]}

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2021-170 LHCb-PAPER-2021-024

Measurement of the $W$ boson mass

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

The $W$ boson mass is measured using proton-proton collision data at
$\sqrt{s}=13\text{\,Te\kern-1.00006ptV}$ corresponding to an integrated
luminosity of 1.7$\text{\,fb}^{-1}$ recorded during 2016 by the LHCb
experiment. With a simultaneous fit of the muon $q/p_{\mathrm{T}}$
distribution of a sample of $W\rightarrow{\mu}{\nu}$ decays and the $\phi^{*}$
distribution of a sample of $Z\rightarrow{\mu}{\mu}$ decays the $W$ boson mass
is determined to be

$m_{W}=\text{80354}\pm 23_{\rm stat}\pm 10_{\rm exp}\pm 17_{\rm theory}\pm
9_{\rm PDF}~{}\mathrm{MeV},$

where uncertainties correspond to contributions from statistical, experimental
systematic, theoretical and parton distribution function sources. This is an
average of results based on three recent global parton distribution function
sets. The measurement agrees well with the prediction of the global
electroweak fit and with previous measurements.

Published in JHEP 01 (2022) 036.

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

##  1 Introduction

The $W$ boson mass ($m_{W}$) is directly related to electroweak (EW) symmetry
breaking in the Standard Model (SM) [1, 2, 3]. At tree level, $m_{W}=gv/2$
where $g$ is the weak-isospin coupling and $v$ is the vacuum expectation value
of the Higgs field. Going beyond tree level the boson masses and couplings
receive loop corrections. The value of $m_{W}$ is related to the precisely
measured fine-structure constant ($\alpha$), the mass of the $Z$ boson
($m_{Z}$) and the Fermi constant ($G_{F}$), as [4, 5]

$m_{W}^{2}\left(1-\frac{m_{W}^{2}}{m_{Z}^{2}}\right)=\frac{\pi\alpha}{\sqrt{2}G_{F}}(1+\Delta),$
(1)

where $\Delta$ encapsulates the loop-level corrections. A global fit of EW
observables, excluding direct measurements of $m_{W}$, yields a prediction of
$m_{W}=80354\pm 7\text{\,Me\kern-1.00006ptV}$ [6].111Throughout this paper
natural units with $c=1$ are used. This can be compared with direct
measurements to test for possible beyond SM contributions to $\Delta$ in Eq.
1. The 2020 PDG average of direct measurements is $m_{W}=80379\pm
12\text{\,Me\kern-1.00006ptV}$ [7]. The sensitivity of the global EW fit to
physics beyond the SM is primarily limited by the precision of the direct
measurements of $m_{W}$ [6]. Furthermore, the uncertainty in the prediction is
expected to reduce as the top-quark mass, which is the leading source of the
parametric uncertainty, is determined more precisely in the future.

The value of $m_{W}$ was measured to a precision of 33 MeV at the Large
Electron-Positron (LEP) collider [8] at CERN and to a precision of 16 MeV in
an average [9] of measurements by the CDF [10] and D0 [11] experiments at the
Fermilab Tevatron collider. The first measurement at the LHC was performed by
the ATLAS collaboration and has an uncertainty of 19 MeV [12]. The hadron
collider measurements are based on three observables in leptonic $W$ boson
decays, namely the transverse mass, missing transverse momentum and charged
lepton transverse momentum ($p_{\mathrm{T}}$). At hadron colliders, the lepton
$p_{\mathrm{T}}$ is measured with good resolution but it is strongly
influenced by the $W$ boson transverse momentum distribution, the modelling of
which is a potential source of a limiting systematic uncertainty. However, the
resolution of the transverse mass is degraded by the pile-up of proton-proton
interactions in the same bunch crossing. Therefore, the lepton
$p_{\mathrm{T}}$ was the most sensitive observable in the recent measurement
performed by the ATLAS collaboration. Despite being based on a small subset of
the data recorded to date, the ATLAS measurement of $m_{W}$ is already limited
by uncertainties in modelling $W$ boson production, in particular the parton
distribution functions (PDFs) of the proton.

The potential for a measurement based on the muon $p_{\mathrm{T}}$ with the
LHCb experiment is studied in Ref. [13] It was estimated that LHCb data
collected in LHC Run 2, at a proton-proton ($pp$) centre of mass energy
$\sqrt{s}=13\text{\,Te\kern-1.00006ptV}$, would allow a measurement with a
statistical precision of around 10 MeV. Owing to the complementary
pseudorapidity ($\eta$) coverage of the LHCb experiment with respect to the
ATLAS and CMS experiments, it was demonstrated in Ref. [13] that the PDF
uncertainty could partially cancel in an average of $m_{W}$ measurements by
the LHC experiments.

Figure 1: Muon $q/p_{\mathrm{T}}$ distribution in simulated
$W\rightarrow\mu\nu$ events with variations in (left) $m_{W}$ and (right) the
$A_{3}$ coefficient. The dashed vertical lines indicate the two
$p_{\mathrm{T}}$ ranges that are included in the $m_{W}$ fit.

In this paper a first measurement of $m_{W}$ is presented using
$W\rightarrow\mu\nu$ decays, including both $W$ boson and muon charges,
collected at the LHCb experiment.222The inclusion of charge-conjugate
processes is implied throughout unless otherwise specified. This measurement
considers the muon $q/p_{\mathrm{T}}$ distribution, where $q$ is the muon
charge. Figure 1 (left) illustrates how the shape of the muon
$q/p_{\mathrm{T}}$ distribution in simulated $W$ boson decays is influenced by
variations in $m_{W}$ of $\pm 300\text{\,Me\kern-1.00006ptV}$, which
corresponds to roughly ten times the target precision of the present analysis.
The $q/p_{\mathrm{T}}$ variable allows all muons with
$p_{\mathrm{T}}>24\text{\,Ge\kern-1.00006ptV}$ to be visualised; those with
$28<p_{\mathrm{T}}<52\text{\,Ge\kern-1.00006ptV}$ are used to determine
$m_{W}$, while consistent control of the fit can be demonstrated in the region
$p_{\mathrm{T}}>52\text{\,Ge\kern-1.00006ptV}$.

The $p_{\mathrm{T}}$ of a muon produced by the decay of a $W$ boson has a
strong dependence on the $W$ boson transverse momentum ($p_{\mathrm{T}}^{W}$).
Direct measurements of the $p_{\mathrm{T}}^{W}$ distribution have been
reported by the ATLAS [14] and CMS [15] collaborations but the intervals are
necessarily coarse due to the limited $p_{\mathrm{T}}^{W}$ resolution.
Measurements of the transverse momentum distribution for $Z$ boson production
($p_{\mathrm{T}}^{Z}$) are therefore used to validate the predictions for the
$p_{\mathrm{T}}^{W}$ distribution.333For brevity $Z$ denotes
$Z/\gamma^{\ast}$. The angular variable $\phi^{\ast}$ [16], defined in Eq. 4,
is used in this analysis as a proxy for $p_{T}^{Z}$ since its distribution can
be measured more precisely than that of $p_{T}^{Z}$. Parton-shower programs
such as Pythia [17] can be tuned (e.g. Ref. [18]) to describe the
$p_{\mathrm{T}}^{Z}$ and $\phi^{\ast}$ data at the per cent level but it is
challenging to reliably translate such tunes to $W$ boson production. However,
a $W$-boson-specific tuning of a parton-shower model can be performed
simultaneously with a determination of $m_{W}$ [19].

If electroweak corrections are neglected then the production and leptonic
decay of the $W$ boson factorise such that the differential cross-section can
be written as

$\displaystyle\begin{split}\frac{\mathrm{d}\sigma}{\mathrm{d}p_{\mathrm{T}}^{W}\mathrm{d}y\mathrm{d}M\mathrm{d}\cos{\vartheta}\mathrm{d}{\varphi}}&=\frac{3}{16\pi}\frac{\mathrm{d}\sigma^{\rm
unpol.}}{\mathrm{d}p_{\mathrm{T}}^{W}\mathrm{d}y\mathrm{d}M}\\\
&\big{\\{}(1+\cos^{2}{\vartheta})+A_{0}\frac{1}{2}(1-3\cos^{2}{\vartheta})+A_{1}\sin
2{\vartheta}\cos{\varphi}\\\ &+A_{2}\frac{1}{2}\sin^{2}{\vartheta}\cos
2{\varphi}+A_{3}\sin{\vartheta}\cos{\varphi}+A_{4}\cos{\vartheta}\\\
&+A_{5}\sin^{2}{\vartheta}\sin 2{\varphi}+A_{6}\sin
2{\vartheta}\sin{\varphi}+A_{7}\sin{\vartheta}\sin{\varphi}\big{\\}},\\\
\end{split}$ (2)

where ${\vartheta}$ and ${\varphi}$ are the lepton decay angles defined in a
suitable frame (the Collins-Soper frame [20] is used in this analysis), and
$p_{\mathrm{T}}^{W}$, $y$ and $M$ denote the transverse momentum, rapidity and
mass of the final state lepton pair, respectively. An equivalent expression
applies to $Z\rightarrow\mu\mu$ production. The eight angular coefficients
($A_{i}$) are ratios of helicity cross-sections and depend on
$p_{\mathrm{T}}^{W}$, $y$ and $M$; $\sigma^{\rm unpol.}$ is usually referred
to as the unpolarised cross-section. The coefficients $A_{5}-A_{7}$ are
numerically small because they only arise at second order, or higher, in the
strong coupling constant ($\alpha_{s}$).444Throughout this paper $\alpha_{s}$
denotes the strong coupling at the scale of the $Z$ boson mass. The
coefficient $A_{3}$ is particularly influential on the muon $p_{\mathrm{T}}$
distribution. Figure 1 (right) shows how the $q/p_{\mathrm{T}}$ distribution
in simulated $W$ boson events, after the selection requirements described in
Sect. 3, changes when $A_{3}$ is scaled up and down by 3%.

In this paper, the simulated samples are weighted in the full five-dimensional
phase space of vector boson production and decay, using different models for
the unpolarised cross-section, angular coefficients, and QED final-state
radiation. Several PDF sets are used in the analysis but none of the analysed
data were included in the determination of these PDF sets.

This paper is organised as follows. Section 2 describes the data and simulated
samples. Section 3 details the signal candidate selection requirements.
Section 4 describes charge-dependent curvature corrections that are applied to
the data and simulation. The determination of residual smearing corrections to
the simulation with a simultaneous fit of $Z\rightarrow\mu\mu$ and quarkonia
decays is subsequently described. Section 5 details the measurement of muon
selection efficiencies and subsequent weight-based corrections to the
simulation. Section 6 describes the treatment of background arising from in-
flight decays of light hadrons. Section 7 sets out the modelling of vector
boson production and decay. Section 8 describes the simultaneous fit of the
model to the muon $p_{\mathrm{T}}$ distribution of $W$ boson candidates and
the $\phi^{\ast}$ (defined in Eq. 4) distribution of $Z$ boson candidates to
determine $m_{W}$. Section 9 explains how results based on three different PDF
sets are averaged, and summarises the systematic uncertainties. Several cross-
checks of the measurement are reported. The impact of analysis choices and
systematic variations on $m_{W}$ is discussed throughout the paper. The
conclusions of the analysis are presented in Sect. 10.

##  2 Data sets and event selection

The LHCb detector [21, 22] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the proton-
proton ($pp$) interaction region [23], 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 [24] placed downstream of the magnet. The tracking system provides
a measurement of the momentum, $p$, of charged particles with a relative
uncertainty that varies from 0.5% at low momentum to 1.0% at 200 GeV. The
minimum distance of a track to a primary $pp$ collision vertex (PV), the
impact parameter, is measured with a resolution of $(15\oplus
29/p_{\mathrm{T}})\,\upmu\text{m}$, where $p_{\mathrm{T}}$ is the component of
the momentum transverse to the beam, in  GeV. Different types of charged
hadrons are distinguished using information from two ring-imaging Cherenkov
detectors [25]. Photons, electrons and hadrons are identified by a calorimeter
system consisting of scintillating-pad and preshower detectors, an
electromagnetic calorimeter and a hadronic calorimeter. Muons are identified
by a system composed of alternating layers of iron and multiwire proportional
chambers [26]. The online event selection is performed by a trigger [27],
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.

This analysis uses a data sample of $pp$ collisions at $\sqrt{s}=13$ TeV
recorded during 2016, corresponding to an integrated luminosity of about
$1.7\text{\,fb}^{-1}$. Roughly half of the data were recorded in each of the
dipole magnet polarity configurations, resulting in a large degree of
cancellation of charge-dependent curvature biases and their associated
uncertainties. These data correspond to an average number of proton-proton
interactions per bunch-crossing event of $\mathcal{O}(1)$.

During Run 2 the LHCb detector was aligned and calibrated in real-time [28].
The alignment of the tracking system is based on a $\chi^{2}$ minimisation of
the residuals of the clusters of tracker hits evaluated with a Kalman filter
that takes into account multiple scattering and energy loss [29]. The
alignment algorithm also permits mass and vertex constraints [30]. An
optimised offline alignment, which includes ${Z}\rightarrow\mu\mu$ events with
a mass constraint that accounts for the natural width of the $Z$ boson, is
used to determine the track parameters. Since the real-time alignment is not
optimised for the analysis of high-$p_{\mathrm{T}}$ final states, this
realignment improves the ${Z}\rightarrow\mu\mu$ mass resolution by around 30%.

All signal and background processes are simulated using an LHCb specific tune
[31] of Pythia version 8.186 [17]. The interaction of the generated particles
with the detector, and its response, are implemented using the Geant4 toolkit
[32, *Agostinelli:2002hh] as described in Ref. [34]. Events are simulated with
both polarity configurations and weights are assigned to events in each
polarity such that the polarity distribution matches the recorded data.

##  3 Selection of $W$ boson, $Z$ boson and quarkonia signal candidates

Tracks are identified as muons if they are matched to hits in either three or
all four of the most downstream muon stations depending on their momentum.
They are then considered in this analysis if they are within the range
$1.7<\eta<5.0$, and have a momentum of less than
$2\text{\,Te\kern-1.00006ptV}$. The tracks must have a good fit quality and a
relative momentum uncertainty of less than 6%.

Candidate $W\rightarrow\mu\nu$ events are selected online by requiring that
one identified muon satisfies the requirements of all stages of the trigger.
At the hardware stage a $p_{\mathrm{T}}$ of at least 6 GeV is required. The
isolation of a muon is defined as the scalar sum of the transverse momenta of
all charged and neutral particles, as selected by a particle-flow algorithm,
described in Ref. [35], within $(\Delta\eta)^{2}+(\Delta\phi)^{2}<0.4^{2}$
around the muon, where $\Delta\eta$ and $\Delta\phi$ denote the separation in
$\eta$ and azimuthal angle around the beam direction ($\phi$), respectively.
Hadronic background contributions are suppressed by requiring the muon to have
an isolation of less than 4 GeV. For the $W\rightarrow\mu\nu$ selection the
$\eta$ range is tightened to $2.2<\eta<4.4$ so that the area of the isolation
cone is fully instrumented. The muon must satisfy ${\chi^{2}_{\rm IP}}<9$
where $\chi^{2}_{\rm IP}$ is defined by the difference in the vertex fit
$\chi^{2}$ of the PV with and without including the muon. Background from $Z$
boson events is suppressed by rejecting events that contain a second muon with
$p_{\mathrm{T}}>25\text{\,Ge\kern-1.00006ptV}$ and an opposite charge to that
of the primary muon candidate. Roughly $2.4$ million $W\rightarrow\mu\nu$
candidates are selected in the range $28<p_{\mathrm{T}}<52$ GeV.

Candidate $Z\rightarrow\mu\mu$ events are reconstructed from combinations of
two oppositely charged identified muons associated to the same PV with an
invariant mass within $\pm 14$ GeV of the known $Z$ boson mass [7]. At least
one muon must be matched to a single muon selection at all stages of the
trigger. Both muons must have $p_{\mathrm{T}}>20$ GeV, an isolation value
below 10 GeV, and an impact parameter significance of less than ten standard
deviations. Roughly $190$ thousand $Z\rightarrow\mu\mu$ candidates are
selected.

Candidate ${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\rightarrow\mu\mu$ and
${\mathchar 28935\relax}(1S)\rightarrow\mu\mu$ events, which are primarily
used to calibrate the modelling of the momentum measurement, are required to
have a pair of oppositely charged identified muons. Both muons must have a
transverse momentum above 3 GeV and satisfy a tighter muon identification
requirement. In order to specifically select
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\rightarrow\mu\mu$ candidates originating
from $b$-hadron decays the decay vertices must be displaced from the nearest
PV with a significance of at least three standard deviations. These selections
retain roughly $1.0$ million ${\mathchar 28935\relax}(1S)\rightarrow\mu\mu$
candidates and $220$ thousand
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\rightarrow\mu\mu$ candidates.

##  4 Momentum calibration and modelling

The momentum scale can be precisely determined from the mass measurements of
various resonances, including those that decay to muon pairs. However, charge-
dependent curvature biases that shift $q/p$ are challenging to estimate
because their effect largely cancels in the mass of the resonances. They are
also particularly important for the high momentum muons from $W$ and $Z$ boson
decays. In Ref. [36] it was proposed to determine corrections using the so-
called _pseudomass_ variable in ${Z}\rightarrow\mu\mu$ events

$\mathcal{M}^{\pm}=\sqrt{2p^{\pm}p_{\mathrm{T}}^{\pm}\frac{p^{\mp}}{p_{\mathrm{T}}^{\mp}}(1-\cos\theta)},$
(3)

where $p^{\pm}$ and $p_{\mathrm{T}}^{\pm}$ are the momenta and transverse
momenta of the $\mu^{\pm}$, respectively. The opening angle between the two
muons is denoted $\theta$. Crucially, the value of $\mathcal{M^{\pm}}$ is
independent of the magnitude of the momentum of the $\mu^{\mp}$ and is
therefore directly sensitive to curvature biases affecting the $\mu^{\pm}$
candidate. The pseudomass is an approximation of the dimuon mass under the
assumption that the dimuon system has zero momentum transverse to the bisector
of the two lepton transverse momenta. The $\phi^{\ast}$ observable is defined
as [16]

$\phi^{\ast}=\frac{\tan((\pi-\Delta\phi)/2)}{\cosh(\Delta\eta/2)}\sim\frac{p_{\mathrm{T}}^{Z}}{M},$
(4)

where $\Delta\phi$ is the azimuthal opening angle between the two leptons and
$\Delta\eta$ is the difference between the pseudorapidities of the negatively
and positively charged lepton. In events with small values of $\phi^{\ast}$
the pseudomass better approximates the dimuon mass. The pseudomass
distributions for events with $\phi^{\ast}<0.05$ are studied in intervals of
$\phi$ and $\eta$ of the $\mu^{\pm}$ candidate, with a further categorisation
into candidates traversing the silicon strip or straw drift tube detectors
downstream of the magnet. A maximum likelihood fit of the $\mathcal{M^{\pm}}$
distributions is performed for each of these detector regions. The signal
shapes are described by the sum of a resonant Crystal-Ball [37] component and
a nonresonant component represented by an exponential function. The means of
the $\mathcal{M^{\pm}}$ Crystal-Ball functions are parameterised as
$\bar{M}(1\pm\mathcal{A})$, where $\mathcal{A}$ and $\bar{M}$ are freely
varying asymmetry and mass parameters, respectively. The resulting $q/p$
corrections, which are given by $\mathcal{A}/\bar{p}$ where $\bar{p}$ is the
average muon momentum for a given interval in $\eta$ and $\phi$, are presented
for the data and simulation for both polarity configurations in Fig. 2.

Figure 2: Curvature corrections as a function of the detector region index
(depends on $\eta$, $\phi$ and tracking detector, as described in the text)
for (left) data and (right) simulation. The corrections are shown for both
polarity configurations. The periodic pattern corresponds to a dependence on
pseudorapidity that repeats in the intervals of the azimuthal angle.

After the curvature corrections are applied to the data and simulation, the
momenta of the simulated muons are smeared to match those in the data, as
described below, according to

$\frac{q}{p}\rightarrow\frac{q}{p\cdot\mathcal{N}(1+\alpha,{\sigma_{\text{MS}}})}+\mathcal{N}\left(\delta,\frac{{\sigma_{\delta}}}{\cosh\eta}\right),$
(5)

where $\mathcal{N}(a,b)$ represents a random number sampled from a Gaussian
distribution with mean $a$ and width $b$. The $\sigma_{\text{MS}}$ and
$\sigma_{\delta}$ parameters correspond to the multiple scattering and
curvature measurement contributions to the resolution, respectively. The
smearing model includes six parameters in total. There are two momentum scale
parameters $\alpha$ corresponding to the $2.2<\eta<4.4$ region, which
coincides with the selection of $W$ boson candidates, and the $\eta<2.2$
region. A single $\delta$ parameter, corresponding to a curvature bias, covers
the region $2.2<\eta<4.4$, while the value of $\delta$ is fixed to zero in the
region $\eta<2.2$. There are two ${\sigma_{\delta}}$ parameters corresponding
to the $2.2<\eta<4.4$ and $\eta<2.2$ regions, while a single
${\sigma_{\text{MS}}}$ parameter is found to adequately cover all $\eta$
values. The empirical $1/\\!\cosh\eta$ dependence of the second term in Eq. 5
improves the modelling of the $\eta$ dependence in the ${Z}\rightarrow\mu\mu$
mass distribution. As a further correction to Eq. 5, the value of
$\sigma_{\text{MS}}$ is increased by a factor of 1.5 in the region $\eta>3.3$
since this improves the agreement between data and simulation in the $\eta$
dependence of the quarkonia mass distributions.

The six smearing parameters are determined in a simultaneous fit of
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\rightarrow\mu\mu$, ${\mathchar
28935\relax}(1S)\rightarrow\mu\mu$ and $Z\rightarrow\mu\mu$ candidates in data
and simulation. A total of 36 dimuon invariant mass distributions are used in
the fit. First, there are three $\eta$ regions covering $\eta<2.2$,
$2.2<\eta<3.3$ and $3.3<\eta<4.4$, which result in six categories that depend
on the $\eta$ regions of the two muons. The quarkonia mass distributions are
only used in categories with both muons having $\eta>2.2$. In the subset of
the $\eta$ categories with both muons in $\eta>2.2$, the
${Z}\rightarrow\mu\mu$ data are split into three intervals of the asymmetry
between the momenta of the two muons, which provides a first order sensitivity
to the $\delta$ parameters. Finally, all categories are divided by magnet
polarity.

As in previous studies of ${Z}\rightarrow\mu\mu$ production with the LHCb
experiment [38], the background under the ${Z}\rightarrow\mu\mu$ peak is low
enough to be neglected but the fit includes exponential functions for the
background contributions under the quarkonia resonance peaks. The fractions
and slopes of these exponential components vary freely in the fit.

The total $\chi^{2}$ from the fit is 1862 for 2082 degrees of freedom. Table 1
shows the fit values of the six parameters in the smearing model. The $\delta$
values are close to zero as expected given the curvature corrections that have
already been applied. Figure 3 shows the dimuon mass distributions for the
$Z$, ${\mathchar 28935\relax}(1S)$ and ${J\mskip-3.0mu/\mskip-2.0mu\psi}$
samples after combining all categories with both muons in the $2.2<\eta<4.4$
range.

Table 1: Parameters in the momentum smearing model where the uncertainties quoted are statistical. Parameter | Fit value
---|---
$\alpha$ ($\eta<2.2$) | $(0.58\pm 0.10)\times 10^{-3}$
$\alpha$ ($2.2<\eta<4.4$) | $(-0.0054\pm 0.0025)\times 10^{-3}$
$\delta$ | $(-0.48\pm 0.37)\times 10^{-6}~{}\text{\,Ge\kern-1.00006ptV}^{-1}$
${\sigma_{\delta}}$ ($\eta<2.2$) | $(17.7\pm 1.2)\text{\,ke\kern-1.00006ptV}^{-1}$
${\sigma_{\delta}}$ ($2.2<\eta<4.4$) | $(14.9\pm 0.9)\text{\,ke\kern-1.00006ptV}^{-1}$
${\sigma_{\text{MS}}}$ | $(2.015\pm 0.019)\times 10^{-3}$

The statistical uncertainties in the smearing parameters result in an
uncertainty in $m_{W}$ of $3\text{\,Me\kern-1.00006ptV}$. The uncertainty in
the world average of the ${\mathchar 28935\relax{(1S)}}$ mass [7] leads to an
uncertainty of $2\text{\,Me\kern-1.00006ptV}$. The uncertainty in the
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}$ mass is negligible compared to that in
the ${\mathchar 28935\relax{(1S)}}$ mass. The ${\mathchar 28935\relax{(1S)}}$
and the $Z$ masses have comparable relative uncertainties but the latter has a
negligible effect given the limited size of the $Z$ boson sample. The amount
of material in the detector, which affects the modelling of energy losses, is
varied by 10%, leading to an uncertainty of $3\text{\,Me\kern-1.00006ptV}$. A
total uncertainty of $5\text{\,Me\kern-1.00006ptV}$ is attributed to the
shifts in $m_{W}$ corresponding to: an alternative form of the
$1/\\!\cosh\eta$ factor in Eq. 5; and a variation in the $\eta$ region over
which $\sigma_{\text{MS}}$ is scaled. An uncertainty of
$2\text{\,Me\kern-1.00006ptV}$ is attributed to the modelling of the radiative
tails in the $\mathchar 28935\relax{(1S)}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi}$ simulation, using the methods described in
Ref. [39]. The total uncertainty attributed to the modelling of the momentum
scale and resolution is $7\text{\,Me\kern-1.00006ptV}$.

Figure 3: Dimuon mass distributions for selected $J/\psi$, ${\mathchar
28935\relax}(1S)$ and $Z$ boson candidates. All categories with both muons in
the $2.2<\eta<4.4$ region are combined. The data are compared with the fit
model. The red histogram delineates the model before the application of the
smearing.

##  5 Efficiency corrections

Corrections to the simulation are required for the muon trigger,
identification, tracking and isolation efficiencies. The efficiencies are
measured using a combination of ${Z}\rightarrow\mu\mu$ and, in the case of the
trigger efficiency, ${\mathchar 28935\relax{(1S)}}\rightarrow\mu\mu$ samples.
Positively and negatively charged muons are analysed separately but the
results are combined since any charge asymmetries are verified to have a
negligible effect.

The trigger efficiency, which accounts for the hardware and software stages,
is measured using a combination of ${Z}\rightarrow\mu\mu$ and ${\mathchar
28935\relax{(1S)}}\rightarrow\mu\mu$ events in which one so-called tag muon is
required to match a positive decision in the hardware trigger and the first
stage of the software-level trigger such that the other muon can be regarded
as an unbiased probe of the trigger efficiency. Events are categorised as
either matched or unmatched depending on whether the probe muon is matched to
a positive trigger decision in the event data record. The
${Z}\rightarrow\mu\mu$ sample is verified to be sufficiently pure that the
efficiencies can be measured by simply counting the matched and unmatched
events with invariant masses within $\pm 15$ GeV of the known $Z$ boson mass
[7]. The efficiencies are determined in four uniform $\phi$ intervals and
eight uniform $\eta$ intervals in the range $2.2<\eta<4.4$. There are two
additional $\eta$ intervals in the region $\eta<2.2$ and one in the region
$\eta>4.4$. The ${\mathchar 28935\relax{(1S)}}\rightarrow\mu\mu$ sample
requires background subtraction by fitting the dimuon invariant mass
distribution with a parametric model of the signal and background components.

Three $p_{\mathrm{T}}$ intervals, in the range $7.0<p_{\mathrm{T}}<12.5$ GeV,
are used for the probe muons from ${\mathchar
28935\relax{(1S)}}\rightarrow\mu\mu$ decays while for the $Z\rightarrow\mu\mu$
candidates an adaptive algorithm is employed to determine the $p_{\mathrm{T}}$
intervals. The ratios of the trigger efficiencies in data relative to those in
the simulation are shown as a function of $1/p_{\mathrm{T}}$ of the muon in
Fig. 4 for each of the intervals in $\eta$ and $\phi$. These are overlaid with
a linear function of $p_{\mathrm{T}}$, from which correction weights for the
simulated events are evaluated. The weights for the $W$ boson model only rely
on these functions but the weights for the $Z$ boson model also require a
parameterisation of the absolute efficiency in the simulation such that the
efficiency can be correctly modelled for $Z$ boson candidates with one or two
muons matched to a trigger decision. The absolute efficiency is described by
an error function that captures the $p_{\mathrm{T}}$ threshold (roughly 6 GeV)
of the hardware trigger [28].

Figure 4: Trigger efficiency ratios in data relative to simulation in
intervals of $\eta$ for a single interval of $\phi$. The points are presented
as a function of $1/p_{\mathrm{T}}$ and are overlaid with best-fit linear
functions of $p_{\mathrm{T}}$. Points from each $\eta$ interval are separated
for readability by an offset of half the integer index $i$ for that interval.
The uncertainties on most of the points are too small for the error bars to be
visible.

The muon identification efficiency is treated in a similar manner to the
trigger efficiencies, using ${Z}\rightarrow\mu\mu$ events. The resulting event
weights, which are applied to the simulated events, are within a few per cent
from unity. The tracking efficiency is determined as in previous measurements
of $W$ and $Z$ boson production at LHCb [38] using ${Z}\rightarrow\mu\mu$
candidates where the probe muons are reconstructed by combining hits from the
muon stations and the large-area silicon-strip detector located upstream of
the magnet [40]. As neither of these detectors are used in the primary track
reconstruction algorithms, the probes can be used to measure the tracking
efficiency. Correction factors are evaluated using a similar approach to those
of the muon identification efficiency, except that the corrections are assumed
to be independent of $p_{\mathrm{T}}$.

The statistical uncertainties in the muon trigger, tracking and identification
efficiency corrections are evaluated by rerunning the relevant steps of the
analysis, up to and including the $m_{W}$ determination, with random
fluctuations in the underlying efficiency values. The RMS of the resulting
variations in the $m_{W}$ value is regarded as an uncertainty. A systematic
uncertainty is attributed to the dependence of the results on the scheme for
$\eta$ and $\phi$ intervals. This includes restricting to a single interval in
$\phi$, reducing the number of $\eta$ intervals (within $2.2<\eta<4.4$) by a
factor of two, varying the number of $p_{\mathrm{T}}$ intervals between two
and ten, and using the simulation rather than the data to control the adaptive
algorithm. Further systematic uncertainties are attributed to variations in
the isolation and $p_{\mathrm{T}}$ requirements on the tag muons, the mass
windows used to determine the ${Z}\rightarrow\mu\mu$ signal yields, and the
functions of the $p_{\mathrm{T}}$ dependence of the efficiency ratios. As the
probe muons for the tracking efficiency are reconstructed using minimal
tracking information, they have a significantly lower momentum resolution and
so a dedicated momentum smearing is applied by default. A variation in the
size of this smearing is included in the systematic uncertainty evaluation.
The total uncertainty associated to the muon trigger, identification and
tracking uncertainties is $6\text{\,Me\kern-1.00006ptV}$.

The efficiency of the isolation requirement is measured with
${Z}\rightarrow\mu\mu$ events. The isolation variable receives contributions
from pile-up, the underlying event and the recoil component of the hard
process. The recoil projection for each muon is defined by

$u=\frac{\vec{p}^{\;V}_{\rm T}\cdot\vec{p}^{\;\mu}_{\rm T}}{p_{\rm T}^{\mu}},$
(6)

where $\vec{p}^{\;\mu}_{\rm T}$ and $\vec{p}^{\;V}_{\rm T}$ are the two-
dimensional momentum vectors of the muon and the parent vector boson in the
transverse plane. Figure 5 (left) shows the isolation requirement efficiency
as a function of $u$ in the ${Z}\rightarrow\mu\mu$ data and simulation. The
efficiency is around 80% at positive values of $u$ where the underlying event
contribution dominates. At negative values of $u$, corresponding to large
recoil, the efficiency drops to around 70%. The full reconstruction of the $Z$
boson in ${Z}\rightarrow\mu\mu$ events allows the determination of corrections
as a function of $u$ defined at reconstruction level and consistently applied
to $W$ and $Z$ boson events as a function of $u$ defined at generator level,
with this approach validated using the $Z$ boson sample. A map of relative
efficiencies between data and simulation is determined in intervals of $u$ and
$\eta$ and is used to evaluate weights for the simulated events. Figure 5
(right) shows that the dependence of the isolation efficiency on
$1/p_{\mathrm{T}}$ in data is accurately described by the simulation after the
corrections. The statistical uncertainties in the isolation efficiency
corrections are treated as a source of systematic uncertainty. This is
combined in quadrature with a systematic uncertainty that accounts for
variations in the $u$ and $\eta$ intervals and in a smoothing procedure
applied to enhance the effective statistical precision of the correction map.
The total uncertainty attributed to the isolation efficiency modelling is
$4\text{\,Me\kern-1.00006ptV}$.

Figure 5: Isolation efficiencies as a function of the observables (left) $u$
and (right) $1/p_{\mathrm{T}}$ of the muon. The grey histograms indicate the,
arbitrarily normalised, shapes of each distribution in simulated $W$ boson
events. In the lower panels the ratios of the isolation efficiency with
respect to the uncorrected simulation are shown.

The modelling of the impact parameter and track fit $\chi^{2}$ variables in
simulation is improved in two stages, both of which make use of
${Z}\rightarrow\mu\mu$ events. Initially, the values of the variables in the
simulation are smeared and shifted to match the data. Subsequently, weights
are applied to the simulated events to correct for small residual differences
in the efficiencies of the selection requirements between data and simulation.
In order for the impact parameter modelling to be reliably transported between
$Z$ and $W$ boson events, the PVs are refitted with all signal muons removed.
The three-dimensional impact parameter is then decomposed into its individual
components and these are smeared according to a normal distribution in six
intervals in $\eta$ and seven intervals in $\phi$. A similar procedure is used
to improve the modelling of the track fit $\chi^{2}$ distribution. These
corrections are followed by smaller corrections applied to account for the
efficiencies of the impact parameter and track $\chi^{2}$ requirements. The
efficiency weights are also determined with ${Z}\rightarrow\mu\mu$ events and
are typically within a few per mille from unity. Neither the statistical
uncertainties nor reasonable variations in the $\eta$ and $\phi$ interval
schemes are found to have significant impact on the $m_{W}$ value. Therefore,
no systematic uncertainty is considered.

##  6 QCD background model

Figure 6: Weighted $p_{\mathrm{T}}$ spectra of the samples of (left)
positively and (right) negatively charged hadron candidates. The fit results
are overlaid.

A small background from in-flight decays of pions and kaons into muons is
present in the sample of $W\rightarrow\mu\nu$ boson candidates. This
background cannot be modelled with high enough accuracy using full detector
simulation. It is therefore modelled using a sample of high-$p_{\mathrm{T}}$
tracks, selected by dedicated triggers without muon identification
requirements. The $W$ boson selection requirements are applied to this sample
but with the muon identification requirement inverted. The resulting sample is
verified in simulation to be a pure sample of charged hadrons, composed of
roughly 60% pions, 30% kaons and 10% protons, produced directly at the $pp$
interaction vertex. In particular, the impact parameter requirements suppress
the heavy flavour hadron content to a negligible level.

The probability of an unstable hadron of mass $m$, lifetime $\tau$, and
momentum $p$ to decay within a detector of length $d$ is

$1-\exp\left(-\frac{md}{\tau p}\right)\approx\frac{md}{\tau p}.$ (7)

Similar kinematic distributions are predicted for pions, kaons and protons in
the simulation. Therefore, the in-flight decay background can be modelled by
the data with weights of $1/p$. The majority of the in-flight decays occur
outside the magnetic field region and therefore have minimal influence on the
measured momentum. The absolute normalisation is not needed because this
background component is allowed to vary freely in the $m_{W}$ fit.

The weighted $p_{\mathrm{T}}$ spectra for both charges are shown in Fig. 6 and
are overlaid with best-fit functions of the form [41],

$\left(1+\frac{p_{\mathrm{T}}}{a}\right)^{-n},$ (8)

where $a$ and $n$ are empirical parameters that are determined in the fits. In
addition to giving a good fit to the data, this functional form is verified to
describe the pion and kaon spectra in simulation. The fit functions are
sampled to generate background candidates for inclusion in the $m_{W}$ fit. A
charge asymmetry of $\mathcal{O}$(10%), favouring positively charged hadrons
due to the $pp$ initial state, is observed and is included in the sampling.

The uncertainty in the hadronic background model is dominated by systematic
sources. Three different systematic uncertainties, which are combined in
quadrature, are assessed. The first accounts for assuming that the data sample
can be treated as containing a single hadron species. The second accounts for
a small bias from the inverted muon identification requirements. The third
accounts for a small dependence on the range of $p_{\mathrm{T}}$ values used
in the fits to Eq. 8. The combined systematic uncertainty is
$2\text{\,Me\kern-1.00006ptV}$.

##  7 Modelling $W$ and $Z$ boson production

The emulation of different $m_{W}$ hypotheses is achieved by assigning event
weights based on a relativistic Breit-Wigner function with a mass-dependent
width. Further weight-based corrections are also applied to simulated $W$ and
$Z$ boson events to improve upon the limited formal accuracy of Pythia. A
weighting to model QCD effects is applied in the basis of Eq. 2. A further
weighting to model QED effects is applied as a function of the logarithm of
the relative energy difference between the dimuon system before and after QED
final-state radiation.

Higher order electroweak corrections are not included in the model. Instead,
an uncertainty of 5 MeV is attributed to these missing corrections using
samples of events generated using POWHEGBoxV2 [42, 43, 44] with and without
electroweak corrections. These events are interfaced with Pythia and the
uncertainty is evaluated using the data challenge methods described in Sect.
8.1.

###  7.1 Candidate QCD programs

Five software programs, or combinations of these, are evaluated as potential
candidates for the weighting of the simulated $W$ and $Z$ boson events.

  1. 1.

Pythia: Events are generated using Pythia version 8.235 [17] with several
values of the intrinsic transverse momentum ($k_{\rm T}^{\rm intr}$) of the
initial state partons and $\alpha_{s}$, closely following the work of Ref.
[19]. The NNPDF23_lo_as_0130_qed [45] PDFs are used in the event generation.
The events are weighted using the methods described in Ref. [46] to the
NNPDF31_lo_as_0118 [47] and CT09MCS [48] PDFs.

  2. 2.

POWHEGPythia: Events are generated using POWHEGBoxV2 [49] with the
NNPDF31_nlo_as_0118 PDFs and are subsequently showered with Pythia version
8.244 [17]. The event generation with POWHEGBoxV2 is repeated with different
values of $\alpha_{s}$. The default Monash [18] tune of Pythia is used but
event samples are generated with different values of $k_{\rm T}^{\rm intr}$,
and with the same value of $\alpha_{s}$ as used in POWHEGBoxV2. This results
in a grid of predictions with different $\alpha_{s}$ and $k_{\rm T}^{\rm
intr}$ values.

  3. 3.

POWHEGHerwig: Events are generated equivalently to those from POWHEGPythia but
substituting Pythia with Herwig [50] for the parton-shower stage.

  4. 4.

Herwig: These events are also equivalent to those of POWHEGPythia except that
the hard process and the parton shower are both fully implemented in Herwig
[50].

  5. 5.

DYTurbo: The cross-sections and angular coefficients are computed at
$\mathcal{O}(\alpha_{s}^{2})$ accuracy using DYTurbo [51] with the
NNPDF31_nnlo_as_0118 PDFs [47]. Predictions for the unpolarised cross-section
include resummation to next-to-next-to-leading logarithms and are produced
with several values of the $g$ parameter that controls nonperturbative
effects.

Histograms of the unpolarised cross-section in Eq. 2 and the angular
coefficients are produced for all combinations of programs and tuning
parameters. These histograms, which are used to determine event weights, have
intervals in the transverse momentum, rapidity and mass of the vector boson.

###  7.2 QCD weighting and transverse momentum model

The simulated samples described in Sect. 2 can be weighted in the full five-
dimensional phase space of vector boson decays, according to Eq. 2, to provide
predictions based on different models of QCD. For the unpolarised cross-
section such weights are found by interpolating between the generated
histograms described above.

A detailed measurement of the angular coefficients in $pp\rightarrow
Z\rightarrow\mu\mu$ at $\sqrt{s}=8$ TeV was reported by ATLAS [52].
Predictions based on parton showers are generally found to be unreliable in
predicting the angular coefficients. However, the ATLAS data are reasonably
well described by $\mathcal{O}(\alpha_{s}^{2})$ predictions from DYNNLO [53],
on which DYTurbo is based. An exception is the difference between $A_{0}$ and
$A_{2}$, for which $\mathcal{O}(\alpha_{s}^{2})$ is effectively only leading
order, but the present measurement has a negligible sensitivity to this
particular detail. Hereafter DYTurbo is used in the modelling of the angular
coefficients.

Since the prediction of each angular coefficient relies on separate numerator
and denominator calculations, there are four independent renormalisation and
factorisation scales that are varied to assess the uncertainty associated with
missing higher orders in $\alpha_{s}$. In Ref. [54] it is argued that fully
correlating the scale variations between the numerator and denominator, which
leads to a large degree of cancellation, may result in inadequate uncertainty
coverage. The present analysis therefore follows the recommendation of Ref.
[54], which is to vary the four scales independently by factors of
$\frac{1}{2}$ and 2 with the constraint that all ratios that could be
constructed from the four scales are between $\frac{1}{2}$ and $2$. This
results in an envelope of 31 values of $m_{W}$ that sets the associated
uncertainty.

Figure 7 compares the $p_{\mathrm{T}}^{Z}$ distribution in the data with the
Pythia simulation weighted to the different unpolarised cross-section
predictions before and after tuning them to the data. Table 2 lists the
$\chi^{2}$ and the preferred parameter values for the fits with each model.
DYTurbo gives a reasonable prediction but overestimates the number of events
with large $p_{\mathrm{T}}^{Z}$ even with tuning of the $g$ parameter. A
reasonable initial description is provided by Pythia, which is to be expected
since it has already been tuned to $p_{\mathrm{T}}^{Z}$ data. The
POWHEGPythia, POWHEGHerwig and Herwig predictions poorly describe the shape of
the $p_{\mathrm{T}}^{Z}$ distribution with their default values of
$\alpha_{s}=0.118$. Their descriptions of the $p_{\mathrm{T}}^{Z}$
distribution are greatly improved when their $\alpha_{s}$ and $k_{\rm T}^{\rm
intr}$ parameters are tuned. Of these programs, POWHEGPythia gives the most
reliable description of the data, with a preferred $\alpha_{s}$ value of
around $0.125$. Large values of $\alpha_{s}$ are also favoured by other models
and in other studies of the $p_{\mathrm{T}}^{Z}$ distribution [55].555The
spread in $\alpha_{s}$ values between the different models also means that
this fit result should not be interpreted as a precise and accurate
determination of the value of $\alpha_{s}$. Therefore, POWHEGPythia with
freely varying $\alpha_{s}$ and $k_{\rm T}^{\rm intr}$ values is selected for
the default fitting model. The systematic uncertainty in the description of
the $p_{\mathrm{T}}^{Z}$ and $p_{\mathrm{T}}^{W}$ shapes is evaluated with
alternative predictions from: Pythia with the CT09MCS and leading-order
NNPDF31 PDFs; Herwig and POWHEGHerwig with the next-to-leading-order NNPDF31
PDFs. The envelope of shifts in $m_{W}$ obtained from using these alternative
descriptions is found to be $11\text{\,Me\kern-1.00006ptV}$, providing the
dominant contribution to the systematic uncertainty associated with the
modelling of the vector boson transverse momentum.

All of the event generator predictions can be weighted at leading-order to
emulate event generation based on different PDFs. As discussed in Ref. [46]
this weighting is not completely valid for events generated at next-to-leading
order. Since POWHEGPythia allows in situ computation of next-to-leading order
PDF weights, it is possible to directly estimate the inaccuracy of the
leading-order approximation. With five NNPDF31_nlo_as_0118 replicas it is
verified that the differences between the leading-order and next-to-leading-
order weighting approaches are smaller than 1 MeV in the $m_{W}$ fit.

Since it is computationally expensive to determine fully the PDF uncertainty
in the DYTurbo angular coefficients, the variations in the next-to-leading
order PDFs used in the POWHEGPythia model of the unpolarised cross-section are
coherently propagated to the angular coefficients. The DYTurbo angular
coefficients are shifted by the differences in values predicted by
POWHEGPythia in the default (NNPDF31_nlo_as_0118) PDF compared to the target
PDF in the uncertainty assessment.

Separate measurements of $m_{W}$ based on the NNPDF3.1 [47], CT18 [56] and
MSHT20 [57] PDF sets, each with their own PDF uncertainty estimate, are
reported. However, since these three sets are based on almost the same data,
the central result of this analysis is a simple arithmetic average of the
three results, under the assumption that the three PDF uncertainties are fully
correlated.

Figure 7: Distributions of $p_{\mathrm{T}}^{Z}$ (left) before and (right) after the fit for the different candidate models of the unpolarised cross-sections. The fit only considers the region $p_{\mathrm{T}}^{Z}<30\text{\,Ge\kern-1.00006ptV}$, indicated by the dashed vertical line. In the lower panels the ratios with respect to the POWHEGPythia model are shown. Table 2: Results of fits of different models to the $p_{\mathrm{T}}^{Z}$ distribution. The uncertainties quoted are statistical, and the $\chi^{2}$ comparison of the different models to the data is evaluated considering only statistical uncertainties. The right-hand column lists the fit values of the $k_{\rm T}^{\rm intr}$ parameter or, for DYTurbo, the analogous $g$ parameter. The fit with DYTurbo has one more degree of freedom than the fits with the other models since only one tuning parameter ($g$) is used for DYTurbo.  Program |  $\chi^{2}$/ndf | $\alpha_{s}$ |
---|---|---|---
DYTurbo | 208.1/13 | $0.1180$ | $g\phantom{{}^{\rm intr}}=0.523\pm 0.047\text{\,Ge\kern-1.00006ptV}^{2}$
POWHEGPythia | 30.3/12 | $0.1248\pm 0.0004$ | ${k_{\rm T}^{\rm intr}}=1.470\pm 0.130\text{\,Ge\kern-1.00006ptV}$
POWHEGHerwig | 55.6/12 | $0.1361\pm 0.0001$ | ${k_{\rm T}^{\rm intr}}=0.802\pm 0.053\text{\,Ge\kern-1.00006ptV}$
Herwig | 41.8/12 | $0.1352\pm 0.0002$ | ${k_{\rm T}^{\rm intr}}=0.753\pm 0.052\text{\,Ge\kern-1.00006ptV}$
Pythia, CT09MCS | 69.0/12 | $0.1287\pm 0.0004$ | ${k_{\rm T}^{\rm intr}}=2.113\pm 0.032\text{\,Ge\kern-1.00006ptV}$
Pythia, NNPDF31 | 62.1/12 | $0.1289\pm 0.0004$ | ${k_{\rm T}^{\rm intr}}=2.109\pm 0.032\text{\,Ge\kern-1.00006ptV}$

###  7.3 Angular scale factors

The uncertainties in the angular coefficients from DYTurbo would lead to an
uncertainty of $\mathcal{O}(30)\text{\,Me\kern-1.00006ptV}$ in $m_{W}$, with
the dominant contribution attributed to the $A_{3}$ coefficient. The
importance of $A_{3}$ can be understood by inspection of Eq. 2: an increase in
$A_{3}$ enhances the cross-section for events with large $\sin{\vartheta}$ and
$\cos{\varphi}$. The contribution to the muon $p_{T}$ from the $W$ boson mass
scales with $\sin{\vartheta}$ while the contribution from the transverse
momentum of the $W$ boson scales with $\pm\cos{\varphi}$ for $W^{\pm}$ boson
production. By allowing a single $A_{3}$ scaling factor, which is shared
between the $W^{+}$ and $W^{-}$ processes, to vary freely in the $m_{W}$ fit
the angular coefficient uncertainty is reduced by roughly a factor of three,
to $10\text{\,Me\kern-1.00006ptV}$. Effectively the resulting model only
depends on DYTurbo for the kinematic dependence of $A_{3}$, while all other
coefficients are fully modelled by DYTurbo.

###  7.4 Parametric correction at high transverse momentum

While POWHEGPythia is shown in Sect. 7 to describe the $p_{\mathrm{T}}^{Z}$
distribution in the region below 30 GeV, it systematically underestimates the
cross-section at higher $p_{\mathrm{T}}^{Z}$. This is expected due to the
missing matrix elements for the production of a weak boson and more than one
jet. Figure 8 compares the $p_{\mathrm{T}}^{Z}$ distribution in the data with
the model prediction having set $\alpha_{s}$ and $k_{\rm T}^{\rm intr}$ to be
close to the final fit values. For $p_{\mathrm{T}}^{Z}\geq 40$ GeV the model
starts to underestimate the cross-section, reaching the ten per cent level at
$p_{\mathrm{T}}^{Z}\sim 100$ GeV. In the lower panel of Fig. 8 the data to
prediction ratio is overlaid with a function of the form

$(1+p_{0}+p_{0}\mathrm{Erf}(p_{1}(p_{\mathrm{T}}^{V}-p_{2})))\times(1+p_{3}p_{\mathrm{T}}^{V}).$
(9)

Since the universality of this correction between $W$ and $Z$ boson processes
is not well controlled, an uncertainty of 100% of this correction is included
as an additional systematic uncertainty associated with the vector boson
$p_{\mathrm{T}}$ model. This contributes an uncertainty in the $m_{W}$ value
smaller than $1\text{\,Me\kern-1.00006ptV}$.

Figure 8: Distribution of $p_{\mathrm{T}}^{Z}$ compared to the POWHEGPythia
model prior to the parametric correction, which is delineated by the red line
in the lower panel.

###  7.5 QED weighting

Figure 9: Logarithm of the relative energy loss of the dilepton system due to
final-state radiation for (left) $W$ boson events and (right) $Z$ boson
events. An energy loss of below $10^{-6}$ is considered unresolvable and is
accounted for in the underflow bin to the left of the dashed vertical line. In
the lower panel the ratio with respect to Pythia is shown.

The effect of the QED final-state radiation is largely characterised by the
energy difference between the final-state lepton system before and after
radiation. The logarithm of this energy difference as described by Pythia,
Herwig, and an alternative configuration of Pythia with the final-state
radiation modelled by Photos [58] is shown in Fig. 9. Event-by-event weights
are evaluated for the Herwig and Photos models relative to Pythia and applied
to the simulated samples used in the analysis. The default model uses the
arithmetic average of the Herwig, Photos and Pythia weights, where the Pythia
weights are equal to unity. The systematic uncertainty, amounting to
$7\text{\,Me\kern-1.00006ptV}$, is taken from the envelope of $m_{W}$ values
corresponding to each of the three models taken individually.

##  8 $W$ boson mass fit

The $m_{W}$ value is determined through a simultaneous fit of the
$q/p_{\mathrm{T}}$ distribution of $W$ boson candidates and the $\phi^{\ast}$
distribution of $Z$ boson candidates. The $\phi^{\ast}$ variable is preferred
over $p_{\mathrm{T}}^{Z}$ because it is less susceptible to several of the
modelling uncertainties, while still being sensitive to the parameters
affecting the predicted $p_{\mathrm{T}}^{Z}$ and $p_{\mathrm{T}}^{W}$
distributions. The $\phi^{\ast}$ distribution extends to $\phi^{\ast}=0.5$
while the $q/p_{\mathrm{T}}$ distribution includes two fit regions covering
$28<p_{\mathrm{T}}<52$ GeV. Projections of the $q/p_{\mathrm{T}}$ distribution
cover a wider interval with $p_{\mathrm{T}}>24$ GeV that includes regions
outside the fit. The model of both distributions is based on simulated event
samples with event-by-event weights. The fit minimises the sum of two negative
log-likelihood terms, associated with the $q/p_{\mathrm{T}}$ and $\phi^{\ast}$
distributions, that are computed using the Beeston-Barlow-Lite prescription
[59], which accounts for the finite size of the simulated samples. The sum of
these terms multiplied by a factor of two is denoted as the $\chi^{2}$.

The $\phi^{\ast}$ distribution is modelled including background contributions
from $Z\rightarrow\tau\tau$ and top quarks. The model of the
$q/p_{\mathrm{T}}$ distribution includes the dominant $W\rightarrow\mu\nu$
signal component and several background sources. The largest background, with
a fraction of around $7\times 10^{-2}$, is attributed to
${Z}\rightarrow\mu\mu$, which is simulated with true invariant masses above 20
GeV. The $W\rightarrow\tau\nu$ and hadronic background components each
contribute at the $\mathcal{O}(10^{-2})$ level. A combination of rarer
background sources including $Z\rightarrow\tau\tau$, top quarks, vector boson
pairs, and heavy flavour hadrons gives a total contribution below $10^{-2}$.

The fractions of the $W^{+}$ and $W^{-}$ signal components and the hadronic
background are allowed to vary freely. The $W\rightarrow\tau\nu$ component is
constrained, using the known $\tau\rightarrow\mu\nu\bar{\nu}$ branching
fraction [7], relative to the $W\rightarrow\mu\nu$ component. All other
component fractions are fixed relative to the observed number of $Z$ boson
candidates in the $\phi^{\ast}$ distribution using the fiducial cross-sections
for the corresponding processes relative to that of $Z$ boson production. The
fiducial selections for $W$ and $Z$ boson processes are the same as used in
measurements of the corresponding cross-sections [60]. For all other processes
the fiducial regions correspond to the requirement of a single muon in the
region $p_{\mathrm{T}}>20$ GeV and $2<\eta<4.5$. The measured cross-section
for $Z\rightarrow\mu\mu$ is used [38]. The cross-sections for the rare
background processes are determined using Powheg with the next-to-leading-
order NNPDF3.1 PDF sets.

The shapes of background components arising from the decay of electroweak
bosons are determined from the same models used to describe the signal
component. Systematic variations of the model used to describe $W$ boson
production therefore also simultaneously provide systematic variations
associated with the shapes of background contributions from the decay of
electroweak bosons. The uncertainty in $m_{W}$ from varying the predicted
cross-sections for the rarer background within their uncertainties is
negligible.

The default physics model is based on the simulated samples set out in Sect.
2, fully weighted using a combination of POWHEGPythia and DYTurbo for the QCD
description and a combination of Pythia, Photos and Herwig for the QED
description. The fit is configured to determine the following parameters:

  1. 1.

the value of $m_{W}$,

  2. 2.

the fraction of $W^{+}$ signal,

  3. 3.

the fraction of $W^{-}$ signal,

  4. 4.

the fraction of QCD background,

  5. 5.

the value of $\alpha_{s}$ for the $Z$ boson processes ($\alpha_{s}^{Z}$),

  6. 6.

an independent $\alpha_{s}$ value that is shared for the $W^{+}$ and $W^{-}$
signals ($\alpha_{s}^{W}$),

  7. 7.

a shared $k_{\rm T}^{\rm intr}$ value for all $W$ and $Z$ boson processes,

  8. 8.

and an $A_{3}$ scale factor that is shared by the $W^{+}$ and $W^{-}$ signals.

###  8.1 Data challenge tests

In Sect. 7 it is concluded that POWHEGPythia describes the
$p_{\mathrm{T}}^{Z}$ distribution, in the $p_{\mathrm{T}}^{Z}\leq 30$ GeV
region, better than the other candidate models. It is important to demonstrate
that the fit can reliably determine $m_{W}$ if $W$ boson production is better
described by one or more of the other models. Several pseudodata samples are
prepared in which the underlying Pythia events, without detector simulation,
are weighted to match the default DYTurbo and POWHEGPythia model but with the
$p_{\mathrm{T}}^{V}$ distribution modified to match an alternative model. The
$m_{W}$ fit is configured with a simplified model, without background
components, using a statistically independent sample of the same Pythia events
without detector simulation. Figure 10 shows the resulting $q/p_{\mathrm{T}}$
and $\phi^{*}$ distributions of these pseudodata samples. Variations of up to
five per cent are seen in the shape of the $\phi^{\ast}$ distribution. Within
the fit regions in $q/p_{\mathrm{T}}$ variations of several per cent can be
seen, while the variations exceed $\mathcal{O}(10^{-1})$ in the
high-$p_{\mathrm{T}}$ control region. However, the fit model is able to absorb
these differences in the $\alpha_{s}$ and $k_{\rm T}^{\rm intr}$ nuisance
parameters with variations in the preferred $m_{W}$ value of no more than 10
MeV. Table 3 lists the results of the fits to these pseudodata samples. The
observed variation in $m_{W}$ is consistent with the uncertainty due to
modelling the vector boson transverse momentum distribution in the fit to LHCb
data, as discussed in Sect. 7.

Figure 10: Projections of the (left) $q/p_{\mathrm{T}}$ and (right) $\phi^{*}$
distributions for the challenge datasets. The four dashed vertical lines
indicate the two fit regions in the $q/p_{\mathrm{T}}$ distribution. Table 3:
Fit results with the same default fit model used for the templates but with
different models used for the pseudodata. The POWHEGPythia pseudodata
correspond to $\alpha_{s}=0.125$ and ${k_{\rm T}^{\rm
intr}}=1.8\text{\,Ge\kern-1.00006ptV}$. The Herwig and POWHEGHerwig pseudodata
correspond to $\alpha_{s}=0.136$ and $k_{\rm T}^{\rm intr}$ =1.3 GeV. The
Pythia pseudodata correspond to $\alpha_{s}=0.127$ and ${k_{\rm T}^{\rm
intr}}=2.7\text{\,Ge\kern-1.00006ptV}$. The DYTurbo pseudodata correspond to
$\alpha_{s}=0.118$ and $g=1\text{\,Ge\kern-1.00006ptV}^{2}$. The contributions
to the total $\chi^{2}$ from the $q/p_{\mathrm{T}}$ and $\phi^{*}$
distributions are denoted $\chi^{2}_{W}$ and $\chi^{2}_{Z}$, respectively. The
shift in the $m_{W}$ value with respect to the POWHEGPythia pseudodata is
denoted $\delta m_{W}$. The uncertainties quoted are statistical.

Data config. | $\chi^{2}_{W}$ | $\chi^{2}_{Z}$ |  $\delta m_{W}$ [MeV] | $\alpha_{s}^{Z}$ | $\alpha_{s}^{W}$ |  $A_{3}$ scaling
---|---|---|---|---|---|---
POWHEGPythia | $64.8$ | $34.2$ | – | $0.1246\pm 0.0002$ | $0.1245\pm 0.0003$ | $0.979\pm 0.029$
Herwig | $71.9$ | $600.4$ | $1.6$ | $0.1206\pm 0.0002$ | $0.1218\pm 0.0003$ | $1.001\pm 0.029$
POWHEGHerwig | $64.0$ | $118.6$ | $2.7$ | $0.1206\pm 0.0002$ | $0.1226\pm 0.0003$ | $0.991\pm 0.029$
Pythia, CT09MCS | $71.0$ | $215.8$ | $-2.4$ | $0.1239\pm 0.0002$ | $0.1243\pm 0.0003$ | $0.983\pm 0.029$
Pythia, NNPDF31 | $66.9$ | $156.2$ | $-10.4$ | $0.1225\pm 0.0002$ | $0.1223\pm 0.0003$ | $0.967\pm 0.029$
DYTurbo | $83.0$ | $428.5$ | $4.3$ | $0.1305\pm 0.0001$ | $0.1321\pm 0.0003$ | $0.982\pm 0.028$

###  8.2 Fit results

The fit to the data, with the NNPDF31_nlo_as_0118 PDF set, returns a total
$\chi^{2}$ of 105 for 102 degrees of freedom. Figure 11 compares the
$q/p_{\mathrm{T}}$ and $\phi^{\ast}$ distributions from the data with the fit
model overlaid. The model is in good agreement with the data within the fit
ranges but it underestimates the high-$p_{\mathrm{T}}$ control region of the
$q/p_{\mathrm{T}}$ distribution by up to ten per cent. This underestimation is
within the band of modelling uncertainty, which is dominated by the
high-$p_{\mathrm{T}}^{V}$ parametric correction in that region. The values of
the eight parameters determined from the fit are listed in Table 4. The
$\alpha_{s}$ value for the $W$ boson events is roughly $0.002$ higher than for
the $Z$ boson events. If the fit is configured with a shared $\alpha_{s}$
value for the $W$ and $Z$ boson events the value of $m_{W}$ changes by
$+39\text{\,Me\kern-1.00006ptV}$ but the $\chi^{2}$ is increased by more than
20 units, which strongly favours the configuration with independent
$\alpha_{s}$ values. Furthermore, similar variations between the $\alpha_{s}$
values for $W$ and $Z$ boson events are found in the data challenge tests, as
shown in Table 3. The $A_{3}$ scaling factor is statistically consistent with
unity, which suggests that the $\mathcal{O}(\alpha_{s}^{2})$ predictions from
DYTurbo, with the central scale choices, are compatible with the data.

Table 4: Values of the parameters determined in the $m_{W}$ fit with the NNPDF31_nlo_as_0118 PDF set. The uncertainties quoted are statistical. Parameter | Value |
---|---|---
Fraction of $W^{+}\rightarrow\mu^{+}\nu$ |  $0.5288$ $\pm$ | $0.0006$
Fraction of $W^{-}\rightarrow\mu^{-}\nu$ |  $0.3508$ $\pm$ | $0.0005$
Fraction of hadron background |  $0.0146$ $\pm$ | $0.0007$
$\alpha_{s}^{Z}$ |  $0.1243$ $\pm$ | $0.0004$
$\alpha_{s}^{W}$ |  $0.1263$ $\pm$ | $0.0003$
$k_{\rm T}^{\rm intr}$ |  $1.57$ $\pm$ |  $0.14$ GeV
$A_{3}$ scaling |  $0.975$ $\pm$ | $0.026$
$m_{W}$ | 80362 $\pm$ | 23 MeV

Figure 11: Distributions of (left) $q/p_{\mathrm{T}}$ and (right)
$\phi^{\ast}$ compared to the model after the $m_{W}$ fit.

Figure 12 (left) shows the projection of the $q/p_{\mathrm{T}}$ distribution
in the $Z$ boson sample, where the final state muon is only included if it
satisfies the $W$ boson selection requirements. The model is in good agreement
with the data. Figure 12 (right) shows that the $Z$ boson rapidity
distribution is well described by the model.

Figure 12: Projections of the (left) $q/p_{\mathrm{T}}$ and (right) rapidity
distributions for the $Z$ boson selection. A final state muon is only included
in the $q/p_{\mathrm{T}}$ distribution if it satisfies the $W$ boson selection
requirements.

##  9 Systematic uncertainties and cross-checks

Table 5 lists the PDF uncertainties evaluated for fits based on the NNPDF3.1,
CT18 and MSHT20 PDF sets. The $m_{W}$ values agree within an envelope of
$12\text{\,Me\kern-1.00006ptV}$, which supports the choice to report an
arithmetic average of the three. With respect to the central value obtained
with the NNPDF31 PDFs, the $m_{W}$ values obtained with the CT18 and MSHT20
PDF sets differ by $-12$ MeV and $-11$ MeV, respectively. The uncertainties
are evaluated according to the specific methods for the three groups. The
NNPDF3.1 uncertainty is evaluated as the RMS of $m_{W}$ values according to
100 replicas, whereas the other two sets use fixed numbers of eigenvectors.
The CT18 uncertainty is corrected from 90% confidence level (CL) to 68% CL to
be consistent with all other uncertainties in this analysis. For each PDF set,
the uncertainty from the replica variations is added in quadrature to the
uncertainty from variations in the $\alpha_{s}$ used in the PDF fits. Values
of $0.116<\alpha_{s}<0.120$ are considered, and the uncertainty in $m_{W}$ is
taken as half of the absolute difference between the corresponding shifts in
$m_{W}$ [61].666The variations in $\alpha_{s}$ are larger than the $\pm
0.0015$ variations recommended in Ref. [61] but this choice avoids further ad
hoc scaling. The PDF uncertainty on the arithmetic average of the three
results is taken as the arithmetic average of the three uncertainties, in
accordance with the assumption that the uncertainties are fully correlated.

Table 5: Uncertainties for the NNPDF3.1, CT18 and MSHT20 sets. The contributions from the PDF uncertainty with fixed $\alpha_{s}$ and from the $\alpha_{s}$ variation are quoted separately as is their sum in quadrature, which defines the total uncertainty for each PDF set. Set |  $\sigma_{\rm PDF,base}$ [ MeV ] |  $\sigma_{\text{PDF},\alpha_{s}}$ [ MeV ] |  $\sigma_{\rm PDF}$ [ MeV ]
---|---|---|---
NNPDF3.1 | $8.3$ | $2.4$ | $8.6$
CT18 | $11.5$ | $1.4$ | $11.6$
MSHT20 | $6.5$ | $2.1$ | $6.8$

Table 6 lists each contribution to the systematic uncertainty in the final
result, after averaging results based on the three PDF sets. The systematic
uncertainty is split into three orthogonal components that are combined in
quadrature. The uncertainty due to the description of the parton distribution
functions is $9$ MeV. The remaining theory uncertainty in the modelling of $W$
and $Z$ boson production is $17$ MeV, as described in Sect. 7, with the
largest contribution arising from variations of the transverse momentum model.
The experimental uncertainty is $10$ MeV, with the different contributions
discussed in Sects. 4, 5, and 6.

Table 6: Contributions to the systematic uncertainty in $m_{W}$. Negligible contributions below $1\text{\,Me\kern-1.00006ptV}$ are not listed. Source | Size [ MeV ]
---|---
Parton distribution functions | $9$
Theory (excl. PDFs) total | $17$
Transverse momentum model | $11$
Angular coefficients | $10$
QED FSR model | $7$
Additional electroweak corrections | $5$
Experimental total | $10$
Momentum scale and resolution modelling | $7$
Muon ID, trigger and tracking efficiency | $6$
Isolation efficiency | $4$
QCD background | $2$
Statistical | $23$
Total | $32$

Independently of the systematic uncertainty evaluation, several cross-checks
of the measurement are performed.

  * •

Consistency of orthogonal subsets: The data and simulation are split into
orthogonal subsets by magnet polarity, the product of the muon charge and
polarity, and the $\phi$ and $\eta$ of the muon in the $W$ boson selection.
These results are reported in Table 7. Considering the statistical
uncertainties only, all differences are within, or just outside, two standard
deviations, which was predefined as a criterion for this test.

  * •

Fit range: The minimum and maximum $p_{\mathrm{T}}$ of the fit range in the
$q/p_{\mathrm{T}}$ distribution are varied around their default values of $28$
GeV and $52$ GeV, respectively. The results are reported in Table 8.
Considering the variations in the statistical uncertainties in $m_{W}$ this
test shows that the fit results are stable with respect to variations in the
fit range.

  * •

Fit model freedom: The choice of parameters that are determined in the fit is
varied and the results are reported in Table 9. The default fit determines one
$\alpha_{s}$ parameter for the $Z$ processes and a second that is shared
between $W^{+}$ and $W^{-}$ processes. With three $\alpha_{s}$ parameters
there is only a small change in $m_{W}$ and the fit quality. The default fit
determines a single floating $k_{\rm T}^{\rm intr}$ parameter that is shared
among all three processes. Neither the $m_{W}$ value nor the $\chi^{2}$ are
strongly affected by allowing two (with one shared between the $W^{+}$ and
$W^{-}$ processes) or three $k_{\rm T}^{\rm intr}$ parameters to vary freely.
If the $A_{3}$ scaling factor is fixed to unity the value of $m_{W}$ shifts by
7 MeV and the $\chi^{2}$ increases by a few units. In summary the $m_{W}$ fit
seems to be rather insensitive to all of these variations, except that the
data strongly prefer independent POWHEGPythia tunes for the $W$ and $Z$ boson
production processes.

  * •

Use of NNLO PDF sets: The PDF set used for the analysis is varied from
NNPDF31_nlo_as_0118 to NNPDF31_nnlo_as_0118. The shift in $m_{W}$ is
$1\text{\,Me\kern-1.00006ptV}$.

  * •

Separate $m_{W}$ values for $W^{+}$ and $W^{-}$ bosons: an additional
parameter is included in the fit, allowing for separate values of $m_{W}$ for
$W^{+}$ and $W^{-}$ bosons. This mass difference is found to be consistent
with zero within one standard deviation.

  * •

$W$-like measurement of the $Z$ boson mass: the same methods are applied to
the $Z$ boson sample alone, to perform a $W$-like measurement of the $Z$ boson
mass. The values measured with positive and negatively charged muons agree
within one standard deviation and their average is consistent with the PDG
average [7] within one standard deviation.

Table 7: Fit results where the data and simulation samples are split into two orthogonal subsets. For a given split, the first row is defined as the reference with respect to which the difference in $m_{W}$, denoted by $\delta m_{W}$, is defined. The uncertainties quoted on $\delta m_{W}$ are statistical. Subset |  $\chi^{2}_{\mathrm{tot}}$/ndf |  $\delta m_{W}$ [ MeV ]
---|---|---
Polarity $=-1$ | $92.5/102$ | –
Polarity $=+1$ | $97.3/102$ | $-57.5\pm 45.4$
$\eta>3.3$ | $115.4/102$ | –
$\eta<3.3$ | $85.9/102$ | $+56.9\pm 45.5$
Polarity $\times$ $q=+1$ | $95.9/102$ | –
Polarity $\times$ $q=-1$ | $98.2/102$ | $+16.1\pm 45.4$
$|\phi|>\pi/2$ | $98.8/102$ | –
$|\phi|<\pi/2$ | $115.0/102$ | $+66.7\pm 45.5$
$\phi<0$ | $91.8/102$ | –
$\phi>0$ | $103.0/102$ | $-100.5\pm 45.3$
Table 8: Fit results with variations in the fit range around the default $p_{\mathrm{T}}^{\rm min}=28$ GeV and $p_{\mathrm{T}}^{\rm min}=52$ GeV. The second column lists the $\chi^{2}$ values, the third column lists the shifts in $m_{W}$ with respect to the default fit and the third column lists the statistical uncertainties in $m_{W}$. Change to fit range |  $\chi^{2}_{\mathrm{tot}}$/ndf |  $\delta m_{W}$ [ MeV ] |  $\sigma(m_{W})$ [ MeV ]
---|---|---|---
$p_{\mathrm{T}}^{\rm min}=24$ GeV | $96.5/102$ | $+6.8$ | $19.7$
$p_{\mathrm{T}}^{\rm min}=26$ GeV | $97.7/102$ | $+9.6$ | $20.9$
$p_{\mathrm{T}}^{\rm min}=30$ GeV | $102.7/102$ | $+3.0$ | $25.7$
$p_{\mathrm{T}}^{\rm min}=32$ GeV | $84.9/102$ | $-21.6$ | $30.8$
$p_{\mathrm{T}}^{\rm max}=48$ GeV | $105.3/102$ | $-3.8$ | $23.2$
$p_{\mathrm{T}}^{\rm max}=50$ GeV | $103.0/102$ | $-2.1$ | $23.0$
$p_{\mathrm{T}}^{\rm max}=54$ GeV | $96.3/102$ | $-8.6$ | $22.6$
$p_{\mathrm{T}}^{\rm max}=56$ GeV | $103.7/102$ | $-14.3$ | $22.4$
Table 9: Fit results with variations in which physics parameters are varying freely. Configuration change |  $\chi^{2}_{\mathrm{tot}}$/ndf |  $\delta m_{W}$ [ MeV ] |  $\sigma(m_{W})$ [ MeV ]
---|---|---|---
$2\rightarrow 3$ $\alpha_{s}$ parameters | $103.4/101$ | $-6.0$ | $\pm 23.1$
$2\rightarrow 1$ $\alpha_{s}$ and $1\rightarrow 2$ $k_{\rm T}^{\rm intr}$ parameters | $116.1/102$ | $+13.9$ | $\pm 22.4$
$1\rightarrow 2$ $k_{\rm T}^{\rm intr}$ parameters | $104.0/101$ | $+0.4$ | $\pm 22.7$
$1\rightarrow 3$ $k_{\rm T}^{\rm intr}$ parameters | $102.8/100$ | $-2.7$ | $\pm 22.9$
No $A_{3}$ scaling | $106.0/103$ | $+4.4$ | $\pm 22.2$
Varying QCD background asymmetry | $103.8/101$ | $-0.7$ | $\pm 22.7$
Figure 13: Measured value of $m_{W}$ compared to those from the ALEPH [62],
DELPHI [63], L3 [64], OPAL [65], CDF [10], D0 [11] and ATLAS [12] experiments.
The current prediction of $m_{W}$ from the global electroweak fit is also
included.

##  10 Summary and Conclusion

This paper reports the first measurement of $m_{W}$ with the LHCb experiment.
A data sample of $pp$ collisions at $\sqrt{s}=13\text{\,Te\kern-1.00006ptV}$
corresponding to an integrated luminosity of 1.7 fb-1 is analysed. The
measurement is based on the shape of the $p_{\mathrm{T}}$ distribution of
muons from $W$ boson decays. A simultaneous fit of the $q/p_{\mathrm{T}}$
distribution of $W$ boson decay candidates and of the $\phi^{\ast}$
distribution of $Z$ boson decay candidates is verified to reliably determine
$m_{W}$. This method has reduced sensitivity to the uncertainties in modelling
the $W$ boson transverse momentum distribution compared to previous
determinations of $m_{W}$ at hadron colliders. The following results are
obtained

$\displaystyle m_{W}$ $\displaystyle=80362\pm 23_{\rm stat}\pm 10_{\rm exp}\pm
17_{\rm theory}\pm 9_{\rm PDF}\text{\,Me\kern-1.00006ptV},$ $\displaystyle
m_{W}$ $\displaystyle=80350\pm 23_{\rm stat}\pm 10_{\rm exp}\pm 17_{\rm
theory}\pm 12_{\rm PDF}\text{\,Me\kern-1.00006ptV},$ $\displaystyle m_{W}$
$\displaystyle=80351\pm 23_{\rm stat}\pm 10_{\rm exp}\pm 17_{\rm theory}\pm
7_{\rm PDF}\text{\,Me\kern-1.00006ptV},$

with the NNPDF3.1, CT18 and MSHT20 PDF sets, respectively. The first
uncertainty is statistical, the second is due to experimental systematic
uncertainties, and the third and fourth are due to uncertainties in the
theoretical modelling and the description of the PDFs, respectively. Treating
the three PDF sets equally results in the following arithmetic average

$m_{W}=80354\pm 23_{\rm stat}\pm 10_{\rm exp}\pm 17_{\rm theory}\pm 9_{\rm
PDF}\text{\,Me\kern-1.00006ptV}.$

This result agrees with the current PDG average of direct measurements [7] and
the indirect prediction from the global EW fit [6], and is compared to
previous measurements in Fig. 13. This measurement also serves as a first
proof-of-principle of a measurement of $m_{W}$ with the LHCb experiment. In
Ref. [66] it was demonstrated that the PDF uncertainty in a measurement of
$m_{W}$ by LHCb can be strongly reduced by using in situ constraints and by
fitting the doubly differential distribution of $p_{\mathrm{T}}$ and $\eta$,
similar to the measurement by the CMS Collaboration [67], instead of the
singly differential $p_{\mathrm{T}}$ distribution. An approximately three
times larger data sample is already available for analysis but particular
attention should be paid to reducing the dominant source of systematic
uncertainty, which is the modelling of $W$ boson production.

## Acknowledgements

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

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LHCb collaboration

R. Aaij32, A.S.W. Abdelmotteleb56, C. Abellán Beteta50, T. Ackernley60, B.
Adeva46, M. Adinolfi54, H. Afsharnia9, C. Agapopoulou13, C.A. Aidala86, S.
Aiola25, Z. Ajaltouni9, S. Akar65, J. Albrecht15, F. Alessio48, M.
Alexander59, A. Alfonso Albero45, Z. Aliouche62, G. Alkhazov38, P. Alvarez
Cartelle55, S. Amato2, J.L. Amey54, Y. Amhis11, L. An48, L. Anderlini22, A.
Andreianov38, M. Andreotti21, F. Archilli17, A. Artamonov44, M. Artuso68, K.
Arzymatov42, E. Aslanides10, M. Atzeni50, B. Audurier12, S. Bachmann17, M.
Bachmayer49, J.J. Back56, P. Baladron Rodriguez46, V. Balagura12, W.
Baldini21, J. Baptista Leite1, M. Barbetti22, R.J. Barlow62, S. Barsuk11, W.
Barter61, M. Bartolini24, F. Baryshnikov83, J.M. Basels14, S. Bashir34, G.
Bassi29, B. Batsukh68, A. Battig15, A. Bay49, A. Beck56, M. Becker15, F.
Bedeschi29, I. Bediaga1, A. Beiter68, V. Belavin42, S. Belin27, V. Bellee50,
K. Belous44, I. Belov40, I. Belyaev41, G. Bencivenni23, E. Ben-Haim13, A.
Berezhnoy40, R. Bernet50, D. Berninghoff17, H.C. Bernstein68, C. Bertella48,
A. Bertolin28, C. Betancourt50, F. Betti48, Ia. Bezshyiko50, S. Bhasin54, J.
Bhom35, L. Bian73, M.S. Bieker15, S. Bifani53, P. Billoir13, M. Birch61,
F.C.R. Bishop55, A. Bitadze62, A. Bizzeti22,k, M. Bjørn63, M.P. Blago48, T.
Blake56, F. Blanc49, S. Blusk68, D. Bobulska59, J.A. Boelhauve15, O. Boente
Garcia46, T. Boettcher65, A. Boldyrev82, A. Bondar43, N. Bondar38,48, S.
Borghi62, M. Borisyak42, M. Borsato17, J.T. Borsuk35, S.A. Bouchiba49, T.J.V.
Bowcock60, A. Boyer48, C. Bozzi21, M.J. Bradley61, S. Braun66, A. Brea
Rodriguez46, M. Brodski48, J. Brodzicka35, A. Brossa Gonzalo56, D. Brundu27,
A. Buonaura50, L. Buonincontri28, A.T. Burke62, C. Burr48, A. Bursche72, A.
Butkevich39, J.S. Butter32, J. Buytaert48, W. Byczynski48, S. Cadeddu27, H.
Cai73, R. Calabrese21,f, L. Calefice15,13, L. Calero Diaz23, S. Cali23, R.
Calladine53, M. Calvi26,j, M. Calvo Gomez85, P. Camargo Magalhaes54, P.
Campana23, A.F. Campoverde Quezada6, S. Capelli26,j, L. Capriotti20,d, A.
Carbone20,d, G. Carboni31, R. Cardinale24, A. Cardini27, I. Carli4, P.
Carniti26,j, L. Carus14, K. Carvalho Akiba32, A. Casais Vidal46, G. Casse60,
M. Cattaneo48, G. Cavallero48, S. Celani49, J. Cerasoli10, D. Cervenkov63,
A.J. Chadwick60, M.G. Chapman54, M. Charles13, Ph. Charpentier48, G.
Chatzikonstantinidis53, C.A. Chavez Barajas60, M. Chefdeville8, C. Chen3, S.
Chen4, A. Chernov35, V. Chobanova46, S. Cholak49, M. Chrzaszcz35, A.
Chubykin38, V. Chulikov38, P. Ciambrone23, M.F. Cicala56, X. Cid Vidal46, G.
Ciezarek48, P.E.L. Clarke58, M. Clemencic48, H.V. Cliff55, J. Closier48, J.L.
Cobbledick62, V. Coco48, J.A.B. Coelho11, J. Cogan10, E. Cogneras9, L.
Cojocariu37, P. Collins48, T. Colombo48, L. Congedo19,c, A. Contu27, N.
Cooke53, G. Coombs59, I. Corredoira 46, G. Corti48, C.M. Costa Sobral56, B.
Couturier48, D.C. Craik64, J. Crkovská67, M. Cruz Torres1, R. Currie58, C.L.
Da Silva67, S. Dadabaev83, L. Dai71, E. Dall’Occo15, J. Dalseno46, C.
D’Ambrosio48, A. Danilina41, P. d’Argent48, J.E. Davies62, A. Davis62, O. De
Aguiar Francisco62, K. De Bruyn79, S. De Capua62, M. De Cian49, J.M. De
Miranda1, L. De Paula2, M. De Serio19,c, D. De Simone50, P. De Simone23, J.A.
de Vries80, C.T. Dean67, D. Decamp8, V. Dedu10, L. Del Buono13, B. Delaney55,
H.-P. Dembinski15, A. Dendek34, V. Denysenko50, D. Derkach82, O. Deschamps9,
F. Desse11, F. Dettori27,e, B. Dey77, A. Di Cicco23, P. Di Nezza23, S.
Didenko83, L. Dieste Maronas46, H. Dijkstra48, V. Dobishuk52, C. Dong3, A.M.
Donohoe18, F. Dordei27, A.C. dos Reis1, L. Douglas59, A. Dovbnya51, A.G.
Downes8, M.W. Dudek35, L. Dufour48, V. Duk78, P. Durante48, J.M. Durham67, D.
Dutta62, A. Dziurda35, A. Dzyuba38, S. Easo57, U. Egede69, V. Egorychev41, S.
Eidelman43,v, S. Eisenhardt58, S. Ek-In49, L. Eklund59,w, S. Ely68, A. Ene37,
E. Epple67, S. Escher14, J. Eschle50, S. Esen13, T. Evans48, A. Falabella20,
J. Fan3, Y. Fan6, B. Fang73, S. Farry60, D. Fazzini26,j, M. Féo48, A.
Fernandez Prieto46, J.M. Fernandez-tenllado Arribas45, A.D. Fernez66, F.
Ferrari20,d, L. Ferreira Lopes49, F. Ferreira Rodrigues2, S. Ferreres Sole32,
M. Ferrillo50, M. Ferro-Luzzi48, S. Filippov39, R.A. Fini19, M. Fiorini21,f,
M. Firlej34, K.M. Fischer63, D.S. Fitzgerald86, C. Fitzpatrick62, T.
Fiutowski34, A. Fkiaras48, F. Fleuret12, M. Fontana13, F. Fontanelli24,h, R.
Forty48, D. Foulds-Holt55, V. Franco Lima60, M. Franco Sevilla66, M. Frank48,
E. Franzoso21, G. Frau17, C. Frei48, D.A. Friday59, J. Fu25, Q. Fuehring15, E.
Gabriel32, A. Gallas Torreira46, D. Galli20,d, S. Gambetta58,48, Y. Gan3, M.
Gandelman2, P. Gandini25, Y. Gao5, M. Garau27, L.M. Garcia Martin56, P. Garcia
Moreno45, J. García Pardiñas26,j, B. Garcia Plana46, F.A. Garcia Rosales12, L.
Garrido45, C. Gaspar48, R.E. Geertsema32, D. Gerick17, L.L. Gerken15, E.
Gersabeck62, M. Gersabeck62, T. Gershon56, D. Gerstel10, Ph. Ghez8, L.
Giambastiani28, V. Gibson55, H.K. Giemza36, A.L. Gilman63, M. Giovannetti23,p,
A. Gioventù46, P. Gironella Gironell45, L. Giubega37, C. Giugliano21,f,48, K.
Gizdov58, E.L. Gkougkousis48, V.V. Gligorov13, C. Göbel70, E. Golobardes85, D.
Golubkov41, A. Golutvin61,83, A. Gomes1,a, S. Gomez Fernandez45, F. Goncalves
Abrantes63, M. Goncerz35, G. Gong3, P. Gorbounov41, I.V. Gorelov40, C.
Gotti26, E. Govorkova48, J.P. Grabowski17, T. Grammatico13, L.A. Granado
Cardoso48, E. Graugés45, E. Graverini49, G. Graziani22, A. Grecu37, L.M.
Greeven32, N.A. Grieser4, L. Grillo62, S. Gromov83, B.R. Gruberg Cazon63, C.
Gu3, M. Guarise21, M. Guittiere11, P. A. Günther17, E. Gushchin39, A. Guth14,
Y. Guz44, T. Gys48, T. Hadavizadeh69, G. Haefeli49, C. Haen48, J.
Haimberger48, T. Halewood-leagas60, P.M. Hamilton66, J.P. Hammerich60, Q.
Han7, X. Han17, T.H. Hancock63, S. Hansmann-Menzemer17, N. Harnew63, T.
Harrison60, C. Hasse48, M. Hatch48, J. He6,b, M. Hecker61, K. Heijhoff32, K.
Heinicke15, A.M. Hennequin48, K. Hennessy60, L. Henry48, J. Heuel14, A.
Hicheur2, D. Hill49, M. Hilton62, S.E. Hollitt15, R. Hou7, Y. Hou6, J. Hu17,
J. Hu72, W. Hu7, X. Hu3, W. Huang6, X. Huang73, W. Hulsbergen32, R.J.
Hunter56, M. Hushchyn82, D. Hutchcroft60, D. Hynds32, P. Ibis15, M. Idzik34,
D. Ilin38, P. Ilten65, A. Inglessi38, A. Ishteev83, K. Ivshin38, R.
Jacobsson48, H. Jage14, S. Jakobsen48, E. Jans32, B.K. Jashal47, A.
Jawahery66, V. Jevtic15, M. Jezabek35, F. Jiang3, M. John63, D. Johnson48,
C.R. Jones55, T.P. Jones56, B. Jost48, N. Jurik48, S.H. Kalavan Kadavath34, S.
Kandybei51, Y. Kang3, M. Karacson48, M. Karpov82, F. Keizer48, D.M. Keller68,
M. Kenzie56, T. Ketel33, B. Khanji15, A. Kharisova84, S. Kholodenko44, T.
Kirn14, V.S. Kirsebom49, O. Kitouni64, S. Klaver32, N. Kleijne29, K.
Klimaszewski36, M.R. Kmiec36, S. Koliiev52, A. Kondybayeva83, A.
Konoplyannikov41, P. Kopciewicz34, R. Kopecna17, P. Koppenburg32, M.
Korolev40, I. Kostiuk32,52, O. Kot52, S. Kotriakhova21,38, P. Kravchenko38, L.
Kravchuk39, R.D. Krawczyk48, M. Kreps56, F. Kress61, S. Kretzschmar14, P.
Krokovny43,v, W. Krupa34, W. Krzemien36, W. Kucewicz35,t, M. Kucharczyk35, V.
Kudryavtsev43,v, H.S. Kuindersma32,33, G.J. Kunde67, T. Kvaratskheliya41, D.
Lacarrere48, G. Lafferty62, A. Lai27, A. Lampis27, D. Lancierini50, J.J.
Lane62, R. Lane54, G. Lanfranchi23, C. Langenbruch14, J. Langer15, O.
Lantwin83, T. Latham56, F. Lazzari29,q, R. Le Gac10, S.H. Lee86, R. Lefèvre9,
A. Leflat40, S. Legotin83, O. Leroy10, T. Lesiak35, B. Leverington17, H. Li72,
P. Li17, S. Li7, Y. Li4, Y. Li4, Z. Li68, X. Liang68, T. Lin61, R. Lindner48,
V. Lisovskyi15, R. Litvinov27, G. Liu72, H. Liu6, S. Liu4, A. Lobo Salvia45,
A. Loi27, J. Lomba Castro46, I. Longstaff59, J.H. Lopes2, S. Lopez Solino46,
G.H. Lovell55, Y. Lu4, C. Lucarelli22, D. Lucchesi28,l, S. Luchuk39, M. Lucio
Martinez32, V. Lukashenko32, Y. Luo3, A. Lupato62, E. Luppi21,f, O. Lupton56,
A. Lusiani29,m, X. Lyu6, L. Ma4, R. Ma6, S. Maccolini20,d, F. Machefert11, F.
Maciuc37, V. Macko49, P. Mackowiak15, S. Maddrell-Mander54, O. Madejczykt,
L.R. Madhan Mohan54, O. Maev38, A. Maevskiy82, D. Maisuzenko38, M.W.
Majewskit, J.J. Malczewski35, S. Malde63, B. Malecki48, A. Malinin81, T.
Maltsev43,v, H. Malygina17, G. Manca27,e, G. Mancinelli10, D. Manuzzi20,d, D.
Marangotto25,i, J. Maratas9,s, J.F. Marchand8, U. Marconi20, S. Mariani22,g,
C. Marin Benito48, M. Marinangeli49, J. Marks17, A.M. Marshall54, P.J.
Marshall60, G. Martelli78, G. Martellotti30, L. Martinazzoli48,j, M.
Martinelli26,j, D. Martinez Santos46, F. Martinez Vidal47, A. Massafferri1, M.
Materok14, R. Matev48, A. Mathad50, Z. Mathe48, V. Matiunin41, C. Matteuzzi26,
K.R. Mattioli86, A. Mauri32, E. Maurice12, J. Mauricio45, M. Mazurek48, M.
McCann61, L. Mcconnell18, T.H. Mcgrath62, N.T. Mchugh59, A. McNab62, R.
McNulty18, J.V. Mead60, B. Meadows65, G. Meier15, N. Meinert76, D.
Melnychuk36, S. Meloni26,j, M. Merk32,80, A. Merli25, L. Meyer Garcia2, M.
Mikhasenko48, D.A. Milanes74, E. Millard56, M. Milovanovic48, M.-N. Minard8,
A. Minotti26,j, L. Minzoni21,f, S.E. Mitchell58, B. Mitreska62, D.S. Mitzel48,
A. Mödden 15, R.A. Mohammed63, R.D. Moise61, T. Mombächer46, I.A. Monroy74, S.
Monteil9, M. Morandin28, G. Morello23, M.J. Morello29,m, J. Moron34, A.B.
Morris75, A.G. Morris56, R. Mountain68, H. Mu3, F. Muheim58,48, M. Mulder48,
D. Müller48, K. Müller50, C.H. Murphy63, D. Murray62, P. Muzzetto27,48, P.
Naik54, T. Nakada49, R. Nandakumar57, T. Nanut49, I. Nasteva2, M. Needham58,
I. Neri21, N. Neri25,i, S. Neubert75, N. Neufeld48, R. Newcombe61, T.D.
Nguyen49, C. Nguyen-Mau49,x, E.M. Niel11, S. Nieswand14, N. Nikitin40, N.S.
Nolte64, C. Normand8, C. Nunez86, A. Oblakowska-Mucha34, V. Obraztsov44, T.
Oeser14, D.P. O’Hanlon54, S. Okamura21, R. Oldeman27,e, F. Oliva58, M.E.
Olivares68, C.J.G. Onderwater79, R.H. O’neil58, A. Ossowska35, J.M. Otalora
Goicochea2, T. Ovsiannikova41, P. Owen50, A. Oyanguren47, K.O. Padeken75, B.
Pagare56, P.R. Pais48, T. Pajero63, A. Palano19, M. Palutan23, Y. Pan62, G.
Panshin84, A. Papanestis57, M. Pappagallo19,c, L.L. Pappalardo21,f, C.
Pappenheimer65, W. Parker66, C. Parkes62, B. Passalacqua21, G. Passaleva22, A.
Pastore19, M. Patel61, C. Patrignani20,d, C.J. Pawley80, A. Pearce48, A.
Pellegrino32, M. Pepe Altarelli48, S. Perazzini20, D. Pereima41, A. Pereiro
Castro46, P. Perret9, I. Petrenko52, M. Petric59,48, K. Petridis54, A.
Petrolini24,h, A. Petrov81, S. Petrucci58, M. Petruzzo25, T.T.H. Pham68, L.
Pica29,m, M. Piccini78, B. Pietrzyk8, G. Pietrzyk49, M. Pili63, D. Pinci30, F.
Pisani48, M. Pizzichemi48, Resmi P.K10, V. Placinta37, J. Plews53, M. Plo
Casasus46, F. Polci13, M. Poli Lener23, M. Poliakova68, A. Poluektov10, N.
Polukhina83,u, I. Polyakov68, E. Polycarpo2, S. Ponce48, D. Popov6,48, S.
Popov42, S. Poslavskii44, K. Prasanth35, L. Promberger48, C. Prouve46, V.
Pugatch52, V. Puill11, H. Pullen63, G. Punzi29,n, H. Qi3, W. Qian6, J. Qin6,
N. Qin3, R. Quagliani13,54, B. Quintana8, N.V. Raab18, R.I. Rabadan Trejo6, B.
Rachwal34, J.H. Rademacker54, M. Rama29, M. Ramos Pernas56, M.S. Rangel2, F.
Ratnikov42,82, G. Raven33, M. Reboud8, F. Redi49, F. Reiss62, C. Remon
Alepuz47, Z. Ren3, V. Renaudin63, R. Ribatti29, S. Ricciardi57, K. Rinnert60,
P. Robbe11, G. Robertson58, A.B. Rodrigues49, E. Rodrigues60, J.A. Rodriguez
Lopez74, E. Rodriguez Rodriguez46, A. Rollings63, P. Roloff48, V.
Romanovskiy44, M. Romero Lamas46, A. Romero Vidal46, J.D. Roth86, M.
Rotondo23, M.S. Rudolph68, T. Ruf48, R.A. Ruiz Fernandez46, J. Ruiz Vidal47,
A. Ryzhikov82, J. Ryzka34, J.J. Saborido Silva46, N. Sagidova38, N. Sahoo56,
B. Saitta27,e, M. Salomoni48, D. Sanchez Gonzalo45, C. Sanchez Gras32, R.
Santacesaria30, C. Santamarina Rios46, M. Santimaria23, E. Santovetti31,p, D.
Saranin83, G. Sarpis59, M. Sarpis75, A. Sarti30, C. Satriano30,o, A. Satta31,
M. Saur15, D. Savrina41,40, H. Sazak9, L.G. Scantlebury Smead63, A.
Scarabotto13, S. Schael14, S. Scherl60, M. Schiller59, H. Schindler48, M.
Schmelling16, B. Schmidt48, S. Schmitt14, O. Schneider49, A. Schopper48, M.
Schubiger32, S. Schulte49, M.H. Schune11, R. Schwemmer48, B. Sciascia23,48, S.
Sellam46, A. Semennikov41, M. Senghi Soares33, A. Sergi24, N. Serra50, L.
Sestini28, A. Seuthe15, Y. Shang5, D.M. Shangase86, M. Shapkin44, I.
Shchemerov83, L. Shchutska49, T. Shears60, L. Shekhtman43,v, Z. Shen5, V.
Shevchenko81, E.B. Shields26,j, Y. Shimizu11, E. Shmanin83, J.D. Shupperd68,
B.G. Siddi21, R. Silva Coutinho50, G. Simi28, S. Simone19,c, N. Skidmore62, T.
Skwarnicki68, M.W. Slater53, I. Slazyk21,f, J.C. Smallwood63, J.G. Smeaton55,
A. Smetkina41, E. Smith50, M. Smith61, A. Snoch32, M. Soares20, L. Soares
Lavra9, M.D. Sokoloff65, F.J.P. Soler59, A. Solovev38, I. Solovyev38, F.L.
Souza De Almeida2, B. Souza De Paula2, B. Spaan15, E. Spadaro Norella25, P.
Spradlin59, F. Stagni48, M. Stahl65, S. Stahl48, S. Stanislaus63, O.
Steinkamp50,83, O. Stenyakin44, H. Stevens15, S. Stone68, M. Straticiuc37, D.
Strekalina83, F. Suljik63, J. Sun27, L. Sun73, Y. Sun66, P. Svihra62, P.N.
Swallow53, K. Swientek34, A. Szabelski36, T. Szumlak34, M. Szymanski48, S.
Taneja62, A.R. Tanner54, M.D. Tat63, A. Terentev83, F. Teubert48, E. Thomas48,
D.J.D. Thompson53, K.A. Thomson60, V. Tisserand9, S. T’Jampens8, M. Tobin4, L.
Tomassetti21,f, X. Tong5, D. Torres Machado1, D.Y. Tou13, M.T. Tran49, E.
Trifonova83, C. Trippl49, G. Tuci29,n, A. Tully49, N. Tuning32,48, A.
Ukleja36, D.J. Unverzagt17, E. Ursov83, A. Usachov32, A. Ustyuzhanin42,82, U.
Uwer17, A. Vagner84, V. Vagnoni20, A. Valassi48, G. Valenti20, N. Valls
Canudas85, M. van Beuzekom32, M. Van Dijk49, E. van Herwijnen83, C.B. Van
Hulse18, M. van Veghel79, R. Vazquez Gomez46, P. Vazquez Regueiro46, C.
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P. Vincent13, F.C. Volle11, D. Vom Bruch10, A. Vorobyev38, V. Vorobyev43,v, N.
Voropaev38, K. Vos80, R. Waldi17, J. Walsh29, C. Wang17, J. Wang5, J. Wang4,
J. Wang3, J. Wang73, M. Wang3, R. Wang54, Y. Wang7, Z. Wang50, Z. Wang3, Z.
Wang6, J.A. Ward56, N.K. Watson53, S.G. Weber13, D. Websdale61, C. Weisser64,
B.D.C. Westhenry54, D.J. White62, M. Whitehead54, A.R. Wiederhold56, D.
Wiedner15, G. Wilkinson63, M. Wilkinson68, I. Williams55, M. Williams64,
M.R.J. Williams58, F.F. Wilson57, W. Wislicki36, M. Witek35, L. Witola17, G.
Wormser11, S.A. Wotton55, H. Wu68, K. Wyllie48, Z. Xiang6, D. Xiao7, Y. Xie7,
A. Xu5, J. Xu6, L. Xu3, M. Xu7, Q. Xu6, Z. Xu5, Z. Xu6, D. Yang3, S. Yang6, Y.
Yang6, Z. Yang5, Z. Yang66, Y. Yao68, L.E. Yeomans60, H. Yin7, J. Yu71, X.
Yuan68, O. Yushchenko44, E. Zaffaroni49, M. Zavertyaev16,u, M. Zdybal35, O.
Zenaiev48, M. Zeng3, D. Zhang7, L. Zhang3, S. Zhang71, S. Zhang5, Y. Zhang5,
Y. Zhang63, A. Zharkova83, A. Zhelezov17, Y. Zheng6, T. Zhou5, X. Zhou6, Y.
Zhou6, V. Zhovkovska11, X. Zhu3, X. Zhu7, Z. Zhu6, V. Zhukov14,40, J.B.
Zonneveld58, Q. Zou4, S. Zucchelli20,d, D. Zuliani28, G. Zunica62.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4Institute Of High Energy Physics (IHEP), Beijing, China
5School of Physics State Key Laboratory of Nuclear Physics and Technology,
Peking University, Beijing, China
6University of Chinese Academy of Sciences, Beijing, China
7Institute of Particle Physics, Central China Normal University, Wuhan, Hubei,
China
8Univ. 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
12Laboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole Polytechnique, Institut
Polytechnique de Paris, Palaiseau, France
13LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3,
Paris, France
14I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
15Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
16Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
17Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
18School of Physics, University College Dublin, Dublin, Ireland
19INFN Sezione di Bari, Bari, Italy
20INFN Sezione di Bologna, Bologna, Italy
21INFN Sezione di Ferrara, Ferrara, Italy
22INFN Sezione di Firenze, Firenze, Italy
23INFN Laboratori Nazionali di Frascati, Frascati, Italy
24INFN Sezione di Genova, Genova, Italy
25INFN Sezione di Milano, Milano, Italy
26INFN Sezione di Milano-Bicocca, Milano, Italy
27INFN Sezione di Cagliari, Monserrato, Italy
28Universita degli Studi di Padova, Universita e INFN, Padova, Padova, Italy
29INFN Sezione di Pisa, Pisa, Italy
30INFN Sezione di Roma La Sapienza, Roma, Italy
31INFN Sezione di Roma Tor Vergata, Roma, Italy
32Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
33Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, Netherlands
34AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
35Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
36National Center for Nuclear Research (NCBJ), Warsaw, Poland
37Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
38Petersburg Nuclear Physics Institute NRC Kurchatov Institute (PNPI NRC KI),
Gatchina, Russia
39Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
40Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
41Institute of Theoretical and Experimental Physics NRC Kurchatov Institute
(ITEP NRC KI), Moscow, Russia
42Yandex School of Data Analysis, Moscow, Russia
43Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
44Institute for High Energy Physics NRC Kurchatov Institute (IHEP NRC KI),
Protvino, Russia, Protvino, Russia
45ICCUB, Universitat de Barcelona, Barcelona, Spain
46Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
47Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain
48European Organization for Nuclear Research (CERN), Geneva, Switzerland
49Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
50Physik-Institut, Universität Zürich, Zürich, Switzerland
51NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
52Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
53University of Birmingham, Birmingham, United Kingdom
54H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
55Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
56Department of Physics, University of Warwick, Coventry, United Kingdom
57STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
58School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
59School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
60Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
61Imperial College London, London, United Kingdom
62Department of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
63Department of Physics, University of Oxford, Oxford, United Kingdom
64Massachusetts Institute of Technology, Cambridge, MA, United States
65University of Cincinnati, Cincinnati, OH, United States
66University of Maryland, College Park, MD, United States
67Los Alamos National Laboratory (LANL), Los Alamos, United States
68Syracuse University, Syracuse, NY, United States
69School of Physics and Astronomy, Monash University, Melbourne, Australia,
associated to 56
70Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
71Physics and Micro Electronic College, Hunan University, Changsha City,
China, associated to 7
72Guangdong Provincial Key Laboratory of Nuclear Science, Guangdong-Hong Kong
Joint Laboratory of Quantum Matter, Institute of Quantum Matter, South China
Normal University, Guangzhou, China, associated to 3
73School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
74Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 13
75Universität Bonn - Helmholtz-Institut für Strahlen und Kernphysik, Bonn,
Germany, associated to 17
76Institut für Physik, Universität Rostock, Rostock, Germany, associated to 17
77Eotvos Lorand University, Budapest, Hungary, associated to 48
78INFN Sezione di Perugia, Perugia, Italy, associated to 21
79Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 32
80Universiteit Maastricht, Maastricht, Netherlands, associated to 32
81National Research Centre Kurchatov Institute, Moscow, Russia, associated to
41
82National Research University Higher School of Economics, Moscow, Russia,
associated to 42
83National University of Science and Technology “MISIS”, Moscow, Russia,
associated to 41
84National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
41
85DS4DS, La Salle, Universitat Ramon Llull, Barcelona, Spain, associated to 45
86University of Michigan, Ann Arbor, United States, associated to 68

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bHangzhou Institute for Advanced Study, UCAS, Hangzhou, China
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, Italy
hUniversità di Genova, Genova, Italy
iUniversità degli Studi di Milano, Milano, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Modena e Reggio Emilia, Modena, Italy
lUniversità di Padova, Padova, Italy
mScuola Normale Superiore, Pisa, Italy
nUniversità di Pisa, Pisa, Italy
oUniversità della Basilicata, Potenza, Italy
pUniversità di Roma Tor Vergata, Roma, Italy
qUniversità di Siena, Siena, Italy
rUniversità di Urbino, Urbino, Italy
sMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
tAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
uP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
vNovosibirsk State University, Novosibirsk, Russia
wDepartment of Physics and Astronomy, Uppsala University, Uppsala, Sweden
xHanoi University of Science, Hanoi, Vietnam