EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2021-062 LHCb-PAPER-2020-017

Search for $C\\!P$ violation

in ${{\mathchar 28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$
decays

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

A search for $C\\!P$ violation in charmless three-body ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ decays is performed
using $pp$ collision data recorded with the LHCb detector, corresponding to
integrated luminosities of $1\text{\,fb}^{-1}$ at a centre-of-mass energy
$\sqrt{s}=7\text{\,Te\kern-1.00006ptV}$, $2\text{\,fb}^{-1}$ at
$\sqrt{s}=8\text{\,Te\kern-1.00006ptV}$ and $2\text{\,fb}^{-1}$ at
$\sqrt{s}=13\text{\,Te\kern-1.00006ptV}$. A good description of the phase-
space distribution is obtained with an amplitude model containing
contributions from ${\mathchar 28934\relax}(1385)$, ${\mathchar
28931\relax}(1405)$, ${\mathchar 28931\relax}(1520)$, ${\mathchar
28931\relax}(1670)$, ${\mathchar 28934\relax}(1775)$ and ${\mathchar
28934\relax}(1915)$ resonances. The model allows for $C\\!P$-violation
effects, which are found to be consistent with zero. The branching fractions
of ${{\mathchar 28932\relax}^{-}_{b}}\\!\rightarrow{\mathchar
28934\relax}(1385){{K}^{-}}$, ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{\mathchar 28931\relax}(1405){{K}^{-}}$,
${{\mathchar 28932\relax}^{-}_{b}}\\!\rightarrow{\mathchar
28931\relax}(1520){{K}^{-}}$, ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{\mathchar 28931\relax}(1670){{K}^{-}}$,
${{\mathchar 28932\relax}^{-}_{b}}\\!\rightarrow{\mathchar
28934\relax}(1775){{K}^{-}}$ and ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{\mathchar 28934\relax}(1915){{K}^{-}}$
decays are also reported. In addition, an upper limit is placed on the product
of ratios of ${\mathchar 28938\relax}^{-}_{b}$ and ${\mathchar
28932\relax}^{-}_{b}$ fragmentation fractions and the ${{\mathchar
28938\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ and ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ branching fractions.

Published in Phys. Rev. D 104, 052010

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

##  1 Introduction

In the Standard Model (SM), $C\\!P$ violation, defined as the breaking of
symmetry under the combined charge conjugation and parity operations, owes its
origin to a single irreducible complex phase in the Cabibbo–Kobayashi–Maskawa
(CKM) matrix [1, 2]. All effects of $C\\!P$ violation in particle decays
observed so far are consistent with this paradigm. However, the degree of
$C\\!P$ violation permitted in the SM is inconsistent with the observed
matter-antimatter asymmetry in the Universe [3, 4]. This motivates further
searches for sources of $C\\!P$ violation beyond the SM.

Interference between two amplitudes with different weak and strong phases
leads to $C\\!P$ violation in decay, where weak phases are those that change
sign under $C\\!P$ conjugation while strong phases do not. In the SM, weak
phases are associated with the complex elements of the CKM matrix and strong
phases are associated with hadronic final-state effects. Two such amplitudes
are potentially present in decays of $b$ hadrons to final states that do not
contain charm quarks, which therefore provide fertile ground for studies of
$C\\!P$ violation. Significant asymmetries have been observed between $B$ and
$\kern 1.79993pt\overline{\kern-1.79993ptB}$ partial widths in ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$
[5, 6, 7, 8, 9] ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$
[5, 6, 10] and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$
[7, 8] decays. Even larger $C\\!P$-violation effects have been observed in
regions of the phase space of ${{B}^{-}}$ decays to
${{\pi}^{+}}{{\pi}^{-}}{{\pi}^{-}}$, ${{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$,
${{K}^{+}}{{K}^{-}}{{K}^{-}}$ and ${{K}^{+}}{{K}^{-}}{{\pi}^{-}}$ final states
[11, 12, 13, 14, 15, 16].

Breaking of $C\\!P$ symmetry has not yet been observed in the properties of
any baryon. Tests of this symmetry have been performed through studies of
${\mathchar 28931\relax}^{0}_{b}$ baryon decays to ${p}{{\pi}^{-}}$,
${p}{{K}^{-}}$ [7, 17], ${{K}^{0}_{\mathrm{S}}}{p}{{\pi}^{-}}$ [18],
${\mathchar 28931\relax}{{K}^{+}}{{K}^{-}}$, ${\mathchar
28931\relax}{{K}^{+}}{{\pi}^{-}}$ [19],
${p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$, ${p}{{\pi}^{-}}{{K}^{+}}{{K}^{-}}$,
${p}{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ and ${p}{{K}^{-}}{{K}^{+}}{{K}^{-}}$ [20,
21, 22] final states, as well as ${\mathchar 28932\relax}^{0}_{b}$ decays to
${p}{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ and ${p}{{K}^{-}}{{\pi}^{+}}{{K}^{-}}$
[21, 22]. No significant evidence of $C\\!P$ violation has been found in any
of these studies, nor in measurements of the properties of charm baryon decays
[37]. In light of the large $C\\!P$-violation effects observed in three-body
charmless decays of $B$ mesons, it is of great interest to extend the range of
searches in $b$-baryon decays. In particular, the recently observed
${{\mathchar 28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ decay
[23] provides an interesting new opportunity to search for $C\\!P$-violation
effects.

In this paper, the first amplitude analysis of ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ decays is reported.
This is also the first amplitude analysis of any $b$-baryon decay mode
allowing for $C\\!P$-violation effects. A search for the previously unobserved
${{\mathchar 28938\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ decay
is also presented. The analysis reported here is performed using proton-proton
($pp$) collision data recorded with the LHCb detector, corresponding to
integrated luminosities of $1\text{\,fb}^{-1}$ at a centre-of-mass energy of
$\sqrt{s}=7\text{\,Te\kern-1.00006ptV}$ collected in 2011, $2\text{\,fb}^{-1}$
at $\sqrt{s}=8\text{\,Te\kern-1.00006ptV}$ in 2012 and $2\text{\,fb}^{-1}$ at
$\sqrt{s}=13\text{\,Te\kern-1.00006ptV}$ in 2015 and 2016. The data-taking
period of 2011 and 2012 is referred to hereafter as Run 1 and that of 2015 and
2016 as Run 2. The inclusion of charge-conjugate processes is implied
throughout the paper, except where asymmetries are discussed.

This paper is organised as follows. Section 2 gives a brief description of the
LHCb detector, trigger requirements and simulation software. The signal
candidate selection procedure is set out in Sec. 3. In Sec. 4, the procedure
for estimating the signal and background yields that enter the amplitude fit
is explained. Section 5 covers the modelling of the distribution of decays
across the phase space. Sections 6 and 7 contain a description of the
systematic uncertainties associated with the analysis procedure and a
presentation of the results, respectively. A brief summary of the analysis is
given in Sec. 8.

##  2 Detector, trigger and simulation

The LHCb detector [24, 25] 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 GeV.111Natural units with $\hbar=c=1$ are used
throughout this paper. The minimum distance of a track to a primary $pp$
collision vertex (PV), the impact parameter (IP), is measured with a
resolution of $(15+29/p_{\mathrm{T}})\,\upmu\text{m}$, where $p_{\mathrm{T}}$
is the component of the momentum transverse to the beam direction, in  GeV.
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 hadronic calorimeter. Muons are
identified by a system composed of alternating layers of iron and multiwire
proportional chambers. The magnetic field deflects oppositely charged
particles in opposite directions and this can lead to detection asymmetries.
Periodically reversing the magnetic field polarity throughout the data-taking
reduces this effect to a negligible level. Approximately 60% of 2011 data, 50%
of 2012 data, 61% of 2015 data and 53% of 2016 data were collected in the
“down” polarity configuration, and the rest in the “up” configuration.

The online event selection is performed by a trigger, which consists of a
hardware stage, based on information from the calorimeter and muon systems,
followed by a software stage, which applies a full event reconstruction.
During offline analysis, reconstructed candidates are associated with trigger
decisions. Events considered in the analysis are required to have been
triggered at the hardware level in one of two ways: either through one of the
final-state tracks of the signal decay depositing sufficient energy in the
calorimeter system, or by one of the other tracks in the event, not
reconstructed as part of the signal candidate, fulfilling any hardware trigger
requirement. At the software stage, it is required that at least one charged
particle associated to the $b$-hadron candidate has high $p_{\mathrm{T}}$ and
high $\chi^{2}_{\text{IP}}$, where $\chi^{2}_{\text{IP}}$ is defined as the
difference in PV fit $\chi^{2}$ with and without the inclusion of a specific
particle. A multivariate algorithm [26] is used to identify secondary vertices
consistent with being a two- or three-track $b$-hadron decay. The PVs are
fitted with and without the tracks that comprise the $b$-baryon candidate, and
the PV that gives the smallest $\chi^{2}_{\text{IP}}$ is associated with the
candidate. Finally, the momentum scale for charged particles is calibrated
using samples of
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\\!\rightarrow{\mu^{+}\mu^{-}}$,
${{B}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}$ and
$\mathchar 28931\relax\\!\rightarrow{p}\pi^{-}$ decays collected concurrently
with the data sample used for this analysis [27, 28].

Simulation samples are used to investigate background from other $b$-hadron
decays and to study the detection and reconstruction efficiency of the signal.
In the simulation, $pp$ collisions are generated using Pythia [29,
*Sjostrand:2007gs] with a specific LHCb configuration [31]. Decays of unstable
particles are described by EvtGen [32], in which final-state radiation is
generated using Photos [33]. The interaction of the generated particles with
the detector, and its response, are implemented using the Geant4 toolkit [34,
*Agostinelli:2002hh] as described in Ref. [36].

##  3 Offline selection

The offline selection consists of an initial filtering stage followed by a
requirement on the output of a multivariate algorithm (MVA). Compared to the
procedure applied to select the ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ channel in Ref.
[23], improvements in both stages lead to a significant increase in
efficiency. In particular, the inclusion in the multivariate algorithm of
particle identification (PID) variables that distinguish the final-state
charged hadrons from misidentified particles, is found to separate signal from
background effectively.

In the filtering stage, tracks are required to be of good quality, to satisfy
$p>1500\text{\,Me\kern-1.00006ptV}$ and
$p_{\mathrm{T}}>250\text{\,Me\kern-1.00006ptV}$, and to be displaced from all
PVs. Tracks associated to proton candidates must, at this stage, satisfy a
loose PID requirement and all tracks are required to not be associated to hits
in the muon system. Each $b$-hadron (henceforth denoted as $X^{-}_{b}$)
candidate must form a good-quality decay vertex that is separated
significantly from any PV, and must be consistent with originating from its
associated PV. Only $X^{-}_{b}$ candidates with
$p_{\mathrm{T}}>3500\text{\,Me\kern-1.00006ptV}$ and invariant mass
$5545<m({p}{{K}^{-}}{{K}^{-}})<6470\text{\,Me\kern-1.00006ptV}$ are retained
for further analysis.

In the selected $m({p}{{K}^{-}}{{K}^{-}})$ range there are three main
categories of background that contribute: combinatorial background that
results from random association of unrelated tracks; partially reconstructed
background due to $b$-hadron decays into final states similar to the signal,
but with additional soft particles that are not reconstructed; and cross-feed
background that results from misidentification of one or more final-state
particles. The MVA classifier is designed primarily to reduce combinatorial
background while retaining high signal efficiency, but also has some
discriminating power against the other background sources. It is trained with
a signal sample comprised of simulated ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ decays generated
uniformly across the phase space, and a background sample obtained from
candidates in data in the sideband regions
$5545.0<m({p}{{K}^{-}}{{K}^{-}})<5634.4\text{\,Me\kern-1.00006ptV}$ and
$6209.0<m({p}{{K}^{-}}{{K}^{-}})<6470.0\text{\,Me\kern-1.00006ptV}$. The
latter of these regions is dominated by combinatorial background, as its lower
threshold excludes possible ${{\mathchar
28938\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ decays from the
sample. The former region includes also contributions from sources of
partially reconstructed background such as ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{N}({p}{{\pi}^{0}}){{K}^{-}}{{K}^{-}}$ or
${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{{K}^{*-}}({{K}^{-}}{{\pi}^{0}}){{K}^{-}}{p}$
decays. Potential cross-feed background from
${{{B}^{-}}}\\!\rightarrow{{K}^{+}}{{K}^{-}}{{K}^{-}}$ decays is removed by
assigning the proton candidate the kaon mass and vetoing the
$m({{K}^{+}}{{K}^{-}}{{K}^{-}})$ region within $\pm
45\text{\,Me\kern-1.00006ptV}$ around the known ${{B}^{-}}$ mass [37]. This
veto corresponds to approximately $\pm 3$ times the invariant mass resolution
for ${{{B}^{-}}}\\!\rightarrow{{K}^{+}}{{K}^{-}}{{K}^{-}}$ decays.

Variables that exhibit good discriminating power between the signal and
background samples are chosen as inputs to the MVA. These are: the angle
between the $X^{-}_{b}$ candidate’s momentum vector and the line connecting
its decay vertex to its associated PV; the scalar sum of the $p_{\mathrm{T}}$
of all final-state tracks; the $\chi^{2}_{\text{IP}}$ of the highest
$p_{\mathrm{T}}$ final-state track and of the $X^{-}_{b}$ candidate; the
square of the significance of the distance between the $X^{-}_{b}$ decay
vertex and its associated PV; the vertex fit $\chi^{2}$ per degree of freedom
of the $X^{-}_{b}$ candidate; the minimum change in the $X^{-}_{b}$ candidate
vertex fit $\chi^{2}$ when including an additional track; variables that
characterise the PID information of the proton and kaon candidates; and a
variable that quantifies the isolation of the $X^{-}_{b}$ candidate. The last
of these is defined as the $p_{\mathrm{T}}$ asymmetry between the $X^{-}_{b}$
candidate and the tracks within a circle, centred on the $X^{-}_{b}$ candidate
(but excluding its decay products), with a radius
$\sqrt{\delta\eta^{2}+\delta\phi^{2}}<1.7$ in the space of pseudorapidity
$\eta$ and azimuthal angle $\phi$ (in radians) around the beam direction [38].

To describe accurately the proton and kaon PID variables, the quantities in
simulation are resampled according to values obtained from data calibration
samples of ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{-}}$,
${{D}^{+}_{s}}\\!\rightarrow\phi{{\pi}^{+}}$ and
${{D}^{*+}}\\!\rightarrow{{D}^{0}}{{\pi}^{+}}$ decays [39]. The procedure
accounts for correlations between the variables associated to a particular
track, as well as the dependence of the PID response on $p_{\mathrm{T}}$,
$\eta$, and the multiplicity of tracks in the event. All other MVA input
variables show good agreement between simulation and data, as validated with a
control sample of ${{{B}^{-}}}\\!\rightarrow{p}{\overline{p}}{{K}^{-}}$
decays. The MVA input variables are also found to not be correlated strongly
either with $X^{-}_{b}$ candidate mass or with position in the phase space of
the decay.

Several types of MVA classifier are investigated, with a gradient boosted
decision tree algorithm giving the best performance [40]. Four classifiers are
trained separately with samples separated by data-taking period (Run 1 or Run
2) and by even or odd event numbers. The event number identifies the proton-
proton bunch crossing, from which the $X^{-}_{b}$ candidate was recorded, in a
certain operational period of the experiment. To avoid possible MVA
overtraining, for each data-taking period the classifier trained on the sample
with even event numbers is validated and employed on the sample with odd event
numbers, and vice versa.

A threshold on the output of the MVA is chosen to maximise ${\cal
N}_{S}/\sqrt{{\cal N}_{S}+{\cal N}_{B}}$, where ${\cal N}_{S}$ and ${\cal
N}_{B}$ represent the estimated numbers of ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ signal and
combinatorial background candidates, respectively, within a signal region of
$\pm 40\text{\,Me\kern-1.00006ptV}$ around the ${\mathchar
28932\relax}^{-}_{b}$ mass from Ref. [41]. This range corresponds to
approximately $\pm 2.5$ times the ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ invariant mass
resolution. The value of ${\cal N}_{S}$ is estimated using the signal
efficiency evaluated from simulation, multiplied by the ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ branching fraction,
the ${\mathchar 28932\relax}^{-}_{b}$ fragmentation fraction, the
${b}{\overline{b}}$ production cross-section [42] and the integrated
luminosity for the relevant data-taking period. The product of the
${{\mathchar 28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$
branching fraction and the ${\mathchar 28932\relax}^{-}_{b}$ fragmentation
fraction is obtained from the results of Ref. [23], where the
${{{B}^{-}}}\\!\rightarrow{{K}^{+}}{{K}^{-}}{{K}^{-}}$ channel is used for
normalisation, by multiplying the ${{B}^{-}}$ fragmentation fraction in the
relevant kinematic range [43] and the
${{{B}^{-}}}\\!\rightarrow{{K}^{+}}{{K}^{-}}{{K}^{-}}$ branching fraction
[37]. The value of ${\cal N}_{B}$ is estimated from data by fitting the region
$6125<m({p}{{K}^{-}}{{K}^{-}})<6470\text{\,Me\kern-1.00006ptV}$ with a linear
function and extrapolating the result into the signal region. The MVA output
requirements have efficiencies of about $52\%$ and $61\%$ for Run 1 and Run 2,
respectively, with combinatorial background rejection of about $98\%$ for both
data-taking periods. The choice of ${\cal N}_{S}/\sqrt{{\cal N}_{S}+{\cal
N}_{B}}$ as the figure of merit is intended to obtain a sufficiently large
data sample to make an amplitude analysis viable. After all offline selection
requirements are applied, each selected event contains a single $X^{-}_{b}$
candidate.

The variables describing the phase space of the decay, which are used in the
amplitude analysis, are calculated following a kinematic fit in which the
$X^{-}_{b}$ candidate mass is fixed to the ${\mathchar 28932\relax}^{-}_{b}$
mass from Ref. [41]. This procedure improves resolution of these variables and
ensures that all decays remain within the phase-space boundary. The difference
between the ${\mathchar 28932\relax}^{-}_{b}$ mass value used in this fit and
recent more precise results [44, 45, 46] has negligible impact on the
analysis. The experimental resolution of the $m({p}{{K}^{-}})$ invariant mass,
in the region with the narrowest resonance considered in this analysis, the
$\mathchar 28931\relax(1520)$ state, is expected to be around
$1.5\text{\,Me\kern-1.00006ptV}$. This is smaller than the $\mathchar
28931\relax(1520)$ width, and therefore effects related to finite resolution
in the phase-space variables are not considered further.

The expected ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ signal efficiency,
assuming uniform distribution of decays across the phase space and taking into
account the LHCb detector acceptance, reconstruction and both online and
offline selection criteria, is $\left(1.159\pm 0.005\right)\%$ for Run 1 and
$\left(1.748\pm 0.006\right)\%$ for Run 2. The corresponding ${{\mathchar
28938\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ signal efficiencies
are $\left(1.257\pm 0.005\right)\%$ and $\left(1.921\pm 0.006\right)\%$. The
quoted uncertainties are due to the limited size of the simulation samples
only.

##  4 $\boldsymbol{X^{-}_{b}}$ candidate mass fit

Distributions of $m({p}{{K}^{-}}{{K}^{-}})$ for selected $X^{-}_{b}$
candidates are shown in Fig. 1 for Run 1 and Run 2 separately. The signal
yields are obtained from unbinned extended maximum-likelihood fits to these
distributions. The fit model is composed of signal and background components
whose shape parameters are mostly obtained from fits to the corresponding
simulation samples, after imposing the same selection requirements as on the
data. One exception is the combinatorial background component, which is
modelled by an exponential function with slope parameter allowed to vary
freely in the fit to data.

Signal ${{\mathchar 28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$
and ${{\mathchar 28938\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$
components are each modelled with the sum of two Crystal Ball (CB) functions
[47] where the core width and peak position are shared and with independent
power-law tails on both sides. The tail parameters and the relative
normalisation of the CB functions are determined from simulation. The peak
positions are fixed to the ${\mathchar 28932\relax}^{-}_{b}$ mass from Ref.
[44] and the known ${\mathchar 28938\relax}^{-}_{b}$ mass [37], and a scale
factor relating the width in data to that in simulation is introduced.

A possible cross-feed background contribution from ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{-}}$ decays [23], where
the pion is misidentified as a kaon, is modelled with the sum of two CB
functions. All shape parameters of this function are fixed according to the
values obtained from a fit to simulation but the width is scaled by the same
factor as the signal components. The phase-space distribution of these decays
is not known, and the simulation sample is weighted according to a model,
inspired by the $m({p}{{K}^{-}})$ and $m({p}{{\pi}^{-}})$ mass spectra
observed in ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{p}{{K}^{-}}$
[48] and ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{p}{{\pi}^{-}}$
[49] decays, which consists of the ${\mathchar 28931\relax}(1405)$,
${\mathchar 28931\relax}(1520)$, ${\mathchar 28931\relax}(1690)$, $N(1440)$,
$N(1520)$, $N(1535)$ and $N(1650)$ resonances. The yield of the ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{-}}$ cross-feed
component is expressed relative to the ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ signal yield and
constrained within uncertainty according to the previous branching fraction
ratio measurement [23] and relative selection efficiency. The expected
relative yields are $0.15\pm 0.06$ and $0.14\pm 0.03$ for Run 1 and Run 2,
respectively.

Partially reconstructed and combinatorial background contributions are also
included in the fit model. It is found that the $m({p}{{K}^{-}}{{K}^{-}})$
distributions of various potential sources of partially reconstructed
background, such as ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{N}({p}{{\pi}^{0}}){{K}^{-}}{{K}^{-}}$ or
${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{{K}^{*-}}({{K}^{-}}{{\pi}^{0}}){{K}^{-}}{p}$
decays, are very similar [48]. Therefore, the baseline fit model includes a
single partially reconstructed background component, which is modelled from
simulated ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{{K}^{*-}}({{K}^{-}}{{\pi}^{0}}){{K}^{-}}{p}$
decays with an ARGUS function [50] convolved with a Gaussian function. The
threshold of the ARGUS function is fixed to the known value of $m_{{{\mathchar
28932\relax}^{-}_{b}}}-m_{{{\pi}^{0}}}$ [37, 44], and the width parameter of
the Gaussian function is taken from the fit to simulation and scaled by the
same factor as the signal components. Negligible contributions are expected
from partially reconstructed ${\mathchar 28938\relax}^{-}_{b}$ decays, such as
${{\mathchar
28938\relax}^{-}_{b}}\\!\rightarrow{{K}^{*-}}({{K}^{-}}{{\pi}^{0}}){{K}^{-}}{p}$.

The results of the fits to Run 1 and Run 2 data are shown in Table 1 and Fig.
1. The free parameters of each fit are the two signal yields, the partially
reconstructed and combinatorial background yields, the width scale factor and
the exponential shape parameter of the combinatorial background, while the
cross-feed background yield is constrained to its expectation relative to the
${{\mathchar 28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ signal
yield.

Figure 1: Distributions of ${p}{{K}^{-}}{{K}^{-}}$ invariant mass for $X^{-}_{b}$ candidates in (left) Run 1 and (right) Run 2 data with results of the unbinned extended maximum-likelihood fits superimposed. The total fit result is shown as the blue solid curve, with individual components shown as indicated in the legend.  Table 1: Yields obtained from fits to the $m({p}{{K}^{-}}{{K}^{-}})$ distributions, in the full invariant mass range. The quoted uncertainties are statistical only.  Parameter | Run 1 | Run 2
---|---|---
${{\mathchar 28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ yield | 193$\,\pm\,$ | 21 | 297$\,\pm\,$ | 23
${{\mathchar 28938\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ yield |  $-4$$\,\pm\,$ | 6 | 15$\,\pm\,$ | 9
Partially reconstructed background yield | 231$\,\pm\,$ | 34 | 442$\,\pm\,$ | 36
Combinatorial background yield | 721$\,\pm\,$ | 50 | 775$\,\pm\,$ | 51

Only candidates in the $m({p}{{K}^{-}}{{K}^{-}})$ signal region of $\pm
40\text{\,Me\kern-1.00006ptV}$ around the ${\mathchar 28932\relax}^{-}_{b}$
mass from Ref. [41] are retained for the amplitude analysis. In this region,
the yields of the signal, cross-feed and combinatorial components are $N_{\rm
sig}=181\pm 20$, $N_{\rm cf}=16\pm 7$ and $N_{\rm comb}=90\pm 6$ for Run 1,
and $N_{\rm sig}=278\pm 21$, $N_{\rm cf}=25\pm 6$ and $N_{\rm comb}=95\pm 6$
for Run 2, where the quoted uncertainties are statistical only. These
correspond to signal purities of $(63\pm 3)\%$ and $(70\pm 2)\%$ for Run 1 and
Run 2, respectively. The contribution from partially reconstructed background
in the signal region is negligible.

No significant signal from the ${{\mathchar
28938\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ decay is observed.
The results of the fits are used to set limits on the product of its branching
fraction with the fragmentation fraction for ${\mathchar 28938\relax}^{-}_{b}$
production, normalised to the corresponding quantities for ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ decay, i.e.

$\mathcal{R}=\frac{f_{{{\mathchar 28938\relax}^{-}_{b}}}}{f_{{{\mathchar
28932\relax}^{-}_{b}}}}\times\frac{{\cal B}({{\mathchar
28938\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}})}{{\cal
B}({{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}})}=\frac{\epsilon({{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}})}{\epsilon({{\mathchar
28938\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}})}\times\frac{N({{\mathchar
28938\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}})}{N({{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}})}\,,$ (1)

where $N$ and $\epsilon$ denote yield and efficiency, respectively, for the
indicated mode, while $f_{{{\mathchar 28932\relax}^{-}_{b}}}$ and
$f_{{{\mathchar 28938\relax}^{-}_{b}}}$ are the ${\mathchar
28932\relax}^{-}_{b}$ and ${\mathchar 28938\relax}^{-}_{b}$ fragmentation
fractions. Results for the ratio $\mathcal{R}$ are reported, both for Run 1
and Run 2 separately and combined, in Sec. 7.

##  5 Amplitude analysis

The phase space of the three-body decay of a, potentially polarised, $b$
baryon has five degrees of freedom. A baseline assumption is made that
${\mathchar 28932\relax}^{-}_{b}$ baryons produced in ${p}{p}$ collisions
within the LHCb acceptance have negligible polarisation, as observed for
${\mathchar 28931\relax}^{0}_{b}$ baryons [51, 52]. As a result, the phase
space of the ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{K}^{-}_{1}{K}^{-}_{2}$ decay is
characterised by two independent kinematic variables (subscripts here
distinguish the two kaons in the final state). Since no resonances are
expected to decay to ${K}^{-}_{1}{K}^{-}_{2}$, these variables chosen are the
squared invariant masses $m^{2}({p}{K}^{-}_{1})$ and $m^{2}({p}{K}^{-}_{2})$.
The presence of two identical kaons in the final state imposes a Bose symmetry
that the decay amplitudes must be invariant under the exchange of these two
particles. As a result, the two-dimensional distribution of
$m^{2}({p}{K}^{-}_{1})$ and $m^{2}({p}{K}^{-}_{2})$ has a symmetry under
interchange of the variables. This motivates the use of the variables $m_{\rm
low}^{2}$ and $m_{\rm high}^{2}$, which denote the lower and higher of
$m^{2}({p}{K}^{-}_{1})$ and $m^{2}({p}{K}^{-}_{2})$, respectively, effectively
removing a duplicated half of the
$\left(m^{2}({p}{K}^{-}_{1}),m^{2}({p}{K}^{-}_{2})\right)$ plane. References
hereafter to the Dalitz plot (DP) of ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ decays refer to the
two-dimensional $\left(m_{\rm low}^{2},m_{\rm high}^{2}\right)$ distribution.
The DP distributions of selected candidates in Run 1 and Run 2 are shown in
the top row of Fig. 2.

It is common practice in amplitude analysis to use the so-called “square”
Dalitz plot (SDP) variables [53, 15], which in this case are defined as

$m^{\prime}=\frac{1}{\pi}\arccos{\left(2\frac{m({{K}^{-}}{{K}^{-}})-m_{\rm\text{min}}({{K}^{-}}{{K}^{-}})}{m_{\rm\text{max}}({{K}^{-}}{{K}^{-}})-m_{\rm\text{min}}({{K}^{-}}{{K}^{-}})}\right)}\quad\text{and}\quad\theta^{\prime}=\frac{1}{\pi}\theta({{K}^{-}}{{K}^{-}})\,.$
(2)

Here $m_{\rm\text{min}}({{K}^{-}}{{K}^{-}})=2m_{K}$ and
$m_{\rm\text{max}}({{K}^{-}}{{K}^{-}})=m_{{\mathchar 28932\relax}_{b}}-m_{p}$
represent the kinematic limits of $m({{K}^{-}}{{K}^{-}})$ for ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ decay and
$\theta({{K}^{-}}{{K}^{-}})$ is the angle between one ${K}^{-}$ direction in
the ${{K}^{-}}{{K}^{-}}$ centre-of-mass frame and the direction of the
${{K}^{-}}{{K}^{-}}$ system in the ${\mathchar 28932\relax}^{-}_{b}$ centre-
of-mass frame. The symmetry of the final state requires that distributions are
symmetric with respect to $\theta^{\prime}=0.5$, so only the region
$\theta^{\prime}\in\left[0,0.5\right]$ is considered. These SDP variables
provide improved granularity, when using uniform binning, in the regions close
to the DP boundaries that tend to be populated most densely. This is
beneficial for example in the modelling of the signal efficiency. Furthermore,
the mapping to a square space aligns the bin boundaries to the kinematic
boundaries of the phase space. As such, all efficiencies and background
distributions in the analysis are obtained as functions of the SDP variables.
The SDP distributions of selected candidates in Run 1 and Run 2 are shown in
the bottom row of Fig. 2.

Figure 2: Distributions of selected candidates from (left) Run 1 and (right)
Run 2 data in the (top) Dalitz-plot and (bottom) square Dalitz-plot
representations of the phase space.

###  5.1 Modelling of the signal component

The probability density function (PDF) for the signal component is expressed
as

$\mathcal{P}^{Q}_{\rm
sig}(\Omega)=\frac{\epsilon^{Q}(\Omega)}{\Gamma}\frac{d\Gamma^{Q}}{d\Omega}\,,$
(3)

where $Q=+1$ for ${\mathchar 28932\relax}^{-}_{b}$ decays and $Q=-1$ for
${\kern 1.79993pt\overline{\kern-1.79993pt{\mathchar 28932\relax}}}{}^{+}_{b}$
decays and $\Omega$ denotes the phase space in terms of the DP variables. The
efficiency is denoted by $\epsilon^{Q}(\Omega)$, and can differ for $Q=+1$ and
$-1$ to accommodate efficiency asymmetries; as described in Sec. 5.2, the
efficiency maps are determined using SDP coordinates, denoted
$\Omega^{\prime}$, but at any point in the phase space
$\epsilon^{Q}(\Omega)=\epsilon^{Q}(\Omega^{\prime})$. The term
$d\Gamma^{Q}/d\Omega$ describes the differential decay densities for
${\mathchar 28932\relax}^{-}_{b}$ and ${\kern
1.79993pt\overline{\kern-1.79993pt{\mathchar 28932\relax}}}{}^{+}_{b}$ decays,
including both local and overall rate asymmetries and the normalisation factor
$\Gamma$ is

$\Gamma=\int_{\Omega}\left(\epsilon^{Q=+1}(\Omega)\frac{d\Gamma^{Q=+1}}{d\Omega}+\epsilon^{Q=-1}(\Omega)\frac{d\Gamma^{Q=-1}}{d\Omega}\right)d\Omega\,.$
(4)

Equations (3) and (4) assume no asymmetry in the production rates of
${\mathchar 28932\relax}^{-}_{b}$ and ${\kern
1.79993pt\overline{\kern-1.79993pt{\mathchar 28932\relax}}}{}^{+}_{b}$ baryons
produced within the LHCb acceptance from high-energy $pp$ collisions,
consistent with measurement [46]. The effect of such a production asymmetry
would, in this analysis, mimic a global (i.e. phase-space independent)
difference between $\epsilon^{Q=+1}$ and $\epsilon^{Q=-1}$, and therefore the
systematic uncertainty due to this assumption can be evaluated
straightforwardly.

The differential decay density is expressed as

$\frac{d\Gamma^{Q}}{d\Omega}=\frac{1}{\left(8\pi m_{{\mathchar
28932\relax}_{b}}\right)^{3}}\sum_{M_{{\mathchar
28932\relax}_{b}},\lambda_{{p}}}\left|\sum_{R}A^{Q}_{R,M_{{\mathchar
28932\relax}_{b}},\lambda_{{p}}}(\Omega)\right|^{2}\,,$ (5)

where $A^{Q}_{R,M_{{\mathchar 28932\relax}_{b}},\lambda_{{p}}}$ denotes the
symmetrised decay amplitude for a given intermediate state $R$, ${\mathchar
28932\relax}^{-}_{b}$ spin component along a chosen quantisation axis
$M_{{\mathchar 28932\relax}_{b}}$, and proton helicity $\lambda_{{p}}$. The
quantisation axis is chosen to be the direction opposite to the proton
momentum in the ${\mathchar 28932\relax}^{-}_{b}$ rest frame, and the proton
helicity is defined in the rest frame of the ${{K}^{-}}{{K}^{-}}$ system to
ensure explicit symmetry between the ${p}{K}^{-}_{\rm low}$ and
${p}{K}^{-}_{\rm high}$ decay chains. Here ${K}^{-}_{\rm low}$ is the kaon
whose four-momentum is used in the definition of $m_{\rm low}^{2}$ and
${K}^{-}_{\rm high}$ denotes the other kaon. The amplitude in Eq. (5) has been
summed incoherently over the spins of the initial and final states
(corresponding to an average over initial states) and coherently over all
contributing intermediate states.

The helicity formalism is used to parametrise the decay dynamics. A detailed
description of this formalism can be found in Refs. [49, 54, 48, 55, 56]. In
particular, the Dalitz-plot decomposition procedure [56] is followed to
express the symmetrised decay amplitude as

$A^{Q}_{R,M_{{\mathchar 28932\relax}_{b}},\lambda_{p}}(m_{\rm low}^{2},m_{\rm
high}^{2})=T^{Q}_{R,M_{{\mathchar 28932\relax}_{b}},\lambda_{p}}(m_{\rm
low}^{2},m_{\rm high}^{2})+(-1)^{M_{{\mathchar
28932\relax}_{b}}+\lambda_{p}}T^{Q}_{R,M_{{\mathchar
28932\relax}_{b}},\lambda_{p}}(m_{\rm high}^{2},m_{\rm low}^{2})\,.$ (6)

The first term corresponds to the amplitude for the weak decay ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow R{K}^{-}_{\rm high}$, where $R$ decays to
${p}{K}^{-}_{\rm low}$ via the strong interaction. This decay amplitude is
expressed as

$\displaystyle\begin{split}T^{Q}_{R,M_{{\mathchar
28932\relax}_{b}},\lambda_{p}}(m_{\rm low}^{2},m_{\rm
high}^{2})=\sum_{\lambda_{R},\lambda_{p}^{\prime}}\Big{(}d_{M_{{\mathchar
28932\relax}_{b}},\lambda_{R}}^{J_{{\mathchar
28932\relax}_{b}}}(\theta_{R})&d_{\lambda_{R},\lambda_{p}^{\prime}}^{J_{R}}(\theta_{p})d_{\lambda_{p}^{\prime},\lambda_{p}}^{J_{p}}(\zeta)\\\
&\times\eta_{\lambda_{p}^{\prime}}(-1)^{\lambda_{p}^{\prime}-\lambda_{p}}h^{Q}_{R,\lambda_{R}}R(m_{\rm
low}^{2})\Big{)}\,,\end{split}$ (7)

where the amplitude is summed coherently over the allowed helicities of the
intermediate state, $\lambda_{R}$, and of the proton, $\lambda_{p}^{\prime}$,
defined in the rest frames of the ${\mathchar 28932\relax}^{-}_{b}$ and the
${p}{K}^{-}_{\rm low}$ systems, respectively. The ${\mathchar
28932\relax}^{-}_{b}$, $R$ and proton spins are denoted by $J_{{\mathchar
28932\relax}_{b}}$, $J_{R}$ and $J_{p}$, respectively.

The three functions of the form $d^{J}_{\lambda,\lambda^{\prime}}$ in Eq. (7)
are the small Wigner d-matrix elements [57] that impose angular momentum
conservation giving rise to the condition $|\lambda_{R}|\leq 1/2$. As a
result, for intermediate states with any half-integer spin, only helicities
corresponding to $\lambda_{R}=\pm 1/2$ contribute to the amplitude. The three
angles $\theta_{R}$, $\theta_{p}$ and $\zeta$ are functions of the DP
variables. The angle $\theta_{R}$, defined in the ${\mathchar
28932\relax}^{-}_{b}$ rest frame, is formed between the direction opposite to
the proton momentum and the combined momentum of the ${p}{K}^{-}_{\rm low}$
system. The angle $\theta_{p}$ is between the direction opposite to the
${K}^{-}_{\rm high}$ momentum in the ${\mathchar 28932\relax}^{-}_{b}$ rest
frame and the proton momentum in the rest frame of the ${p}{K}^{-}_{\rm low}$
system. The angle $\zeta$ gives the Wigner rotation that is required to relate
the proton helicity state, $|\lambda_{p}^{\prime}\rangle$, defined in the
${p}{K}^{-}_{\rm low}$ rest frame to the proton helicity state,
$|\lambda_{p}\rangle$, defined in the ${{K}^{-}}{{K}^{-}}$ rest frame. This
angle, computed in the proton rest frame, is formed between the momenta of
${K}^{-}_{\rm low}$ and of the ${{K}^{-}}{{K}^{-}}$ system. Mathematical
definitions of these three angles, each defined in the range $[0,\pi]$, are

$\displaystyle\cos\theta_{R}$ $\displaystyle=\frac{(m_{{\mathchar
28932\relax}_{b}}^{2}+m_{K}^{2}-m_{\rm low}^{2})(m_{{\mathchar
28932\relax}_{b}}^{2}+m_{p}^{2}-m_{{K}{K}}^{2})-2m_{{\mathchar
28932\relax}_{b}}^{2}(m_{\rm high}^{2}-m_{K}^{2}-m_{p}^{2})}{\sqrt{{\cal
K}(m_{{\mathchar 28932\relax}_{b}}^{2},m_{p}^{2},m_{{K}{K}}^{2})}\sqrt{{\cal
K}(m_{{\mathchar 28932\relax}_{b}}^{2},m_{\rm low}^{2},m_{K}^{2})}},$ (8)
$\displaystyle\cos\theta_{p}$ $\displaystyle=\frac{2m_{\rm low}^{2}(m_{\rm
high}^{2}-m_{K}^{2}-m_{p}^{2})-(m_{\rm
low}^{2}+m_{p}^{2}-m_{K}^{2})(m_{{\mathchar 28932\relax}_{b}}^{2}-m_{\rm
low}^{2}-m_{K}^{2})}{\sqrt{{\cal K}(m_{{\mathchar
28932\relax}_{b}}^{2},m_{K}^{2},m_{\rm low}^{2})}\sqrt{{\cal K}(m_{\rm
low}^{2},m_{p}^{2},m_{K}^{2})}}\,,$ (9) $\displaystyle\cos\zeta$
$\displaystyle=\frac{2m_{p}^{2}(m_{\rm high}^{2}-m_{{\mathchar
28932\relax}_{b}}^{2}-m_{K}^{2})+(m_{{\mathchar
28932\relax}_{b}}^{2}+m_{p}^{2}-m_{{K}{K}}^{2})(m_{\rm
low}^{2}-m_{p}^{2}-m_{K}^{2})}{\sqrt{{\cal K}(m_{{\mathchar
28932\relax}_{b}}^{2},m_{p}^{2},m_{{K}{K}}^{2})}\sqrt{{\cal K}(m_{\rm
low}^{2},m_{p}^{2},m_{K}^{2})}}\,,$ (10)

where $m_{{K}{K}}^{2}=m_{{\mathchar
28932\relax}_{b}}^{2}+2m_{K}^{2}+m_{p}^{2}-m_{\rm low}^{2}-m_{\rm high}^{2}$,
the Källén function is given by ${\cal
K}(a,b,c)=a^{2}+b^{2}+c^{2}-2(ab+ac+bc)$ and the ${\mathchar
28932\relax}^{-}_{b}$, $K$ and $p$ masses are denoted by $m_{{\mathchar
28932\relax}_{b}}$, $m_{K}$ and $m_{p}$, respectively.

The second term in Eq. (6) corresponds to the weak decay of ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow R\,{K}^{-}_{\rm low}$ where $R$ now decays
to $R\\!\rightarrow{p}{K}^{-}_{\rm high}$. The expression for this amplitude
can be obtained by interchanging $m_{\rm low}^{2}\leftrightarrow m_{\rm
high}^{2}$ in Eqs. (7)–(10).

The term $\eta_{\lambda_{p}^{\prime}}$, in Eq. (7), arises as a consequence of
parity conservation in the strong decay of the intermediate state $R$. It is
defined as

$\displaystyle\eta_{\lambda_{p}^{\prime}}\,=\,\begin{cases}1\text{ when
}\lambda_{p}^{\prime}=1/2\,,\\\ (-1)^{\frac{3}{2}-J_{R}}\eta_{R}\text{ when
}\lambda_{p}^{\prime}=-1/2\,,\end{cases}$

where $\eta_{R}$ is the intrinsic parity of $R$.

The complex coefficient $h^{Q}_{R,\lambda_{R}}$, in Eq. (7), encapsulates the
combined couplings of the weak decay of the initial state and the strong decay
of the intermediate state. This coefficient, subsequently referred to as the
helicity coupling, can be expressed as

$h^{Q}_{R,\lambda_{R}}=(x_{R,\lambda_{R}}+Q\,\delta
x_{R,\lambda_{R}})+i\,(y_{R,\lambda_{R}}+Q\,\delta y_{R,\lambda_{R}})\,,$ (11)

where $x_{R,\lambda_{R}}$ and $y_{R,\lambda_{R}}$ denote the real and
imaginary components of the $C\\!P$-conserving part of the coupling, while
$\delta x_{R,\lambda_{R}}$ and $\delta y_{R,\lambda_{R}}$ are
$C\\!P$-violating parameters.

The term $R(m_{\rm low}^{2})$, in Eq. (7), describes the lineshape of each
resonant or nonresonant contribution. Resonances are parametrised with
relativistic Breit–Wigner (RBW) functions, $F_{\rm RBW}$, that are modified by
Blatt–Weisskopf barrier factors, $B_{L_{{{\mathchar 28932\relax}_{b}}}}$ and
$B_{L_{R}}$, and are given by

$R(m^{2}_{x})=B_{L_{{{\mathchar
28932\relax}_{b}}}}(p|p_{0},d)\left(\frac{p}{m_{{{\mathchar
28932\relax}_{b}}}}\right)^{L_{{{\mathchar 28932\relax}_{b}}}}\;F_{\rm
RBW}(m^{2}_{x}|m_{0},\Gamma_{0})\;B_{L_{R}}(q|q_{0},d)\left(\frac{q}{m_{0}}\right)^{L_{R}}.$
(12)

Here $m^{2}_{x}$ is either $m_{\rm low}^{2}$ or $m_{\rm high}^{2}$, while $p$
is the magnitude of the resonance momentum in the ${{\mathchar
28932\relax}^{-}_{b}}$ centre-of-mass frame and $q$ is the magnitude of the
proton momentum in the resonance centre-of-mass frame. The symbols $p_{0}$ and
$q_{0}$ denote the values of these quantities at the resonance peak, i.e. when
$m_{x}=m_{0}$. The orbital angular momentum released in the ${\mathchar
28932\relax}^{-}_{b}$ decay is denoted $L_{{{\mathchar 28932\relax}_{b}}}$
while that in the resonance decay is denoted $L_{R}$. Angular momentum
conservation in the ${\mathchar 28932\relax}^{-}_{b}$ decay imposes the
condition $J_{R}-1/2\leq L_{{{\mathchar 28932\relax}_{b}}}\leq J_{R}+1/2$. The
minimal value $L_{{{\mathchar 28932\relax}_{b}}}=J_{R}-1/2$ is assumed when
calculating $R(m^{2}_{x})$. Angular momentum conservation in the resonance
decay limits $L_{R}$ to $J_{R}\pm\frac{1}{2}$, which is then uniquely defined
by parity conservation in the decay, $\eta_{R}=(-1)^{L_{R}+1}$.

The Blatt–Weisskopf barrier functions are

$\displaystyle B_{0}(k|k_{0},d)$ $\displaystyle=$ $\displaystyle 1\,,$ (13)
$\displaystyle B_{1}(k|k_{0},d)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1+(k_{0}\,d)^{2}}{1+(k\phantom{{}_{0}}\,d)^{2}}}\,,$
(14) $\displaystyle B_{2}(k|k_{0},d)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{9+3(k_{0}\,d)^{2}+(k_{0}\,d)^{4}}{9+3(k\phantom{{}_{0}}\,d)^{2}+(k\phantom{{}_{0}}\,d)^{4}}}\,,$
(15) $\displaystyle B_{3}(k|k_{0},d)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{225+45(k_{0}\,d)^{2}+6(k_{0}\,d)^{4}+(k_{0}\,d)^{6}}{225+45(k\phantom{{}_{0}}\,d)^{2}+6(k\phantom{{}_{0}}\,d)^{4}+(k\phantom{{}_{0}}\,d)^{6}}}\,,$
(16)

and account for suppression creating high values of the orbital angular
momentum $L$, which depends on the momentum of one of the decay products, $k$,
in the centre-of-mass frame of the decaying particle and on the size of the
decaying particle given by the constant $d$. The value
$d=5.0\text{\,Ge\kern-1.00006ptV}^{-1}$ is used for ${{\mathchar
28932\relax}^{-}_{b}}$ decays while $1.5\text{\,Ge\kern-1.00006ptV}^{-1}$ is
used for resonances [55].

The relativistic Breit–Wigner amplitude is given by

$F_{\rm
RBW}(m^{2}_{x}|m_{0},\Gamma_{0})=\frac{1}{m_{0}^{2}-m^{2}_{x}-im_{0}\Gamma(m_{x})}\,,$
(17)

where

$\Gamma(m_{x})=\Gamma_{0}\left(\frac{q}{q_{0}}\right)^{2\,L_{R}+1}\;\frac{m_{0}}{m_{x}}\;B^{\prime}_{L_{R}}(q,q_{0},d)^{2}\,.$
(18)

Here $m_{0}$ and $\Gamma_{0}$ denote the pole mass and width of the resonance,
respectively. In the case of the ${\mathchar 28931\relax}(1405)$ resonance,
which peaks below the ${p}{{K}^{-}}$ threshold, $m_{0}$ is replaced by an
effective mass in the kinematically allowed region [58],

$m^{\rm eff}_{0}=m^{\rm min}+(m^{\rm max}-m^{\rm
min})\left(1+\tanh{\left(\frac{m_{0}-\frac{m^{\rm max}+m^{\rm min}}{2}}{m^{\rm
max}-m^{\rm min}}\right)}\right)\,,$ (19)

where $m^{\rm max}$ and $m^{\rm min}$ are the upper and lower limits of the
kinematically allowed range, respectively. In this case, the $q_{0}$ value in
Eq. (18) is the value of $q$ at $m=m^{\rm eff}_{0}$. This parameterisation
ensures that only the tail of the RBW function enters the fit model as a
virtual contribution. Nonresonant components are modelled using an exponential
lineshape,

$R_{\rm NR}(m^{2}_{x})=\left(\frac{p}{m_{{{\mathchar
28932\relax}_{b}}}}\right)^{L_{{{\mathchar
28932\relax}_{b}}}^{R}}\left(\frac{q}{m_{0}}\right)^{L_{R}}\exp(-\alpha
m^{2}_{x})\,,$ (20)

where $\alpha$ is a slope parameter that is determined from the fit and
$m_{0}$ is fixed to be the midpoint of the $m_{\rm low}$ range, i.e.
$2.83\text{\,Ge\kern-1.00006ptV}$.

The primary outputs of the amplitude analysis are the $C\\!P$-conserving and
$C\\!P$-violating components of the helicity couplings introduced in Eq. (11).
However, since these depend on the choice of phase convention, amplitude
formalism and normalisation, they can be difficult to compare between
analyses. It is therefore more useful to report the fit fractions
$\mathcal{F}_{\rm i}$ for each intermediate component $i$ of the fit model,
defined by

$\mathcal{F}_{\rm
i}=\frac{\int_{\Omega}(d\Gamma^{+}_{i}/d\Omega+d\Gamma^{-}_{i}/d\Omega)d\Omega}{\int_{\Omega}(d\Gamma^{+}/d\Omega+d\Gamma^{-}/d\Omega)d\Omega}\,,$
(21)

where

$\frac{d\Gamma^{Q}_{i}}{d\Omega}=\frac{1}{\left(8\pi m_{{\mathchar
28932\relax}_{b}}\right)^{3}}\sum_{M_{{\mathchar
28932\relax}_{b}},\lambda_{{p}}}\left|A^{Q}_{i,M_{{\mathchar
28932\relax}_{b}},\lambda_{{p}}}(\Omega)\right|^{2}\,.$ (22)

It is also useful to report the interference fit fractions $\mathcal{I}_{ij}$
between the two intermediate components $i$ and $j$, defined by

$\mathcal{I}_{ij}=\frac{\int_{\Omega}(d\Gamma^{+}_{ij}/d\Omega+d\Gamma^{-}_{ij}/d\Omega)d\Omega}{\int_{\Omega}(d\Gamma^{+}/d\Omega+d\Gamma^{-}/d\Omega)d\Omega}\,,$
(23)

where

$\frac{d\Gamma^{Q}_{ij}}{d\Omega}=\frac{1}{\left(8\pi m_{{\mathchar
28932\relax}_{b}}\right)^{3}}\sum_{M_{{\mathchar
28932\relax}_{b}},\lambda_{{p}},i,j}2\,{\rm Re}\left(A^{Q}_{i,M_{{\mathchar
28932\relax}_{b}},\lambda_{{p}}}(A^{Q}_{j,M_{{\mathchar
28932\relax}_{b}},\lambda_{{p}}})^{*}\right)\,.$ (24)

The parameters of $C\\!P$ violation, $A^{{C\\!P}}_{i}$, associated with each
component $i$ of the model are also reported. These are defined as

$A^{{C\\!P}}_{i}=\frac{\int_{\Omega}(d\Gamma^{+}_{i}/d\Omega-d\Gamma^{-}_{i}/d\Omega)d\Omega}{\int_{\Omega}(d\Gamma^{+}_{i}/d\Omega+d\Gamma^{-}_{i}/d\Omega)d\Omega}\,.$
(25)

###  5.2 Modelling of signal efficiency and background distributions

The detector geometry and the online and offline selection procedure can
induce variation in the signal efficiency across the phase space of the decay.
This is accounted for, as shown in Eq. (3), by determining the efficiency as a
function of the SDP variables. The efficiency maps are obtained from
simulation, but with effects related to PID calibrated using data as outlined
in Sec. 3. The efficiency maps for ${\mathchar 28932\relax}^{-}_{b}$ and
${\kern 1.79993pt\overline{\kern-1.79993pt{\mathchar 28932\relax}}}{}^{+}_{b}$
decays can be seen in Fig. 3 separately for Run 1 and Run 2. These maps are
obtained by employing a uniform $10\times 10$ binning scheme and smoothing
with a two-dimensional cubic spline to mitigate effects of discontinuity at
the bin edges. No significant detection asymmetry is observed.

Figure 3: Efficiency as a function of square Dalitz-plot position for (left)
${\mathchar 28932\relax}^{-}_{b}$ and (right) ${\kern
1.61993pt\overline{\kern-1.61993pt{\mathchar 28932\relax}}}{}^{+}_{b}$ decays,
for (top) Run 1 and (bottom) Run 2.

Candidates selected in data in the sideband
$5890<m({p}{{K}^{-}}{{K}^{-}})<6470\text{\,Me\kern-1.00006ptV}$ are used to
model the SDP distribution of the combinatorial background, which dominates
this region as discussed in Sec. 4. The effect of the ${\mathchar
28932\relax}^{-}_{b}$ mass constraint used when calculating the SDP variables
causes a distortion of the distribution from that of combinatorial background
in the signal region. This is accounted for using a method [59] in which the
unnormalised function describing the $m({p}{{K}^{-}}{{K}^{-}})$ and
$\Omega^{\prime}=(m^{\prime},\theta^{\prime})$ space is expressed as

$F(m({p}{{K}^{-}}{{K}^{-}}),\Omega^{\prime})=\big{|}f_{0}(\Omega^{\prime})+\exp\left(-\beta\,m({p}{{K}^{-}}{{K}^{-}})\right)\times
f_{1}(\Omega^{\prime})\big{|}^{2}$ (26)

where $\beta$ is a free parameter determined from a fit to the sideband
candidates. The functions $f_{0}$ and $f_{1}$ are modelled using neural
networks that are trained using candidates from the data sideband region. This
model is then extrapolated to predict the PDF of the combinatorial background
at the ${\mathchar 28932\relax}^{-}_{b}$ mass, i.e. $\mathcal{P}_{\rm
comb}(\Omega^{\prime})=F(m_{{{\mathchar
28932\relax}_{b}}},\Omega^{\prime})/N$, where the normalisation factor
$N=\int_{\Omega^{\prime}}F(m_{{{\mathchar
28932\relax}_{b}}},\Omega^{\prime})d\Omega^{\prime}$. The PDF in terms of DP
variables is obtained using $\mathcal{P}_{\rm
comb}(\Omega)=|J|\mathcal{P}_{\rm comb}(\Omega^{\prime})$, where $|J|$ is the
Jacobian determinant of the transformation between variables,
$d\Omega=|J|d\Omega^{\prime}$. The SDP distributions of the combinatorial
background for Run 1 and Run 2 are shown in the top row of Fig. 4.

After imposing the selection criteria and splitting the data sample by the
initial state charge, too few candidates are available in the sideband region
to train the neural networks. As a result, the neural networks are trained
using the combined sample of ${\mathchar 28932\relax}^{-}_{b}$ and ${\kern
1.79993pt\overline{\kern-1.79993pt{\mathchar 28932\relax}}}{}^{+}_{b}$
candidates, and no asymmetry in the shape of the combinatorial background SDP
distribution is assumed in the baseline model.

Figure 4: SDP distributions of (top) combinatorial and (bottom) cross-feed
background components for (left) Run 1 and (right) Run 2.

The SDP distribution of cross-feed background from misidentified ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{-}}$ decays that enter
the signal region is modelled using simulation, and is shown in the bottom row
of Fig. 4 separately for Run 1 and Run 2. These distributions are described in
terms of a uniform $10\times 10$ binned SDP histogram, smoothed with a two-
dimensional cubic spline. As described in Sec. 4, the simulation is weighted
to reproduce resonance structures expected in the phase space of ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{-}}$ decays.
Differences in selection requirements, together with the limited statistics of
the ${{\mathchar 28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{-}}$
simulation samples, cause the PDFs to differ between Run 1 and Run 2. In the
baseline fit it is assumed that there is no asymmetry between ${\mathchar
28932\relax}^{-}_{b}$ and ${\kern 1.79993pt\overline{\kern-1.79993pt{\mathchar
28932\relax}}}{}^{+}_{b}$ candidates in the cross-feed yields or SDP
distributions.

###  5.3 Fitting procedure

The total PDF that is used to model the phase-space distributions of
${{\mathchar 28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ and its
conjugate decay is

$\displaystyle\mathcal{P}^{Q}_{\rm tot}(\Omega)=\frac{1}{N_{\rm
tot}}\Big{[}N_{\rm sig}\mathcal{P}^{Q}_{\rm sig}(\Omega)$
$\displaystyle+N_{\rm comb}\frac{(1-QA_{\rm comb})}{2}\mathcal{P}_{\rm
comb}(\Omega)$ $\displaystyle+\frac{N_{\rm cf}}{2}\mathcal{P}_{\rm
cf}(\Omega)\Big{]}\,.$ (27)

The yields of signal, combinatorial background and cross-feed background
components are denoted by $N_{\rm sig}$, $N_{\rm comb}$ and $N_{\rm cf}$,
respectively, and are obtained as described in Sec. 4, separately for Run 1
and Run 2. The quantity $N_{\rm tot}=N_{\rm sig}+N_{\rm comb}+N_{\rm cf}$ is
the total yield in the signal region. The PDFs for the signal, combinatorial
background and cross-feed background components are denoted by
$\mathcal{P}^{Q}_{\rm sig}(\Omega)$, $\mathcal{P}_{\rm comb}(\Omega)$ and
$\mathcal{P}_{\rm cf}(\Omega)$, respectively, where the former is given in Eq.
(3) and the latter two are displayed in Fig. 4 in terms of the SDP variables.
In the baseline model, only the signal PDF can differ for ${\mathchar
28932\relax}^{-}_{b}$ and ${\kern 1.79993pt\overline{\kern-1.79993pt{\mathchar
28932\relax}}}{}^{+}_{b}$ candidates, although a possible global combinatorial
background asymmetry, $A_{\rm comb}$, is a free parameter of the model.

An unbinned maximum-likelihood fit is performed to the combined sample of
candidates for ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ and its conjugate
decay to determine the parameters of the model, which are the
$C\\!P$-conserving and $C\\!P$-violating coefficients of the helicity
couplings of Eq. (11). The fit is performed simultaneously to the Run 1 and
Run 2 data samples, which have separate efficiency and background models as
described above. The fit model is implemented in a fitting package based on
TensorFlow [60], interfaced with the Minuit function minimisation algorithm
[61, 62]. The function that is minimised, twice the negative log-likelihood,
is

$-2\ln\mathcal{L}=-2\sum_{r}\left(\sum_{i}^{N_{r}}\ln\left(\mathcal{P}^{Q}_{\rm
tot}(\Omega_{i})\right)\right)\,,$ (28)

where the index $i$ runs over the $N_{r}$ candidates in the data sample from
run period $r$ (Run 1 or Run 2), and $\Omega_{i}$ denotes the DP coordinates
of candidate $i$.

###  5.4 Model selection

The ${{\mathchar 28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$
decay can proceed via intermediate ${p}{{K}^{-}}$ resonances. Various
${\mathchar 28931\relax}^{*}$ and ${\mathchar 28934\relax}^{*}$ resonances
that are known to decay to ${p}{{K}^{-}}$ are considered as potential
components of the signal model.222 All ${\mathchar 28934\relax}^{*}$
resonances considered are neutral; the conventional charge superscripts are
omitted for brevity of notation. The Particle Data Group (PDG) [37] reports a
large number of such states; those that are sufficiently well established are
considered in this study and are shown in Table 2. Masses and widths of all
resonance components are fixed to either the central value or the midpoint in
the range of values quoted in Table 2. Nonresonant components, labelled ${\rm
NR}(J^{P})$, with spin-parity
$J^{P}=\frac{1}{2}^{+},\frac{1}{2}^{-},\frac{3}{2}^{+},\frac{3}{2}^{-}$ are
also considered. Resonances with spin $J\geq 7/2$ are excluded from
consideration as these would require $L_{{{\mathchar 28932\relax}_{b}}}\geq 3$
and are therefore expected to be significantly suppressed.

Table 2: Summary of the considered ${\mathchar 28931\relax}^{*}$ and ${\mathchar 28934\relax}^{*}$ resonances, ranked either **** or *** by the PDG [37]. Note that the ${p}{{K}^{-}}$ threshold is at $1432\text{\,Me\kern-0.90005ptV}$. Resonances marked $\dagger$ are included in the baseline model, as described in the text. The spin-parity of the ${\mathchar 28934\relax}(2250)$ is not known and is assumed to be $\frac{3}{2}^{+}$. For many of these states, the PDG does not report masses and widths with central values and uncertainties, but rather gives real and imaginary parts of the pole position. This reflects the fact that a simple Breit–Wigner parameterisation of these resonances may not fully describe their lineshapes; however, more sophisticated parametrisations are beyond the scope of the current analysis.  | Name | $J^{P}$ | Mass $(\text{\,Me\kern-1.00006ptV})$ | Width $(\text{\,Me\kern-1.00006ptV})$ | Main decay channels
---|---|---|---|---|---
| ****
$\dagger$ | ${\mathchar 28931\relax}(1405)$ | $\frac{1}{2}^{-}$ | $1405.1\,^{+1.3}_{-1.0}$ | $50.5\pm 2.0$ | ${\mathchar 28934\relax}\pi$
$\dagger$ | ${\mathchar 28931\relax}(1520)$ | $\frac{3}{2}^{-}$ | 1518 to 1520 | 15 to 17 |  $N{\kern 1.79993pt\overline{\kern-1.79993ptK}}$, ${\mathchar 28934\relax}\pi$
$\dagger$ | ${\mathchar 28931\relax}(1670)$ | $\frac{1}{2}^{-}$ | 1660 to 1680 | 25 to 50 |  $N{\kern 1.79993pt\overline{\kern-1.79993ptK}}$, ${\mathchar 28934\relax}\pi$, ${\mathchar 28931\relax}\eta$
| ${\mathchar 28931\relax}(1690)$ | $\frac{3}{2}^{-}$ | 1685 to 1695 | 50 to 70 |  $N{\kern 1.79993pt\overline{\kern-1.79993ptK}}$, ${\mathchar 28934\relax}\pi$, ${\mathchar 28931\relax}\pi\pi$, ${\mathchar 28934\relax}\pi\pi$
| ${\mathchar 28931\relax}(1820)$ | $\frac{5}{2}^{+}$ | 1815 to 1825 | 70 to 90 | $N{\kern 1.79993pt\overline{\kern-1.79993ptK}}$
| ${\mathchar 28931\relax}(1830)$ | $\frac{5}{2}^{-}$ | 1810 to 1830 |  160 to 110 | ${\mathchar 28934\relax}\pi$
| ${\mathchar 28931\relax}(1890)$ | $\frac{3}{2}^{+}$ | 1850 to 1910 |  160 to 200 | $N{\kern 1.79993pt\overline{\kern-1.79993ptK}}$
$\dagger$ | ${\mathchar 28934\relax}(1385)$ | $\frac{3}{2}^{+}$ | $1383.7\pm 1$ | $36\pm 5$ | ${\mathchar 28931\relax}\pi,{\mathchar 28934\relax}\pi$
| ${\mathchar 28934\relax}(1670)$ | $\frac{3}{2}^{-}$ | 1665 to 1685 | 40 to 80 | ${\mathchar 28934\relax}\pi$
$\dagger$ | ${\mathchar 28934\relax}(1775)$ | $\frac{5}{2}^{-}$ | 1770 to 1780 | 105 to 135 |  $N{\kern 1.79993pt\overline{\kern-1.79993ptK}}$, ${\mathchar 28931\relax}^{(*)}\pi$
$\dagger$ | ${\mathchar 28934\relax}(1915)$ | $\frac{5}{2}^{+}$ | 1900 to 1935 |  180 to 160 | not clear
| ***
| ${\mathchar 28931\relax}(1600)$ | $\frac{1}{2}^{+}$ | 1560 to 1700 |  150 to 250 |  $N{\kern 1.79993pt\overline{\kern-1.79993ptK}}$, ${\mathchar 28934\relax}\pi$
| ${\mathchar 28931\relax}(1800)$ | $\frac{1}{2}^{-}$ | 1720 to 1850 | 200 to 400 |  $N{\kern 1.79993pt\overline{\kern-1.79993ptK}}^{(*)}$, ${\mathchar 28934\relax}\pi$, ${\mathchar 28931\relax}\eta$
| ${\mathchar 28931\relax}(1810)$ | $\frac{1}{2}^{+}$ | 1750 to 1850 |  150 to 250 |  $N{\kern 1.79993pt\overline{\kern-1.79993ptK}}^{(*)}$, ${\mathchar 28934\relax}\pi$, ${\mathchar 28931\relax}\eta$, $\Xi{K}$
| ${\mathchar 28931\relax}(2110)$ | $\frac{5}{2}^{+}$ | 2090 to 2140 | 150 to 250 |  $N{\kern 1.79993pt\overline{\kern-1.79993ptK}}^{(*)}$, ${\mathchar 28934\relax}\pi$, ${\mathchar 28931\relax}\Omega$
| ${\mathchar 28934\relax}(1660)$ | $\frac{1}{2}^{-}$ | 1630 to 1690 |  140 to 200 |  $N{\kern 1.79993pt\overline{\kern-1.79993ptK}}$, ${\mathchar 28934\relax}\pi$, ${\mathchar 28931\relax}\pi$
| ${\mathchar 28934\relax}(1750)$ | $\frac{1}{2}^{-}$ | 1730 to 1800 |  160 to 160 |  $N{\kern 1.79993pt\overline{\kern-1.79993ptK}}$, ${\mathchar 28934\relax}\pi$, ${\mathchar 28931\relax}\pi$, ${\mathchar 28934\relax}\eta$
| ${\mathchar 28934\relax}(1940)$ | $\frac{3}{2}^{-}$ | 1900 to 1950 | 150 to 300 |  $N{\kern 1.79993pt\overline{\kern-1.79993ptK}}$, ${\mathchar 28934\relax}\pi$, ${\mathchar 28931\relax}\pi$
| ${\mathchar 28934\relax}(2250)$ | $?^{?}$ | 2210 to 2280 |  160 to 150 |  $N{\kern 1.79993pt\overline{\kern-1.79993ptK}}$, ${\mathchar 28934\relax}\pi$, ${\mathchar 28931\relax}\pi$

In this analysis, as the normalisation of the decay density is arbitrary, the
${\mathchar 28931\relax}(1520)$ resonance is chosen as the reference
component. This implies that the coupling with the positive helicity of the
${\mathchar 28931\relax}(1520)$ resonance $a^{Q}_{R,\lambda_{R}=+1/2}$ is
real. Explicitly, for the reference ${\mathchar 28931\relax}(1520)$ resonance,
$y_{R,\lambda_{R}=+1/2}=\delta y_{R,\lambda_{R}=+1/2}=0$ and
$x_{R,\lambda_{R}=+1/2}=1$, while $\delta x_{R,\lambda_{R}=+1/2}$ is free to
vary in the fit to allow for $C\\!P$ violation in the ${\mathchar
28931\relax}(1520)$ amplitude. The analysis is found to be insensitive to the
coupling of the ${\mathchar 28931\relax}(1520)$ component with negative
helicity, and therefore $x_{R,\lambda_{R}=-1/2}=\delta
x_{R,\lambda_{R}=-1/2}=y_{R,\lambda_{R}=-1/2}=\delta y_{R,\lambda_{R}=-1/2}=0$
for the reference resonance. The helicity couplings of all other resonant and
nonresonant components are left free to vary in the fit.

To establish a baseline fit model, the ${\mathchar 28931\relax}(1520)$
component alone is initially included in the model, with additional components
added iteratively in the order that maximises the change in $-2\ln\mathcal{L}$
obtained from fits to the data with prospective models. Components with
different spin and parity should have zero interference fit fractions due to
the orthogonality relation satisfied by the small Wigner d-matrix elements.
However, the symmetrisation of the Dalitz plot can lead to non-zero values for
such interference fit fractions in this analysis. As a result, in establishing
the baseline model, it is possible to encounter “unphysical” interference fit
fractions ($>40\%$) between two components. When such a case occurs, the
component that gives the minimal change in $-2\ln\mathcal{L}$ when removed
from the fit model is discarded. The procedure is terminated when the change
in $-2\ln\mathcal{L}$ from including any further contribution is less than 9
units, limiting the potential for the model to be influenced by statistical
fluctuations. This approach leads to a model that contains ${\mathchar
28934\relax}(1385)$, ${\mathchar 28931\relax}(1405)$, ${\mathchar
28931\relax}(1520)$, ${\mathchar 28931\relax}(1670)$, ${\mathchar
28934\relax}(1775)$ and ${\mathchar 28934\relax}(1915)$ components. The
potential for additional components to be present in the true underlying model
is considered as a source of systematic uncertainty.

###  5.5 Fit to data

In an attempt to find the global minimum, a large number of fits to data are
performed, where the initial values of the helicity couplings are randomised.
The baseline results are obtained from the fit that returns the smallest
$-2\ln\mathcal{L}$ value out of this ensemble. This procedure is found to
converge successfully to the global minimum without any secondary minima.

The Dalitz-plot distribution of the combined Run 1 and Run 2 data sample is
compared to the model obtained from the fit to data, separately for
${\mathchar 28932\relax}^{-}_{b}$ and ${\kern
1.79993pt\overline{\kern-1.79993pt{\mathchar 28932\relax}}}{}^{+}_{b}$
candidates, in Fig. 5. Projections of the fit results onto $m_{\rm low}$ and
$m_{\rm high}$ are compared to the data in Fig. 6. Further comparisons of the
fit result and the data in regions of the phase space are presented in
Appendix A. There is no indication of $C\\!P$ violation, i.e. no significant
difference between ${\mathchar 28932\relax}^{-}_{b}$ and ${\kern
1.79993pt\overline{\kern-1.79993pt{\mathchar 28932\relax}}}{}_{b}^{+}$ decays,
in the distributions.

The overall agreement between the data and the model is good, with unbinned
goodness-of-fit tests using the mixed-sample and point-to-point dissimilarity
approaches [63] giving $p$-values of 0.20 and 0.25, respectively. In Fig. 6,
there is an apparent discrepancy between the model and the data at $m_{\rm
high}$ between $3.4\text{\,Ge\kern-1.00006ptV}$ to
$3.7\text{\,Ge\kern-1.00006ptV}$, predominantly in the ${\kern
1.79993pt\overline{\kern-1.79993pt{\mathchar 28932\relax}}}{}_{b}^{+}$ sample.
However, due to the symmetry of the final state, any structure that appears in
$m_{\rm high}$ should also appear in $m_{\rm low}$, where no such structure is
observed. Addition of extra components to the fit model does not significantly
improve the data description. Moreover, the apparent discrepancy in Fig. 6
does not take into account the systematic uncertainty in the mismodelling of
the combinatorial background, which is the largest component at $m_{\rm
high}\sim 3.7\text{\,Ge\kern-1.00006ptV}$. Therefore, this feature is not
considered to be significant and is not investigated further.

Figure 5: Dalitz-plot distributions from (left) data and (right) the fit model
for (top) ${\mathchar 28932\relax}^{-}_{b}$ and (bottom) ${\kern
1.61993pt\overline{\kern-1.61993pt{\mathchar 28932\relax}}}{}^{+}_{b}$
candidates.

Figure 6: Distributions of (top) $m_{\rm low}$ and (bottom) $m_{\rm high}$ for
(left) ${\mathchar 28932\relax}^{-}_{b}$ and (right) ${\kern
1.61993pt\overline{\kern-1.61993pt{\mathchar 28932\relax}}}{}^{+}_{b}$
candidates, with results of the fits superimposed. The total fit result is
shown as the blue solid curve, with contributions from individual signal
components and from combinatorial (Comb) and cross-feed (Crsfd) background
shown as indicated in the legend.

##  6 Systematic uncertainties

The outcomes of the analysis are the ratio $\mathcal{R}$ of ${\mathchar
28938\relax}^{-}_{b}$ and ${\mathchar 28932\relax}^{-}_{b}$ branching
fractions and fragmentation fractions (see Eq. (1)), as well as the fit
fractions, interference fit fractions and $C\\!P$-asymmetry parameters
obtained from the amplitude analysis. Various sources of systematic
uncertainty can affect these measurements. These are discussed one by one in
this section, concluding with a summary. Systematic uncertainties are
evaluated separately for Run 1 and Run 2, where appropriate.

###  6.1 Invariant mass fits

The fits to the $m({p}{{K}^{-}}{{K}^{-}})$ invariant mass distributions
determine the signal and background yields, which are used in both the
calculation of $\mathcal{R}$ and the amplitude analysis. Four sources of
systematic uncertainty arising from these fits are considered. The first is
due to the limited size of the data sample. This enters the calculation of
$\mathcal{R}$ as statistical uncertainty, but is a source of systematic
uncertainty in the amplitude analysis where the signal and background yields
are fixed parameters. To evaluate the associated systematic uncertainty, these
yields are varied according to the covariance matrix obtained from the
$m({p}{{K}^{-}}{{K}^{-}})$ fit and for each variation the fit to the phase-
space distribution is repeated. The root mean square (RMS) of the distribution
of the change in each result of the amplitude analysis is assigned as the
corresponding systematic uncertainty.

The second source relates to the $m({p}{{K}^{-}}{{K}^{-}})$ fit model. Models
for each of the components are varied to evaluate associated systematic
uncertainties: the signal model is replaced with a Hypatia function [64]; the
combinatorial background model is replaced with a second-order Chebyshev
polynomial; the cross-feed and partially reconstructed background models are
replaced with kernel density estimates; additional partially reconstructed
background components are included; the model used to describe the phase-space
distribution of the ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{-}}$ cross-feed
background is varied. In each case, the change in each result from its
baseline value is taken as the systematic uncertainty.

The third source concerns fixed parameters in the $m({p}{{K}^{-}}{{K}^{-}})$
fit that are taken from simulation. An ensemble of pseudoexperiments is
generated using the nominal values of these parameters, and each is then
fitted many times with parameters fixed to alternative values obtained using
the covariance matrices of the fits to the simulation samples. The standard
deviation of the change in each result is evaluated for every
pseudoexperiment, and its average value over the ensemble is assigned as the
systematic uncertainty.

Finally, potential fit bias is investigated by generating multiple
pseudoexperiments with yield and fit parameters obtained from the nominal
$m({p}{{K}^{-}}{{K}^{-}})$ fit. The difference between the mean fit result of
the ensemble and the nominal value is assigned as the associated systematic
uncertainty.

###  6.2 Selection efficiency maps

The efficiency maps are altered to evaluate systematic uncertainties from six
sources. In each case the difference between the results obtained using the
alternative efficiency maps and that with the baseline efficiency maps is
assigned as the systematic uncertainty.

The first source reflects uncertainties in the $p_{\mathrm{T}}$ distribution
of ${\mathchar 28932\relax}^{-}_{b}$ baryons produced in the LHCb acceptance.
Alternative efficiency maps are obtained where the simulation samples are
weighted so that the $p_{\mathrm{T}}$ distribution matches that of the
background-subtracted data. Since there is no significant signal of
${\mathchar 28938\relax}^{-}_{b}$ baryons, they are assumed to have the same
$p_{\mathrm{T}}$ distribution as ${\mathchar 28932\relax}^{-}_{b}$ baryons and
the ${{\mathchar 28938\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$
efficiency map is altered in the same way.

The second source is a possible mismatch of the hardware trigger efficiency
between simulation and data, which could arise due to miscalibration of
transverse energy measurements from the calorimeter. Alternative efficiency
maps are obtained by applying corrections that are calculated, as a function
of track $p_{\mathrm{T}}$, using control samples of kaons from
${{D}^{*+}}\\!\rightarrow{{D}^{0}}({{K}^{-}}{{\pi}^{+}}){{\pi}^{+}}$ decays
and protons from ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{{\mathchar
28931\relax}^{+}_{c}}({p}{{K}^{-}}{{\pi}^{+}}){{\pi}^{-}}$ decays.

The third source is due to uncertainty arising from binning the phase space
when evaluating the efficiency maps. Alternative efficiency maps are obtained
employing a different SDP binning scheme. An additional systematic uncertainty
is associated to the efficiency of ${{\mathchar
28938\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ decays, and arises
from their unknown phase-space distribution. The standard deviation of the
variation of the efficiency across the binned SDP histogram is assigned as the
corresponding uncertainty.

The remaining sources relate to particle identification. The PID variables
used in the MVA are drawn from data calibration samples accounting for
dependence on the signal kinematics. Systematic uncertainties in this
procedure arise from the limited statistics of both the simulation and
calibration samples, and the modelling of the PID variables in the calibration
samples. The limitations due to both simulation and calibration sample size
are evaluated by bootstrapping to create multiple samples, and repeating the
procedure for each sample. The impact of potential mismodelling of the PID
variables in the calibration samples is evaluated by describing the
corresponding distributions using density estimates with different kernel
widths. For each of these cases, alternative efficiency maps are produced to
determine the associated uncertainties on the results of the analysis.

In principle, mismodelling of the proton and kaon reconstruction efficiencies,
and associated asymmetries, could be a source of systematic uncertainty.
However, such effects are known to be negligible at the level of precision
achieved in this analysis [LHCb-PAPER-2014-013, LHCb-PAPER-2021-016], and
therefore are not accounted for explicitly.

###  6.3 Background shapes

The combinatorial background SDP distribution is obtained by extrapolating
from an $m({p}{{K}^{-}}{{K}^{-}})$ sideband region, and has uncertainties
related to the available yield in the sideband and the extrapolation procedure
itself. The former is evaluated by bootstrapping to create multiple
combinatorial background samples, and repeating the amplitude fit with each.
The RMS of the distribution of the change in each result is taken as the
systematic uncertainty. The latter is evaluated by changing the architecture
of the neural network, with the change in each result with respect to its
baseline value assigned as the associated systematic uncertainty.

In the baseline fit, the ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{-}}$ cross-feed
background is described with a model consisting of ${\mathchar
28931\relax}(1405)$, ${\mathchar 28931\relax}(1520)$, ${\mathchar
28931\relax}(1690)$, $N(1440)$, $N(1520)$, $N(1535)$ and $N(1650)$ resonances.
To evaluate the systematic uncertainty arising from this assumption, the model
is modified by adding ${\mathchar 28931\relax}(1600)$, ${\mathchar
28931\relax}(1670)$, ${\mathchar 28931\relax}(1800)$ and $N(1720)$ components,
and removing the $N(1520)$ component. The change in each result with respect
to its baseline value is assigned as the associated systematic uncertainty.

###  6.4 Background asymmetry

In the baseline model, it is assumed that there is no local asymmetry in the
combinatorial background as described in Sec. 5.2. The associated systematic
uncertainty is evaluated by considering separate background distributions for
${\mathchar 28932\relax}^{-}_{b}$ and ${\kern
1.79993pt\overline{\kern-1.79993pt{\mathchar 28932\relax}}}{}^{+}_{b}$
candidates. In order to obtain sufficiently large background samples to
determine these separate distributions, the MVA output requirement for
candidates in the sideband region is relaxed.

A possible global combinatorial background asymmetry is accounted for in the
baseline fit, while cross-feed background is assumed to have no asymmetry. A
fit allowing for a global cross-feed background asymmetry is performed, and
the differences between the results in this fit and their nominal values is
assigned as the systematic uncertainty arising from this assumption.

###  6.5 Production asymmetry

The baseline fit model assumes no asymmetry in the ${\mathchar
28932\relax}^{-}_{b}$ production rates in the LHCb acceptance, consistent with
measurements [46]. To evaluate the associated systematic uncertainty, the
model is adjusted to include production asymmetries within the experimentally
allowed range by introducing a global asymmetry in the efficiency maps. An
ensemble of fits with varied ${\mathchar 28932\relax}^{-}_{b}$ production
asymmetries is performed, and the RMS of the distribution of the change in
each result with respect to its baseline value is assigned as the systematic
uncertainty.

###  6.6 Polarisation

The transverse polarisation of ${\mathchar 28932\relax}^{-}_{b}$ baryons
produced in $pp$ collisions is assumed to be consistent with zero, as observed
for ${\mathchar 28931\relax}^{0}_{b}$ baryons [51, 52]. The distributions of
$m_{\rm low}^{2}$ and $m_{\rm high}^{2}$ are independent of the ${\mathchar
28932\relax}^{-}_{b}$ polarisation. However, if the ${\mathchar
28932\relax}^{-}_{b}$ baryons produced in LHC collisions are polarised,
efficiency variation across additional phase-space variables should be
considered in the analysis. To evaluate the systematic uncertainty due to
potential ${\mathchar 28932\relax}^{-}_{b}$ polarisation, two sets of
pseudoexperiments are generated, with the ${\mathchar 28932\relax}^{-}_{b}$
polarisation set to $20\%$ in one case and $-8\%$ in the other. This
corresponds to the $\pm 2\sigma$ range measured for the ${\mathchar
28931\relax}^{0}_{b}$ baryon in Ref. [51, 52], where $\sigma$ indicates the
Gaussian standard deviation. A conservatively broad range is taken to allow
for differences between ${\mathchar 28932\relax}^{-}_{b}$ and ${\mathchar
28931\relax}^{0}_{b}$ polarisation. The pseudoexperiments are generated using
a signal model whose helicity couplings are set to the values from the nominal
fit and where the measured efficiency variation over the additional phase-
space observables is introduced. A fit to the Dalitz-plot variables is then
performed using the baseline model. The largest deviation of the parameters
from the nominal case is assigned as the systematic uncertainty.

###  6.7 Modelling of the lineshapes

Each resonant contribution has fixed parameters in the amplitude fit. These
include masses, widths and Blatt–Weisskopf radius parameters. An ensemble of
fits is obtained varying the masses and widths of all resonances within the
range of values quoted by the PDG and given in Table 2. The Blatt–Weisskopf
radius parameter associated with the ${\mathchar 28932\relax}^{-}_{b}$ baryon
is varied in the range $3$–$7\text{\,Ge\kern-1.00006ptV}^{-1}$ and that
associated with the resonances is varied in the range
$0$–$3.5\text{\,Ge\kern-1.00006ptV}^{-1}$. The RMS of the distribution of the
change in each fitted parameter is taken as the systematic uncertainty.

For the ${\mathchar 28931\relax}(1405)$ and ${\mathchar 28934\relax}(1385)$
resonances that peak below the $m_{\rm low}$ threshold, the effective RBW used
in the baseline fit is replaced with a lineshape equivalent to the Flatté
parameterisation as done in Ref. [48]. The total width is modified to account
for the $\mathchar 28934\relax^{+}\pi^{-}$ channel, i.e.
$\Gamma(m)=\Gamma_{pK^{-}}(m)+\Gamma_{\mathchar 28934\relax^{+}\pi^{-}}(m)$,
assuming equal couplings to both channels. For each of these lineshape
variations, the differences in the results between fits with the alternative
and baseline models are assigned as the associated systematic uncertainties.

###  6.8 Alternative fit model

The effect of including additional signal components in the fit model is
examined to assign systematic uncertainty due to the composition of the
baseline model. The ${\mathchar 28931\relax}(1690)$, ${\mathchar
28931\relax}(1820)$, ${\mathchar 28934\relax}(1670)$ and ${\rm
NR}(\frac{3}{2}^{+})$ components are added to the nominal model individually.
These modifications of the model are chosen since they improve
$-2\ln\mathcal{L}$, although not by a significant amount. The largest
deviation, among the four cases, from the nominal value of each measured
quantity is taken as the associated systematic uncertainty.

###  6.9 Summary of systematic uncertainties

#### Ratio of fragmentation and branching fractions:

Since separate fits are performed to the $m({p}{{K}^{-}}{{K}^{-}})$
distributions from the Run 1 and Run 2 samples, and the signal efficiencies
are also determined separately, results for $\mathcal{R}$ in each of the two
samples are obtained. Systematic uncertainties on $\mathcal{R}$ are considered
as being either completely uncorrelated or 100% correlated between the two
results. The systematic uncertainties that are uncorrelated between Run 1 and
Run 2 are folded into their respective likelihood functions, by convolution
with a Gaussian of appropriate width. The correlated systematic uncertainties
are later folded into the combined likelihood that is obtained by multiplying
the likelihood functions of the two samples.

The uncorrelated systematic uncertainties are those that are related to the
fixed parameters in the fit model, the fit bias and the impact on the
efficiency of the ${\mathchar 28932\relax}^{-}_{b}$ production kinematics, and
the descriptions of the hardware trigger and particle identification response.
The correlated systematic uncertainties are those related to knowledge of the
phase-space distributions of the decays and the fit model choice. Slightly
different procedures are used to obtain the total uncertainty for the two
sources of correlated systematic uncertainty. That related to knowledge of the
phase-space distribution affects the efficiency, and hence is a constant
relative uncertainty. The method to evaluate the uncertainty due to fit model
choice gives different relative uncertainties between Run 1 and Run 2. Since
the two samples have approximately equal statistical weight in the
combination, the average of the relative uncertainties is taken and assigned
to the combined result. Table 3 summarises the systematic uncertainties on
$\mathcal{R}$.

Table 3: Absolute systematic uncertainties on $\mathcal{R}$, in units of
$10^{-3}$, from (top) uncorrelated and (bottom) correlated sources. The total
is the sum in quadrature of all contributions.

Uncorrelated sources | Run 1 | Run 2
---|---|---
${\mathchar 28932\relax}^{-}_{b}$ $p_{\mathrm{T}}$ distribution | ${<\,}0.1\phantom{\,<}$ | 0.7
Hardware trigger efficiency | 0.1 | 1.6
PID efficiency | 0.1 | 0.6
Fixed parameters | 0.8 | 0.5
Fit bias | 0.5 | ${<\,}0.1\phantom{\,<}$
Total | 1.0 | 1.9

Correlated sources | Run 1 | Run 2 | Combined
---|---|---|---
Phase-space distribution | 8.9 | 22.5 | 10.6
Fit model choice | 9.1 | 13.1 |  18.6
Total | – | – | 13.6

#### Amplitude analysis:

The results of the amplitude analysis are the $C\\!P$ asymmetry parameters
$A^{{C\\!P}}$, defined in Eq. (25), the fit fractions $\mathcal{F}$ defined in
Eq. (21) and the interference fit fractions $\mathcal{I}$ defined in Eq. (23).
A summary of the systematic uncertainties on these quantities is shown in
Table 4.

The most precise results are those related to the ${\mathchar
28931\relax}(1520)$ and ${\mathchar 28931\relax}(1670)$ resonances. For
${\mathchar 28931\relax}(1520)$, the dominant systematic uncertainty on
$A^{{C\\!P}}$ is due to ignoring the efficiency variation over angular
variables if ${\mathchar 28932\relax}^{-}_{b}$ baryons are produced polarised,
whereas for $\mathcal{F}$ it is due to the limited size of the sample used for
combinatorial background modelling and the variation of the Blatt–Weisskopf
radius parameters. For ${\mathchar 28931\relax}(1670)$, the largest systematic
uncertainty on $A^{{C\\!P}}$ is due to variation of the ${\mathchar
28931\relax}(1405)$ lineshape which has the same spin and parity as this
component, whereas for $\mathcal{F}$ it is due to use of an alternate fit
model. For ${\mathchar 28934\relax}(1385)$, the dominant systematic
uncertainty on $A^{{C\\!P}}$ is due to use of an alternate fit model, whereas
for $\mathcal{F}$ it is due to variation of the Blatt–Weisskopf radius
parameters. For ${\mathchar 28931\relax}(1405)$, the largest systematic
uncertainty on $A^{{C\\!P}}$ is due to use of an alternate fit model, whereas
for $\mathcal{F}$ it is due to variation in its lineshape. For ${\mathchar
28934\relax}(1775)$, the dominant systematic uncertainty on $A^{{C\\!P}}$ is
due to the limited size of the sample used for modelling of the combinatorial
background, whereas for $\mathcal{F}$ it is due to use of an alternate fit
model. For ${\mathchar 28934\relax}(1915)$, the dominant systematic
uncertainty on both $A^{{C\\!P}}$ and $\mathcal{F}$ is due to use of an
alternate fit model.

For interference fit fractions, the largest systematic uncertainties are
mainly due to the use of an alternate fit model, the limited size of the
sample used for modelling of the combinatorial background, variation of the
resonance lineshapes and variation of Blatt–Weisskopf radius parameters.

Table 4: Summary of absolute systematic uncertainties, in units of $10^{-2}$,
on the results of the amplitude analysis: (top) $C\\!P$ asymmetry parameters
$A^{{C\\!P}}$ and fit fractions $\mathcal{F}$, (bottom) interference fit
fractions $\mathcal{I}$. The total is the sum in quadrature of all individual
sources.

Component & Parameter | Mass fits | Bkg shapes | Bkg asym | Eff | Prod asym | Polarisation | RBW params | Lineshapes | Alt fit model | Total
---|---|---|---|---|---|---|---|---|---|---
${\mathchar 28934\relax}(1385)$ | $A^{{C\\!P}}$ | 3.3 | 20.62 | 4.4 | 8.2 | 6.9 | 15.01 | 7.0 | 12.61 | 65.56 | 72.77
2-12 | ${\cal F}$ | 1.4 | 3.1 | 0.5 | 0.7 | 0.2 | 1.0 | 5.0 | 0.6 | 4.4 | 7.6
${\mathchar 28931\relax}(1405)$ | $A^{{C\\!P}}$ | 2.4 | 9.6 | 2.7 | 5.1 | 5.1 | 5.5 | 5.1 | 19.51 | 20.62 | 31.93
2-12 | ${\cal F}$ | 0.3 | 1.4 | 0.3 | 0.8 | ${<\,}0.1\phantom{\,<}$ | 0.3 | 0.3 | 2.3 | 0.9 | 3.0
${\mathchar 28931\relax}(1520)$ | $A^{{C\\!P}}$ | 0.3 | 0.9 | 0.6 | 2.9 | 4.3 | 5.0 | 0.7 | 1.2 | 1.3 | 7.6
2-12 | ${\cal F}$ | 1.1 | 1.8 | ${<\,}0.1\phantom{\,<}$ | 1.3 | ${<\,}0.1\phantom{\,<}$ | 0.6 | 1.8 | 0.6 | 1.7 | 3.6
${\mathchar 28931\relax}(1670)$ | $A^{{C\\!P}}$ | 1.8 | 4.2 | 1.4 | 2.9 | 4.4 | 3.3 | 3.7 | 4.9 | 1.6 | 10.11
2-12 | ${\cal F}$ | 0.8 | 2.3 | 0.1 | 0.7 | 0.1 | 0.6 | 1.4 | 1.8 | 4.4 | 5.6
${\mathchar 28934\relax}(1775)$ | $A^{{C\\!P}}$ | 2.5 | 7.8 | 1.7 | 3.1 | 3.4 | 7.0 | 3.8 | 4.7 | 3.7 | 13.81
2-12 | ${\cal F}$ | 0.5 | 1.5 | 0.1 | 0.4 | 0.2 | 0.2 | 1.0 | 0.9 | 3.5 | 4.1
${\mathchar 28934\relax}(1915)$ | $A^{{C\\!P}}$ | 2.5 | 6.7 | 5.0 | 6.4 | 4.8 | 5.2 | 10.51 | 2.1 | 13.91 | 21.82
2-12 | ${\cal F}$ | 0.2 | 2.3 | 0.1 | 1.3 | 0.2 | 0.2 | 2.2 | 1.5 | 8.4 | 9.2
${\mathchar 28931\relax}(1405),{\mathchar 28931\relax}(1520)$ ${\cal I}$ | 0.2 | 0.6 | 0.1 | 0.1 | ${<\,}0.1\phantom{\,<}$ | 0.1 | 0.2 | 0.2 | 0.4 | 0.8
${\mathchar 28931\relax}(1405),{\mathchar 28931\relax}(1670)$ ${\cal I}$ | 0.3 | 0.9 | 0.1 | 0.5 | ${<\,}0.1\phantom{\,<}$ | ${<\,}0.1\phantom{\,<}$ | 1.2 | 1.6 | 0.9 | 2.4
${\mathchar 28931\relax}(1405),{\mathchar 28934\relax}(1385)$ ${\cal I}$ | 0.2 | 0.4 | 0.1 | 0.2 | ${<\,}0.1\phantom{\,<}$ | 0.1 | 0.5 | 0.7 | 1.8 | 2.0
${\mathchar 28931\relax}(1405),{\mathchar 28934\relax}(1775)$ ${\cal I}$ | ${<\,}0.1\phantom{\,<}$ | 0.3 | ${<\,}0.1\phantom{\,<}$ | 0.1 | ${<\,}0.1\phantom{\,<}$ | 0.1 | 0.4 | 0.3 | 1.1 | 1.2
${\mathchar 28931\relax}(1405),{\mathchar 28934\relax}(1915)$ ${\cal I}$ | 0.1 | 0.4 | ${<\,}0.1\phantom{\,<}$ | 0.1 | ${<\,}0.1\phantom{\,<}$ | ${<\,}0.1\phantom{\,<}$ | 0.1 | 0.3 | 0.6 | 0.8
${\mathchar 28931\relax}(1520),{\mathchar 28931\relax}(1670)$ ${\cal I}$ | 0.1 | 0.2 | ${<\,}0.1\phantom{\,<}$ | ${<\,}0.1\phantom{\,<}$ | ${<\,}0.1\phantom{\,<}$ | ${<\,}0.1\phantom{\,<}$ | 0.2 | 0.1 | 0.8 | 0.8
${\mathchar 28931\relax}(1520),{\mathchar 28934\relax}(1385)$ ${\cal I}$ | 0.7 | 0.9 | 0.2 | 0.2 | 0.2 | 0.1 | 1.6 | 0.9 | 3.6 | 4.2
${\mathchar 28931\relax}(1520),{\mathchar 28934\relax}(1775)$ ${\cal I}$ | 0.3 | 0.6 | 0.1 | 0.1 | 0.1 | 0.2 | 0.5 | 0.5 | 1.6 | 1.9
${\mathchar 28931\relax}(1520),{\mathchar 28934\relax}(1915)$ ${\cal I}$ | ${<\,}0.1\phantom{\,<}$ | 0.1 | ${<\,}0.1\phantom{\,<}$ | ${<\,}0.1\phantom{\,<}$ | ${<\,}0.1\phantom{\,<}$ | ${<\,}0.1\phantom{\,<}$ | 0.1 | ${<\,}0.1\phantom{\,<}$ | 0.1 | 0.2
${\mathchar 28931\relax}(1670),{\mathchar 28934\relax}(1385)$ ${\cal I}$ | 0.2 | 0.3 | 0.1 | 0.1 | ${<\,}0.1\phantom{\,<}$ | ${<\,}0.1\phantom{\,<}$ | 0.4 | 0.6 | 0.6 | 1.0
${\mathchar 28931\relax}(1670),{\mathchar 28934\relax}(1775)$ ${\cal I}$ | 0.1 | 0.1 | ${<\,}0.1\phantom{\,<}$ | ${<\,}0.1\phantom{\,<}$ | ${<\,}0.1\phantom{\,<}$ | ${<\,}0.1\phantom{\,<}$ | 0.1 | 0.1 | 0.2 | 0.3
${\mathchar 28931\relax}(1670),{\mathchar 28934\relax}(1915)$ ${\cal I}$ | ${<\,}0.1\phantom{\,<}$ | 0.1 | ${<\,}0.1\phantom{\,<}$ | ${<\,}0.1\phantom{\,<}$ | ${<\,}0.1\phantom{\,<}$ | ${<\,}0.1\phantom{\,<}$ | 0.3 | 0.1 | 0.4 | 0.5
${\mathchar 28934\relax}(1385),{\mathchar 28934\relax}(1775)$ ${\cal I}$ | 0.1 | 0.3 | ${<\,}0.1\phantom{\,<}$ | ${<\,}0.1\phantom{\,<}$ | ${<\,}0.1\phantom{\,<}$ | ${<\,}0.1\phantom{\,<}$ | 0.2 | 0.1 | 0.2 | 0.4
${\mathchar 28934\relax}(1385),{\mathchar 28934\relax}(1915)$ ${\cal I}$ | 0.2 | 0.6 | ${<\,}0.1\phantom{\,<}$ | 0.2 | ${<\,}0.1\phantom{\,<}$ | 0.1 | 0.6 | 0.2 | 1.0 | 1.3
${\mathchar 28934\relax}(1775),{\mathchar 28934\relax}(1915)$ ${\cal I}$ | 0.1 | 0.4 | ${<\,}0.1\phantom{\,<}$ | 0.1 | ${<\,}0.1\phantom{\,<}$ | 0.1 | 0.7 | 0.2 | 1.0 | 1.3

##  7 Results

#### Ratio of fragmentation and branching fractions:

The results for the ratio $\mathcal{R}$ of the relative fragmentation and
branching fractions for ${{\mathchar
28938\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ and ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ decays are

$\displaystyle\mathcal{R}$ $\displaystyle=$ $\displaystyle({-20}\pm
30\text{\,(stat)}\pm 1\,(\text{uncorr syst}))\times 10^{-3}~{}~{}~{}\text{for
Run~{}1 and}$ $\displaystyle\mathcal{R}$ $\displaystyle=$
$\displaystyle({\phantom{-}51}\pm 28\text{\,(stat)}\pm 2\,(\text{uncorr
syst}))\times 10^{-3}~{}~{}~{}\text{for Run~{}2}\,,$

where the first uncertainty is statistical and the second includes only the
uncorrelated systematic effects presented in Table 3. The negative log-
likelihood functions for these two results are added to obtain a combined
result,

$\mathcal{R}\equiv\frac{f_{{{\mathchar 28938\relax}^{-}_{b}}}}{f_{{{\mathchar
28932\relax}^{-}_{b}}}}\times\frac{{\cal B}({{\mathchar
28938\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}})}{{\cal
B}({{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}})}=(24\pm
21\text{\,(stat)}\pm 14\text{\,(syst)})\times 10^{-3}\,,$

where both uncorrelated and correlated systematic uncertainties are included.
In the combined result, it is implied that $f_{{{\mathchar
28938\relax}^{-}_{b}}}/f_{{{\mathchar 28932\relax}^{-}_{b}}}$, which may vary
with centre-of-mass energy of the LHC $pp$ collisions, is an effective value
averaged over the Run 1 and Run 2 data samples. This result is found to be
consistent with, and more precise than, the previous measurement [23]. No
significant evidence of the ${{\mathchar
28938\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ decay is found, and
therefore an upper limit on $\mathcal{R}$ is calculated at 90 (95) %
confidence level by integrating the likelihood in the physical region of non-
negative branching fraction,

$\mathcal{R}\equiv\frac{f_{{{\mathchar 28938\relax}^{-}_{b}}}}{f_{{{\mathchar
28932\relax}^{-}_{b}}}}\times\frac{{\cal B}({{\mathchar
28938\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}})}{{\cal
B}({{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}})}<62~{}(71)\times
10^{-3}\,.$

#### Amplitude analysis:

The results for the $C\\!P$-asymmetry parameters for each component of the
signal model are shown in Table 5. No significant $C\\!P$ asymmetry is
observed. The fit fraction matrix is reported in Table 6. The diagonal
elements correspond to the fit fractions of the respective components, and the
off-diagonal elements are the interference fit fractions. These results are
derived from the helicity couplings that are the free parameters of the
amplitude fit. Their statistical uncertainties are evaluated from an ensemble
of pseudoexperiments, while systematic uncertainties are obtained as described
in Sec. 6.

Table 5: Results for the $C\\!P$-asymmetry parameters. The statistical uncertainties are obtained from pseudoexperiments while the systematic uncertainties are obtained following the procedure described in Sec. 6.  Component |  $A^{{C\\!P}}$ ($10^{-2}$)
---|---
${\mathchar 28934\relax}(1385)$ |  $-27$$\,\pm\,$ | 34 (stat) $\,\pm\,$ | 73 (syst)
${\mathchar 28931\relax}(1405)$ |  $-1$$\,\pm\,$ | 24 (stat) $\,\pm\,$ | 32 (syst)
${\mathchar 28931\relax}(1520)$ |  $-5$$\,\pm\,$ |  19 (stat) $\,\pm\,$ | 8 (syst)
${\mathchar 28931\relax}(1670)$ |  $3$$\,\pm\,$ | 14 (stat) $\,\pm\,$ | 10 (syst)
${\mathchar 28934\relax}(1775)$ |  $-47$$\,\pm\,$ | 26 (stat) $\,\pm\,$ | 14 (syst)
${\mathchar 28934\relax}(1915)$ |  $11$$\,\pm\,$ | 26 (stat) $\,\pm\,$ | 22 (syst)

Table 6: Results for the fit fractions (diagonal elements) and interference fit fractions (off-diagonal elements) obtained from the amplitude analysis. Identical values for the interference fit fractions in the upper triangle are omitted. All values are in units of $10^{-2}$, with the first uncertainty being statistical and the second systematic.  Component | ${\mathchar 28934\relax}(1385)$ | ${\mathchar 28931\relax}(1405)$ | ${\mathchar 28931\relax}(1520)$ | ${\mathchar 28931\relax}(1670)$ | ${\mathchar 28934\relax}(1775)$ | ${\mathchar 28934\relax}(1915)$
---|---|---|---|---|---|---
${\mathchar 28934\relax}(1385)$ |  $11.4$$\,\pm\,$ | 4.9 $\,\pm\,$ | 7.6 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
${\mathchar 28931\relax}(1405)$ |  $-1.3$$\,\pm\,$ | 0.8 $\,\pm\,$ | 2.0 |  $8.1$$\,\pm\,$ | 2.7 $\,\pm\,$ | 3.0 |  |  |  |  |  |  |  |  |  |  |  |
${\mathchar 28931\relax}(1520)$ |  $3.4$$\,\pm\,$ | 1.6 $\,\pm\,$ | 4.2 |  $0.1$$\,\pm\,$ | 0.5 $\,\pm\,$ | 0.8 |  $33.0$$\,\pm\,$ | 4.1 $\,\pm\,$ | 3.6 |  |  |  |  |  |  |  |  |
${\mathchar 28931\relax}(1670)$ |  $-0.1$$\,\pm\,$ | 0.6 $\,\pm\,$ | 1.0 |  $3.0$$\,\pm\,$ | 1.8 $\,\pm\,$ | 2.4 |  $-0.1$$\,\pm\,$ | 0.4 $\,\pm\,$ | 0.8 |  $19.5$$\,\pm\,$ | 3.2 $\,\pm\,$ | 5.6 |  |  |  |  |  |
${\mathchar 28934\relax}(1775)$ |  $0.1$$\,\pm\,$ | 0.3 $\,\pm\,$ | 0.4 |  $-0.7$$\,\pm\,$ | 0.5 $\,\pm\,$ | 1.2 |  $1.1$$\,\pm\,$ | 1.0 $\,\pm\,$ | 1.9 |  $-0.3$$\,\pm\,$ | 0.2 $\,\pm\,$ | 0.3 |  $9.7$$\,\pm\,$ | 3.5 $\,\pm\,$ | 4.1 |  |  |
${\mathchar 28934\relax}(1915)$ |  $0.6$$\,\pm\,$ | 0.6 $\,\pm\,$ | 1.3 |  $0.3$$\,\pm\,$ | 0.3 $\,\pm\,$ | 0.8 |  $0.1$$\,\pm\,$ | 0.1 $\,\pm\,$ | 0.2 |  $-0.1$$\,\pm\,$ | 0.2 $\,\pm\,$ | 0.5 |  $1.0$$\,\pm\,$ | 0.5 $\,\pm\,$ | 1.3 |  $11.3$$\,\pm\,$ | 3.7 $\,\pm\,$ | 9.2

The significance of each component in the baseline model is evaluated using
pseudoexperiments. These are generated, each with a sample size corresponding
to the data, according to the best fit model with the component of interest
removed from the model. They are then fitted both with the model used to
generate and with the model including the component of interest. Twice the
difference between the negative log-likelihood values obtained in these two
fits ($-2\Delta\ln\mathcal{L}$) is used as a test statistic. A $p$-value,
corresponding to the probability of observing $-2\Delta\ln\mathcal{L}$ values
as large or larger than that found in the fit to data, is found by
extrapolating the tail of the distribution obtained from the ensemble of
pseudoexperiments. In order to account for dominant systematic uncertainties,
this procedure is performed for the alternative model that gives the smallest
value of $-2\ln\mathcal{L}$ in fits to data. The outcome is that the
${\mathchar 28931\relax}(1520)$ and ${\mathchar 28931\relax}(1670)$ components
have $p$-values corresponding to $12.0\,\sigma$ and $6.1\,\sigma$,
respectively. All other components have significance below $3.5\,\sigma$.

The branching fraction of each quasi-two-body contribution to the ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ decay, corresponding
to an intermediate resonance $R$, can be obtained from its fit fraction
$\mathcal{F}_{i}$,

${\cal B}({{\mathchar 28932\relax}^{-}_{b}}\\!\rightarrow R{{K}^{-}})={\cal
B}({{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}})\times\mathcal{F}_{i}\,.$
(29)

The branching fraction of ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ has not been
measured directly, but the ratio of fragmentation and branching fractions
relative to the ${{{B}^{-}}}\rightarrow{{K}^{+}}{{K}^{-}}{{K}^{-}}$ decay is
known [23]. This can be combined with the known values of ${\cal
B}\left({{{B}^{-}}}\rightarrow{{K}^{+}}{{K}^{-}}{{K}^{-}}\right)$ [65, 66,
37], $f_{{{\mathchar 28932\relax}^{-}_{b}}}/f_{{{\mathchar
28931\relax}^{0}_{b}}}$ [46] and $f_{{{\mathchar
28931\relax}^{0}_{b}}}/\left(f_{u}+f_{d}\right)$ [67], assuming that
$f_{u}=f_{d}$, to obtain

${\cal B}\left({{\mathchar
28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{K}^{-}}\right)=\left(2.3\pm
0.9\right)\times 10^{-6}\,,$

where the dominant uncertainty is that due to possible SU(3)-breaking effects
which affect $f_{{{\mathchar 28932\relax}^{-}_{b}}}/f_{{{\mathchar
28931\relax}^{0}_{b}}}$ [46]. Consequently, the values of the quasi-two-body
branching fractions are found to be

$\displaystyle{\cal B}\left({{\mathchar
28932\relax}^{-}_{b}}\rightarrow{\mathchar 28934\relax}(1385){{K}^{-}}\right)$
$\displaystyle=$ $\displaystyle\left(0.26\pm 0.11\pm 0.17\pm 0.10\right)\times
10^{-6}\,,$ $\displaystyle{\cal B}\left({{\mathchar
28932\relax}^{-}_{b}}\rightarrow{\mathchar 28931\relax}(1405){{K}^{-}}\right)$
$\displaystyle=$ $\displaystyle\left(0.19\pm 0.06\pm 0.07\pm 0.07\right)\times
10^{-6}\,,$ $\displaystyle{\cal B}\left({{\mathchar
28932\relax}^{-}_{b}}\rightarrow{\mathchar 28931\relax}(1520){{K}^{-}}\right)$
$\displaystyle=$ $\displaystyle\left(0.76\pm 0.09\pm 0.08\pm 0.30\right)\times
10^{-6}\,,$ $\displaystyle{\cal B}\left({{\mathchar
28932\relax}^{-}_{b}}\rightarrow{\mathchar 28931\relax}(1670){{K}^{-}}\right)$
$\displaystyle=$ $\displaystyle\left(0.45\pm 0.07\pm 0.13\pm 0.18\right)\times
10^{-6}\,,$ $\displaystyle{\cal B}\left({{\mathchar
28932\relax}^{-}_{b}}\rightarrow{\mathchar 28934\relax}(1775){{K}^{-}}\right)$
$\displaystyle=$ $\displaystyle\left(0.22\pm 0.08\pm 0.09\pm 0.09\right)\times
10^{-6}\,,$ $\displaystyle{\cal B}\left({{\mathchar
28932\relax}^{-}_{b}}\rightarrow{\mathchar 28934\relax}(1915){{K}^{-}}\right)$
$\displaystyle=$ $\displaystyle\left(0.26\pm 0.09\pm 0.21\pm 0.10\right)\times
10^{-6}\,,$

where the uncertainties are statistical, systematic and due to the knowledge
of ${\cal B}\left({{\mathchar
28932\relax}^{-}_{b}}\rightarrow{p}{{K}^{-}}{{K}^{-}}\right)$, respectively.

##  8 Summary

The structure of ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ decays has been
studied through an amplitude analysis. This is the first amplitude analysis of
any $b$-baryon decay mode allowing for $C\\!P$-violation effects. The analysis
uses $pp$ collision data recorded with the LHCb detector, corresponding to
integrated luminosities of $1\text{\,fb}^{-1}$ at
$\sqrt{s}=7\text{\,Te\kern-1.00006ptV}$, $2\text{\,fb}^{-1}$ at
$\sqrt{s}=8\text{\,Te\kern-1.00006ptV}$ and $2\text{\,fb}^{-1}$ at
$\sqrt{s}=13\text{\,Te\kern-1.00006ptV}$. Due to the inclusion of more data
and significantly improving the selection procedure compared to the previous
study of this channel [23], a yield of about 460 signal decays within the
$m({p}{{K}^{-}}{{K}^{-}})$ signal region is obtained, with a signal to
background ratio of about $2:1$. A good description of the data is obtained
with an amplitude model containing contributions from ${\mathchar
28934\relax}(1385)$, ${\mathchar 28931\relax}(1405)$, ${\mathchar
28931\relax}(1520)$, ${\mathchar 28931\relax}(1670)$, ${\mathchar
28934\relax}(1775)$ and ${\mathchar 28934\relax}(1915)$ resonances. The
$C\\!P$ asymmetry for each contributing component is evaluated and no
significant $C\\!P$-violation effect is observed. The ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{\mathchar 28931\relax}(1520){{K}^{-}}$ and
${{\mathchar 28932\relax}^{-}_{b}}\\!\rightarrow{\mathchar
28931\relax}(1670){{K}^{-}}$ decays are observed with significance greater
than $5\,\sigma$, and their branching fractions measured, together with those
of ${{\mathchar 28932\relax}^{-}_{b}}\\!\rightarrow{\mathchar
28934\relax}(1385){{K}^{-}}$, ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{\mathchar 28931\relax}(1405){{K}^{-}}$,
${{\mathchar 28932\relax}^{-}_{b}}\\!\rightarrow{\mathchar
28934\relax}(1775){{K}^{-}}$ and ${{\mathchar
28932\relax}^{-}_{b}}\\!\rightarrow{\mathchar 28934\relax}(1915){{K}^{-}}$
decays. No significant signal for ${{\mathchar
28938\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ decays is found and
an upper limit on the ratio of fragmentation and branching fractions of
${{\mathchar 28938\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ and
${{\mathchar 28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$ decays
is set. With the substantially larger samples that are anticipated following
the upgrade of LHCb [68, 69], it will be possible to reduce both statistical
and systematic uncertainties on $C\\!P$-violation observables in three-body
$b$-baryon decays, and thereby test the Standard Model using the methods
pioneered in this study.

##  Appendix A Fit projections

Projections of the fit result are compared to the data in slices of $m_{\rm
low}$ in Figs. 7–11. Similar projections in slices of $m_{\rm high}$ are shown
in Figs. 12–14.

Figure 7: Distributions of (top) $m_{\rm low}$ and (bottom) $m_{\rm high}$,
with $1.43<m_{\rm low}<1.60\text{\,Ge\kern-0.90005ptV}$, for (left)
${\mathchar 28932\relax}^{-}_{b}$ and (right) ${\kern
1.61993pt\overline{\kern-1.61993pt{\mathchar 28932\relax}}}{}^{+}_{b}$
candidates, with results of the fits superimposed. The total fit result is
shown as the blue solid curve, with contributions from individual signal
components and from combinatorial (Comb) and cross-feed (Crsfd) background
shown as indicated in the legend.

Figure 8: Distributions of (top) $m_{\rm low}$ and (bottom) $m_{\rm high}$,
with $1.6<m_{\rm low}<1.8\text{\,Ge\kern-0.90005ptV}$, for (left) ${\mathchar
28932\relax}^{-}_{b}$ and (right) ${\kern
1.61993pt\overline{\kern-1.61993pt{\mathchar 28932\relax}}}{}^{+}_{b}$
candidates, with results of the fits superimposed. The total fit result is
shown as the blue solid curve, with contributions from individual signal
components and from combinatorial (Comb) and cross-feed (Crsfd) background
shown as indicated in the legend.

Figure 9: Distributions of (top) $m_{\rm low}$ and (bottom) $m_{\rm high}$,
with $1.8<m_{\rm low}<2.2\text{\,Ge\kern-0.90005ptV}$, for (left) ${\mathchar
28932\relax}^{-}_{b}$ and (right) ${\kern
1.61993pt\overline{\kern-1.61993pt{\mathchar 28932\relax}}}{}^{+}_{b}$
candidates, with results of the fits superimposed. The total fit result is
shown as the blue solid curve, with contributions from individual signal
components and from combinatorial (Comb) and cross-feed (Crsfd) background
shown as indicated in the legend.

Figure 10: Distributions of (top) $m_{\rm low}$ and (bottom) $m_{\rm high}$,
with $2.2<m_{\rm low}<2.8\text{\,Ge\kern-0.90005ptV}$, for (left) ${\mathchar
28932\relax}^{-}_{b}$ and (right) ${\kern
1.61993pt\overline{\kern-1.61993pt{\mathchar 28932\relax}}}{}^{+}_{b}$
candidates, with results of the fits superimposed. The total fit result is
shown as the blue solid curve, with contributions from individual signal
components and from combinatorial (Comb) and cross-feed (Crsfd) background
shown as indicated in the legend.

Figure 11: Distributions of (top) $m_{\rm low}$ and (bottom) $m_{\rm high}$,
with $m_{\rm low}>2.8\text{\,Ge\kern-0.90005ptV}$, for (left) ${\mathchar
28932\relax}^{-}_{b}$ and (right) ${\kern
1.61993pt\overline{\kern-1.61993pt{\mathchar 28932\relax}}}{}^{+}_{b}$
candidates, with results of the fits superimposed. The total fit result is
shown as the blue solid curve, with contributions from individual signal
components and from combinatorial (Comb) and cross-feed (Crsfd) background
shown as indicated in the legend.

Figure 12: Distributions of (top) $m_{\rm low}$ and (bottom) $m_{\rm high}$,
with $m_{\rm high}<3.15\text{\,Ge\kern-0.90005ptV}$ , for (left) ${\mathchar
28932\relax}^{-}_{b}$ and (right) ${\kern
1.61993pt\overline{\kern-1.61993pt{\mathchar 28932\relax}}}{}^{+}_{b}$
candidates, with results of the fits superimposed. The total fit result is
shown as the blue solid curve, with contributions from individual signal
components and from combinatorial (Comb) and cross-feed (Crsfd) background
shown as indicated in the legend.

Figure 13: Distributions of (top) $m_{\rm low}$ and (bottom) $m_{\rm high}$,
with $3.15<m_{\rm high}<3.95\text{\,Ge\kern-0.90005ptV}$, for (left)
${\mathchar 28932\relax}^{-}_{b}$ and (right) ${\kern
1.61993pt\overline{\kern-1.61993pt{\mathchar 28932\relax}}}{}^{+}_{b}$
candidates, with results of the fits superimposed. The total fit result is
shown as the blue solid curve, with contributions from individual signal
components and from combinatorial (Comb) and cross-feed (Crsfd) background
shown as indicated in the legend.

Figure 14: Distributions of (top) $m_{\rm low}$ and (bottom) $m_{\rm high}$,
with $m_{\rm high}>3.95\text{\,Ge\kern-0.90005ptV}$, for (left) ${\mathchar
28932\relax}^{-}_{b}$ and (right) ${\kern
1.61993pt\overline{\kern-1.61993pt{\mathchar 28932\relax}}}{}^{+}_{b}$
candidates, with results of the fits superimposed. The total fit result is
shown as the blue solid curve, with contributions from individual signal
components and from combinatorial (Comb) and cross-feed (Crsfd) background
shown as indicated in the legend.

## 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); MICINN (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC
(United Kingdom); DOE NP and NSF (USA). We acknowledge the computing resources
that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN
(Italy), SURF (Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and
Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-
GRID (Poland) and NERSC (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 ARC and ARDC (Australia); AvH Foundation
(Germany); EPLANET, Marie Skłodowska-Curie Actions and ERC (European Union);
A*MIDEX, ANR, IPhU and Labex P2IO, and Région Auvergne-Rhône-Alpes (France);
Key Research Program of Frontier Sciences of CAS, CAS PIFI, CAS CCEPP,
Fundamental Research Funds for the Central Universities, and Sci. & Tech.
Program of Guangzhou (China); RFBR, RSF and Yandex LLC (Russia); GVA, XuntaGal
and GENCAT (Spain); the Leverhulme Trust, the Royal Society and UKRI (United
Kingdom).

## References

  * [1] N. Cabibbo, _Unitary symmetry and leptonic decays_ , Phys. Rev. Lett. 10 (1963) 531
  * [2] M. Kobayashi and T. Maskawa, _$C\\!P$ -violation in the renormalizable theory of weak interaction_, Prog. Theor. Phys. 49 (1973) 652
  * [3] A. D. Sakharov, _Violation of CP invariance, C asymmetry, and baryon asymmetry of the universe_ , Pisma Zh. Eksp. Teor. Fiz. 5 (1967) 32
  * [4] M. E. Shaposhnikov, _Standard model solution of the baryogenesis problem_ , Phys. Lett. B277 (1992) 324, Erratum ibid. B282 (1992) 483
  * [5] BaBar collaboration, J. P. Lees et al., _Measurement of $C\\!P$ asymmetries and branching fractions in charmless two-body $B$-meson decays to pions and kaons_, Phys. Rev. D87 (2013) 052009, arXiv:1206.3525
  * [6] Belle collaboration, Y.-T. Duh et al., _Measurements of branching fractions and direct CP asymmetries for $B\rightarrow K\pi$, $B\rightarrow\pi\pi$ and $B\rightarrow KK$ decays_, Phys. Rev. D87 (2013) 031103, arXiv:1210.1348
  * [7] CDF collaboration, T. A. Aaltonen et al., _Measurements of direct $C\\!P$-violating asymmetries in charmless decays of bottom baryons_, Phys. Rev. Lett. 113 (2014) 242001, arXiv:1403.5586
  * [8] LHCb collaboration, R. Aaij et al., _First observation of $C\\!P$ violation in the decays of ${B}^{0}_{s}$ mesons_, Phys. Rev. Lett. 110 (2013) 221601, arXiv:1304.6173
  * [9] LHCb collaboration, R. Aaij et al., _Measurement of $C\\!P$ asymmetries in two-body ${B}^{0}_{\mathrm{(}s)}$-meson decays to charged pions and kaons_, Phys. Rev. D98 (2018) 032004, arXiv:1805.06759
  * [10] Belle collaboration, I. Adachi et al., _Measurement of the CP violation parameters in $B^{0}\rightarrow\pi^{+}\pi^{-}$ decays_, Phys. Rev. D88 (2013) 092003, arXiv:1302.0551
  * [11] LHCb collaboration, R. Aaij et al., _Measurement of $C\\!P$ violation in the phase space of ${{B}^{\pm}}\\!\rightarrow{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}$ and ${{B}^{\pm}}\\!\rightarrow{{K}^{\pm}}{{K}^{+}}{{K}^{-}}$ decays_, Phys. Rev. Lett. 111 (2013) 101801, arXiv:1306.1246
  * [12] LHCb collaboration, R. Aaij et al., _Measurement of $C\\!P$ violation in the phase space of ${{B}^{\pm}}\\!\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{\pm}}$ and ${{B}^{\pm}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{\pm}}$ decays_, Phys. Rev. Lett. 112 (2014) 011801, arXiv:1310.4740
  * [13] LHCb collaboration, R. Aaij et al., _Measurement of $C\\!P$ violation in the three-body phase space of charmless ${B}^{\pm}$ decays_, Phys. Rev. D90 (2014) 112004, arXiv:1408.5373
  * [14] LHCb collaboration, R. Aaij et al., _Amplitude analysis of ${{B}^{\pm}}\\!\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}$ decays_, Phys. Rev. Lett. 123 (2019) 231802, arXiv:1905.09244
  * [15] LHCb collaboration, R. Aaij et al., _Amplitude analysis of the ${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ decay_, Phys. Rev. D101 (2020) 012006, arXiv:1909.05211
  * [16] LHCb collaboration, R. Aaij et al., _Observation of several sources of $C\\!P$ violation in ${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ decays_, Phys. Rev. Lett. 124 (2020) 031801, arXiv:1909.05211
  * [17] LHCb collaboration, R. Aaij et al., _Search for $C\\!P$ violation in ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow p{{K}^{-}}$ and ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow p{{\pi}^{-}}$ decays_, Phys. Lett. B784 (2018) 101, arXiv:1807.06544
  * [18] LHCb collaboration, R. Aaij et al., _Searches for ${\mathchar 28931\relax}^{0}_{b}$ and ${\mathchar 28932\relax}^{0}_{b}$ decays to ${{K}^{0}_{\mathrm{S}}}{p}{{\pi}^{-}}$ and ${{K}^{0}_{\mathrm{S}}}{p}{{K}^{-}}$ final states with first observation of the ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{{K}^{0}_{\mathrm{S}}}{p}{{\pi}^{-}}$ decay_, JHEP 04 (2014) 087, arXiv:1402.0770
  * [19] LHCb collaboration, R. Aaij et al., _Observations of ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar 28931\relax}{{K}^{+}}{{\pi}^{-}}$ and ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar 28931\relax}{{K}^{+}}{{K}^{-}}$ decays and searches for other ${\mathchar 28931\relax}^{0}_{b}$ and ${\mathchar 28932\relax}^{0}_{b}$ decays to ${\mathchar 28931\relax}h^{+}h^{-}$ final states_, JHEP 05 (2016) 081, arXiv:1603.00413
  * [20] LHCb collaboration, R. Aaij et al., _Search for $C\\!P$ violation and observation of $P$ violation in ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow p{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decays_, Phys. Rev. D102 (2020) 051101, arXiv:1912.10741
  * [21] LHCb collaboration, R. Aaij et al., _Measurement of $C\\!P$ asymmetries in charmless four-body ${\mathchar 28931\relax}^{0}_{b}$ and ${\mathchar 28932\relax}^{0}_{b}$ decays_, Eur. Phys. J. C79 (2019) 745, arXiv:1903.06792
  * [22] LHCb collaboration, R. Aaij et al., _Search for $C\\!P$ violation using triple product asymmetries in ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow p{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$, ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow p{{K}^{-}}{{K}^{+}}{{K}^{-}}$, and ${\mathchar 28932\relax}^{0}_{b}\\!\rightarrow p{{K}^{-}}{{K}^{-}}{{\pi}^{+}}$ decays_, JHEP 08 (2018) 039, arXiv:1805.03941
  * [23] LHCb collaboration, R. Aaij et al., _Observation of the decay ${{\mathchar 28932\relax}^{-}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{-}}$_, Phys. Rev. Lett. 118 (2017) 071801, arXiv:1612.02244
  * [24] LHCb collaboration, A. A. Alves Jr. et al., _The LHCb detector at the LHC_, JINST 3 (2008) S08005
  * [25] LHCb collaboration, R. Aaij et al., _LHCb detector performance_ , Int. J. Mod. Phys. A30 (2015) 1530022, arXiv:1412.6352
  * [26] V. V. Gligorov and M. Williams, _Efficient, reliable and fast high-level triggering using a bonsai boosted decision tree_ , JINST 8 (2013) P02013, arXiv:1210.6861
  * [27] LHCb collaboration, R. Aaij et al., _Measurements of the ${\mathchar 28931\relax}^{0}_{b}$, ${\mathchar 28932\relax}^{-}_{b}$, and ${\mathchar 28938\relax}^{-}_{b}$ baryon masses_, Phys. Rev. Lett. 110 (2013) 182001, arXiv:1302.1072
  * [28] LHCb collaboration, R. Aaij et al., _Precision measurement of $D$ meson mass differences_, JHEP 06 (2013) 065, arXiv:1304.6865
  * [29] T. Sjöstrand, S. Mrenna, and P. Skands, _PYTHIA 6.4 physics and manual_ , JHEP 05 (2006) 026, arXiv:hep-ph/0603175
  * [30] T. Sjöstrand, S. Mrenna, and P. Skands, _A brief introduction to PYTHIA 8.1_ , Comput. Phys. Commun. 178 (2008) 852, arXiv:0710.3820
  * [31] I. Belyaev et al., _Handling of the generation of primary events in Gauss, the LHCb simulation framework_ , J. Phys. Conf. Ser. 331 (2011) 032047
  * [32] D. J. Lange, _The EvtGen particle decay simulation package_ , Nucl. Instrum. Meth. A462 (2001) 152
  * [33] N. Davidson, T. Przedzinski, and Z. Was, _PHOTOS interface in C++: Technical and physics documentation_ , Comp. Phys. Comm. 199 (2016) 86, arXiv:1011.0937
  * [34] Geant4 collaboration, J. Allison et al., _Geant4 developments and applications_ , IEEE Trans. Nucl. Sci. 53 (2006) 270
  * [35] Geant4 collaboration, S. Agostinelli et al., _Geant4: A simulation toolkit_ , Nucl. Instrum. Meth. A506 (2003) 250
  * [36] M. Clemencic et al., _The LHCb simulation application, Gauss: Design, evolution and experience_, J. Phys. Conf. Ser. 331 (2011) 032023
  * [37] Particle Data Group, P. A. Zyla et al., _Review of particle physics_ , Prog. Theor. Exp. Phys. 2020 (2020) 083C01
  * [38] LHCb collaboration, R. Aaij et al., _Observation of $C\\!P$ violation in ${{B}^{\pm}}\\!\rightarrow{D}{{K}^{\pm}}$ decays_, Phys. Lett. B712 (2012) 203, Erratum ibid. B713 (2012) 351, arXiv:1203.3662
  * [39] R. Aaij et al., _Selection and processing of calibration samples to measure the particle identification performance of the LHCb experiment in Run 2_ , Eur. Phys. J. Tech. Instr. 6 (2019) 1, arXiv:1803.00824
  * [40] T. Chen and C. Guestrin, _XGBoost: A scalable tree boosting system_ , in Proceedings of the 22Nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 16 785, 2016, arXiv:1603.02754
  * [41] Particle Data Group, K. A. Olive et al., _Review of particle physics_ , Chin. Phys. C38 (2014) 090001
  * [42] LHCb collaboration, R. Aaij et al., _Measurement of the $b$-quark production cross-section in 7 and 13 TeV ${p}{p}$ collisions_, Phys. Rev. Lett. 118 (2017) 052002, Erratum ibid. 119 (2017) 169901, arXiv:1612.05140
  * [43] LHCb collaboration, R. Aaij et al., _Measurement of $B$ meson production cross-sections in proton-proton collisions at $\sqrt{s}=$7 TeV_, JHEP 08 (2013) 117, arXiv:1306.3663
  * [44] LHCb collaboration, R. Aaij et al., _Precision measurement of the mass and lifetime of the ${\mathchar 28932\relax}^{-}_{b}$ baryon_, Phys. Rev. Lett. 113 (2014) 242002, arXiv:1409.8568
  * [45] LHCb collaboration, R. Aaij et al., _Observation of the ${{\mathchar 28932\relax}^{-}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{\mathchar 28931\relax}{{K}^{-}}$ decay_, Phys. Lett. B772 (2017) 265, arXiv:1701.05274
  * [46] LHCb collaboration, R. Aaij et al., _Measurement of the mass and production rate of ${\mathchar 28932\relax}^{-}_{b}$ baryons_, Phys. Rev. D99 (2019) 052006, arXiv:1901.07075
  * [47] T. Skwarnicki, A study of the radiative cascade transitions between the Upsilon-prime and Upsilon resonances, PhD thesis, Institute of Nuclear Physics, Krakow, 1986, DESY-F31-86-02
  * [48] LHCb collaboration, R. Aaij et al., _Observation of ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}{p}$ resonances consistent with pentaquark states in ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{p}{{K}^{-}}$ decays_, Phys. Rev. Lett. 115 (2015) 072001, arXiv:1507.03414
  * [49] LHCb collaboration, R. Aaij et al., _Evidence for exotic hadron contributions to ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{p}{{\pi}^{-}}$ decays_, Phys. Rev. Lett. 117 (2016) 082003, arXiv:1606.06999
  * [50] ARGUS collaboration, H. Albrecht et al., _Search for hadronic $b\rightarrow u$ decays_, Phys. Lett. B241 (1990) 278
  * [51] LHCb collaboration, R. Aaij et al., _Measurements of the ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{\mathchar 28931\relax}$ decay amplitudes and the ${\mathchar 28931\relax}^{0}_{b}$ polarisation in $p$ $p$ collisions at $\sqrt{s}=$7 TeV_, Phys. Lett. B724 (2013) 27, arXiv:1302.5578
  * [52] LHCb collaboration, R. Aaij et al., _Measurement of the ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{\mathchar 28931\relax}$ angular distribution and the $\mathchar 28931\relax$ polarisation in $pp$ collisions_, JHEP 06 (2020) 110, arXiv:2004.10563
  * [53] BaBar collaboration, B. Aubert et al., _An amplitude analysis of the decay $B^{\pm}\rightarrow\pi^{\pm}\pi^{\pm}\pi^{\mp}$_, Phys. Rev. D72 (2005) 052002, arXiv:hep-ex/0507025
  * [54] LHCb collaboration, R. Aaij et al., _Amplitude analysis of ${{{B}^{+}}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}\phi{{K}^{+}}$ decays_, Phys. Rev. D95 (2017) 012002, arXiv:1606.07898
  * [55] LHCb collaboration, R. Aaij et al., _Study of the ${{D}^{0}}{p}$ amplitude in ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{{D}^{0}}{p}{{\pi}^{-}}$ decays_, JHEP 05 (2017) 030, arXiv:1701.07873
  * [56] JPAC collaboration, M. Mikhasenko et al., _Dalitz-plot decomposition for three-body decays_ , Phys. Rev. D101 (2020) 034033, arXiv:1910.04566
  * [57] E. P. Wigner, Group theory and its application to the quantum mechanics of atomic spectra, Pure Appl. Phys., Academic Press, New York, NY, 1959
  * [58] LHCb collaboration, R. Aaij et al., _Dalitz plot analysis of ${{B}^{0}_{s}}\\!\rightarrow{{\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}{{K}^{-}}{{\pi}^{+}}$ decays_, Phys. Rev. D90 (2014) 072003, arXiv:1407.7712
  * [59] A. Mathad, D. O’Hanlon, A. Poluektov, and R. Rabadan, _Efficient description of experimental effects in amplitude analyses_ , arXiv:1902.01452, to appear in JINST
  * [60] M. Abadi et al., _TensorFlow: Large-scale machine learning on heterogeneous systems_ , arXiv:1603.04467, software available from tensorflow.org
  * [61] F. James and M. Roos, _Minuit: A system for function minimization and analysis of the parameter errors and correlations_ , Comput. Phys. Commun. 10 (1975) 343
  * [62] R. Brun and F. Rademakers, _ROOT: An object oriented data analysis framework_ , Nucl. Instrum. Meth. A389 (1997) 81
  * [63] M. Williams, _How good are your fits? Unbinned multivariate goodness-of-fit tests in high energy physics_ , JINST 5 (2010) P09004, arXiv:1006.3019
  * [64] D. Martínez Santos and F. Dupertuis, _Mass distributions marginalized over per-event errors_ , Nucl. Instrum. Meth. A764 (2014) 150, arXiv:1312.5000
  * [65] Belle collaboration, A. Garmash et al., _Dalitz analysis of the three-body charmless decays ${{{B}^{+}}}\rightarrow{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ and ${{{B}^{+}}}\rightarrow{{K}^{+}}{{K}^{+}}{{K}^{-}}$_, Phys. Rev. D71 (2005) 092003, arXiv:hep-ex/0412066
  * [66] BaBar collaboration, J. P. Lees et al., _Study of CP violation in Dalitz-plot analyses of ${{B}^{0}}\rightarrow{{K}^{+}}{{K}^{-}}{{K}^{0}_{\mathrm{S}}}$, ${{{B}^{+}}}\rightarrow{{K}^{+}}{{K}^{-}}{{K}^{+}}$, and ${{{B}^{+}}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{K}^{0}_{\mathrm{S}}}{{K}^{+}}$_, Phys. Rev. D85 (2012) 112010, arXiv:1201.5897
  * [67] LHCb collaboration, R. Aaij et al., _Measurement of $b$-hadron fractions in 13 TeV $p$ $p$ collisions_, Phys. Rev. D100 (2019) 031102(R), arXiv:1902.06794
  * [68] LHCb collaboration, _Framework TDR for the LHCb Upgrade: Technical Design Report_ , CERN-LHCC-2012-007, 2012
  * [69] LHCb collaboration, _Expression of Interest for a Phase-II LHCb Upgrade: Opportunities in flavour physics, and beyond, in the HL-LHC era_ , CERN-LHCC-2017-003, 2017

LHCb collaboration

R. Aaij32, C. Abellán Beteta50, T. Ackernley60, B. Adeva46, M. Adinolfi54, H.
Afsharnia9, C.A. Aidala86, S. Aiola25, Z. Ajaltouni9, S. Akar65, J.
Albrecht15, F. Alessio48, M. Alexander59, A. Alfonso Albero45, Z. Aliouche62,
G. Alkhazov38, P. Alvarez Cartelle55, S. Amato2, Y. Amhis11, L. An48, L.
Anderlini22, A. Andreianov38, M. Andreotti21, F. Archilli17, A. Artamonov44,
M. Artuso68, K. Arzymatov42, E. Aslanides10, M. Atzeni50, B. Audurier12, S.
Bachmann17, M. Bachmayer49, J.J. Back56, P. Baladron Rodriguez46, V.
Balagura12, W. Baldini21, J. Baptista Leite1, R.J. Barlow62, S. Barsuk11, W.
Barter61, M. Bartolini24, F. Baryshnikov83, J.M. Basels14, G. Bassi29, B.
Batsukh68, A. Battig15, A. Bay49, M. Becker15, F. Bedeschi29, I. Bediaga1, A.
Beiter68, V. Belavin42, S. Belin27, V. Bellee49, K. Belous44, I. Belov40, I.
Belyaev41, G. Bencivenni23, E. Ben-Haim13, A. Berezhnoy40, R. Bernet50, D.
Berninghoff17, H.C. Bernstein68, C. Bertella48, A. Bertolin28, C.
Betancourt50, F. Betti48, Ia. Bezshyiko50, S. Bhasin54, J. Bhom35, L. Bian73,
M.S. Bieker15, S. Bifani53, P. Billoir13, M. Birch61, F.C.R. Bishop55, A.
Bitadze62, A. Bizzeti22,k, M. Bjørn63, M.P. Blago48, T. Blake56, F. Blanc49,
S. Blusk68, D. Bobulska59, J.A. Boelhauve15, O. Boente Garcia46, T.
Boettcher65, A. Boldyrev82, A. Bondar43, N. Bondar38,48, S. Borghi62, M.
Borisyak42, M. Borsato17, J.T. Borsuk35, S.A. Bouchiba49, T.J.V. Bowcock60, A.
Boyer48, C. Bozzi21, M.J. Bradley61, S. Braun66, A. Brea Rodriguez46, M.
Brodski48, J. Brodzicka35, A. Brossa Gonzalo56, D. Brundu27, A. Buonaura50, C.
Burr48, A. Bursche72, A. Butkevich39, J.S. Butter32, J. Buytaert48, W.
Byczynski48, S. Cadeddu27, H. Cai73, R. Calabrese21,f, L. Calefice15,13, L.
Calero Diaz23, S. Cali23, R. Calladine53, M. Calvi26,j, M. Calvo Gomez85, P.
Camargo Magalhaes54, P. Campana23, A.F. Campoverde Quezada6, S. Capelli26,j,
L. Capriotti20,d, A. Carbone20,d, G. Carboni31, R. Cardinale24, A. Cardini27,
I. Carli4, P. Carniti26,j, L. Carus14, K. Carvalho Akiba32, A. Casais Vidal46,
G. Casse60, M. Cattaneo48, G. Cavallero48, S. Celani49, J. Cerasoli10, A.J.
Chadwick60, M.G. Chapman54, M. Charles13, Ph. Charpentier48, G.
Chatzikonstantinidis53, C.A. Chavez Barajas60, M. Chefdeville8, C. Chen3, S.
Chen4, A. Chernov35, V. Chobanova46, S. Cholak49, M. Chrzaszcz35, A.
Chubykin38, V. Chulikov38, P. Ciambrone23, M.F. Cicala56, X. Cid Vidal46, G.
Ciezarek48, P.E.L. Clarke58, M. Clemencic48, H.V. Cliff55, J. Closier48, J.L.
Cobbledick62, V. Coco48, J.A.B. Coelho11, J. Cogan10, E. Cogneras9, L.
Cojocariu37, P. Collins48, T. Colombo48, L. Congedo19,c, A. Contu27, N.
Cooke53, G. Coombs59, G. Corti48, C.M. Costa Sobral56, B. Couturier48, D.C.
Craik64, J. Crkovská67, M. Cruz Torres1, R. Currie58, C.L. Da Silva67, S.
Dadabaev83, E. Dall’Occo15, J. Dalseno46, C. D’Ambrosio48, A. Danilina41, P.
d’Argent48, A. Davis62, O. De Aguiar Francisco62, K. De Bruyn79, S. De
Capua62, M. De Cian49, J.M. De Miranda1, L. De Paula2, M. De Serio19,c, D. De
Simone50, P. De Simone23, J.A. de Vries80, C.T. Dean67, D. Decamp8, L. Del
Buono13, B. Delaney55, H.-P. Dembinski15, A. Dendek34, V. Denysenko50, D.
Derkach82, O. Deschamps9, F. Desse11, F. Dettori27,e, B. Dey77, A. Di Cicco23,
P. Di Nezza23, S. Didenko83, L. Dieste Maronas46, H. Dijkstra48, V.
Dobishuk52, A.M. Donohoe18, F. Dordei27, A.C. dos Reis1, L. Douglas59, A.
Dovbnya51, A.G. Downes8, K. Dreimanis60, M.W. Dudek35, L. Dufour48, V. Duk78,
P. Durante48, J.M. Durham67, D. Dutta62, A. Dziurda35, A. Dzyuba38, S. Easo57,
U. Egede69, V. Egorychev41, S. Eidelman43,v, S. Eisenhardt58, S. Ek-In49, L.
Eklund59,w, S. Ely68, A. Ene37, E. Epple67, S. Escher14, J. Eschle50, S.
Esen13, T. Evans48, A. Falabella20, J. Fan3, Y. Fan6, B. Fang73, S. Farry60,
D. Fazzini26,j, M. Féo48, A. Fernandez Prieto46, J.M. Fernandez-tenllado
Arribas45, A.D. Fernez66, F. Ferrari20,d, L. Ferreira Lopes49, F. Ferreira
Rodrigues2, S. Ferreres Sole32, M. Ferrillo50, M. Ferro-Luzzi48, S.
Filippov39, R.A. Fini19, M. Fiorini21,f, M. Firlej34, K.M. Fischer63, D.S.
Fitzgerald86, C. Fitzpatrick62, T. Fiutowski34, A. Fkiaras48, F. Fleuret12, M.
Fontana13, F. Fontanelli24,h, R. Forty48, V. Franco Lima60, M. Franco
Sevilla66, M. Frank48, E. Franzoso21, G. Frau17, C. Frei48, D.A. Friday59, J.
Fu25, Q. Fuehring15, W. Funk48, E. Gabriel32, T. Gaintseva42, A. Gallas
Torreira46, D. Galli20,d, S. Gambetta58,48, Y. Gan3, M. Gandelman2, P.
Gandini25, Y. Gao5, M. Garau27, L.M. Garcia Martin56, P. Garcia Moreno45, J.
García Pardiñas26,j, B. Garcia Plana46, F.A. Garcia Rosales12, L. Garrido45,
C. Gaspar48, R.E. Geertsema32, D. Gerick17, L.L. Gerken15, E. Gersabeck62, M.
Gersabeck62, T. Gershon56, D. Gerstel10, Ph. Ghez8, V. Gibson55, H.K.
Giemza36, M. Giovannetti23,p, A. Gioventù46, P. Gironella Gironell45, L.
Giubega37, C. Giugliano21,f,48, K. Gizdov58, E.L. Gkougkousis48, V.V.
Gligorov13, C. Göbel70, E. Golobardes85, D. Golubkov41, A. Golutvin61,83, A.
Gomes1,a, S. Gomez Fernandez45, F. Goncalves Abrantes63, M. Goncerz35, G.
Gong3, P. Gorbounov41, I.V. Gorelov40, C. Gotti26, E. Govorkova48, J.P.
Grabowski17, T. Grammatico13, L.A. Granado Cardoso48, E. Graugés45, E.
Graverini49, G. Graziani22, A. Grecu37, L.M. Greeven32, P. Griffith21,f, L.
Grillo62, S. Gromov83, B.R. Gruberg Cazon63, C. Gu3, M. Guarise21, P. A.
Günther17, E. Gushchin39, A. Guth14, Y. Guz44, T. Gys48, T. Hadavizadeh69, G.
Haefeli49, C. Haen48, J. Haimberger48, T. Halewood-leagas60, P.M. Hamilton66,
J.P. Hammerich60, Q. Han7, X. Han17, T.H. Hancock63, S. Hansmann-Menzemer17,
N. Harnew63, T. Harrison60, C. Hasse48, M. Hatch48, J. He6,b, M. Hecker61, K.
Heijhoff32, K. Heinicke15, A.M. Hennequin48, K. Hennessy60, L. Henry48, J.
Heuel14, A. Hicheur2, D. Hill49, M. Hilton62, S.E. Hollitt15, J. Hu17, J.
Hu72, W. Hu7, X. Hu3, W. Huang6, X. Huang73, W. Hulsbergen32, R.J. Hunter56,
M. Hushchyn82, D. Hutchcroft60, D. Hynds32, P. Ibis15, M. Idzik34, D. Ilin38,
P. Ilten65, A. Inglessi38, A. Ishteev83, K. Ivshin38, R. Jacobsson48, S.
Jakobsen48, E. Jans32, B.K. Jashal47, A. Jawahery66, V. Jevtic15, M.
Jezabek35, F. Jiang3, M. John63, D. Johnson48, C.R. Jones55, T.P. Jones56, B.
Jost48, N. Jurik48, S. Kandybei51, Y. Kang3, M. Karacson48, M. Karpov82, F.
Keizer48, M. Kenzie56, T. Ketel33, B. Khanji15, A. Kharisova84, S.
Kholodenko44, T. Kirn14, V.S. Kirsebom49, O. Kitouni64, S. Klaver32, K.
Klimaszewski36, S. Koliiev52, A. Kondybayeva83, A. Konoplyannikov41, P.
Kopciewicz34, R. Kopecna17, P. Koppenburg32, M. Korolev40, I. Kostiuk32,52, O.
Kot52, S. Kotriakhova21,38, P. Kravchenko38, L. Kravchuk39, R.D. Krawczyk48,
M. Kreps56, F. Kress61, S. Kretzschmar14, P. Krokovny43,v, W. Krupa34, W.
Krzemien36, W. Kucewicz35,t, M. Kucharczyk35, V. Kudryavtsev43,v, H.S.
Kuindersma32,33, G.J. Kunde67, T. Kvaratskheliya41, D. Lacarrere48, G.
Lafferty62, A. Lai27, A. Lampis27, D. Lancierini50, J.J. Lane62, R. Lane54, G.
Lanfranchi23, C. Langenbruch14, J. Langer15, O. Lantwin50, T. Latham56, F.
Lazzari29,q, R. Le Gac10, S.H. Lee86, R. Lefèvre9, A. Leflat40, S. Legotin83,
O. Leroy10, T. Lesiak35, B. Leverington17, H. Li72, L. Li63, P. Li17, S. Li7,
Y. Li4, Y. Li4, Z. Li68, X. Liang68, T. Lin61, R. Lindner48, V. Lisovskyi15,
R. Litvinov27, G. Liu72, H. Liu6, S. Liu4, A. Loi27, J. Lomba Castro46, I.
Longstaff59, J.H. Lopes2, G.H. Lovell55, Y. Lu4, D. Lucchesi28,l, S. Luchuk39,
M. Lucio Martinez32, V. Lukashenko32, Y. Luo3, A. Lupato62, E. Luppi21,f, O.
Lupton56, A. Lusiani29,m, X. Lyu6, L. Ma4, R. Ma6, S. Maccolini20,d, F.
Machefert11, F. Maciuc37, V. Macko49, P. Mackowiak15, S. Maddrell-Mander54, O.
Madejczyk34, L.R. Madhan Mohan54, O. Maev38, A. Maevskiy82, D. Maisuzenko38,
M.W. Majewski34, J.J. Malczewski35, S. Malde63, B. Malecki48, A. Malinin81, T.
Maltsev43,v, H. Malygina17, G. Manca27,e, G. Mancinelli10, D. Manuzzi20,d, D.
Marangotto25,i, J. Maratas9,s, J.F. Marchand8, U. Marconi20, S. Mariani22,g,
C. Marin Benito48, M. Marinangeli49, J. Marks17, A.M. Marshall54, P.J.
Marshall60, G. Martellotti30, L. Martinazzoli48,j, M. Martinelli26,j, D.
Martinez Santos46, F. Martinez Vidal47, A. Massafferri1, M. Materok14, R.
Matev48, A. Mathad50, Z. Mathe48, V. Matiunin41, C. Matteuzzi26, K.R.
Mattioli86, A. Mauri32, E. Maurice12, J. Mauricio45, M. Mazurek48, M.
McCann61, L. Mcconnell18, T.H. Mcgrath62, A. McNab62, R. McNulty18, J.V.