EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2020-069 LHCb-PAPER-2019-026 21 July 2020

Search for $C\\!P$ violation in ${{\mathchar
28932\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ decays using
model-independent techniques

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

A first search for $C\\!P$ violation in the Cabibbo-suppressed ${{\mathchar
28932\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ decay is performed
using both a binned and an unbinned model-independent technique in the Dalitz
plot. The studies are based on a sample of proton-proton collision data,
corresponding to an integrated luminosity of $3.0\text{\,fb}^{-1}$, and
collected by the LHCb experiment at centre-of-mass energies of $7$ and
$8\text{\,Te\kern-1.00006ptV}$. The data are consistent with the hypothesis of
no $C\\!P$ violation.

Submitted to Eur. Phys. J. C

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

##  1 Introduction

The non-invariance of fundamental interactions under the combination of charge
conjugation and parity transformation, known as $C\\!P$ violation ($C\\!PV$),
is a key requirement for the generation of the baryon-antibaryon asymmetry in
the early Universe [1, 2]. In the Standard Model (SM) of particle physics,
$C\\!PV$ is included through the introduction of a single irreducible complex
phase in the Cabibbo–Kobayashi–Maskawa (CKM) quark-mixing matrix [3, 4]. The
amount of $C\\!PV$ predicted by the CKM mechanism is not sufficient to explain
a matter-dominated universe [5, 6] and other sources of $C\\!PV$ are required.
The realization of $C\\!PV$ in nature has been well established in the $K$\-
and $B$-meson systems by several experiments [7, 8, 9, 10, 11, 12, 13]. The
LHCb experiment has observed for the first time $C\\!PV$ in the charm-meson
sector as the difference of the $C\\!P$ asymmetries between the two-body
decays $D^{0}\rightarrow K^{-}K^{+}$ and $D^{0}\rightarrow\pi^{-}\pi^{+}$
[14]. A similar study using ${\mathchar 28931\relax}^{+}_{c}$ to $p$ ${K}^{-}$
${K}^{+}$ and $p$ ${\pi}^{-}$ ${\pi}^{+}$ found no evidence for $C\\!PV$ [15].
Indeed, so far, $C\\!PV$ has never been observed in any baryon system.
Evidence for $C\\!PV$ in the $b$ baryon sector reported by the LHCb
collaboration in [16] has not been confirmed with more data [17]. Further
measurements of processes involving the decay of charm hadrons can shed light
on the origin and magnitude of $C\\!PV$ mechanisms within the SM and beyond.

In two-body decays of charm hadrons, $C\\!PV$ can manifest itself as an
asymmetry between partial decay rates. Multi-body decays offer access to more
observables that are sensitive to $C\\!P$-violating effects. For a three-body
baryon decay the kinematics can be characterised by three Euler angles and two
squared invariant masses, which form a Dalitz plot [18]. The Euler angles are
redundant if all initial spin states are integrated over. Interference effects
in the Dalitz plot probe $C\\!P$ asymmetries in both the magnitudes and phases
of amplitudes. In three-body decays there can be large local $C\\!P$
asymmetries in the Dalitz plot, even when no significant global $C\\!PV$
exists. A recent example has been measured in the decay
${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}$ [19].

In the SM, $C\\!PV$ asymmetries in the charm sector are expected at the order
of $10^{-3}$ or less [20] for singly Cabibbo-suppressed (SCS) decays. New
physics (NP) contributions can enhance $C\\!P$-violating effects up to
$10^{-2}$ [21, 22, 23, 24, 25, 26, 27, 28, 29]. Searches for $C\\!PV$ in
${\mathchar 28932\relax}^{+}_{c}$ baryon decays111Unless stated explicitly,
the inclusion of charge-conjugate states is implied throughout. provide a test
of the SM and place constraints on NP parameters [30, 31, 32, 33, 34]. In
contrast to SCS decays, in Cabibbo-favoured (CF) charm-quark transitions, such
as ${{\mathchar 28931\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$
decays, there is only one dominant amplitude in the SM, resulting in no
$C\\!P$-violating effects. However this could change with NP, as argued above
in the case of SCS decays.

This article describes searches for direct $C\\!PV$ in the SCS decay
${{\mathchar 28932\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$, for
${{\mathchar 28932\relax}^{+}_{c}}$ baryons produced promptly in $p$ $p$
collisions. The ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ decay is used as a
control mode to study in data the level of experimental asymmetries that
pollute the measurement. In this paper, the symbol $H_{c}^{+}$ is used to
refer to both ${\mathchar 28932\relax}^{+}_{c}$ and ${\mathchar
28931\relax}^{+}_{c}$. It is assumed that the polarisation of charm baryons
produced in $p$ $p$ collisions is sufficiently small, as it is for $b$-baryons
[35], to justify the integration over the Euler angles. This measurement uses
$p$ $p$ collision data, corresponding to an integrated luminosity of
$3\text{\,fb}^{-1}$, recorded by the LHCb detector in 2011
($1\text{\,fb}^{-1}$) at a centre-of-mass energy of
$7\text{\,Te\kern-1.00006ptV}$, and in 2012 ($2\text{\,fb}^{-1}$) at a centre-
of-mass energy of $8\text{\,Te\kern-1.00006ptV}$. The magnetic field polarity
is reversed regularly during the data taking in order to minimise effects of
charged particle and antiparticle detection asymmetries. Approximately half of
the data are collected with each polarity.

There is presently no successful method for computing decay amplitudes in
multi-body charm decays, which could provide reliable predictions on how the
$C\\!P$ asymmetries vary over the phase space of the decay. This situation
favours a model-independent approach, which looks for differences between
multivariate density distributions for baryons and antibaryons. Therefore, in
this article searches for $C\\!PV$ are performed through a direct comparison
between the Dalitz plots of ${\mathchar 28932\relax}^{+}_{c}$ and ${\mathchar
28932\relax}^{-}_{c}$ decays using a binned significance ($S_{{C\\!P}}$)
method [36] and an unbinned k-nearest neighbour method (kNN) [37, 38, 39, 40],
both of which are model independent.

##  2 Detector and simulation

The LHCb detector [41, 42] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$. It is designed for the study of particles
containing $b$ and $c$ 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 $\rm 4~{}Tm$, 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 $200~{}{\rm GeV}/c$. The minimum distance of a track to a
primary vertex (PV), the impact parameter (IP), is measured with a resolution
of $(15+29/p_{\mathrm{T}})~{}{\rm\mu m}$, where $p_{\mathrm{T}}$ is the
component of the momentum transverse to the beam, in ${\rm GeV}/c$. Different
types of charged hadrons are distinguished using information from two ring-
imaging Cherenkov detectors. Photons, electrons and hadrons are identified by
a calorimeter system consisting of scintillating-pad and preshower detectors,
an electromagnetic and a hadron calorimeter. Muons are identified by a system
composed of alternating layers of iron and multiwire proportional chambers.

Samples of simulated events are used to optimise the signal selection, to
derive the angular efficiency and to correct the decay-time efficiency. In the
simulation, $p$ $p$ collisions are generated using PYTHIA [43] with a specific
LHCb configuration [44]. Decays of hadronic particles are described by EVTGEN
[45], in which final-state radiation is generated using PHOTOS [46]. The
interaction of the generated particles with the detector, and its response,
are implemented using the GEANT4 toolkit [47] as described in Ref. [48].

##  3 Selection of signal candidates

The online event selection is performed by a trigger consisting of a hardware
stage, based on information from the calorimeter and muon systems, followed by
two software stages. At the hardware trigger stage, events are required to
have either muons with high $p_{\mathrm{T}}$ or hadrons, photons or electrons
with a high transverse-energy deposit in the calorimeters. For hadrons, the
transverse energy threshold is approximately $3.5~{}{\rm GeV}/c^{2}$. In the
first software trigger stage at least one good-quality track with
$p_{\mathrm{T}}>300~{}{\rm MeV}/c$ is required. In the second software trigger
stage an $H^{+}_{c}$ candidate is fully reconstructed from three high-quality
tracks not pointing to any PV. The three tracks should form a secondary vertex
(SV) which must be well separated from any PV. A momentum
$p>3\text{\,Ge\kern-1.00006ptV\\!/}c$ for each track and the scalar sum of
$p_{\mathrm{T}}$ for the three tracks
$p_{\mathrm{T}}>2\text{\,Ge\kern-1.00006ptV\\!/}c$ are required. The combined
invariant mass of the three tracks is required to be in the range
$2190-2570~{}\text{\,Me\kern-1.00006ptV\\!/}c^{2}$. Requirements are also
placed on the particle identification criteria of the tracks and on the angle
between the vector from the associated PV to the SV and the $H_{c}^{+}$
momentum. The associated PV is the one with smallest difference in vertex fit
$\chi^{2}$ when performed with and without the $H_{c}^{+}$ candidate.

In the offline analysis, tighter selection requirements are placed on the
track-reconstruction quality, the $p$ and $p_{\mathrm{T}}$ of the final-state
particles. For protons $10<p<100~{}\text{\,Ge\kern-1.00006ptV\\!/}c$ is
required, while kaons and pions momentum satisfies
$3<p<150\text{\,Ge\kern-1.00006ptV\\!/}c$. Only $H_{c}^{+}$ candidates with
$p_{\mathrm{T}}$ in the range
$4<p_{\mathrm{T}}<16~{}\text{\,Ge\kern-1.00006ptV}$ are retained. Additional
requirements are also made on the SV fit quality, and the minimum significance
of the displacement from the SV to any PV in the event. This reduces the
contribution of charm baryons from $b$-hadron decays to less than 5% of the
prompt signal. Reconstructed particles are accepted if their momenta are
within a region defined by $|p_{x}|<0.2p_{z}$ and $|p_{x}|>0.01p_{z}$, where
$p_{x}$ and $p_{z}$ are the momentum components along the $x$ and $z$
axes222The LHCb coordinate system is right-handed, with the $z$ axis pointing
along the beam axis, $y$ the vertical direction, and $x$ the horizontal
direction. The $(x,z)$ plane is the bending plane of the dipole magnet.. This
requirement has a signal loss of 25%, and is imposed to avoid large detection
asymmetries that are present in the excluded kinematic regions. Differences
between particles and antiparticles in reconstruction efficiencies are also
observed for $H^{+}_{c}$ candidates where $p<20~{}{\rm GeV}/c$ for all charged
tracks. These differences do not cancel by simply averaging the data acquired
with opposite magnet polarities. To minimise the reconstruction asymmetry, the
momentum of all tracks is required to be greater than $20~{}{\rm GeV}/c$. This
requirement rejects about 20% of the selected charm-baryon candidates.

The distributions of the invariant-mass, $M(pK^{-}\pi^{+})$, of selected
${\mathchar 28931\relax}^{+}_{c}$ and ${\mathchar 28932\relax}^{+}_{c}$
candidates are presented in Figs. 1 and 2, respectively, with fit curves
overlaid. The fit model comprises a sum of two Gaussian functions describing
the signal and a second-order Chebyshev polynomial function describing the
combinatorial background. No additional source of background is found to
contribute significantly, according to studies in data reconstructed with
different mass hypotheses.

Figure 1: Invariant-mass, $M({p}{{K}^{-}}{{\pi}^{+}})$, distributions of
selected ${\mathchar 28931\relax}^{+}_{c}$ candidates are shown in the (left)
2011 and (right) 2012 data samples. Data points are in black. The overlaid
fitted model (blue continuous line) is a sum of two Gaussian functions with
the same mean and different widths (red dashed line) and a second-order
Chebyshev polynomial function (green dotted line) describing the signal and
background components.

Figure 2: Invariant-mass, $M({p}{{K}^{-}}{{\pi}^{+}})$, distributions of
selected ${\mathchar 28932\relax}^{+}_{c}$ candidates are shown in the (left)
2011 and (right) 2012 data samples. Data points are in black. The overlaid
fitted model (blue continuous line) is a sum of two Gaussian functions with
the same mean and different widths (red dashed line) and a second-order
Chebyshev polynomial function (green dotted line) describing the signal and
background components.

The final samples used for the $C\\!PV$ search comprise all candidates with
$M(pK^{-}\pi^{+})$ within $\pm 3\sigma$ around $m({{\mathchar
28931\relax}^{+}_{c}})$ or $m({{\mathchar 28932\relax}^{+}_{c}})$, where
$\sigma$ is the weighted average of the two fitted Gaussian widths and
$m({{\mathchar 28931\relax}^{+}_{c}})$ and $m({{\mathchar
28932\relax}^{+}_{c}})$ are the masses of the ${\mathchar
28931\relax}^{+}_{c}$ and ${\mathchar 28932\relax}^{+}_{c}$ baryons [49].
There are approximatly 2.0 million ${\mathchar 28931\relax}^{+}_{c}$
candidates (0.4 million in the 2011 and 1.6 million in the 2012 data sample)
and 0.25 million ${\mathchar 28932\relax}^{+}_{c}$ candidates (0.05 million in
the 2011 and 0.2 million in the 2012 data sample). The purity for ${\mathchar
28931\relax}^{+}_{c}$ decays is 94% for 2011 and 98% for 2012 and that for
${\mathchar 28932\relax}^{+}_{c}$ decays is 77% for 2011 and 78% for 2012,
where purity is defined as the number of signal candidates obtained from the
fit to the invariant-mass distribution divided by the total number of
candidates.

##  4 Methods

The Dalitz plot for $H_{c}^{+}\rightarrow pK^{-}\pi^{+}$ is formed by the
squares of the invariant masses of two pairs of the decay products:
$M^{2}(K^{-}\pi^{+})$ and $M^{2}(pK^{-})$. Comparisons of the Dalitz plots of
$H_{c}^{+}$ and $H_{c}^{-}$ candidates are performed using the binned
$S_{C\\!P}$ and the unbinned kNN methods, described in the following. For both
the binned $S_{{C\\!P}}$ and unbinned kNN methods, a signal of $C\\!PV$ is
established if a p-value lower then $3\times 10^{-7}$ is found, corresponding
to an exclusion of $C\\!P$ symmetry with a significance of five standard
deviations. However, in case that no $C\\!PV$ is found, there is no model-
independent mechanism for setting an upper limit on the amount of $C\\!PV$ in
the Dalitz plot.

###  4.1 Binned $S_{C\\!P}$ method

The $S_{C\\!P}$ method [36] has been used before for searches of $C\\!PV$
testing in charm and beauty decays [50, 51, 52, 40, 53]. This method is used
to search for localised asymmetries in the phase space of the decay
$H_{c}^{+}\rightarrow pK^{-}\pi^{+}$ and is based on a bin-by-bin comparison
between the Dalitz plots of baryons, $H_{c}^{+}$, and antibaryons,
$H_{c}^{-}$. The Dalitz plots of $H_{c}^{+}$ and $H_{c}^{-}$ are divided using
an identical binning. For each bin $i$ of the Dalitz plot, the significance of
the difference between the number of $H_{c}^{+}$ ($n^{i}_{+}$) and $H_{c}^{-}$
($n^{i}_{-}$) candidates, is computed as

$S^{i}_{C\\!P}=\frac{n^{i}_{+}-\alpha
n^{i}_{-}}{\sqrt{\alpha(n^{i}_{+}+n^{i}_{-})}},$ (1)

where the factor $\alpha$ is defined as $\alpha=\frac{n_{+}}{n_{-}}$ and
$n_{+}$, $n_{-}$ are the total number of $H_{c}^{+}$, $H_{c}^{-}$ candidates.
This factor accounts for asymmetries arising in the production of $H_{c}^{+}$
baryons, as well as in the detection of the final-state particles. The
production and global detection asymmetries do not to depend on the Dalitz
plot position.

A numerical comparison between the Dalitz plots of the $H_{c}^{+}$ and
$H_{c}^{-}$ candidates is made using a $\chi^{2}$ test defined as

$\chi^{2}\equiv\Sigma(S^{i}_{C\\!P})^{2}.$ (2)

A p-value for the hypothesis of no $C\\!PV$ is obtained from the $\chi^{2}$
distribution considering that the number of degrees of freedom is equal to the
total number of bins minus one, due to the constraint on the factor $\alpha$
of the overall $H_{c}^{+}$ and $H_{c}^{-}$ normalisation.

In the hypothesis of no $C\\!PV$, the $S_{C\\!P}$ values are expected to be
distributed according to the normal distribution with a mean of zero and a
standard deviation of unity. The test is performed using only bins with a
minimum of 10 $H_{c}^{+}$ and 10 $H_{c}^{-}$ candidates. In case of $C\\!PV$,
a deviation from the normal distribution is expected, generating a $p$-value
close to zero.

###  4.2 Unbinned kNN method

The kNN method is based on the concept of a set of nearest neighbour
candidates ($n_{k}$) in a combined sample of two data sets: baryons and
antibaryons. As an unbinned method, the kNN approach is more sensitive to a
$C\\!PV$ search in a sample with limited data, compared to that of the binned
$S_{{C\\!P}}$ method. The kNN method is used here to test whether baryons and
antibaryons share the same parent distribution function [37, 38, 39]. To find
the $n_{k}$ nearest neighbour events of each $H_{c}^{+}$ or $H_{c}^{-}$
candidate, an Euclidean distance between closest points in the Dalitz plot is
used. A test statistic $T$ for the null hypothesis is defined as

$T=\frac{1}{n_{k}(n_{+}+n_{-})}\sum\limits_{i=1}^{n_{+}+n_{-}}\sum\limits_{k=1}^{n_{k}}I(i,k),$
(3)

where $I(i,k)=1$ if the $i^{th}$ candidate and its $k^{th}$ nearest neighbour
have the same charge and $I(i,k)=0$ otherwise.

The test statistic $T$ is the mean fraction of like-charged neighbour pairs in
the sample of $H_{c}^{+}$ and $H_{c}^{-}$ decays. The advantage of the kNN
method, in comparison with other proposed methods for unbinned analyses [37],
is that the calculation of $T$ is simple and fast and the expected
distribution of $T$ is well known. Under the hypothesis of no $C\\!PV$, $T$
follows a normal distribution with a mean, $\mu_{T}$, and a variance,
$\sigma_{T}$, where

$\mu_{T}=\frac{n_{+}(n_{+}-1)+n_{-}(n_{-}-1)}{n(n-1)},$ (4)
$\lim_{n,n_{k},D\rightarrow\infty}\sigma_{T}^{2}=\frac{1}{nn_{k}}\left(\frac{n_{+}n_{-}}{n^{2}}+4\frac{n_{+}^{2}n_{-}^{2}}{n^{4}}\right),$
(5)

with $n=n_{+}+n_{-}$ and $D=2$ is the dimensionality of the tested
distribution. A good approximation of $\sigma_{T}$ is obtained even for $D=2$
for the current values of $n_{+}$, $n_{-}$ and $n_{k}$ [37].

For $n_{+}=n_{-}$ the mean $\mu_{T}$ can be expressed as

$\mu_{TR}=\frac{1}{2}\left(\frac{n-2}{n-1}\right)$ (6)

and is called the reference value, $\mu_{TR}$. For large $n$, $\mu_{TR}$
asymptotically tends to $0.5$.

To increase the power of the kNN method, the Dalitz plot is divided into
regions defined around the expected resonances. It can provide one of the
necessary conditions for observation of $C\\!PV$: large relative strong phases
in the final states of interfering amplitudes of the intermediate resonance
states. The Dalitz plot is partitioned into six regions for the decays of the
${\mathchar 28931\relax}^{+}_{c}$ control mode and eleven regions for signal
${\mathchar 28932\relax}^{+}_{c}$ decays according to the present of
resonances of the phase space, as shown in Fig. 3. The definitions of the
regions are also given in Tables 1 and 2 for ${\mathchar 28931\relax}^{+}_{c}$
and ${\mathchar 28932\relax}^{+}_{c}$ baryons, respectively. For ${\mathchar
28931\relax}^{+}_{c}$ decays the $K^{*}(892)$, $K^{*}(1430)$, ${\mathchar
28929\relax}(1232)$, ${\mathchar 28931\relax}(1520)$, ${\mathchar
28931\relax}(1670)$, ${\mathchar 28931\relax}(1690)$ resonances are seen in
data, whilst for ${\mathchar 28932\relax}^{+}_{c}$ decays additional
resonances are seen, namely ${\mathchar 28931\relax}(1520)$, ${\mathchar
28931\relax}(1600)$, ${\mathchar 28931\relax}(1710)$, ${\mathchar
28931\relax}(1800)$, ${\mathchar 28931\relax}(1810)$, ${\mathchar
28931\relax}(1820)$, ${\mathchar 28931\relax}(1830)$, ${\mathchar
28931\relax}(1890)$, ${\mathchar 28929\relax}(1600)$, ${\mathchar
28929\relax}(1620)$ and ${\mathchar 28929\relax}(1700)$. For ${\mathchar
28931\relax}^{+}_{c}$ decays there are four independent regions (R1–R4),
whilst the region R2 is further split into the high $M^{2}(pK^{-})$ region
(R6) and the low $M^{2}(pK^{-})$ region (R5). For ${\mathchar
28932\relax}^{+}_{c}$ there are seven independent regions (R1–R7), whilst the
region R2 is split in mass $M^{2}(pK^{-})$ in two regions at larger mass (R9)
and smaller mass (R8), R2=R8$\cup$R9, similarly for R10 and R11, where
R10=R4$\cup$R5, and R11=R4$\cup$R5$\cup$R6$\cup$R7. Region R0 is the full
Dalitz plot.

Figure 3: Definition of the Dalitz plot regions for (left) ${{\mathchar 28931\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ and (right) ${{\mathchar 28932\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ decays. Additional regions are defined by combining regions. For ${{\mathchar 28931\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ R2=R5$\cup$R6 and for ${{\mathchar 28932\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ R2=R8$\cup$R9, R10=R4$\cup$R5 and R11=R4$\cup$R5$\cup$R6$\cup$R7. The presented distributions correspond to the 2012 data sample. Table 1: Definitions of the Dalitz plot regions for the control mode, ${{\mathchar 28931\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$. Region | Definition
---|---
R0 | Full Dalitz plot
R1 | $M^{2}(K^{-}\pi^{+})<0.7{\rm~{}GeV^{2}}/c^{4}$
R2 | $0.7\leq M^{2}(K^{-}\pi^{+})<0.9{\rm~{}GeV^{2}}/c^{4}$
R3 |  $M^{2}(K^{-}\pi^{+})\geq 0.9{\rm~{}GeV^{2}}/c^{4}$, $M^{2}(pK^{-})<2.8{\rm~{}GeV^{2}}/c^{4}$
R4 |  $M^{2}(K^{-}\pi^{+})\geq 0.9{\rm~{}GeV^{2}}/c^{4}$, $M^{2}(pK^{-})\geq 2.8{\rm~{}GeV^{2}}/c^{4}$
R5 |  $0.7\leq M^{2}(K^{-}\pi^{+})<0.9{\rm~{}GeV^{2}}/c^{4}$, $M^{2}(pK^{-})<3.2{\rm~{}GeV^{2}}/c^{4}$
R6 |  $0.7\leq M^{2}(K^{-}\pi^{+})<0.9{\rm~{}GeV^{2}}/c^{4}$, $M^{2}(pK^{-})\geq 3.2{\rm~{}GeV^{2}}/c^{4}$
Table 2: Definitions of the Dalitz plot regions for ${{\mathchar 28932\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ decays. Region | Definition
---|---
R0 | Full Dalitz plot
R1 | $M^{2}(K^{-}\pi^{+})<0.7{\rm~{}GeV^{2}}/c^{4}$
R2 | $0.7\leq M^{2}(K^{-}\pi^{+})<0.9{\rm~{}GeV^{2}/c^{4}}$
R3 | $0.9\leq M^{2}(K^{-}\pi^{+})<1.3{\rm~{}GeV^{2}}/c^{4}$
R4 |  $M^{2}(K^{-}\pi^{+})\geq 1.3{\rm~{}GeV^{2}}/c^{4}$, $M^{2}(pK^{-})<2.4{\rm~{}GeV^{2}}/c^{4}$
R5 |  $M^{2}(K^{-}\pi^{+})\geq 1.3{\rm~{}GeV^{2}}/c^{4}$, $2.4\leq M^{2}(pK^{-})<3.2{\rm~{}GeV^{2}}/c^{4}$
R6 |  $M^{2}(K^{-}\pi^{+})\geq 1.3{\rm~{}GeV^{2}}/c^{4}$, $3.2\leq M^{2}(pK^{-})<3.8{\rm~{}GeV^{2}}/c^{4}$
R7 |  $M^{2}(K^{-}\pi^{+})\geq 1.3{\rm~{}GeV^{2}}/c^{4}$, $M^{2}(pK^{-})\geq 3.8{\rm~{}GeV^{2}}/c^{4}$
R8 |  $0.7\leq M^{2}(K^{-}\pi^{+})<0.9{\rm~{}GeV^{2}}/c^{4}$, $M^{2}(pK^{-})<4{\rm~{}GeV^{2}}/c^{4}$
R9 |  $0.7\leq M^{2}(K^{-}\pi^{+})<0.9{\rm~{}GeV^{2}}/c^{4}$, $M^{2}(pK^{-})\geq 4{\rm~{}GeV^{2}}/c^{4}$
R10 |  $M^{2}(K^{-}\pi^{+})\geq 1.3{\rm~{}GeV^{2}}/c^{4}$, $M^{2}(pK^{-})<3.2{\rm~{}GeV^{2}}/c^{4}$
R11 | $M^{2}(K^{-}\pi^{+})\geq 1.3{\rm~{}GeV^{2}}/c^{4}$

##  5 Control mode, background and sensitivity studies

The $S_{{C\\!P}}$ and kNN methods are tested using the ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ control mode where
the $C\\!P$ asymmetry is expected to be null [21, 22, 23, 24, 25, 26, 27, 28,
29]. The sidebands of ${{\mathchar
28932\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ candidates in the
mass regions $2320<M({p}{{K}^{-}}{{\pi}^{+}})<2445~{}{\rm MeV}/c^{2}$ and
$2490<M({p}{{K}^{-}}{{\pi}^{+}})<2650~{}{\rm MeV}/c^{2}$ are used to check
that the background does not introduce spurious asymmetries.

The measured total raw asymmetry is defined as

$A_{\rm Raw}=\frac{n_{-}-n_{+}}{n_{-}+n_{+}},$ (7)

and it depends on the production asymmetry of $H_{c}^{+}$ baryons and on the
detection asymmetries that arise through charge-dependent selection
efficiencies due to track reconstruction, trigger selection and particle
identification. The measured value of $A_{\rm Raw}$ in each region of the
Dalitz plot of ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ decays is
presented in Fig. 4. The measured $A_{\rm Raw}$ value integrated over the
Dalitz plot equals $-0.0230\pm 0.0016$ and $-0.0188\pm 0.0008$ in the 2011 and
2012 data samples, where the uncertainties are statistical only. Within
uncertainties, $A_{\rm Raw}$ in all regions amounts to about $-2$%. There is
no significant difference in the measurement of $A_{\rm Raw}$ between the 2011
and 2012 data samples. Since the production and detection asymmetries of
${\mathchar 28931\relax}^{+}_{c}$ baryons can depend on the baryon
pseudorapidity, $\eta$, and $p_{\mathrm{T}}$, the dependence of $A_{\rm Raw}$
in regions of the Dalitz plot is checked in bins of $\eta$ and
$p_{\mathrm{T}}$ of the ${{\mathchar 28931\relax}^{+}_{c}}$ baryon. It is
observed that the value of $A_{\rm Raw}$ globally changes from bin to bin of
$\eta$ and $p_{\mathrm{T}}$ of the ${\mathchar 28931\relax}_{c}$ candidates,
but for a given bin of $\eta$ and $p_{\mathrm{T}}$ a constant behaviour of
$A_{\rm Raw}$ in regions of the Dalitz plot is maintained.

Figure 4: Measured values of $A_{\rm Raw}$ in regions of ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ candidate decays
for 2011 (stars) and 2012 (dots) data samples. R0 corresponds to full Dalitz
plot and R2 is separated into R5 and R6, and these regions are correlated and
separated by dashed lines.

In the $S_{C\\!P}$ method the production asymmetry and all global effects are
considered by introducing the $\alpha$ factor, following the strategy
described in Sec. 4.1. The p-values obtained are larger than 58%, consistent
with the absence of localised asymmetries. As an example, Fig. 5 shows the
distribution of $S^{i}_{C\\!P}$ for ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ decays considering
uniform binning, and for two granularities of the Daliz plot: 28 and 106 bins
in the 2012 sample. Alternatively the Dalitz plot is divided into different
size bins with the same number of events in each bin. The $p$-values obtained
are larger than 34%, consistent with the hypothesis of absence of localised
asymmetries.

Figure 5: Distributions of $S_{C\\!P}^{i}$ and corresponding one-dimensional
distributions for ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ decays for the
data collected in the 2012 data sample: (top row) 28 same-size bins and
(bottom row) 106 same-size bins of the Dalitz plot. The number of analysed
bins, nbins, and the p-values are given.

Following the strategy described in Sec. 4.2, the results of the kNN method in
regions of the Dalitz plot for the ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ control mode are
presented in Fig. 6, for $n_{k}=50$. The pulls,
$(\mu_{T}-\mu_{TR})/\Delta(\mu_{T}-\mu_{TR})$, where
$\Delta(\mu_{T}-\mu_{TR})$ is the statistical uncertainty on the difference
$(\mu_{T}-\mu_{TR})$, are different from zero in all regions. The largest pull
value is observed when integrated over the full Dalitz plot. This asymmetry is
the result of the nonzero production asymmetry that is presented in Fig. 4 and
discussed above. Pulls of the test statistic $T$, $((T-\mu_{T})/\sigma_{T})$,
vary within $-3$ and $+3$, consistent with the hypothesis of absence of
localised asymmetries in any region. The difference among data-taking years
are consistent with statistical fluctuations. Figure 6 illustrates how the
larger 2012 data sample improves the power of the kNN method. In Run 2 (years
of data taking 2016, 2017 and 2018) the yield is expected to be about three
times larger than that from Run 1.

Figure 6: (Top left) pulls, $(\mu_{T}-\mu_{TR})/\Delta(\mu_{T}-\mu_{TR})$, and
(top right) the corresponding p-values, (bottom left) pull values of the test
statistic $T$ and (bottom right) the corresponding p-values in regions for
control ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ candidate decays
obtained using the kNN method with $n_{k}=50$ for data collected in 2011
(stars) and 2012 (dots). The horizontal lines in the left figures represent -3
and +3 pull values. R0 corresponds to full Dalitz plot and R2 is separated
into R5 and R6, and these regions are correlated and separated by dashed
lines.

The interaction cross-section of charged hadrons with matter depends on the
charged hadron momentum. As such, the detection asymmetries of the proton and
kaon-pion systems are momentum dependent. Pseudoexperiments are performed to
check whether the detection asymmetries related to particles reconstructed in
the final state can generate a spurious $C\\!P$ asymmetry. The proton
detection asymmetry varies from about 5% at low momentum to 1% at $100~{}{\rm
GeV}/c$ and is estimated using simulations. The kaon-pion detection asymmetry
is measured to vary from $-1.4$% at low momentum to $-0.7$% at $60~{}{\rm
GeV}/c$ [54]. The combined effect of the two asymmetries is found to cancel
approximately and does not generate a spurious $C\\!P$ asymmetry in the Dalitz
plot.

These studies are repeated using the candidates in the sideband of the
${{\mathchar 28932\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ mass
distribution. No spurious $C\\!P$ asymmetry is found for both methods. For
further cross-checks, the control samples are divided according to the
polarity of the magnetic field. The p-values are found to be distributed
uniformly.

The expected statistical powers of both methods are obtained by performing
pseudoexperiments. One hundred samples of ${{\mathchar
28932\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ decays are
generated, each with a yield and purity equivalent to that observed in the
combined 2011 and 2012 data samples, resulting in 200 000 ${\mathchar
28932\relax}^{+}_{c}$ decays generated in each pseudoexperiment. In this
model, the two-dimensional Dalitz plots are generated assuming that the
${\mathchar 28932\relax}^{+}_{c}$ baryons are produced unpolarised. This model
is built by including the resonances observed in the data, using the same
software as in Ref. [55]. The same resonances as described in Sec. 4.2 are
included. The statistical powers of the two methods are found to be
comparable. Both methods are sensitive to a 5% $C\\!P$ asymmetry in the
$K^{*}(892)$ and ${\mathchar 28929\relax}(1232)$ resonance regions with 3 and
5 sigma significances that would be observed in 69% and 10% of the cases for
the kNN method and 17% and 10% of the cases for the $S_{C\\!P}$ method,
respectively.

##  6 Results

###  6.1 Binned $S_{C\\!P}$ method

The binned $S_{C\\!P}$ method is applied to look for local $C\\!P$ asymmetries
in ${{\mathchar 28932\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$
decays following the strategy described in Sec. 4.1. The distribution of
$S_{C\\!P}^{i}$ for ${{\mathchar
28932\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ decays considering
uniform binning, and for two granularities of the Daliz plot: 29 and 111 bins
are shown in Fig. 7 for the combined 2011 and 2012 data samples. The
normalization factor $\alpha$, defined in Eq. 1, is determined to be $1.029\pm
0.004$. The measured p-values using a $\chi^{2}$ test are larger than 32%,
consistent with no evidence for $C\\!PV$. The obtained $S_{C\\!P}$
distributions agree with a normal distribution. It is also checked that the
results in the 2011 and 2012 data samples are consistent with each other.

Figure 7: Distributions of $S_{C\\!P}^{i}$ and corresponding one-dimensional
distributions for ${{\mathchar
28932\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ decays for the
combined data collected 2011 and 2012: (top row) 29 uniform bins and (bottom
row) 111 uniform bins of the Dalitz plot. The number of analysed bins and the
p-values are given.

###  6.2 Unbinned kNN method

The unbinned kNN method is applied to look for $C\\!P$ asymmetry in
${{\mathchar 28932\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$
decays, following the strategy described in Sec. 4.2. The results are
presented in Fig. 8 for $n_{k}=50$ for the merged 2011 and 2012 data samples.
The measured pull values, $((\mu_{T}-\mu_{TR})/\Delta(\mu_{T}-\mu_{TR}))$, are
different from zero. The largest value of pull is observed integrated over the
full Dalitz plot. This is due to the expected nonzero production and detector
asymmetries, that is presented in Fig. 9. The measured $A_{\rm Raw}$ is
constant within uncertainties in all regions.

The pulls of the test statistic $T$, $((T-\mu_{T})/\sigma_{T})$, shown in Fig.
8 vary within $-3$ and $+3$, consistent with the hypothesis of absence of
localised asymmetries. To check for any systematic effects the kNN test is
repeated for the individual 2011 and 2012 data samples as well as for samples
separated according to the polarity of the magnetic field. All obtained
results are compatible within uncertainties and no systematic effects are
observed.

Since the sensitivity of the method can depend on the $n_{k}$ parameter, the
analysis is repeated with different values of $n_{k}$ from 10 up to 3000. Only
$T$ and $\sigma_{T}$ depend on $n_{k}$. Pulls of the statistic $T$ for the
entire Dalitz plot are shown in Fig. 10. All results show no significant
deviation from the hypothesis of $C\\!P$ symmetry.

Figure 8: (Top left) pulls, $(\mu_{T}-\mu_{TR})/\Delta(\mu_{T}-\mu_{TR})$, and
(top right) the corresponding p-values; (bottom left) pull values of the test
statistic $T$ and (bottom right) the corresponding p-values in regions for
signal ${{\mathchar
28932\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ candidate decays
obtained using the kNN method with $n_{k}=50$ for combined data collected 2011
and 2012. The horizontal lines in the left figures represent $-3$ and $+3$
pull values. R0 corresponds to full Dalitz plot and R2 is separated into R8
and R9, R10 is separated into R4 and R5, R11 is separated into R4, R5, R6 and
R7, and these regions are correlated and separated by dashed lines. Figure 9:
The measured $A_{\rm Raw}$ in regions in signal ${{\mathchar
28932\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ candidate decays
for the combined data collected in 2011 and 2012. R0 corresponds to full
Dalitz plot and R2 is separated into R8 and R9, R10 is separated into R4 and
R5, R11 is separated into R4, R5, R6 and R7, and these regions are correlated
and separated by dashed lines. Figure 10:  (Left) the pull values of the test
statistic $T$ and (right) the corresponding p-value dependence on the $n_{k}$
parameter for the whole Dalitz plot (region R0) for ${{\mathchar
28932\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ candidate decays
obtained using the kNN method for the combined data collected in 2011 and
2012. The horizontal lines in the left figures represent $-3$ and $+3$ pull
values. The points are determined with different $n_{k}$ using same data
sample, therefore are correlated.

##  7 Conclusions

Model-independent searches for $C\\!P$ violation in ${{\mathchar
28932\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ decays are
presented using the binned $S_{C\\!P}$ and the unbinned kNN methods. The
${{\mathchar 28931\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$
candidates and the sideband regions of ${{\mathchar
28932\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ candidates are
used to ensure that no spurious charge asymmetries affect the methods. Both
methods are sensitive to $C\\!P$ asymmetry larger than a 5% in the regions
around the $K^{*}(892)$ and the ${\mathchar 28929\relax}$(1232). The obtained
results are consistent with the absence of $C\\!P$ violation in ${{\mathchar
28932\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ decays.

## 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).

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LHCb collaboration

R. Aaij31, C. Abellán Beteta49, T. Ackernley59, B. Adeva45, M. Adinolfi53, H.
Afsharnia9, C.A. Aidala79, S. Aiola25, Z. Ajaltouni9, S. Akar64, P.
Albicocco22, J. Albrecht14, F. Alessio47, M. Alexander58, A. Alfonso Albero44,
G. Alkhazov37, P. Alvarez Cartelle60, A.A. Alves Jr45, S. Amato2, Y. Amhis11,
L. An21, L. Anderlini21, G. Andreassi48, M. Andreotti20, F. Archilli16, J.
Arnau Romeu10, A. Artamonov43, M. Artuso67, K. Arzymatov41, E. Aslanides10, M.
Atzeni49, B. Audurier26, S. Bachmann16, J.J. Back55, S. Baker60, V.
Balagura11,b, W. Baldini20,47, A. Baranov41, R.J. Barlow61, S. Barsuk11, W.
Barter60, M. Bartolini23,h, F. Baryshnikov76, J.M. Basels13, G. Bassi28, V.
Batozskaya35, B. Batsukh67, A. Battig14, V. Battista48, A. Bay48, M. Becker14,
F. Bedeschi28, I. Bediaga1, A. Beiter67, L.J. Bel31, V. Belavin41, S. Belin26,
N. Beliy5, V. Bellee48, K. Belous43, I. Belyaev38, G. Bencivenni22, E. Ben-
Haim12, S. Benson31, S. Beranek13, A. Berezhnoy39, R. Bernet49, D.
Berninghoff16, H.C. Bernstein67, C. Bertella47, E. Bertholet12, A. Bertolin27,
C. Betancourt49, F. Betti19,e, M.O. Bettler54, Ia. Bezshyiko49, S. Bhasin53,
J. Bhom33, M.S. Bieker14, S. Bifani52, P. Billoir12, A. Birnkraut14, A.
Bizzeti21,u, M. Bjørn62, M.P. Blago47, T. Blake55, F. Blanc48, S. Blusk67, D.
Bobulska58, V. Bocci30, O. Boente Garcia45, T. Boettcher63, A. Boldyrev77, A.
Bondar42,x, N. Bondar37, S. Borghi61,47, M. Borisyak41, M. Borsato16, J.T.
Borsuk33, M. Boubdir13, T.J.V. Bowcock59, C. Bozzi20,47, M.J. Bradley60, S.
Braun16, A. Brea Rodriguez45, M. Brodski47, J. Brodzicka33, A. Brossa
Gonzalo55, D. Brundu26,47, E. Buchanan53, A. Büchler-Germann49, A. Buonaura49,
C. Burr47, A. Bursche26, A. Butkevich40, J.S. Butter31, J. Buytaert47, W.
Byczynski47, S. Cadeddu26, H. Cai71, R. Calabrese20,g, L. Calero Diaz22, S.
Cali22, R. Calladine52, M. Calvi24,i, M. Calvo Gomez44,m, P. Camargo
Magalhaes53, A. Camboni44,m, P. Campana22, D.H. Campora Perez47, A.F.
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Cardinale23,h, A. Cardini26, I. Carli6, P. Carniti24,i, K. Carvalho Akiba31,
A. Casais Vidal45, G. Casse59, M. Cattaneo47, G. Cavallero23, S. Celani48, R.
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Dean66, W. Dean79, D. Decamp8, L. Del Buono12, B. Delaney54, H.-P.
Dembinski15, M. Demmer14, A. Dendek34, V. Denysenko49, D. Derkach77, O.
Deschamps9, F. Desse11, F. Dettori26,f, B. Dey7, A. Di Canto47, P. Di Nezza22,
S. Didenko76, H. Dijkstra47, V. Dobishuk51, F. Dordei26, M. Dorigo28,y, A.C.
dos Reis1, A. Dosil Suárez45, L. Douglas58, A. Dovbnya50, K. Dreimanis59, M.W.
Dudek33, G. Dujany12, P. Durante47, J.M. Durham66, D. Dutta61, R.
Dzhelyadin43,†, M. Dziewiecki16, A. Dziurda33, A. Dzyuba37, S. Easo56, U.
Egede60, V. Egorychev38, S. Eidelman42,x, S. Eisenhardt57, R. Ekelhof14, S.
Ek-In48, L. Eklund58, S. Ely67, A. Ene36, E. Epple66, S. Escher13, S. Esen31,
T. Evans47, A. Falabella19, J. Fan3, Y. Fan5, N. Farley52, S. Farry59, D.
Fazzini11, P. Fedin38, M. Féo47, P. Fernandez Declara47, A. Fernandez
Prieto45, F. Ferrari19,e, L. Ferreira Lopes48, F. Ferreira Rodrigues2, S.
Ferreres Sole31, M. Ferrillo49, M. Ferro-Luzzi47, S. Filippov40, R.A. Fini18,
M. Fiorini20,g, M. Firlej34, K.M. Fischer62, C. Fitzpatrick47, T. Fiutowski34,
F. Fleuret11,b, M. Fontana47, F. Fontanelli23,h, R. Forty47, V. Franco Lima59,
M. Franco Sevilla65, M. Frank47, C. Frei47, D.A. Friday58, J. Fu25,q, Q.
Fuehring14, W. Funk47, E. Gabriel57, A. Gallas Torreira45, D. Galli19,e, S.
Gallorini27, S. Gambetta57, Y. Gan3, M. Gandelman2, P. Gandini25, Y. Gao4,
L.M. Garcia Martin46, J. García Pardiñas49, B. Garcia Plana45, F.A. Garcia
Rosales11, J. Garra Tico54, L. Garrido44, D. Gascon44, C. Gaspar47, G.
Gazzoni9, D. Gerick16, E. Gersabeck61, M. Gersabeck61, T. Gershon55, D.
Gerstel10, Ph. Ghez8, V. Gibson54, A. Gioventù45, O.G. Girard48, P. Gironella
Gironell44, L. Giubega36, C. Giugliano20,g, K. Gizdov57, V.V. Gligorov12, C.
Göbel69, E. Golobardes44,m, D. Golubkov38, A. Golutvin60,76, A. Gomes1,a, P.
Gorbounov38,47, I.V. Gorelov39, C. Gotti24,i, E. Govorkova31, J.P.
Grabowski16, R. Graciani Diaz44, T. Grammatico12, L.A. Granado Cardoso47, E.
Graugés44, E. Graverini48, G. Graziani21, A. Grecu36, R. Greim31, P.
Griffith20,g, L. Grillo61, L. Gruber47, B.R. Gruberg Cazon62, C. Gu3, P. A.
Günther16, X. Guo70, E. Gushchin40, A. Guth13, Yu. Guz43,47, T. Gys47, T.
Hadavizadeh62, C. Hadjivasiliou9, G. Haefeli48, C. Haen47, S.C. Haines54, P.M.
Hamilton65, Q. Han7, X. Han16, T.H. Hancock62, S. Hansmann-Menzemer16, N.
Harnew62, T. Harrison59, R. Hart31, C. Hasse47, M. Hatch47, J. He5, M.
Hecker60, K. Heijhoff31, K. Heinicke14, A. Heister14, A.M. Hennequin47, K.
Hennessy59, L. Henry46, M. Heß73, J. Heuel13, A. Hicheur68, R. Hidalgo
Charman61, D. Hill62, M. Hilton61, P.H. Hopchev48, J. Hu16, W. Hu7, W. Huang5,
Z.C. Huard64, W. Hulsbergen31, T. Humair60, R.J. Hunter55, M. Hushchyn77, D.
Hutchcroft59, D. Hynds31, P. Ibis14, M. Idzik34, P. Ilten52, A. Inglessi37, A.
Inyakin43, K. Ivshin37, R. Jacobsson47, S. Jakobsen47, J. Jalocha62, E.
Jans31, B.K. Jashal46, A. Jawahery65, V. Jevtic14, F. Jiang3, M. John62, D.
Johnson47, C.R. Jones54, B. Jost47, N. Jurik62, S. Kandybei50, M. Karacson47,
J.M. Kariuki53, S. Karodia58, N. Kazeev77, M. Kecke16, F. Keizer54, M.
Kelsey67, M. Kenzie54, T. Ketel32, B. Khanji47, A. Kharisova78, C.
Khurewathanakul48, K.E. Kim67, T. Kirn13, V.S. Kirsebom48, S. Klaver22, K.
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Kopciewicz34, R. Kopecna16, P. Koppenburg31, M. Korolev39, I. Kostiuk31,51, O.
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Kreps55, F. Kress60, S. Kretzschmar13, P. Krokovny42,x, W. Krupa34, W.
Krzemien35, W. Kucewicz33,l, M. Kucharczyk33, V. Kudryavtsev42,x, H.S.
Kuindersma31, G.J. Kunde66, A.K. Kuonen48, T. Kvaratskheliya38, D.
Lacarrere47, G. Lafferty61, A. Lai26, D. Lancierini49, J.J. Lane61, G.
Lanfranchi22, C. Langenbruch13, O. Lantwin49, T. Latham55, F. Lazzari28,v, C.
Lazzeroni52, R. Le Gac10, R. Lefèvre9, A. Leflat39, F. Lemaitre47, O. Leroy10,
T. Lesiak33, B. Leverington16, H. Li70, L. Li62, P.-R. Li5, X. Li66, Y. Li6,
Z. Li67, X. Liang67, R. Lindner47, P. Ling70, F. Lionetto49, V. Lisovskyi11,
G. Liu70, X. Liu3, D. Loh55, A. Loi26, J. Lomba Castro45, I. Longstaff58, J.H.
Lopes2, G. Loustau49, G.H. Lovell54, Y. Lu6, D. Lucchesi27,o, M. Lucio
Martinez31, Y. Luo3, A. Lupato27, E. Luppi20,g, O. Lupton55, A. Lusiani28,t,
X. Lyu5, R. Ma70, S. Maccolini19,e, F. Machefert11, F. Maciuc36, V. Macko48,
P. Mackowiak14, S. Maddrell-Mander53, L.R. Madhan Mohan53, O. Maev37,47, A.
Maevskiy77, K. Maguire61, D. Maisuzenko37, M.W. Majewski34, S. Malde62, B.
Malecki47, A. Malinin75, T. Maltsev42,x, H. Malygina16, G. Manca26,f, G.
Mancinelli10, R. Manera Escalero44, D. Manuzzi19,e, D. Marangotto25,q, J.
Maratas9,w, J.F. Marchand8, U. Marconi19, S. Mariani21, C. Marin Benito11, M.
Marinangeli48, P. Marino48, J. Marks16, P.J. Marshall59, G. Martellotti30, L.
Martinazzoli47, M. Martinelli47,24,i, D. Martinez Santos45, F. Martinez
Vidal46, A. Massafferri1, M. Materok13, R. Matev47, A. Mathad49, Z. Mathe47,
V. Matiunin38, C. Matteuzzi24, K.R. Mattioli79, A. Mauri49, E. Maurice11,b, M.
McCann60,47, L. Mcconnell17, A. McNab61, R. McNulty17, J.V. Mead59, B.
Meadows64, C. Meaux10, G. Meier14, N. Meinert73, D. Melnychuk35, S.
Meloni24,i, M. Merk31, A. Merli25, E. Michielin27, M. Mikhasenko47, D.A.
Milanes72, E. Millard55, M.-N. Minard8, O. Mineev38, L. Minzoni20,g, S.E.
Mitchell57, B. Mitreska61, D.S. Mitzel47, A. Mödden14, A. Mogini12, R.D.
Moise60, T. Mombächer14, I.A. Monroy72, S. Monteil9, M. Morandin27, G.
Morello22, M.J. Morello28,t, J. Moron34, A.B. Morris10, A.G. Morris55, R.
Mountain67, H. Mu3, F. Muheim57, M. Mukherjee7, M. Mulder31, D. Müller47, J.
Müller14, K. Müller49, V. Müller14, C.H. Murphy62, D. Murray61, P. Muzzetto26,
P. Naik53, T. Nakada48, R. Nandakumar56, A. Nandi62, T. Nanut48, I. Nasteva2,
M. Needham57, N. Neri25,q, S. Neubert16, N. Neufeld47, R. Newcombe60, T.D.
Nguyen48, C. Nguyen-Mau48,n, E.M. Niel11, S. Nieswand13, N. Nikitin39, N.S.
Nolte47, C. Nunez79, A. Oblakowska-Mucha34, V. Obraztsov43, S. Ogilvy58, D.P.
O’Hanlon19, R. Oldeman26,f, C.J.G. Onderwater74, J. D. Osborn79, A.
Ossowska33, J.M. Otalora Goicochea2, T. Ovsiannikova38, P. Owen49, A.
Oyanguren46, P.R. Pais48, T. Pajero28,t, A. Palano18, M. Palutan22, G.
Panshin78, A. Papanestis56, M. Pappagallo57, L.L. Pappalardo20,g, W. Parker65,
C. Parkes61,47, G. Passaleva21,47, A. Pastore18, M. Patel60, C.
Patrignani19,e, A. Pearce47, A. Pellegrino31, G. Penso30, M. Pepe Altarelli47,
S. Perazzini19, D. Pereima38, P. Perret9, L. Pescatore48, K. Petridis53, A.
Petrolini23,h, A. Petrov75, S. Petrucci57, M. Petruzzo25,q, B. Pietrzyk8, G.
Pietrzyk48, M. Pikies33, M. Pili62, D. Pinci30, J. Pinzino47, F. Pisani47, A.
Piucci16, V. Placinta36, S. Playfer57, J. Plews52, M. Plo Casasus45, F.
Polci12, M. Poli Lener22, M. Poliakova67, A. Poluektov10, N. Polukhina76,c, I.
Polyakov67, E. Polycarpo2, G.J. Pomery53, S. Ponce47, A. Popov43, D. Popov52,
S. Poslavskii43, K. Prasanth33, L. Promberger47, C. Prouve45, V. Pugatch51, A.
Puig Navarro49, H. Pullen62, G. Punzi28,p, W. Qian5, J. Qin5, R. Quagliani12,
B. Quintana9, N.V. Raab17, R.I. Rabadan Trejo10, B. Rachwal34, J.H.
Rademacker53, M. Rama28, M. Ramos Pernas45, M.S. Rangel2, F. Ratnikov41,77, G.
Raven32, M. Ravonel Salzgeber47, M. Reboud8, F. Redi48, S. Reichert14, F.
Reiss12, C. Remon Alepuz46, Z. Ren3, V. Renaudin62, S. Ricciardi56, D.S.
Richards56, S. Richards53, K. Rinnert59, P. Robbe11, A. Robert12, A.B.
Rodrigues48, E. Rodrigues64, J.A. Rodriguez Lopez72, M. Roehrken47, S.
Roiser47, A. Rollings62, V. Romanovskiy43, M. Romero Lamas45, A. Romero
Vidal45, J.D. Roth79, M. Rotondo22, M.S. Rudolph67, T. Ruf47, J. Ruiz Vidal46,
A. Ryzhikov77, J. Ryzka34, J.J. Saborido Silva45, N. Sagidova37, N. Sahoo55,
B. Saitta26,f, C. Sanchez Gras31, C. Sanchez Mayordomo46, B. Sanmartin
Sedes45, R. Santacesaria30, C. Santamarina Rios45, M. Santimaria22,47, E.
Santovetti29,j, G. Sarpis61, A. Sarti30, C. Satriano30,s, A. Satta29, M.
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Schellenberg14, M. Schiller58, H. Schindler47, M. Schmelling15, T.
Schmelzer14, B. Schmidt47, O. Schneider48, A. Schopper47, H.F. Schreiner64, M.
Schubiger31, S. Schulte48, M.H. Schune11, R. Schwemmer47, B. Sciascia22, A.
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Serrano10, L. Sestini27, A. Seuthe14, P. Seyfert47, D.M. Shangase79, M.
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E. Shmanin76, J.D. Shupperd67, B.G. Siddi20, R. Silva Coutinho49, L. Silva de
Oliveira2, G. Simi27,o, S. Simone18,d, I. Skiba20,g, N. Skidmore16, T.
Skwarnicki67, M.W. Slater52, J.G. Smeaton54, A. Smetkina38, E. Smith13, I.T.
Smith57, M. Smith60, A. Snoch31, M. Soares19, L. Soares Lavra1, M.D.
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S. Stefkova60, O. Steinkamp49, S. Stemmle16, O. Stenyakin43, M. Stepanova37,
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Zonneveld57, S. Zucchelli19,e.

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

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kUniversità di Roma La Sapienza, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mDS4DS, 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
vUniversità di Siena, Siena, Italy
wMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
xNovosibirsk State University, Novosibirsk, Russia
yINFN Sezione di Trieste, Trieste, Italy
zPhysics and Micro Electronic College, Hunan University, Changsha City, China
†Deceased