EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2018-245 LHCb-PAPER-2018-034 December 18, 2018

Evidence for an $\eta_{c}(1S)\pi^{-}$ resonance in
$B^{0}\rightarrow\eta_{c}(1S)K^{+}\pi^{-}$ decays

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

A Dalitz plot analysis of
${{B}^{0}}\\!\rightarrow\eta_{c}(1S){{K}^{+}}{{\pi}^{-}}$ decays is performed
using data samples of $pp$ collisions collected with the LHCb detector at
centre-of-mass energies of $\sqrt{s}=7,~{}8$ and
$13\mathrm{\,Te\kern-1.00006ptV}$, corresponding to a total integrated
luminosity of $4.7\mbox{\,fb}^{-1}$. A satisfactory description of the data is
obtained when including a contribution representing an exotic
$\eta_{c}(1S)\pi^{-}$ resonant state. The significance of this exotic
resonance is more than three standard deviations, while its mass and width are
$4096\pm 20~{}^{+18}_{-22}\mathrm{\,Me\kern-1.00006ptV}$ and $152\pm
58~{}^{+60}_{-35}\mathrm{\,Me\kern-1.00006ptV}$, respectively. The spin-parity
assignments $J^{P}=0^{+}$ and $J^{P}=1^{-}$ are both consistent with the data.
In addition, the first measurement of the
${{B}^{0}}\\!\rightarrow\eta_{c}(1S){{K}^{+}}{{\pi}^{-}}$ branching fraction
is performed and gives

$\displaystyle\mathcal{B}({{B}^{0}}\\!\rightarrow\eta_{c}(1S){{K}^{+}}{{\pi}^{-}})=(5.73\pm
0.24\pm 0.13\pm 0.66)\times 10^{-4}$,

where the first uncertainty is statistical, the second systematic, and the
third is due to limited knowledge of external branching fractions.

Published in Eur. Phys. J. C78 (2018) 1019

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

##  1 Introduction

Since the discovery of the $X(3872)$ state in 2003 [1], several exotic hadron
candidates have been observed, as reported in recent reviews [2, 3, 4, 5, 6,
7].111The $X(3872)$ state has been recently renamed $\chi_{c1}(3872)$ in Ref.
[8] The decay modes of these states indicate that they must contain a heavy
quark-antiquark pair in their internal structure; however, they cannot easily
be accommodated as an unassigned charmonium or bottomonium state due to either
their mass, decay properties or electric charge, which are inconsistent with
those of pure charmonium or bottomonium states. Different interpretations have
been proposed about their nature [2, 3, 4], including their quark composition
and binding mechanisms. In order to improve the understanding of these
hadrons, it is important to search for new exotic candidates, along with new
production mechanisms and decay modes of already observed unconventional
states.

The $Z_{c}(3900)^{-}$ state, discovered by the BESIII collaboration in the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\pi^{-}$ final state [9], and
confirmed by the Belle [10] and CLEO [11] collaborations, can be interpreted
as a hadrocharmonium state, where the compact heavy quark-antiquark pair
interacts with the surrounding light quark mesonic excitation by a QCD
analogue of the van der Waals force [12]. This interpretation of the
$Z_{c}(3900)^{-}$ state predicts an as-yet-unobserved charged charmonium-like
state with a mass of approximately $3800\,\mathrm{MeV}$ whose dominant decay
mode is to the ${\eta_{c}}{{\pi}^{-}}$ system.222Natural units with
$\hbar=c=1$ and the simplified notation ${\eta_{c}}$ to refer to the
$\eta_{c}(1S)$ state are used throughout. In addition, the inclusion of
charge-conjugate processes is always implied. Alternatively, states like the
$Z_{c}(3900)^{-}$ meson could be interpreted as analogues of quarkonium
hybrids, where the excitation of the gluon field (the valence gluon) is
replaced by an isospin-1 excitation of the gluon and light-quark fields [13].
This interpretation, which is based on lattice QCD, predicts different
multiplets of charmonium tetraquarks, comprising states with quantum numbers
allowing the decay into the ${\eta_{c}}{{\pi}^{-}}$ system. The
${\eta_{c}}{{\pi}^{-}}$ system carries isospin $I=1$, $G$-parity $G=-1$, spin
$J=L$ and parity $P=(-1)^{L}$, where $L$ is the orbital angular momentum
between the $\eta_{c}$ and the ${\pi}^{-}$ mesons. Lattice QCD calculations
[14, 15] predict the mass and quantum numbers of these states, comprising a
$I^{G}(J^{P})=1^{-}(0^{+})$ state of mass $4025\pm 49\,\mathrm{MeV}$, a
$I^{G}(J^{P})=1^{-}(1^{-})$ state of mass $3770\pm 42\,\mathrm{MeV}$, and a
$I^{G}(J^{P})=1^{-}(2^{+})$ state of mass $4045\pm 44\,\mathrm{MeV}$. The
$Z_{c}(4430)^{-}$ resonance, discovered by the Belle collaboration [16] and
confirmed by LHCb [17, 18], could also fit into this scenario. Another
prediction of a possible exotic candidate decaying to the
${\eta_{c}}{{\pi}^{-}}$ system is provided by the diquark model [19], where
quarks and diquarks are the fundamental units to build a rich spectrum of
hadrons, including the exotic states observed thus far. The diquark model
predicts a $J^{P}=0^{+}$ candidate below the open-charm threshold that could
decay into the ${\eta_{c}}{{\pi}^{-}}$ final state. Therefore, the discovery
of a charged charmonium-like meson in the ${\eta_{c}}{{\pi}^{-}}$ system would
provide important input towards understanding the nature of exotic hadrons.

In this article, the ${{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}}$
decay is studied for the first time, with the $\eta_{c}$ meson reconstructed
using the $p$ $\overline{p}$ decay mode. The decay is expected to proceed
through ${{K}^{*0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ intermediate states,
where ${K}^{*0}$ refers to any neutral kaon resonance, following the diagram
shown in Fig. 1. If the decay also proceeds through exotic resonances in the
${\eta_{c}}{{\pi}^{-}}$system, denoted by $Z_{c}^{-}$ states in the following,
a diagram like that shown in Fig. 1 would contribute. The
${{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}}$ decay involves only
pseudoscalar mesons, hence it is fully described by two independent kinematic
quantities. Therefore, the Dalitz plot (DP) analysis technique [20] can be
used to completely characterise the decay.

Figure 1: Feynman diagrams for 1 $B^{0}\rightarrow\eta_{c}K^{*0}$ and 1
$B^{0}\rightarrow Z_{c}^{-}K^{+}$ decay sequences.

The data sample used corresponds to an integrated luminosity of
$4.7\mbox{\,fb}^{-1}$ of $p$ $p$ collision data collected with the LHCb
detector at centre-of-mass energies of $\sqrt{s}=7,~{}8$ and
$13\mathrm{\,Te\kern-1.00006ptV}$ in 2011, 2012 and 2016, respectively. Data
collected in 2011 and 2012 are referred to as Run 1 data, while data collected
in 2016 are referred to as Run 2 data.

This paper is organised as follows. A brief description of the LHCb detector
as well as the reconstruction and simulation software is given in Sec. 2. The
selection of ${{B}^{0}}\\!\rightarrow{p}{\overline{p}}{{K}^{+}}{{\pi}^{-}}$
candidates is described in Sec. 3, and the first measurement of the
${{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}}$ branching fraction is
presented in Sec. 4. An overview of the DP analysis formalism is given in Sec.
5. Details of the implementation of the DP fit are presented in Sec. 6. The
evaluation of systematic uncertainties is given in Sec. 7. The results are
summarised in Sec. 8.

##  2 Detector and simulation

The LHCb detector [21, 22] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\mathrm{\,Tm}}$, and three
stations of silicon-strip detectors and straw drift tubes placed downstream of
the magnet. The tracking system provides a measurement of the momentum, $p$,
of charged particles with a relative uncertainty that varies from 0.5% at low
momentum to 1.0% at 200$\mathrm{\,Ge\kern-1.00006ptV}$. The minimum distance
of a track to a primary vertex (PV), the impact parameter, 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.

The online event selection is performed by a trigger [23], which consists of a
hardware stage, based on information from the calorimeter and muon systems,
followed by a software stage, which applies a full event reconstruction. At
the hardware trigger stage, events are required to have a hadron with high
transverse energy in the calorimeters. The software trigger requires a two-,
three- or four-tracks secondary vertex with a significant displacement from
any PV. At least one charged particle must have a large transverse momentum
and be inconsistent with originating from a PV. A multivariate algorithm [24,
25] is used to identify secondary vertices that are consistent with $b$-hadron
decays.

Simulated events, generated uniformly in the phase space of the
${{B}^{0}}\\!\rightarrow{p}{\overline{p}}{{K}^{+}}{{\pi}^{-}}$ or
${{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}}$ decay modes, are used
to develop the selection, to validate the fit models and to evaluate the
efficiencies entering the branching fraction measurement and the DP analysis.
In the simulation, $pp$ collisions are generated using Pythia [26,
*Sjostrand:2006za] with a specific LHCb configuration [28]. Decays of hadronic
particles are described by EvtGen [29], in which final-state radiation is
generated using Photos [30]. The interaction of the generated particles with
the detector, and its response, are implemented using the Geant4 toolkit [31,
*Agostinelli:2002hh] as described in Ref. [33].

##  3 Selection

An initial offline selection comprising loose criteria is applied to
reconstructed particles, where the associated trigger decision was due to the
${B}^{0}$ candidate. The final-state tracks are required to have
$p>1500\,\mathrm{MeV}$, $\mbox{$p_{\mathrm{T}}$}>300\,\mathrm{MeV}$, and to be
inconsistent with originating from any PV in the event. Loose particle
identification (PID) criteria are applied, requiring the particles to be
consistent with either the proton, kaon or pion hypothesis. All tracks are
required to be within the acceptance of the RICH detectors ($2.0<\eta<4.9$).
Moreover, protons and antiprotons are required to have momenta larger than
$8\,\mathrm{GeV}$ ($11\,\mathrm{GeV}$) to avoid kinematic regions where
proton-kaon separation is limited for Run 1 (Run 2) data.

The ${{B}^{0}}$ candidates are required to have a small $\chi^{2}_{\text{IP}}$
with respect to a PV, where $\chi^{2}_{\text{IP}}$ is defined as the
difference in the vertex-fit $\chi^{2}$ of a given PV reconstructed with and
without the candidate under consideration. The PV providing the smallest
$\chi^{2}_{\text{IP}}$ value is associated to the ${{B}^{0}}$ candidate. The
${{B}^{0}}$ candidate is required to be consistent with originating from this
PV by applying a criterion on the direction angle (DIRA) between the
${{B}^{0}}$ candidate momentum vector and the distance vector between the PV
to the ${{B}^{0}}$ decay vertex. When building the ${{B}^{0}}$ candidates, the
resolution on kinematic quantities such as the $m({p}{\overline{p}})$
distribution, and the Dalitz variables that will be defined in Sec. 5, is
improved by performing a kinematic fit [34] in which the ${{B}^{0}}$ candidate
is constrained to originate from its associated PV, and its reconstructed
invariant mass is constrained to the known ${{B}^{0}}$ mass [8].

A boosted decision tree (BDT) [35, 36] algorithm is used to further suppress
the combinatorial background that arises when unrelated particles are combined
to form a ${B}^{0}$ candidate. The training of the BDT is performed using
simulated ${{B}^{0}}\\!\rightarrow{p}{\overline{p}}{{K}^{+}}{{\pi}^{-}}$
decays as the signal sample and candidates from the high-mass data sideband as
the background sample, defined as the
$5450<m({p}{\overline{p}}{{K}^{+}}{{\pi}^{-}})<5550\mathrm{\,Me\kern-1.00006ptV}$
range. The input variables to the BDT classifiers are the same for Run 1 and 2
samples and comprise typical discriminating variables of $b$-hadron decays:
the vertex-fit $\chi^{2}_{\text{vtx}}$, $\chi^{2}_{\text{IP}}$, DIRA and
flight distance significance of the reconstructed ${{B}^{0}}$ candidates; the
maximum distance of closest approach between final-state particles; and the
maximum and minimum $p$ and $p_{\mathrm{T}}$ of the proton and antiproton.

The requirements placed on the output of the BDT algorithm and PID variables
are simultaneously optimised to maximise the figure of merit defined as
$S/\sqrt{S+B}$. Here $S$ is the observed
${{B}^{0}}\\!\rightarrow{p}{\overline{p}}{{K}^{+}}{{\pi}^{-}}$ yield before
any BDT selection multiplied by the efficiency of the BDT requirement
evaluated using simulated decays, while $B$ is the the combinatorial
background yield. The training of the BDT and the optimisation of the
selection are performed separately for Run 1 and 2 data to accommodate for
differences in the two data-taking periods.

##  4 Branching fraction measurement

The measurement of the ${{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}}$
branching fraction is performed relative to that of the
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{\pi}^{-}}$ normalisation channel, where the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson is also reconstructed in
the $p$ $\overline{p}$ decay mode. A two-stage fit procedure to the combined
Run 1 and 2 data sample is used. In the first stage, an extended unbinned
maximum-likelihood (UML) fit is performed to the
$m({p}{\overline{p}}{{K}^{+}}{{\pi}^{-}})$ distribution in order to separate
the ${{B}^{0}}\\!\rightarrow{p}{\overline{p}}{{K}^{+}}{{\pi}^{-}}$ and
background contributions. The RooFit package [37] is used to perform the fit,
and the sPlot technique [38] is applied to assign weights for each candidate
to subtract the background contributions. In the second stage, a weighted UML
fit to the $p$ $\overline{p}$ invariant-mass spectrum is performed to
disentangle the $\eta_{c}$, ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$,
and nonresonant (NR) contributions. The efficiency-corrected yield ratio is

$R=\frac{N_{{\eta_{c}}}}{N_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}}\times\frac{\epsilon_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}}{\epsilon_{{\eta_{c}}}},$ (1)

where $N_{{\eta_{c}}}$ and $N_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}$ are the observed $\eta_{c}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ yields, while
$\epsilon_{{\eta_{c}}}$ and $\epsilon_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}$ are the total efficiencies, which are obtained from a combination of
simulated and calibration samples. The
${{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}}$ branching fraction is
determined as

$\mathcal{B}({{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}})=R\times\mathcal{B}({{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{\pi}^{-}})\times\frac{\mathcal{B}({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\\!\rightarrow{p}{\overline{p}})}{\mathcal{B}({\eta_{c}}\\!\rightarrow{p}{\overline{p}})},$
(2)

where
$\mathcal{B}({{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{\pi}^{-}})=(1.15\pm 0.05)\times 10^{-3}$,
$\mathcal{B}({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\\!\rightarrow{p}{\overline{p}})=(2.121\pm 0.029)\times 10^{-3}$ and
$\mathcal{B}({\eta_{c}}\\!\rightarrow{p}{\overline{p}})=(1.52\pm 0.16)\times
10^{-3}$ are the external branching fractions taken from Ref. [8].

###  4.1 Signal and normalisation yields

The first-stage UML fit to the $m({p}{\overline{p}}{{K}^{+}}{{\pi}^{-}})$
distribution is performed in the 5180–5430$\mathrm{\,Me\kern-1.00006ptV}$
range. The ${{B}^{0}}\\!\rightarrow{p}{\overline{p}}{{K}^{+}}{{\pi}^{-}}$
signal decays,
${{B}^{0}_{s}}\\!\rightarrow{p}{\overline{p}}{{K}^{+}}{{\pi}^{-}}$ decays and
various categories of background are present in this range. In addition to the
combinatorial background, partially reconstructed backgrounds are present
originating from $b$-hadron decays with additional particles that are not part
of the reconstructed decay chain, such as a ${\pi}^{0}$ meson or a photon.
Another source of background is $b$-hadron decays where one of the final-state
particles has been incorrectly identified, which includes the decays
${{B}^{0}}\\!\rightarrow{p}{\overline{p}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{p}{\overline{p}}{{K}^{+}}{{K}^{-}}$. The ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$
and ${{\kern 1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}}{}^{-}_{c}}\\!\rightarrow{\overline{p}}{{K}^{+}}{{\pi}^{-}}$
decays are removed by excluding the mass range
1845–1885$\mathrm{\,Me\kern-1.00006ptV}$ in the $m({{K}^{+}}{{\pi}^{-}})$
distribution and the range 2236–2336$\mathrm{\,Me\kern-1.00006ptV}$ in the
$m({\overline{p}}{{K}^{+}}{{\pi}^{-}})$ distribution, respectively. The latter
veto also removes partially reconstructed $b$-hadron decays.

Both the ${{B}^{0}}\\!\rightarrow{p}{\overline{p}}{{K}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{p}{\overline{p}}{{K}^{+}}{{\pi}^{-}}$ components
are modelled by Hypatia functions [39]. The Hypatia distribution is a
generalisation of the Crystall Ball function [40], where the Gaussian core of
the latter is replaced by a hyperbolic core to take into account the
distortion on the measured mass due to different sources of uncertanty. The
Hypatia functions share a common resolution parameter, while the tail
parameters are fixed to the values obtained from the corresponding simulated
sample. The distributions of the misidentified
${{B}^{0}}\\!\rightarrow{p}{\overline{p}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{p}{\overline{p}}{{K}^{+}}{{K}^{-}}$ backgrounds
are described by Crystal Ball functions, with parameters fixed to the values
obtained from simulation. The combinatorial background is modelled using an
exponential function. The masses of the ${B}^{0}$ and ${B}^{0}_{s}$ mesons,
the resolution parameter of the Hypatia functions, the slope of the
exponential function, and all the yields, are free to vary in the fit to the
data. Using the information from the fit to the
$m({p}{\overline{p}}{{K}^{+}}{{\pi}^{-}})$ distribution, shown in Fig. 2,
${{B}^{0}}\\!\rightarrow{p}{\overline{p}}{{K}^{+}}{{\pi}^{-}}$ signal weights
are computed and the background components are subtracted using the sPlot
technique [38]. About $3.0\times 10^{4}$ ${B}^{0}$ decays are observed.
Correlations between the $p$ $\overline{p}$ and $p$ $\overline{p}$ ${K}^{+}$
${\pi}^{-}$ invariant-mass variables for both signal and background are found
to be negligible.

Figure 2: Distribution of the $p$ $\overline{p}$ ${K}^{+}$ ${\pi}^{-}$
invariant mass. The solid blue curve is the projection of the total fit
result. The components are shown in the legend.

The second-stage UML fit is then performed to the weighted $p$ $\overline{p}$
invariant-mass distribution in the mass range
2700–3300$\mathrm{\,Me\kern-1.00006ptV}$, which includes $\eta_{c}$,
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, and NR
${{B}^{0}}\\!\rightarrow{p}{\overline{p}}{{K}^{+}}{{\pi}^{-}}$ contributions.
The $p$ $\overline{p}$ invariant-mass distribution of $\eta_{c}$ candidates is
described by the convolution of a nonrelativistic Breit–Wigner function and a
Gaussian function describing resolution effects. Using simulated samples, the
$p$ $\overline{p}$ invariant-mass resolution is found to be $\approx
5\,\mathrm{MeV}$. Given the width $\Gamma_{{\eta_{c}}}=32.0\pm
0.8\,\mathrm{MeV}$ [8], the impact of the detector resolution on the
$\eta_{c}$ lineshape is small. The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ resonance, having a small natural width, is parametrised using an
Hypatia function, with tail parameters fixed to the values obtained from the
corresponding simulated sample. The same resolution parameter is used for the
$\eta_{c}$ and ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ contributions,
which is free to vary in the fit to the data. The $\eta_{c}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ masses are also floating, while
the $\eta_{c}$ natural width is Gaussian constrained to the known value [8].
The NR ${{B}^{0}}\\!\rightarrow{p}{\overline{p}}{{K}^{+}}{{\pi}^{-}}$
contribution is parametrised with an exponential function with the slope free
to vary in the fit. All yields are left unconstrained in the fit. A possible
term describing the interference between the $\eta_{c}$ resonance and the NR
$p$ $\overline{p}$ S-wave is investigated and found to be negligible. The
result of the fit to the weighted $p$ $\overline{p}$ invariant-mass
distribution is shown in Fig. 3. The yields of the
${{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}}$ and
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{\pi}^{-}}$ fit components, entering Eq.(1), are $2105\pm 75$
and $5899\pm 86$, respectively.

Figure 3: Distribution of the $p$ $\overline{p}$ invariant mass in (left)
linear and (right) logarithmic vertical-axis scale for weighted
${{B}^{0}}\\!\rightarrow{p}{\overline{p}}{{K}^{+}}{{\pi}^{-}}$ candidates
obtained by using the sPlot technique. The solid blue curve is the projection
of the total fit result. The full azure, tight-cross-hatched red and dashed-
black line areas show the $\eta_{c}$, ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ and NR $p$ $\overline{p}$ contributions, respectively.

###  4.2 Ratio of efficiencies

The ratio of efficiencies of Eq. (1) is obtained from
${{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}}$ and
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{\pi}^{-}}$ simulated samples, both selected using the same
criteria used in data. Since these decays have the same final-state particles
and similar kinematic distributions, the ratio of efficiencies is expected to
be close to unity. The efficiencies are computed as the product of the
geometrical acceptance of the LHCb detector, the reconstruction efficiency and
the efficiency of the offline selection criteria, including the trigger and
PID requirements. The efficiency of the PID requirements is obtained using
calibration samples of pions, kaons and protons, as a function of the particle
momentum, pseudorapidity and the multiplicity of the event, e.g. the number of
charged particles in the event [41]. The final ratio of efficiencies is given
by

$\frac{\epsilon_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}}{\epsilon_{{\eta_{c}}}}=1.000\pm 0.013,$ (3)

which is compatible with unity as expected.

###  4.3 Systematic uncertainties

Table 1 summarises the systematic uncertainties on the measurement of the
ratio $R$ of Eq. (1). Since the kinematic distributions of the signal and
normalisation channel are similar, the uncertainties corresponding to the
reconstruction and selection efficiencies largely cancel in the ratio of
branching fractions. A new value of the ratio $R$ is computed for each source
of systematic uncertainty, and its difference with the nominal value is taken
as the associated systematic uncertainty. The overall systematic uncertainty
is assigned by combining all contributions in quadrature.

The systematic uncertainty arising from fixing the shape parameters of the
Hypatia functions used to parametrise the ${B}^{0}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ components is evaluated by
repeating the fits and varying all shape parameters simultaneously. These
shape parameters are varied according to normal distributions, taking into
account the correlations between the parameters and with variances related to
the size of the simulated samples.

To assign a systematic uncertainty arising from the model used to describe the
detector resolution, the fits are repeated for each step replacing the Hypatia
functions by Crystal Ball functions, whose parameters are obtained from
simulation.

The systematic uncertainty associated to the parametrisation of the NR
${{B}^{0}}\\!\rightarrow{p}{\overline{p}}{{K}^{+}}{{\pi}^{-}}$ contribution is
determined by replacing the exponential function with a linear function.

The systematic uncertainty associated to the determination of the efficiency
involves contributions arising from the weighting procedure of the calibration
samples used to determine the PID efficiencies. The granularity of the binning
in the weighting procedure is halved and doubled.

The free shape parameters in the first stage UML fit lead to uncertainties
that are not taken into account by the sPlot technique. In order to estimate
this effect, these parameters are varied within their uncertainties and the
signal weights are re-evaluated. The variations on the ratio $R$ resulting
from the second stage UML fit are found to be negligible.

Table 1: Relative systematic uncertainties on the ratio $R$ of Eq. (1). The total systematic uncertainty is obtained from the quadratic sum of the individual sources. Source | Systematic uncertainty (%)
---|---
Fixed shape parameters | 0.8
Resolution model | 0.3
NR $p\bar{p}$ model | 1.7
Efficiency ratio | 1.1
Total | 2.2

###  4.4 Results

The ratio $R$ is determined to be

$R=0.357\pm 0.015\pm 0.008,$

where the first uncertainty is statistical and the second systematic. The
statistical uncertainty includes contributions from the per-candidate weights
obtained using the sPlot technique. The value of $R$ is used to compute the
${{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}}$ branching fraction
using Eq. (2) which gives

$\mathcal{B}({{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}})=(5.73\pm
0.24\pm 0.13\pm 0.66)\times 10^{-4},$

where the first uncertainty is statistical, the second systematic, and the
third is due to the limited knowledge of the external branching fractions.

##  5 Dalitz plot formalism

The phase space for a three-body decay involving only pseudoscalar particles
can be represented in a DP, where two of the three possible two-body
invariant-mass-squared combinations, here $m^{2}({{K}^{+}}{{\pi}^{-}})$ and
$m^{2}({\eta_{c}}{{\pi}^{-}})$, are used to define the DP axes. However, given
the sizeable natural width of the $\eta_{c}$ meson, the invariant mass
$m({p}{\overline{p}})$ is used instead of the known value of the $\eta_{c}$
mass [8] to compute the kinematic quantities such as
$m^{2}({\eta_{c}}{{K}^{+}})$, $m^{2}({\eta_{c}}{{\pi}^{-}})$ and the helicity
angles.

The isobar model [42, 43, 44] is used to write the decay amplitude as a
coherent sum of amplitudes from resonant and NR intermediate processes as

$\mathcal{A}[m^{2}({{K}^{+}}{{\pi}^{-}}),m^{2}({\eta_{c}}{{\pi}^{-}})]=\sum_{j=1}^{N}c_{j}\mathcal{F}_{j}[m^{2}({{K}^{+}}{{\pi}^{-}}),m^{2}({\eta_{c}}{{\pi}^{-}})],$
(4)

where $c_{j}$ are complex coefficients giving the relative contribution of
each intermediate process. The
$\mathcal{F}_{j}[m^{2}({{K}^{+}}{{\pi}^{-}}),m^{2}({\eta_{c}}{{\pi}^{-}})]$
complex functions describe the resonance dynamics and are normalised such that
the integral of their squared magnitude over the DP is unity

$\int_{\text{DP}}|\mathcal{F}_{j}[m^{2}({{K}^{+}}{{\pi}^{-}}),m^{2}({\eta_{c}}{{\pi}^{-}})]|^{2}\text{d}m^{2}({{K}^{+}}{{\pi}^{-}})~{}\text{d}m^{2}({\eta_{c}}{{\pi}^{-}})=1.$
(5)

Each
$\mathcal{F}_{j}[m^{2}({{K}^{+}}{{\pi}^{-}}),m^{2}({\eta_{c}}{{\pi}^{-}})]$
contribution is composed of the product of several factors. For a
${{K}^{+}}{{\pi}^{-}}$ resonance, for instance,

$\mathcal{F}[m^{2}({{K}^{+}}{{\pi}^{-}}),m^{2}({\eta_{c}}{{\pi}^{-}})]=\mathcal{N}\times
X(|\vec{p}|r_{\text{BW}})\times X(|\vec{q}|r_{\text{BW}})\times
Z(\vec{p},\vec{q})\times T[m({{K}^{+}}{{\pi}^{-}})],$ (6)

where $\displaystyle\mathcal{N}$ is a normalisation constant and $\vec{p}$ and
$\vec{q}$ are the momentum of the accompanying particle (the $\eta_{c}$ meson
in this case) and the momentum of one of the resonance decay products,
respectively, both evaluated in the ${{K}^{+}}{{\pi}^{-}}$ rest frame. The
$X(z)$ terms are the Blatt–Weisskopf barrier factors [45] reported in Appendix
A. The barrier radius, $r_{\text{BW}}$, is taken to be $4\,\mathrm{GeV^{-1}}$
(corresponding to $\sim 0.8\mathrm{\,fm}$) for all resonances. The
$Z(\vec{p},\vec{q})$ term describes the angular probability distribution in
the Zemach tensor formalism [46, 47], given by the equations reported in
Appendix B. The function $T[m({{K}^{+}}{{\pi}^{-}})]$ of Eq. (6) is the mass
lineshape. Most of the resonant contributions are described by the
relativistic Breit–Wigner (RBW) function

$T(m)=\frac{1}{m_{0}^{2}-m^{2}-im_{0}\Gamma(m)},$ (7)

where the mass-dependent width is given by

$\Gamma(m)=\Gamma_{0}\left(\frac{|\vec{q}|}{q_{0}}\right)^{(2L+1)}\left(\frac{m_{0}}{m}\right)X^{2}(|\vec{q}|r_{\text{BW}})$
(8)

and $q_{0}$ is the value of $|\vec{q}|$ for $m=m_{0}$, $m_{0}$ being the pole
mass of the resonance.

The amplitude parametrisations using RBW functions lead to unitarity violation
within the isobar model if there are overlapping resonances or if there is a
significant interference with a NR component, both in the same partial wave
[48]. This is the case for the ${{K}^{+}}{{\pi}^{-}}$ S-wave at low
${{K}^{+}}{{\pi}^{-}}$ mass, where the $K^{*}_{0}(1430)^{0}$ resonance
interferes strongly with a slowly varying NR S-wave component. Therefore, the
${{K}^{+}}{{\pi}^{-}}$ S-wave at low mass is modelled using a modified LASS
lineshape [49], given by

$T(m)=\frac{m}{|\vec{q}|\cot\delta_{B}-i|\vec{q}|}+e^{2i\delta_{B}}\frac{m_{0}\Gamma_{0}\frac{m_{0}}{q_{0}}}{m_{0}^{2}-m^{2}-im_{0}\Gamma_{0}\frac{|\vec{q}|}{m}\frac{m_{0}}{q_{0}}},$
(9)

with

$\cot\delta_{B}=\frac{1}{a|\vec{q}|}+\frac{1}{2}r|\vec{q}|,$ (10)

and where $m_{0}$ and $\Gamma_{0}$ are the pole mass and width of the
$K^{*}_{0}(1430)^{0}$ state, and $a$ and $r$ are the scattering length and the
effective range, respectively. The parameters $a$ and $r$ depend on the
production mechanism and hence on the decay under study. The slowly varying
part (the first term in Eq. (9)) is not well modelled at high masses and it is
set to zero for $m({{K}^{+}}{{\pi}^{-}})$ values above $1.7\,\mathrm{GeV}$.

The probability density function for signal events across the DP, neglecting
reconstruction effects, can be written as

$\mathcal{P}_{\text{sig}}[m^{2}({{K}^{+}}{{\pi}^{-}}),m^{2}({\eta_{c}}{{\pi}^{-}})]=\frac{|\mathcal{A}|^{2}}{\int_{\text{DP}}|\mathcal{A}|^{2}\text{d}m^{2}({{K}^{+}}{{\pi}^{-}})~{}\text{d}m^{2}({\eta_{c}}{{\pi}^{-}})},$
(11)

where the dependence of $\mathcal{A}$ on the DP position has been suppressed
for brevity. The natural width of the $\eta_{c}$ meson is set to zero when
computing the DP normalisation shown in the denominator of Eq. (11). The
effect of this simplification is determined when assessing the systematic
uncertainties as described in Sec. 7.

The complex coefficients, given by $c_{j}$ in Eq. (4), depend on the choice of
normalisation, phase convention and amplitude formalism. Fit fractions and
interference fit fractions are convention-independent quantities that can be
directly compared between different analyses. The fit fraction is defined as
the integral of the amplitude for a single component squared divided by that
of the coherent matrix element squared for the complete DP,

$\text{FF}_{i}=\frac{\int_{\text{DP}}|c_{i}\mathcal{F}_{i}[m^{2}({{K}^{+}}{{\pi}^{-}}),m^{2}({\eta_{c}}{{\pi}^{-}})]|^{2}\text{d}m^{2}({{K}^{+}}{{\pi}^{-}})~{}\text{d}m^{2}({\eta_{c}}{{\pi}^{-}})}{\int_{\text{DP}}|\mathcal{A}[m^{2}({{K}^{+}}{{\pi}^{-}}),m^{2}({\eta_{c}}{{\pi}^{-}})]|^{2}\text{d}m^{2}({{K}^{+}}{{\pi}^{-}})~{}\text{d}m^{2}({\eta_{c}}{{\pi}^{-}})}.$
(12)

In general, the fit fractions do not sum to unity due to the possible presence
of net constructive or destructive interference over the whole DP area. This
effect can be described by interference fit fractions defined for $i<j$ by

$\text{FF}_{ij}=\frac{\int_{\text{DP}}2\mathcal{R}e\left[c_{i}c_{j}^{*}\mathcal{F}_{i}\mathcal{F}_{j}^{*}\right]\text{d}m^{2}({{K}^{+}}{{\pi}^{-}})~{}\text{d}m^{2}({\eta_{c}}{{\pi}^{-}})}{\int_{\text{DP}}|\mathcal{A}|^{2}\text{d}m^{2}({{K}^{+}}{{\pi}^{-}})~{}\text{d}m^{2}({\eta_{c}}{{\pi}^{-}})},$
(13)

where the dependence of $\mathcal{F}_{i}^{(*)}$ and $\mathcal{A}$ on the DP
position is omitted.

##  6 Dalitz plot fit

The Laura++ package [50] is used to perform the unbinned DP fit, with the Run
1 and 2 subsamples fitted simultaneously using the JFIT framework [51]. The
free parameters in the amplitude fit are in common between the two subsamples,
while the signal and background yields and the maps describing the efficiency
variations across the phase space, are different. Within the DP fit, the
signal corresponds to ${{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}}$
decays, while the background comprises both combinatorial background and NR
${{B}^{0}}\\!\rightarrow{p}{\overline{p}}{{K}^{+}}{{\pi}^{-}}$ contributions.
The likelihood function is given by

$\mathcal{L}=\prod_{i}^{N_{c}}\left[\sum_{k}N_{k}\mathcal{P}_{k}[m_{i}^{2}(K^{+}\pi^{-}),m_{i}^{2}(\eta_{c}\pi^{-})]\right],$
(14)

where the index $i$ runs over the $N_{c}$ candidates, $k$ runs over the signal
and background components, and $N_{k}$ is the yield of each component. The
procedure to determine the signal and background yields is described in Sec.
6.1. The probability density function for the signal,
$\mathcal{P}_{\text{sig}}$, is given by Eq. (11) where the
$|\mathcal{A}[m^{2}({{K}^{+}}{{\pi}^{-}}),m^{2}({\eta_{c}}{{\pi}^{-}})]|^{2}$
term is multiplied by the efficiency function described in Sec. 6.3. In order
to avoid problems related to the imperfect parametrisation of the efficiencies
at the DP borders, a veto of $\pm 70\,\mathrm{MeV}$ is applied around the DP,
i.e. to the phase space boundaries of the $m({{K}^{+}}{{\pi}^{-}})$,
$m({\eta_{c}}{{\pi}^{-}})$ and $m({\eta_{c}}{{K}^{+}})$ distributions. This
veto is used when determining the signal and background yields, and the
probability density functions for the background, obtained as described in
Sec. 6.2. The ${{K}^{+}}{{\pi}^{-}}$ mass resolution is $\approx
5\,\mathrm{MeV}$, which is much smaller than the $K^{*}(892)^{0}$ meson width
$\Gamma_{K^{*}(892)^{0}}\approx 50\,\mathrm{MeV}$, the narrowest contribution
to the DP; therefore, the resolution has negligible effects and is not
considered further. The amplitude fits are repeated many times with randomised
initial values to ensure the absolute minimum is found.

###  6.1 Signal and background yields

There is a non-negligible fraction of NR
${{B}^{0}}\\!\rightarrow{p}{\overline{p}}{{K}^{+}}{{\pi}^{-}}$ decays in the
region of the $\eta_{c}$ meson. In order to separate the contributions of
${{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}}$ and NR
${{B}^{0}}\\!\rightarrow{p}{\overline{p}}{{K}^{+}}{{\pi}^{-}}$ decays, a two-
dimensional (2D) UML fit to the $m({p}{\overline{p}}{{K}^{+}}{{\pi}^{-}})$ and
$m({p}{\overline{p}})$ distributions is performed in the domain
$5220<m({p}{\overline{p}}{{K}^{+}}{{\pi}^{-}})<5340\mathrm{\,Me\kern-1.00006ptV}$
and $2908<m({p}{\overline{p}})<3058\mathrm{\,Me\kern-1.00006ptV}$. These
ranges are chosen to avoid the misidentified decays reported in Sec. 4.1, and
they also define the DP fit domain. The Run 1 and 2 2D mass fits are performed
separately. The $m({p}{\overline{p}}{{K}^{+}}{{\pi}^{-}})$ distributions of
${{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}}$ signal and NR
${{B}^{0}}\\!\rightarrow{p}{\overline{p}}{{K}^{+}}{{\pi}^{-}}$ decays are
described by Hypatia functions. The $m({p}{\overline{p}}{{K}^{+}}{{\pi}^{-}})$
distribution of the combinatorial background is parametrised using an
exponential function. The $m({p}{\overline{p}})$ distribution of
${{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}}$ signal decays is
described by the same model described in Sec. 4.1. A possible component where
genuine $\eta_{c}$ mesons are combined with random kaons and pions from the PV
is investigated but found to be negligible. The ${B}^{0}$ meson mass, the
$m({p}{\overline{p}}{{K}^{+}}{{\pi}^{-}})$ resolution, the value of
$m_{\eta_{c}}$, the slopes of the exponential functions, and the yields, are
free to vary in the 2D mass fits. The $m({p}{\overline{p}})$ resolution and
the $\eta_{c}$ meson natural width are Gaussian constrained to the value
obtained in the fit to the weighted $m({p}{\overline{p}})$ distribution of
Sec. 4.1, and to the known value [8], respectively.

Table 2: Yields of the components in the 2D mass fit to the joint [$m({p}{\overline{p}}{{K}^{+}}{{\pi}^{-}})$, $m({p}{\overline{p}})$] distribution for the Run 1 and 2 subsamples. Component | Run 1 | Run 2
---|---|---
${{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}}$ | $805\hskip 2.84526pt\pm\hskip 2.84526pt48$ | $1065\hskip 2.84526pt\pm\hskip 2.84526pt56$
${{B}^{0}}\\!\rightarrow{p}{\overline{p}}{{K}^{+}}{{\pi}^{-}}$ (NR) | $234\hskip 2.84526pt\pm\hskip 2.84526pt48$ | $273\hskip 2.84526pt\pm\hskip 2.84526pt56$
Combinatorial background | $409\hskip 2.84526pt\pm\hskip 2.84526pt36$ | $498\hskip 2.84526pt\pm\hskip 2.84526pt41$

The yields of all fit components are reported in Table 2. Figure 4 shows the
result of the 2D mass fits for the Run 1 and 2 subsamples that yield a total
of approximately 2000 ${{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}}$
decays. The total yield of the
${{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}}$ component is lower
than that reported in Sec. 4.1 since the fit ranges are reduced. The goodness
of fit is validated using pseudoexperiments to determine the 2D pull, i.e. the
difference between the fit model and data divided by the uncertainty.

Figure 4: Results of the 2D mass fit to the joint
[$m({p}{\overline{p}}{{K}^{+}}{{\pi}^{-}})$, $m({p}{\overline{p}})$]
distribution for the (a) Run 1 $m({p}{\overline{p}}{{K}^{+}}{{\pi}^{-}})$
projection, (b) Run 1 $m({p}{\overline{p}})$ projection, (c) Run 2
$m({p}{\overline{p}}{{K}^{+}}{{\pi}^{-}})$ projection, and (d) Run 2
$m({p}{\overline{p}})$ projection. The legend is shown in the top left plot.

###  6.2 Parametrisation of the backgrounds

The probability density functions for the combinatorial and NR background
categories are obtained from the DP distribution of each background source,
represented with a uniformly binned 2D histogram. In order to avoid artefacts
related to the curved boundaries of the DP, the histograms are built in terms
of the Square Dalitz plot (SDP) parametrised by the variables $m^{\prime}$ and
$\theta^{\prime}$ which are defined in the range 0 to 1 and are given by

$\displaystyle m^{\prime}$
$\displaystyle\equiv\frac{1}{\pi}\arccos\left(2\frac{m({{K}^{+}}{{\pi}^{-}})-m^{\min}_{{{K}^{+}}{{\pi}^{-}}}}{m^{\max}_{{{K}^{+}}{{\pi}^{-}}}-m^{\min}_{{{K}^{+}}{{\pi}^{-}}}}-1\right),$
(15) $\displaystyle\theta^{\prime}$
$\displaystyle\equiv\frac{1}{\pi}\theta({{K}^{+}}{{\pi}^{-}}),$ (16)

where $\displaystyle m^{\max}_{{{K}^{+}}{{\pi}^{-}}}=m_{B^{0}}-m_{\eta_{c}}$,
$\displaystyle m^{\min}_{{{K}^{+}}{{\pi}^{-}}}=m_{K^{+}}+m_{\pi^{-}}$ are the
kinematic boundaries of $m({{K}^{+}}{{\pi}^{-}})$ allowed in the
${{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}}$ decay, and
$\displaystyle\theta({{K}^{+}}{{\pi}^{-}})$ is the helicity angle of the
${{K}^{+}}{{\pi}^{-}}$ system (the angle between the $K^{+}$ and the
$\eta_{c}$ mesons in the ${{K}^{+}}{{\pi}^{-}}$ rest frame).

The combinatorial and NR background histograms are filled using the weights
obtained by applying the sPlot technique to the joint
[$m({p}{\overline{p}}{{K}^{+}}{{\pi}^{-}})$, $m({p}{\overline{p}})$]
distribution, merging the Run 1 and 2 data samples. Each histogram is scaled
for the corresponding yield in the two subsamples. The combinatorial and NR
background histograms for the Run 2 subsample are shown in Fig. 5. Statistical
fluctuations in the histograms due to the limited size of the samples are
smoothed by applying a 2D cubic spline interpolation.

The 2D mass fit described in Sec. 6.1 is repeated to the combined Run 1 and 2
data sample, and the sPlot technique is applied to determine the background-
subtracted DP and SDP distributions shown in Fig. 6.

Figure 5: SDP distributions used in the DP fit to the Run 2 subsample for (a)
combinatorial background and (b) NR
${{B}^{0}}\\!\rightarrow{p}{\overline{p}}{{K}^{+}}{{\pi}^{-}}$ background.

Figure 6: Background-subtracted (top) DP and (bottom) SDP distributions
corresponding to the total data sample used in the analysis. The structure
corresponding to the $K^{*}(892)^{0}$ resonance is evident. The veto of
${{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}}$ decays in the ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ region is visible in the DP.

###  6.3 Signal efficiency

Efficiency variation across the SDP is caused by the detector acceptance and
by the trigger and offline selection requirements. The efficiency variation is
evaluated with simulated samples generated uniformly across the SDP.
Corrections are applied for known differences between data and simulation in
PID efficiencies. The effect of the vetoes in the phase space is separately
accounted for by the Laura++ package, setting to zero the signal efficiency
within the vetoed regions. Therefore, the vetoes corresponding to the ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ meson and the phase-space
border are not applied when constructing the numerator of the efficiency
histogram. The efficiency is studied separately for the Run 1 and 2
subsamples, and the resulting efficiency maps are shown in Fig. 7. Lower
efficiency in regions with a low-momentum track is due to geometrical effects.
Statistical fluctuations in the histograms due to the limited size of the
simulated samples are smoothed by applying a 2D cubic spline interpolation.

Figure 7: ${{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}}$ signal
efficiency across the SDP for the (a) Run 1 and (b) Run 2 samples.

###  6.4 Amplitude model with only ${{K}^{+}}{{\pi}^{-}}$ contributions

In the absence of contributions from exotic resonances, only
${{K}^{+}}{{\pi}^{-}}$ resonances are expected as intermediate states. The
established ${{K}^{*0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$ mesons reported in
Ref. [8] with $m({{K}^{*0}})\lesssim m({{B}^{0}})-m({\eta_{c}})$, i.e. with
masses within or slightly above the phase space boundary in
${{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}}$ decays, are used as a
guide when building the model. Only those amplitudes providing significant
improvements in the description of the data are included. This model is
referred to as the baseline model and comprises the resonances shown in Table
3.

Table 3: Resonances included in the baseline model, where parameters and uncertainties are taken from Ref. [52]. The LASS lineshape also parametrise the ${{K}^{+}}{{\pi}^{-}}$ S-wave in ${{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}}$ NR decays. Resonance |  Mass $[\mathrm{MeV}]$ |  Width $[\mathrm{MeV}]$ | $J^{P}$ | Model
---|---|---|---|---
$K^{*}(892)^{0}$ | $895.55\hskip 2.84526pt\pm\hskip 2.84526pt0.20$ | $47.3\hskip 2.84526pt\pm\hskip 2.84526pt0.5$ | $1^{-}$ | RBW
$K^{*}(1410)^{0}$ | $1414\hskip 2.84526pt\pm\hskip 2.84526pt15$ | $232\hskip 2.84526pt\pm\hskip 2.84526pt21$ | $1^{-}$ | RBW
$K^{*}_{0}(1430)^{0}$ | $1425\hskip 2.84526pt\pm\hskip 2.84526pt50$ | $270\hskip 2.84526pt\pm\hskip 2.84526pt80$ | $0^{+}$ | LASS
$K^{*}_{2}(1430)^{0}$ | $1432.4\hskip 2.84526pt\pm\hskip 2.84526pt1.3$ | $109\hskip 2.84526pt\pm\hskip 2.84526pt5$ | $2^{+}$ | RBW
$K^{*}(1680)^{0}$ | $1717\hskip 2.84526pt\pm\hskip 2.84526pt27$ | $322\hskip 2.84526pt\pm\hskip 2.84526pt110$ | $1^{-}$ | RBW
$K^{*}_{0}(1950)^{0}$ | $1945\hskip 2.84526pt\pm\hskip 2.84526pt22$ | $201\hskip 2.84526pt\pm\hskip 2.84526pt90$ | $0^{+}$ | RBW

The S-wave at low ${{K}^{+}}{{\pi}^{-}}$ mass is modelled with the LASS
probability density function. The real and imaginary parts of the complex
coefficients $c_{j}$ introduced in Eq. (4) are free parameters of the fit,
except for the $K^{*}(892)^{0}$ component, which is taken as the reference
amplitude. Other free parameters in the fit are the scattering length ($a$)
and the effective range ($r$) parameters of the LASS function, defined in Eq.
(9). The mass and width of the $K^{*}_{0}(1430)^{0}$ meson are Gaussian
constrained to the known values [8].

While it is possible to describe the $m({{K}^{+}}{{\pi}^{-}})$ and
$m({\eta_{c}}{{K}^{+}})$ distributions well with ${{K}^{+}}{{\pi}^{-}}$
contributions alone, the fit projection onto the $m({\eta_{c}}{{\pi}^{-}})$
distribution does not provide a good description of data, as shown in Fig. 8.
In particular, a discrepancy around $m({\eta_{c}}{{\pi}^{-}})\approx
4.1\,\mathrm{GeV}$ is evident.

Figure 8: Projections of the data and amplitude fit using the baseline model
onto (a) $m({{K}^{+}}{{\pi}^{-}})$, (c) $m({\eta_{c}}{{\pi}^{-}})$ and (e)
$m({\eta_{c}}{{K}^{+}})$, with the same projections shown in (b), (d) and (f)
with a logarithmic vertical-axis scale. The veto of
${{B}^{0}}\\!\rightarrow{p}{\overline{p}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ decays is visible in plot (b).
The ${{K}^{+}}{{\pi}^{-}}$ S-wave component comprises the LASS and
$K^{*}_{0}(1950)^{0}$ meson contributions. The components are described in the
legend at the bottom.

A $\chi^{2}$ variable is computed as a quantitative determination of the fit
quality, using an adaptive 2D binning schema to obtain 144 equally populated
bins. The baseline model yields a $\chi^{2}/\text{ndof}=195/129=1.5$ value,
where ndof is the number of degrees of freedom. Including additional
${{K}^{+}}{{\pi}^{-}}$ resonant states does not lead to significant
improvements in the description of the data. These include established states
such as the $K^{*}_{3}(1780)^{0}$ and $K^{*}_{4}(2045)^{0}$ mesons, the high
mass $K^{*}_{5}(2380)^{0}$ resonance which falls outside the phase space
limits, and the $K^{*}_{2}(1980)^{0}$ state which has not been seen in the
${{K}^{+}}{{\pi}^{-}}$ final state thus far. The unestablished P-, D- and
F-wave ${{K}^{+}}{{\pi}^{-}}$ states predicted by the Godfrey–Isgur model [53]
to decay into the ${{K}^{+}}{{\pi}^{-}}$ final state were also tested.

###  6.5 Amplitude model with ${{K}^{+}}{{\pi}^{-}}$ and
${\eta_{c}}{{\pi}^{-}}$ contributions

A better description of the data is obtained by adding an exotic
$Z_{c}^{-}\rightarrow{\eta_{c}}{{\pi}^{-}}$ component to the
${{K}^{+}}{{\pi}^{-}}$ contributions of Table 3. The resulting signal model
consists of eight amplitudes: seven resonances and one NR term. The
${{K}^{+}}{{\pi}^{-}}$ amplitudes are modelled in the same way as in the
baseline model. Alternative models for the ${{K}^{+}}{{\pi}^{-}}$ S-wave are
used to assign systematic uncertainties as discussed in Sec. 7. In addition to
the free parameters used in the baseline model, the isobar coefficients, mass
and width of the $Z^{-}_{c}$ resonance are left floating.

A likelihood-ratio test is used to discriminate between any pair of amplitude
models based on the log-likelihood difference $\Delta(-2\ln\mathcal{L})$ [54].
Three quantum number hypotheses are probed for the $Z_{c}^{-}$ resonance,
repeating the amplitude fit for the
$J^{P}=0^{+}~{},1^{-}~{}\text{and}~{}2^{+}$ assignments. The variations of the
$\Delta(-2\ln\mathcal{L})$ value with respect to the baseline model are
$\Delta(-2\ln\mathcal{L})=22.8,~{}41.4$, and 7.0, respectively. The model
providing the best description of the data, referred to below as the nominal
fit model, is obtained with the addition of a $Z^{-}_{c}$ candidate with
$J^{P}=1^{-}$. The $J^{P}=2^{+}$ assignment is not considered further given
the small variation in $\ln\mathcal{L}$ with respect to the additional four
free parameters.

The LASS parameters obtained in the nominal fit model are
$m_{K^{*}_{0}(1430)^{0}}=1427\pm 21\,\mathrm{MeV}$,
$\Gamma_{K^{*}_{0}(1430)^{0}}=256\pm 33\,\mathrm{MeV}$, $a=3.1\pm
1.0\,\mathrm{GeV^{-1}}$ and $r=7.0\pm 2.4\,\mathrm{GeV^{-1}}$. The parameters
of the $Z_{c}^{-}$ candidate obtained in the nominal fit model are
$m_{Z_{c}^{-}}=4096\pm 20\,\mathrm{MeV}$ and $\Gamma_{Z_{c}^{-}}=152\pm
58\,\mathrm{MeV}$. The values of the complex coefficients and fit fractions
returned by the nominal fit model are shown in Table 4. The statistical
uncertainties on all parameters of interest are calculated using large samples
of simulated pseudoexperiments generated from the fit results in order to take
into account the correlations between parameters and to guarantee the correct
coverage of the uncertainties.

Table 4: Complex coefficients and fit fractions determined from the DP fit using the nominal model. Uncertainties are statistical only. Amplitude | Real part | Imaginary part | Fit fraction (%)
---|---|---|---
$B^{0}\rightarrow\eta_{c}K^{*}(892)^{0}$ |  1 (fixed) |  0 (fixed) | $51.4\hskip 2.84526pt\pm\hskip 2.84526pt1.9$
$B^{0}\rightarrow\eta_{c}K^{*}(1410)^{0}$ | $0.17\hskip 2.84526pt\pm\hskip 2.84526pt0.07$ | $0.11\hskip 2.84526pt\pm\hskip 2.84526pt0.08$ | $2.1\hskip 2.84526pt\pm\hskip 2.84526pt1.1$
$B^{0}\rightarrow\eta_{c}K^{+}\pi^{-}$ (NR) | $-0.45\hskip 2.84526pt\pm\hskip 2.84526pt0.08$ | $0.01\hskip 2.84526pt\pm\hskip 2.84526pt0.09$ | $10.3\hskip 2.84526pt\pm\hskip 2.84526pt1.4$
$B^{0}\rightarrow\eta_{c}K^{*}_{0}(1430)^{0}$ | $-0.62\hskip 2.84526pt\pm\hskip 2.84526pt0.09$ | $-0.33\hskip 2.84526pt\pm\hskip 2.84526pt0.25$ | $25.3\hskip 2.84526pt\pm\hskip 2.84526pt3.5$
$B^{0}\rightarrow\eta_{c}K^{*}_{2}(1430)^{0}$ | $0.16\hskip 2.84526pt\pm\hskip 2.84526pt0.06$ | $-0.23\hskip 2.84526pt\pm\hskip 2.84526pt0.05$ | $4.1\hskip 2.84526pt\pm\hskip 2.84526pt1.5$
$B^{0}\rightarrow\eta_{c}K^{*}(1680)^{0}$ | $-0.11\hskip 2.84526pt\pm\hskip 2.84526pt0.08$ | $-0.18\hskip 2.84526pt\pm\hskip 2.84526pt0.06$ | $2.2\hskip 2.84526pt\pm\hskip 2.84526pt2.0$
$B^{0}\rightarrow\eta_{c}K^{*}_{0}(1950)^{0}$ | $0.27\hskip 2.84526pt\pm\hskip 2.84526pt0.04$ | $0.04\hskip 2.84526pt\pm\hskip 2.84526pt0.14$ | $3.8\hskip 2.84526pt\pm\hskip 2.84526pt1.8$
$B^{0}\rightarrow Z_{c}(4100)^{-}K^{+}$ | $-0.25\hskip 2.84526pt\pm\hskip 2.84526pt0.04$ | $-0.01\hskip 2.84526pt\pm\hskip 2.84526pt0.08$ | $3.3\hskip 2.84526pt\pm\hskip 2.84526pt1.1$

Figure 9 shows the projections of the nominal fit model and the data onto
$m({{K}^{+}}{{\pi}^{-}})$, $m({\eta_{c}}{{\pi}^{-}})$ and
$m({\eta_{c}}{{K}^{+}})$ invariant masses. A good agreement between the
nominal fit model and the data is obtained. The value of the
$\chi^{2}/\text{ndof}$ is $164/125=1.3$ for the nominal fit model. The fit
quality is further discussed in Appendix C, where a comparison is reported of
the unnormalised Legendre moments between data, the baseline and nominal
models. The 2D pull distributions for the baseline and nominal models are
reported as well.

Figure 9: Projections of the data and amplitude fit using the nominal model
onto (a) $m({{K}^{+}}{{\pi}^{-}})$, (c) $m({\eta_{c}}{{\pi}^{-}})$ and (e)
$m({\eta_{c}}{{K}^{+}})$, with the same projections shown in (b), (d) and (f)
with a logarithmic vertical-axis scale. The veto of
${{B}^{0}}\\!\rightarrow{p}{\overline{p}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ decays is visible in plot (b).
The ${{K}^{+}}{{\pi}^{-}}$ S-wave component comprises the LASS and
$K^{*}_{0}(1950)^{0}$ meson contributions. The components are described in the
legend at the bottom.

The significance of the $Z_{c}^{-}$ candidate, referred to as the
$Z_{c}(4100)^{-}$ state in the following, is evaluated from the change in the
likelihood of the fits with and without the $Z_{c}(4100)^{-}$ component,
assuming that this quantity, $\Delta(-2\ln\mathcal{L})$, follows a $\chi^{2}$
distribution with a number of degrees of freedom equal to twice the number of
free parameters in its parametrisation [17, 55, 56, 57]. This assumption takes
into account the look-elsewhere effect due to the floating mass and width of
the $Z_{c}(4100)^{-}$. The validity of this assumption is verified using
pseudoexperiments to predict the distribution of $\Delta(-2\ln\mathcal{L})$
under the no-$Z_{c}(4100)^{-}$ hypothesis, which is found to be well described
by a $\chi^{2}$ probability density function with ndof = 8\. The statistical
significance of the $Z_{c}(4100)^{-}$ is $4.8\sigma$ in the nominal fit model.
The quoted significance does not include the contribution from systematic
uncertainties.

To discriminate between various $J^{P}$ assignments, fits are performed under
alternative $J^{P}$ hypotheses. A lower limit on the significance of rejection
of the $J^{P}=0^{+}$ hypothesis is determined from the change in the log-
likelihood from the preferred hypothesis, assuming a $\chi^{2}$ distribution
with one degree of freedom. The validity of this assumption is verified using
pseudoexperiments to predict the distribution of $\Delta(-2\ln\mathcal{L})$
under the disfavoured $J^{P}=0^{+}$ hypothesis. The statistical rejection of
the $J^{P}=0^{+}$ hypothesis with respect to the $J^{P}=1^{-}$ hypothesis is
$4.3\sigma$.

Systematic effects must be taken into account to report the significance of
the $Z_{c}(4100)^{-}$ contribution and the discrimination of its quantum
numbers. The fit variations producing the largest changes in the values of the
mass, width or isobar coefficients of the exotic candidate are used to probe
the sensitivity of the significance of the $Z_{c}(4100)^{-}$ state to
systematic effects, and to determine its quantum numbers, as described in Sec.
7.

##  7 Systematic uncertainties

Systematic uncertainties can be divided into two categories: experimental and
model uncertainties. Among the experimental uncertainties, the largest changes
in the values of the parameters of the $Z_{c}(4100)^{-}$ candidate are due to
the signal and background yields used in the amplitude fit, the SDP
distributions of the background components, and the phase-space border veto
applied on the parametrisation of the efficiencies. Among the model
uncertainties, the largest effects are due to the treatment of the natural
width of the $\eta_{c}$ meson within the DP fit and to the
${{K}^{+}}{{\pi}^{-}}$ S-wave parametrisation. The DP fits using the baseline
and nominal varied models are used to recompute the significance.

The signal and background yields used in the amplitude fit are fixed to the
values obtained from the 2D mass fit. The statistical uncertainties on the
yields are introduced into the amplitude fit by Gaussian constraining the
yields within their statistical uncertainties and by repeating the fit.

The systematic uncertainties associated to the parametrisation of the
background distributions are evaluated by varying the value in each bin within
the statistical uncertainty prior to the spline interpolation. About 300 new
background histograms are produced for both the combinatorial and NR
background components. The resulting $\Delta(-2\ln\mathcal{L})$ distribution
follows a Gaussian distribution. The most pessimistic background
parametrisation, corresponding to a $\Delta(-2\ln\mathcal{L})$ value that is
below $3\sigma$ of the Gaussian distribution, is considered when quoting the
effect of this source on the significance of the $Z_{c}(4100)^{-}$ state.

The phase-space border veto applied on the parametrisation of the efficiencies
is removed to check the veto does not significantly affect the result.

The natural width of the $\eta_{c}$ meson is set to zero when computing the DP
normalisations, calculated using the $\eta_{c}$ meson mass values resulting
from the 2D UML fits described in Sec. 6.1. In order to associate a systematic
uncertainty to the sizeable $\eta_{c}$ natural width, the amplitude fits are
repeated computing the DP normalisations by using the
$m_{{\eta_{c}}}+\Gamma_{{\eta_{c}}}$ and $m_{{\eta_{c}}}-\Gamma_{{\eta_{c}}}$
values, where $m_{{\eta_{c}}}$ and $\Gamma_{{\eta_{c}}}$ are the mass and
natural width of the $\eta_{c}$ meson, respectively, obtained from the 2D UML
fits.

The LASS model used to parametrise the low mass ${{K}^{+}}{{\pi}^{-}}$ S-wave
in the nominal fit is replaced with $K^{*}_{0}(1430)^{0}$ and
$K^{*}_{0}(700)^{0}$ resonances parametrised with RBW functions, and a NR
S-wave ${{K}^{+}}{{\pi}^{-}}$ component parametrised with a uniform amplitude
within the DP.

The effect of the separate systematic sources to the significance of the
$Z_{c}(4100)^{-}$ are reported in Table 5. When including the most important
systematic effect, corresponding to the pessimistic background
parametrisation, the lowest significance for the $Z_{c}(4100)^{-}$ candidate
is given by $3.4\sigma$. In order to evaluate the effect of possible
correlated or anti-correlated sources of systematic uncertainty, the fits are
repeated using the pessimistic background parametrisation together with the
alternative ${{K}^{+}}{{\pi}^{-}}$ S-wave model, and with mass values of the
$\eta_{c}$ meson varied within the corresponding statistical uncertainty
resulting from the 2D UML fit. The lower limit on the significance of the
$Z_{c}(4100)^{-}$ state is found to be $3.2\sigma$.

Table 5: Significance of the $Z_{c}(4100)^{-}$ contribution for the systematic effects producing the largest variations in the parameters of the $Z_{c}(4100)^{-}$ candidate. The values obtained in the nominal amplitude fit are shown in the first row. Source | $\Delta(-2\ln\mathcal{L})$ | Significance
---|---|---
Nominal fit | $41.4$ | $4.8\sigma$
Fixed yields | $45.8$ | $5.2\sigma$
Phase-space border veto | $44.6$ | $5.1\sigma$
$\eta_{c}$ width | $36.6$ | $4.3\sigma$
${{K}^{+}}{{\pi}^{-}}$ S-wave | $31.8$ | $3.9\sigma$
Background | $27.4$ | $3.4\sigma$

The discrimination between the $J^{P}=0^{+}$ and $J^{P}=1^{-}$ assignments is
not significant when systematic uncertainties are taken into account, as
reported in Table 6. When the S-wave model is varied, the two spin-parity
hypotheses only differ by $1.2\sigma$.

Table 6: Rejection level of the $J^{P}=0^{+}$ hypothesis with respect to the $J^{P}=1^{-}$ hypothesis for the systematic variations producing the largest variations in the parameters of the $Z_{c}(4100)^{-}$ candidate. The values obtained in the nominal amplitude fit are shown in the first row. Source | $\Delta(-2\ln\mathcal{L})$ | Significance
---|---|---
Default | $18.6$ | $4.3\sigma$
Fixed yields | $23.8$ | $4.9\sigma$
Phase-space border veto | $24.4$ | $4.9\sigma$
$\eta_{c}$ width | $4.2$ | $2.0\sigma$
Background | $3.4$ | $1.8\sigma$
${{K}^{+}}{{\pi}^{-}}$ S-wave | $1.4$ | $1.2\sigma$

Additional sources of systematic uncertainties are considered when evaluating
the uncertainty on the mass and width of the $Z_{c}(4100)^{-}$ resonance, and
on the fit fractions obtained with the nominal model. These additional sources
are: the efficiency variation across the SDP and a possible bias due to the
fitting procedure, contributing to the experimental systematic uncertainties
category; and the fixed parameters of the resonances in the amplitude model
and the addition or removal of marginal amplitudes, contributing to the model
systematic uncertainties category. For each source, the systematic uncertainty
assigned to each quantity is taken as the difference between the value
returned by the modified amplitude fit and nominal model fit result. The
uncertainties due to all these sources are obtained by combining positive and
negative deviations in quadrature separately.

The bin contents of the histograms describing the efficiency variation across
the SDP are varied within their uncertainties prior to the spline
interpolation, as is done for the systematic uncertainty associated to the
background parametrisations. A possible source of systematic effects in the
efficiency histograms is due to neighbouring bins varying in a correlated way.
In order to evaluate this systematic uncertainty, 10 bins of the efficiency
histograms are varied within their statistical uncertainty, and the
neighbouring bins are varied by linear interpolation. The binning scheme of
the control sample used to evaluate the PID performance is varied.

Pseudoexperiments are generated from the fit results using the nominal model
in order to assign a systematic uncertainty due to possible amplitude fit
bias.

Systematic uncertainties due to fixed parameters in the fit model are
determined by repeating the fit and varying these parameters. The fixed masses
and widths of the ${{K}^{+}}{{\pi}^{-}}$ contributions are varied 100 times
assigning a random number within the range defined by the corresponding
uncertainties reported in Table 3. The Blatt–Weisskopf barrier radii,
$r_{\text{BW}}$, are varied independently for ${{K}^{+}}{{\pi}^{-}}$ and
${\eta_{c}}{{\pi}^{-}}$ resonances between $3$ and $5\,\mathrm{GeV^{-1}}$.

Systematic uncertainties are assigned from the changes in the results when the
amplitudes due to the established $K^{*}_{3}(1780)^{0}$ and
$K^{*}_{4}(2045)^{0}$ resonances, not contributing significantly in the
baseline and nominal models, are included.

The total systematic uncertainties for the fit fractions are given together
with the results in Sec. 8. The dominant experimental systematic uncertainty
is due to either the phase-space border veto, related to the efficiency
parametrisation, or the background distributions across the SDP, while the
model uncertainties are dominated by the description of the
${{K}^{+}}{{\pi}^{-}}$ S-wave.

The stability of the fit results is confirmed by several cross-checks. The
addition of further high-mass $K^{*0}$ states to the nominal model does not
improve the quality of the fit. An additional amplitude decaying to
${\eta_{c}}{{\pi}^{-}}$ is not significant, nor is an additional exotic
amplitude decaying to ${\eta_{c}}{{K}^{+}}$. The $\eta_{c}$ meson resonant
phase motion due to the sizeable natural width could affect the overall
amplitude of Eq. (4), introducing interference effects with the NR $p\bar{p}$
contribution. In order to investigate this effect, the data sample is divided
in two parts, containing candidates with masses below and above the $\eta_{c}$
meson peak, respectively. The results are compatible with those reported in
Sec. 6 using the full data sample, supporting the argument that the effects
due to the variation of the $\eta_{c}$ phase are negligible.

##  8 Results and summary

In summary, the first measurement of the
${{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}}$ branching fraction is
reported and gives

$\mathcal{B}({{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}})=(5.73\pm
0.24\pm 0.13\pm 0.66)\times 10^{-4},$

where the first uncertainty is statistical, the second systematic, and the
third is due to limited knowledge of external branching fractions. The first
Dalitz plot analysis of the
${{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}}$ decay is performed. A
good description of data is obtained when including a charged charmonium-like
resonance decaying to the ${\eta_{c}}{{\pi}^{-}}$ final state with
$m_{Z_{c}^{-}}=4096\pm 20~{}^{+18}_{-22}\,\mathrm{MeV}$ and
$\Gamma_{Z_{c}^{-}}=152\pm 58~{}^{+60}_{-35}\,\mathrm{MeV}$. The fit fractions
are reported in Table 7. The fit fractions for resonant and nonresonant
contributions are converted into quasi-two-body branching fractions by
multiplying by the ${{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}}$
branching fraction. The corresponding results are shown in Table 8. The
$B^{0}\rightarrow\eta_{c}K^{*}(892)^{0}$ branching fraction is compatible with
the world-average value [8], taking into account the
$K^{*}(892)^{0}\rightarrow K^{+}\pi^{-}$ branching fraction. The values of the
interference fit fractions are given in Table 9.

The significance of the $Z_{c}(4100)^{-}$ candidate is more than three
standard deviations when including systematic uncertainties. This is the first
evidence for an exotic state decaying into two pseudoscalars. The favoured
spin-parity assignments, $J^{P}=0^{+}$ and $J^{P}=1^{-}$, cannot be
discriminated once systematic uncertainties are taken into account, which
prohibits unambiguously assigning the $Z_{c}(4100)^{-}$ as one of the states
foreseen by the models described in Sec. 1. Furthermore, the mass value of the
$Z_{c}(4100)^{-}$ charmonium-like state is above the open-charm threshold, in
contrast with the predictions of such models. More data will be required to
conclusively determine the nature of the $Z_{c}(4100)^{-}$ candidate.

Table 7: Fit fractions and their uncertainties. The quoted uncertainties are statistical and systematic, respectively. Amplitude |  Fit fraction (%)
---|---
$B^{0}\rightarrow\eta_{c}K^{*}(892)^{0}$ | $51.4\hskip 2.84526pt\pm\hskip 2.84526pt1.9~{}^{+1.7}_{-4.8}$
$B^{0}\rightarrow\eta_{c}K^{*}(1410)^{0}$ | $2.1\hskip 2.84526pt\pm\hskip 2.84526pt1.1~{}^{+1.1}_{-1.1}$
$B^{0}\rightarrow\eta_{c}K^{+}\pi^{-}$ (NR) | $10.3\hskip 2.84526pt\pm\hskip 2.84526pt1.4~{}^{+1.0}_{-1.2}$
$B^{0}\rightarrow\eta_{c}K^{*}_{0}(1430)^{0}$ | $25.3\hskip 2.84526pt\pm\hskip 2.84526pt3.5~{}^{+3.5}_{-2.8}$
$B^{0}\rightarrow\eta_{c}K^{*}_{2}(1430)^{0}$ | $4.1\hskip 2.84526pt\pm\hskip 2.84526pt1.5~{}^{+1.0}_{-1.6}$
$B^{0}\rightarrow\eta_{c}K^{*}(1680)^{0}$ | $2.2\hskip 2.84526pt\pm\hskip 2.84526pt2.0~{}^{+1.5}_{-1.7}$
$B^{0}\rightarrow\eta_{c}K^{*}_{0}(1950)^{0}$ | $3.8\hskip 2.84526pt\pm\hskip 2.84526pt1.8~{}^{+1.4}_{-2.5}$
$B^{0}\rightarrow Z_{c}(4100)^{-}K^{+}$ | $3.3\hskip 2.84526pt\pm\hskip 2.84526pt1.1~{}^{+1.2}_{-1.1}$
Table 8: Branching fraction results. The four quoted uncertainties are statistical, ${{B}^{0}}\\!\rightarrow{\eta_{c}}{{K}^{+}}{{\pi}^{-}}$ branching fraction systematic (not including the contribution from the uncertainty associated to the efficiency ratio, to avoid double counting the systematic uncertainty associated to the evaluation of the efficiencies), fit fraction systematic and external branching fractions uncertainties, respectively. Decay mode |  Branching fraction ($10^{-5}$)
---|---
$B^{0}\rightarrow\eta_{c}K^{*}(892)^{0}(\rightarrow K^{+}\pi^{-})$ | $29.5\hskip 2.84526pt\pm\hskip 5.69054pt1.6\hskip 2.84526pt\pm\hskip 2.84526pt0.6\hskip 9.95845pt^{+1.0}_{-2.8}\hskip 9.95845pt\pm 3.4$
$B^{0}\rightarrow\eta_{c}K^{*}(1410)^{0}(\rightarrow K^{+}\pi^{-})$ | $1.20\hskip 2.84526pt\pm\hskip 2.84526pt0.63\pm 0.02\pm 0.63\pm 0.14$
$B^{0}\rightarrow\eta_{c}K^{+}\pi^{-}$ (NR) | $5.90\hskip 2.84526pt\pm\hskip 2.84526pt0.84\pm 0.11\hskip 7.11317pt^{+0.57}_{-0.69}\hskip 5.69054pt\pm 0.68$
$B^{0}\rightarrow\eta_{c}K^{*}_{0}(1430)^{0}(\rightarrow K^{+}\pi^{-})$ | $14.50\hskip 2.84526pt\pm\hskip 2.84526pt2.10\pm 0.28\hskip 7.11317pt^{+2.01}_{-1.60}\hskip 5.69054pt\pm 1.67$
$B^{0}\rightarrow\eta_{c}K^{*}_{2}(1430)^{0}(\rightarrow K^{+}\pi^{-})$ | $2.35\hskip 2.84526pt\pm\hskip 2.84526pt0.87\pm 0.05\hskip 7.11317pt^{+0.57}_{-0.92}\hskip 5.69054pt\pm 0.27$
$B^{0}\rightarrow\eta_{c}K^{*}(1680)^{0}(\rightarrow K^{+}\pi^{-})$ | $1.26\hskip 2.84526pt\pm\hskip 2.84526pt1.15\pm 0.02\hskip 4.2679pt~{}^{+0.86}_{-0.97}\hskip 4.2679pt\pm 0.15$
$B^{0}\rightarrow\eta_{c}K^{*}_{0}(1950)^{0}(\rightarrow K^{+}\pi^{-})$ | $2.18\hskip 2.84526pt\pm\hskip 2.84526pt1.04\pm 0.04\hskip 7.11317pt^{+0.80}_{-1.43}\hskip 5.69054pt\pm 0.25$
$B^{0}\rightarrow Z_{c}(4100)^{-}K^{+}$ | $1.89\hskip 2.84526pt\pm\hskip 2.84526pt0.64\pm 0.04\hskip 7.11317pt^{+0.69}_{-0.63}\hskip 5.69054pt\pm 0.22$

Table 9: Symmetric matrix of the fit fractions (%) from the amplitude fit using the nominal model. The quoted uncertainties are statistical and systematic, respectively. The diagonal elements correspond to the values reported in Table 7. | $K^{*}(892)^{0}$ | $K^{*}(1410)^{0}$ | LASS NR | $K^{*}_{0}(1430)^{0}$ | $K^{*}_{2}(1430)^{0}$ | $K^{*}(1680)^{0}$ | $K^{*}_{0}(1950)^{0}$ | $Z_{c}(4100)^{-}$
---|---|---|---|---|---|---|---|---
$K^{*}(892)^{0}$ | $51.4\pm 1.9~{}^{+1.7}_{-4.8}$ | $1.7\pm 1.9~{}^{+2.4}_{-1.4}$ | $0$ | $0$ | $0$ | $-2.1\pm 1.1~{}^{+1.4}_{-1.5}$ | $0$ | $1.4\pm 1.0~{}^{+1.2}_{-1.1}$
$K^{*}(1410)^{0}$ |  | $2.1\pm 1.1~{}^{+1.1}_{-1.1}$ | $0$ | $0$ | $0$ | $-2.5\pm 1.6~{}^{+1.9}_{-1.7}$ | $0$ | $-0.4\pm 0.4~{}^{+0.7}_{-0.5}$
LASS NR |  |  | $10.3\pm 1.4~{}^{+1.0}_{-1.2}$ | $-5.8\pm 1.3~{}^{+2.2}_{-2.0}$ | $0$ | $0$ | $-3.2\pm 2.8~{}^{+4.9}_{-1.4}$ | $1.11\pm 0.23~{}^{+0.54}_{-0.35}$
$K^{*}_{0}(1430)^{0}$ |  |  |  | $25.3\pm 3.5~{}^{+3.5}_{-2.8}$ | $0$ | $0$ | $4.7\pm 0.7~{}^{+1.3}_{-1.5}$ | $2.8\pm 0.4~{}^{+0.6}_{-0.4}$
$K^{*}_{2}(1430)^{0}$ |  |  |  |  | $4.1\pm 1.5~{}^{+1.0}_{-1.6}$ | $0$ | $0$ | $0.00\pm 0.31~{}^{+0.76}_{-0.26}$
$K^{*}(1680)^{0}$ |  |  |  |  |  | $2.2\pm 2.0~{}^{+1.5}_{-1.7}$ | $0$ | $0.7\pm 0.5~{}^{+0.5}_{-0.9}$
$K^{*}_{0}(1950)^{0}$ |  |  |  |  |  |  | $3.8\pm 1.8~{}^{+1.4}_{-2.5}$ | $0.6\pm 0.5~{}^{+0.8}_{-1.1}$
$Z_{c}(4100)^{-}$ |  |  |  |  |  |  |  | $3.3\pm 1.1~{}^{+1.2}_{-1.1}$

## Acknowledgements

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

Appendix

##  Appendix A Blatt-Weisskopf barrier factors

The Blatt–Weisskopf barrier factors [45] $X(z)$, where
$z=|\vec{q}|r_{\text{BW}}$ or $|\vec{p}|r_{\text{BW}}$ with $r_{\text{BW}}$
being the barrier radius, are given by

$\displaystyle L=0:~{}X(z)=1,$ (17) $\displaystyle
L=1:~{}X(z)=\sqrt{\frac{1+z_{0}^{2}}{1+z^{2}}},$ (18) $\displaystyle
L=2:~{}X(z)=\sqrt{\frac{z_{0}^{4}+3z_{0}^{2}+9}{z^{4}+3z^{2}+9}},$ (19)
$\displaystyle
L=3:~{}X(z)=\sqrt{\frac{z_{0}^{6}+6z_{0}^{4}+45z_{0}^{2}+225}{z^{6}+6z^{4}+45z^{2}+225}},$
(20) $\displaystyle
L=4:~{}X(z)=\sqrt{\frac{z_{0}^{8}+10z_{0}^{6}+135z_{0}^{4}+1575z_{0}^{2}+11025}{z^{8}+10z^{6}+135z^{4}+1575z^{2}+11025}},$
(21)

where $z_{0}$ is the value of $z$ when the invariant mass is equal to the pole
mass of the resonance and $L$ is the orbital angular momentum between the
resonance children. Since the latter are scalars, $L$ is equal to the spin of
the resonance. Since the ${B}^{0}$ meson and the accompanying particle in the
decay are scalars as well, $L$ is also equal to the orbital angular momentum
between the resonance and the accompanying particle in the decay.

##  Appendix B Angular probability distributions

Using the Zemach tensor formalism [46, 47], the angular probability
distributions $Z(\vec{p},\vec{q})$ are given by

$\displaystyle L=0:~{}Z(\vec{p},\vec{q})=1,$ (22) $\displaystyle
L=1:~{}Z(\vec{p},\vec{q})=-2\vec{p}\cdot\vec{q},$ (23) $\displaystyle
L=2:~{}Z(\vec{p},\vec{q})=\frac{4}{3}\left[3(\vec{p}\cdot\vec{q})^{2}-(|\vec{p}||\vec{q}|)^{2}\right],$
(24) $\displaystyle
L=3:~{}Z(\vec{p},\vec{q})=-\frac{8}{5}\left[5(\vec{p}\cdot\vec{q})^{3}-3(\vec{p}\cdot\vec{q})(|\vec{p}||\vec{q}|)^{2}\right],$
(25) $\displaystyle
L=4:~{}Z(\vec{p},\vec{q})=\frac{16}{35}\left[35(\vec{p}\cdot\vec{q})^{4}-30(\vec{p}\cdot\vec{q})^{2}(|\vec{p}||\vec{q}|)^{2}+3\right(|\vec{p}||\vec{q}|)^{4}],$
(26)

##  Appendix C Investigation of the fit quality

Comparisons of the first four Legendre moments determined from background-
subtracted data and from the amplitude fit results using the baseline and
nominal model are reported in Figs. 10, 11 and 12 for the
$m({{K}^{+}}{{\pi}^{-}})$, $m({\eta_{c}}{{\pi}^{-}})$ and
$m({\eta_{c}}{{K}^{+}})$ projections, respectively.
The 2D pull distributions for the baseline and nominal models are reported in
Figs. 13 and 14, respectively.

Figure 10: Comparison of the first four ${{K}^{+}}{{\pi}^{-}}$ Legendre
moments determined from background-subtracted data (black points) and from the
results of the amplitude fit using the baseline model (red triangles) and
nominal model (blue triangles) as a function of $m({{K}^{+}}{{\pi}^{-}})$.

Figure 11: Comparison of the first four ${\eta_{c}}{{\pi}^{-}}$ Legendre
moments determined from background-subtracted data (black points) and from the
results of the amplitude fit using the baseline model (red triangles) and
nominal model (blue triangles) as a function of $m({\eta_{c}}{{\pi}^{-}})$.

Figure 12: Comparison of the first four ${\eta_{c}}{{K}^{+}}$ Legendre moments
determined from background-subtracted data (black points) and from the results
of the amplitude fit using the baseline model (red triangles) and nominal
model (blue triangles) as a function of $m({\eta_{c}}{{K}^{+}})$. Figure 13:
2D pull distribution for to the baseline model. Figure 14: 2D pull
distribution for to the nominal model.

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LHCb collaboration

R. Aaij28, C. Abellán Beteta45, B. Adeva42, M. Adinolfi49, C.A. Aidala77, Z.
Ajaltouni6, S. Akar60, P. Albicocco19, J. Albrecht11, F. Alessio43, M.
Alexander54, A. Alfonso Albero41, G. Alkhazov34, P. Alvarez Cartelle56, A.A.
Alves Jr42, S. Amato2, S. Amerio24, Y. Amhis8, L. An3, L. Anderlini18, G.
Andreassi44, M. Andreotti17, J.E. Andrews61, R.B. Appleby57, F. Archilli28, P.
d’Argent13, J. Arnau Romeu7, A. Artamonov40, M. Artuso62, K. Arzymatov38, E.
Aslanides7, M. Atzeni45, B. Audurier23, S. Bachmann13, J.J. Back51, S.
Baker56, V. Balagura8,b, W. Baldini17, A. Baranov38, R.J. Barlow57, S.
Barsuk8, W. Barter57, F. Baryshnikov73, V. Batozskaya32, B. Batsukh62, A.
Battig11, V. Battista44, A. Bay44, J. Beddow54, F. Bedeschi25, I. Bediaga1, A.
Beiter62, L.J. Bel28, S. Belin23, N. Beliy65, V. Bellee44, N. Belloli21,i, K.
Belous40, I. Belyaev35, E. Ben-Haim9, G. Bencivenni19, S. Benson28, S.
Beranek10, A. Berezhnoy36, R. Bernet45, D. Berninghoff13, E. Bertholet9, A.
Bertolin24, C. Betancourt45, F. Betti16,43, M.O. Bettler50, M. van Beuzekom28,
Ia. Bezshyiko45, S. Bhasin49, J. Bhom30, S. Bifani48, P. Billoir9, A.
Birnkraut11, A. Bizzeti18,u, M. Bjørn58, M.P. Blago43, T. Blake51, F. Blanc44,
S. Blusk62, D. Bobulska54, V. Bocci27, O. Boente Garcia42, T. Boettcher59, A.
Bondar39,w, N. Bondar34, S. Borghi57,43, M. Borisyak38, M. Borsato42, F.
Bossu8, M. Boubdir10, T.J.V. Bowcock55, C. Bozzi17,43, S. Braun13, M.
Brodski43, J. Brodzicka30, A. Brossa Gonzalo51, D. Brundu23,43, E. Buchanan49,
A. Buonaura45, C. Burr57, A. Bursche23, J. Buytaert43, W. Byczynski43, S.
Cadeddu23, H. Cai67, R. Calabrese17,g, R. Calladine48, M. Calvi21,i, M. Calvo
Gomez41,m, A. Camboni41,m, P. Campana19, D.H. Campora Perez43, L. Capriotti16,
A. Carbone16,e, G. Carboni26, R. Cardinale20, A. Cardini23, P. Carniti21,i, L.
Carson53, K. Carvalho Akiba2, G. Casse55, L. Cassina21, M. Cattaneo43, G.
Cavallero20, R. Cenci25,p, D. Chamont8, M.G. Chapman49, M. Charles9, Ph.
Charpentier43, G. Chatzikonstantinidis48, M. Chefdeville5, V. Chekalina38, C.
Chen3, S. Chen23, S.-G. Chitic43, V. Chobanova42, M. Chrzaszcz43, A.
Chubykin34, P. Ciambrone19, X. Cid Vidal42, G. Ciezarek43, P.E.L. Clarke53, M.
Clemencic43, H.V. Cliff50, J. Closier43, V. Coco43, J.A.B. Coelho8, J. Cogan7,
E. Cogneras6, L. Cojocariu33, P. Collins43, T. Colombo43, A. Comerma-
Montells13, A. Contu23, G. Coombs43, S. Coquereau41, G. Corti43, M. Corvo17,g,
C.M. Costa Sobral51, B. Couturier43, G.A. Cowan53, D.C. Craik59, A.
Crocombe51, M. Cruz Torres1, R. Currie53, C. D’Ambrosio43, F. Da Cunha
Marinho2, C.L. Da Silva78, E. Dall’Occo28, J. Dalseno49, A. Danilina35, A.
Davis3, O. De Aguiar Francisco43, K. De Bruyn43, S. De Capua57, M. De Cian44,
J.M. De Miranda1, L. De Paula2, M. De Serio15,d, P. De Simone19, C.T. Dean54,
D. Decamp5, L. Del Buono9, B. Delaney50, H.-P. Dembinski12, M. Demmer11, A.
Dendek31, D. Derkach38, O. Deschamps6, F. Desse8, F. Dettori55, B. Dey68, A.
Di Canto43, P. Di Nezza19, S. Didenko73, H. Dijkstra43, F. Dordei43, M.
Dorigo43,x, A. Dosil Suárez42, L. Douglas54, A. Dovbnya46, K. Dreimanis55, L.
Dufour28, G. Dujany9, P. Durante43, J.M. Durham78, D. Dutta57, R.
Dzhelyadin40, M. Dziewiecki13, A. Dziurda30, A. Dzyuba34, S. Easo52, U.
Egede56, V. Egorychev35, S. Eidelman39,w, S. Eisenhardt53, U. Eitschberger11,
R. Ekelhof11, L. Eklund54, S. Ely62, A. Ene33, S. Escher10, S. Esen28, T.
Evans60, A. Falabella16, N. Farley48, S. Farry55, D. Fazzini21,43,i, L.
Federici26, P. Fernandez Declara43, A. Fernandez Prieto42, F. Ferrari16, L.
Ferreira Lopes44, F. Ferreira Rodrigues2, M. Ferro-Luzzi43, S. Filippov37,
R.A. Fini15, M. Fiorini17,g, M. Firlej31, C. Fitzpatrick44, T. Fiutowski31, F.
Fleuret8,b, M. Fontana43, F. Fontanelli20,h, R. Forty43, V. Franco Lima55, M.
Frank43, C. Frei43, J. Fu22,q, W. Funk43, C. Färber43, M. Féo Pereira Rivello
Carvalho28, E. Gabriel53, A. Gallas Torreira42, D. Galli16,e, S. Gallorini24,
S. Gambetta53, Y. Gan3, M. Gandelman2, P. Gandini22, Y. Gao3, L.M. Garcia
Martin76, B. Garcia Plana42, J. García Pardiñas45, J. Garra Tico50, L.
Garrido41, D. Gascon41, C. Gaspar43, L. Gavardi11, G. Gazzoni6, D. Gerick13,
E. Gersabeck57, M. Gersabeck57, T. Gershon51, D. Gerstel7, Ph. Ghez5, S.
Gianì44, V. Gibson50, O.G. Girard44, P. Gironella Gironell41, L. Giubega33, K.
Gizdov53, V.V. Gligorov9, D. Golubkov35, A. Golutvin56,73, A. Gomes1,a, I.V.
Gorelov36, C. Gotti21,i, E. Govorkova28, J.P. Grabowski13, R. Graciani Diaz41,
L.A. Granado Cardoso43, E. Graugés41, E. Graverini45, G. Graziani18, A.
Grecu33, R. Greim28, P. Griffith23, L. Grillo57, L. Gruber43, B.R. Gruberg
Cazon58, O. Grünberg70, C. Gu3, E. Gushchin37, A. Guth10, Yu. Guz40,43, T.
Gys43, C. Göbel64, T. Hadavizadeh58, C. Hadjivasiliou6, G. Haefeli44, C.
Haen43, S.C. Haines50, B. Hamilton61, X. Han13, T.H. Hancock58, S. Hansmann-
Menzemer13, N. Harnew58, S.T. Harnew49, T. Harrison55, C. Hasse43, M. Hatch43,
J. He65, M. Hecker56, K. Heinicke11, A. Heister11, K. Hennessy55, L. Henry76,
E. van Herwijnen43, J. Heuel10, M. Heß70, A. Hicheur63, R. Hidalgo Charman57,
D. Hill58, M. Hilton57, P.H. Hopchev44, W. Hu68, W. Huang65, Z.C. Huard60, W.
Hulsbergen28, T. Humair56, M. Hushchyn38, D. Hutchcroft55, D. Hynds28, P.
Ibis11, M. Idzik31, P. Ilten48, K. Ivshin34, R. Jacobsson43, J. Jalocha58, E.
Jans28, A. Jawahery61, F. Jiang3, M. John58, D. Johnson43, C.R. Jones50, C.
Joram43, B. Jost43, N. Jurik58, S. Kandybei46, M. Karacson43, J.M. Kariuki49,
S. Karodia54, N. Kazeev38, M. Kecke13, F. Keizer50, M. Kelsey62, M. Kenzie50,
T. Ketel29, E. Khairullin38, B. Khanji43, C. Khurewathanakul44, K.E. Kim62, T.
Kirn10, S. Klaver19, K. Klimaszewski32, T. Klimkovich12, S. Koliiev47, M.
Kolpin13, R. Kopecna13, P. Koppenburg28, I. Kostiuk28, S. Kotriakhova34, M.
Kozeiha6, L. Kravchuk37, M. Kreps51, F. Kress56, P. Krokovny39,w, W. Krupa31,
W. Krzemien32, W. Kucewicz30,l, M. Kucharczyk30, V. Kudryavtsev39,w, A.K.
Kuonen44, T. Kvaratskheliya35,43, D. Lacarrere43, G. Lafferty57, A. Lai23, D.
Lancierini45, G. Lanfranchi19, C. Langenbruch10, T. Latham51, C. Lazzeroni48,
R. Le Gac7, A. Leflat36, J. Lefrançois8, R. Lefèvre6, F. Lemaitre43, O.
Leroy7, T. Lesiak30, B. Leverington13, P.-R. Li65, T. Li3, Y. Li4, Z. Li62, X.
Liang62, T. Likhomanenko72, R. Lindner43, F. Lionetto45, V. Lisovskyi8, G.
Liu66, X. Liu3, D. Loh51, A. Loi23, I. Longstaff54, J.H. Lopes2, G.H.
Lovell50, D. Lucchesi24,o, M. Lucio Martinez42, A. Lupato24, E. Luppi17,g, O.
Lupton43, A. Lusiani25, X. Lyu65, F. Machefert8, F. Maciuc33, V. Macko44, P.
Mackowiak11, S. Maddrell-Mander49, O. Maev34,43, K. Maguire57, D.
Maisuzenko34, M.W. Majewski31, S. Malde58, B. Malecki30, A. Malinin72, T.
Maltsev39,w, G. Manca23,f, G. Mancinelli7, D. Marangotto22,q, J. Maratas6,v,
J.F. Marchand5, U. Marconi16, C. Marin Benito8, M. Marinangeli44, P. Marino44,
J. Marks13, P.J. Marshall55, G. Martellotti27, M. Martin7, M. Martinelli43, D.
Martinez Santos42, F. Martinez Vidal76, A. Massafferri1, M. Materok10, R.
Matev43, A. Mathad51, Z. Mathe43, C. Matteuzzi21, A. Mauri45, E. Maurice8,b,
B. Maurin44, A. Mazurov48, M. McCann56,43, A. McNab57, R. McNulty14, J.V.
Mead55, B. Meadows60, C. Meaux7, N. Meinert70, D. Melnychuk32, M. Merk28, A.
Merli22,q, E. Michielin24, D.A. Milanes69, E. Millard51, M.-N. Minard5, L.
Minzoni17,g, D.S. Mitzel13, A. Mogini9, R.D. Moise56, J. Molina Rodriguez1,y,
T. Mombächer11, I.A. Monroy69, S. Monteil6, M. Morandin24, G. Morello19, M.J.
Morello25,t, O. Morgunova72, J. Moron31, A.B. Morris7, R. Mountain62, F.
Muheim53, M. Mulder28, C.H. Murphy58, D. Murray57, A. Mödden 11, D. Müller43,
J. Müller11, K. Müller45, V. Müller11, P. Naik49, T. Nakada44, R.
Nandakumar52, A. Nandi58, T. Nanut44, I. Nasteva2, M. Needham53, N. Neri22, S.
Neubert13, N. Neufeld43, M. Neuner13, R. Newcombe56, T.D. Nguyen44, C. Nguyen-
Mau44,n, S. Nieswand10, R. Niet11, N. Nikitin36, A. Nogay72, N.S. Nolte43,
D.P. O’Hanlon16, A. Oblakowska-Mucha31, V. Obraztsov40, S. Ogilvy19, R.
Oldeman23,f, C.J.G. Onderwater71, A. Ossowska30, J.M. Otalora Goicochea2, P.
Owen45, A. Oyanguren76, P.R. Pais44, T. Pajero25,t, A. Palano15, M. Palutan19,
G. Panshin75, A. Papanestis52, M. Pappagallo53, L.L. Pappalardo17,g, W.
Parker61, C. Parkes57, G. Passaleva18,43, A. Pastore15, M. Patel56, C.
Patrignani16,e, A. Pearce43, A. Pellegrino28, G. Penso27, M. Pepe Altarelli43,
S. Perazzini43, D. Pereima35, P. Perret6, L. Pescatore44, K. Petridis49, A.
Petrolini20,h, A. Petrov72, S. Petrucci53, M. Petruzzo22,q, B. Pietrzyk5, G.
Pietrzyk44, M. Pikies30, M. Pili58, D. Pinci27, J. Pinzino43, F. Pisani43, A.
Piucci13, V. Placinta33, S. Playfer53, J. Plews48, M. Plo Casasus42, F.
Polci9, M. Poli Lener19, A. Poluektov51, N. Polukhina73,c, I. Polyakov62, E.
Polycarpo2, G.J. Pomery49, S. Ponce43, A. Popov40, D. Popov48,12, S.
Poslavskii40, C. Potterat2, E. Price49, J. Prisciandaro42, C. Prouve49, V.
Pugatch47, A. Puig Navarro45, H. Pullen58, G. Punzi25,p, W. Qian65, J. Qin65,
R. Quagliani9, B. Quintana6, B. Rachwal31, J.H. Rademacker49, M. Rama25, M.
Ramos Pernas42, M.S. Rangel2, F. Ratnikov38,ab, G. Raven29, M. Ravonel
Salzgeber43, M. Reboud5, F. Redi44, S. Reichert11, A.C. dos Reis1, F. Reiss9,
C. Remon Alepuz76, Z. Ren3, V. Renaudin8, S. Ricciardi52, S. Richards49, K.
Rinnert55, P. Robbe8, A. Robert9, A.B. Rodrigues44, E. Rodrigues60, J.A.
Rodriguez Lopez69, M. Roehrken43, S. Roiser43, A. Rollings58, V.
Romanovskiy40, A. Romero Vidal42, M. Rotondo19, M.S. Rudolph62, T. Ruf43, J.
Ruiz Vidal76, J.J. Saborido Silva42, N. Sagidova34, B. Saitta23,f, V.
Salustino Guimaraes64, C. Sanchez Gras28, C. Sanchez Mayordomo76, B. Sanmartin
Sedes42, R. Santacesaria27, C. Santamarina Rios42, M. Santimaria19,43, E.
Santovetti26,j, G. Sarpis57, A. Sarti19,k, C. Satriano27,s, A. Satta26, M.
Saur65, D. Savrina35,36, S. Schael10, M. Schellenberg11, M. Schiller54, H.
Schindler43, M. Schmelling12, T. Schmelzer11, B. Schmidt43, O. Schneider44, A.
Schopper43, H.F. Schreiner60, M. Schubiger44, M.H. Schune8, R. Schwemmer43, B.
Sciascia19, A. Sciubba27,k, A. Semennikov35, E.S. Sepulveda9, A. Sergi48,43,
N. Serra45, J. Serrano7, L. Sestini24, A. Seuthe11, P. Seyfert43, M.
Shapkin40, Y. Shcheglov34,†, T. Shears55, L. Shekhtman39,w, V. Shevchenko72,
E. Shmanin73, B.G. Siddi17, R. Silva Coutinho45, L. Silva de Oliveira2, G.
Simi24,o, S. Simone15,d, I. Skiba17, N. Skidmore13, T. Skwarnicki62, M.W.
Slater48, J.G. Smeaton50, E. Smith10, I.T. Smith53, M. Smith56, M. Soares16,
l. Soares Lavra1, M.D. Sokoloff60, F.J.P. Soler54, B. Souza De Paula2, B.
Spaan11, E. Spadaro Norella22,q, P. Spradlin54, F. Stagni43, M. Stahl13, S.
Stahl43, P. Stefko44, S. Stefkova56, O. Steinkamp45, S. Stemmle13, O.
Stenyakin40, M. Stepanova34, H. Stevens11, A. Stocchi8, S. Stone62, B.
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S. Strokov75, J. Sun3, L. Sun67, K. Swientek31, A. Szabelski32, T. Szumlak31,
M. Szymanski65, S. T’Jampens5, Z. Tang3, A. Tayduganov7, T. Tekampe11, G.
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R. Tourinho Jadallah Aoude1, E. Tournefier5, M. Traill54, M.T. Tran44, A.
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T. Williams48, F.F. Wilson52, M. Winn8, J. Wishahi11, W. Wislicki32, M.
Witek30, G. Wormser8, S.A. Wotton50, K. Wyllie43, D. Xiao68, Y. Xie68, A. Xu3,
M. Xu68, Q. Xu65, Z. Xu3, Z. Xu5, Z. Yang3, Z. Yang61, Y. Yao62, L.E.
Yeomans55, H. Yin68, J. Yu68,aa, X. Yuan62, O. Yushchenko40, K.A. Zarebski48,
M. Zavertyaev12,c, D. Zhang68, L. Zhang3, W.C. Zhang3,z, Y. Zhang8, A.
Zhelezov13, Y. Zheng65, X. Zhu3, V. Zhukov10,36, J.B. Zonneveld53, S.
Zucchelli16.

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

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kUniversità di Roma La Sapienza, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
nHanoi University of Science, Hanoi, Vietnam
oUniversità di Padova, Padova, Italy
pUniversità di Pisa, Pisa, Italy
qUniversità degli Studi di Milano, Milano, Italy
rUniversità di Urbino, Urbino, Italy
sUniversità della Basilicata, Potenza, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità di Modena e Reggio Emilia, Modena, Italy
vMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
wNovosibirsk State University, Novosibirsk, Russia
xSezione INFN di Trieste, Trieste, Italy
yEscuela Agrícola Panamericana, San Antonio de Oriente, Honduras
zSchool of Physics and Information Technology, Shaanxi Normal University
(SNNU), Xi’an, China
aaPhysics and Micro Electronic College, Hunan University, Changsha City, China
abNational Research University Higher School of Economics, Moscow, Russia
†Deceased