EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2020-051 LHCb-PAPER-2020-005 22 April 2020

Measurement of the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}$ angular distribution and the ${\mathchar
28931\relax}^{0}_{b}$ polarisation in $pp$ collisions

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

This paper presents an analysis of the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}$ angular distribution and the transverse
production polarisation of ${\mathchar 28931\relax}^{0}_{b}$ baryons in
proton-proton collisions at centre-of-mass energies of 7, 8 and 13 TeV. The
measurements are performed using data corresponding to an integrated
luminosity of 4.9$\text{\,fb}^{-1}$, collected with the LHCb experiment. The
polarisation is determined in a fiducial region of ${\mathchar
28931\relax}^{0}_{b}$ transverse momentum and pseudorapidity of
$1<p_{\mathrm{T}}<20\text{\,Ge\kern-1.00006ptV\\!/}c$ and $2<\eta<5$,
respectively. The data are consistent with ${\mathchar 28931\relax}^{0}_{b}$
baryons being produced unpolarised in this region. The parity-violating
asymmetry parameter of the ${\mathchar 28931\relax}\\!\rightarrow
p{{\pi}^{-}}$ decay is also determined from the data and its value is found to
be consistent with a recent measurement by the BES III collaboration.

Published in J. High Energ. Phys. 2020, 110 (2020)

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

##  1 Introduction

Studies of the production and decay of heavy-flavour hadrons are an important
part of contemporary particle physics. The spin-$\tfrac{1}{2}$ ${\mathchar
28931\relax}^{0}_{b}$ baryon can provide information about the production of
hadrons containing $b$ quarks. For example, the ${\mathchar
28931\relax}^{0}_{b}$ polarisation is closely related to that of the $b$ quark
[1]. Heavy-quark effective theory (HQET) predicts that ${\mathchar
28931\relax}^{0}_{b}$ baryons originating from energetic $b$ quarks retain a
large fraction of the transverse $b$-quark polarisation [2, 3]. The
longitudinal polarisation is expected to vanish in $pp$ collisions due to
parity conservation in strong interactions and the term polarisation is used
to refer to the transverse polarisation of particles in this paper. The
authors of Ref. [4] estimate that the $b$-quark polarisation is of the order
of 10%. This leads to an estimate that the polarisation of the ${\mathchar
28931\relax}^{0}_{b}$ baryon can be around 10% with possible values up to 20%
[5, 1]. Measurements of $\mathchar 28931\relax$ polarisation at fixed-target
experiments [6, 7, 8] find that the polarisation strongly depends on
Feynman-$x$, $x_{\rm F}$, with polarisation vanishing at $x_{\rm F}=0$. The
variable $x_{\rm F}$ is defined by $x_{\rm F}=2p_{\rm L}/\sqrt{s}$, where
$p_{\rm L}$ is the longitudinal momentum of the baryon with respect to the
beam line and $\sqrt{s}$ is the centre-of-mass energy of the collision. If a
similar $x_{\rm F}$-dependence is present in ${\mathchar
28931\relax}^{0}_{b}$-baryon production, a negligible polarisation would be
expected at the LHC since the experiments mostly cover the phase-space region
close to $x_{\rm F}=0$. In addition, several heavy $b$-baryon states are
observed experimentally [9, 10, 11, 12, 13]. In the production of ${\mathchar
28931\relax}^{0}_{b}$ baryons from decays of these states, the connection
between the ${\mathchar 28931\relax}^{0}_{b}$ and the $b$-quark polarisation
can be further diluted due to the interaction of the $b$ quark with the light
quarks in the heavy $b$-baryon [1, 3]. The fraction of the $b$-quark
polarisation transferred to the ${\mathchar 28931\relax}^{0}_{b}$ baryon is
estimated to be around 75% in Ref. [1].

The decay ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}$, where the $\mathchar 28931\relax$ baryon
decays to ${p}{{\pi}^{-}}$ and the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ meson decays to $\mu^{+}\mu^{-}$, can be used to measure the
polarisation of the ${\mathchar 28931\relax}^{0}_{b}$ baryon as well as to
test the theoretical understanding of hadronic decays of ${\mathchar
28931\relax}^{0}_{b}$ baryons.111The inclusion of charge-conjugate processes
is implied throughout this paper except when stated otherwise. The angular
distribution of the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}$ decay is described by the polarisation of the
${\mathchar 28931\relax}^{0}_{b}$ baryon, $P_{b}$, four decay amplitudes and
by the parity-violating asymmetry parameter of the $\mathchar 28931\relax$
baryon decay, $\alpha_{\mathchar 28931\relax}$. The decay parameter
$\alpha_{\mathchar 28931\relax}$ arises due to the V$-$A nature of the weak
interaction [14]. The four decay amplitudes, $A(\lambda_{{\mathchar
28931\relax}},\lambda_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}})$
correspond to different $\mathchar 28931\relax$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ helicities,
$\lambda_{{\mathchar 28931\relax}}$ and
$\lambda_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}$. The notation
$a_{\pm}=A(\pm\tfrac{1}{2},0)$ and $b_{\pm}=A(\mp\tfrac{1}{2},\pm 1)$ is used
in this paper.

In the naive heavy-quark and light-diquark limit, the $u$ and $d$ quark in the
baryon form a spin- and isospin-zero spectator system. The left-handed nature
of the charged-current interaction then implies that the $\mathchar
28931\relax$-baryon helicity is $-\tfrac{1}{2}$, such that
$|a_{+}|\approx|b_{-}|\approx 0$. Several theoretical approaches have been
used to predict the ${\mathchar 28931\relax}^{0}_{b}$ parity-violating decay
parameter

$\displaystyle\alpha_{b}=\frac{|a_{+}|^{2}-|a_{-}|^{2}+|b_{+}|^{2}-|b_{-}|^{2}}{|a_{+}|^{2}+|a_{-}|^{2}+|b_{+}|^{2}+|b_{-}|^{2}}~{},$
(1)

which is the analogue of $\alpha_{{\mathchar 28931\relax}}$ but applied to the
${\mathchar 28931\relax}^{0}_{b}$ decay. The value of $\alpha_{b}$ is
predicted to be in the range from $-0.2$ to $-0.1$ within a factorisation
approximation [15, 16, 17], around $-0.2$ in the covariant oscillator quark
model [18] or light-front quark model [19] and in the range from $-0.17$ to
$-0.14$ in approaches based on perturbative QCD [20]. In contrast, a
prediction based on HQET yields a value of $\alpha_{b}\sim 0.8$ [5]. The
covariant quark model has recently been used to predict $\alpha_{b}\sim-0.07$
and the magnitudes of the four helicity amplitudes [21, 22]. The amplitudes
predicted by this model agree with the naive expectation that $|a_{+}|$ and
$|b_{-}|$ are small, while $|a_{-}|$ and $|b_{+}|$ are of similar size.

The polarisation of ${\mathchar 28931\relax}^{0}_{b}$ baryons was previously
measured at LEP in $Z$ decays [23, 24, 25] and at the LHC in $pp$ collisions
[26, 27]. The values measured at the LHC are

$\begin{split}P_{b}&=0.06\pm 0.07\pm 0.02\quad({\rm LHCb})~{},\\\
P_{b}&=0.00\pm 0.06\pm 0.02\quad({\rm CMS})~{}.\end{split}$

Both measurements were performed using an angular analysis of the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}$ decay. The LHCb measurement used data
collected at $\sqrt{s}=7\text{\,Te\kern-1.00006ptV}$, while the CMS
measurement used data from both 7 and 8 TeV $pp$ collisions. A similar
analysis was performed by the ATLAS collaboration [28] but assuming $P_{b}=0$
and measuring only magnitudes of the decay amplitudes. While all three
measurements are compatible, the LHCb and CMS results are unphysical; the LHCb
value of $|b_{-}|^{2}$ and the CMS value of $|a_{+}|^{2}$ are negative. This
is likely to be due to the use of a now outdated value of $\alpha_{\mathchar
28931\relax}=0.642\pm 0.013$ from an earlier Particle Data Group average of
the results of Refs. [29, 30, 31, 32, 33] that is no longer used. This value
is significantly smaller than that measured by the BES III collaboration using
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\\!\rightarrow{\mathchar
28931\relax}{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}$
decays [34]. In their analysis, the BES III collaboration determine
$\alpha_{\mathchar 28931\relax}$ and $\alpha_{\kern
1.25995pt\overline{\kern-1.25995pt\mathchar 28931\relax}}$, for the
${\mathchar 28931\relax}\\!\rightarrow{p}{{\pi}^{-}}$ and ${\kern
1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}\\!\rightarrow{\overline{p}}{{\pi}^{+}}$ decays, to be
$\alpha_{{\mathchar 28931\relax}}=0.750\pm 0.009\pm 0.004$ and $\alpha_{{\kern
1.25995pt\overline{\kern-1.25995pt\mathchar 28931\relax}}}=-0.758\pm 0.010\pm
0.007$. The BES III measurement is supported by a reanalysis of CLAS
$\gamma{p}\rightarrow{{K}^{+}}{\mathchar 28931\relax}$ scattering data in Ref.
[35], which gives $\alpha_{\mathchar 28931\relax}=0.721\pm 0.006\pm 0.005$.
The polarisation of ${\mathchar 28931\relax}^{0}_{b}$ baryons has also been
determined to be $P_{b}=(0\pm 5)\%$ in the LHCb acceptance using ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar 28931\relax}{\mu^{+}\mu^{-}}$
decays, under the assumption that the polarisation is independent of
$\sqrt{s}$ [36].

This paper describes a measurement of the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}$ angular distribution using data collected with
the LHCb experiment during Run 1 and Run 2 of the LHC. The data set
corresponds to 1.0, 2.0 and 1.9$\text{\,fb}^{-1}$ of integrated luminosity
collected at $\sqrt{s}=7$, 8 and 13 TeV in 2011, 2012 and 2015–2016,
respectively. A measurement of the polarisation and the decay amplitudes is
made, using the BES III value of $\alpha_{{\mathchar 28931\relax}}$ as an
input. The polarisation of ${\mathchar 28931\relax}^{0}_{b}$ baryons is
measured for the first time at $\sqrt{s}=13\text{\,Te\kern-1.00006ptV}$.

The paper starts by describing the angular formalism used in the analysis in
Section 2. Section 3 introduces the LHCb detector. Section 4 describes the
selection of candidates from the LHCb data set. The yields of ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}$ decays in the different data sets are obtained
in Section 5. Section 6 describes the procedure used to correct the data for
the nonuniformity of the reconstruction and selection. The production
polarisation and decay amplitudes are obtained through a two-step procedure
described in Sections 7 and 9. Section 8 discusses sources of systematic
uncertainty in the measurement. Finally, conclusions are presented in Section
10.

##  2 Angular formalism

The kinematics of the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}$ decay, including the subsequent decays of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson and the $\mathchar
28931\relax$ baryon, can be parameterised by five decay angles and a unit
vector in the direction transverse to the production plane, $\hat{n}$, against
which the polarisation is measured [37]. The unit vector is defined as
$\hat{n}=(\vec{p}_{\rm beam}\times\vec{p}_{{{\mathchar
28931\relax}^{0}_{b}}})/|\vec{p}_{\rm beam}\times\vec{p}_{{{\mathchar
28931\relax}^{0}_{b}}}|$, where $\vec{p}_{{{\mathchar 28931\relax}^{0}_{b}}}$
and $\vec{p}_{\rm beam}$ are vectors in the direction of the ${\mathchar
28931\relax}^{0}_{b}$ baryon and the beam in the centre-of-mass frame of the
$pp$ collision. In the case of the LHCb detector, $\vec{p}_{\rm beam}$ is the
direction of the beam that points into the detector from the collision point.
The four-momentum of each particle is boosted into the centre-of-mass frame to
account for the small beam-crossing angle of the LHC collisions before
$\hat{n}$ is calculated. The five decay angles are: the angle, $\theta$,
between $\hat{n}$ and the $\mathchar 28931\relax$ flight direction in the
${\mathchar 28931\relax}^{0}_{b}$ rest frame; the polar, $\theta_{b}$, and
azimuthal, $\phi_{b}$, angles of the proton in the $\mathchar 28931\relax$
rest frame; and the polar, $\theta_{l}$, and azimuthal, $\phi_{l}$, angles of
the $\mu^{+}$ in the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ rest
frame. The angles $\theta$, $\theta_{l}$ and $\theta_{b}$ are defined in the
range $[0,\pi]$, while $\phi_{l}$ and $\phi_{b}$ are defined over
$[-\pi,+\pi]$. A visual depiction of the angular basis is given in Fig. 1. The
decay angles for the ${\kern 1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{0}_{b}$ decay are defined assuming no $C\\!P$ violation in
the ${\mathchar 28931\relax}^{0}_{b}$ or $\mathchar 28931\relax$ decay, such
that the distributions of ${\mathchar 28931\relax}^{0}_{b}$ and ${\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{0}_{b}$ decays
are identical.

Figure 1:  Definition of the five decay angles, $\theta$, $\theta_{b}$,
$\phi_{b}$, $\theta_{l}$ and $\phi_{l}$ used to describe the kinematics of the
${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}$ decay. The angles are described in the text.

The angular distribution of the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}$ decay can be expressed as [38]

$\displaystyle\frac{\mathrm{d}^{5}\Gamma}{\mathrm{d}\vec{\Omega}}$
$\displaystyle=\frac{3}{32\pi^{2}}\sum\limits_{i}J_{i}(a_{+},a_{-},b_{+},b_{-},\alpha_{{\mathchar
28931\relax}},P_{b})f_{i}(\vec{\Omega})~{},$ (2)

where
$\vec{\Omega}=(\cos\theta,\cos\theta_{b},\phi_{b},\cos\theta_{l},\phi_{l})$.
The angular terms, $J_{i}$, and the angular functions, $f_{i}$, are given in
Table 1. The ${\mathchar 28931\relax}^{0}_{b}$ polarisation is accessible
through terms $J_{11}$–$J_{34}$.

Table 1:  Angular functions parameterising the ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{\mathchar 28931\relax}$ angular distribution. The numbering scheme is the same as in Ref. [37].  $i$ | $J_{i}$ | $f_{i}(\vec{\Omega})$
---|---|---
1 | $\tfrac{1}{4}(2|a_{+}|^{2}+2|a_{-}|^{2}+|b_{+}|^{2}+|b_{-}|^{2})$ | $\sin^{2}\theta_{l}$
2 | $\tfrac{1}{2}|b_{+}|^{2}+\tfrac{1}{2}|b_{-}|^{2}$ | $\cos^{2}\theta_{l}$
4 | $\tfrac{1}{4}\alpha_{{\mathchar 28931\relax}}(2|a_{+}|^{2}-2|a_{-}|^{2}-|b_{+}|^{2}+|b_{-}|^{2})$ | $\sin^{2}\theta_{l}\cos\theta_{b}$
5 | $\tfrac{1}{2}\alpha_{{\mathchar 28931\relax}}(|b_{-}|^{2}-|b_{+}|^{2})$ | $\cos^{2}\theta_{l}\cos\theta_{b}$
7 | $\tfrac{1}{\sqrt{2}}\alpha_{{\mathchar 28931\relax}}{\rm Re}(-b_{+}^{*}a_{+}+b_{-}a_{-}^{*})$ | $\sin\theta_{l}\cos\theta_{l}\sin\theta_{b}\cos\left(\phi_{b}+\phi_{l}\right)$
9 | $\tfrac{1}{\sqrt{2}}\alpha_{{\mathchar 28931\relax}}{\rm Im}(b_{+}^{*}a_{+}-b_{-}a_{-}^{*})$ | $\sin\theta_{l}\cos\theta_{l}\sin\theta_{b}\sin\left(\phi_{b}+\phi_{l}\right)$
11 | $\tfrac{1}{4}P_{b}(2|a_{+}|^{2}-2|a_{-}|^{2}+|b_{+}|^{2}-|b_{-}|^{2})$ | $\sin^{2}\theta_{l}\cos\theta$
12 | $\tfrac{1}{2}P_{b}(|b_{+}|^{2}-|b_{-}|^{2})$ | $\cos^{2}\theta_{l}\cos\theta$
14 | $\tfrac{1}{4}P_{b}\alpha_{{\mathchar 28931\relax}}(2|a_{+}|^{2}+2|a_{-}|^{2}-|b_{+}|^{2}-|b_{-}|^{2})$ | $\sin^{2}\theta_{l}\cos\theta_{b}\cos\theta$
15 | $-\tfrac{1}{2}P_{b}\alpha_{{\mathchar 28931\relax}}(|b_{+}|^{2}+|b_{-}|^{2})$ | $\cos^{2}\theta_{l}\cos\theta_{b}\cos\theta$
17 | $-\tfrac{1}{\sqrt{2}}P_{b}\alpha_{{\mathchar 28931\relax}}{\rm Re}(b_{+}^{*}a_{+}+b_{-}a_{-}^{*})$ | $\sin\theta_{l}\cos\theta_{l}\sin\theta_{b}\cos\left(\phi_{b}+\phi_{l}\right)\cos\theta$
19 | $\tfrac{1}{\sqrt{2}}P_{b}\alpha_{{\mathchar 28931\relax}}{\rm Im}(b_{+}^{*}a_{+}+b_{-}a_{-}^{*})$ | $\sin\theta_{l}\cos\theta_{l}\sin\theta_{b}\sin\left(\phi_{b}+\phi_{l}\right)\cos\theta$
21 | $-\tfrac{1}{\sqrt{2}}P_{b}{\rm Im}(b_{+}^{*}a_{-}-b_{-}a_{+}^{*})$ | $\sin\theta_{l}\cos\theta_{l}\sin\phi_{l}\sin\theta$
23 | $\tfrac{1}{\sqrt{2}}P_{b}{\rm Re}(b_{+}^{*}a_{-}-b_{-}a_{+}^{*})$ | $\sin\theta_{l}\cos\theta_{l}\cos\phi_{l}\sin\theta$
25 | $\tfrac{1}{\sqrt{2}}P_{b}\alpha_{{\mathchar 28931\relax}}{\rm Im}(b_{+}^{*}a_{-}+b_{-}a_{+}^{*})$ | $\sin\theta_{l}\cos\theta_{l}\cos\theta_{b}\sin\phi_{l}\sin\theta$
27 | $-\tfrac{1}{\sqrt{2}}P_{b}\alpha_{{\mathchar 28931\relax}}{\rm Re}(b_{+}^{*}a_{-}+b_{-}a_{+}^{*})$ | $\sin\theta_{l}\cos\theta_{l}\cos\theta_{b}\cos\phi_{l}\sin\theta$
30 | $P_{b}\alpha_{{\mathchar 28931\relax}}{\rm Im}(a_{+}a_{-}^{*})$ | $\sin^{2}\theta_{l}\sin\theta_{b}\sin\phi_{b}\sin\theta$
32 | $-P_{b}\alpha_{{\mathchar 28931\relax}}{\rm Re}(a_{+}a_{-}^{*})$ | $\sin^{2}\theta_{l}\sin\theta_{b}\cos\phi_{b}\sin\theta$
33 | $-\tfrac{1}{2}P_{b}\alpha_{{\mathchar 28931\relax}}{\rm Re}(b_{+}^{*}b_{-})$ | $\sin^{2}\theta_{l}\sin\theta_{b}\cos(2\phi_{l}+\phi_{b})\sin\theta$
34 | $\tfrac{1}{2}P_{b}\alpha_{{\mathchar 28931\relax}}{\rm Im}(b_{+}^{*}b_{-})$ | $\sin^{2}\theta_{l}\sin\theta_{b}\sin(2\phi_{l}+\phi_{b})\sin\theta$

##  3 Detector and simulation

The LHCb detector [39, 40] 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 [41], 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 [42, 43] 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$\text{\,Ge\kern-1.00006ptV\\!/}c$.
The minimum distance of a track to a primary $pp$ collision vertex (PV), the
impact parameter (IP), is measured with a resolution of
$(15+(29\text{\,Ge\kern-1.00006ptV\\!/}c)/p_{\mathrm{T}})\,\upmu\text{m}$,
where $p_{\mathrm{T}}$ is the component of the momentum transverse to the
beam. Muons are identified by a system composed of alternating layers of iron
and multiwire proportional chambers [44]. The online event selection is
performed by a trigger [45], which consists of a hardware stage, based on
information from the muon system and calorimeters, followed by a software
stage, which applies a full event reconstruction.

Samples of simulated events are required to model the effects of the detector
acceptance and the imposed selection requirements on the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}$ angular distribution. In the simulation, $pp$
collisions are generated using Pythia [46] with a specific LHCb configuration
[47]. Decays of unstable particles are described by EvtGen [48], in which
final-state radiation is generated using Photos [49]. The interaction of the
generated particles with the detector, and its response, are implemented using
the Geant4 toolkit [50, *Agostinelli:2002hh] as described in Ref. [52]. The
$p_{\mathrm{T}}$ distribution of the simulated ${\mathchar
28931\relax}^{0}_{b}$ baryons is weighted to match the spectrum observed in
Ref. [53].

##  4 Candidate selection

Signal candidates are formed by combining a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$-meson candidate with a
$\mathchar 28931\relax$-baryon candidate. The
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates are reconstructed
from two oppositely charged tracks that have been identified as muons. The
muons are required to have a significant IP with respect to all PVs in the
event and form a common vertex with a good vertex-fit quality. The dimuon mass
is required to be in the range
$2900<m({\mu^{+}\mu^{-}})<3150\text{\,Me\kern-1.00006ptV\\!/}c^{2}$. The
$\mathchar 28931\relax$ candidates are reconstructed in two categories:
$\mathchar 28931\relax$ baryons that decay early enough for the proton and
pion to be reconstructed in the vertex detector; and $\mathchar 28931\relax$
baryons that decay later, such that they cannot be reconstructed in the vertex
detector. These categories are referred to as long and downstream,
respectively. The $\mathchar 28931\relax$ candidates in the long category have
a better mass, momentum and vertex resolution than those in the downstream
category. Approximately two thirds of the candidates are reconstructed in the
downstream category. For both categories, the proton and pion are required to
be significantly displaced from all PVs in the event and form a common vertex
with a good vertex-fit quality. The $\mathchar 28931\relax$ candidates are
also required to have an invariant mass within
30$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ of the known $\mathchar
28931\relax$-baryon mass [54], a decay time larger than 2 ps and a decay
vertex at $z<2350\text{\,mm}$. The $z$-axis is aligned with the LHC beam line,
with positive $z$ in the direction of the LHCb detector acceptance, where
$z=0$ corresponds approximately to the centre of the $pp$ interaction region.
The vertex position requirement is imposed to remove background from material
interactions in front of the large-area silicon-strip detector. The
${\mathchar 28931\relax}^{0}_{b}$ candidate is associated with the PV relative
to which it has smallest $\chi^{2}_{\text{IP}}$, where $\chi^{2}_{\text{IP}}$
is defined as the difference in the vertex-fit $\chi^{2}$ of a given PV
reconstructed with and without a considered particle. The ${\mathchar
28931\relax}^{0}_{b}$ candidate is required to have a good vertex-fit quality,
to be consistent with originating from its associated PV and to have a vertex
position that is significantly displaced from that PV. A kinematic fit is then
performed, constraining the masses of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\mathchar 28931\relax$
candidates to their known values [54] and constraining the ${\mathchar
28931\relax}^{0}_{b}$ candidate to originate from its associated PV.

The signal candidates are required to have passed a hardware trigger that
selects either a single muon with a large transverse momentum or a pair of
muons with a large product of their individual transverse momenta. The
software trigger requires a candidate to be at least partially reconstructed
with a secondary vertex that has a significant displacement from any PV. At
least one charged particle must have a large $p_{\mathrm{T}}$ and be
inconsistent with originating from a PV. A multivariate discriminator [55] is
used for the identification of secondary vertices consistent with the decay of
a $b$ hadron.

A neural network [56, 57] is trained to reject background from events where
tracks have been mistakenly combined to form a signal candidate (combinatorial
background). The network is trained using simulated ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}$ decays as a signal sample and candidates from
the data with a ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{\mathchar
28931\relax}$ invariant mass, $m({{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}})$, larger than
$5900\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ as a background sample. The neural
network uses the following inputs: the ${\mathchar 28931\relax}^{0}_{b}$ decay
time and $p_{\mathrm{T}}$; the $\mathchar 28931\relax$ mass, decay time and
$p_{\mathrm{T}}$; the $\chi^{2}$ of the fitted ${\mathchar
28931\relax}^{0}_{b}$ decay vertex; the angle between the ${\mathchar
28931\relax}^{0}_{b}$ momentum direction and the vector connecting the primary
and ${\mathchar 28931\relax}^{0}_{b}$ decay vertices; and the
$\chi^{2}_{\text{IP}}$ of the final-state hadron and muon with the largest
$p_{\mathrm{T}}$ with respect to its associated PV. Separate classifiers are
trained for data taken at different collision energies. A single neural
network is used for both long and downstream candidates, with the $\mathchar
28931\relax$ category used as an input to the network. The working point of
the neural network is chosen to maximise
$\varepsilon_{S}S/\sqrt{\varepsilon_{S}S+\varepsilon_{B}B}$. Here, $S$ and $B$
are the number of signal and background decays within
14$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ of the known ${\mathchar
28931\relax}^{0}_{b}$ mass [54] (about twice the resolution on the invariant
mass) before the application of the classifier, $\varepsilon_{S}$ and
$\varepsilon_{B}$ are the efficiencies of the classifier requirement evaluated
on the signal and background training samples.

The ${\mathchar 28931\relax}^{0}_{b}$ candidates are required to be in the
fiducial region, $1<p_{\mathrm{T}}<20\text{\,Ge\kern-1.00006ptV\\!/}c$ and
$2<\eta<5$. The mean of the $x_{\rm F}$ distribution of the selected
${\mathchar 28931\relax}^{0}_{b}$ signal decays varies between 0.015 at
$\sqrt{s}=13\text{\,Te\kern-1.00006ptV}$ and 0.028 at 7 TeV. The corresponding
standard deviations of these distributions are 0.008 and 0.014.

Several sources of specific background have been considered. The largest
specific background originates from
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{S}}}$ decays, where one of the pions from the
${{K}^{0}_{\mathrm{S}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decay is
reconstructed as a proton. Background from partially reconstructed $b$-baryon
decays such as ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}(1520)$, ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\mathchar 28934\relax}{}^{0}}$ or ${\mathchar
28932\relax}_{b}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28932\relax}$ decays, where the ${\mathchar
28931\relax}(1520)$, ${{\mathchar 28934\relax}{}^{0}}$ and ${\mathchar
28932\relax}$ subsequently decay to a $\mathchar 28931\relax$ baryon, give a
negligible contribution to the selected sample.

##  5 Signal yields

The yield of ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}$ decays in each data set and in each $\mathchar
28931\relax$ category is determined by performing an extended unbinned
maximum-likelihood fit to the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}$ mass distribution. The signal is parameterised
by the sum of two Crystal Ball (CB) functions [58] combined with a Gaussian
function. The two CB functions have a common peak position and width; one has
a power-law tail on the lower side of the peak, the other on the upper side of
the peak. The Gaussian function shares the same peak position as the two CB
functions. The tail parameters and the relative fractions of the three signal
components are fixed, for each data set, from fits to simulated ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}$ decays. The combinatorial background is
described by an exponential function. The background from
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{S}}}$ decays is described by a CB function with
parameters fixed from simulated decays. Figure 2 shows the
$m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{\mathchar 28931\relax})$
distribution and the result of the fits for each of the four data-taking
years, with the two $\mathchar 28931\relax$ categories combined. The signal
yields in the long and downstream categories of the 2011, 2012, 2015 and 2016
data are given in Table 2.

Figure 2:  Mass distribution of selected ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{\mathchar 28931\relax}$ candidates in (top-left) the 2011, (top-right) the 2012, (bottom-left) the 2015 and (bottom-right) the 2016 data sets. The long and downstream categories have been combined. The results of fits to the distributions are overlaid.  Table 2:  Signal yields in the long and downstream categories of the 2011, 2012, 2015 and 2016 data sets.  | 2011 | 2012 | 2015 | 2016
---|---|---|---|---
Long | $1\,792\pm 46$ | $4\,099\pm 74$ | $\phantom{0}\,925\pm 34$ | $\phantom{0}6\,291\pm\phantom{0}88$
Downstream | $3\,030\pm 59$ | $7\,904\pm 96$ | $1\,722\pm 47$ | $12\,809\pm 125$

##  6 Angular efficiency

Both the detector acceptance and candidate selection affect the observed
angular distribution of the candidates. As described in Ref. [59], the largest
distortions of the angular distribution arise from kinematic requirements in
the reconstruction and in the trigger. Corrections for the nonuniformity of
the angular efficiency are determined using samples of simulated ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}$ decays. The simulated samples are generated
with isotropic decays of the ${\mathchar 28931\relax}^{0}_{b}$ baryon, the
$\mathchar 28931\relax$ baryon and the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ meson. The resulting angular distribution is uniform in each of the
five decay angles. After the selection procedure is applied, the angular
distribution of the simulated decays is proportional to the full
reconstruction and selection efficiency. A full five-dimensional description
is used to parameterise the angular distribution. The parameterisation
exploits the orthogonality of Legendre polynomials, $L_{j}(x)$, and of cosine
functions. In its most general form, the distribution and hence the efficiency
can be described by the sum

$\displaystyle\varepsilon(\vec{\Omega})=\sum_{rstuv}c_{rstuv}L_{r}(\cos\theta)L_{s}(\cos\theta_{l})L_{t}(\cos\theta_{b})L_{u}(\phi_{b}/\pi)\cos(v\phi_{l})~{}.$
(3)

The coefficients $c_{rstuv}$ are determined by performing a moment analysis of
the simulated sample.

To describe the efficiency shape accurately, a large number of terms is needed
in each dimension. An absolute normalisation of the efficiency is not needed
in this analysis. To reduce the complexity of the parameterisation, an
iterative approach is used, where the efficiency model is constructed in
stages. At the first stage, each dimension is parameterised independently and
the simulated decays are corrected by the inverse of this simplified
efficiency model. At the second stage, three-dimensional corrections are
determined separately for ($\cos\theta_{l}$, $\phi_{l}$, $\cos\theta$) and for
($\cos\theta_{b}$, $\phi_{b}$, $\cos\theta$), which are subsequently applied
to the simulated decays. Finally, a five-dimensional correction is applied
according to Eq. 3 with $r$, $s$, $t$, $u$ and $v$ between zero and two. Since
the $\mu^{+}$ and $\mu^{-}$ from the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ have almost identical interactions in the detector, the
parameterisation is required to be symmetric in $\cos\theta_{l}$ and
$\phi_{l}$ about zero such that only terms with even values of $s$ and $v$ are
used in the efficiency model. This assumption is validated on simulated
${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}$ decays, generated with a more realistic decay
model. A separate efficiency correction is derived for the long and downstream
$\mathchar 28931\relax$ categories in each data-taking year.

##  7 Angular moments

The values of the angular terms normalised to the total rate,
$M_{i}=J_{i}/(2J_{1}+J_{2})$, can be determined from the data by a moment
analysis,

$\displaystyle\begin{split}M_{i}&=\frac{1}{2J_{1}+J_{2}}\int\limits_{\Omega}\frac{\mathrm{d}^{5}\Gamma}{\mathrm{d}\vec{\Omega}}g_{i}(\vec{\Omega})\mathrm{d}\vec{\Omega}~{},\end{split}$
(4)

through an appropriate choice of the functions $g_{i}(\vec{\Omega})$ [37]. The
integral can be estimated by a sum over the observed candidates, $c$,

$\displaystyle
M_{i}=\left(\sum\limits_{c=1}^{N}w_{c}g_{i}(\vec{\Omega}_{c})\right)\Big{/}\left(\sum\limits_{c=1}^{N}w_{c}\right)~{},$
(5)

where the weights, $w_{c}$, are used to account for both background
contamination and the non-uniform angular efficiency of the detector
acceptance and the candidate selection and $N$ is the number of observed
candidates. The background contamination is subtracted using the sPlot
technique [60] with $m({{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}})$ as a discriminating variable.

The analysis procedure is validated on
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{S}}}$ decays, where the ${K}^{0}_{\mathrm{S}}$ meson
subsequently decays to ${{\pi}^{+}}{{\pi}^{-}}$. This decay has a similar
topology to that of the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}$ decay but has an angular dependence that is
uniform in $\cos\theta$, $\cos\theta_{b}$, $\phi_{l}$ and $\phi_{b}$ and
depends only on $\sin^{2}\theta_{l}$, resulting in $M_{1}=\tfrac{1}{2}$ and
the remaining moments being zero. The
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{S}}}$ candidates are selected in data in an analogous
way to the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}$ candidates. The measured moments for the
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{S}}}$ decay are consistent with expectation and a
$\chi^{2}$ comparison of the moments with their expected values yields a
$p$-value of 12%.

The values of the moments for the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}$ decay at the three different centre-of-mass
energies are given in Table 3. The results from the long and downstream
categories are compatible and are combined in the table. Systematic
uncertainties on the moments are discussed in Section 8. The values of moments
$M_{11}$ to $M_{34}$ are consistent with zero, indicating a small production
polarisation. The statistical covariance matrices for the moments are
determined by bootstrapping the data set (cf. Ref. [61]) and repeating the
analysis procedure. The correlation matrices for the moments are provided in
Appendix A. Figure 3 shows the background-subtracted angular projections of
the five decay angles for the selected candidates. Good agreement is seen
between the data and the result of the moment analysis. The values of the
moments are also found to be in good agreement between ${\mathchar
28931\relax}^{0}_{b}$ and ${\kern 1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{0}_{b}$ baryons, indicating that there is no significant
difference in the production polarisation or decays of the ${\mathchar
28931\relax}^{0}_{b}$ and ${\kern 1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{0}_{b}$ baryons. The numerical values of all moments and the
corresponding covariance matrices are available as supplementary material to
this article.

Table 3:  Values of the 20 moments, $M_{i}$, measured in the data collected at 7, 8 and 13 TeV centre-of-mass energies. The long and downstream categories have been combined. The first and second uncertainties are statistical and systematic, respectively.  | 7 TeV | 8 TeV | 13 TeV
---|---|---|---
$M_{1}$ | $\phantom{+}0.374\pm 0.007\pm 0.003$ | $\phantom{+}0.373\pm 0.004\pm 0.002$ | $\phantom{+}0.380\pm 0.003\pm 0.001$
$M_{2}$ | $\phantom{+}0.253\pm 0.014\pm 0.005$ | $\phantom{+}0.254\pm 0.008\pm 0.003$ | $\phantom{+}0.239\pm 0.006\pm 0.002$
$M_{4}$ | $-0.286\pm 0.017\pm 0.008$ | $-0.268\pm 0.011\pm 0.009$ | $-0.273\pm 0.008\pm 0.006$
$M_{5}$ | $-0.157\pm 0.025\pm 0.008$ | $-0.181\pm 0.015\pm 0.007$ | $-0.179\pm 0.011\pm 0.005$
$M_{7}$ | $\phantom{+}0.051\pm 0.029\pm 0.005$ | $\phantom{+}0.025\pm 0.018\pm 0.003$ | $\phantom{+}0.022\pm 0.013\pm 0.002$
$M_{9}$ | $-0.017\pm 0.029\pm 0.005$ | $-0.011\pm 0.018\pm 0.003$ | $-0.027\pm 0.013\pm 0.002$
$M_{11}$ | $\phantom{+}0.005\pm 0.014\pm 0.004$ | $\phantom{+}0.003\pm 0.009\pm 0.004$ | $-0.005\pm 0.006\pm 0.002$
$M_{12}$ | $-0.004\pm 0.018\pm 0.005$ | $\phantom{+}0.010\pm 0.011\pm 0.004$ | $\phantom{+}0.006\pm 0.008\pm 0.003$
$M_{14}$ | $\phantom{+}0.007\pm 0.025\pm 0.007$ | $-0.015\pm 0.016\pm 0.007$ | $-0.009\pm 0.012\pm 0.003$
$M_{15}$ | $-0.027\pm 0.032\pm 0.008$ | $\phantom{+}0.009\pm 0.021\pm 0.008$ | $-0.006\pm 0.016\pm 0.005$
$M_{17}$ | $\phantom{+}0.008\pm 0.039\pm 0.006$ | $-0.002\pm 0.025\pm 0.004$ | $\phantom{+}0.011\pm 0.018\pm 0.003$
$M_{19}$ | $-0.006\pm 0.038\pm 0.004$ | $-0.015\pm 0.025\pm 0.004$ | $-0.003\pm 0.018\pm 0.002$
$M_{21}$ | $-0.015\pm 0.037\pm 0.008$ | $\phantom{+}0.007\pm 0.022\pm 0.005$ | $-0.032\pm 0.016\pm 0.005$
$M_{23}$ | $-0.001\pm 0.028\pm 0.007$ | $-0.022\pm 0.017\pm 0.003$ | $\phantom{+}0.018\pm 0.012\pm 0.002$
$M_{25}$ | $-0.029\pm 0.064\pm 0.010$ | $-0.001\pm 0.038\pm 0.008$ | $\phantom{+}0.044\pm 0.029\pm 0.006$
$M_{27}$ | $\phantom{+}0.059\pm 0.051\pm 0.007$ | $\phantom{+}0.014\pm 0.030\pm 0.005$ | $\phantom{+}0.038\pm 0.023\pm 0.006$
$M_{30}$ | $-0.000\pm 0.023\pm 0.004$ | $-0.028\pm 0.014\pm 0.005$ | $\phantom{+}0.008\pm 0.010\pm 0.003$
$M_{32}$ | $-0.001\pm 0.021\pm 0.005$ | $\phantom{+}0.013\pm 0.014\pm 0.004$ | $-0.022\pm 0.010\pm 0.003$
$M_{33}$ | $\phantom{+}0.019\pm 0.021\pm 0.005$ | $-0.017\pm 0.013\pm 0.003$ | $-0.007\pm 0.009\pm 0.002$
$M_{34}$ | $\phantom{+}0.017\pm 0.021\pm 0.004$ | $\phantom{+}0.033\pm 0.013\pm 0.004$ | $\phantom{+}0.008\pm 0.009\pm 0.002$

Figure 3:  Angular distributions of $\cos\theta_{b}$, $\cos\theta_{l}$,
$\cos\theta$, $\phi_{b}$ and $\phi_{l}$ for the background-subtracted
candidates. The long and downstream categories for the different data-taking
years have been combined. The result of the moment analysis, folded with the
angular efficiency, is overlaid. The contribution from the long and downstream
categories are indicated by the green and red lines, respectively.

##  8 Systematic uncertainties

Sources of systematic uncertainty are considered if they either impact the fit
to the $m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{\mathchar
28931\relax})$ distribution, and the subsequent background subtraction, or
would directly bias the measured angular distribution. The various sources of
systematic uncertainty on this measurement are discussed below and summarised
in Table 4.

A systematic uncertainty is assigned to cover the knowledge of the signal
lineshape parameters by repeating the analysis 1000 times, varying the
lineshape parameters within their uncertainties. The resulting systematic
uncertainty is given by the standard deviation of the moments evaluated with
the different variations.

The impact of statistical uncertainty on the efficiency model, due to the
limited size of the simulated samples, is determined by bootstrapping the
simulated samples 1000 times and rederiving the efficiency models. For each
bootstrap, a new set of efficiency coefficients, $c_{rstuv}$, is determined
and the angular moments are reevaluated. For each moment, the standard
deviation of the distribution of the difference between the new and the
nominal values is assigned as a systematic uncertainty.

To evaluate the impact of the limited number of terms used for the efficiency
model, a new parameterisation is determined that allows for higher-order terms
in each dimension. Pseudoexperiments are then generated from the higher-order
model and the values of the moments determined from each pseudoexperiment
using the nominal model. The average bias on the determined value of the
moments and its uncertainty are added in quadrature and are assigned as a
systematic uncertainty.

A systematic uncertainty is assigned to cover the choice of criteria used to
match reconstructed and true particles in the simulation. This uncertainty is
evaluated using pseudoexperiments that are generated from an efficiency model
derived with a less strict set of matching requirements. The moments are then
evaluated with the nominal model. As before, the average bias on the
determined value of the moments and its uncertainty are added in quadrature
and are assigned as a systematic uncertainty.

The impact of neglecting the detector’s angular resolution in the analysis is
explored using pseudoexperiments in which the simulated angles are smeared
according to the resolution. The resolution, determined using simulated
decays, is approximately $3\text{\,mrad}$ in $\theta$ and $\theta_{l}$,
$20\text{\,mrad}$ in $\theta_{b}$, $10\text{\,mrad}$ in $\phi_{l}$ and
$45\text{\,mrad}$ in $\phi_{b}$. The resolution of the long and downstream
categories are similar after constraining the masses of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\mathchar 28931\relax$
candidates to their known values. The angular moments are then determined from
the pseudoexperiments, neglecting the resolution. The average bias on the
determined value of the moments and its uncertainty are added in quadrature
and are assigned as a systematic uncertainty. The analysis procedure also
assumes that the mass and angular variables factorise for both the signal and
the background. No significant correlation is found between the mass and
angular distribution of simulated ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}$ decays. The variables are also found to be
uncorrelated for the combinatorial background. However, a correlation is seen
between the mass and angular distributions of misidentified
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{S}}}$ decays. The impact of neglecting this
correlation is tested using pseudoexperiments, with the mass and angular
distributions of the
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{S}}}$ decays taken from a detailed simulation. The
values of the moments are then determined neglecting the correlation and the
resulting bias is taken as a systematic uncertainty. In principle there is
also an effect arising from neglecting the precession of the $\mathchar
28931\relax$-baryon spin in the external magnetic field of the experiment. The
precession is small due to the small size of the integrated field between the
production and decay points of the $\mathchar 28931\relax$ baryon.

The track-reconstruction and muon-identification efficiency of the LHCb
detector are determined from data, in bins of $p_{\mathrm{T}}$ and $\eta$,
using a tag-and-probe approach with ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\\!\rightarrow{\mu^{+}\mu^{-}}$ decays [40]. The resulting corrections
to the simulation are small and are neglected in the analysis. The impact of
neglecting these corrections is evaluated using pseudoexperiments. The
pseudoexperiments are generated from an efficiency model that takes into
account the corrections. The moments are then determined using a model that
neglects the corrections and a systematic uncertainty is assigned based on the
average bias on the moments and its uncertainty.

The trigger efficiency of the hardware trigger is also determined in data, as
a function of the muon $p_{\mathrm{T}}$, using the method described in Ref.
[45]. The impact of the resulting corrections to the simulation is again
investigated with pseudoexperiments. The pseudoexperiments are generated
taking into account corrections to the trigger efficiency and the moments are
determined neglecting the corrections. The resulting uncertainty is assigned
based on the average bias and its uncertainty.

A systematic uncertainty is assigned to the kinematic weighting of the
simulated samples using pseudoexperiments. The pseudoexperiments are generated
using the nominal model from which moments are determined using an efficiency
model that neglects the kinematic corrections. Again, the average bias and its
uncertainty are added in quadrature and are assigned as the systematic
uncertainty.

Finally, a systematic uncertainty is evaluated to cover the uncertainty on the
beam crossing angle at the LHCb interaction point. This is estimated using
simulated events in which the crossing angle is varied. The resulting
systematic uncertainty is negligible.

The total systematic uncertainty on each moment is determined by summing the
individual sources of uncertainty in quadrature. The resulting values are
given in Table 3. The systematic uncertainty is typically less than half the
size of the statistical uncertainty on a given moment. Correlated systematic
uncertainties between different moments are found to be small as are
correlations between the different data sets. Correlations between systematic
uncertainties are therefore neglected when determining the decay amplitudes
and production polarisation.

Table 4:  Systematic uncertainties on the angular moments. The largest value amongst the moments is given for each source. The total systematic uncertainty varies from 0.002 to 0.010, depending on the moment considered. The sources are described in the text.  Source | Uncertainty
---|---
Mass model | $0.003$
Simulation sample size | $0.006$
Polynomial order | $0.004$
Truth matching criteria | $0.007$
Angular resolution | $0.003$
Factorisation of mass and angles | $0.003$
Tracking and muon-identification efficiency | $0.005$
Trigger efficiency modelling | $0.003$
Kinematic weighting | $0.006$
Beam-crossing angle | $0.001$

##  9 Decay amplitudes and production polarisation

The decay amplitudes and the production polarisation are determined from the
moments using a Bayesian analysis. The marginalisation over unwanted
parameters is performed using Markov Chain Monte Carlo, with the Metropolis-
Hastings algorithm employed to sample points in the parameter space [62, 63].
The likelihood at each point in the parameter space is given by

$\displaystyle L=\left[\prod\limits_{{\rm data\,set}\,j}{\rm
exp}(-\tfrac{1}{2}\vec{D}_{j}^{\rm T}C^{-1}_{j}\vec{D}_{j})\right]\times{\rm
exp}\left(-\frac{1}{2}\left(\frac{\alpha_{{\mathchar
28931\relax}}-\alpha_{{\mathchar
28931\relax}}^{\text{BES}}}{\sigma(\alpha^{\text{BES}}_{{\mathchar
28931\relax}})}\right)^{2}\right)~{},$ (6)

where $\vec{D}_{j}$ is a vector representing the difference between the
measured values of the moments and the values of the moments at that point in
the parameter space and $C_{j}$ is the covariance matrix combining the
statistical and systematic uncertainties on the moments. The last term in the
likelihood originates from the external constraints from BES III on the value
of $\alpha_{{\mathchar 28931\relax}}$. In this analysis, the recent BES III
result [34] for the $\mathchar 28931\relax$ asymmetry parameter is used.
Averaging the BES III values for $\mathchar 28931\relax$ and $\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}$ decays yields
$\alpha_{{\mathchar 28931\relax}}^{\text{BES}}=0.754$ with an uncertainty
$\sigma(\alpha^{\text{BES}}_{{\mathchar 28931\relax}})=0.003$. The value of
$\alpha_{{\mathchar 28931\relax}}$ and the values of the complex amplitudes
$a_{\pm}$ and $b_{\pm}$ are shared between the different data sets but the
polarisation is allowed to differ between different centre-of-mass energies.
The Bayesian analysis procedure has been validated for both small and large
values of the polarisation using pseudoexperiments.

The resulting marginal posterior distributions for the amplitudes and
polarisation are shown in Figure 4. The magnitude and phase of $b_{+}$ are
fixed to be $|b_{+}|=1$ and ${\rm arg}(b_{+})=0$. This amplitude is one of the
two amplitudes that are expected to be large. The remaining amplitudes are
measured relative to $b_{+}$. A uniform prior is assumed on their magnitudes
and phases and on $P_{b}$. The priors use the ranges $[-1,+1]$ for $P_{b}$,
$[-\pi,+\pi]$ for the phases, and the range $[0,20]$ for the magnitudes of the
amplitudes. The values of the amplitudes and the polarisations are given in
Table 5. The 95% credibility intervals are provided in Table 6 of the
Appendix. Figure 5 shows $P_{b}$ as a function of the $\sqrt{s}$ of the data
set. The resulting ${\mathchar 28931\relax}^{0}_{b}$ polarisation at each
centre-of-mass energy is found to be consistent with zero.

Figure 4:  Posterior probability distributions of $|a_{\pm}|$, ${\rm arg}(a_{\pm})$, $|b_{-}|$, ${\rm arg}(b_{-})$ and the transverse production polarisation of the ${\mathchar 28931\relax}^{0}_{b}$ baryons, $P_{b}$, at centre-of-mass energies of 7, 8 and 13 TeV assuming uniform priors. The shaded regions indicate the 68% and 95% credibility intervals.  Table 5:  Estimates for the magnitude and phase of the decay amplitudes and the transverse production polarisation of the ${\mathchar 28931\relax}^{0}_{b}$ baryons, extracted using the Bayesian analysis. The most probable value (MPV) and the shortest 68% interval containing the most probable value are given.  Observable | MPV | Interval
---|---|---
$|a_{+}|$ | $\phantom{+}0.129$ | $[\phantom{+}0.033,\phantom{+}0.163]$
$|a_{-}|$ | $\phantom{+}1.021$ | $[\phantom{+}0.998,\phantom{+}1.041]$
$|b_{-}|$ | $\phantom{+}0.145$ | $[\phantom{+}0.060,\phantom{+}0.188]$
${\rm arg}(a_{+})$ [rad] | $-2.523$ | $[-\pi,-1.131]~{}\text{or}~{}[2.117,\pi]$
${\rm arg}(a_{-})$ [rad] | $\phantom{+}1.122$ | $[-2.633,-1.759]~{}\text{or}~{}[0.101,2.224]$
${\rm arg}(b_{-})$ [rad] | $\phantom{+}1.788$ | $[-\pi,-2.275]~{}\text{or}~{}[0.232,\pi]$
$P_{b}$ (7 TeV) | $-0.004$ | $[-0.064,\phantom{+}0.051]$
$P_{b}$ (8 TeV) | $\phantom{+}0.001$ | $[-0.035,\phantom{+}0.045]$
$P_{b}$ (13 TeV) | $\phantom{+}0.032$ | $[-0.011,\phantom{+}0.065]$
$\alpha_{b}$ | $-0.022$ | $[-0.048,\phantom{+}0.005]$
Figure 5:  Measured transverse production polarisation of the ${\mathchar
28931\relax}^{0}_{b}$ baryons, $P_{b}$, as a function of the centre-of-mass
energy, $\sqrt{s}$, of the data set. The points indicate the most probable
value and the shaded regions the 68% and 95% credibility level intervals.

The Markov chain finds two almost-degenerate solutions, which correspond to a
change in sign of the polarisation accompanied by a change in sign of the
decay amplitudes. This occurs due to the small size of two of the amplitudes.
The degeneracy is most visible in the posterior distribution of $P_{b}$
determined at $\sqrt{s}$ of 13 TeV, leading to an asymmetric distribution. Due
to the small size of polarisation, there is little sensitivity to the phases
of the amplitudes. The magnitudes of the amplitudes $a_{+}$ and $b_{-}$ are
consistent with zero at the 95% credibility level, as expected in the heavy-
quark limit. The magnitudes of $a_{-}$ and $b_{+}$ are found to be similar in
size. Figure 6 shows the posterior distribution of the parity-violating
asymmetry parameter, $\alpha_{b}$, from the Bayesian analysis. The most
probable value of $\alpha_{b}$ is $-0.022$. The 68% credibility interval
around the most probable value is $[-0.048,0.005]$. This measurement is
consistent with, but more precise than, previous measurements of $\alpha_{b}$
by the ATLAS, CMS and LHCb collaborations [28, 27, 26].

The posterior distribution of $\alpha_{{\mathchar 28931\relax}}$ with the
constraint on $\alpha_{{\mathchar 28931\relax}}$ removed, assuming a uniform
prior in the range $[-1,+1]$, is shown in Fig. 7. The most probable value of
$\alpha_{{\mathchar 28931\relax}}$ is 0.74. The 68% credibility interval spans
$[0.71,0.78]$. The data strongly favour the larger $\alpha_{{\mathchar
28931\relax}}$ value reported by the BES III collaboration [34] over the
values from older secondary scattering measurements [29, 30, 31, 32, 33],
which are excluded with high significance. Small values of $\alpha_{{\mathchar
28931\relax}}$ are excluded by the large ${p}{{\pi}^{-}}$ forward-backward
asymmetry observed in the $\cos\theta_{b}$ distribution. Larger values of
$\alpha_{{\mathchar 28931\relax}}$ can be accommodated by changing the
magnitudes of the decay amplitudes to reduce the asymmetry.

Figure 6:  Posterior probability distribution of the parity-violating
asymmetry parameter, $\alpha_{b}$. The shaded regions indicate the 68% and 95%
credibility intervals.  Figure 7:  Posterior probability distribution for
$\alpha_{{\mathchar 28931\relax}}$, assuming a uniform prior, with all
external constraints removed. The shaded regions indicate the 68% and 95%
credibility intervals.

##  10 Summary

This paper presents a measurement of the decay amplitudes parameterising the
${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}$ angular distribution, and a measurement of the
transverse production polarisation of the ${\mathchar 28931\relax}^{0}_{b}$
baryons at $\sqrt{s}$ of 7, 8 and 13 TeV, using data collected with the LHCb
experiment. The measurements are performed in a fiducial region of ${\mathchar
28931\relax}^{0}_{b}$ transverse momentum and pseudorapidity of
$1<p_{\mathrm{T}}<20\text{\,Ge\kern-1.00006ptV\\!/}c$ and $2<\eta<5$,
respectively. The magnitudes of two of the four decay amplitudes are found to
be small. One of these amplitudes corresponds to $\mathchar 28931\relax$
helicity of $+\tfrac{1}{2}$ and ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
helicity of $0$ and the other to $\mathchar 28931\relax$ helicity of
$-\tfrac{1}{2}$ and ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ helicity of
$-1$. This is consistent with the expectation from the heavy-quark limit and
the left-handed nature of the weak interaction. The parity-violating parameter
$\alpha_{b}$ is found to be consistent with zero, with a 68% credibility
interval from $-0.048$ to $0.005$. The small negative value of $\alpha_{b}$
favoured by the data is consistent with most theoretical predictions but is
inconsistent with the prediction based on HQET in Ref. [5]. The ${\mathchar
28931\relax}^{0}_{b}$ production polarisation is found to be consistent with
zero, with 68% credibility level intervals of $[-0.06,0.05]$, $[-0.04,0.05]$
and $[-0.01,0.07]$ at $\sqrt{s}$ of 7, 8 and 13 TeV, respectively. The results
in this paper supersede those of Ref. [26] and are largely consistent with the
previous measurements [26, 27, 28]. Differences between the results presented
in this paper and the previous measurements can be attributed to the value of
$\alpha_{{\mathchar 28931\relax}}$ used in those measurements. The data
strongly support the recent BES III measurement of $\alpha_{{\mathchar
28931\relax}}$ over the previous value from secondary scattering data. With
the old value of $\alpha_{{\mathchar 28931\relax}}$, it is not possible to
describe the data with a physical set of amplitudes.

## Acknowledgements

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

## Appendices

##  Appendix A Correlation matrices

The statistical correlations between the different moments determined at the
three different centre-of-mass energies are shown in Figs. 8, 9 and 10. The
correlation coefficients are determined by bootstrapping the data set. The
covariance matrices are available as supplementary material to this article.

Figure 8: Statistical correlation between the moments determined at $\sqrt{s}$
of 7 TeV. Figure 9: Statistical correlation between the moments determined at
$\sqrt{s}$ of 8 TeV. Figure 10: Statistical correlation between the moments
determined at $\sqrt{s}$ of 13 TeV.

##  Appendix B Intervals at 95% credibility level

The 95% credibility level intervals on the decay amplitudes and production
polarisation from the Bayesian analysis of the moments are given in Table 6.
The 95% intervals on $\alpha_{b}$ and on $\alpha_{\mathchar 28931\relax}$ are
also provided. The interval on $\alpha_{\mathchar 28931\relax}$ is evaluated
after removing the external constraint on that parameter.

Table 6:  Intervals at 95% credibility level on the amplitudes, the polarisation and $\alpha_{b}$ from the Bayesian analysis. The interval on $\alpha_{{\mathchar 28931\relax}}$, with the external constraint removed, is also provided.  Observable | Interval
---|---
$|a_{+}|$ | $[\phantom{+}0.000,\phantom{+}0.200]$
$|a_{-}|$ | $[\phantom{+}0.978,\phantom{+}1.063]$
$|b_{-}|$ | $[\phantom{+}0.000,\phantom{+}0.208]$
${\rm arg}(a_{+})$ [rad] | $[-\pi,\phantom{+}0.251]~{}\text{or}~{}[\phantom{+}0.848,\pi]$
${\rm arg}(a_{-})$ [rad] | $[-\pi,-1.137]~{}\text{or}~{}[-0.459,\pi]$
${\rm arg}(b_{-})$ [rad] | $[-\pi,-0.396]~{}\text{or}~{}[\phantom{+}0.013,\pi]$
$P_{b}$ (7 TeV) | $[-0.119,\phantom{+}0.107]$
$P_{b}$ (8 TeV) | $[-0.071,\phantom{+}0.085]$
$P_{b}$ (13 TeV) | $[-0.052,\phantom{+}0.091]$
$\alpha_{b}$ | $[-0.071,\phantom{+}0.031]$
$\alpha_{\mathchar 28931\relax}$ | $[\phantom{+}0.700,\phantom{+}0.921]$

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LHCb collaboration

R. Aaij31, C. Abellán Beteta49, T. Ackernley59, B. Adeva45, M. Adinolfi53, H.
Afsharnia9, C.A. Aidala81, S. Aiola25, Z. Ajaltouni9, S. Akar66, J.
Albrecht14, F. Alessio47, M. Alexander58, A. Alfonso Albero44, G. Alkhazov37,
P. Alvarez Cartelle60, A.A. Alves Jr45, S. Amato2, Y. Amhis11, L. An21, L.
Anderlini21, G. Andreassi48, A. Andreianov37, M. Andreotti20, F. Archilli16,
A. Artamonov43, M. Artuso67, K. Arzymatov41, E. Aslanides10, M. Atzeni49, B.
Audurier11, S. Bachmann16, J.J. Back55, S. Baker60, V. Balagura11,b, W.
Baldini20, J. Baptista Leite1, R.J. Barlow61, S. Barsuk11, W. Barter60, M.
Bartolini23,47,h, F. Baryshnikov78, J.M. Basels13, G. Bassi28, V.
Batozskaya35, B. Batsukh67, A. Battig14, A. Bay48, M. Becker14, F. Bedeschi28,
I. Bediaga1, A. Beiter67, V. Belavin41, S. Belin26, V. Bellee48, K. Belous43,
I. Belyaev38, G. Bencivenni22, E. Ben-Haim12, S. Benson31, A. Berezhnoy39, R.
Bernet49, D. Berninghoff16, H.C. Bernstein67, C. Bertella47, E. Bertholet12,
A. Bertolin27, C. Betancourt49, F. Betti19,e, M.O. Bettler54, Ia. Bezshyiko49,
S. Bhasin53, J. Bhom33, M.S. Bieker14, S. Bifani52, P. Billoir12, A.
Bizzeti21,t, M. Bjørn62, M.P. Blago47, T. Blake55, F. Blanc48, S. Blusk67, D.
Bobulska58, V. Bocci30, O. Boente Garcia45, T. Boettcher63, A. Boldyrev79, A.
Bondar42,w, N. Bondar37,47, S. Borghi61, M. Borisyak41, M. Borsato16, J.T.
Borsuk33, T.J.V. Bowcock59, A. Boyer47, C. Bozzi20, M.J. Bradley60, S.
Braun65, A. Brea Rodriguez45, M. Brodski47, J. Brodzicka33, A. Brossa
Gonzalo55, D. Brundu26, E. Buchanan53, A. Büchler-Germann49, A. Buonaura49, C.
Burr47, A. Bursche26, A. Butkevich40, J.S. Butter31, J. Buytaert47, W.
Byczynski47, S. Cadeddu26, H. Cai72, R. Calabrese20,g, L. Calero Diaz22, S.
Cali22, R. Calladine52, M. Calvi24,i, M. Calvo Gomez44,l, P. Camargo
Magalhaes53, A. Camboni44,l, P. Campana22, D.H. Campora Perez31, A.F.
Campoverde Quezada5, L. Capriotti19,e, A. Carbone19,e, G. Carboni29, R.
Cardinale23,h, A. Cardini26, I. Carli6, P. Carniti24,i, K. Carvalho Akiba31,
A. Casais Vidal45, G. Casse59, M. Cattaneo47, G. Cavallero47, S. Celani48, R.
Cenci28,o, J. Cerasoli10, M.G. Chapman53, M. Charles12, Ph. Charpentier47, G.
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Ferrari19,e, L. Ferreira Lopes48, F. Ferreira Rodrigues2, S. Ferreres Sole31,
M. Ferrillo49, M. Ferro-Luzzi47, S. Filippov40, R.A. Fini18, M. Fiorini20,g,
M. Firlej34, K.M. Fischer62, C. Fitzpatrick61, T. Fiutowski34, F. Fleuret11,b,
M. Fontana47, F. Fontanelli23,h, R. Forty47, V. Franco Lima59, M. Franco
Sevilla65, M. Frank47, C. Frei47, D.A. Friday58, J. Fu25,p, Q. Fuehring14, W.
Funk47, E. Gabriel57, T. Gaintseva41, A. Gallas Torreira45, D. Galli19,e, S.
Gallorini27, S. Gambetta57, Y. Gan3, M. Gandelman2, P. Gandini25, Y. Gao4,
L.M. Garcia Martin46, J. García Pardiñas49, B. Garcia Plana45, F.A. Garcia
Rosales11, L. Garrido44, D. Gascon44, C. Gaspar47, D. Gerick16, E.
Gersabeck61, M. Gersabeck61, T. Gershon55, D. Gerstel10, Ph. Ghez8, V.
Gibson54, A. Gioventù45, P. Gironella Gironell44, L. Giubega36, C.
Giugliano20,g, K. Gizdov57, V.V. Gligorov12, C. Göbel70, E. Golobardes44,l, D.
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Gotti24,i, E. Govorkova31, J.P. Grabowski16, R. Graciani Diaz44, T.
Grammatico12, L.A. Granado Cardoso47, E. Graugés44, E. Graverini48, G.
Graziani21, A. Grecu36, R. Greim31, P. Griffith20,g, L. Grillo61, L. Gruber47,
B.R. Gruberg Cazon62, C. Gu3, M. Guarise20, P. A. Günther16, E. Gushchin40, A.
Guth13, Yu. Guz43,47, T. Gys47, T. Hadavizadeh62, G. Haefeli48, C. Haen47,
S.C. Haines54, P.M. Hamilton65, Q. Han7, X. Han16, T.H. Hancock62, S.
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K. Hennessy59, L. Henry25,46, J. Heuel13, A. Hicheur68, D. Hill62, M.
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Jurik62, S. Kandybei50, M. Karacson47, J.M. Kariuki53, N. Kazeev79, M.
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Maciuc36, V. Macko48, P. Mackowiak14, S. Maddrell-Mander53, L.R. Madhan
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Palutan22, G. Panshin80, A. Papanestis56, M. Pappagallo57, L.L.
Pappalardo20,g, C. Pappenheimer64, W. Parker65, C. Parkes61, C.J. Parkinson45,
G. Passaleva21,47, A. Pastore18, M. Patel60, C. Patrignani19,e, A. Pearce47,
A. Pellegrino31, M. Pepe Altarelli47, S. Perazzini19, D. Pereima38, P.
Perret9, K. Petridis53, A. Petrolini23,h, A. Petrov77, S. Petrucci57, M.
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M. Plo Casasus45, F. Polci12, M. Poli Lener22, M. Poliakova67, A. Poluektov10,
N. Polukhina78,c, I. Polyakov67, E. Polycarpo2, G.J. Pomery53, S. Ponce47, A.
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Trejo10, B. Rachwal34, J.H. Rademacker53, M. Rama28, M. Ramos Pernas45, M.S.
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1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4School of Physics State Key Laboratory of Nuclear Physics and Technology,
Peking University, Beijing, China
5University of Chinese Academy of Sciences, Beijing, China
6Institute Of High Energy Physics (IHEP), Beijing, China
7Institute of Particle Physics, Central China Normal University, Wuhan, Hubei,
China
8Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy,
France
9Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
10Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
11Université Paris-Saclay, CNRS/IN2P3, IJCLab, Orsay, France
12LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3,
Paris, France
13I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
14Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
15Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
16Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
17School of Physics, University College Dublin, Dublin, Ireland
18INFN Sezione di Bari, Bari, Italy
19INFN Sezione di Bologna, Bologna, Italy
20INFN Sezione di Ferrara, Ferrara, Italy
21INFN Sezione di Firenze, Firenze, Italy
22INFN Laboratori Nazionali di Frascati, Frascati, Italy
23INFN Sezione di Genova, Genova, Italy
24INFN Sezione di Milano-Bicocca, Milano, Italy
25INFN Sezione di Milano, Milano, Italy
26INFN Sezione di Cagliari, Monserrato, Italy
27INFN Sezione di Padova, Padova, Italy
28INFN Sezione di Pisa, Pisa, Italy
29INFN Sezione di Roma Tor Vergata, Roma, Italy
30INFN Sezione di Roma La Sapienza, Roma, Italy
31Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
32Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, Netherlands
33Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
34AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
35National Center for Nuclear Research (NCBJ), Warsaw, Poland
36Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
37Petersburg Nuclear Physics Institute NRC Kurchatov Institute (PNPI NRC KI),
Gatchina, Russia
38Institute of Theoretical and Experimental Physics NRC Kurchatov Institute
(ITEP NRC KI), Moscow, Russia, Moscow, Russia
39Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
40Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
41Yandex School of Data Analysis, Moscow, Russia
42Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
43Institute for High Energy Physics NRC Kurchatov Institute (IHEP NRC KI),
Protvino, Russia, Protvino, Russia
44ICCUB, Universitat de Barcelona, Barcelona, Spain
45Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
46Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain
47European Organization for Nuclear Research (CERN), Geneva, Switzerland
48Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
49Physik-Institut, Universität Zürich, Zürich, Switzerland
50NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
51Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
52University of Birmingham, Birmingham, United Kingdom
53H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
54Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
55Department of Physics, University of Warwick, Coventry, United Kingdom
56STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
57School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
58School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
59Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
60Imperial College London, London, United Kingdom
61Department of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
62Department of Physics, University of Oxford, Oxford, United Kingdom
63Massachusetts Institute of Technology, Cambridge, MA, United States
64University of Cincinnati, Cincinnati, OH, United States
65University of Maryland, College Park, MD, United States
66Los Alamos National Laboratory (LANL), Los Alamos, United States
67Syracuse University, Syracuse, NY, United States
68Laboratory of Mathematical and Subatomic Physics , Constantine, Algeria,
associated to 2
69School of Physics and Astronomy, Monash University, Melbourne, Australia,
associated to 55
70Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
71Guangdong Provencial Key Laboratory of Nuclear Science, Institute of Quantum
Matter, South China Normal University, Guangzhou, China, associated to 3
72School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
73Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 12
74Institut für Physik, Universität Rostock, Rostock, Germany, associated to 16
75Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 31
76Universiteit Maastricht, Maastricht, Netherlands, associated to 31
77National Research Centre Kurchatov Institute, Moscow, Russia, associated to
38
78National University of Science and Technology “MISIS”, Moscow, Russia,
associated to 38
79National Research University Higher School of Economics, Moscow, Russia,
associated to 41
80National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
38
81University of Michigan, Ann Arbor, United States, associated to 67

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