\right]\,{\cal{K}}_{88|KK}^{(0,0)}\right\\}\;.$
(C.4)

## References

  * [1] LHCb collaboration, R. Aaij et al., _Measurement of the branching fractions of the decays $D^{+}\rightarrow K^{-}K^{+}K^{+}$, $D^{+}\rightarrow\pi^{-}\pi^{+}K^{+}$ and $D^{+}_{s}\rightarrow\pi^{-}K^{+}K^{+}$_, arXiv:1810.03138, Submitted to JHEP
  * [2] D. Asner, _Charm Dalitz plot analysis formalism and results_ , arXiv:hep-ex/0410014
  * [3] R. T. Aoude, P. C. Magalhães, A. C. dos Reis, and M. R. Robilotta, _Multimeson model for the $D^{+}\rightarrow K^{-}K^{+}K^{+}$ decay amplitude_, Phys. Rev. D98 (2018) 056021, arXiv:1805.11764
  * [4] G. Grayer et al., _High statistics study of the reaction $\pi^{-}p\rightarrow\pi^{-}\pi^{+}n$: apparatus, method of analysis, and general features of results at 17 GeV/c_, Nucl. Phys. B75 (1974) 189
  * [5] E135 collaboration, K. Takamatsu et al., _$\pi^{0}\pi^{0}$ scattering amplitudes and phase shifts obtained by the $\pi^{-}p$ charge exchange process_, Nucl. Phys. A675 (2000) 312C
  * [6] D. Aston et al., _A study of $K^{-}\pi^{+}$ scattering in the reaction $K^{-}p\rightarrow K^{-}\pi^{+}n$ at 11 GeV/c_, Nucl. Phys. B296 (1988) 493
  * [7] A. J. Pawlicki et al., _High-statistics study of the reactions ${\pi}^{-}{p}\rightarrow{K}^{-}{K}^{+}{p}$ at 6 GeV/${c}$_, Phys. Rev. D15 (1977) 3196
  * [8] D. Cohen et al., _Amplitude analysis of the ${K}^{-}{K}^{+}$ system produced in the reactions ${\pi}^{-}{p}\rightarrow{K}^{-}{K}^{+}{n}$ and ${\pi}^{+}{n}\rightarrow{K}^{-}{K}^{+}{p}$ at 6 GeV/${c}$_, Phys. Rev. D22 (1980) 2595
  * [9] OBELIX collaboration, M. Bargiotti et al., _Coupled channel analysis of ${\pi}^{-}{\pi}^{+}{\pi}^{0}$, ${K}^{-}{K}^{+}{\pi}^{0}$ and ${K}^{0}_{S}{K}^{\pm}{\pi}^{\mp}$ from $\bar{p}p$ annihilation at rest in hydrogen target at three densities_, Eur. Phys. J. C26 (2003) 371
  * [10] LHCb collaboration, A. A. Alves Jr. et al., _The LHCb detector at the LHC_, JINST 3 (2008) S08005
  * [11] LHCb collaboration, R. Aaij et al., _LHCb detector performance_ , Int. J. Mod. Phys. A30 (2015) 1530022, arXiv:1412.6352
  * [12] M. Adinolfi et al., _Performance of the LHCb RICH detector at the LHC_, Eur. Phys. J. C73 (2013) 2431, arXiv:1211.6759
  * [13] T. Sjöstrand, S. Mrenna, and P. Skands, _PYTHIA 6.4 physics and manual_ , JHEP 05 (2006) 026, arXiv:hep-ph/0603175
  * [14] T. Sjöstrand, S. Mrenna, and P. Skands, _A brief introduction to PYTHIA 8.1_ , Comput. Phys. Commun. 178 (2008) 852, arXiv:0710.3820
  * [15] I. Belyaev et al., _Handling of the generation of primary events in Gauss, the LHCb simulation framework_ , J. Phys. Conf. Ser. 331 (2011) 032047
  * [16] D. J. Lange, _The EvtGen particle decay simulation package_ , Nucl. Instrum. Meth. A462 (2001) 152
  * [17] P. Golonka and Z. Was, _PHOTOS Monte Carlo: A precision tool for QED corrections in $Z$ and $W$ decays_, Eur. Phys. J. C45 (2006) 97, arXiv:hep-ph/0506026
  * [18] Geant4 collaboration, J. Allison et al., _Geant4 developments and applications_ , IEEE Trans. Nucl. Sci. 53 (2006) 270
  * [19] Geant4 collaboration, S. Agostinelli et al., _Geant4: A simulation toolkit_ , Nucl. Instrum. Meth. A506 (2003) 250
  * [20] M. Clemencic et al., _The LHCb simulation application, Gauss: Design, evolution and experience_, J. Phys. Conf. Ser. 331 (2011) 032023
  * [21] L. Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stone, Classification and regression trees, Wadsworth international group, Belmont, California, USA, 1984
  * [22] Y. Freund and R. E. Schapire, _A decision-theoretic generalization of on-line learning and an application to boosting_ , J. Comput. Syst. Sci. 55 (1997) 119
  * [23] Particle Data Group, M. Tanabashi et al., _Review of particle physics_ , Phys. Rev. D98 (2018) 030001
  * [24] L. Anderlini et al., _The PIDCalib package_ , LHCb-PUB-2016-021
  * [25] A. James and M. Roos, _A System for Function Minimization and Analysis of the Parameter Errors and Correlations_ , Comput. Phys. Commun. 10 (1975) 343
  * [26] J. Blatt and J. Weisskopf, Theoretical nuclear physics, John Wiley and Sons, 1952
  * [27] C. Zemach, _Three-Pion decays of unstable particles_ , Phys. Rev. B133 (1964) 1201
  * [28] BES collaboration, M. Ablikim et al., _Resonances in $J/\psi\rightarrow\phi\pi^{+}\pi^{-}$ and $\phi K^{+}K^{-}$_, Phys. Lett. B607 (2005) 243, arXiv:hep-ex/0411001
  * [29] G. Breit and E. Wigner, _Capture of Slow Neutrons_ , Phys. Rev. 49 (1936) 519
  * [30] S. M. Flatté, _Coupled-channel analysis of the $\pi\eta$ and $K\bar{K}$ systems near $K\bar{K}$ threshold_, Phys. Lett. 63 (1976) B224

LHCb collaboration

R. Aaij30, C. Abellán Beteta48, B. Adeva45, M. Adinolfi52, C.A. Aidala80, Z.
Ajaltouni7, S. Akar63, P. Albicocco21, J. Albrecht12, F. Alessio46, M.
Alexander57, A. Alfonso Albero44, G. Alkhazov36, P. Alvarez Cartelle59, A.A.
Alves Jr45, S. Amato2, S. Amerio26, Y. Amhis9, L. An4, L. Anderlini20, G.
Andreassi47, M. Andreotti19, J.E. Andrews64, F. Archilli30, P. d’Argent14, J.
Arnau Romeu8, A. Artamonov43, M. Artuso65, K. Arzymatov40, E. Aslanides8, M.
Atzeni48, B. Audurier25, S. Bachmann14, J.J. Back54, S. Baker59, V.
Balagura9,b, W. Baldini19, A. Baranov40, R.J. Barlow60, G.C. Barrand9, S.
Barsuk9, W. Barter60, M. Bartolini22, F. Baryshnikov76, V. Batozskaya34, B.
Batsukh65, A. Battig12, V. Battista47, A. Bay47, J. Beddow57, F. Bedeschi27,
I. Bediaga1, A. Beiter65, L.J. Bel30, S. Belin25, N. Beliy68, V. Bellee47, N.
Belloli23,i, K. Belous43, I. Belyaev37, E. Ben-Haim10, G. Bencivenni21, S.
Benson30, S. Beranek11, A. Berezhnoy38, R. Bernet48, D. Berninghoff14, E.
Bertholet10, A. Bertolin26, C. Betancourt48, F. Betti18,46, M.O. Bettler53, M.
van Beuzekom30, Ia. Bezshyiko48, S. Bhasin52, J. Bhom32, S. Bifani51, P.
Billoir10, A. Birnkraut12, A. Bizzeti20,u, M. Bjørn61, M.P. Blago46, T.
Blake54, F. Blanc47, S. Blusk65, D. Bobulska57, V. Bocci29, O. Boente
Garcia45, T. Boettcher62, A. Bondar42,x, N. Bondar36, S. Borghi60,46, M.
Borisyak40, M. Borsato45, F. Bossu9, M. Boubdir11, T.J.V. Bowcock58, C.
Bozzi19,46, S. Braun14, M. Brodski46, J. Brodzicka32, A. Brossa Gonzalo54, D.
Brundu25,46, E. Buchanan52, A. Buonaura48, C. Burr60, A. Bursche25, J.
Buytaert46, W. Byczynski46, S. Cadeddu25, H. Cai70, R. Calabrese19,g, R.
Calladine51, M. Calvi23,i, M. Calvo Gomez44,m, A. Camboni44,m, P. Campana21,
D.H. Campora Perez46, L. Capriotti18, A. Carbone18,e, G. Carboni28, R.
Cardinale22, A. Cardini25, P. Carniti23,i, L. Carson56, K. Carvalho Akiba2, G.
Casse58, L. Cassina23, M. Cattaneo46, G. Cavallero22, R. Cenci27,p, D.
Chamont9, M.G. Chapman52, M. Charles10, Ph. Charpentier46, G.
Chatzikonstantinidis51, M. Chefdeville6, V. Chekalina40, C. Chen4, S. Chen25,
S.-G. Chitic46, V. Chobanova45, M. Chrzaszcz46, A. Chubykin36, P. Ciambrone21,
X. Cid Vidal45, G. Ciezarek46, F. Cindolo18, P.E.L. Clarke56, M. Clemencic46,
H.V. Cliff53, J. Closier46, V. Coco46, J.A.B. Coelho9, J. Cogan8, E.
Cogneras7, L. Cojocariu35, P. Collins46, T. Colombo46, A. Comerma-Montells14,
A. Contu25, G. Coombs46, S. Coquereau44, G. Corti46, M. Corvo19,g, C.M. Costa
Sobral54, B. Couturier46, G.A. Cowan56, D.C. Craik62, A. Crocombe54, M. Cruz
Torres1, R. Currie56, C. D’Ambrosio46, F. Da Cunha Marinho2, C.L. Da Silva81,
E. Dall’Occo30, J. Dalseno45,v, A. Danilina37, A. Davis4, O. De Aguiar
Francisco46, K. De Bruyn46, S. De Capua60, M. De Cian47, J.M. De Miranda1, L.
De Paula2, M. De Serio17,d, P. De Simone21, C.T. Dean57, D. Decamp6, L. Del
Buono10, B. Delaney53, H.-P. Dembinski13, M. Demmer12, A. Dendek33, D.
Derkach41, O. Deschamps7, F. Desse9, F. Dettori58, B. Dey71, A. Di Canto46, P.
Di Nezza21, S. Didenko76, H. Dijkstra46, F. Dordei46, M. Dorigo46,y, A. Dosil
Suárez45, L. Douglas57, A. Dovbnya49, K. Dreimanis58, L. Dufour30, G.
Dujany10, P. Durante46, J.M. Durham81, D. Dutta60, R. Dzhelyadin43, M.
Dziewiecki14, A. Dziurda32, A. Dzyuba36, S. Easo55, U. Egede59, V.
Egorychev37, S. Eidelman42,x, S. Eisenhardt56, U. Eitschberger12, R.
Ekelhof12, L. Eklund57, S. Ely65, A. Ene35, S. Escher11, S. Esen30, T.
Evans63, A. Falabella18, N. Farley51, S. Farry58, D. Fazzini23,46,i, L.
Federici28, P. Fernandez Declara46, A. Fernandez Prieto45, F. Ferrari18, L.
Ferreira Lopes47, F. Ferreira Rodrigues2, M. Ferro-Luzzi46, S. Filippov39,
R.A. Fini17, M. Fiorini19,g, M. Firlej33, C. Fitzpatrick47, T. Fiutowski33, F.
Fleuret9,b, M. Fontana46, F. Fontanelli22,h, R. Forty46, V. Franco Lima58, M.
Frank46, C. Frei46, J. Fu24,q, W. Funk46, C. Färber46, M. Féo30, E. Gabriel56,
A. Gallas Torreira45, D. Galli18,e, S. Gallorini26, S. Gambetta56, Y. Gan4, M.
Gandelman2, P. Gandini24, Y. Gao4, L.M. Garcia Martin78, B. Garcia Plana45, J.
García Pardiñas48, J. Garra Tico53, L. Garrido44, D. Gascon44, C. Gaspar46, L.
Gavardi12, G. Gazzoni7, D. Gerick14, E. Gersabeck60, M. Gersabeck60, T.
Gershon54, D. Gerstel8, Ph. Ghez6, V. Gibson53, O.G. Girard47, P. Gironella
Gironell44, L. Giubega35, K. Gizdov56, V.V. Gligorov10, D. Golubkov37, A.
Golutvin59,76, A. Gomes1,a, I.V. Gorelov38, C. Gotti23,i, E. Govorkova30, J.P.
Grabowski14, R. Graciani Diaz44, L.A. Granado Cardoso46, E. Graugés44, E.
Graverini48, G. Graziani20, A. Grecu35, R. Greim30, P. Griffith25, L.
Grillo60, L. Gruber46, B.R. Gruberg Cazon61, O. Grünberg73, C. Gu4, E.
Gushchin39, A. Guth11, Yu. Guz43,46, T. Gys46, C. Göbel67, T. Hadavizadeh61,
C. Hadjivasiliou7, G. Haefeli47, C. Haen46, S.C. Haines53, B. Hamilton64, X.
Han14, T.H. Hancock61, S. Hansmann-Menzemer14, N. Harnew61, S.T. Harnew52, T.
Harrison58, C. Hasse46, M. Hatch46, J. He68, M. Hecker59, K. Heinicke12, A.
Heister12, K. Hennessy58, L. Henry78, E. van Herwijnen46, J. Heuel11, M.
Heß73, A. Hicheur66, R. Hidalgo Charman60, D. Hill61, M. Hilton60, P.H.
Hopchev47, J. Hu14, W. Hu71, W. Huang68, Z.C. Huard63, W. Hulsbergen30, T.
Humair59, M. Hushchyn41, D. Hutchcroft58, D. Hynds30, P. Ibis12, M. Idzik33,
P. Ilten51, A. Inyakin43, K. Ivshin36, R. Jacobsson46, J. Jalocha61, E.
Jans30, B.K. Jashal78, A. Jawahery64, F. Jiang4, M. John61, D. Johnson46, C.R.
Jones53, C. Joram46, B. Jost46, N. Jurik61, S. Kandybei49, M. Karacson46, J.M.
Kariuki52, S. Karodia57, N. Kazeev41, M. Kecke14, F. Keizer53, M. Kelsey65, M.
Kenzie53, T. Ketel31, E. Khairullin40, B. Khanji46, C. Khurewathanakul47, K.E.
Kim65, T. Kirn11, S. Klaver21, K. Klimaszewski34, T. Klimkovich13, S.
Koliiev50, M. Kolpin14, R. Kopecna14, P. Koppenburg30, I. Kostiuk30, S.
Kotriakhova36, M. Kozeiha7, L. Kravchuk39, M. Kreps54, F. Kress59, P.
Krokovny42,x, W. Krupa33, W. Krzemien34, W. Kucewicz32,l, M. Kucharczyk32, V.
Kudryavtsev42,x, A.K. Kuonen47, T. Kvaratskheliya37,46, D. Lacarrere46, G.
Lafferty60, A. Lai25, D. Lancierini48, G. Lanfranchi21, C. Langenbruch11, T.
Latham54, C. Lazzeroni51, R. Le Gac8, A. Leflat38, J. Lefrançois9, R.
Lefèvre7, F. Lemaitre46, O. Leroy8, T. Lesiak32, B. Leverington14, P.-R.
Li68,ab, Y. Li5, Z. Li65, X. Liang65, T. Likhomanenko75, R. Lindner46, F.
Lionetto48, V. Lisovskyi9, G. Liu69, X. Liu4, D. Loh54, A. Loi25, I.
Longstaff57, J.H. Lopes2, G.H. Lovell53, D. Lucchesi26,o, M. Lucio Martinez45,
A. Lupato26, E. Luppi19,g, O. Lupton46, A. Lusiani27, X. Lyu68, F. Machefert9,
F. Maciuc35, V. Macko47, P. Mackowiak12, S. Maddrell-Mander52, O. Maev36,46,
P. C. Magalhães15, K. Maguire60, D. Maisuzenko36, M.W. Majewski33, S. Malde61,
B. Malecki32, A. Malinin75, T. Maltsev42,x, G. Manca25,f, G. Mancinelli8, D.
Marangotto24,q, J. Maratas7,w, J.F. Marchand6, U. Marconi18, C. Marin Benito9,
M. Marinangeli47, P. Marino47, J. Marks14, P.J. Marshall58, G. Martellotti29,
M. Martin8, M. Martinelli46, D. Martinez Santos45, F. Martinez Vidal78, A.
Massafferri1, M. Materok11, R. Matev46, A. Mathad54, Z. Mathe46, C.
Matteuzzi23, A. Mauri48, E. Maurice9,b, B. Maurin47, A. Mazurov51, M.
McCann59,46, A. McNab60, R. McNulty16, J.V. Mead58, B. Meadows63, C. Meaux8,
N. Meinert73, D. Melnychuk34, M. Merk30, A. Merli24,q, E. Michielin26, D.A.
Milanes72, E. Millard54, M.-N. Minard6, L. Minzoni19,g, D.S. Mitzel14, A.
Mogini10, R.D. Moise59, T. Mombächer12, I.A. Monroy72, S. Monteil7, M.
Morandin26, G. Morello21, M.J. Morello27,t, O. Morgunova75, J. Moron33, A.B.
Morris8, R. Mountain65, F. Muheim56, M. Mulder30, C.H. Murphy61, D. Murray60,
A. Mödden 12, D. Müller46, J. Müller12, K. Müller48, V. Müller12, P. Naik52,
T. Nakada47, R. Nandakumar55, A. Nandi61, T. Nanut47, I. Nasteva2, M.
Needham56, N. Neri24,q, S. Neubert14, N. Neufeld46, M. Neuner14, R.
Newcombe59, T.D. Nguyen47, C. Nguyen-Mau47,n, S. Nieswand11, R. Niet12, N.
Nikitin38, A. Nogay75, N.S. Nolte46, D.P. O’Hanlon18, A. Oblakowska-Mucha33,
V. Obraztsov43, S. Ogilvy57, R. Oldeman25,f, C.J.G. Onderwater74, A.
Ossowska32, J.M. Otalora Goicochea2, T. Ovsiannikova37, P. Owen48, A.
Oyanguren78, P.R. Pais47, T. Pajero27,t, A. Palano17, M. Palutan21, G.
Panshin77, A. Papanestis55, M. Pappagallo56, L.L. Pappalardo19,g, W. Parker64,
C. Parkes60,46, G. Passaleva20,46, A. Pastore17, M. Patel59, C.
Patrignani18,e, A. Pearce46, A. Pellegrino30, G. Penso29, M. Pepe Altarelli46,
S. Perazzini46, D. Pereima37, P. Perret7, L. Pescatore47, K. Petridis52, A.
Petrolini22,h, A. Petrov75, S. Petrucci56, M. Petruzzo24,q, B. Pietrzyk6, G.
Pietrzyk47, M. Pikies32, M. Pili61, D. Pinci29, J. Pinzino46, F. Pisani46, A.
Piucci14, V. Placinta35, S. Playfer56, J. Plews51, M. Plo Casasus45, F.
Polci10, M. Poli Lener21, A. Poluektov54, N. Polukhina76,c, I. Polyakov65, E.
Polycarpo2, G.J. Pomery52, S. Ponce46, A. Popov43, D. Popov51,13, S.
Poslavskii43, C. Potterat2, E. Price52, J. Prisciandaro45, C. Prouve52, V.
Pugatch50, A. Puig Navarro48, H. Pullen61, G. Punzi27,p, W. Qian68, J. Qin68,
R. Quagliani10, B. Quintana7, N.V. Raab16, B. Rachwal33, J.H. Rademacker52, M.
Rama27, M. Ramos Pernas45, M.S. Rangel2, F. Ratnikov40,41, G. Raven31, M.
Ravonel Salzgeber46, M. Reboud6, F. Redi47, S. Reichert12, A.C. dos Reis1, F.
Reiss10, C. Remon Alepuz78, Z. Ren4, V. Renaudin9, S. Ricciardi55, S.
Richards52, K. Rinnert58, P. Robbe9, A. Robert10, M. R. Robilotta3, A.B.
Rodrigues47, E. Rodrigues63, J.A. Rodriguez Lopez72, M. Roehrken46, S.
Roiser46, A. Rollings61, V. Romanovskiy43, A. Romero Vidal45, M. Rotondo21,
M.S. Rudolph65, T. Ruf46, J. Ruiz Vidal78, J.J. Saborido Silva45, N.
Sagidova36, B. Saitta25,f, V. Salustino Guimaraes67, C. Sanchez Gras30, C.
Sanchez Mayordomo78, B. Sanmartin Sedes45, R. Santacesaria29, C. Santamarina
Rios45, M. Santimaria21,46, E. Santovetti28,j, G. Sarpis60, A. Sarti21,k, C.
Satriano29,s, A. Satta28, M. Saur68, D. Savrina37,38, S. Schael11, M.
Schellenberg12, M. Schiller57, H. Schindler46, M. Schmelling13, T.
Schmelzer12, B. Schmidt46, O. Schneider47, A. Schopper46, H.F. Schreiner63, M.
Schubiger47, M.H. Schune9, R. Schwemmer46, B. Sciascia21, A. Sciubba29,k, A.
Semennikov37, E.S. Sepulveda10, A. Sergi51, N. Serra48, J. Serrano8, L.
Sestini26, A. Seuthe12, P. Seyfert46, M. Shapkin43, Y. Shcheglov36,†, T.
Shears58, L. Shekhtman42,x, V. Shevchenko75, E. Shmanin76, B.G. Siddi19, R.
Silva Coutinho48, L. Silva de Oliveira2, G. Simi26,o, S. Simone17,d, I.
Skiba19, N. Skidmore14, T. Skwarnicki65, M.W. Slater51, J.G. Smeaton53, E.
Smith11, I.T. Smith56, M. Smith59, M. Soares18, l. Soares Lavra1, M.D.
Sokoloff63, F.J.P. Soler57, B. Souza De Paula2, B. Spaan12, E. Spadaro
Norella24,q, P. Spradlin57, F. Stagni46, M. Stahl14, S. Stahl46, P. Stefko47,
S. Stefkova59, O. Steinkamp48, S. Stemmle14, O. Stenyakin43, M. Stepanova36,
H. Stevens12, A. Stocchi9, S. Stone65, B. Storaci48, S. Stracka27, M.E.
Stramaglia47, M. Straticiuc35, U. Straumann48, S. Strokov77, J. Sun4, L.
Sun70, K. Swientek33, A. Szabelski34, T. Szumlak33, M. Szymanski68, S.
T’Jampens6, Z. Tang4, A. Tayduganov8, T. Tekampe12, G. Tellarini19, F.
Teubert46, E. Thomas46, J. van Tilburg30, M.J. Tilley59, V. Tisserand7, M.
Tobin33, S. Tolk46, L. Tomassetti19,g, D. Tonelli27, D.Y. Tou10, R. Tourinho
Jadallah Aoude1, E. Tournefier6, M. Traill57, M.T. Tran47, A. Trisovic53, A.
Tsaregorodtsev8, G. Tuci27,p, A. Tully53, N. Tuning30,46, A. Ukleja34, A.
Usachov9, A. Ustyuzhanin40, U. Uwer14, A. Vagner77, V. Vagnoni18, A.
Valassi46, S. Valat46, G. Valenti18, R. Vazquez Gomez46, P. Vazquez
Regueiro45, S. Vecchi19, M. van Veghel30, J.J. Velthuis52, M. Veltri20,r, G.
Veneziano61, A. Venkateswaran65, M. Vernet7, M. Veronesi30, M. Vesterinen61,
J.V. Viana Barbosa46, D. Vieira68, M. Vieites Diaz45, H. Viemann73, X.
Vilasis-Cardona44,m, A. Vitkovskiy30, M. Vitti53, V. Volkov38, A. Vollhardt48,
D. Vom Bruch10, B. Voneki46, A. Vorobyev36, V. Vorobyev42,x, N. Voropaev36,
J.A. de Vries30, C. Vázquez Sierra30, R. Waldi73, J. Walsh27, J. Wang5, M.
Wang4, Y. Wang71, Z. Wang48, D.R. Ward53, H.M. Wark58, N.K. Watson51, D.
Websdale59, A. Weiden48, C. Weisser62, M. Whitehead11, J. Wicht54, G.
Wilkinson61, M. Wilkinson65, I. Williams53, M.R.J. Williams60, M. Williams62,
T. Williams51, F.F. Wilson55, M. Winn9, W. Wislicki34, M. Witek32, G.
Wormser9, S.A. Wotton53, K. Wyllie46, D. Xiao71, Y. Xie71, A. Xu4, M. Xu71, Q.
Xu68, Z. Xu4, Z. Xu6, Z. Yang4, Z. Yang64, Y. Yao65, L.E. Yeomans58, H. Yin71,
J. Yu71,aa, X. Yuan65, O. Yushchenko43, K.A. Zarebski51, M. Zavertyaev13,c, D.
Zhang71, L. Zhang4, W.C. Zhang4,z, Y. Zhang9, A. Zhelezov14, Y. Zheng68, X.
Zhu4, V. Zhukov11,38, J.B. Zonneveld56, S. Zucchelli18.

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

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
kUniversità di Roma La Sapienza, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
nHanoi University of Science, Hanoi, Vietnam
oUniversità di Padova, Padova, Italy
pUniversità di Pisa, Pisa, Italy
qUniversità degli Studi di Milano, Milano, Italy
rUniversità di Urbino, Urbino, Italy
sUniversità della Basilicata, Potenza, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità di Modena e Reggio Emilia, Modena, Italy
vH.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom
wMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
xNovosibirsk State University, Novosibirsk, Russia
ySezione INFN di Trieste, Trieste, Italy
zSchool of Physics and Information Technology, Shaanxi Normal University
(SNNU), Xi’an, China
aaPhysics and Micro Electronic College, Hunan University, Changsha City, China
abLanzhou University, Lanzhou, China
†Deceased