EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2018-197 LHCb-PAPER-2018-029 August 1, 2018

Angular moments of the decay ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar 28931\relax}{\mu^{+}\mu^{-}}$ at
low hadronic recoil

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

An analysis of the angular distribution of the decay ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar 28931\relax}{\mu^{+}\mu^{-}}$ is
presented, using data collected with the LHCb detector between 2011 and 2016
and corresponding to an integrated luminosity of approximately
5$\mbox{\,fb}^{-1}$. Angular observables are determined using a moment
analysis of the angular distribution at low hadronic recoil, corresponding to
the dimuon invariant mass squared range
$15<{q^{2}}<20{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$. The full basis of
observables is measured for the first time. The lepton-side, hadron-side and
combined forward-backward asymmetries of the decay are determined to be

$\begin{split}A^{\ell}_{\mathrm{FB}}&=-0.39\pm 0.04\mathrm{\,(stat)}\,\pm
0.01\mathrm{\,(syst)}~{},\\\ A^{h}_{\mathrm{FB}}&=-0.30\pm
0.05\mathrm{\,(stat)}\,\pm 0.02\mathrm{\,(syst)}~{},\\\ A^{\ell
h}_{\mathrm{FB}}&=+0.25\pm 0.04\mathrm{\,(stat)}\,\pm
0.01\mathrm{\,(syst)}~{}.\end{split}$

The measurements are consistent with Standard Model predictions.

Published in JHEP 09 (2018) 146

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

##  1 Introduction

In the Standard Model of particle physics (SM), the decay ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar 28931\relax}{\mu^{+}\mu^{-}}$
proceeds via a $b$ to $s$ quark flavour-changing neutral-current transition.
The decay is consequently rare in the SM, with a branching fraction of order
$10^{-6}$ [1]. In extensions of the SM the branching fraction and angular
distribution of the decay can be modified significantly, with the latter
providing a large number of particularly sensitive observables (see e.g. Ref.
[2]). The rate and angular distribution of corresponding $B$ meson decays have
been studied by the $B$-factory experiments, CDF at the TeVatron and the
ATLAS, CMS and LHCb experiments at the LHC. A global analyses of the
measurements favours a modification of the coupling strengths of the $b$ to
$s$ transition from their SM values at the level of 4 to 5 standard deviations
[3, 4, 5, 6, 7]. The decay ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar 28931\relax}{\mu^{+}\mu^{-}}$
has several important phenomenological differences to the $B$ meson decays:
the ${\mathchar 28931\relax}^{0}_{b}$ baryon is a spin-half particle and could
be produced polarised; the transition involves a diquark system as a
spectator, rather than a single quark; and the $\mathchar 28931\relax$ baryon
decays weakly resulting in observables related to the hadronic part of the
decay that are not present in the meson decays. The decay ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar 28931\relax}{\mu^{+}\mu^{-}}$
therefore provides an important additional test of the SM predictions, which
can be used to improve our understanding of the nature of the anomalies seen
in the $B$ meson decays.

The decay ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar
28931\relax}{\mu^{+}\mu^{-}}$ was first observed by the CDF collaboration [8].
The LHCb collaboration has subsequently studied the rate of the decay as a
function of the dimuon invariant mass squared, $q^{2}$, in Refs. [9, 10]. In
the LHCb analysis, evidence for a signal was only found at low hadronic recoil
(corresponding to the range
$15<{q^{2}}<20\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$). This is consistent
with recent SM predictions based on Lattice QCD calculations of the form
factors of the decay [1]. The angular distribution of the decay was studied
for the first time in Ref. [10], using two projections of the five-dimensional
angular distribution of the decay and a data set corresponding to an
integrated luminosity of 3$\mbox{\,fb}^{-1}$. The analysis measured two
angular asymmetries using the hadronic and leptonic parts of the decay in the
range $15<{q^{2}}<20\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$.

This paper presents the first measurement of the full basis of angular
observables for the decay ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar 28931\relax}{\mu^{+}\mu^{-}}$ in
the range $15<{q^{2}}<20\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$.111The
inclusion of charge-conjugated processes is implicit throughout.  The
measurement uses $pp$ collision data, corresponding to an integrated
luminosity of approximately 5$\mbox{\,fb}^{-1}$, collected between 2011 and
2016 at centre-of-mass energies of 7, 8 and 13$\mathrm{\,Te\kern-1.00006ptV}$.
The paper is organised as follows: Sec. 2 introduces the moment analysis used
to characterise the angular observables; Sec. 3 describes the LHCb detector;
Sec. 4 outlines the selection of ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar 28931\relax}{\mu^{+}\mu^{-}}$
candidates, where the $\mathchar 28931\relax$ is reconstructed in the $p$
${\pi}^{-}$ final state; Sec. 5 presents the fit to the invariant-mass
distribution of $p{{\pi}^{-}}{\mu^{+}\mu^{-}}$ candidates, from which the
yield of the ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar
28931\relax}{\mu^{+}\mu^{-}}$ signal is obtained; results are given in Sec. 7;
Section 8 summarises potential sources of systematic uncertainty; and
conclusions are presented in Sec. 9.

##  2 Moments of the angular distribution

The angular distribution of the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar 28931\relax}{\mu^{+}\mu^{-}}$
decay can be described using a normal unit-vector, $\hat{n}$, defined by the
vector product of the beam direction and the ${\mathchar 28931\relax}^{0}_{b}$
momentum vector, and five angles [11]: the angle, $\theta$, between $\hat{n}$
and the $\mathchar 28931\relax$ baryon direction in the rest frame of the
${\mathchar 28931\relax}^{0}_{b}$ baryon; polar and azimuthal angles
$\theta_{\ell}$ and $\phi_{\ell}$ describing the decay of the dimuon system;
and polar and azimuthal angles $\theta_{b}$ and $\phi_{b}$ describing the
decay of the $\mathchar 28931\relax$ baryon. An explicit definition of the
angular basis is provided in Appendix A. The beam direction is assumed to be
aligned with the positive $z$ direction in the LHCb coordinate system
[12].222The coordinate system is defined with the centre of the LHCb vertex
detector as the origin and positive $z$ pointing along the beam-line in the
direction of the detector’s dipole magnet. The small crossing angle of the
colliding beams is neglected in the analysis but is considered as a source of
systematic uncertainty. If the ${\mathchar 28931\relax}^{0}_{b}$ baryon is
produced without any preferred polarisation, the angular distribution only
depends on the angles $\theta_{\ell}$ and $\theta_{b}$ and on the angle
between the decay planes of the $\mathchar 28931\relax$ baryon and the dimuon
system ($\phi_{\ell}+\phi_{b}$). An illustration of this angular basis can be
found in Ref. [11].

The full angular distribution, averaged over the range
$15<{q^{2}}<20\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$, can be described by
the sum of 34 $q^{2}$-dependent angular terms [11],

$\displaystyle\frac{\mathrm{d}^{5}\Gamma}{\mathrm{d}\vec{\Omega}}=\frac{3}{32\pi^{2}}\sum\limits_{i}^{34}K_{i}f_{i}(\vec{\Omega})~{},$
(1)

where
$\vec{\Omega}\equiv(\cos\theta,\cos\theta_{\ell},\phi_{\ell},\cos\theta_{b},\phi_{b})$
and the $f_{i}(\vec{\Omega})$ functions have different dependencies on the
angles. The $K_{i}$ parameters depend on the underlying short-distance physics
and on the form factors governing the ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{\mathchar 28931\relax}$ transition. The full
form of the distribution is given in Appendix B. Equation 1 is normalised such
that $2K_{1}+K_{2}=1$. Twenty-four of the observables, $K_{11}$ to $K_{34}$,
are proportional to the ${\mathchar 28931\relax}^{0}_{b}$ production
polarisation and are zero if the ${\mathchar 28931\relax}^{0}_{b}$ baryons are
produced unpolarised. The reduced form of the angular distribution in the case
of zero production polarisation can be found in Refs. [13, 2].

The $K_{i}$ parameters can be determined from data by means of a maximum-
likelihood fit or via a moment analysis [14, 15]. The latter is preferred in
this analysis due to the small size of the available data sample and the large
number of unknown parameters. To determine the values of the $K_{i}$
parameters, weighting functions $g_{i}(\vec{\Omega})$ are chosen to project
out individual angular observables. The $g_{i}(\vec{\Omega})$ functions, which
are orthogonal to the $f_{j}(\vec{\Omega})$ functions, are normalised such
that

$\displaystyle K_{i}$
$\displaystyle=\int\frac{\mathrm{d}^{5}\Gamma}{\mathrm{d}\vec{\Omega}}g_{i}(\vec{\Omega})\mathrm{d}\vec{\Omega}~{}.$
(2)

The set of weighting functions used in this analysis can be found in Refs.
[11, 15] and listed in Appendix B. For the case of ideal detector response and
in the absence of background, the $K_{i}$ parameters can be estimated from
data by summing over the observed candidates. In realistic scenarios, per-
candidate weights are necessary to compensate for nonuniform selection
efficiency and background contamination. The $K_{i}$ parameters are then
estimated as

$\displaystyle
K_{i}=\sum\limits_{n}w_{n}\,g_{i}(\vec{\Omega}_{n})\Big{/}\sum\limits_{n}w_{n}~{},$
(3)

where $w_{n}$ is the product of the two weights associated with candidate $n$.
The background is subtracted using weights based on the _sPlot_ technique [16,
17]. The efficiency to reconstruct and select the candidates is determined
using samples of simulated events. The small effects of finite angular
resolution are neglected in the analysis but are considered as a source of
systematic uncertainty.

##  3 Detector and simulation

The LHCb detector [12, 18] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector (VELO) surrounding the
$pp$ interaction region [19], 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 [20]
placed downstream of the magnet. The tracking system provides a measurement of
the momentum, $p$, of charged particles with a relative uncertainty that
varies from 0.5% at low momentum to 1.0% at
200${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The minimum distance between a
track and a primary $pp$ interaction vertex (PV), the impact parameter (IP),
is measured with a resolution of
$(15+29/\mbox{$p_{\mathrm{T}}$}){\,\upmu\mathrm{m}}$, where $p_{\mathrm{T}}$
is the component of the momentum transverse to the beam, in
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors
(RICH1 and RICH2) [21]. Photons, electrons and hadrons are identified by a
calorimeter system consisting of scintillating-pad and preshower detectors, an
electromagnetic calorimeter and a hadronic calorimeter. Muons are identified
with a system composed of alternating layers of iron and multiwire
proportional chambers [22].

The online event selection is performed by a trigger [23], which consists of a
hardware stage, based on information from the calorimeter and muon systems,
followed by a software stage, which applies a full event reconstruction. The
signal candidates are required to pass through a hardware trigger that selects
events containing at least one muon with large $p_{\mathrm{T}}$ or a pair of
muons with a large product of their transverse momenta. The $p_{\mathrm{T}}$
threshold of the single muon trigger varied in the range between 1 and
2${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, depending on the data-taking
conditions. The subsequent software trigger requires a two-, three- or four-
track secondary vertex with a significant displacement from any PV. At least
one of the tracks must have a transverse momentum
$\mbox{$p_{\mathrm{T}}$}>1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and be
inconsistent with originating from a PV. A multivariate algorithm [24] is used
to identify whether the secondary vertex is consistent with the decay of a $b$
hadron.

Samples of simulated ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar 28931\relax}{\mu^{+}\mu^{-}}$
events are used to develop an offline event selection and to quantify the
effects of detector response, candidate reconstruction and selection on the
measured angular distribution. In the simulation, $pp$ collisions are
generated using Pythia [25, 26] with a specific LHCb configuration [27].
Decays of hadrons are described by EvtGen [28], in which final-state radiation
is generated using Photos [29]. The interaction of the generated particles
with the detector, and its response, are implemented using the Geant4 toolkit
[30, *Agostinelli:2002hh] as described in Ref. [32]. The samples of simulated
data are corrected to account for observed differences relative to data in
detector occupancy, vertex quality and the production kinematics of the
${\mathchar 28931\relax}^{0}_{b}$ baryon. The particle identification
performance of the detector is measured using calibration samples of data.

##  4 Candidate selection

Signal candidates are formed by combining a $\mathchar 28931\relax$ baryon
candidate with two oppositely charged particles that are identified as muons
by the muon system and have track segments in the VELO. Only muon pairs with
$q^{2}$ in the range
$15<{q^{2}}<20{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$, where the
majority of the ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar
28931\relax}{\mu^{+}\mu^{-}}$ signal is expected to be observed, are
considered. Candidates in the range
$8<{q^{2}}<11{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$, which
predominantly consist of decays via an intermediate
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson that subsequently decays
to $\mu^{+}\mu^{-}$, are also retained and used to cross-check various aspects
of the analysis.

Candidate $\mathchar 28931\relax$ decays are reconstructed in the ${\mathchar
28931\relax}\\!\rightarrow{p}{{\pi}^{-}}$ decay mode from two oppositely
charged tracks. The tracks are reconstructed in one of two categories,
depending on where the $\mathchar 28931\relax$ decayed in the detector. The
two tracks either both include information from the VELO (long candidates) or
both do not include information from the VELO (downstream
candidates).333Tracks with information from the VELO typically have a better
momentum resolution and are associated with $\mathchar 28931\relax$ baryons
with shorter lifetimes. The $\mathchar 28931\relax$ candidates must also have:
a vertex fit with a good $\chi^{2}$; a decay time of at least
2${\mathrm{\,ps}}$; an invariant mass within
30${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known $\mathchar
28931\relax$ mass [33]; and a decay vertex at $z<2350\mathrm{\,mm}$. The
requirement on the decay position removes background from hadronic
interactions in the material at the exit of the RICH1 detector. The $\mathchar
28931\relax$ baryon and the dimuon pair are required to form a vertex with a
good fit quality. The resulting ${\mathchar 28931\relax}^{0}_{b}$ candidate is
required to be consistent with originating from one of the PVs in the event
and to have a vertex position that is significantly displaced from that PV.

An artificial neural network is trained to further suppress combinatorial
background, in which tracks from an event are mistakenly combined to form a
candidate. The neural network uses simulated ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar 28931\relax}{\mu^{+}\mu^{-}}$
decays as a proxy for the signal and candidates from the upper mass sideband
of the data, with a ${\mathchar 28931\relax}{\mu^{+}\mu^{-}}$ invariant mass
greater than 5670${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, for the
background. The inputs to the neural network are: the $\chi^{2}$ of the vertex
fit to the ${\mathchar 28931\relax}^{0}_{b}$ candidate; the ${\mathchar
28931\relax}^{0}_{b}$ decay-time and the angle between the ${\mathchar
28931\relax}^{0}_{b}$ momentum vector and the vector between the PV and the
${\mathchar 28931\relax}^{0}_{b}$ decay vertex; the $\mathchar 28931\relax$
flight distance from the PV, its $p_{\mathrm{T}}$ and reconstructed mass; the
IP of the muon with the highest $p_{\mathrm{T}}$; the IP of either the pion or
proton from the $\mathchar 28931\relax$, depending on which has the highest
$p_{\mathrm{T}}$; and a measure of the isolation of the ${\mathchar
28931\relax}^{0}_{b}$ baryon in the detector. The working point of the neural
network is chosen to maximise the expected significance of the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar 28931\relax}{\mu^{+}\mu^{-}}$
signal in the $15<{q^{2}}<20{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$
region, assuming the branching fraction measured in Ref. [10]. It is checked
that selecting events based on their neural network response does not
introduce any significant bias in the reconstructed
$p{{\pi}^{-}}{\mu^{+}\mu^{-}}$ mass distribution,
$m(p{{\pi}^{-}}{\mu^{+}\mu^{-}})$.

##  5 Candidate yields

Figure 1 shows the $p{{\pi}^{-}}{\mu^{+}\mu^{-}}$ mass distribution of the
selected candidates in the Run 1 and Run 2 data sets, separated into the long-
track and downstream-track $p{{\pi}^{-}}$ categories. The candidates comprise
a mixture of ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar
28931\relax}{\mu^{+}\mu^{-}}$ decays, combinatorial background and a
negligible contribution from other $b$-hadron decays. The largest single
component of the latter arises from the decay
${{B}^{0}}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{\mu^{+}\mu^{-}}$, where the ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$
meson decays to ${{\pi}^{+}}{{\pi}^{-}}$ and is mis-reconstructed as a
$\mathchar 28931\relax$ baryon.

The yield of ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar
28931\relax}{\mu^{+}\mu^{-}}$ decays is determined by performing an unbinned
extended maximum-likelihood fit to $m(p{{\pi}^{-}}{\mu^{+}\mu^{-}})$. In the
fit, the signal is described by the sum of two modified Gaussian functions,
one with a power-law tail on the low-mass side and the other with a power-law
tail on the high-mass side of the distribution. The two Gaussian functions
have a common peak position and width parameter but different tail parameters
and relative fractions. The tail parameters and the relative fraction of the
two functions is fixed from fits performed to simulated ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar 28931\relax}{\mu^{+}\mu^{-}}$
decays. The mean and width are determined from fits to ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}$ candidates in the data. A small correction is
applied to the width parameter to account for a $q^{2}$ dependence of the
resolution seen in the simulation. Combinatorial background is described by an
exponential function, with a slope parameter that is determined from data. The
parameters describing the signal and the background are determined separately
for each data-taking period and for the long- and the downstream-track
$p{{\pi}^{-}}$ categories.

The fits result in yields of $120\pm 13$ ($175\pm 15$) and $126\pm 13$
($189\pm 16$ ) decays in the long (downstream) $p{{\pi}^{-}}$ category of the
Run 1 and Run 2 data, respectively. These fits are used to the determine the
weights needed to subtract the background in the moment analysis. The yields
are consistent with those expected based on the estimated signal efficiency,
the recorded integrated luminosity and the scaling of the ${\mathchar
28931\relax}^{0}_{b}$ production cross-section with centre-of-mass energy.

Figure 1:  Distribution of $p{{\pi}^{-}}{\mu^{+}\mu^{-}}$ invariant mass for
(left) long- and (right) downstream-track $p{{\pi}^{-}}$ categories in the
(top) Run 1 data and (bottom) Run 2 data. The result of the fit to each sample
of data is indicated by the solid blue line. The signal and background
components are illustrated by the dotted green and dashed red lines,
respectively.

##  6 Angular efficiency

The trigger, reconstruction and the selection process distort the measured
angular distribution of the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar 28931\relax}{\mu^{+}\mu^{-}}$
decays. The largest distortions are found to be the result of kinematic
requirements in the reconstruction, most notably due to an implicit momentum
threshold applied by requiring that the muons traverse the detector and reach
the muon system. The angular efficiency is parameterised in six dimensions
taking into account the correlations between the different angles and the
$q^{2}$-dependence of the angular efficiency.

$\displaystyle\varepsilon(\vec{\Omega},{q^{2}})=\sum_{ijmnrs}c_{ijmnrs}\,L_{i}(\cos\theta)\,L_{j}(\cos\theta_{\ell})\,L_{m}(\phi_{\ell}/\pi)\,L_{n}(\cos\theta_{b})\,L_{r}(\phi_{b}/\pi)\,L_{s}({q^{2}})~{},$
(4)

where the $L_{t}(x)$ denote a Legendre polynomial of order $t$ in variable
$x$, and the $q^{2}$ range considered has been rescaled linearly between $-1$
and $+1$. The coefficients $c_{ijmnrs}$ are determined by performing a moment
analysis of ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar
28931\relax}{\mu^{+}\mu^{-}}$ decays simulated according to a phase-space
model. The simulated decays are weighted such that they are uniformly
distributed in $q^{2}$ and in the five angles, after which the angular
distribution of the selected decays is proportional to the efficiency.

To achieve a good parameterisation of the efficiency, a large number of terms
is required. The number of terms is reduced using an iterative approach. As a
first step, the efficiency projection of each variable is parameterised
independently using the sum of Legendre polynomials of up to eighth order. As
a second step, correlations between pairs of angles and between individual
angles and $q^{2}$ are accounted for in turn. These corrections are
parameterised by sums involving pairs of polynomials that run up to sixth
order in each variable. As a final step, a six-dimensional correction is
applied allowing for polynomials of up to first order in the angles and
$q^{2}$. Before each step, the simulated decays are corrected to remove the
effects parameterised in the previous step. Small differences in the
efficiency to reconstruct $p$/$\overline{p}$ and ${\pi}^{+}$/${\pi}^{-}$ are
neglected.

The angular efficiency model is cross-checked in data using ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}$ and
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ decays, with
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\\!\rightarrow{\mu^{+}\mu^{-}}$. These decays have a similar topology
to the ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar
28931\relax}{\mu^{+}\mu^{-}}$ decay and well known angular distributions. For
the ${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ decay, where the
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ decays to ${\pi}^{+}$ ${\pi}^{-}$,
the parameter $K_{1}$ is one-half and the remaining observables are equal to
zero. 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 compatible with the measurements in
Refs. [34, 35, 36].

##  7 Results

The angular observables are obtained using a moment analysis of the angular
distribution, weighting candidates as described in Sec. 6 to account for their
detection efficiency. Background is subtracted using weights obtained from the
sPlot technique from the fits described in Sec. 5. The weights used to correct
for the efficiency and subtract the background are determined separately for
each data-taking period and for the long-track and downstream-track
$p{{\pi}^{-}}$ categories. The $K_{i}$ parameters are then determined from a
data set that combines the two reconstruction categories. As the polarisation
of the ${\mathchar 28931\relax}^{0}_{b}$ baryons at production may vary with
centre-of-mass energy between the Run 1 data, collected at $\sqrt{s}=7$ and
8$\mathrm{\,Te\kern-1.00006ptV}$, and the Run 2 data, collected at
$\sqrt{s}=13\mathrm{\,Te\kern-1.00006ptV}$, these two data sets are initially
treated independently. The results for the two data-taking periods are given
in Appendix C. The statistical uncertainties on the various $K_{i}$ parameters
are determined using a bootstrapping technique [37]. In each step of the
bootstrap, the process of subtracting the background and the weighting of the
candidates is repeated.

A $\chi^{2}$ comparison of the results from the two data-taking periods,
taking into account the correlations between the observables, yields a
$\chi^{2}$ of 35.0 with 33 degrees of freedom. This indicates an excellent
agreement between the two data sets and suggests that the production
polarisation is consistent for the centre-of-mass energies studied. The Run 1
and Run 2 data samples are therefore combined and the observables are
determined on the combined sample. The results are given in Table 1. The
correlation between the angular observables is presented in Appendix D. Figure
2 shows the one-dimensional angular projections of $\cos\theta_{\ell}$,
$\cos\theta_{b}$, $\cos\theta$, $\phi_{\ell}$ and $\phi_{b}$ for the
background-subtracted candidates. The data are described well by the product
of the angular distributions obtained from the moment analysis and the
efficiency model.

Figure 3 compares the measured observables with their corresponding SM
predictions, obtained from the EOS software [38] using the values of the
${\mathchar 28931\relax}^{0}_{b}$ production polarisation measured in Ref.
[34]. The values of the observables $K_{11}$ to $K_{34}$ are consistent with
zero. This is expected from measurements of the angular distribution of the
decay ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mathchar 28931\relax}$ by CMS [36] and LHCb [34], which indicate that
the production polarisation of ${\mathchar 28931\relax}^{0}_{b}$ baryons is
small in $pp$ collisions at 7 and 8$\mathrm{\,Te\kern-1.00006ptV}$. The
measurements are consistent with the SM predictions for $K_{1}$ to $K_{10}$.
The largest discrepancy is seen in $K_{6}$, which is 2.6 standard deviations
from the SM prediction. The angular observables result in an angular
distribution that is not positive for all values of the angles. To obtain a
physical angular distribution, $K_{6}$ has to move closer to its SM value. The
measured $K_{i}$ values are also consistent with the values predicted by new
physics scenarios favoured by global fits to data from $b$ to $s$ quark
transitions [3, 4, 5, 6, 7]. These new physics scenarios result in only a
small change of $K_{1}$ to $K_{10}$ in the low-recoil region.

The $K_{i}$ observables can be combined to determine the angular asymmetries

$\begin{split}A^{\ell}_{\mathrm{FB}}=\tfrac{3}{2}K_{3}&=-0.39\pm 0.04\,\pm
0.01~{},\\\ A^{h}_{\mathrm{FB}}=K_{4}+\tfrac{1}{2}K_{5}&=-0.30\pm 0.05\,\pm
0.02~{},\\\ A^{\ell h}_{\mathrm{FB}}=\tfrac{3}{4}K_{6}&=+0.25\pm 0.04\,\pm
0.01~{},\end{split}$

where the first uncertainties are statistical and the second are the
systematic uncertainties that are discussed in the following section. The
forward-backward asymmetries $A^{\ell}_{\mathrm{FB}}$ and
$A^{h}_{\mathrm{FB}}$ are in good agreement with the SM predictions. The
asymmetry $A^{\ell h}_{\mathrm{FB}}$, which is proportional to $K_{6}$, is 2.6
standard deviations from its SM prediction. The value of $A^{h}_{\mathrm{FB}}$
is consistent with that measured in Ref. [10]. The value of
$A^{\ell}_{\mathrm{FB}}$ is not comparable due to an inconsistency in the
definition of $\theta_{\ell}$ in that reference.444 Under the definition of
$\theta_{\ell}$ used in Ref. [10], $A^{\ell}_{\mathrm{FB}}$ measured the
asymmetry difference between ${\mathchar 28931\relax}^{0}_{b}$ and ${\kern
1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}}{}^{0}_{b}$ decays
rather than the average of the asymmetries.

Table 1:  Angular observables combining the results of the moments obtained from Run 1 and Run 2 data, as well as candidates reconstructed in the long- and downstream-track ${p}{{\pi}^{-}}$ categories. The first and second uncertainties are statistical and systematic, respectively.  Obs. | Value | Obs. | Value
---|---|---|---
$K_{1}$ | $\phantom{+}0.346\pm 0.020\pm 0.004$ | $K_{18}$ | $-0.108\pm 0.058\pm 0.008$
$K_{2}$ | $\phantom{+}0.308\pm 0.040\pm 0.008$ | $K_{19}$ | $-0.151\pm 0.122\pm 0.022$
$K_{3}$ | $-0.261\pm 0.029\pm 0.006$ | $K_{20}$ | $-0.116\pm 0.056\pm 0.008$
$K_{4}$ | $-0.176\pm 0.046\pm 0.016$ | $K_{21}$ | $-0.041\pm 0.105\pm 0.020$
$K_{5}$ | $-0.251\pm 0.081\pm 0.016$ | $K_{22}$ | $-0.014\pm 0.045\pm 0.007$
$K_{6}$ | $\phantom{+}0.329\pm 0.055\pm 0.012$ | $K_{23}$ | $-0.024\pm 0.077\pm 0.012$
$K_{7}$ | $-0.015\pm 0.084\pm 0.013$ | $K_{24}$ | $\phantom{+}0.005\pm 0.033\pm 0.005$
$K_{8}$ | $-0.099\pm 0.037\pm 0.012$ | $K_{25}$ | $-0.226\pm 0.176\pm 0.030$
$K_{9}$ | $\phantom{+}0.005\pm 0.084\pm 0.012$ | $K_{26}$ | $\phantom{+}0.140\pm 0.074\pm 0.014$
$K_{10}$ | $-0.045\pm 0.037\pm 0.006$ | $K_{27}$ | $\phantom{+}0.016\pm 0.140\pm 0.025$
$K_{11}$ | $-0.007\pm 0.043\pm 0.009$ | $K_{28}$ | $\phantom{+}0.032\pm 0.058\pm 0.009$
$K_{12}$ | $-0.009\pm 0.063\pm 0.014$ | $K_{29}$ | $-0.127\pm 0.097\pm 0.016$
$K_{13}$ | $\phantom{+}0.024\pm 0.045\pm 0.010$ | $K_{30}$ | $\phantom{+}0.011\pm 0.061\pm 0.011$
$K_{14}$ | $\phantom{+}0.010\pm 0.082\pm 0.013$ | $K_{31}$ | $\phantom{+}0.180\pm 0.094\pm 0.015$
$K_{15}$ | $\phantom{+}0.158\pm 0.117\pm 0.027$ | $K_{32}$ | $-0.009\pm 0.055\pm 0.008$
$K_{16}$ | $\phantom{+}0.050\pm 0.084\pm 0.023$ | $K_{33}$ | $\phantom{+}0.022\pm 0.060\pm 0.009$
$K_{17}$ | $-0.000\pm 0.120\pm 0.022$ | $K_{34}$ | $\phantom{+}0.060\pm 0.058\pm 0.009$

Figure 2:  One-dimensional projections of the angular distributions of the
candidates (black points), combining Run 1 and Run 2 data, as well as
candidates reconstructed in the long- and downstream-track $p{{\pi}^{-}}$
categories. The background is subtracted from the data but no efficiency
correction is applied. The projection of each angular distribution obtained
from the moment analysis multiplied by the efficiency distribution is
superimposed. The large variation in $\phi_{\ell}$ is primarily due to the
angular acceptance. Figure 3: Angular observables combining the results for
the moments obtained from Run 1 and Run 2 data, as well as candidates
reconstructed in the long- and downstream-track ${p}{{\pi}^{-}}$ categories.
The blue line represents the SM predictions obtained using the EOS software.
The thickness of the light-blue band represents the uncertainty on the SM
predictions.

##  8 Systematic uncertainties

The angular observables may be sensitive to systematic effects arising from
imperfect modelling of either the angular efficiency or the
$m(p{{\pi}^{-}}{\mu^{+}\mu^{-}})$ distribution. Where possible, systematic
uncertainties are estimated using pseudoexperiments. These are generated from
a systematically varied model and the observables are then estimated using the
nominal analysis, neglecting the variation in the generation. The sources of
systematic uncertainty considered are listed in Table 2. In general,
systematic uncertainties are found to be small compared to the statistical
uncertainties on the measurements.

The largest systematic uncertainties in modelling the angular efficiency are
from the size of the simulated data samples and the order of the Legendre
polynomials used to parameterise the efficiency. The former is determined by
bootstrapping the simulated sample and re-evaluating the model. The latter is
estimated by increasing the order of polynomials used in the efficiency
parameterisation by up to two orders. By default, the efficiency model is
chosen to have the minimum number of terms needed to get a good description of
both the simulated and the data control samples. Increasing further the number
of terms results in an overfitting of statistical fluctuations in the
simulated data used to determine the efficiency model (due to the limited size
of the simulated data set).

A systematic uncertainty due to the modelling of the data by the simulation is
estimated by varying the tracking and muon identification efficiencies, and by
applying an additional correction to the $p_{\mathrm{T}}$ and $\eta$ spectra
of the ${\mathchar 28931\relax}^{0}_{b}$ baryons.

The impact of neglecting angular resolution when determining the angular
observables is estimated by smearing pseudoexperiments according to the
resolution determined using simulated data. The angular resolution is poorest
for $\theta$, $\theta_{b}$ and $\phi_{b}$ in the downstream $p{{\pi}^{-}}$
category, at around 90$\mathrm{\,mrad}$ for $\theta$ and $\theta_{b}$ and
150$\mathrm{\,mrad}$ for $\phi_{b}$.

In the calculation of the angular basis, the crossing angle of the LHC beams
is neglected. The impact of this is estimated by generating pseudoexperiments
with the correct crossing angle and neglecting this when the angular
observables are determined.

The systematic uncertainty due to modelling the shape of the signal mass
distribution is small. The main contribution to this uncertainty comes from
the modelling of the tails of the signal mass distribution. The factorisation
of the mass model and the angular distribution, which is a requirement of the
sPlot technique, is also tested and results in a negligible systematic
uncertainty.

Table 2: Sources of systematic uncertainty on the $K_{i}$ angular observables, together with the mean and the range of uncertainty values assigned to the 34 $K_{i}$ parameters in each case. The variation of each source of systematic uncertainty between the different observables depends on the structure of the weighting functions used to extract the observable and its correlation with the angular efficiency.  Source | Uncertainty [$10^{-3}$]
---|---
| Range among $K_{i}$ | Mean
Simulated sample size | 3–22 | 9
Efficiency parameterisation | 1–13 | 4
Data-simulation differences | 2–16 | 6
Angular resolution | 1–11 | 4
Beam crossing angle | 1–8 | 4
Signal mass model | 1–4 | 2

##  9 Summary

An analysis of the angular distribution of the decay ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar 28931\relax}{\mu^{+}\mu^{-}}$ in
the dimuon invariant mass squared range
$15<{q^{2}}<20\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$ is reported. Using data
collected with the LHCb detector between 2011 and 2016, the full basis of
angular observables is measured for the first time. From the measured
observables, the lepton-side, hadron-side and combined forward-backward
asymmetries of the decay are determined to be

$\begin{split}A^{\ell}_{\mathrm{FB}}&=-0.39\pm 0.04\mathrm{\,(stat)}\,\pm
0.01\mathrm{\,(syst)}~{},\\\ A^{h}_{\mathrm{FB}}&=-0.30\pm
0.05\mathrm{\,(stat)}\,\pm 0.02\mathrm{\,(syst)}~{},\\\ A^{\ell
h}_{\mathrm{FB}}&=+0.25\pm 0.04\mathrm{\,(stat)}\,\pm
0.01\mathrm{\,(syst)}~{}.\end{split}$

The results presented here supersede the results for angular observables in
Ref. [10] (see discussion in Sec. 7). The measured angular observables are
compatible with the SM predictions obtained using the EOS software [38], where
the ${\mathchar 28931\relax}^{0}_{b}$ production polarisation is set to the
value obtained by the LHCb collaboration in $pp$ collisions at a centre-of-
mass energy of $7$ TeV [34].

## 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); MinES
and FASO (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine);
STFC (United Kingdom); NSF (USA). We acknowledge the computing resources that
are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy),
SURF (Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and Yandex LLC
(Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-GRID
(Poland) and OSC (USA). We are indebted to the communities behind the multiple
open-source software packages on which we depend. Individual groups or members
have received support from AvH Foundation (Germany); EPLANET, Marie
Skłodowska-Curie Actions and ERC (European Union); ANR, Labex P2IO and OCEVU,
and Région Auvergne-Rhône-Alpes (France); Key Research Program of Frontier
Sciences of CAS, CAS PIFI, and the Thousand Talents Program (China); RFBR, RSF
and Yandex LLC (Russia); GVA, XuntaGal and GENCAT (Spain); the Royal Society
and the Leverhulme Trust (United Kingdom); Laboratory Directed Research and
Development program of LANL (USA).

Appendices

##  Appendix A Angular basis

The angular distribution of the decay ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar 28931\relax}{\mu^{+}\mu^{-}}$ is
described by five angles, $\theta$, $\theta_{\ell}$, $\phi_{\ell}$,
$\theta_{b}$ and $\phi_{b}$ defined with respect to the normal-vector

$\displaystyle\hat{n}=\frac{\vec{p}_{\rm beam}^{\,\\{{\rm
lab}\\}}\times\vec{p}_{{{\mathchar 28931\relax}^{0}_{b}}}^{\,\\{{\rm
lab}\\}}}{|\vec{p}_{\rm beam}^{\,\\{{\rm lab}\\}}\times\vec{p}_{{{\mathchar
28931\relax}^{0}_{b}}}^{\,\\{{\rm lab}\\}}|}~{},$ (5)

where $\hat{p}=\vec{p}/|\vec{p}|$ and the parentheses refer to the rest frame
the momentum is measured in. The angle $\theta$ is defined by the angle
between $\hat{n}$ and the $\mathchar 28931\relax$ baryon momentum in the
${\mathchar 28931\relax}^{0}_{b}$ baryon rest frame, i.e.

$\displaystyle\cos\theta=\hat{n}\cdot\hat{p}_{{\mathchar
28931\relax}}^{\\{{{\mathchar 28931\relax}^{0}_{b}}\\}}~{}.$ (6)

The decay of the $\mathchar 28931\relax$ baryon and the dimuon system can be
described by coordinate systems with $\hat{z}_{b}=\hat{p}_{{\mathchar
28931\relax}}^{\\{{{\mathchar 28931\relax}^{0}_{b}}\\}}$ and
$\hat{z}_{\ell}=\hat{p}_{{\mu^{+}\mu^{-}}}^{\\{{{\mathchar
28931\relax}^{0}_{b}}\\}}$, $\hat{y}_{b,\ell}=\hat{n}\times\hat{z}_{b,\ell}$
and $\hat{x}_{b,\ell}=\hat{z}_{b,\ell}\times\hat{y}_{b,\ell}$. The angles
$\theta_{b}$ and $\phi_{b}$ ($\theta_{\ell}$ and $\phi_{\ell}$) are the polar
and azimuthal angle of the proton ($\mu^{+}$) in the $\mathchar 28931\relax$
baryon (dimuon) rest frame. The angles are defined by

$\displaystyle\begin{split}\cos\theta_{b,\ell}&=\hat{z}_{b,\ell}\cdot\hat{p}_{b,\ell}~{},~{}\cos\phi_{b,\ell}=\hat{x}_{b,\ell}\cdot\hat{p}_{\perp
b,\ell}~{},~{}\sin\phi_{b,\ell}=\hat{y}_{b,\ell}\cdot\hat{p}_{\perp
b,\ell}~{},\end{split}$ (7)

where $\hat{p}_{b}$ $(\hat{p}_{\ell})$ is the direction of the proton
($\mu^{+}$) and $\hat{p}_{\perp b}$ $(\hat{p}_{\perp\ell})$ is a unit vector
corresponding to the component perpendicular to the $\hat{z}_{b}$
$(\hat{z}_{\ell})$ axis. For the ${\kern
1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}}{}^{0}_{b}$ decay,
the angular variables are transformed such that
$\theta_{\ell}\rightarrow\pi-\theta_{\ell}$,
$\phi_{\ell}\rightarrow\pi-\phi_{\ell}$ and $\phi_{b}\rightarrow-\phi_{b}$.
This ensures that, in the absence of $C\\!P$ violating effects, the $K_{i}$
observables are the same for ${\mathchar 28931\relax}^{0}_{b}$ and ${\kern
1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}}{}^{0}_{b}$ decays.

##  Appendix B Angular distribution and weighting functions

The full form of the angular distribution of the decay ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar 28931\relax}{\mu^{+}\mu^{-}}$ is
given by

$\displaystyle\frac{\mathrm{d}^{5}\Gamma}{\mathrm{d}\vec{\Omega}}=\frac{3}{32\pi^{2}}\Big{(}$
$\displaystyle\left(K_{1}\sin^{2}\theta_{\ell}+K_{2}\cos^{2}\theta_{\ell}+K_{3}\cos\theta_{\ell}\right)+$
(8)
$\displaystyle\left(K_{4}\sin^{2}\theta_{\ell}+K_{5}\cos^{2}\theta_{\ell}+K_{6}\cos\theta_{\ell}\right)\cos\theta_{b}+$
$\displaystyle\left(K_{7}\sin\theta_{\ell}\cos\theta_{\ell}+K_{8}\sin\theta_{\ell}\right)\sin\theta_{b}\cos\left(\phi_{b}+\phi_{\ell}\right)+$
$\displaystyle\left(K_{9}\sin\theta_{\ell}\cos\theta_{\ell}+K_{10}\sin\theta_{\ell}\right)\sin\theta_{b}\sin\left(\phi_{b}+\phi_{\ell}\right)+$
$\displaystyle\left(K_{11}\sin^{2}\theta_{\ell}+K_{12}\cos^{2}\theta_{\ell}+K_{13}\cos\theta_{\ell}\right)\cos\theta+$
$\displaystyle\left(K_{14}\sin^{2}\theta_{\ell}+K_{15}\cos^{2}\theta_{\ell}+K_{16}\cos\theta_{\ell}\right)\cos\theta_{b}\cos\theta+$
$\displaystyle\left(K_{17}\sin\theta_{\ell}\cos\theta_{\ell}+K_{18}\sin\theta_{\ell}\right)\sin\theta_{b}\cos\left(\phi_{b}+\phi_{\ell}\right)\cos\theta+$
$\displaystyle\left(K_{19}\sin\theta_{\ell}\cos\theta_{\ell}+K_{20}\sin\theta_{\ell}\right)\sin\theta_{b}\sin\left(\phi_{b}+\phi_{\ell}\right)\cos\theta+$
$\displaystyle\left(K_{21}\cos\theta_{\ell}\sin\theta_{\ell}+K_{22}\sin\theta_{\ell}\right)\sin\phi_{\ell}\sin\theta+$
$\displaystyle\left(K_{23}\cos\theta_{\ell}\sin\theta_{\ell}+K_{24}\sin\theta_{\ell}\right)\cos\phi_{\ell}\sin\theta+$
$\displaystyle\left(K_{25}\cos\theta_{\ell}\sin\theta_{\ell}+K_{26}\sin\theta_{\ell}\right)\sin\phi_{\ell}\cos\theta_{b}\sin\theta+$
$\displaystyle\left(K_{27}\cos\theta_{\ell}\sin\theta_{\ell}+K_{28}\sin\theta_{\ell}\right)\cos\phi_{\ell}\cos\theta_{b}\sin\theta+$
$\displaystyle\left(K_{29}\cos^{2}\theta_{\ell}+K_{30}\sin^{2}\theta_{\ell}\right)\sin\theta_{b}\sin\phi_{b}\sin\theta+$
$\displaystyle\left(K_{31}\cos^{2}\theta_{\ell}+K_{32}\sin^{2}\theta_{\ell}\right)\sin\theta_{b}\cos\phi_{b}\sin\theta+$
$\displaystyle\left(K_{33}\sin^{2}\theta_{\ell}\right)\sin\theta_{b}\cos\left(2\phi_{\ell}+\phi_{b}\right)\sin\theta+$
$\displaystyle\left(K_{34}\sin^{2}\theta_{\ell}\right)\sin\theta_{b}\sin\left(2\phi_{\ell}+\phi_{b}\right)\sin\theta\Big{)}~{}.$

The individual $K_{i}$ parameters can be determined using a moment analysis of
the angular distribution with the weighting functions

$\displaystyle g_{1}(\vec{\Omega})=$
$\displaystyle\tfrac{1}{4}(3-5\cos^{2}\theta_{\ell})~{},$ $\displaystyle
g_{6}(\vec{\Omega})~{}=3\cos\theta_{\ell}\cos\theta_{b}~{},$ (9)
$\displaystyle g_{2}(\vec{\Omega})=$
$\displaystyle\tfrac{1}{2}(5\cos^{2}\theta_{\ell}-1)~{},$ $\displaystyle
g_{7}(\vec{\Omega})~{}=\tfrac{15}{2}\cos\theta_{\ell}\sin\theta_{\ell}\sin\theta_{b}\cos(\phi_{\ell}+\phi_{b})~{},$
$\displaystyle g_{3}(\vec{\Omega})=$ $\displaystyle\cos\theta_{\ell}~{},$
$\displaystyle
g_{8}(\vec{\Omega})~{}=\tfrac{3}{2}\sin\theta_{\ell}\sin\theta_{b}\cos(\phi_{\ell}+\phi_{b})~{},$
$\displaystyle g_{4}(\vec{\Omega})=$
$\displaystyle\tfrac{3}{4}(3-5\cos^{2}\theta_{\ell})\cos\theta_{b}~{},$
$\displaystyle
g_{9}(\vec{\Omega})~{}=\tfrac{15}{2}\cos\theta_{\ell}\sin\theta_{\ell}\sin\theta_{b}\sin(\phi_{\ell}+\phi_{b})~{},$
$\displaystyle g_{5}(\vec{\Omega})=$
$\displaystyle\tfrac{3}{2}(5\cos^{2}\theta_{\ell}-1)\cos\theta_{b}~{},$
$\displaystyle
g_{10}(\vec{\Omega})=\tfrac{3}{2}\sin\theta_{\ell}\sin\theta_{b}\sin(\phi_{\ell}+\phi_{b})~{},$

from Ref. [15] and the weighting functions

$\displaystyle g_{11}(\vec{\Omega})=$
$\displaystyle\tfrac{3}{4}(3-5\cos^{2}\theta_{\ell})\cos\theta~{},$
$\displaystyle
g_{23}(\vec{\Omega})=\tfrac{15}{2}\cos\theta_{\ell}\sin\theta_{\ell}\sin\theta\cos\phi_{\ell}~{},$
(10) $\displaystyle g_{12}(\vec{\Omega})=$
$\displaystyle\tfrac{3}{2}(5\cos^{2}\theta_{\ell}-1)\cos\theta~{},$
$\displaystyle
g_{24}(\vec{\Omega})=\tfrac{3}{2}\sin\theta\sin\theta_{\ell}\cos\phi_{\ell}~{},$
$\displaystyle g_{13}(\vec{\Omega})=$ $\displaystyle
3\cos\theta_{\ell}\cos\theta~{},$ $\displaystyle
g_{25}(\vec{\Omega})=\tfrac{45}{2}\cos\theta_{\ell}\sin\theta_{\ell}\cos\theta_{b}\sin\theta\sin\phi_{\ell}~{},$
$\displaystyle g_{14}(\vec{\Omega})=$
$\displaystyle\tfrac{9}{4}(3-5\cos^{2}\theta_{\ell})\cos\theta_{b}\cos\theta~{},$
$\displaystyle
g_{26}(\vec{\Omega})=\tfrac{9}{2}\sin\theta\sin\theta_{\ell}\cos\theta_{b}\sin\phi_{\ell}~{},$
$\displaystyle g_{15}(\vec{\Omega})=$
$\displaystyle\tfrac{9}{2}(5\cos^{2}\theta_{\ell}-1)\cos\theta_{b}\cos\theta~{},$
$\displaystyle
g_{27}(\vec{\Omega})=\tfrac{45}{2}\cos\theta_{\ell}\sin\theta_{\ell}\cos\theta_{b}\sin\theta\cos\phi_{\ell}~{},$
$\displaystyle g_{16}(\vec{\Omega})=$ $\displaystyle
9\cos\theta\cos\theta_{\ell}\cos\theta_{b}~{},$ $\displaystyle
g_{28}(\vec{\Omega})=\tfrac{9}{2}\sin\theta\sin\theta_{\ell}\cos\theta_{b}\cos\phi_{\ell}~{},$
$\displaystyle g_{17}(\vec{\Omega})=$
$\displaystyle\tfrac{45}{2}\cos\theta_{\ell}\sin\theta_{\ell}\sin\theta_{b}\cos\theta\cos(\phi_{\ell}+\phi_{b})~{},$
$\displaystyle
g_{29}(\vec{\Omega})=\tfrac{9}{4}(5\cos^{2}\theta_{\ell}-1)\sin\theta_{b}\sin\theta\sin\phi_{b}~{},$
$\displaystyle g_{18}(\vec{\Omega})=$
$\displaystyle\tfrac{9}{2}\sin\theta_{\ell}\sin\theta_{b}\cos\theta\cos(\phi_{\ell}+\phi_{b})~{},$
$\displaystyle
g_{30}(\vec{\Omega})=\tfrac{9}{8}(3-5\cos^{2}\theta_{\ell})\sin\theta_{b}\sin\theta\sin\phi_{b}~{},$
$\displaystyle g_{19}(\vec{\Omega})=$
$\displaystyle\tfrac{45}{2}\cos\theta_{\ell}\sin\theta_{\ell}\sin\theta_{b}\cos\theta\sin(\phi_{\ell}+\phi_{b})~{},$
$\displaystyle
g_{31}(\vec{\Omega})=\tfrac{9}{4}(5\cos^{2}\theta_{\ell}-1)\sin\theta_{b}\sin\theta\cos\phi_{b}~{},$
$\displaystyle g_{20}(\vec{\Omega})=$
$\displaystyle\tfrac{9}{2}\sin\theta_{\ell}\sin\theta_{b}\cos\theta\sin(\phi_{\ell}+\phi_{b})~{},$
$\displaystyle
g_{32}(\vec{\Omega})=\tfrac{9}{8}(3-5\cos^{2}\theta_{\ell})\sin\theta_{b}\sin\theta\cos\phi_{b}~{},$
$\displaystyle g_{21}(\vec{\Omega})=$
$\displaystyle\tfrac{15}{2}\cos\theta_{\ell}\sin\theta_{\ell}\sin\theta\sin\phi_{\ell}~{},$
$\displaystyle
g_{33}(\vec{\Omega})=\tfrac{9}{4}\sin\theta_{b}\sin\theta\cos(2\phi_{\ell}+\phi_{b})~{},$
$\displaystyle g_{22}(\vec{\Omega})=$
$\displaystyle\tfrac{3}{2}\sin\theta\sin\theta_{\ell}\sin\phi_{\ell}~{},$
$\displaystyle
g_{34}(\vec{\Omega})=\tfrac{9}{4}\sin\theta_{b}\sin\theta\sin(2\phi_{\ell}+\phi_{b})~{},$

from Ref. [11].

##  Appendix C Results separated by data-taking period

Tables 3 and 4 show the values of the observables for each of the two data-
taking periods. Table 3 shows the values of the observables combining the 2011
data, collected at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, and the 2012
data, collected at $\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$. Table 4 shows
the values of the observables in the Run 2 data, collected at
$\sqrt{s}=13\mathrm{\,Te\kern-1.00006ptV}$.

Table 3: Measured values for the angular observables from the Run 1 data combining the results of the moments obtained from the candidates reconstructed in the long- and downstream-track ${p}{{\pi}^{-}}$ categories. The first and second uncertainties are statistical and systematic, respectively.  Obs. | Value | Obs. | Value
---|---|---|---
$K_{1}$ | $\phantom{+}0.376\pm 0.029\pm 0.006$ | $K_{18}$ | $-0.081\pm 0.081\pm 0.015$
$K_{2}$ | $\phantom{+}0.248\pm 0.057\pm 0.012$ | $K_{19}$ | $-0.023\pm 0.165\pm 0.031$
$K_{3}$ | $-0.241\pm 0.041\pm 0.008$ | $K_{20}$ | $-0.156\pm 0.078\pm 0.019$
$K_{4}$ | $-0.212\pm 0.070\pm 0.013$ | $K_{21}$ | $-0.050\pm 0.150\pm 0.032$
$K_{5}$ | $-0.123\pm 0.117\pm 0.020$ | $K_{22}$ | $\phantom{+}0.032\pm 0.064\pm 0.014$
$K_{6}$ | $\phantom{+}0.247\pm 0.079\pm 0.017$ | $K_{23}$ | $\phantom{+}0.038\pm 0.104\pm 0.018$
$K_{7}$ | $-0.027\pm 0.124\pm 0.022$ | $K_{24}$ | $\phantom{+}0.004\pm 0.047\pm 0.008$
$K_{8}$ | $-0.081\pm 0.054\pm 0.010$ | $K_{25}$ | $-0.107\pm 0.254\pm 0.046$
$K_{9}$ | $-0.123\pm 0.115\pm 0.018$ | $K_{26}$ | $\phantom{+}0.130\pm 0.106\pm 0.024$
$K_{10}$ | $\phantom{+}0.021\pm 0.051\pm 0.009$ | $K_{27}$ | $-0.200\pm 0.190\pm 0.035$
$K_{11}$ | $-0.030\pm 0.062\pm 0.014$ | $K_{28}$ | $\phantom{+}0.058\pm 0.084\pm 0.015$
$K_{12}$ | $-0.114\pm 0.092\pm 0.022$ | $K_{29}$ | $-0.172\pm 0.142\pm 0.027$
$K_{13}$ | $\phantom{+}0.059\pm 0.064\pm 0.016$ | $K_{30}$ | $-0.060\pm 0.088\pm 0.014$
$K_{14}$ | $\phantom{+}0.122\pm 0.126\pm 0.026$ | $K_{31}$ | $\phantom{+}0.252\pm 0.126\pm 0.022$
$K_{15}$ | $\phantom{+}0.247\pm 0.171\pm 0.042$ | $K_{32}$ | $-0.074\pm 0.075\pm 0.011$
$K_{16}$ | $-0.193\pm 0.116\pm 0.029$ | $K_{33}$ | $-0.010\pm 0.081\pm 0.014$
$K_{17}$ | $-0.119\pm 0.178\pm 0.033$ | $K_{34}$ | $\phantom{+}0.140\pm 0.088\pm 0.012$
Table 4: Measured values for the angular observables from the Run 2 data combining the results of the moments obtained from the candidates reconstructed in the long- and downstream-track ${p}{{\pi}^{-}}$ categories. The first and second uncertainties are statistical and systematic, respectively.  Obs. | Value | Obs. | Value
---|---|---|---
$K_{1}$ | $\phantom{+}0.318\pm 0.028\pm 0.007$ | $K_{18}$ | $-0.134\pm 0.081\pm 0.014$
$K_{2}$ | $\phantom{+}0.364\pm 0.056\pm 0.013$ | $K_{19}$ | $-0.273\pm 0.178\pm 0.040$
$K_{3}$ | $-0.279\pm 0.042\pm 0.010$ | $K_{20}$ | $-0.078\pm 0.082\pm 0.017$
$K_{4}$ | $-0.143\pm 0.063\pm 0.012$ | $K_{21}$ | $-0.033\pm 0.142\pm 0.023$
$K_{5}$ | $-0.372\pm 0.113\pm 0.024$ | $K_{22}$ | $-0.058\pm 0.062\pm 0.008$
$K_{6}$ | $\phantom{+}0.407\pm 0.076\pm 0.017$ | $K_{23}$ | $-0.082\pm 0.111\pm 0.018$
$K_{7}$ | $-0.004\pm 0.114\pm 0.018$ | $K_{24}$ | $\phantom{+}0.005\pm 0.046\pm 0.008$
$K_{8}$ | $-0.116\pm 0.051\pm 0.011$ | $K_{25}$ | $-0.339\pm 0.243\pm 0.042$
$K_{9}$ | $\phantom{+}0.126\pm 0.124\pm 0.017$ | $K_{26}$ | $\phantom{+}0.150\pm 0.101\pm 0.017$
$K_{10}$ | $-0.108\pm 0.054\pm 0.008$ | $K_{27}$ | $\phantom{+}0.221\pm 0.203\pm 0.036$
$K_{11}$ | $\phantom{+}0.014\pm 0.060\pm 0.009$ | $K_{28}$ | $\phantom{+}0.008\pm 0.083\pm 0.015$
$K_{12}$ | $\phantom{+}0.091\pm 0.085\pm 0.023$ | $K_{29}$ | $-0.085\pm 0.135\pm 0.025$
$K_{13}$ | $-0.009\pm 0.063\pm 0.016$ | $K_{30}$ | $\phantom{+}0.079\pm 0.084\pm 0.014$
$K_{14}$ | $-0.096\pm 0.105\pm 0.019$ | $K_{31}$ | $\phantom{+}0.113\pm 0.140\pm 0.021$
$K_{15}$ | $\phantom{+}0.073\pm 0.159\pm 0.046$ | $K_{32}$ | $\phantom{+}0.053\pm 0.080\pm 0.012$
$K_{16}$ | $\phantom{+}0.280\pm 0.120\pm 0.034$ | $K_{33}$ | $\phantom{+}0.052\pm 0.088\pm 0.011$
$K_{17}$ | $\phantom{+}0.112\pm 0.166\pm 0.033$ | $K_{34}$ | $-0.015\pm 0.079\pm 0.012$

##  Appendix D Correlation matrices

Figures 4 and 5 shows the statistical correlation between the angular
observables determined using bootstrapped samples. The correlation
coefficients are typically small but can be as large as 30–40% between pairs
of observables. The observables $K_{1}$ and $K_{2}$ are fully anticorrelated
due to the normalisation of the observables, which requires $2K_{1}+K_{2}=1$.
The correlation matrices in numerical form are attached as supplementary
material to this article.

Figure 4: Correlation coefficients between observables for the combined Run 1
and Run 2 data. The correlations are estimated using bootstrapped samples.

Figure 5: Correlation coefficients between the observables (top) in the Run 1
data and (bottom) in the Run 2 data. The correlations are estimated using
bootstrapped samples.

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Trisovic49, A. Tsaregorodtsev6, G. Tuci24, A. Tully49, N. Tuning27,42, A.
Ukleja31, A. Usachov7, A. Ustyuzhanin37, U. Uwer12, A. Vagner71, V. Vagnoni15,
A. Valassi42, S. Valat42, G. Valenti15, R. Vazquez Gomez42, P. Vazquez
Regueiro41, S. Vecchi16, M. van Veghel27, J.J. Velthuis48, M. Veltri17,r, G.
Veneziano57, A. Venkateswaran61, T.A. Verlage9, M. Vernet5, M. Veronesi27,
N.V. Veronika13, M. Vesterinen57, J.V. Viana Barbosa42, D. Vieira63, M.
Vieites Diaz41, H. Viemann67, X. Vilasis-Cardona40,m, A. Vitkovskiy27, M.
Vitti49, V. Volkov35, A. Vollhardt44, B. Voneki42, A. Vorobyev33, V.
Vorobyev38,w, J.A. de Vries27, C. Vázquez Sierra27, R. Waldi67, J. Walsh24, J.
Wang61, M. Wang3, Y. Wang65, Z. Wang44, D.R. Ward49, H.M. Wark54, N.K.
Watson47, D. Websdale55, A. Weiden44, C. Weisser58, M. Whitehead9, J. Wicht50,
G. Wilkinson57, M. Wilkinson61, I. Williams49, M.R.J. Williams56, M.
Williams58, T. Williams47, F.F. Wilson51,42, J. Wimberley60, M. Winn7, J.
Wishahi10, W. Wislicki31, M. Witek29, G. Wormser7, S.A. Wotton49, K. Wyllie42,
D. Xiao65, Y. Xie65, A. Xu3, M. Xu65, Q. Xu63, Z. Xu3, Z. Xu4, Z. Yang3, Z.
Yang60, Y. Yao61, L.E. Yeomans54, H. Yin65, J. Yu65,ab, X. Yuan61, O.
Yushchenko39, K.A. Zarebski47, M. Zavertyaev11,c, D. Zhang65, L. Zhang3, W.C.
Zhang3,aa, Y. Zhang7, A. Zhelezov12, Y. Zheng63, X. Zhu3, V. Zhukov9,35, J.B.
Zonneveld52, S. Zucchelli15.

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

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