EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2020-086 LHCb-PAPER-2020-008 March 11, 2021

Study of the lineshape of the ${\chi_{{c}1}}(3872)$ state

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

A study of the lineshape of the ${\chi_{{c}1}}(3872)$ state is made using a
data sample corresponding to an integrated luminosity of $3\mbox{\,fb}^{-1}$
collected in $pp$ collisions at centre-of-mass energies of 7 and 8 TeV with
the LHCb detector. Candidate ${\chi_{{c}1}}(3872)$ and ${\psi{(2S)}}$ mesons
from ${b}$-hadron decays are selected in the $J/\psi\pi^{+}\pi^{-}$ decay
mode. Describing the lineshape with a Breit–Wigner function, the mass
splitting between the ${\chi_{{c}1}}(3872)$ and ${\psi{(2S)}}$ states, $\Delta
m$, and the width of the ${\chi_{{c}1}}(3872)$ state, $\Gamma_{\mathrm{BW}}$,
are determined to be

$\displaystyle\Delta m$ $\displaystyle=$ $\displaystyle 185.598\pm 0.067\pm
0.068\mathrm{\,Me\kern-1.00006ptV}\,,$ $\displaystyle\Gamma_{\mathrm{BW}}$
$\displaystyle=$ $\displaystyle\phantom{00}1.39\phantom{0}\pm
0.24\phantom{0}\pm 0.10\phantom{0}\mathrm{\,Me\kern-1.00006ptV}\,,$

where the first uncertainty is statistical and the second systematic. Using a
Flatté-inspired model, the mode and full width at half maximum of the
lineshape are determined to be

$\displaystyle\mathrm{mode}$
$\displaystyle=3871.69^{\,+\,0.00\,+\,0.05}_{-0.04\,-\,0.13}$
$\displaystyle\mathrm{\,Me\kern-1.00006ptV}$ $\displaystyle\mathrm{FWHM}$
$\displaystyle=0.22^{\,+\,0.07\,+\,0.11}_{\,-\,0.06\,-\,0.13}$
$\displaystyle\mathrm{\,Me\kern-1.00006ptV}.$

An investigation of the analytic structure of the Flatté amplitude reveals a
pole structure, which is compatible with a quasi-bound ${{D}^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ state but a quasi-virtual
state is still allowed at the level of $2$ standard deviations.

Published in Phys. Rev. D102 (2020) 092005

© CERN on behalf of the LHCb collaboration, licence CC BY 4.0.

##  1 Introduction

The last two decades have seen a resurgence of interest in the spectroscopy of
non-conventional (exotic) charmonium states [1] starting with the observation
of the charmonium-like ${\chi_{{c}1}}(3872)$ state by the Belle collaboration
[2]. Though the existence of the ${\chi_{{c}1}}(3872)$ particle has been
confirmed by many experiments [3, 4, 5, 6, 7] with quantum numbers measured to
be $1^{++}$ [8, 9], its nature is still uncertain. Several exotic
interpretations have been suggested: e.g. a tetraquark [10], a loosely bound
deuteron-like ${{D}^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ molecule [11] or a
charmonium-molecule mixture [12].

A striking feature of the ${\chi_{{c}1}}(3872)$ state is the proximity of its
mass to the sum of the ${{D}^{*0}}$ and ${{D}^{0}}$ meson masses. Accounting
for correlated uncertainties due to the knowledge of the kaon mass, this sum
is evaluated to be $m_{{{D}^{0}}}+m_{{{D}^{*0}}}=3871.70\pm
0.11\mathrm{\,Me\kern-1.00006ptV}$ 111 Natural units with $c=\hbar=1$ are used
through this paper. . The molecular interpretation of the
${\chi_{{c}1}}(3872)$ state requires it to be a bound state. Assuming a
Breit–Wigner lineshape, this implies that $\delta E\equiv
m_{{{D}^{0}}}+m_{{{D}^{*0}}}-m_{{\chi_{{c}1}}(3872)}>0$. Current knowledge of
$\delta E$ is limited by the uncertainty on the ${\chi_{{c}1}}(3872)$ mass,
motivating a more precise determination of this quantity. The nature of the
${\chi_{{c}1}}(3872)$ state can also be elucidated by studies of its
lineshape. This has been analysed by several experiments assuming a
Breit–Wigner function [3, 5, 13]. The current upper limit on the natural
width, $\Gamma_{\mathrm{BW}}$, is $1.2\mathrm{\,Me\kern-1.00006ptV}$ at $90\%$
confidence level [14].

In this analysis a sample of ${\chi_{{c}1}}(3872)\rightarrow
J/\psi\pi^{+}\pi^{-}$ candidates produced in inclusive ${b}$-hadron decays is
used to measure precisely the mass and to determine the lineshape of the
${\chi_{{c}1}}(3872)$ meson. Studies are made assuming both a Breit–Wigner
lineshape and a Flatté-inspired model that accounts for the opening up of the
${{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{D}^{*0}}$ threshold
[15, 16]. The analysis uses a data sample corresponding to an integrated
luminosity of $3\mbox{\,fb}^{-1}$ of data collected in $pp$ collisions at
centre-of-mass energies of 7 TeV and 8 TeV during 2011 and 2012 using the LHCb
detector.

##  2 Detector and simulation

The LHCb detector [17, 18] 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 [19], a large-area silicon-strip detector (TT) 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 [20]
placed downstream of the magnet. The tracking system provides a measurement of
momentum, $p$, of charged particles with a relative uncertainty that varies
from 0.5% at low momentum to 1.0% at 200$\mathrm{\,Ge\kern-1.00006ptV}$. As
described in Refs. [21, 22] the momentum scale is calibrated using samples of
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rightarrow{\mu^{+}}{\mu^{-}}$
and ${{B}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}$ decays collected concurrently with the data sample used for
this analysis. The relative accuracy of this procedure is estimated to be
$3\times 10^{-4}$ using samples of other fully reconstructed ${b}$ hadrons,
$\mathchar 28935\relax$ and ${{K}^{0}_{\mathrm{S}}}$ mesons. 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_{\mathrm{T}}$}){\,\upmu\mathrm{m}}$, where $p_{\mathrm{T}}$
is the component of the momentum transverse to the beam, in
$\mathrm{\,Ge\kern-1.00006ptV}$.

Different types of charged hadrons are distinguished using information from
two ring-imaging Cherenkov (RICH) detectors. Photons, electrons and hadrons
are identified by a calorimeter system consisting of scintillating-pad and
preshower detectors, an electromagnetic calorimeter and a hadronic
calorimeter. Muons are identified by a system composed of alternating layers
of iron and multiwire proportional chambers [23].

The online event selection is performed by a trigger [24], which consists of a
hardware stage, based on information from the calorimeter and muon systems,
followed by a software stage, where a full event reconstruction is made.
Candidate events are required to pass the hardware trigger, which selects muon
and dimuon candidates with high $p_{\mathrm{T}}$ based upon muon system
information. The subsequent software trigger is composed of two stages. The
first performs a partial event reconstruction and requires events to have two
well-identified oppositely charged muons and that the mass of the pair is
larger than $2.7\mathrm{\,Ge\kern-1.00006ptV}$. The second stage performs a
full event reconstruction. Events are retained for further processing if they
contain a displaced ${\mu^{+}}{\mu^{-}}$ vertex. The decay vertex is required
to be well separated from each reconstructed PV of the proton-proton
interaction by requiring the distance between the PV and the
${\mu^{+}}{\mu^{-}}$ vertex divided by its uncertainty to be greater than
three.

To study the properties of the signal and the most important backgrounds,
simulated sampels of $pp$ collisions are generated using Pythia [25,
*Sjostrand:2007gs] with a specific LHCb configuration [27]. Decays of hadronic
particles are described by EvtGen [28], in which final-state radiation is
generated using Photos [29]. The interaction of the generated particles with
the detector, and its response, are implemented using the Geant4 toolkit [30,
*Agostinelli:2002hh] as described in Ref. [32]. For the study of the lineshape
it is important that the simulation models well the mass resolution. The
simulation used in this study reproduces the observed mass resolution for
selected samples of
${{B}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}$,
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{\pi}^{-}}$,
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$
and ${{B}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ decays within $5\%$. To further
improve the agreement for the mass resolution between data and simulation,
scale factors are determined using a large sample of
${\psi{(2S)}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ decays collected concurrently with the
${\chi_{{c}1}}(3872)$ sample. This will be discussed in detail below.

##  3 Selection

The selection of
${\chi_{{c}1}}(3872)\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ candidates from ${b}$-hadron decays is
performed in two steps. First, loose selection criteria are applied that
reduce the background from random combinations of tracks significantly while
retaining high signal efficiency. Subsequently, a multivariate selection is
used to further reduce this combinatorial background. In both steps, the
selection requirements are chosen to reduce background whilst selecting well
reconstructed candidates. The requirements are optimised using simulated
signal decays together with a sample of selected candidates in data where the
charged pions have the same sign. The latter sample is found to be a good
proxy to describe the background shape. Though the selection criteria are
tuned using the ${\chi_{{c}1}}(3872)$ simulation sample, the
${\psi{(2S)}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\pi^{+}\pi^{-}$ decay mode is also selected with high efficiency and
used for calibration.

The selection starts from a pair of oppositely charged particles, identified
as muons. Incorrectly reconstructed tracks are suppressed by imposing a
requirement on the output of a neural network trained to discriminate between
these and trajectories from real particles. To select
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rightarrow{\mu^{+}}{\mu^{-}}$
candidates, the two muons are required to originate from a common vertex that
is significantly displaced from any PV. The difference between the
reconstructed invariant mass of the pair and the known value of the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ mass [33] is required to be
within three times the uncertainty on the reconstructed mass of the
${\mu^{+}}{\mu^{-}}$ pair.

Pion candidates are selected using the same track-quality requirements as the
muons. Information from the muon system is used to reject pions that decayed
in the spectrometer since these pions tend to have poorly reconstructed
trajectories which result in ${\chi_{{c}1}}(3872)$ candidates with worse mass
resolution. Combinatorial background is suppressed by requiring that the
$\chi^{2}_{\text{IP}}$ of the pion candidates, defined as the difference
between the $\chi^{2}$ of the PV reconstructed with and without the considered
particle, is larger than four for all PVs. Good pion identification is ensured
by applying a requirement on a variable that combines information from the
RICH detectors with kinematic and track quality information. Since the pions
produced in ${\chi_{{c}1}}(3872)$ decays have relatively small
$p_{\mathrm{T}}$, only a loose requirement on the transverse momentum
($\mbox{$p_{\mathrm{T}}$}>200\mathrm{\,Me\kern-1.00006ptV}$) is imposed. In
addition, the pion candidates are required to have
$p<50\mathrm{\,Ge\kern-1.00006ptV}$. This requirement rejects candidates with
poor momentum resolution and has an efficiency of $99.5\,\%$.

To create ${\chi_{{c}1}}(3872)$ candidates, $J/\psi$ candidates are combined
with pairs of oppositely charged pions. To improve the mass resolution a
kinematic vertex fit [34] is made which constrains the $J/\psi$ invariant mass
to its known value [33]. The reduced $\chi^{2}$ of the fit,
$\chi^{2}_{\mathrm{fit}}/\mathrm{ndf}$, is required to be less than five.
Candidates with a mass uncertainty greater than
$5.0\mathrm{\,Me\kern-1.00006ptV}$ are rejected. Finally, requiring the
$Q$-value of the decay to be below 200$\mathrm{\,Me\kern-1.00006ptV}$
substantially reduces the background whilst retaining $96\%$ of the
${\chi_{{c}1}}(3872)$ signal. Here the $Q$-value is defined as $Q\equiv
m_{{\mu^{+}}{\mu^{-}}{{\pi}^{+}}{{\pi}^{-}}}-m_{{\mu^{+}}{\mu^{-}}}-m_{{{\pi}^{+}}{{\pi}^{-}}}$
where $m_{{\mu^{+}}{\mu^{-}}{{\pi}^{+}}{{\pi}^{-}}}$, $m_{{\mu^{+}}{\mu^{-}}}$
and $m_{{{\pi}^{+}}{{\pi}^{-}}}$ are the reconstructed masses of the final
state combinations.

The final step of the selection process is based on a neural network
classifier [35, 36, 37, 38, *TMVA4]. This is trained on a simulated sample of
inclusive ${b}\rightarrow{\chi_{{c}1}}(3872)X$ decays and the same-sign pion
sample in data. Simulated samples are corrected to reproduce kinematical
distributions of the $\psi{(2S)}$ mesons observed on data. The training is
performed separately for the 2011 and 2012 data samples. Twelve variables that
give good separation between signal and background are considered: the
pseudorapidity and transverse momentum of the two pion candidates, the
$\chi^{2}_{\text{IP}}$ for each of the two pions, the pseudorapidity and
transverse momentum of the ${\chi_{{c}1}}(3872)$ candidate, the $\chi^{2}$ of
the two-track vertex fit for the pions, the
$\chi^{2}_{\mathrm{fit}}/\mathrm{ndf}$, the flight distance $\chi^{2}$ of the
candidate calculated using the reconstructed primary and secondary vertices,
and the total number of hits in the TT detector. All these variables show good
agreement between simulation and data. The optimal cut on the classifier
output is chosen using pseudoexperiments so as to minimise the uncertainty on
the measured ${\chi_{{c}1}}(3872)$ mass.

##  4 Mass model

The observed invariant mass distribution of the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ system,
$m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{\pi}^{+}}{{\pi}^{-}}}$,
for the ${\psi{(2S)}}$ and ${\chi_{{c}1}}(3872)$ resonances is a convolution
of the natural lineshape with the detector resolution. For the ${\psi{(2S)}}$
resonance the lineshape is well described by a Breit–Wigner function. The
situation for the ${\chi_{{c}1}}(3872)$ meson is more complex. Previous
measurements have assumed a Breit–Wigner resonance shape. However, as
discussed in Refs. [15, 16, 12], this is not well motivated due to the
proximity of the ${{D}^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ threshold. Several other
alternative lineshapes have been proposed in the literature [15, 16, 40, 41].
In this analysis two lineshapes for the ${\chi_{{c}1}}(3872)$ meson are
considered in detail, a Breit–Wigner and a Flatté-inspired model [15, 16].
These models are investigated in the next sections. The S-wave threshold
resonance model described in Ref. [40, 41], that accounts for the non-zero
width of the $D^{*0}$ meson, was considered but did not fit the data well. If
the mass is close to the ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{D}^{*0}}$ threshold, this
model is not able to accommodate a value of the natural width much larger than
$\Gamma_{{{D}^{*0}}}=65.5\pm 15.4\mathrm{\,ke\kern-1.00006ptV}$ [40]. As will
be discussed below, the study presented here favours larger values of the
natural width.

The analysis proceeds in two steps. First, unbinned maximum-likelihood fits
are made to the $m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}}$ distribution in the region around the
${\psi{(2S)}}$ mass. These measured values of the ${\psi{(2S)}}$ mass and mass
resolution are used to control systematic uncertainties in the subsequent fits
to the $m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}}$ distribution in the ${\chi_{{c}1}}(3872)$ mass
region. For both sets of fits the natural lineshape is convolved with a
resolution model developed using the simulation. The application of the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ mass constraint in the fit
[34] results in the mass resolution being dominated by the kinematics of the
pion pair. In particular, the resolution is worse for higher values of the
total momentum of the pion pair, $p_{{{\pi}^{+}}{{\pi}^{-}}}$. Consequently,
the analysis is performed in three $p_{{{\pi}^{+}}{{\pi}^{-}}}$ bins chosen to
contain an approximately equal number of signal candidates:
$p_{{{\pi}^{+}}{{\pi}^{-}}}<12\mathrm{\,Ge\kern-1.00006ptV}$, $12\leq
p_{{{\pi}^{+}}{{\pi}^{-}}}<20\mathrm{\,Ge\kern-1.00006ptV}$ and $20\leq
p_{{{\pi}^{+}}{{\pi}^{-}}}<50\mathrm{\,Ge\kern-1.00006ptV}$. The core mass
resolution for the ${\chi_{{c}1}}(3872)$ state varies monotonically between
$2.4\mathrm{\,Me\kern-1.00006ptV}$ and $3.0\mathrm{\,Me\kern-1.00006ptV}$
between the lowest-$p_{{{\pi}^{+}}{{\pi}^{-}}}$ and
highest-$p_{{{\pi}^{+}}{{\pi}^{-}}}$ bin. Possible differences in data taking
conditions are allowed for by dividing the data according to the year of
collection resulting in a total of six data samples.

The resolution model is studied using simulation. In each
$p_{{{\pi}^{+}}{{\pi}^{-}}}$ bin the mass resolution is modelled with the sum
of a narrow Crystal Ball function [42] combined with a wider Gaussian
function. The Crystal Ball function has a Gaussian core and two parameters
that describe the power-law tail. The simulation is also used to determine the
value of the transition point between the core and the power law tail, $a$, as
a multiple of the width, $\sigma$, of the Gaussian core. The value of the
exponent of the power law, $n$, is allowed to vary in the data fits with a
Gaussian constraint to the value obtained in the simulation applied. When
fitting the ${\chi_{{c}1}}(3872)$ mass region in data the values of the core
resolution, $\sigma$, for the Gaussian and Crystal Ball functions are taken
from simulation up to an overall scale factor, $s_{f}$, that accounts for
residual discrepancies between data and simulation. For each
$p_{{{\pi}^{+}}{{\pi}^{-}}}$ data sample the value of $s_{f}$ is determined in
the corresponding fit to the ${\psi{(2S)}}$ mass region and applied as a
Gaussian constraint. The systematic uncertainty associated with the choice of
the signal model is assessed by replacing the nominal model with the sum of
either two Crystal Ball or Gaussian functions.

The shape of the combinatorial background is studied using the same-sign data
sample as well as samples of simulated inclusive $b\rightarrow J/\psi X$
decays. Based upon these studies, the background is modelled by the form
$(m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}}-m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}-2m_{\pi^{\pm}})^{c_{0}}\,e^{-m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}}/c_{1}}$, where $c_{0}$ is fixed to 3.6 based on
fits to the same-sign data. Variations of this functional form together with
other models (e.g. exponential or polynomial functions) are used as systematic
variations. In total, seven different background forms are considered.

##  5 $\psi{(2S)}$ mass

Since the $\psi{(2S)}$ state is narrow and away from the phase-space limits, a
spin-0 relativistic Breit-Wigner function is used to model the lineshape. A
spin-1 Breit-Wigner function is considered as part of the systematic
uncertainties and found to give identical results. This lineshape is convolved
with the default resolution model and a fit to
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ mass is
performed in each of the six $p_{{{\pi}^{+}}{{\pi}^{-}}}$ data samples. The
natural width of the ${\psi{(2S)}}$ is fixed to the known value [33]. Figure 1
shows the $m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}}$ distributions and fit projections for each
data sample and Table 1 summarises the resulting parameters of interest.
Binning the data and calculating the $\chi^{2}$ probability of consistency
with the fit model gives the values greater than 5% for all fits. The fitted
values of the $\psi{(2S)}$ mass agree with the known value [33] within the
uncertainty of the calibration procedure. The values of $s_{f}$ are consistent
with the expectation that the simulation reproduces the mass resolution in
data at the level of $5\%$ or better. When applied as constraint in the fit to
the ${\chi_{{c}1}}(3872)$ region, additional uncertainties on $s_{f}$ are
considered. Accounting for the finite size of the simulation samples, the
background modelling and the assumption that the $\psi(2S)$ calibration factor
can be applied to the ${\chi_{{c}1}}(3872)$ candidates, the uncertainty on
$s_{f}$ is 0.02, independent of the bin. The values of $s_{f}$ in Table 1 are
applied as Gaussian constraints in the fits to the ${\chi_{{c}1}}(3872)$
region with an uncertainty of 0.02.

Figure 1: Mass distributions for ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ candidates in the ${\psi{(2S)}}$ region for (top) the low, (middle) mid and (bottom) high $p_{{{\pi}^{+}}{{\pi}^{-}}}$ bins. The left (right)-hand plot is for 2011 (2012) data. The projection of the fit described in the text is superimposed.  Table 1: Results of the ${\psi{(2S)}}$ mass and scale factor $s_{f}$ obtained for the nominal fit model. The quoted uncertainties on the ${\psi{(2S)}}$ mass and $s_{f}$ are statistical. Year | $p_{{{\pi}^{+}}{{\pi}^{-}}}~{}\left[\\!\mathrm{\,Ge\kern-1.00006ptV}\right]$ | $m_{{\psi{(2S)}}}~{}\left[\\!\mathrm{\,Me\kern-1.00006ptV}\right]$ | $s_{f}$
---|---|---|---
2011 | $\phantom{00\leq}p_{{{\pi}^{+}}{{\pi}^{-}}}<12$ | $3685.97\pm 0.02$ | $1.03\pm 0.01$
2011 | $12\leq p_{{{\pi}^{+}}{{\pi}^{-}}}<20$ | $3685.98\pm 0.02$ | $1.05\pm 0.01$
2011 | $20\leq p_{{{\pi}^{+}}{{\pi}^{-}}}<50$ | $3686.10\pm 0.03$ | $1.04\pm 0.01$
2012 | $\phantom{00\leq}p_{{{\pi}^{+}}{{\pi}^{-}}}<12$ | $3686.01\pm 0.01$ | $1.03\pm 0.01$
2012 | $12\leq p_{{{\pi}^{+}}{{\pi}^{-}}}<20$ | $3686.02\pm 0.01$ | $1.05\pm 0.01$
2012 | $20\leq p_{{{\pi}^{+}}{{\pi}^{-}}}<50$ | $3686.09\pm 0.02$ | $1.01\pm 0.01$

##  6 Breit–Wigner mass and width of the ${\chi_{{c}1}}(3872)$ state

To extract the Breit–Wigner lineshape parameters of the ${\chi_{{c}1}}(3872)$
meson, a fit is made to the mass range
$3832<m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}}<3912\mathrm{\,Me\kern-1.00006ptV}$ in each of
the six $p_{{{\pi}^{+}}{{\pi}^{-}}}$ data samples described above. A spin-0
relativistic Breit-Wigner is used, as in Ref. [9],

For each data sample the mass difference between the ${\psi{(2S)}}$ and
${\chi_{{c}1}}(3872)$ meson, $\Delta m$, is measured relative to the measured
mass of the $\psi{(2S)}$ state rather than the absolute mass. This minimises
the systematic uncertainty due to the momentum scale. The fit in each bin has
seven free parameters: $\Delta m$, the natural width $\Gamma_{\mathrm{BW}}$,
the background parameter $c_{1}$, the resolution scale factor $s_{f}$, the
tail parameter $n$, and the signal and background yields. Again a Gaussian
constraint is applied to $n$ based on the simulation. The parameter $s_{f}$ is
constrained to the result of the fit to the ${\psi{(2S)}}$ data. The fit
procedure is validated using both the simulation and pseudoexperiments. No
significant bias is found and the uncertainties estimated by the fit agree
with the spread observed in the pseudoexperiments. These studies show that,
values of $\Gamma_{\mathrm{BW}}$ larger than
$0.6\mathrm{\,Me\kern-1.00006ptV}$ can reliably be determined.

For the six $p_{{{\pi}^{+}}{{\pi}^{-}}}$ data samples the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ mass
distributions in the ${\chi_{{c}1}}(3872)$ region and fits are shown in Fig. 2
and the results summarised in Table 2. Binning the data and calculating the
$\chi^{2}$ probability of consistency with the fit model gives values much
larger than $5\%$ for all bins apart from the high-momentum bin in the 2012
data where the probability is $2\%$. The values of $\Delta m$ and
$\Gamma_{\mathrm{BW}}$ are consistent between the bins giving confidence in
the results.

Figure 2: Mass distributions for ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ candidates in the ${\chi_{{c}1}}(3872)$ region for (top) the low, (middle) mid and (bottom) high $p_{{{\pi}^{+}}{{\pi}^{-}}}$ bins. The left (right)-hand plot is for 2011 (2012) data. The projection of the fit described in the text is superimposed.  Table 2: Results for $\Delta m$ and $\Gamma_{\mathrm{BW}}$ and ${\chi_{{c}1}}(3872)$ signal yields. The quoted uncertainties are statistical.  Year | $p_{{{\pi}^{+}}{{\pi}^{-}}}~{}\left[\\!\mathrm{\,Ge\kern-1.00006ptV}\right]$ | $\Delta m~{}\left[\\!\mathrm{\,Me\kern-1.00006ptV}\right]$ | $\Gamma_{\mathrm{BW}}~{}\left[\\!\mathrm{\,Me\kern-1.00006ptV}\right]$ | $N_{\mathrm{sig}}~{}\left[10^{3}\right]$
---|---|---|---|---
2011 | $\phantom{00\leq}p_{{{\pi}^{+}}{{\pi}^{-}}}<12$ | $185.32\phantom{0}\pm 0.20\phantom{0}$ | $1.88\pm 0.74$ | $\phantom{0}1.78\pm 0.13$
2011 | $12\leq p_{{{\pi}^{+}}{{\pi}^{-}}}<20$ | $185.78\phantom{0}\pm 0.21\phantom{0}$ | $1.53\pm 0.74$ | $\phantom{0}1.79\pm 0.13$
2011 | $20\leq p_{{{\pi}^{+}}{{\pi}^{-}}}<50$ | $185.46\phantom{0}\pm 0.21\phantom{0}$ | $1.03\pm 0.82$ | $\phantom{0}1.68\pm 0.13$
2012 | $\phantom{00\leq}p_{{{\pi}^{+}}{{\pi}^{-}}}<12$ | $185.63\phantom{0}\pm 0.13\phantom{0}$ | $1.23\pm 0.47$ | $\phantom{0}3.24\pm 0.18$
2012 | $12\leq p_{{{\pi}^{+}}{{\pi}^{-}}}<20$ | $185.47\phantom{0}\pm 0.14\phantom{0}$ | $1.48\pm 0.48$ | $\phantom{0}3.70\pm 0.18$
2012 | $20\leq p_{{{\pi}^{+}}{{\pi}^{-}}}<50$ | $185.81\phantom{0}\pm 0.15\phantom{0}$ | $1.15\pm 0.57$ | $\phantom{0}3.26\pm 0.17$
Total | $185.588\pm 0.067$ | $1.39\pm 0.24$ | $15.63\pm 0.38$

A simultaneous fit is made to the six data samples with $\Delta m$ and
$\Gamma_{\mathrm{BW}}$ as shared parameters. This gives $\Delta m=185.588\pm
0.067\mathrm{\,Me\kern-1.00006ptV}$ and $\Gamma_{\mathrm{BW}}=1.39\pm
0.24\mathrm{\,Me\kern-1.00006ptV}$, where the uncertainties are statistical.
Consistent values are found when these parameters are determined through a
weighted average of the six individual bins, or by summing the likelihood
profiles returned by the fit.

The dominant systematic uncertainty on the mass difference $\Delta m$ arises
from the $3\times 10^{-4}$ relative uncertainty on the momentum scale. Its
effect is evaluated by adjusting the four-vectors of the pions by this amount
and repeating the analysis. The bias on $\Delta m$ from QED radiative
corrections is determined to be $(-10\pm 14)\mathrm{\,ke\kern-1.00006ptV}$
using the simulation, which uses Photos [29] to model this effect. The
measured value of $\Delta m$ is corrected by this value and the uncertainty
considered as a systematic error. The small uncertainty on the fitted values
of the ${\psi{(2S)}}$ mass is also propagated to the $\Delta m$ value. Biases
arising from the modelling of the resolution and the treatment of the
background shape are evaluated to be $2\mathrm{\,ke\kern-1.00006ptV}$ using
the discrete profiling method described in Ref. [43]. The uncertainties on the
$\Delta m$ measurement are summarised in Table 3. Combining all uncertainties,
the mass splitting between the ${\chi_{{c}1}}(3872)$ and ${\psi{(2S)}}$ mesons
is determined as

$\Delta m=185.598\pm 0.067\pm 0.068\mathrm{\,Me\kern-1.00006ptV}\,,$

where the first uncertainty is statistical and the second is systematic. The
value of $\Delta m$ can be translated into an absolute measurement of the
${\chi_{{c}1}}(3872)$ mass using $m_{{\psi{(2S)}}}=3686.097\pm
0.010\mathrm{\,Me\kern-1.00006ptV}$ from Ref. [33], yielding

$m_{{\chi_{{c}1}}(3872)}=3871.695\pm 0.067\pm 0.068\pm
0.010\mathrm{\,Me\kern-1.00006ptV}\,,$

where the third uncertainty is due to the knowledge of the $\psi{(2S)}$ mass.
For these measurements it is assumed that interference effects with other
partially reconstructed ${b}$-hadron decays do not affect the lineshape. This
assumption is reasonable since many exclusive ${b}$-hadron decays contribute
to the final sample, and the ${\chi_{{c}1}}(3872)$ state is narrow. This
assumption has been explored in pseudoexperiments varying the composition and
phases of the possible decay amplitudes that are likely to contribute to the
observed data set. These studies conservatively limit the size of any possible
effect on $m_{{\chi_{{c}1}}(3872)}$ to be less than
$40\mathrm{\,ke\kern-1.00006ptV}$.

Table 3: Systematic uncertainties on the measurement of the mass difference $\Delta m$.  Source | Uncertainty $\left[\\!\mathrm{\,Me\kern-1.00006ptV}\right]$
---|---
Momentum scale | 0.066
Radiative corrections | 0.014
Fitted ${\psi{(2S)}}$ mass uncertainty | 0.007
Signal + background model | 0.002
Sum in quadrature | 0.068

The uncertainties from the knowledge of $s_{f}$ and $n$ are already included
in the statistical uncertainty of $\Gamma_{\mathrm{BW}}$ via the Gaussian
constraints. Their contribution to the statistical uncertainty is estimated to
be $0.05\mathrm{\,Me\kern-1.00006ptV}$ by comparison to a fit with these
parameters fixed. Further uncertainties arise from the choice of signal and
background model. These are evaluated using the discrete profiling method with
the alternative models described above. Based upon these studies an
uncertainty of $0.10\mathrm{\,Me\kern-1.00006ptV}$ is assigned. The
uncertainty due to possible differences in the $p_{\mathrm{T}}$ distribution
between data and simulation is evaluated by weighting the simulation to
achieve better agreement and lead to a $0.01\mathrm{\,Me\kern-1.00006ptV}$
uncertainty. Summing these values in quadrature gives a total uncertainty of
$0.1\mathrm{\,Me\kern-1.00006ptV}$.

The value of $\Gamma_{\mathrm{BW}}$, including systematic uncertainties,

$\Gamma_{\mathrm{BW}}=1.39\pm 0.24\pm 0.10\mathrm{\,Me\kern-1.00006ptV}\,,$

differs from zero by more than $5$ standard deviations. Fits were also made
fixing $\Gamma_{\mathrm{BW}}$ to zero and allowing $s_{f}$ to float in each
bin without constraint. The value of $s_{f}$ obtained is between 1.2 and 1.25
depending on the bin, much larger than can be reasonably explained by
differences in the mass resolution between data and simulation after the
calibration using the ${\psi{(2S)}}$ data.

Care is needed in the interpretation of the measured $\Gamma_{\mathrm{BW}}$
and $m_{{\chi_{{c}1}}(3872)}$ parameters since $\left|m_{{{D}^{0}}}+m_{{{\kern
1.39998pt\overline{\kern-1.39998ptD}{}}{}^{*0}}}-m_{{\chi_{{c}1}}(3872)}\right|<\Gamma_{\mathrm{BW}}$.
The Breit–Wigner parameterisation may not be valid since it neglects the
opening of the ${{D}^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ channel.

##  7 Flatté model

###  7.1 The Flatté lineshape model

The proximity of the ${\chi_{{c}1}}(3872)$ mass to the ${D}^{0}$ ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}$ threshold distorts the
lineshape from the simple Breit–Wigner form. This has to be taken into account
explicitly. The general solution to this problem requires a full understanding
of the analytic structure of the coupled-channel scattering amplitude.
However, if the relevant threshold is close to the resonance, simplified
parametrisations are available and have been used to describe the
${\chi_{{c}1}}(3872)$ lineshape [15, 16].

In the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$
channel the ${\chi_{{c}1}}(3872)$ lineshape as a function of the energy with
respect to the ${{D}^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ threshold, $E\equiv
m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}}-\left(m_{{{D}^{0}}}+m_{{{D}^{*0}}}\right)$, can
be written as

$\dfrac{dR({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}})}{dE}\propto\dfrac{\Gamma_{\rho}(E)}{\left|D(E)\right|^{2}},$
(1)

where $\Gamma_{\rho}(E)$ is the contribution of the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ channel
to the width of the ${\chi_{{c}1}}(3872)$ state. The complex-valued
denominator function, taking into account the ${{D}^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ and ${{D}^{+}}{{D}^{*-}}$
two-body thresholds, and the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{0}}$ channels, is given by

$D(E)=E-E_{f}+\frac{i}{2}\left[g\left(k_{1}+k_{2}\right)+\Gamma_{\rho}(E)+\Gamma_{\omega}(E)+\Gamma_{0}\right].$
(2)

The Flatté energy parameter, $E_{f}$, is related to a mass parameter, $m_{0}$,
via the relation $E_{f}=m_{0}-(m_{{{D}^{0}}}+m_{{{D}^{*0}}})$. The width
$\Gamma_{0}$ is introduced in Ref. [16] to represent further open channels,
such as radiative decays. The model assumes an isoscalar assignment of the
${\chi_{{c}1}}(3872)$ state, using the same effective coupling, $g$, for both
channels. The relative momenta of the decay products in the rest frame of the
two-body system, $k_{1}$ for ${{D}^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ and $k_{2}$ for the
${{D}^{+}}{{D}^{*-}}$ channel, are given by

$k_{1}=\sqrt{2\mu_{1}E},\ \quad k_{2}=\sqrt{2\mu_{2}(E-\delta)}\,,$ (3)

where $\delta=8.2\mathrm{\,Me\kern-1.00006ptV}$ is the isospin splitting
between the two channels. The reduced masses are given by
$\mu_{1}=\tfrac{m_{{{D}^{0}}}m_{{{D}^{*0}}}}{(m_{{{D}^{0}}}+m_{{{D}^{*0}}})}$
and
$\mu_{2}=\tfrac{m_{{{D}^{+}}}m_{{{D}^{*-}}}}{(m_{{{D}^{+}}}+m_{{{D}^{*-}}})}$.
For $m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{\pi}^{+}}{{\pi}^{-}}}$
masses below the ${{D}^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ and ${{D}^{+}}{{D}^{*-}}$
thresholds these momenta become imaginary and thus their contribution to the
denominator will be real. The energy dependence of the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{0}}$ partial widths is given by [16]

$\displaystyle\Gamma_{\rho}(E)=$ $\displaystyle
f_{\rho}\int\limits_{2m_{\pi}}^{M(E)}\dfrac{dm^{\prime}}{2\pi}\dfrac{q(m^{\prime},E)\,\Gamma_{\rho}}{(m^{\prime}-m_{\rho})^{2}+\Gamma_{\rho}^{2}/4}\,,$
(4) $\displaystyle\Gamma_{\omega}(E)=$ $\displaystyle
f_{\omega}\int\limits_{3m_{\pi}}^{M(E)}\dfrac{dm^{\prime}}{2\pi}\dfrac{q(m^{\prime},E)\,\Gamma_{\omega}}{(m^{\prime}-m_{\omega})^{2}+\Gamma_{\omega}^{2}/4}\,.$
(5)

The known values for masses $m_{\rho}$, $m_{\omega}$ and widths
$\Gamma_{\rho}$, $\Gamma_{\omega}$ [33] are used and the lineshapes are
approximated with fixed-width Breit–Wigner functions. The partial widths are
parameterised by the respective effective couplings $f_{\rho}$ and
$f_{\omega}$ and the phase space of these decays, where intermediate
resonances $\rho^{0}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ and
$\omega\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{0}}$ are assumed. The
dependence on $E$ is given by the upper boundary of the integrals
$M(E)=E+(m_{{{D}^{0}}}+m_{{{D}^{*0}}})-m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}$. The momentum of the two- or three-pion system in the rest frame of
the ${\chi_{{c}1}}(3872)$ is given by

$q(m^{\prime},E)=\sqrt{\dfrac{\left[M^{2}(E)-(m^{\prime}+m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}})^{2}\right]\left[M^{2}(E)-(m^{\prime}-m_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}})^{2}\right]}{4M^{2}(E)}}.$ (6)

The model as specified contains five free parameters: $m_{0},g,\Gamma_{0}$ and
the effective couplings $f_{\rho}$ and $f_{\omega}$. In contrast to the
Breit–Wigner lineshape, the parameters of the Flatté model cannot be easily
interpreted in terms of the mass and width of the state. Instead it is
necessary to determine the location of the poles of the amplitude. The
analysis proceeds with a fit of the Flatté amplitude to the data and
subsequent search for the poles.

The resulting Flatté lineshape replaces the Breit–Wigner function and is
convolved with the resolution models described in the first part of the paper.
The Flatté parameters are estimated from a simultaneous unbinned likelihood
fit to the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ mass distribution in the six
$p_{{{\pi}^{+}}{{\pi}^{-}}}$ data samples. The data points are corrected for
the observed shifts of the reconstructed mass of the $\psi{(2S)}$ in each bin.

###  7.2 Fits of the Flatté lineshape to the data

In order to obtain stable results when using the coupled channel model to
describe the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ mass spectrum, a relation between the effective
couplings $f_{\rho}$ and $f_{\omega}$ is imposed. This relation requires that
the branching fractions of the ${\chi_{{c}1}}(3872)$ state to
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rho^{0}$ and
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\omega^{0}$ final states are
equal, which is consistent with experimental data [5, 44, 14], thus
eliminating one free parameter in the fit. Furthermore, a Gaussian fit
constraint is applied on the ratio of branching fractions

$R_{{D}{{\kern
1.39998pt\overline{\kern-1.39998ptD}{}}{}^{*}}}=\dfrac{\Gamma({\chi_{{c}1}}(3872)\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}})}{\Gamma({\chi_{{c}1}}(3872)\\!\rightarrow{{D}^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}})}=0.11\pm 0.03\,.$ (7)

The value used here is obtained as the weighted average of the results from
the BaBar [5] and Belle [44, 14] collaborations, as listed in Ref. [45]. The
Flatté model reduces to the Breit–Wigner model as a special case, namely when
there is no additional decay channel available near the resonance. However,
the $R_{{D}{{\kern 1.39998pt\overline{\kern-1.39998ptD}{}}{}^{*}}}$ constraint
enforces a large coupling to the ${{D}^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ channel and the lineshape
will be different from the Breit–Wigner function in the region of interest.

For large couplings to the two-body channel the Flatté parameterisation
exhibits a scaling property [46] that prohibits the unique determination of
all free parameters on the given data set. Almost identical lineshapes are
obtained when the parameters $E_{f}$, $g$, $f_{\rho}$ and $\Gamma_{0}$ are
scaled appropriately. In particular, it is possible to counterbalance a lower
value of $E_{f}$ with a linear increase in the coupling to the ${D}{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}^{*}$ channels $g$. While this is not a
true symmetry of the parameterisation — there are subtle differences in the
tails of the lineshape — in practice, within the experimental precision this
effect leads to strong correlations between the parameters.

Figure 3 illustrates the scaling behavior in the data. The black points show
the best-fit result for the parameter $g$ evaluated at fixed $E_{f}$,
optimising the remaining parameters at every step. To a good approximation $g$
depends linearly on $E_{f}$ with

$\dfrac{dg}{dE_{f}}=(-15.11\pm 0.16)\mathrm{\,Ge\kern-1.00006ptV}^{-1}\,.$ (8)

The red points show the negative log likelihood relative to is minimum value
$\Delta\mathrm{LL}$ for each of these fits, revealing a shallow minimum around
$m_{0}=3860\mathrm{\,Me\kern-1.00006ptV}$. At lower $E_{f}$ values
$\Delta\mathrm{LL}$ raises very slowly, reaching a value of 1 around
$-270\mathrm{\,Me\kern-1.00006ptV}$. Values of $E_{f}$ approaching the
${{D}^{0}}{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ threshold
are disfavoured, though. In particular good quality fits are obtained only for
negative values of $E_{f}$. A similar phenomenon has been observed in the
previous analyses of BaBar and Belle data and is discussed in Ref. [16]. As in
those studies, for the remainder of the paper the practical solution of fixing
$m_{0}=3864.5\mathrm{\,Me\kern-1.00006ptV}$, corresponding to
$E_{f}=-7.2\mathrm{\,Me\kern-1.00006ptV}$, is adopted. The remaining model
parameters are evaluated with this constraint applied. This procedure has been
validated using pseudoexperiments and no significant bias is found. For $g$
and $\Gamma_{0}$ the uncertainties estimated by the fit agree with the spread
of the pseudoexperiments. For $f_{\rho}$ an uncertainty which is 10% larger
than what is found in the pseudoexperiments is observed and this conservative
estimate is reported. The measured values for $g$, $f_{\rho}$ and $\Gamma_{0}$
are presented in Table 4. In order to fulfill the constraint on the branching
ratios, Eq. (7), the effective coupling, $f_{\omega}$, is found to be $0.01$.

Figure 3: The coupling to the ${D}{\kern 1.79997pt\overline{\kern-1.79997ptD}{}}^{*}$ channels $g$ as a function of Flatté energy parameter $E_{f}$ (black points with error bars). The corresponding change in negative log likelihood, $\Delta\mathrm{LL}$ is shown as well (red dots). Table 4: Results from the constrained Flatté fit. The uncertainties are statistical.  $g$ | $f_{\rho}\times 10^{3}$ | $\Gamma_{0}\,\left[\\!\mathrm{\,Me\kern-1.00006ptV}\right]$ | $m_{0}\,\left[\\!\mathrm{\,Me\kern-1.00006ptV}\right]$
---|---|---|---
$0.108\pm 0.003$ | $1.8\pm 0.6$ | $1.4\pm 0.4$ | $3864.5\,{\mathrm{(fixed)}}$
Table 5: Systematic uncertainty on the measurement of the Flatté parameters.  Systematic | $g$ | $f_{\rho}\times 10^{3}$ | $\Gamma_{0}~{}\left[\\!\mathrm{\,Me\kern-1.00006ptV}\right]$
---|---|---|---
Model | $~{}~{}+0.003$ | $-0.004~{}~{}$ | $~{}~{}+0.6$ | $-0.5~{}~{}$ | $~{}~{}+0.5$ | $-0.4~{}~{}$
Momentum scale | $~{}~{}+0.003$ | $-0.003~{}~{}$ | $~{}~{}+0.1$ | $-0.2~{}~{}$ | $~{}~{}+0.1$ | $-0.2~{}~{}$
Threshold mass | $~{}~{}+0.003$ | $-0.003~{}~{}$ | $~{}~{}+0.2$ | $-0.2~{}~{}$ | $~{}~{}+0.2$ | $-0.3~{}~{}$
${D}^{*0}$ width |  | $-0.001~{}~{}$ |  |  |  | $-0.2~{}~{}$
Sum in quadrature | $~{}~{}+0.005$ | $-0.006~{}~{}$ | $~{}~{}+0.7$ | $-0.6~{}~{}$ | $~{}~{}+0.6$ | $-0.6~{}~{}$

The systematic uncertainties on the Flatté parameters are summarised in Table
5 and discussed below. The systematic uncertainties introduced by the
background and resolution parameterisations are evaluated in the same way as
for the Breit–Wigner analysis, using discrete profiling. The impact of the
momentum scale uncertainty is investigated by shifting the data points by
$66\mathrm{\,ke\kern-1.00006ptV}$ and repeating the fit. Further systematic
uncertainties are particular to the Flatté parameterisation. The location of
the ${{D}^{0}}{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$
threshold is known to a precision of $0.11\mathrm{\,Me\kern-1.00006ptV}$ [33].
Varying the threshold by this amount and repeating the fit leads to an
uncertainty on the parameters which is similar to that introduced by the
momentum scale. Finally, the ${{D}^{*0}}$ meson has a finite natural width,
for which an upper limit of
$\Gamma_{{{D}^{*0}}}<2.1\mathrm{\,Me\kern-1.00006ptV}$ [33] has been measured.
However, theoretical predictions estimate $\Gamma_{D^{*}}=65.5\pm
15.4\,\mathrm{\,ke\kern-1.00006ptV}$ [40], based on the measured width of the
${{D}^{*+}}$ meson. Modified lineshape models taking into account the finite
width of the ${{D}^{*0}}$ are available. In particular, Refs. [40, 47] suggest
replacing $k_{1}(E)$ in Eq. (3) with

$k^{\prime}_{1}(E)=\sqrt{-2\mu\left(E-E_{R}+i\Gamma_{{{D}^{*0}}}/2\right)}\,,$
(9)

where $E_{R}\equiv m_{{{D}^{*0}}}-m_{{{D}^{0}}}-m_{{{\pi}^{0}}}$. The reduced
mass, $\mu$, is calculated as
$\tfrac{m_{{{D}^{0}}}(m_{{{D}^{0}}}+m_{{{\pi}^{0}}})}{(2m_{{{D}^{0}}}+m_{{{\pi}^{0}}})}$.
With this modification there is always a contribution to both the imaginary
and real part of the denominator function in Eq. (2). Repeating the fit
results in a similar but worse fit quality with a log-likelihood difference of
$0.1$. The width $\Gamma_{0}$ is reduced by
$0.2\,\mathrm{\,Me\kern-1.00006ptV}$, which is the smallest systematic
uncertainty on this parameter.

###  7.3 Comparison between Breit–Wigner and Flatté lineshapes

Figure 4 shows the comparison between the Breit–Wigner and the Flatté
lineshapes. While in both cases the signal peaks at the same mass, the Flatté
model results in a signifcantly narrower lineshape. However, after folding
with the resolution function and adding the background, the observable
distributions are indistinguishable.

Figure 4: Comparison of the Flatté (solid, red) and Breit–Wigner (dotted, black) lineshapes. The left plot shows the raw lineshapes for the default fits. The location of the ${D}^{0}$ ${\kern 1.79997pt\overline{\kern-1.79997ptD}{}}{}^{*0}$ threshold is indicated by the blue vertical line. On the right the distributions are shown after applying smearing with the resolution function and adding background. Table 6: Results of the fit with the Flatté lineshape including statistical and systematic uncertainties. The Flatté mass parameter $m_{0}=3864.5\mathrm{\,Me\kern-0.90005ptV}$ is used.  Mode $\left[\\!\mathrm{\,Me\kern-1.00006ptV}\right]$ | Mean $\left[\\!\mathrm{\,Me\kern-1.00006ptV}\right]$ | FWHM $\left[\\!\mathrm{\,Me\kern-1.00006ptV}\right]$
---|---|---
$3871.69^{\,+\,0.00\,+\,0.05}_{\,-\,0.04\,-\,0.13}$ |  $3871.66^{\,+\,0.07\,+\,0.11}_{\,-\,0.06\,-\,0.13}$ | $0.22^{\,+\,0.06\,+\,0.25}_{\,-\,0.08\,-\,0.17}$

To quantify this comparison the fit results for the mode, the mean and the
full width at half-maximum (FWHM) of the Flatté model and their uncertainties
are summarised in Table 6. The mode of the Flatté distribution agrees within
uncertainties with the Breit–Wigner solution. However, the FWHM of the Flatté
model is a factor of five smaller than the Breit–Wigner width. To check the
consistency of these seemingly contradictory results, pseudoexperiments
generated with the Flatté model and folded with the known resolution function
are analysed with the Breit–Wigner model. Figure 5 shows the resulting
distribution of the Breit–Wigner width determined from the pseudoexperiments,
which is in good agreement with the value observed in the data. This
demonstrates that the value obtained for the Breit–Wigner width, after taking
into account the experimental resolution, is consistent with the expectation
of the Flatté model. The result highlights the importance of a proper
lineshape parameterisation for a measurement of the location of the pole.

Figure 5: Distribution of the FWHM obtained for simulated experiments
generated from the result of the Flatté model and fitted with the Breit–Wigner
model (filled histogram). Both models account for the experimental resolution.
The dashed red line shows the FWHM of the Flatté lineshape, while the solid
blue line indicates the value of the Breit–Wigner width observed in data.

###  7.4 Pole search

The amplitude as a function of the energy defined by Eq. (2) can be continued
analytically to complex values of the energy $E$. This continuation is valid
up to singularities of the amplitude. There are two types of singularities,
which are relevant here: poles and branch points. Poles of the amplitude in
the complex energy plane are identified with hadronic states. The pole
location is a unique property of the respective state, which is independent of
the production process and the observed decay mode. In the absence of nearby
thresholds the real part of the pole is located at the mass of the hadron and
the imaginary part at half the width of the state. Branch point singularities
occur at the threshold of every coupled channel and lead to branch cuts in the
Riemann surface on which the amplitude is defined. Each branch cut corresponds
to two Riemann sheets. Through Eq. (2) the amplitude will inherit the analytic
structure of the square root functions of Eq. (3) that describe the momenta of
the decay products in the rest frame of the two-body system. The square root
is a two-sheeted function of complex energy. In the following, a convention is
used where the two sheets are connected along the negative real axis. An
introduction to this subject can be found in Refs. [48, 49, 50] and a summary
is available in Ref. [51].

For the ${\chi_{{c}1}}(3872)$ state only the Riemann sheets associated to the
${{D}^{0}}{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ channel are
important, since all other thresholds are far from the signal region. The
following convention is adopted to label the relevant sheets:

  1. I:

$E-E_{f}-\dfrac{g}{2}\left(+\sqrt{-2\mu_{1}E}+\sqrt{-2\mu_{2}(E-\delta)}\right)+\dfrac{i}{2}\Gamma(E)$
with $\operatorname{Im}E>0$ ,

  2. II:

$E-E_{f}-\dfrac{g}{2}\left(+\sqrt{-2\mu_{1}E}+\sqrt{-2\mu_{2}(E-\delta)}\right)+\dfrac{i}{2}\Gamma(E)$
with $\operatorname{Im}E<0$ ,

  3. III:

$E-E_{f}-\dfrac{g}{2}\left(-\sqrt{-2\mu_{1}E}+\sqrt{-2\mu_{2}(E-\delta)}\right)+\dfrac{i}{2}\Gamma(E)$
with $\operatorname{Im}E<0$ ,

  4. IV:

$E-E_{f}-\dfrac{g}{2}\left(-\sqrt{-2\mu_{1}E}+\sqrt{-2\mu_{2}(E-\delta)}\right)+\dfrac{i}{2}\Gamma(E)$
with $\operatorname{Im}E>0$ ,

where $\Gamma(E)\equiv\Gamma_{\rho}(E)+\Gamma_{\omega}(E)+\Gamma_{0}$. The
fact that the model contains several coupled channels in addition to the
${{D}^{0}}{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ channel
complicates the analytical structure. The sign in front of the momentum
$\sqrt{-2\mu_{1}E}$ is the same for sheets I and II and therefore they belong
to a single sheet with respect to the ${{D}^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ channel. The two regions are
labelled separately due to the presence of the the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$,
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{0}}$ channels, as well as radiative
decays. Those channels have their associated branch points at smaller masses
than the signal region. The analysis is performed close to the
${{D}^{0}}{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ threshold
and points above and below the real axis lie on different sheets with respect
to those open channels.

Sheets I and II correspond to a physical sheet with respect to the
${{D}^{0}}{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ channel,
where the amplitude is evaluated in order to compute the measurable lineshape
at real energies $E$. Sheets III and IV correspond to an unphysical sheet with
respect to that channel. Sheet II is analytically connected to sheet IV along
the real axis, above the ${{D}^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ threshold.

In the single-channel case, a bound ${{D}^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ state would appear below
threshold on the real axis and on the physical sheet.

A virtual state would appear as well below threshold on the real axis, but on
the unphysical sheet. A resonance would appear on the unphysical sheet in the
complex plane [48, 49, 50]. The presence of inelastic, open channels shifts
the pole into the complex plane and turns both a bound state as well as a
virtual state into resonances. In the implementation of the amplitude used for
the analysis, the branch cut for the ${{D}^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ channel is taken to go from
threshold towards larger energy $E$, while the branch cuts associated to the
open channels $\Gamma(E)$ are chosen to lie along the negative real axis. The
analytic structure around the branch cut associated to the
${{D}^{+}}{{D}^{*-}}$ threshold is also investigated, but no nearby poles are
found on the respective Riemann sheets.

At the best estimate of the Flatté parameters the model exhibits two pole
singularities. The first pole appears on sheet II and is located very close to
the ${{D}^{0}}{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$
threshold . The location of this pole with respect to the branch point,
obtained using the algorithm described in Ref. [52], is
$E_{\mathrm{II}}=(0.06-0.13\>i)\mathrm{\,Me\kern-1.00006ptV}$. Recalling that
the imaginary part of the pole position corresponds to half the visible width,
it is clear that this pole is responsible for the peaking region of the
lineshape. A second pole is found on sheet III. It appears well below the
threshold and is also further displaced from the physical axis at
$E_{\mathrm{III}}=(-3.58-1.22\>i)\mathrm{\,Me\kern-1.00006ptV}$.

Figure 6: The phase of the Flatté amplitude obtained from the fit to the data
with $m_{0}=3864.5\mathrm{\,Me\kern-0.90005ptV}$ on sheets I (for
$\operatorname{Im}E>0$) and II (for $\operatorname{Im}E<0$) of the complex
energy plane. The pole singularity is visible at
$E_{\mathrm{II}}=(0.06-0.13\>i)\mathrm{\,Me\kern-0.90005ptV}$. The branch cut
is highlighted with the black line. The trajectory of the pole taken when the
couplings to all but the ${D}{{\kern
1.79997pt\overline{\kern-1.79997ptD}{}}{}^{*}}$ channel are scaled down to
zero is indicated in red.

Figure 6 shows the analytic structure of the Flatté amplitude in the vicinity
of the threshold. The color code corresponds to the phase of the amplitude on
sheets I (for $\operatorname{Im}E>0$) and II (for $\operatorname{Im}E<0$) in
the complex energy plane. The pole on sheet II is visible, as is the
discontinuity along the ${{D}^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ branch cut, which for clarity
is also indicated by the black line. The trajectory followed by the pole when
taking the limit where the couplings to all channels but ${{D}^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ are sent to zero is shown in
red and discussed below.

Figure 7: The phase of the Flatté amplitude as obtained from the fit with a
finite ${D}^{*0}$ width of
$\Gamma_{{D}^{*0}}=65.5\mathrm{\,ke\kern-0.90005ptV}$ on sheets I (for
$\operatorname{Im}E>-\Gamma_{{{D}^{*0}}}/2$) and II (for
$\operatorname{Im}E<-\Gamma_{{{D}^{*0}}}/2$) of the complex energy plane.
Since the ${\kern 1.79997pt\overline{\kern-1.79997ptD}{}}{}^{*0}$ meson is
treated as an unstable particle, the ${D}^{0}$ ${\kern
1.79997pt\overline{\kern-1.79997ptD}{}}{}^{*0}$ branch cut, indicated by the
black solid line, is located at $\operatorname{Im}E=-\Gamma_{{{D}^{*0}}}/2$.
The location of the pole is on the physical sheet with respect to the
${D}^{0}$ ${\kern 1.79997pt\overline{\kern-1.79997ptD}{}}{}^{*0}$ system.
Figure 8: Confidence regions for the pole position on sheets II and IV in the
complex energy plane. The displayed uncertainties include statistical
contributions and the modeling uncertainty. The poles are extracted at a
Flatté mass point of $m_{0}=3864.5\mathrm{\,Me\kern-0.90005ptV}$. The shaded
areas are the 1, 2 and 3$\sigma$ confidence regions. The branch cut is shown
as the blue line. The location of the branch cut singularity is indicated with
a vertical bar at $E=0+0\>i$. The best estimates for the pole position is
indicated by a cross. The black points indicate the samples from the
pseudoexperiments procedure that lie outside the $3\sigma$ region.  Figure 9:
Confidence regions for the pole position on sheet III in the complex energy
plane. The displayed uncertainties include statistical contributions and the
modeling uncertainty. The poles are extracted at a Flatté mass point of
$m_{0}=3864.5\mathrm{\,Me\kern-0.90005ptV}$. The shaded areas are the 1, 2 and
3$\sigma$ confidence regions. The branch cut is shown as the blue line. The
location of the branch cut singularity is indicated with a vertical bar at
$E=0+0\>i$. The best estimate for the pole positions is indicated by a cross.
The confidence region for the pole on sheets II/IV is shown in outline for
comparison. The black points indicate the samples from the pseudoexperiments
procedure that lie outside the $3\sigma$ region.

As shown in Table 5, taking into account the finite width of the ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}$ has a small effect on the
Flatté parameters. However, the analytic structure of the amplitude close to
the threshold is changed such that in this case the branch cut is located in
the complex plane at $\operatorname{Im}E=-\Gamma_{{{D}^{*0}}}/2$. The phase of
the amplitude for this case is shown in Fig. 7. The displaced branch cut is
highlighted in black. The pole is found at
$E^{\prime}_{\mathrm{II}}=(25-140\>i)\mathrm{\,ke\kern-1.00006ptV}$ in a
similar location to the case without taking into account the ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}$ width. In particular, the most
likely pole position is on sheet II, the physical sheet with respect to the
${D}^{0}$ ${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}$ system. The
location of the pole on sheet III is found to be
$E^{\prime}_{\mathrm{III}}=(-3.59-1.05\>i)\mathrm{\,Me\kern-1.00006ptV}$,
similar to the fit that does not account for ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}$ width.

The uncertainties of the Flatté parameters are propagated to the pole position
by generating large sets of pseudoexperiments, sampling from the asymmetric
Gaussian uncertainties that describe the statistical and the systematic
uncertainties introduced through the resolution and background
parameterisation. The systematic uncertainty on the pole position due to the
momentum scale, location of the threshold and the choice of the Flatté mass
parameter are discussed in the following.

The confidence regions for the location of the poles, corresponding to
$68.3\%,95.4\%$ and $99.7\%$ intervals, are shown in Figures 8 and 9. For
large values of $g$ the pole on sheet II moves to sheet IV, which is
analytically connected to the former along the real axis above threshold.
Therefore, sheet II (for $\operatorname{Im}E<0$) and sheet IV (for
$\operatorname{Im}E>0$) are shown together for this pole. While a pole
location on sheet II is preferred by the data, a location on sheet IV is still
allowed at the $2\sigma$ level. The pole on sheet III is located well below
threshold and comparatively deep in the complex plane and is shown in Fig. 9.
For comparison, the location of the confidence region for the first pole on
sheets II and IV are also indicated on sheet III.

The positions of both poles depend on the choice of the Flatté mass parameter
$m_{0}$. The dependence of the lineshape on $m_{0}$ has been explored in the
region below threshold and for the results shown in Fig. 3 the corresponding
pole positions are evaluated. The location of the pole on sheet II extracted
for $-17<E_{f}<0\,\mathrm{\,Me\kern-1.00006ptV}$ are marked by black circles
in Fig. 10. For smaller values of $m_{0}$ the pole moves closer to the real
axis, for values of $m_{0}$ approaching the threshold, the pole moves further
into the complex plane. For all fits performed the best estimate for the
location of the pole is on sheet II.

Figure 10: Confidence regions for the pole on sheet II in the complex energy
plane. The displayed uncertainties include statistical contributions and the
uncertainty from the choice of the Flatté mass parameter $m_{0}$. Modelling
uncertainties are not shown. The shaded areas are the 1, 2 and 3$\sigma$
confidence regions. The branch cut is shown as the blue line. The location of
the branch cut singularity is indicated with a vertical bar at $E=0+0\>i$. The
black circles indicate the best estimates for the pole position for the
different choices of $m_{0}$.

Figure 10 also shows the combined confidence regions, which account for the
explored range of $E_{f}$. For each fit, a sample consisting of $10^{5}$
pseudoexperiments is drawn from the Gaussian distribution described by the
covariance matrix of the fit parameters. Only the statistical uncertainties
obtained for each fit are used for this study. The resulting samples of pole
positions are combined by weighting with their respective likelihood ratios
with respect to the best fit. The preferred location of the pole is on sheet
II. However, a location of the pole on sheet IV is still allowed at the
$2\sigma$ level.

The location of the pole on sheet III, in particular its real part, depends
strongly on the choice of $m_{0}$. For small values of $m_{0}$ the pole moves
away from the threshold and has less impact on the lineshape. For $m_{0}$
approaching the threshold this pole moves closer to the branch point and
closer to the pole on sheet II. Since the asymmetry of the poles with respect
to the threshold contains information on the potential molecular nature of the
state [53], the values of the pole positions are provided for the most extreme
scenario that is still allowed by the data with a likelihood difference of
$\Delta\mathrm{LL}=1$, ($cf.$ Fig. 3) at
$m_{0}=3869.3\mathrm{\,Me\kern-1.00006ptV}$. In this case the two poles are
found at $E_{\mathrm{II}}=(0.09-0.33\>i)\mathrm{\,Me\kern-1.00006ptV}$ and
$E_{\mathrm{III}}=(-0.85-0.97\>i)\mathrm{\,Me\kern-1.00006ptV}$.

The location of the threshold with respect to the observed location of the
peak has a profound impact on the Flatté parameters and therefore on the pole
position. The main uncertainties, which affect on which sheet the pole is
found, are the knowledge of the momentum scale and the location of the
${{D}^{0}}{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ threshold.
As shown in Table 5, both effects are of equal importance. Figure 11 shows the
statistical uncertainties of the pole on sheet II for the case that the mass
scale is shifted up by $66\mathrm{\,ke\kern-1.00006ptV}$. The pole is moving
closer towards the real axis but the preferred location remains on sheet II. A
measurement of the lineshape in the ${{D}^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ channel is needed to further
improve the knowledge on the impact of the threshold location.

Figure 11: Confidence regions for the pole on sheet II in the complex energy
plane, in the case that the mass scale is shifted up by
$0.066\mathrm{\,Me\kern-0.90005ptV}$, due to systematic uncertainty of the
momentum scale. Only the statistical uncertainties are displayed. The shaded
areas are the 1, 2 and 3$\sigma$ confidence regions. The cross indicates the
location of the pole found in the default fit, with the nominal momentum
scale. The branch cut is shown as the blue line. The location of the branch
cut singularity is indicated with a vertical bar at $E=0+0\>i$.

It is possible to study the behavior of the poles in the limit where only the
${D}{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*}}$ channels are
considered. The trajectory traced by the pole on sheet II when the couplings
to the other channels ($f_{\rho}$, $f_{\omega}$, $\Gamma_{0}$) are sent to
zero is indicated by the red curve in Fig. 6. The coupling $g$ and the Flatté
mass parameter $E_{f}$ are kept fixed while taking this limit. For the best
fit solution the pole moves below threshold and reaches the real axis at
$E=-24\mathrm{\,ke\kern-1.00006ptV}$ staying on the physical sheet with
respect to the ${{D}^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ threshold. This location is
consistent with a quasi-bound state in that channel with a binding energy of
$E_{b}=24\,\mathrm{\,ke\kern-1.00006ptV}$. If the pole lies in the allowed
region on sheet IV, taking the same limit also sends the pole onto the real
axis below threshold, but on the unphysical sheet with respect to
${{D}^{0}}{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$. This
situation corresponds to a quasi-virtual state. Both types of solutions are
analytically connected along the real axis through the branch cut singularity.
Therefore, only upper limits on the binding energy can be set. For the bound
state solution and only accounting for statistical uncertainties, the result
is $E_{b}<57\mathrm{\,ke\kern-1.00006ptV}$ at 90% confidence level (CL).
Including the systematic uncertainties due to the choice of the model this
limit becomes $E_{b}<100\mathrm{\,ke\kern-1.00006ptV}$ at $90\%$ CL. Setting
the couplings to the other channels to zero causes the pole on sheet III to
move to the real axis as well, reaching it at
$E=-3.51\mathrm{\,Me\kern-1.00006ptV}$. The corresponding values extracted at
the highest allowed value of $m_{0}=3869.3\mathrm{\,Me\kern-1.00006ptV}$ are
$E_{b}=29\mathrm{\,ke\kern-1.00006ptV}$ for the bound state pole and
$E_{b}=0.73\mathrm{\,Me\kern-1.00006ptV}$ for the pole on the unphysical
sheet.

##  8 Results and discussion

In this paper a large sample of ${\chi_{{c}1}}(3872)$ mesons from ${b}$-hadron
decays collected by LHCb in 2011 and 2012 is exploited to study the lineshape
of the ${\chi_{{c}1}}(3872)$ meson. Describing the lineshape with a
Breit–Wigner function determines the mass splitting between the
${\chi_{{c}1}}(3872)$ and ${\psi{(2S)}}$ states to be

$\Delta m=185.598\pm 0.067\pm 0.068\mathrm{\,Me\kern-1.00006ptV}\,,$

where the first uncertainty is statistical and the second systematic. Using
the known value of the $\psi{(2S)}$ mass [33] this corresponds to

$m_{{\chi_{{c}1}}(3872)}=3871.695\pm 0.067\pm 0.068\pm
0.010\mathrm{\,Me\kern-1.00006ptV}\,,$

where the third uncertainty is due to the knowledge of the $\psi{(2S)}$ mass.
The result is in good agreement with the current world average [33]. The
uncertainty is improved by a factor of two compared to the best previous
measurement by the CDF collaboration [4]. The measured value can also be
compared to the threshold value, $m_{{{D}^{0}}}+m_{{{D}^{*0}}}=3871.70\pm
0.11\mathrm{\,Me\kern-1.00006ptV}$. The ${\chi_{{c}1}}(3872)$ mass, evaluated
from the mean of a fit assuming the Breit–Wigner lineshape, is coincident with
the ${{D}^{0}}{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$
threshold within uncertainties, with $\delta E=0.01\pm
0.14\mathrm{\,Me\kern-1.00006ptV}$. A non-zero Breit-Wigner width of the
${\chi_{{c}1}}(3872)$ state is obtained with a value of

$\Gamma_{\mathrm{BW}}=1.39\pm 0.24\pm 0.10\mathrm{\,Me\kern-1.00006ptV}.$

The values found here for $m_{{\chi_{{c}1}}(3872)}$ and $\Gamma_{\mathrm{BW}}$
are in good agreement with a complementary analysis using fully reconstructed
${{B}^{+}}\rightarrow{\chi_{{c}1}}(3872){{K}^{+}}$ decays presented in Ref.
[54] and combined therein.

Since $\left|\delta E\right|<\Gamma_{\mathrm{BW}}$, the value of
$\Gamma_{\mathrm{BW}}$ needs to be interpreted with caution as coupled channel
effects distort the lineshape. To elucidate this, fits using the Flatté
parameterization discussed in Refs. [15, 16] are performed. The parameters are
found to be

$\displaystyle g$ $\displaystyle=$ $\displaystyle 0.108\pm
0.003^{\,+\,0.005}_{\,-\,0.006}\,,$ $\displaystyle f_{\rho}$ $\displaystyle=$
$\displaystyle\left(1.8\pm 0.6^{\,+\,0.7}_{\,-\,0.6}\right)\times 10^{-3}\,,$
$\displaystyle\Gamma_{0}$ $\displaystyle=$ $\displaystyle 1.4\pm 0.4\pm
0.6\mathrm{\,Me\kern-1.00006ptV}\,,$

with $m_{0}$ fixed at $3864.5\mathrm{\,Me\kern-1.00006ptV}$. The mode of the
Flatté distribution agrees with the mean of the Breit–Wigner lineshape.
However, the determined FWHM is much smaller,
$0.22^{\,+\,0.06\,+\,0.25}_{\,-\,0.08\,-\,0.17}\,\mathrm{\,Me\kern-1.00006ptV}$,
highlighting the importance of a physically well-motivated lineshape
parameterization. The sensitivity of the data to the tails of the mass
distribution limits the extent to which the Flatté parameters can be
determined, as is expected in the case of a strong coupling of the state to
the ${{D}^{0}}{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ channel
[46]. Values of the parameter $E_{f}$ above
$-2.0\mathrm{\,Me\kern-1.00006ptV}$ are excluded at 90% confidence level. The
allowed region below threshold is
$-270<E_{f}<-2.0\mathrm{\,Me\kern-1.00006ptV}$. In this region a linear
dependence between the parameters is observed. The slope $\frac{dg}{dE_{f}}$
is related to the real part of the scattering length [15] and is measured to
be

$\dfrac{dg}{dE_{f}}=\left(-15.11\pm
0.16\right)\mathrm{\,Ge\kern-1.00006ptV}^{-1}\,.$

In order to investigate the nature of the ${\chi_{{c}1}}(3872)$ state, the
analytic structure of the amplitude in the vicinity of the ${{D}^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ threshold is examined. Using
the Flatté amplitude, two poles are found. Both poles appear on unphysical
sheets with respect to the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ channel and formally can be classified as
resonances. With respect to the ${{D}^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ channel, one pole appears on
the physical sheet, the other on the unphysical sheet. This configuration,
corresponding to a quasi-bound ${{D}^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ state, is preferred for all
scenarios studied in this paper. However, within combined statistical and
systematic uncertainties a location of the first pole on the unphysical sheet
is still allowed at the $2\sigma$ level and a quasi-virtual state assignment
for the ${\chi_{{c}1}}(3872)$ state cannot be excluded.

For the preferred quasi-bound state scenario the $90\%$ CL upper limit of the
${{D}^{0}}{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ binding
energy $E_{b}$ is found to be $100\,\mathrm{\,ke\kern-1.00006ptV}$. The
asymmetry of the locations of the two poles, which is found to be substantial,
provides information on the composition of the ${\chi_{{c}1}}(3872)$ state. In
the case of a dominantly molecular nature of a state a single pole close to
threshold is expected, while in the case of a compact state there should be
two nearby poles [55]. The argument is equivalent to the Weinberg composition
criterion [56] in the sense that the asymmetry of the pole location in
momentum space determines the relative fractions of molecular and compact
components in the ${\chi_{{c}1}}(3872)$ wave function [57]

$\dfrac{\left|k_{2}\right|-\left|k_{1}\right|}{\left|k_{1}\right|+\left|k_{2}\right|}=1-Z\,.$

Here $Z$ is the probability to find a compact component in the wave function.
The momentum $\left|k_{1}\right|=6.8\mathrm{\,Me\kern-1.00006ptV}$ is obtained
by inserting the binding energy of the bound state pole into Eq. (3). The
corresponding value for the second pole is
$\left|k_{2}\right|=82\mathrm{\,Me\kern-1.00006ptV}$ and therefore one obtains
$Z=15\%$. The asymmetry of the poles depends on the choice of $m_{0}$. The
asymmetry is reduced as the $m_{0}$ parameter approaches the threshold. The
largest value for $m_{0}$ that is still compatible with the data is
$3869.3\mathrm{\,Me\kern-1.00006ptV}$. In this case one obtains $Z=33\%$ and
therefore the probability to find a compact component in the
${\chi_{{c}1}}(3872)$ wave function is less than a third. It should be noted
that this argument depends on the extrapolation to the single channel case.
For resonances the wave function normalisation used in the Weinberg criterion
is not valid and $Z$ has to be replaced by an integral over the spectral
density [57]. Nevertheless, the value obtained in this work is in agreement
with the results of the analysis of the spectral density using Belle data [44,
58], presented in Ref. [16].

The results for the amplitude parameters and in particular the locations of
the poles, are systematically limited. In the future, a combined analysis of
the ${\chi_{{c}1}}(3872)\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ and
${\chi_{{c}1}}(3872)\\!\rightarrow{{D}^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}$ channels will make possible
improvements to the knowledge on the amplitude parameters.

## Acknowledgements

We thank C. Hanhart and A. Pilloni for useful discussions on the Flatté model
and the analytic structure of the amplitude. 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 (The Netherlands);
MNiSW and NCN (Poland); MEN/IFA (Romania); MinES and FASO (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 (The Netherlands),
PIC (Spain), GridPP (United Kingdom), RRCKI and Yandex LLC (Russia), CSCS
(Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-GRID (Poland) and OSC
(USA). We are indebted to the communities behind the multiple open-source
software packages on which we depend. Individual groups or members have
received support from AvH Foundation (Germany), EPLANET, Marie Skłodowska-
Curie Actions and ERC (European Union), ANR, Labex P2IO and OCEVU, and Région
Auvergne-Rhône-Alpes (France), RFBR, RSF and Yandex LLC (Russia), GVA,
XuntaGal and GENCAT (Spain), Herchel Smith Fund, the Royal Society, the
English-Speaking Union and the Leverhulme Trust (United Kingdom).

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LHCb collaboration

R. Aaij31, C. Abellán Beteta49, T. Ackernley59, B. Adeva45, M. Adinolfi53, H.
Afsharnia9, C.A. Aidala82, S. Aiola25, Z. Ajaltouni9, S. Akar64, J.
Albrecht14, F. Alessio47, M. Alexander58, A. Alfonso Albero44, Z. Aliouche61,
G. Alkhazov37, P. Alvarez Cartelle47, A.A. Alves Jr45, S. Amato2, Y. Amhis11,
L. An21, L. Anderlini21, G. Andreassi48, A. Andreianov37, M. Andreotti20, F.
Archilli16, A. Artamonov43, M. Artuso67, K. Arzymatov41, E. Aslanides10, M.
Atzeni49, B. Audurier11, S. Bachmann16, M. Bachmayer48, J.J. Back55, S.
Baker60, P. Baladron Rodriguez45, V. Balagura11,b, W. Baldini20, J. Baptista
Leite1, R.J. Barlow61, S. Barsuk11, W. Barter60, M. Bartolini23,47,h, F.
Baryshnikov79, J.M. Basels13, G. Bassi28, V. Batozskaya35, B. Batsukh67, A.
Battig14, A. Bay48, M. Becker14, F. Bedeschi28, I. Bediaga1, A. Beiter67, V.
Belavin41, S. Belin26, V. Bellee48, K. Belous43, I. Belyaev38, G.
Bencivenni22, E. Ben-Haim12, S. Benson31, A. Berezhnoy39, R. Bernet49, D.
Berninghoff16, H.C. Bernstein67, C. Bertella47, E. Bertholet12, A. Bertolin27,
C. Betancourt49, F. Betti19,e, M.O. Bettler54, Ia. Bezshyiko49, S. Bhasin53,
J. Bhom33, L. Bian72, M.S. Bieker14, S. Bifani52, P. Billoir12, F.C.R.
Bishop54, A. Bizzeti21,t, M. Bjørn62, M.P. Blago47, T. Blake55, F. Blanc48, S.
Blusk67, D. Bobulska58, V. Bocci30, J.A. Boelhauve14, O. Boente Garcia45, T.
Boettcher63, A. Boldyrev80, A. Bondar42,w, N. Bondar37,47, S. Borghi61, M.
Borisyak41, M. Borsato16, J.T. Borsuk33, S.A. Bouchiba48, T.J.V. Bowcock59, A.
Boyer47, C. Bozzi20, M.J. Bradley60, S. Braun65, A. Brea Rodriguez45, M.
Brodski47, J. Brodzicka33, A. Brossa Gonzalo55, D. Brundu26, E. Buchanan53, A.
Büchler-Germann49, A. Buonaura49, C. Burr47, A. Bursche26, A. Butkevich40,
J.S. Butter31, J. Buytaert47, W. Byczynski47, S. Cadeddu26, H. Cai72, R.
Calabrese20,g, L. Calero Diaz22, S. Cali22, R. Calladine52, M. Calvi24,i, M.
Calvo Gomez44,l, P. Camargo Magalhaes53, A. Camboni44, P. Campana22, D.H.
Campora Perez31, A.F. Campoverde Quezada5, S. Capelli24,i, L. Capriotti19,e,
A. Carbone19,e, G. Carboni29, R. Cardinale23,h, A. Cardini26, I. Carli6, P.
Carniti24,i, K. Carvalho Akiba31, A. Casais Vidal45, G. Casse59, M.
Cattaneo47, G. Cavallero47, S. Celani48, R. Cenci28, J. Cerasoli10, A.J.
Chadwick59, M.G. Chapman53, M. Charles12, Ph. Charpentier47, G.
Chatzikonstantinidis52, M. Chefdeville8, C. Chen3, S. Chen26, A. Chernov33,
S.-G. Chitic47, V. Chobanova45, S. Cholak48, M. Chrzaszcz33, A. Chubykin37, V.
Chulikov37, P. Ciambrone22, M.F. Cicala55, X. Cid Vidal45, G. Ciezarek47, F.
Cindolo19, P.E.L. Clarke57, M. Clemencic47, H.V. Cliff54, J. Closier47, J.L.
Cobbledick61, V. Coco47, J.A.B. Coelho11, J. Cogan10, E. Cogneras9, L.
Cojocariu36, P. Collins47, T. Colombo47, A. Contu26, N. Cooke52, G. Coombs58,
S. Coquereau44, G. Corti47, C.M. Costa Sobral55, B. Couturier47, D.C. Craik63,
J. Crkovská66, M. Cruz Torres1,y, R. Currie57, C.L. Da Silva66, E.
Dall’Occo14, J. Dalseno45, C. D’Ambrosio47, A. Danilina38, P. d’Argent47, A.
Davis61, O. De Aguiar Francisco47, K. De Bruyn47, S. De Capua61, M. De Cian48,
J.M. De Miranda1, L. De Paula2, M. De Serio18,d, D. De Simone49, P. De
Simone22, J.A. de Vries77, C.T. Dean66, W. Dean82, D. Decamp8, L. Del Buono12,
B. Delaney54, H.-P. Dembinski14, A. Dendek34, V. Denysenko49, D. Derkach80, O.
Deschamps9, F. Desse11, F. Dettori26,f, B. Dey7, A. Di Canto47, P. Di Nezza22,
S. Didenko79, H. Dijkstra47, V. Dobishuk51, A.M. Donohoe17, F. Dordei26, M.
Dorigo28,x, A.C. dos Reis1, L. Douglas58, A. Dovbnya50, A.G. Downes8, K.
Dreimanis59, M.W. Dudek33, L. Dufour47, P. Durante47, J.M. Durham66, D.
Dutta61, M. Dziewiecki16, A. Dziurda33, A. Dzyuba37, S. Easo56, U. Egede69, V.
Egorychev38, S. Eidelman42,w, S. Eisenhardt57, S. Ek-In48, L. Eklund58, S.
Ely67, A. Ene36, E. Epple66, S. Escher13, J. Eschle49, S. Esen31, T. Evans47,
A. Falabella19, J. Fan3, Y. Fan5, B. Fang72, N. Farley52, S. Farry59, D.
Fazzini11, P. Fedin38, M. Féo47, P. Fernandez Declara47, A. Fernandez
Prieto45, F. Ferrari19,e, L. Ferreira Lopes48, F. Ferreira Rodrigues2, S.
Ferreres Sole31, M. Ferrillo49, M. Ferro-Luzzi47, S. Filippov40, R.A. Fini18,
M. Fiorini20,g, M. Firlej34, K.M. Fischer62, C. Fitzpatrick61, T. Fiutowski34,
F. Fleuret11,b, M. Fontana47, F. Fontanelli23,h, R. Forty47, V. Franco Lima59,
M. Franco Sevilla65, M. Frank47, E. Franzoso20, G. Frau16, C. Frei47, D.A.
Friday58, J. Fu25,p, Q. Fuehring14, W. Funk47, E. Gabriel57, T. Gaintseva41,
A. Gallas Torreira45, D. Galli19,e, S. Gallorini27, S. Gambetta57, Y. Gan3, M.
Gandelman2, P. Gandini25, Y. Gao4, M. Garau26, L.M. Garcia Martin46, P. Garcia
Moreno44, J. García Pardiñas49, B. Garcia Plana45, F.A. Garcia Rosales11, L.
Garrido44, D. Gascon44, C. Gaspar47, R.E. Geertsema31, D. Gerick16, E.
Gersabeck61, M. Gersabeck61, T. Gershon55, D. Gerstel10, Ph. Ghez8, V.
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Giugliano20,g, K. Gizdov57, V.V. Gligorov12, C. Göbel70, E. Golobardes44,l, D.
Golubkov38, A. Golutvin60,79, A. Gomes1,a, M. Goncerz33, P. Gorbounov38, I.V.
Gorelov39, C. Gotti24,i, E. Govorkova31, J.P. Grabowski16, R. Graciani Diaz44,
T. Grammatico12, L.A. Granado Cardoso47, E. Graugés44, E. Graverini48, G.
Graziani21, A. Grecu36, L.M. Greeven31, P. Griffith20,g, L. Grillo61, L.
Gruber47, B.R. Gruberg Cazon62, C. Gu3, M. Guarise20, P. A. Günther16, E.
Gushchin40, A. Guth13, Yu. Guz43,47, T. Gys47, T. Hadavizadeh69, G. Haefeli48,
C. Haen47, S.C. Haines54, P.M. Hamilton65, Q. Han7, X. Han16, T.H. Hancock62,
S. Hansmann-Menzemer16, N. Harnew62, T. Harrison59, R. Hart31, C. Hasse47, M.
Hatch47, J. He5, M. Hecker60, K. Heijhoff31, K. Heinicke14, A.M. Hennequin47,
K. Hennessy59, L. Henry25,46, J. Heuel13, A. Hicheur68, D. Hill62, M.
Hilton61, S.E. Hollitt14, P.H. Hopchev48, J. Hu16, J. Hu71, W. Hu7, W. Huang5,
W. Hulsbergen31, T. Humair60, R.J. Hunter55, M. Hushchyn80, D. Hutchcroft59,
D. Hynds31, P. Ibis14, M. Idzik34, D. Ilin37, P. Ilten52, A. Inglessi37, K.
Ivshin37, R. Jacobsson47, S. Jakobsen47, E. Jans31, B.K. Jashal46, A.
Jawahery65, V. Jevtic14, F. Jiang3, M. John62, D. Johnson47, C.R. Jones54,
T.P. Jones55, B. Jost47, N. Jurik62, S. Kandybei50, Y. Kang3, M. Karacson47,
J.M. Kariuki53, N. Kazeev80, M. Kecke16, F. Keizer54,47, M. Kelsey67, M.
Kenzie55, T. Ketel32, B. Khanji47, A. Kharisova81, K.E. Kim67, T. Kirn13, V.S.
Kirsebom48, O. Kitouni63, S. Klaver22, K. Klimaszewski35, S. Koliiev51, A.
Kondybayeva79, A. Konoplyannikov38, P. Kopciewicz34, R. Kopecna16, P.
Koppenburg31, M. Korolev39, I. Kostiuk31,51, O. Kot51, S. Kotriakhova37, P.
Kravchenko37, L. Kravchuk40, R.D. Krawczyk47, M. Kreps55, F. Kress60, S.
Kretzschmar13, P. Krokovny42,w, W. Krupa34, W. Krzemien35, W. Kucewicz33,k, M.
Kucharczyk33, V. Kudryavtsev42,w, H.S. Kuindersma31, G.J. Kunde66, T.
Kvaratskheliya38, D. Lacarrere47, G. Lafferty61, A. Lai26, A. Lampis26, D.
Lancierini49, J.J. Lane61, R. Lane53, G. Lanfranchi22, C. Langenbruch13, O.
Lantwin49,79, T. Latham55, F. Lazzari28,u, R. Le Gac10, S.H. Lee82, R.
Lefèvre9, A. Leflat39,47, O. Leroy10, T. Lesiak33, B. Leverington16, H. Li71,
L. Li62, P. Li16, X. Li66, Y. Li6, Y. Li6, Z. Li67, X. Liang67, T. Lin60, R.
Lindner47, V. Lisovskyi14, R. Litvinov26, G. Liu71, H. Liu5, S. Liu6, X. Liu3,
A. Loi26, J. Lomba Castro45, I. Longstaff58, J.H. Lopes2, G. Loustau49, G.H.
Lovell54, Y. Lu6, D. Lucchesi27,n, S. Luchuk40, M. Lucio Martinez31, V.
Lukashenko31, Y. Luo3, A. Lupato61, E. Luppi20,g, O. Lupton55, A. Lusiani28,s,
X. Lyu5, L. Ma6, S. Maccolini19,e, F. Machefert11, F. Maciuc36, V. Macko48, P.
Mackowiak14, S. Maddrell-Mander53, L.R. Madhan Mohan53, O. Maev37, A.
Maevskiy80, D. Maisuzenko37, M.W. Majewski34, S. Malde62, B. Malecki47, A.
Malinin78, T. Maltsev42,w, H. Malygina16, G. Manca26,f, G. Mancinelli10, R.
Manera Escalero44, D. Manuzzi19,e, D. Marangotto25,p, J. Maratas9,v, J.F.
Marchand8, U. Marconi19, S. Mariani21,21,47, C. Marin Benito11, M.
Marinangeli48, P. Marino48, J. Marks16, P.J. Marshall59, G. Martellotti30, L.
Martinazzoli47, M. Martinelli24,i, D. Martinez Santos45, F. Martinez Vidal46,
A. Massafferri1, M. Materok13, R. Matev47, A. Mathad49, Z. Mathe47, V.
Matiunin38, C. Matteuzzi24, K.R. Mattioli82, A. Mauri49, E. Maurice11,b, M.
Mazurek35, M. McCann60, L. Mcconnell17, T.H. Mcgrath61, A. McNab61, R.
McNulty17, J.V. Mead59, B. Meadows64, C. Meaux10, G. Meier14, N. Meinert75, D.
Melnychuk35, S. Meloni24,i, M. Merk31, A. Merli25, L. Meyer Garcia2, M.
Mikhasenko47, D.A. Milanes73, E. Millard55, M.-N. Minard8, O. Mineev38, L.
Minzoni20,g, S.E. Mitchell57, B. Mitreska61, D.S. Mitzel47, A. Mödden14, A.
Mogini12, R.A. Mohammed62, R.D. Moise60, T. Mombächer14, I.A. Monroy73, S.
Monteil9, M. Morandin27, G. Morello22, M.J. Morello28,s, J. Moron34, A.B.
Morris10, A.G. Morris55, R. Mountain67, H. Mu3, F. Muheim57, M. Mukherjee7, M.
Mulder47, D. Müller47, K. Müller49, C.H. Murphy62, D. Murray61, P. Muzzetto26,
P. Naik53, T. Nakada48, R. Nandakumar56, T. Nanut48, I. Nasteva2, M.
Needham57, I. Neri20,g, N. Neri25,p, S. Neubert74, N. Neufeld47, R.
Newcombe60, T.D. Nguyen48, C. Nguyen-Mau48,m, E.M. Niel11, S. Nieswand13, N.
Nikitin39, N.S. Nolte47, C. Nunez82, A. Oblakowska-Mucha34, V. Obraztsov43, S.
Ogilvy58, D.P. O’Hanlon53, R. Oldeman26,f, C.J.G. Onderwater76, J. D.
Osborn82, A. Ossowska33, J.M. Otalora Goicochea2, T. Ovsiannikova38, P.
Owen49, A. Oyanguren46, B. Pagare55, P.R. Pais47, T. Pajero28,47,28,s, A.
Palano18, M. Palutan22, Y. Pan61, G. Panshin81, A. Papanestis56, M.
Pappagallo57, L.L. Pappalardo20,g, C. Pappenheimer64, W. Parker65, C.
Parkes61, C.J. Parkinson45, G. Passaleva21,47, A. Pastore18, M. Patel60, C.
Patrignani19,e, A. Pearce47, A. Pellegrino31, M. Pepe Altarelli47, S.
Perazzini19, D. Pereima38, P. Perret9, K. Petridis53, A. Petrolini23,h, A.
Petrov78, S. Petrucci57, M. Petruzzo25,p, A. Philippov41, L. Pica28, B.
Pietrzyk8, G. Pietrzyk48, M. Pili62, D. Pinci30, J. Pinzino47, F. Pisani19, A.
Piucci16, V. Placinta36, S. Playfer57, J. Plews52, M. Plo Casasus45, F.
Polci12, M. Poli Lener22, M. Poliakova67, A. Poluektov10, N. Polukhina79,c, I.
Polyakov67, E. Polycarpo2, G.J. Pomery53, S. Ponce47, A. Popov43, D. Popov52,
S. Popov41, S. Poslavskii43, K. Prasanth33, L. Promberger47, C. Prouve45, V.
Pugatch51, A. Puig Navarro49, H. Pullen62, G. Punzi28,o, W. Qian5, J. Qin5, R.
Quagliani12, B. Quintana8, N.V. Raab17, R.I. Rabadan Trejo10, B. Rachwal34,
J.H. Rademacker53, M. Rama28, M. Ramos Pernas45, M.S. Rangel2, F.
Ratnikov41,80, G. Raven32, M. Reboud8, F. Redi48, F. Reiss12, C. Remon
Alepuz46, Z. Ren3, V. Renaudin62, R. Ribatti28, S. Ricciardi56, D.S.
Richards56, S. Richards53, K. Rinnert59, P. Robbe11, A. Robert12, G.
Robertson57, A.B. Rodrigues48, E. Rodrigues59, J.A. Rodriguez Lopez73, M.
Roehrken47, A. Rollings62, V. Romanovskiy43, M. Romero Lamas45, A. Romero
Vidal45, J.D. Roth82, M. Rotondo22, M.S. Rudolph67, T. Ruf47, J. Ruiz Vidal46,
A. Ryzhikov80, J. Ryzka34, J.J. Saborido Silva45, N. Sagidova37, N. Sahoo55,
B. Saitta26,f, C. Sanchez Gras31, C. Sanchez Mayordomo46, R. Santacesaria30,
C. Santamarina Rios45, M. Santimaria22, E. Santovetti29,j, G. Sarpis61, M.
Sarpis74, A. Sarti30, C. Satriano30,r, A. Satta29, M. Saur5, D. Savrina38,39,
H. Sazak9, L.G. Scantlebury Smead62, S. Schael13, M. Schellenberg14, M.
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M.H. Schune11, R. Schwemmer47, B. Sciascia22, A. Sciubba22, S. Sellam68, A.
Semennikov38, A. Sergi52,47, N. Serra49, J. Serrano10, L. Sestini27, A.
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Shears59, L. Shekhtman42,w, V. Shevchenko78, E.B. Shields24,i, E. Shmanin79,
J.D. Shupperd67, B.G. Siddi20, R. Silva Coutinho49, L. Silva de Oliveira2, G.
Simi27,n, S. Simone18,d, I. Skiba20,g, N. Skidmore74, T. Skwarnicki67, M.W.
Slater52, J.C. Smallwood62, J.G. Smeaton54, A. Smetkina38, E. Smith13, I.T.
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Stone67, S. Stracka28, M.E. Stramaglia48, M. Straticiuc36, D. Strekalina79, S.
Strokov81, F. Suljik62, J. Sun26, L. Sun72, Y. Sun65, P. Svihra61, P.N.
Swallow52, K. Swientek34, A. Szabelski35, T. Szumlak34, M. Szymanski47, S.
Taneja61, Z. Tang3, T. Tekampe14, F. Teubert47, E. Thomas47, K.A. Thomson59,
M.J. Tilley60, V. Tisserand9, S. T’Jampens8, M. Tobin6, S. Tolk47, L.
Tomassetti20,g, D. Torres Machado1, D.Y. Tou12, E. Tournefier8, M. Traill58,
M.T. Tran48, E. Trifonova79, C. Trippl48, A. Tsaregorodtsev10, G. Tuci28,o, A.
Tully48, N. Tuning31, A. Ukleja35, D.J. Unverzagt16, A. Usachov31, A.
Ustyuzhanin41,80, U. Uwer16, A. Vagner81, V. Vagnoni19, A. Valassi47, G.
Valenti19, M. van Beuzekom31, H. Van Hecke66, E. van Herwijnen79, C.B. Van
Hulse17, M. van Veghel76, R. Vazquez Gomez45, P. Vazquez Regueiro45, C.
Vázquez Sierra31, S. Vecchi20, J.J. Velthuis53, M. Veltri21,q, A.
Venkateswaran67, M. Veronesi31, M. Vesterinen55, J.V. Viana Barbosa47, D.
Vieira64, M. Vieites Diaz48, H. Viemann75, X. Vilasis-Cardona44, E. Vilella
Figueras59, P. Vincent12, G. Vitali28, A. Vitkovskiy31, A. Vollhardt49, D. Vom
Bruch12, A. Vorobyev37, V. Vorobyev42,w, N. Voropaev37, R. Waldi75, J.
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Websdale60, A. Weiden49, C. Weisser63, B.D.C. Westhenry53, D.J. White61, M.
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Zdybal33, O. Zenaiev47, M. Zeng3, D. Zhang7, L. Zhang3, S. Zhang4, Y. Zhang47,
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Zonneveld57, S. Zucchelli19,e, D. Zuliani27, G. Zunica61.

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

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
lDS4DS, La Salle, Universitat Ramon Llull, Barcelona, Spain
mHanoi University of Science, Hanoi, Vietnam
nUniversità di Padova, Padova, Italy
oUniversità di Pisa, Pisa, Italy
pUniversità degli Studi di Milano, Milano, Italy
qUniversità di Urbino, Urbino, Italy
rUniversità della Basilicata, Potenza, Italy
sScuola Normale Superiore, Pisa, Italy
tUniversità di Modena e Reggio Emilia, Modena, Italy
uUniversità di Siena, Siena, Italy
vMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
wNovosibirsk State University, Novosibirsk, Russia
xINFN Sezione di Trieste, Trieste, Italy
yUniversidad Nacional Autonoma de Honduras, Tegucigalpa, Honduras