11institutetext: DESY, Hamburg, Germany 22institutetext: CEA/IRFU, Gif-sur-
Yvette, France 33institutetext: University of Edinburgh, Edinburgh, UK
44institutetext: Massachusetts Institute of Technology, Cambridge,
Massachusetts, USA 55institutetext: University of Liverpool, Liverpool UK
66institutetext: University College Dublin, Dublin, Ireland 77institutetext:
University of Oxford, Oxford, UK 88institutetext: University of Warwick,
Warwick, UK 99institutetext: University of Milan, Milan, Italy
1010institutetext: Johannes Gutenberg University, Mainz, Germany

# Compatibility and combination of world $W$-boson mass measurements

S. Amoroso 11 N. Andari 22 W. Barter 33 J. Bendavid 44 M. Boonekamp 22 S.
Farry 55 M. Grünewald 66 C. Hays 77 R. Hunter 88 J. Kretzschmar 55 O. Lupton
88 M. Pili 77 M. Ramos Pernas 88 B. Tuchming 22 M. Vesterinen 88 A. Vicini 99
C. Wang 1010 M. Xu (LHC-TeV MW Working Group) 88

###### Abstract

The compatibility of $W$-boson mass measurements performed by the ATLAS, LHCb,
CDF, and D0 experiments is studied using a coherent framework with theory
uncertainty correlations. The measurements are combined using a number of
recent sets of parton distribution functions (PDF), and are further combined
with the average value of measurements from the Large Electron-Positron
collider. The considered PDF sets generally have a low compatibility with a
suite of global rapidity-sensitive Drell-Yan measurements. The most compatible
set is CT18 due to its larger uncertainties. A combination of all $m_{W}$
measurements yields a value of $m_{W}=80394.6\pm 11.5$ MeV with the CT18 set,
but has a probability of compatibility of 0.5% and is therefore disfavoured.
Combinations are performed removing each measurement individually, and a 91%
probability of compatibility is obtained when the CDF measurement is removed.
The corresponding value of the $W$ boson mass is $80369.2\pm 13.3$ MeV, which
differs by $3.6\sigma$ from the CDF value determined using the same PDF set.

###### pacs:

14.70.FmProperties of specific particles, W bosons

##  1 Introduction

The $W$-boson mass ($m_{W}$) is an important parameter of the Standard Model
(SM) of particle physics, providing a sensitive test of the model’s
consistency and offering a window to potential new processes. An active
program of measurements at the Tevatron and Large Hadron Collider (LHC)
continues to improve the experimental precision of $m_{W}$, which is
approaching the uncertainty on the SM prediction. Previous measurements from
the Large Electron Positron collider (LEP) together have a precision
comparable to the individual hadron-collider measurements. A combination of
the Tevatron, LHC, and LEP measurements can thus improve the precision on
$m_{W}$ and quantify the compatibility of the measurements. Such a
compatibility study is particularly motivated in light of the discrepancy
between the most recent measurement CDF:2022hxs  from the CDF experiment at
the Tevatron and previous measurements Abazov:2009cp ; D0:2013jba ;
LHCb:2021bjt ; Aaboud:2017svj  from the D0 experiment at the Tevatron, and the
LHCb and ATLAS experiments at the LHC.

At hadron colliders, measurements of $m_{W}$ exploit the kinematic peaks of
distributions observed in leptonic $W$-boson decays. These final-state
distributions carry information about the decaying particle mass, but also
depend on other $W$-boson degrees of freedom such as the $W$-boson rapidity,
transverse momentum, and polarization. Predictions of these distributions are
generally obtained using Monte Carlo (MC) event generators with input parton
distribution functions (PDF). Past measurements have used different generators
and PDF sets, so prior to combining the measurements a coherent treatment is
required to compare measurements and obtain uncertainty correlations. Where
appropriate, small adjustments are thus applied to the measured values or
uncertainties. These adjustments are estimated using a fast detector
simulation developed for this purpose, or using the simulation from the
experimental measurement.

The presentation of the combination begins with an overview of the individual
measurements in Sec. 2, followed by a description of the methods in Sec. 3.
The theoretical treatment of the $W$-boson production and decay is provided in
Sec. 4, along with uncertainties, correlations, and any adjustments to the
measurements. The results of the combination are presented in Sec. 5.2, and
conclusions are given in Sec. 6.

##  2 Overview of the measurements

The combination uses the latest measurements from D0 and CDF at the Tevatron
and ATLAS and LHCb at the LHC. The CMS Collaboration has not yet measured
$m_{W}$, though it has measured differential $W$-boson cross sections on the
path to the measurement CMSW . Prior measurements from the Tevatron and the
CERN Super Proton Synchroton are not included as they are expected to have
negligible impact. The hadron-collider measurements are combined with the
result from the Large Electron Positron collider (LEP) ALEPH:2013dgf ,
$m_{W}=80.376\pm 0.033$ GeV111We use the convention $c\equiv 1$ and work in a
right-handed coordinate system with the origin at the centre of the detector
and the $z$-axis along the beam pipe. Cylindrical coordinates $(r,\phi)$ are
used in the transverse plane, where $\phi$ is the azimuthal angle around the
z-axis. The pseudorapidity is defined in terms of the polar angle $\theta$ as
$\eta=-\ln\tan(\theta/2)$. Transverse momentum is defined as
$p_{\scriptsize\textrm{T}}=p\sin\theta$. .

The kinematic observables used in $m_{W}$ measurements at hadron colliders are
the momentum of the charged lepton from the $W$-boson decay
($p^{\ell}_{\scriptsize\textrm{T}}$) and the recoil transverse momentum
($u_{\scriptsize\mathrm{T}}$) balancing the transverse momentum of the $W$
boson ($p_{\scriptsize\textrm{T}}^{W}$). The recoil is measured by vectorially
summing the momentum of all objects interacting in the detector, except for
the charged lepton. The neutrino momentum is inferred from the net momentum
imbalance,
$\vec{p}^{\nu}_{\scriptsize\mathrm{T}}\equiv-(\vec{p}^{\ell}_{\scriptsize\mathrm{T}}+\vec{u}_{\scriptsize\mathrm{T}})$.
For experiments with sufficiently good recoil resolution the most sensitive
kinematic distribution is the transverse mass,
$m_{\scriptsize\textrm{T}}=\sqrt{2p_{\scriptsize\textrm{T}}^{\ell}p_{\scriptsize\textrm{T}}^{\nu}(1-\cos\Delta\phi)}$,
where $\Delta\phi$ is the angle between the charged lepton and the neutrino in
the transverse plane.

The CDF Collaboration measured $m_{W}$ CDF:2022hxs  using Run 2 data collected
between 2003 and 2011 at the Tevatron collider, corresponding to 8.8 fb-1 of
integrated luminosity from proton-antiproton ($p\bar{p}$) collisions at a
center-of-mass energy of $\sqrt{s}=1.96$ TeV. The mass was obtained from
template fits to the reconstructed distributions of
$p^{\ell}_{\scriptsize\mathrm{T}}$, $m_{\scriptsize\textrm{T}}$, and
$p^{\nu}_{\mathrm{T}}$ in the electron and muon decay channels, yielding
$m_{W}=80433.5\pm 6.4~{}\textrm{(stat.)}\pm 6.9~{}\textrm{(sys.)}$ MeV, or
$80433.5\pm 9.4$ MeV. The quoted value of $m_{W}$ corresponds to the NNPDF3.1
PDF set Ball:2017nwa , with the PDF uncertainty estimated using the largest 25
symmetric eigenvectors constructed through a principal-component analysis from
the full replica set. The direct fit for $m_{W}$ to the data used events from
a version of the ResBos Ladinsky:1993zn  generator referred to as ResBos-C in
this paper. The generation used the CTEQ6M PDF set Pumplin:2002vw  and was
tuned to fit the observed spectrum of $Z$-boson transverse momentum. The
uncertainty on the $W$-boson transverse momentum
$p_{\scriptsize\textrm{T}}^{W}$ was determined using DYqT Bozzi:2008bb ;
Bozzi:2010xn , with a constraint from the observed recoil distribution in
$W$-boson events. The adjustment of the model to the NNPDF3.1 PDF set included
an effective update of the modelling of the leptonic angular distributions, as
discussed in Sec. 4.3.

The D0 Collaboration performed two measurements of $m_{W}$ in Run 2 of the
Tevatron collider. The first used data taken between 2002 and 2006,
corresponding to an integrated luminosity of 1.1 fb-1 Abazov:2009cp , and the
second used 2006-2008 data corresponding to an integrated luminosity of 4.3
fb-1 D0:2013jba . The analysis produced template fits for $m_{W}$ using the
$p^{\ell}_{\scriptsize\textrm{T}}$, $m_{\scriptsize\textrm{T}}$, and
$p^{\nu}_{\scriptsize\textrm{T}}$ kinematic distributions in the electron
decay channel. The initial 1.1 fb-1 measurement combined the results from
these three distributions, while the 4.3 fb-1 measurement removed the
$p^{\nu}_{\scriptsize\textrm{T}}$ result due to its small weight in the
combination. The overall combined result of all measurements is
$m_{W}=80375\pm 13~{}\textrm{(stat.)}\pm 22~{}\textrm{(sys.)}$ MeV, or
$80375\pm 23$ MeV. This value was determined using the CTEQ6.1 Stump:2003yu
(CTEQ6.6 Nadolsky:2008zw ) PDF set for the measurement using 1.1 fb-1 (4.3
fb-1). The uncertainties were evaluated using Pythia6 Sjostrand:2006za  and
the CTEQ6.1 PDF Hessian eigenvectors scaled to reduce the nominal 90% C.L.
coverage to 68% C.L. The $p_{\scriptsize\textrm{T}}^{W}$ modelling used a
version of the ResBos Balazs:1997xd ; Landry:2002ix  generator referred to
here as ResBos-CP.

The $m_{W}$ measurement performed by the ATLAS Collaboration used $\sqrt{s}=7$
TeV proton-proton collision data corresponding to 4.6 fb-1 of integrated
luminosity collected in 2011 during Run 1 of the LHC collider. ATLAS performed
template fits to the $p^{\ell}_{\scriptsize\textrm{T}}$ and
$m_{\scriptsize\textrm{T}}$ distributions in the electron and muon channels
separately for $W^{+}$ and $W^{-}$ events, since in proton-proton ($pp$)
collisions the final-state distributions are different for these processes.
The fits were further subdivided into three (four) pseudorapidity ranges in
the electron (muon) channel, yielding a total of 28 measurements. The
combination of these measurements yields $m_{W}=80370\pm
7~{}\textrm{(stat.)}\pm 18~{}\textrm{(sys.)}$, or $80370\pm 19$ MeV. The
parton distribution functions were modelled with the NNLO CT10 PDF set
Lai:2010vv , with the Hessian uncertainties scaled to 68% C.L. The
$p_{\scriptsize\textrm{T}}^{W}$ modelling relied on the parton shower Monte
Carlo (MC) Pythia8 Sjostrand:2014zea  tuned to match the
$p_{\scriptsize\textrm{T}}^{Z}$ distribution observed in data. The impact of
the PDF uncertainties on the $m_{W}$ measurement was reduced by a simultaneous
fit in different lepton pseudorapidity regions. The PDFs affect both the
$p_{\scriptsize\textrm{T}}^{W}$ and $p_{\scriptsize\textrm{T}}^{Z}$
distributions, and to preserve the agreement with the
$p_{\scriptsize\textrm{T}}^{Z}$ data distribution only the relative variation
between the $p_{\scriptsize\textrm{T}}^{W}$ and
$p_{\scriptsize\textrm{T}}^{Z}$ distributions was propagated in the
uncertainty estimate. Generated events were reweighted according to the
calculation of the leptonic angular distributions in DYNNLO Catani:2007vq ;
Catani:2009sm .

The LHCb Collaboration performed a measurement of $m_{W}$ using Run 2 $pp$ LHC
collision data collected in 2016 at $\sqrt{s}=13$ TeV, corresponding to 1.7
fb-1 of integrated luminosity. The measurement used the
$q/p^{\ell}_{\scriptsize\textrm{T}}$ distribution in the muon decay channels,
where $q$ is the muon charge, giving a result of $m_{W}=80354\pm
23~{}\textrm{(stat.)}\pm 10~{}\textrm{(exp.)}\pm 17~{}\textrm{(th.)}\pm
9~{}\textrm{(PDF)}$ MeV, or $80354\pm 32$ MeV. The LHCb central value of
$m_{W}$ and its uncertainty correspond to an unweighted average of results
using the NNPDF3.1, MSHT2020 Bailey:2020ooq  and CT18 Hou:2019qau  PDF sets,
all at next-to-leading order in the strong coupling and with 68% C.L.
coverage. The $p_{\scriptsize\textrm{T}}^{W}$ distribution was modelled with
Powheg Nason:2004rx ; Frixione:2007vw ; Alioli:2010xd  interfaced to Pythia8,
with a correction at high boson $p_{\scriptsize\textrm{T}}$ derived from the
observed $Z$-boson $p_{\scriptsize\textrm{T}}$ distribution. The leptonic
angular distributions were modelled with exact
$\mathcal{O}(\alpha_{\scriptsize\textrm{S}}^{2})$ predictions from DYTurbo
Camarda:2019zyx  and modified by scaling one of the leptonic angular
coefficients when fitting the data.

The event requirements and fit ranges used in the measurements are summarized
in Table 1. CDF and D0 used similar analysis configurations, while at ATLAS
the looser recoil requirement and wider $m_{\scriptsize\textrm{T}}$ fit range
were a consequence of the lower recoil resolution. The LHCb measurement was
inclusive in recoil, with only a loose requirement on the momentum of the
muon. The ATLAS, CDF, and D0 measurements fit $m_{W}$ only, while LHCb
performed a simultaneous fit for $m_{W}$ and the relative fraction of
$W^{+}$\- to $W^{-}$-boson decays, the hadronic background fraction,
$\alpha_{\scriptsize\textrm{S}}$ in $W$-boson events,
$\alpha_{\scriptsize\textrm{S}}$ in $Z$-boson events, the intrinsic transverse
momentum distribution of partons inside the proton, and the $A_{3}$ leptonic
angular coefficient (see Sec. 4.3).

Experiment | Event requirements | Fit ranges
---|---|---
CDF |  $30<p_{\scriptsize\textrm{T}}^{\ell}<55$ GeV |  $32<p_{\scriptsize\textrm{T}}^{\ell}<48$ GeV
| $|etal.|<1$ |  $32<p_{\scriptsize\textrm{T}}^{\nu}<48$ GeV
|  $30<p_{\scriptsize\textrm{T}}^{\nu}<55$ GeV |  $60<m_{\scriptsize\textrm{T}}<100$ GeV
|  $65<m_{\scriptsize\textrm{T}}<90$ GeV |
|  $u_{\scriptsize\textrm{T}}<15$ GeV |
D0 |  $p_{\scriptsize\textrm{T}}^{e}>25$ GeV |  $32<p_{\scriptsize\textrm{T}}^{e}<48$ GeV
| $|etal.|<1.05$ |  $65<m_{\scriptsize\textrm{T}}<90$ GeV
|  $p_{\scriptsize\textrm{T}}^{\nu}>25$ GeV |
|  $m_{\scriptsize\textrm{T}}>50$ GeV |
|  $u_{\scriptsize\textrm{T}}<15$ GeV |
ATLAS |  $p_{\scriptsize\textrm{T}}^{\ell}>30$ GeV |  $32<p_{\scriptsize\textrm{T}}^{\ell}<45$ GeV
| $|etal.|<2.4$ |  $66<m_{\scriptsize\textrm{T}}<99$ GeV
|  $p_{\scriptsize\textrm{T}}^{\nu}>30$ GeV |
|  $m_{\scriptsize\textrm{T}}>60$ GeV |
|  $u_{\scriptsize\textrm{T}}<30$ GeV |
LHCb |  $p_{\scriptsize\textrm{T}}^{\mu}>24$ GeV |  $28<p_{\scriptsize\textrm{T}}^{\mu}<52$ GeV
| $2.2<\eta_{\mu}<4.4$ |
Table 1: Event requirements and fit ranges for CDF, D0, ATLAS, and LHCb.

##  3 Methods

The combination consists of three steps. First, the results are adjusted to a
common model to allow a consistent comparison of central values and evaluation
of uncertainty correlations. This reference model includes the description of
the $W$-boson production, the Breit-Wigner lineshape, and the $W$-boson
polarization, and is described in Section 4. Second, the correlation of
uncertainties between the experiments is evaluated. The different center-of-
mass energies, initial states, and lepton pseudorapidity coverage make the
correlation non-trivial. Finally, the results are combined for representative
PDF sets, with the compatibility of the measurements determined for each set.
In addition, other $W$ and $Z$ boson measurements at the Tevatron and LHC are
compared to predictions using these PDF sets, in order to study the
reliability of the PDF predictions and uncertainties for the $m_{W}$
measurement.

The first step of adjusting each result to a different theoretical model
requires an emulation of the measurement process, which consists of Monte
Carlo event generation (see Sec. 3.1), detector simulation (see Sec. 3.2),
event selection, and a kinematic fit for $m_{W}$. The Monte Carlo samples are
produced using a reference value $m_{W}^{\scriptsize\textrm{ref}}$ for the
$W$-boson mass and width (${\mathrm{\Gamma}}_{W}$), and different values of
$m_{W}$ are obtained by reweighting events according to a Breit-Wigner
distribution,

$w(m,m_{W},m_{W}^{\scriptsize\textrm{ref}})=\frac{(m^{2}-m_{W}^{2})^{2}+m^{4}{\mathrm{\Gamma}}_{W}^{2}/m_{W}^{2}}{(m^{2}-{m_{W}^{\scriptsize\textrm{ref}}}^{2})^{2}+m^{4}{\mathrm{\Gamma}}_{W}^{2}/{m_{W}^{\scriptsize\textrm{ref}}}^{2}},$
(1)

using the final-state invariant mass $m$. This parameterization uses the
running width scheme in accordance with the published measurement procedures.
The mass reweighting procedure has been checked to give the correct target
mass value within a statistical uncertainty of $\approx 0.2$ MeV.

The detector simulations used in the original ATLAS, CDF, and D0 measurements
are simplified so that large event samples can be simulated for a variety of
PDF sets (see Section 3.2). These simulations do not have the complexity
required for a mass measurement in data but are sufficient for estimating the
impact of small theoretical modifications on the measurement. For the LHCb
measurement no simplification is necessary and the same detector simulation is
used as in the original measurement.

The shift in the value of $m_{W}$ resulting from a change in the generator
model is estimated by producing template distributions using a given
experiment’s model, and the same kinematic distributions for an alternate
model (the “pseudo-data”). The shift is determined by minimizing the negative
log-likelihood between the pseudo-data and the template distributions. In the
following we quote the impact $\delta m_{W}$ of each theoretical shift on a
measurement, i.e. the change in the measured $m_{W}$ value for a given change
in the theoretical model.

A common set of uncertainties and correlations between experiments is obtained
by evaluating $\delta m_{W}$ for a variety of PDF sets within the reference
theoretical model. Summing the theoretical shifts gives a total $\delta m_{W}$
that we add to each experimental measurement to obtain the value to be used in
the combination. For each PDF set the combination is performed using the
method of the best linear unbiased estimator (BLUE) Valassi:2003mu , including
both theoretical and experimental uncertainties. The BLUE method is used by
the individual experiments to combine results from different kinematic
distributions, and the combination procedure is validated by reproducing each
experiment’s published value. Results are presented for a combination of all
experimental measurements, as well as for various measurement subsets.

###  3.1 Monte Carlo event generation

The effects of modifying the $W$-boson production and decay model are studied
using event samples for the $W\rightarrow\ell\nu$ process in $pp$ collisions
at $\sqrt{s}=7$ TeV and $\sqrt{s}=13$ TeV, and for $p\bar{p}$ collisions at
$\sqrt{s}=1.96$ TeV. The event generators include those used by the original
experiments, along with more recent versions with improved calculations. The
PDF sets considered include those from the original measurements (CTEQ6M,
CTEQ6.1, CTEQ6.6, CT10, NNPDF3.1, CT18, and MSHT2020), as well as the
following sets at next-to-next-to-leading order (NNLO) in
$\alpha_{\scriptsize\textrm{S}}$: NNPDF4.0 NNPDF:2021njg , ABMP16
Alekhin:2017kpj , CT14 Dulat:2015mca , and MMHT2014 Harland-Lang:2014zoa .

For the ATLAS, CDF, and D0 experiments the $m_{W}$ shift associated with a
particular NNLO PDF set is evaluated using the NNLO QCD calculation Wj-MiNNLO
Monni:2019whf ; Monni:2020nks  implemented in Powheg-Box-V2 Nason:2004rx ;
Frixione:2007vw ; Alioli:2010xd . The analysis is performed at the Les Houches
event level Alwall2007  without interfacing to a parton shower, allowing for
an efficient and fast processing. The addition of the parton shower has been
confirmed to negligibly affect the shift associated with the PDF set. A large
sample is produced for each PDF set, with weights calculated internally by Wj-
MiNNLO in order to evaluate PDF uncertainties.

The uncertainty associated with a given PDF set is evaluated using the next-
to-leading order (NLO) QCD calculation W_ew-BMNNP Barze:2012tt  implemented in
the Powheg-Box-V2. This calculation is used for efficiency reasons and the
difference in the estimated uncertainty with respect to an NNLO calculation is
expected to be negligible. For LHCb the $m_{W}$ shifts and uncertainties are
evaluated using the same Powheg NLO calculation of $W$-boson production
Alioli:2008gx  as used in the original measurement.

The modelling of the $W$-boson polarization and resonance lineshape are
studied using large ResBos samples corresponding to those from the Tevatron
measurements: ResBos-C Balazs:1997xd , used by CDF in their direct fit to the
data CDF:2022hxs , with an accuracy of NLO and approximate NNLL in QCD
Isaacson:2022rts ; and ResBos-CP Landry:2002ix , used in the D0 $m_{W}$
measurement D0:2013jba , with an accuracy of NNLO+NNLL in QCD. A third sample
is generated using ResBos2 Isaacson:2022rts  with an accuracy of NLO+NNLL in
QCD and including a full resummation of the coefficients describing the
leptonic angular distributions (see Section 4.3). The difference between NLO
and NNLO predictions of these coefficients is studied using the DYNNLO
generator Catani:2007vq ; Catani:2009sm , which has been confirmed to be
consistent with other fixed-order calculations comparisons .

Electroweak corrections, primarily photon radiation in $W$-boson decay, have a
large impact on the final-state distributions but are calculated accurately.
The experiments factorize these corrections from PDF and QCD effects and we
therefore do not include them in the sample generation.

###  3.2 Detector simulations

This subsection provides brief descriptions of the parameterized simulations
used to study the effects of model variations on the combination, and shows
fit distributions comparing the simulations to those used in the experiments.
The simulation of each detector is referred to as the “LHC-TeV MWWG” or “MWWG”
simulation for that detector. A systematic uncertainty due to the simplified
parameterization is conservatively estimated by varying the lepton and recoil
scales and resolutions by $\pm 5$% and determining the effect on $\delta
m_{W}$.

####  3.2.1 CDF response and resolution model

The CDF detector model consists of parameterizations of the electron and
recoil momentum response and resolution. The muon momentum response and
resolution are similar to those of electrons.

The electron fractional momentum resolution
$\sigma_{p_{\tiny\textrm{T}}}/p_{\scriptsize\textrm{T}}=\sqrt{\kappa^{2}+S^{2}/p_{\scriptsize\textrm{T}}}$
is modelled using a sampling term of $S=12.6\%$ GeV1/2 and a constant term of
$\kappa=2\%$. The constant term is larger than that used in the CDF simulation
in order to correct for the lack of final-state radiation in the generated
samples. The leakage of the shower beyond the electromagnetic calorimeter is
parameterized in the same manner as for the CDF measurement, and the reduction
in electron momentum is corrected with a scale factor applied to the electron
momentum.

The recoil response is defined as the ratio $R(p_{\scriptsize\textrm{T}}^{W})$
of the measured recoil $u_{T}$ to the generated
$p_{\scriptsize\textrm{T}}^{W}$, before accounting for effects of underlying
event and additional proton-antiproton interactions (pileup). The CDF response
is parameterized as

$R(p_{\scriptsize\textrm{T}}^{W})=r\log(ap_{\scriptsize\textrm{T}}^{W})/\log(ap_{\scriptsize\textrm{T}}^{\scriptsize\textrm{ref}}),$
(2)

where $r=0.65$, $a=6.7/$GeV, and
$p_{\scriptsize\textrm{T}}^{\scriptsize\textrm{ref}}=15$ GeV.

The jet-like sampling for the recoil resolution is

$\sigma(p_{\scriptsize\textrm{T}}^{W})=s\sqrt{p_{\scriptsize\textrm{T}}^{W}},$
(3)

with $s=0.87$ GeV1/2 and $p_{\scriptsize\textrm{T}}^{W}$ in GeV. The recoil
azimuthal angular resolution $\sigma_{u_{\phi}}$ is parameterized as

$\sigma_{u_{\phi}}(p_{\scriptsize\textrm{T}}^{W})=\alpha-\beta\times
p_{\scriptsize\textrm{T}}^{W},$ (4)

where $\alpha=0.273$ rad and $\beta=0.016$ rad/GeV for
$p_{\scriptsize\textrm{T}}^{W}<p_{\scriptsize\textrm{T}}^{\scriptsize\textrm{ref}}$,
and $\alpha=0.143$ rad and $\beta=0.0044$ rad/GeV for
$p_{\scriptsize\textrm{T}}^{W}\geq
p_{\scriptsize\textrm{T}}^{\scriptsize\textrm{ref}}$.

The contribution of the underlying event to the measured recoil is represented
by a randomly-oriented Gaussian distribution of width 6.2 GeV. Finally, the
removal of lepton calorimeter towers from the recoil reconstruction is
modelled by subtracting 660 MeV from the generated recoil along the direction
of the decay lepton.

The distributions obtained using the MWWG simulation are compared with those
from the CDF simulation in Fig. 1. The agreement is at the percent level in
the range of interest for the measurement. The systematic uncertainties on the
MWWG simulation are estimated by varying the response and resolution by $\pm
5\%$, calculating the $\delta m_{W}$ shifts for fourteen PDFs, and taking the
maximum $|\delta m_{W}|$. The resulting uncertainties are 1.0 MeV for the
$m_{\scriptsize\textrm{T}}$ fit, 0.9 MeV for the
$p_{\scriptsize\textrm{T}}^{\ell}$ fit, and 2.0 MeV for the
$p_{\scriptsize\textrm{T}}^{\nu}$ fit.

Figure 1:  Comparisons between the CDF simulation Aaltonen:2013vwa  and the
LHC-TeV MWWG simulation for the $m_{\scriptsize\textrm{T}}$ (top) and
$p_{\scriptsize\textrm{T}}^{\ell}$ (bottom) distributions.

####  3.2.2 D0 response and resolution model

The LHC-TeV MWWG simulation of the D0 detector includes a model of the
efficiency of the electron reconstruction and selection along with the
response and resolution of the recoil and the electron momentum. The
simulation reproduces distributions from the D0 parameterized Monte Carlo
simulation (PMCS) used for the final D0 measurement based on an integrated
luminosity of 4.3 fb-1 D0:2013jba . The prior measurement based on 1.1 fb-1
Abazov:2009cp  of integrated luminosity has a lower mean number of pileup
events; the corresponding impact on the estimation of mass shifts is within
the applied uncertainty.

The electron energy response is parameterised as:

$E=\alpha(E_{0}-\bar{E_{0}})+\beta+\bar{E_{0}},$ (5)

where $E$ is the calibrated electron energy, $\bar{E_{0}}=43$ GeV is a
reference value corresponding to the electron energy in $Z$-boson events, and
$\alpha$ and $\beta$ are luminosity-dependent energy scale and offset
corrections, respectively. We take $\alpha=1.0164$ and $\beta=0.188$ GeV, the
values determined in Ref. D0:2013jba  for an instantaneous luminosity in the
range (2–4)$\times 36\times 10^{30}$$\textnormal{cm}^{-2}\textnormal{s}^{-1}$
corresponding to the largest fraction of the data. Implementing the
instantaneous luminosity dependence gives results in agreement with the
average response to within a percent.

The electron energy resolution $\sigma_{E}/E$ is simulated using the same
functional form as for CDF, with a constant term of $\kappa=1.997\%$
D0:2013jba  and a sampling term of

$\displaystyle S$ $\displaystyle=$ $\displaystyle
S_{0}\exp\left[S_{1}\left(\frac{1}{\sin\theta}-1\right)\right]+\frac{S_{2}\eta+S_{3}}{\sqrt{E}},$
(6)

where $S_{0}=0.153$ GeV1/2, $S_{1}=1.543$, $S_{2}=-0.025$ GeV, $S_{3}=0.172$
GeV, and $E$ is in GeV. The resulting fractional resolution is increased by 2%
to account for the lack of generated final-state radiation and improve the
agreement with the distributions from the D0 PMCS.

The electron reconstruction and identification efficiency is modeled by the
following function determined using the data points in Fig. 25(b) of Ref.
D0:2013jba :

$\displaystyle\varepsilon(p_{\scriptsize\textrm{T}}^{\ell})$ $\displaystyle=$
$\displaystyle
0.95\left(1-e^{-{0.074p_{\scriptsize\textrm{T}}^{\ell}}}\right),$ (7)

where $p_{\scriptsize\textrm{T}}^{\ell}$ is in GeV.

The recoil is modelled using a migration matrix to obtain a simulated
$u_{\scriptsize\textrm{T}}$ value for a given generated
$p_{\scriptsize\textrm{T}}^{W}$ D0:2020ujb . In order to model the recoil
energy in the electron cone that is not included in the recoil measurement,
150 MeV are subtracted from the recoil component parallel to the decay lepton
D0:2013jba .

Figure 2 shows the $p_{\scriptsize\textrm{T}}^{\ell}$ and
$m_{\scriptsize\textrm{T}}$ distributions from the D0 PMCS and the LHC-TeV
MWWG simulation after reweighting the events to match the
$p_{\scriptsize\textrm{T}}^{W}$ distribution used for the D0 measurement. The
distributions agree to within 2% in the range of interest for the $m_{W}$
extraction. The shifts in $m_{W}$ are studied for the eigenvectors of the
CTEQ6.6 and CT10 PDFs, and the MWWG simulation and D0 PMCS agree within the
statistical precision of $\approx 1$ MeV. Systematic uncertainties are
determined by varying the scales and resolutions, and calculating the effect
on $\delta m_{W}$ for fourteen PDF sets. The resulting uncertainties are 1.0
MeV on the $m_{\scriptsize\textrm{T}}$ fit, 1.0 MeV on the
$p_{\scriptsize\textrm{T}}^{\ell}$ fits, and 2.2 MeV on the
$p_{\scriptsize\textrm{T}}^{\nu}$ fit.

Figure 2: Comparisons of the D0 PMCS and the LHC-TeV MWWG simulation for the
$m_{\scriptsize\textrm{T}}$ (top) and
$p_{\scriptsize\textrm{T}}^{\ell}$(bottom) distributions.

####  3.2.3 ATLAS response and resolution model

The ATLAS recoil response and resolution are parametrized using distributions
Aaboud:2017svj  of the projections of these quantities along and perpendicular
to the lepton direction, as a function of the $W$-boson transverse momentum.
The parameterizations are calibrated using the full ATLAS simulation. The
recoil resolution ranges from 12 to 16 GeV, depending primarily on the amount
of pileup. The electron and muon resolutions are parameterized using the
documented detector performance PERF-2013-05 ; PERF-2014-05 . The resulting
$p_{\scriptsize\textrm{T}}^{\ell}$ and $m_{\scriptsize\textrm{T}}$
distributions are given in Figure 3, which shows that the resolution is
accurately modeled and that residual differences could be improved with lepton
energy scale adjustments and do not significantly affect the results.

Figure 3: Comparison of the published and MWWG simulated
$p_{\scriptsize\textrm{T}}^{\ell}$ (top) and $m_{\scriptsize\textrm{T}}$
(bottom) distributions for ATLAS.

The accuracy of the LHC-TeV MWWG simulation in determining $m_{W}$ shifts is
studied using PDF variations from the ATLAS measurement. With 28 measurement
categories and 25 CT10 PDF eigensets, a statistically accurate comparison is
made between the emulated measurement procedure and the results of the ATLAS
measurement. A root-mean-square spread of 1.5 MeV is found between the
published and emulated shifts in the various categories and eigensets. The
differences dominantly reflect approximations in the Powheg-based reweighting
procedure compared to the kinematic reweighting to NNLO-accurate distributions
implemented in Ref. Aaboud:2017svj . Systematic uncertainties are assessed by
varying the response and resolution by $\pm 5\%$, and are 1.1 MeV for the
$p_{\scriptsize\textrm{T}}^{\ell}$ fit and 1.2 MeV for the
$m_{\scriptsize\textrm{T}}$ fit.

##  4 $W$-boson production and decay

The process of $W$-boson production and decay is similar in $pp$ and
$p\bar{p}$ collisions, with differences arising mainly in the parton
distribution functions. Different PDF sets use different input data sets and
procedures, and the correlation between sets cannot be readily calculated.
Thus the combination is performed by adjusting the $m_{W}$ measurements to a
common PDF set through the addition of a $\delta
m_{W}^{\scriptsize\textrm{PDF}}$ specific to each experimental result. Events
generated with the Wj-MiNNLO Monte Carlo are used to evaluate the
corresponding PDF uncertainty and correlations. A separate shift $\delta
m_{W}^{\scriptsize\textrm{pol}}$ is calculated to update the Resbos-C and
Resbos-CP treatment of the $W$ boson polarization to Resbos2. The line shape
of the dilepton invariant mass is also studied, and adjustments are made for
differences in the spectrum due to the CDF generator-level requirements
($\delta m_{W}^{\scriptsize\textrm{gen}}$) or to the assumed decay width in
the measurements ($\delta m_{W}^{\mathrm{\Gamma}}$). Finally, correlations are
estimated for uncertainties due to electroweak corrections such as final-state
photon radiation.

###  4.1 $W$-boson $p_{\scriptsize\textrm{T}}$ distribution

In the region relevant to the $m_{W}$ measurement, the
$p_{\scriptsize\textrm{T}}^{W}$ distribution is described by a combination of
perturbative fixed-order QCD, soft-gluon resummation, and non-perturbative
effects. The Tevatron experiments use analytical resummation as implemented in
ResBos-C and ResBos-CP, while ATLAS and LHCb use the Pythia8 parton shower
interfaced to Powheg.

Non-perturbative effects influence the very low boson
$p_{\scriptsize\textrm{T}}^{W}$ region, typically
$p_{\scriptsize\textrm{T}}^{W}<5$ GeV, and are generally assumed to be
universal between $W$ and $Z$ production (up to differences in $\sqrt{s}$). In
the absence of precise direct measurements of the $W$-boson
$p_{\scriptsize\textrm{T}}$ distribution, all measurements use $Z$-boson data
to constrain the non-perturbative parameters. The resulting model is then used
for the $W$-boson $p_{\scriptsize\textrm{T}}$ distribution. The associated
uncertainty accounts for the limited precision of the $Z$-boson data and for
differences between the $Z$\- and $W$-boson production mechanisms, in
particular related to the different initial-state parton configurations.

To describe the $p_{\scriptsize\textrm{T}}^{W}$ distribution, ATLAS and LHCb
tune the shower and non-perturbative parameters in Pythia (intrinsic
$k_{\scriptsize\textrm{T}}$ and $\alpha_{\scriptsize\textrm{S}}$), and LHCb
adds an $\alpha_{\scriptsize\textrm{S}}$ tune in Powheg. The ATLAS tunes use
the $p_{\scriptsize\textrm{T}}^{Z}$ distribution while the LHCb tunes use an
angular distribution in $Z\to\mu\mu$ decays as well as the
$q/p_{\scriptsize\textrm{T}}$ distribution used for the $m_{W}$ fit. CDF fits
the non-perturbative resummation parameters $g_{1},g_{2}$ in ResBos-C using
the $p_{\scriptsize\textrm{T}}^{Z}$ distribution, and D0 uses the default
values of these parameters in ResBos-CP. CDF additionally constrains the
region above the peak with a fit for $\alpha_{\scriptsize\textrm{S}}$. The
resulting Tevatron and ATLAS $p_{\scriptsize\textrm{T}}^{W}$ distributions,
after event selection and using the detector simulations described in Section
3.2, are shown in Figure 4.

Theoretical uncertainties in the extrapolation from the
$p_{\scriptsize\textrm{T}}^{Z}$ distribution to the
$p_{\scriptsize\textrm{T}}^{W}$ distribution are considered by the ATLAS and
CDF experiments, which use the observed $W$-boson $p_{\scriptsize\textrm{T}}$
distribution to validate (ATLAS) or further constrain (CDF) the associated
uncertainty in situ. CDF (D0) quotes an uncertainty due to the $W$-boson
$p_{\scriptsize\textrm{T}}$ modelling of 2.2 (2.4) MeV and ATLAS quotes 6.0
MeV. For LHCb an 11 MeV uncertainty is assessed using the envelope of fit
results from Pythia8 (without Powheg), Powheg matched to Pythia8 or Herwig,
and Herwig with its own matrix-element calculation. Since the $W$-boson
$p_{\scriptsize\textrm{T}}$ distributions are modelled with different
generators or parameter values between the experiments, the corresponding
uncertainties are taken to be uncorrelated.

Figure 4: Distributions of generated $p_{\scriptsize\textrm{T}}^{W}$for
$W^{\pm}$ candidate events in $p\bar{p}$ collisions at CDF and D0 (top), and
for $W^{+}$ and $W^{-}$ events at ATLAS (bottom). The distributions represent
the best-fit model resulting from the analysis of $W$\- and $Z$-boson data in
the respective experiments, and are shown after all event selection
requirements.

###  4.2 Parton distribution functions

####  4.2.1 Central values and uncertainty correlations

The $\delta m_{W}^{\scriptsize\textrm{PDF}}$ shift for each PDF set is
evaluated for the Tevatron experiments and ATLAS using the precise NNLO Wj-
MiNNLO calculation. The resulting shifts are are compared to those from the
NLO W_ew-BMNNP and Resbos calculations, and the differences are typically
within the statistical precision of the comparison. For LHCb the PDF shift is
determined with a direct fit to the data as in the original measurement.

All experimental measurements include an in situ constraint on
$p_{\scriptsize\textrm{T}}^{Z}$ and/or $p_{\scriptsize\textrm{T}}^{W}$. We
preserve these constraints by reweighting the relevant boson
$p_{\scriptsize\textrm{T}}$distribution for each PDF set to match that used in
the measurement. For the Tevatron experiments $p_{\scriptsize\textrm{T}}^{W}$
is reweighted, while in the case of ATLAS $p_{\scriptsize\textrm{T}}^{Z}$ is
reweighted since the lower recoil resolution does not provide a significant
$p_{\scriptsize\textrm{T}}^{W}$ constraint from the data. For LHCb a
constraint on $p_{\scriptsize\textrm{T}}^{W}$ is applied as part of the direct
fit to the data for each PDF set.

For each PDF set $\delta m_{W}^{\scriptsize\textrm{PDF}}$ is evaluated using a
common boson $p_{\scriptsize\textrm{T}}$ distribution across PDFs separately
for each experiment. For the Tevatron experiments the $W$ boson
$p_{\scriptsize\textrm{T}}$ is reweighted to match that of the original
measurement, due to the observed agreement between the measured recoil
distribution and the model. In the case of ATLAS the $Z$-boson
$p_{\scriptsize\textrm{T}}$ is reweighted to match the original measurement,
since the lower recoil resolution does not provide a significant $W$-boson
$p_{\scriptsize\textrm{T}}$ constraint from the data. For LHCb the PDF shift
is determined with a direct fit to the data as in the original measurement,
including constraints on $p_{\scriptsize\textrm{T}}^{W}$ and the most relevant
polarization coefficient.

In order to facilitate the evaluation of uncertainty correlations, Hessian
eigenvector sets are used. The upper and lower uncertainties are taken to be

$\displaystyle\sigma_{m_{W}^{+}}=\left[\sum_{i}\left({\sigma_{m_{W}^{i}}}\right)^{2}\right]^{1/2}\textrm{
if }\sigma_{m_{W}^{i}}>0~{}\,\,\,\textrm{and}\,\,\,\,\,\,$
$\displaystyle\sigma_{m_{W}^{-}}=\left[\sum_{i}\left({\sigma_{m_{W}^{i}}}\right)^{2}\right]^{1/2}\textrm{
if }\sigma_{m_{W}^{i}}<0,$ (8)

where $i$ runs over the uncertainty sets. The uncertainties are symmetrized
according to $\sigma_{m_{W}}=(\sigma_{m_{W}^{+}}+\sigma_{m_{W}^{-}})/2$. For
CTEQ PDF sets the translation from 90% C.L. to 68% C.L. assumes a gaussian
distribution, i.e. a division by 1.645. The effect of each PDF eigenset is
correlated across experiment or measurement category, and its contribution to
the covariance between any two measurements $\alpha$ and $\beta$ is given by
$C^{i}_{\alpha\beta}=\sigma_{m_{W\alpha}^{i}}\sigma_{m_{W\beta}^{i}}$.
Accounting for all eigensets of a given PDF, the total uncertainty covariance
and the corresponding uncertainty correlation are calculated as

$C^{\scriptsize\textrm{PDF}}_{\alpha\beta}=\sum_{i}C^{i}_{\alpha\beta},~{}\,\,\,\textrm{and}\,\,\,\,\rho_{\alpha\beta}=\frac{\sum_{i}\sigma_{m_{W\alpha}^{i}}\sigma_{m_{W\beta}^{i}}}{\sigma_{m_{W\alpha}}\sigma_{m_{W\beta}}}.$
(9)

Tables 2 and 3 show $\delta m_{W}^{\scriptsize\textrm{PDF}}$ for each PDF set
and each experiment using distributions based on the transverse mass and the
lepton or neutrino $p_{T}$, respectively. For simplicity the ATLAS shifts are
shown inclusively in lepton $\eta$, though separated by boson charge. The PDF
uncertainties for each measurement are shown in Table 4, and the correlation
matrices for the the most recent PDF sets are shown in Fig. 5. Correlation
matrices for older sets are provided in the Appendix.

PDF set | D0 | CDF | ATLAS $W^{+}$ | ATLAS $W^{-}$
---|---|---|---|---
CTEQ6 | $-14.6$ | 0.0 | – | –
CTEQ6.6 | 0.0 | 14.2 | – | –
CT10 | $-0.5$ | 14.3 | 0.0 | 0.0
CT14 | $-8.7$ | 5.2 | $-0.5$ | $-7.6$
CT18 | $-7.5$ | 6.5 | 13.4 | $-5.5$
ABMP16 | $-17.9$ | $-2.4$ | $-25.7$ | $-7.9$
MMHT2014 | $-10.1$ | 4.5 | $-3.6$ | 9.1
MSHT20 | $-12.9$ | 2.5 | $-22.3$ | 4.2
NNPDF3.1 | $-1.0$ | 13.1 | $-14.6$ | $-6.3$
NNPDF4.0 | 6.2 | 20.1 | $-23.3$ | 4.3
Table 2: Values of $\delta m_{W}^{\scriptsize\textrm{PDF}}$ in MeV for each PDF set using the $m_{\scriptsize\textrm{T}}$ fit distribution, determined using the Wj-MiNNLO calculation.  PDF set | D0 $p_{\scriptsize\textrm{T}}^{\ell}$ | D0 $p_{\scriptsize\textrm{T}}^{\nu}$ | CDF $p_{\scriptsize\textrm{T}}^{\ell}$ | CDF $p_{\scriptsize\textrm{T}}^{\nu}$ | ATLAS $W^{+}$ | ATLAS $W^{-}$ | LHCb
---|---|---|---|---|---|---|---
CTEQ6 | $-17.0$ | $-17.7$ | 0.0 | 0.0 | – | – | –
CTEQ6.6 | 0.0 | 0.0 | 15.0 | 17.0 | – | – | –
CT10 | 0.4 | $-1.3$ | 16.0 | 16.3 | 0.0 | 0.0 | –
CT14 | $-9.7$ | $-10.6$ | 5.8 | 6.8 | -1.2 | $-5.8$ | 1.1
CT18 | $-8.2$ | $-9.3$ | 7.2 | 7.7 | 12.1 | $-2.3$ | $-6.0$
ABMP16 | $-19.6$ | $-21.5$ | $-1.4$ | $-2.4$ | $-22.5$ | $-3.1$ | 7.7
MMHT2014 | $-10.4$ | $-12.7$ | 6.1 | 5.5 | $-2.6$ | 9.9 | $-10.8$
MSHT20 | $-13.7$ | $-15.4$ | 3.6 | 4.1 | $-20.9$ | 4.5 | $-2.0$
NNPDF3.1 | $-1.0$ | $-1.2$ | 14.0 | 15.1 | $-14.1$ | -1.8 | 6.0
NNPDF4.0 | 6.7 | 8.1 | 20.8 | 24.1 | $-22.4$ | 6.9 | 8.3
Table 3: Values of $\delta m_{W}^{\scriptsize\textrm{PDF}}$ in MeV for each PDF set using the $p_{\scriptsize\textrm{T}}^{\ell}$ (all experiments) or $p_{\scriptsize\textrm{T}}^{\nu}$ (CDF and D0) distribution, determined using the Wj-MiNNLO calculation.  PDF set | D0 | CDF | ATLAS | LHCb
---|---|---|---|---
CTEQ6 | – | 14.1 | – | –
CTEQ6.6 | 15.1 | – | – | –
CT10 | – | – | 9.2 | –
CT14 | 13.8 | 12.4 | 11.4 | 10.8
CT18 | 14.9 | 13.4 | 10.0 | 12.2
ABMP16 | 4.5 | 3.9 | 4.0 | 3.0
MMHT2014 | 8.8 | 7.7 | 8.8 | 8.0
MSHT20 | 9.4 | 8.5 | 7.8 | 6.8
NNPDF3.1 | 7.7 | 6.6 | 7.4 | 7.0
NNPDF4.0 | 8.6 | 7.7 | 5.3 | 4.1
Table 4: Uncertainty in MeV for each PDF set after combining the individual
fit categories.

Figure 5: PDF uncertainty correlation matrices for the CT18, MSHT20, NNPDF4.0,
and ABMP16 PDF sets, in order from top to bottom.

####  4.2.2 $W$\- and $Z$-boson production measurements

The various kinematic distributions and fiducial regions used to fit $m_{W}$
in the ATLAS and Tevatron experiments provides some sensitivity to PDF
predictions. Other $W$\- and $Z$-boson measurements from the LHC and the
Tevatron provide more significant PDF constraints and are used in the
determination of the PDF sets. This section compares the compatibility of
these other measurements with the various PDF sets. Some sets have low
compatibility and are not favoured for an $m_{W}$ combination.

The $W$-boson rapidity ($y_{W}$) distribution affects the $m_{W}$ measurement
through the $p_{\scriptsize\textrm{T}}^{\ell}$ distribution: more central $W$
bosons can have more forward-decaying leptons within the detector acceptance,
lowering the mean observed $p_{\scriptsize\textrm{T}}^{\ell}$. Measurements
that probe PDF parameters describing $y_{W}$ include the $Z$ boson rapidity
$y_{Z}$ and the asymmetries in the rapidity distribution between positive and
negative $W$ bosons ($A_{W}$), or similarly the positive and negative charged
leptons from their decay ($A_{\ell}$). These measurements are considered in
this compatibility study, and are shown in Table 5.

The comparison between data and predictions is performed with the xFitter
Bertone:2017tig  framework. A $\chi^{2}$ measure is constructed including all
experimental uncertainties and their correlations, as well as the PDF
uncertainties. Theory predictions are calculated at NNLO in QCD and corrected
to NLO electroweak predictions using multiplicative $k$-factors in each
measurement bin. PDF uncertainties are computed at NLO in QCD using Applgrids
Carli:2010rw  with calculations from MCFM-6.8 Campbell:1999ah . The results
for various PDF sets are shown in Table 6.

Most of the Drell-Yan measurements have good $\chi^{2}$ values for all PDFs.
The most significant outlier is the D0 $W\rightarrow e\nu$ lepton asymmetry
measurement, for which the CT18 set has the lowest $\chi^{2}$ primarily due to
its larger uncertainties. These larger uncertainties also reduce the
correlated $\chi^{2}$, which represents the contribution from correlated
uncertainties alekhin:2015herafitter . The correlated $\chi^{2}$ reduces from
251 to 43 after including PDF uncertainties in the CT18 set; the corresponding
reduction for the NNPDF3.1 set is 110 to 76. The overall probability of
consistency of the combined datasets is 1.5% for the CT18 set, and is much
lower for the other sets. Among the studied PDF sets CT18 is therefore
considered to give the most accurate estimate of the 68% C.L. interval for
combined $W$\- and $Z$-boson measurements.

Exp. | Obs. | Decay | $\sqrt{s}$ | Lum. | bins
---|---|---|---|---|---
CDF CDF:2009cjw  | $A_{W}$ | $e\nu$ | 1.96 TeV | 1 fb-1 | 13
CDF CDF:2010vek  | $y_{Z}$ | $ee$ | 1.96 TeV | 2.1 fb-1 | 28
D0 D0:2007djv  | $y_{Z}$ | $ee$ | 1.96 TeV | 0.4 fb-1 | 28
D0 D0:2013xqc  | $A_{\ell}$ | $\mu\nu$ | 1.96 TeV | 7.3 fb-1 | 12
D0 D0:2014kma  | $A_{\ell}$ | $e\nu$ | 1.96 TeV | 9.7 fb-1 | 13
ATLAS ATLAS:2016nqi  |  $Z$,$W$ |  $\ell\ell$,$\ell\nu$ | 7 TeV | 4.7 fb-1 | 61
Table 5: Drell-Yan measurements used for the PDF compatibility study.  Measurement | NNPDF3.1 | NNPDF4.0 | MMHT14 | MSHT20 | CT14 | CT18 | ABMP16
---|---|---|---|---|---|---|---
CDF $y_{Z}$ | 24 / 28 | 28 / 28 | 30 / 28 | 32 / 28 | 29 / 28 | 27 / 28 | 31 / 28
CDF $A_{W}$ | 11 / 13 | 14 / 13 | 12 / 13 | 28 / 13 | 12 / 13 | 11 / 13 | 21 / 13
D0 $y_{Z}$ | 22 / 28 | 23 / 28 | 23 / 28 | 24 / 28 | 22 / 28 | 22 / 28 | 22 / 28
D0 $W\to e\nu$ $A_{\ell}$ | 22 / 13 | 23 / 13 | 52 / 13 | 42 / 13 | 21 / 13 | 19 / 13 | 26 / 13
D0 $W\to\mu\nu$ $A_{\ell}$ | 12 / 10 | 12 / 10 | 11 / 10 | 11 / 10 | 11 / 10 | 12 / 10 | 11 / 10
ATLAS peak CC $y_{Z}$ | 13 / 12 | 13 / 12 | 58 / 12 | 17 / 12 | 12 / 12 | 11 / 12 | 18 / 12
ATLAS $W^{-}$ $y_{\ell}$ | 12 / 11 | 12 / 11 | 33 / 11 | 16 / 11 | 13 / 11 | 10 / 11 | 14 / 11
ATLAS $W^{+}$ $y_{\ell}$ | 9 / 11 | 9 / 11 | 15 / 11 | 12 / 11 | 9 / 11 | 9 / 11 | 10 / 11
Correlated $\chi^{2}$ | 75 | 62 | 210 | 88 | 81 | 41 | 83
Total $\chi^{2}$ / d.o.f. | 200 / 126 | 196 / 126 | 444 / 126 | 270 / 126 | 210 / 126 | 162 / 126 | 236 / 126
p($\chi^{2},n$) | 0.003% | 0.007% | $<10^{-10}$ | $<10^{-10}$ | $0.0004\%$ | 1.5% | $10^{-8}$
Table 6: $\chi^{2}$ per degree of freedom for the Tevatron $Z$-rapidity and
$W$\- and $l$-asymmetry measurements at $\sqrt{s}=1.96$ TeV, and the LHC
$Z$-rapidity and $W$ lepton-rapidity measurements at $\sqrt{s}=7$ TeV. The
total $\chi^{2}$ is the sum of those quoted for individual measurements along
with a separate contribution for correlated uncertainties, where the latter is
extracted using a nuisance parameter representation of the $\chi^{2}$
alekhin:2015herafitter . The CT14 and CT18 PDF uncertainties correspond to 68%
coverage, obtained by rescaling the eigenvectors by a factor of 1/1.645. The
probability of obtaining a total $\chi^{2}$ at least as high as that observed
is labelled p($\chi^{2},n$).

###  4.3 $W$-boson polarization

The $W$-boson polarization affects the lepton decay angles, and in turn the
transverse momentum of the leptons. A general expression for the fully
differential distribution of the charged lepton is

$\displaystyle\frac{d\sigma}{dp_{\scriptsize\textrm{T}}^{W}dydmd\Omega}$
$\displaystyle=$
$\displaystyle\frac{d\sigma}{dp_{\scriptsize\textrm{T}}^{W}dydm}[(1+\cos^{2}\theta)$
(10) $\displaystyle+$
$\displaystyle\frac{1}{2}A_{0}(1-3\cos^{2}\theta)+A_{1}\sin 2\theta\cos\phi$
$\displaystyle+$ $\displaystyle\frac{1}{2}A_{2}\sin^{2}\theta\cos
2\phi+A_{3}\sin\theta\cos\phi$ $\displaystyle+$ $\displaystyle
A_{4}\cos\theta+A_{5}\sin^{2}\theta\sin 2\phi$ $\displaystyle+$ $\displaystyle
A_{6}\sin 2\theta\sin\phi+A_{7}\sin\theta\sin\phi],$

where the decay angles $\theta,\phi$ are expressed in the Collins-Soper (C-S)
frame Collins:1977iv , and the $A_{i}$ coefficients depend on the
$p_{\scriptsize\textrm{T}}$, rapidity, and invariant mass of the $\ell\nu$
system. The coefficients can be calculated perturbatively in
$\alpha_{\mathrm{S}}$, with $A_{5},~{}A_{6}$, and $A_{7}$ becoming non-zero
only at NNLO in QCD. The $A_{0}$ term primarily reflects the relative
fractions of the $qq\to W$, $qg\to Wq$, and higher-order subprocesses, and has
a significant $p_{\scriptsize\textrm{T}}^{W}$ dependence while being nearly
independent of boson rapidity. The $A_{4}$ term produces a forward-backward
asymmetry, and is thus sensitive to the directions of the incoming quark and
anti-quark in the dominant $q\bar{q}^{\prime}\to W$ process. It depends on
rapidity and on the PDF set used in the calculation, and decreases with
increasing $p_{\scriptsize\textrm{T}}^{W}$.

The ResBos-C and ResBos-CP codes resum a subset of contributions to Equation
10, specifically those affecting the $(1+\cos^{2}\theta)$ and
$A_{4}\cos\theta$ terms. This partial resummation modifies the $A_{0}$–$A_{3}$
terms relative to fixed-order predictions, as demonstrated in Figure 6, where
$A_{0}-A_{3}$ are shown for $W$-boson events generated at $\sqrt{s}=1.96$ TeV
with ResBos-C, ResBos-CP, ResBos2, and DYNNLO. The partial-resummation
predictions differ with respect to measurements performed at the LHC
STDM-2014-10 , which instead agree with fully-resummed calculations such as
ResBos2 or Wj-MiNNLO, and fixed-order calculations such as DYNNLO.

Experimental fits for $m_{W}$ in data use theoretical predictions of the
leptonic angular distributions from ResBos-C for CDF, ResBos-CP for D0, DYNNLO
Catani:2007vq ; Catani:2009sm  for ATLAS, and DYTurbo for LHCb. The CDF
experiment applies a post-fit correction to reproduce the NNPDF3.1 PDF
prediction, and this correction includes the effect of updating the angular
coefficients to those calculated by ResBos2.

In order to achieve a common theoretical treatment of the $W$-boson
polarization, the results of the CDF and D0 fits to the measurement
distributions are adjusted to correspond to the ResBos2 calculation of the
leptonic angular distributions at ${\cal O}(\alpha_{\mathrm{S}})$. Events
generated with ResBos-C or ResBos-CP are reweighted such that the
$A_{0}-A_{4}$ coefficients match those of ResBos2, as functions of
$p_{\scriptsize\textrm{T}}^{W}$ and $y_{W}$. The $W$-boson
$p_{\scriptsize\textrm{T}}$ is fixed to that of the original measurement, in
the same manner as for the $\delta m_{W}^{\scriptsize\textrm{PDF}}$
evaluations in Sec. 4.2.1. The impact of the reweighting on the CDF
$m_{\scriptsize\textrm{T}}$ and $p_{\scriptsize\textrm{T}}^{\ell}$
distributions is shown in Fig. 7, and the $\delta m_{W}^{\mathrm{pol}}$ values
from reweighting the $A_{i}$ coefficients individually and together are given
in Tables 7 and 8 for CDF and D0, respectively. The reweighting procedure
reproduces the direct fit from ResBos-C or ResBos-CP to ResBos2, as expected
since the basis of spherical harmonics is complete and exact. The results of
the reweighting procedure for the D0 configuration, $\delta
m_{W}^{\mathrm{pol}}=-6.4$, $-6.9$, and $-15.8$ MeV for the
$m_{\scriptsize\textrm{T}}$, $p_{\scriptsize\textrm{T}}^{\ell}$, and
$p_{\scriptsize\textrm{T}}^{\nu}$ distributions, respectively, are applied to
the measured $m_{W}$. For CDF, values of $\delta m_{W}^{\mathrm{pol}}=-9.5$,
$-8.4$, and $-12.5$ MeV for the $m_{\scriptsize\textrm{T}}$,
$p_{\scriptsize\textrm{T}}^{\ell}$, and $p_{\scriptsize\textrm{T}}^{\nu}$
distributions, respectively, are applied to events generated with ResBos-C.

ATLAS estimates a 5.8 MeV polarization modelling uncertainty based on the
precision of measurements on the $Z$-boson resonance, while the LHCb
uncertainty of 10 MeV arises from its determination of the $A_{3}$ coefficient
as part of its fit for $m_{W}$. These uncertainties are taken to be
uncorrelated. The Tevatron experiments do not include a corresponding
uncertainty in their measurements. An uncorrelated uncertainty is applied to
the shift calculated for each experiment to account for the limitations of the
parameterized MWWG simulation. This uncertainty is $\approx 1$ MeV and is
similar to that obtained by taking the difference between the NLO and NNLO
fixed-order calculations of the leptonic angular coefficients.

Figure 6: $A_{0}$ to $A_{3}$ as a function of $p_{\scriptsize\textrm{T}}^{W}$
extracted from ResBos-C, ResBos-CP, ResBos2, DYNNLO at ${\cal
O}(\alpha_{\scriptsize\textrm{S}})$, and DYNNLO at ${\cal
O}(\alpha^{2}_{\scriptsize\textrm{S}})$ in $p\bar{p}$ collisions at 1.96 TeV.
The CTEQ6M PDF set is used for all generators except ResBos-CP, for which
CTEQ6.6 is used. The ResBos-C and ResBos2 calculations are at ${\cal
O}(\alpha_{\mathrm{S}})$ in QCD, and ResBos-CP is at ${\cal
O}(\alpha^{2}_{\scriptsize\textrm{S}})$. The difference between DYNNLO at
${\cal O}(\alpha_{\scriptsize\textrm{S}})$ and ${\cal
O}(\alpha^{2}_{\scriptsize\textrm{S}})$ has an ${\cal{O}}$(1 MeV) effect on
$\delta m_{W}^{\mathrm{pol}}$.

Figure 7: Relative effect of reweighting the $A_{0}$–$A_{4}$ coefficients from ResBos-C to ResBos2 on the CDF $m_{\scriptsize\textrm{T}}$ (left) and $p_{\scriptsize\textrm{T}}^{\ell}$ (right) distributions.  Coefficient | $m_{\scriptsize\textrm{T}}$ | $p_{\scriptsize\textrm{T}}^{\ell}$ | $p_{\scriptsize\textrm{T}}^{\nu}$
---|---|---|---
$A_{0}$ | $-6.3$ | $-2.6$ | $-9.1$
$A_{1}$ | 1.1 | 1.3 | 0.3
$A_{2}$ | $-0.7$ | 0.4 | $-3.2$
$A_{3}$ | $-2.1$ | $-4.1$ | 1.0
$A_{4}$ | $-1.4$ | $-3.3$ | $-1.6$
$A_{0}-A_{4}$ | $-9.5$ | $-8.4$ | $-12.5$
ResBos2 | $-10.2\pm 1.1$ | $-7.6\pm 1.2$ | $-11.8\pm 1.4$
Difference | $-0.7\pm 1.1$ | $0.8\pm 1.2$ | $0.7\pm 1.4$
Table 7: Values of $\delta m_{W}^{\mathrm{pol}}$ in MeV associated with reweighting each $A_{i}$ coefficient from Resbos-C to Resbos2 for the CDF detector, as well as the result of a direct fit to ResBos2. The result of the direct fit is consistent with that of the reweighting.  Coefficient | $m_{\scriptsize\textrm{T}}$ | $p_{\scriptsize\textrm{T}}^{\ell}$ | $p_{\scriptsize\textrm{T}}^{\nu}$
---|---|---|---
$A_{0}$ | $-9.8$ | $-7.3$ | $-15.6$
$A_{1}$ | 1.9 | 2.4 | 1.8
$A_{2}$ | 3.0 | 3.3 | $-2.7$
$A_{3}$ | $-1.6$ | $-2.9$ | 0.4
$A_{4}$ | 0.2 | $-2.3$ | 0.5
$A_{0}-A_{4}$ | $-6.4$ | $-6.9$ | $-15.8$
ResBos2 | $-7.8\pm 1.0$ | $-6.6\pm 1.1$ | $-16.5\pm 1.2$
Difference | $-1.4\pm 1.0$ | $0.3\pm 1.1$ | $-0.7\pm 1.2$
Table 8: Values of $\delta m_{W}^{\mathrm{pol}}$ in MeV associated with
reweighting each $A_{i}$ coefficient from ResBos-CP to Resbos2 for the D0
detector, as well as the result of a direct fit to ResBos2. The result of the
direct fit is consistent with that of the reweighting.

###  4.4 $W$-boson resonance

The details of the event generation for the $m_{W}$ measurement differ between
measurements due to the assumed $W$-boson width ${\mathrm{\Gamma}}_{W}$ and to
a restriction on the generated $\ell\nu$ invariant mass range in the CDF
sample. These lead to $\delta m_{W}$ corrections on the direct fits to these
samples.

The assumed ${\mathrm{\Gamma}}_{W}$ values used by the experiments are as
follows: 2089.5 MeV for the CDF measurement; 2099 and 2100 MeV for the D0
measurements with 1.0 fb-1 and 4.3 fb-1, respectively; 2094 MeV for the ATLAS
measurement; and 2085 for the LHCb measurement. Using the SM prediction of
${\mathrm{\Gamma}}_{W}=2089.5\pm 0.6$ MeV leads to $\delta
m_{W}^{\mathrm{\Gamma}}=0.0$, 1.4, 1.5, 0.7, and $-0.7$ MeV for the CDF, D0
1.0 fb-1, D0 4.3 fb-1, ATLAS, and LHCb measurements, respectively.

The ResBos-C events used in the fit to the CDF data includes an $\ell\nu$
invariant mass requirement of $m_{\ell\nu}<150$ GeV. Differences of up to 10%
are observed between Resbos-C and ResBos2 for an invariant mass below 70 GeV,
though these have a negligible effect on the measurement. Using the ResBos2
invariant mass distribution without any requirements leads to $\delta
m_{W}^{\scriptsize\textrm{gen}}=-1.6,-3.4,$ and $-3.2$ MeV for the
$m_{\scriptsize\textrm{T}}$, $p_{\scriptsize\textrm{T}}^{\ell}$, and
$p_{\scriptsize\textrm{T}}^{\nu}$ distributions, respectively, for the CDF fit
results. The measured CDF $m_{W}$ accounts for these effects as part of the
update of the PDF set to NNPDF3.1. Smaller differences are observed between
Resbos-CP and ResBos2, and there is no significant $\delta
m_{W}^{\scriptsize\textrm{gen}}$ from the invariant mass modelling for the D0
measurement.

###  4.5 Electroweak corrections

The dominant electroweak effect on the $m_{W}$ measurement is final-state QED
radiation EWK , which reduces the momentum of the charged lepton from the
$W$-boson decay. The experiments model this radiation using generators that
resum multiple soft photon emissions above an energy threshold. Uncertainties
on the modelling of electroweak corrections include: (1) the perturbative
calculation of photon radiation, including the modelling of single-photon and
multi-photon emission and the matching of the fixed-order and all-orders
descriptions (2) the energy threshold for producing final-state photons; and
(3) higher-order corrections from final-state $e^{+}e^{-}$ pair production.
Tables 9 and 10 list the size of these uncertainties for each experiment in
the electron and muon channels, respectively. The uncertainties are completely
correlated between the decay channels.

Uncertainty | CDF | D0 | ATLAS
---|---|---|---
Perturbative photon rad. | 2.3 (2.3) | 5 (5) | 2.5 (3.3)
Photon energy cutoff | 1 (1) | 2 (1) | —
FSR $e^{+}e^{-}$ | 1 (1) | — | 0.8 (3.6)
Total | 2.7 (2.7) | 7 (7) | 2.6 (4.9)
Table 9: QED uncertainties in MeV on the $m_{W}$ measurement in the electron channel using the $m_{\scriptsize\textrm{T}}$ ($p_{\scriptsize\textrm{T}}$) fit. The uncertainties are uncorrelated except for those due to the perturbative photon radation calculation, which is taken to be 100% correlated between D0 and ATLAS, and to the photon energy cutoff, taken to be 100% correlated between CDF and D0.  Uncertainty | CDF | ATLAS | LHCb
---|---|---|---
Perturbative photon rad. | 2.3 (2.3) | 2.5 (3.5) | 8.6
Photon energy cutoff | 1 (1) | — | —
FSR $e^{+}e^{-}$ | 1 (1) | 0.8 (3.6) | —
Total | 2.7 (2.7) | 2.6 (5.6) | 8.6
Table 10: QED uncertainties in MeV on the $m_{W}$ measurement in the muon
channel for ATLAS and CDF using the $m_{\scriptsize\textrm{T}}$
($p_{\scriptsize\textrm{T}}$) fit, and for LHCb. The uncertainties are taken
to be uncorrelated between the experiments.

To estimate the uncertainty from the limitations of the shower model relative
to the matrix-element calculation, D0 and ATLAS perform a direct comparison
between PHOTOS and WGRAD Baur:1998kt ; Baur:2004ig  or WINHAC Placzek:2003zg ;
Placzek:2009jy ; Placzek:2013moa , respectively. Since ATLAS and D0 use the
same shower model, their uncertainties are considered as correlated. LHCb
estimates the uncertainty with a hybrid approach of comparing Powheg with and
without the NLO EW calculation, and the range of the PHOTOS, Pythia8, and
Herwig shower models. The average of the measurements from the different
shower models is taken as the central value, so the uncertainty is considered
as uncorrelated. CDF uses a third strategy, applying a correction to the
measurement using the HORACE CarloniCalame:2003ux ; CarloniCalame:2006zq ;
CarloniCalame:2007cd  generator, which matches multiple-photon radiation to
the $O(\alpha)$ calculation. The residual uncertainties are largely due to MC
statistics, and are considered as uncorrelated.

The shower model includes a lower threshold on the emitted photon energy,
expressed as a ratio with respect to the energy of the lepton from the $W$
boson decay. CDF uses a threshold of $10^{-5}$ and determines the uncertainty
by increasing the threshold by a factor of 3. D0 uses a similar procedure
except with an increase from $2.5\times 10^{-4}$ to $2\times 10^{-2}$. These
uncertainties are taken to be completely correlated.

To account for the higher-order process of an off-shell final-state photon
splitting into an $e^{+}e^{-}$ pair, CDF applies an effective radiator
approximation to the radiated photons. ATLAS does not apply a correction,
instead taking the uncertainty from a PHOTOS model of this process. The
uncertainties are treated as uncorrelated.

##  5 Combination

###  5.1 Procedures

The combination of $m_{W}$ measurements is performed by first replicating each
experiment’s combination of fit results within the experiment, applying any
relevant $\delta m_{W}$ shifts, and then combining across experiments.

The CDF individual $m_{W}$ values using the $m_{\scriptsize\textrm{T}}$,
$p_{\scriptsize\textrm{T}}^{\ell}$, and $p_{\scriptsize\textrm{T}}^{\nu}$
distributions in the electron and muon channels are combined using the
reported uncertainties and correlations, giving the results shown in Table 11.
The CDF measurement applies $\delta m_{W}$ values of $3.3$, $3.6$, and $3.0$
MeV, respectively, to fits to ResBos-C with the CTEQ6M PDF set. We remove
these $\delta m_{W}$ corrections and add $\delta
m_{W}^{\scriptsize\textrm{pol}}+\delta m_{W}^{\scriptsize\textrm{gen}}=-11.1$,
$-11.8$, and $-15.7$ MeV to the $m_{\scriptsize\textrm{T}}$,
$p_{\scriptsize\textrm{T}}^{\ell}$, and $p_{\scriptsize\textrm{T}}^{\nu}$
results, respectively, corresponding to the ResBos2 calculation of leptonic
angular distributions described in Sec. 4.3 and the removal of the $\ell\nu$
invariant mass requirement discussed in Sec. 4.4. Finally, a shift to the
target PDF set is applied. For the NNPDF3.1 PDF set this procedure gives a
combined CDF value of $m_{W}=80432.1\pm 9.4$ MeV, which is consistent with the
published CDF value of $m_{W}=80433.5\pm 9.4$ MeV within the uncertainty of
the procedure.

The principal component analysis used by CDF to reduce statistical effects in
the PDF uncertainty evaluation is not used in the combination with other
experiments, since different measurements would give different principal
components and complicate the correlation evaluations for the combination.
Instead the Hessian sets provided by the NNPDF collaboration are used to
estimate the PDF uncertainty. The combined CDF value with this uncertainty is
labelled “Combined ($\sigma_{\scriptsize\textrm{PDF}}=6.6$ MeV)” in Table 11
and corresponds to the entry labelled ”NNPDF 3.1” in Table 13 in Sec. 5.2.

| Published $m_{W}$ | Input $m_{W}$ | $\delta m_{W}^{\scriptsize\textrm{pol}}$ | $\delta m_{W}^{\scriptsize\textrm{gen}}$ | $\delta m_{W}^{\scriptsize\textrm{PDF}}$ | LHC-TeV MWWG $m_{W}$
---|---|---|---|---|---|---
$m_{\scriptsize\textrm{T}}$($e,\nu$) | 80429.1 | 80425.8 | $-9.5$ | $-1.6$ | $13.1$ | 80427.8
$p_{\scriptsize\textrm{T}}^{\ell}$($e$) | 80411.4 | 80407.8 | $-8.4$ | $-3.4$ | $14.0$ | 80410.0
$p_{\scriptsize\textrm{T}}^{\nu}$($e$) | 80426.3 | 80423.3 | $-12.5$ | $-3.2$ | $15.1$ | 80422.7
$m_{\scriptsize\textrm{T}}$($\mu,\nu$) | 80446.1 | 80442.8 | $-9.5$ | $-1.6$ | $13.1$ | 80444.8
$p_{\scriptsize\textrm{T}}^{\ell}$($\mu$) | 80428.2 | 80424.6 | $-8.4$ | $-3.4$ | $14.0$ | 80426.8
$p_{\scriptsize\textrm{T}}^{\nu}$($\mu$) | 80428.9 | 80425.9 | $-12.5$ | $-3.2$ | $15.1$ | 80425.3
Combined ($\sigma_{\scriptsize\textrm{PDF}}=3.9$ MeV) | 80433.5 |  |  |  |  | 80432.1
Combined ($\sigma_{\scriptsize\textrm{PDF}}=6.6$ MeV) |  |  |  |  |  | 80433.3
Table 11: Published CDF values and input values to the combination, where the
latter correspond to the results obtained from the direct CDF fits to ResBos-C
with the CTEQ6M PDF set. The combination procedure applies shifts to these
results to update to the ResBos2 calculation ($\delta
m_{W}^{\scriptsize\textrm{pol}}$ and $\delta m_{W}^{\scriptsize\textrm{gen}}$)
and a shift to update to the NNPDF3.1 PDF set ($\delta
m_{W}^{\scriptsize\textrm{PDF}}$). The total statistical and systematic
uncertainties on the shifts are 1.2, 1.1, and 2.1 MeV for the
$m_{\scriptsize\textrm{T}}$, $p_{\scriptsize\textrm{T}}^{\ell}$, and
$p_{\scriptsize\textrm{T}}^{\nu}$ fits respectively. The combined value is
consistent with that obtained by CDF when using the PDF uncertainties
determined by CDF, labelled “Combined ($\sigma_{\scriptsize\textrm{PDF}}=3.9$
MeV)”. When combining the result with other measurements, the uncertainty is
evaluated using NNPDF3.1 eigenvectors to give the result labelled “Combined
($\sigma_{\scriptsize\textrm{PDF}}=6.6$ MeV)”. The difference is due to a
change in the weight of each fit distribution. All units are in MeV.

The individual D0 measurements with the $m_{\scriptsize\textrm{T}}$ and
$p_{\scriptsize\textrm{T}}^{\ell}$ distributions using data sets corresponding
to 1.1 fb-1 and 4.3 fb-1 are combined using the reported uncertainties to give
the result $m_{W}=80375.1\pm 23.1$ MeV, which is the value quoted by D0
rounded to the nearest MeV. Before combining with other measurements, a number
of shifts are applied. First, a shift from CTEQ6.1 to CTEQ6.6 is applied to
the measurement based on 1.1 fb-1 of integrated luminosity. Shifts of $\delta
m_{W}^{\scriptsize\textrm{pol}}=-6.4$, $-6.9$, and $-15.8$ are applied to the
$m_{\scriptsize\textrm{T}}$, $p_{\scriptsize\textrm{T}}^{\ell}$, and
$p_{\scriptsize\textrm{T}}^{\nu}$fit results, respectively, to update the
ResBos-CP leptonic angular distributions to those of ResBos2. Finally, a
$\delta m_{W}^{\mathrm{\Gamma}}$ shift adjusts ${\mathrm{\Gamma}_{W}}$ to that
of the SM prediction. The result with these shifts and the published D0 PDF
uncertainty of $\approx 11$ MeV is labelled “Combined
($\sigma_{\scriptsize\textrm{PDF}}=11$ MeV)” in Table 12. The value with
uncertainties updated to those calculated with Wj-MiNNLO and CTEQ6.6 is
$m_{W}=80377.9\pm 25.5$ MeV and is labelled “Combined
($\sigma_{\scriptsize\textrm{PDF}}=15.1$ MeV)” in the table.

| Published $m_{W}$ | $\delta m_{W}^{\scriptsize\textrm{pol}}$ | $\delta m_{W}^{\scriptsize\textrm{PDF}}$ | $\delta m_{W}^{\mathrm{\Gamma}}$ | LHC-TeV MWWG $m_{W}$
---|---|---|---|---|---
Run 2a $m_{\scriptsize\textrm{T}}$($e,\nu$) | 80401 | $-6.4$ | $14.3$ | 1.4 | 80410.3
Run 2a $p_{\scriptsize\textrm{T}}^{\ell}$($e$) | 80400 | $-6.9$ | $16.7$ | 1.4 | 80411.2
Run 2a $p_{\scriptsize\textrm{T}}^{\nu}$($e$) | 80402 | $-15.8$ | $17.5$ | 1.4 | 80405.1
Run 2b $m_{\scriptsize\textrm{T}}$($e,\nu$) | 80371 | $-6.4$ |  | 1.5 | 80366.1
Run 2b $p_{\scriptsize\textrm{T}}^{\ell}$($e$) | 80343 | $-6.9$ |  | 1.5 | 80337.6
Combined ($\sigma_{\scriptsize\textrm{PDF}}=11$ MeV) | 80375 |  |  |  | 80373.4
Combined ($\sigma_{\scriptsize\textrm{PDF}}=15.1$ MeV) |  |  |  |  | 80377.9
Table 12: Published D0 values corresponding to the CTEQ6M (Run 2a) and CTEQ6.6
(Run 2b) PDF sets, along with the following shifts: modifying the leptonic
angular distributions to match those of ResBos2 ($\delta
m_{W}^{\scriptsize\textrm{pol}}$); modifying the Run 2a result to correspond
to the CTEQ6.6 PDF set ($\delta m_{W}^{\scriptsize\textrm{PDF}}$); and
modifying the $W$ boson width to the Standard Model prediction ($\delta
m_{W}^{\mathrm{\Gamma}}$). The total statistical and systematic uncertainties
on the shifts are 1.2, 1.2, and 2.3 MeV for the $m_{\scriptsize\textrm{T}}$,
$p_{\scriptsize\textrm{T}}^{\ell}$, and $p_{\scriptsize\textrm{T}}^{\nu}$ fits
respectively. The combined result with the published D0 PDF uncertainty
obtained using Pythia and the CTEQ6.1 PDF set is labelled “Combined
($\sigma_{\scriptsize\textrm{PDF}}=11$ MeV)”, and the result with PDF
uncertainties updated to those of CTEQ6.6 calculated with Wj-MiNNLO is
labelled “Combined ($\sigma_{\scriptsize\textrm{PDF}}=15.1$ MeV)”. The results
differ due to different weights of the individual fits to kinematic
distributions. All units are in MeV.

The ATLAS measurement is reproduced using the parameterized simulation to give
a value of $m_{W}=80369.7\pm 18.5$ MeV, which is within a few tenths of an MeV
of the published result. A $\delta m_{W}^{\mathrm{\Gamma}}=0.7$ MeV correction
is added to update ${\mathrm{\Gamma}}_{W}$, and further $\delta
m_{W}^{\scriptsize\textrm{PDF}}$ shifts are applied to provide the central
value for the target PDF set. All LHCb $m_{W}$ values for the combination are
determined from a direct fit to the data using the target PDF set, so only a
$\delta m_{W}^{\mathrm{\Gamma}}=-0.7$ MeV shift is applied to update the value
of the $W$ boson width.

###  5.2 Results

A series of combinations are performed corresponding to the Tevatron Run 2
experiments, the LHC experiments, all experiments including the result from
the LEP combination, and all experiments except one. For each experiment the
central value, uncertainty, and $\chi^{2}$ of the individual measurements is
shown for the ABMP16, CT14, CT18, MMHT2014, MSHT20, NNPDF31, and NNPDF40 PDF
sets. For the combined result of multiple experiments the overall PDF
uncertainty is also shown. The PDF uncertainties for the individual
experiments are given in Table 4.

####  5.2.1 Hadron-collider measurements

Results for the Tevatron Run 2 experiments are listed in Table 13. The
individual combinations of the CDF and D0 fit results are satisfactory for all
PDF sets, with probabilities ranging from 12% to 24%. The Tevatron-wide
combination has a total uncertainty ranging from 8.9 MeV for ABMP16 to 15.9
MeV for CT18, and a $\chi^{2}$ probability of 0.5–0.8%.

As discussed in Section 4.2.1, PDF uncertainties are fully correlated between
CDF and D0. The PDF uncertainty in the combination is therefore close to that
obtained for each experiment, and ranges from 4 MeV for ABMP16 to 13.5 MeV for
CT18. The combined central value ranges from 80408.2 MeV for ABMP16 to 80433.4
MeV for NNPDF4.0. The difference between the NNPDF3.1 and NNPDF4.0
combinations, 8.4 MeV, is similar to the PDF uncertainty of the NNPDF4.0 set
(7.8 MeV). Similar trends are observed for the CDF and D0 measurements
separately. Some variation in the results with PDF set is expected due to
differences in input data sets to the PDFs, and to the differences in the
compatibility with Drell-Yan measurements discussed in Section 4.2.2. Further
understanding of these differences would benefit future combinations.

| CDF (5 d.o.f.) |  | D0 (4 d.o.f.) |  | Tevatron Run 2 (1 d.o.f.) |
---|---|---|---|---|---|---
PDF set | $m_{W}$ | $\chi^{2}$ |  | $m_{W}$ | $\chi^{2}$ |  | $m_{W}$ | $\sigma_{\scriptsize\textrm{PDF}}$ | $\chi^{2}$ | p($\chi^{2},n$)
ABMP16 | $80417.3~{}\pm~{}9.5$ | 8.8 |  | $80355.4\pm 20.9$ | 6.6 |  | $80408.2\pm 8.9$ | 4.0 | 7.7 | 0.6%
CT14 | $80432.1\pm 15.5$ | 7.7 |  | $80370.9\pm 24.9$ | 5.9 |  | $80424.0\pm 15.2$ | 12.6 | 7.2 | 0.7%
CT18 | $80432.0\pm 16.1$ | 7.6 |  | $80372.0\pm 25.5$ | 5.9 |  | $80424.9\pm 15.9$ | 13.5 | 7.0 | 0.8%
MMHT2014 | $80425.7\pm 11.6$ | 7.0 |  | $80364.4\pm 22.3$ | 5.5 |  | $80417.4\pm 11.2$ | 7.8 | 7.6 | 0.6%
MSHT20 | $80424.4\pm 12.2$ | 7.6 |  | $80362.3\pm 22.5$ | 6.1 |  | $80415.9\pm 11.8$ | 8.6 | 7.8 | 0.5%
NNPDF3.1 | $80433.3\pm 10.9$ | 7.6 |  | $80372.7\pm 21.9$ | 5.8 |  | $80425.0\pm 10.5$ | 6.8 | 7.4 | 0.7%
NNPDF4.0 | $80441.8\pm 11.6$ | 7.2 |  | $80381.3\pm 22.2$ | 5.7 |  | $80433.4\pm 11.2$ | 7.8 | 7.4 | 0.7%
Table 13: The CDF and D0 Run 2 $m_{W}$ and $\chi^{2}$ values obtained from a
combination of the individual measurement distributions and decay channels,
along with the combined Tevatron Run 2 $m_{W}$, PDF uncertainty, $\chi^{2}$,
and probability of obtaining this $\chi^{2}$ or larger. Mass units are in MeV.

The LHC results are summarized in Table 14. The $\chi^{2}$ per degree of
freedom of the ATLAS combination ranges from 29/27 (for NNPDF3.1) to 45/27
(for MSHT20). The latter corresponds to a probability of about 2%. The larger
$\chi^{2}$ for MSHT20 is consistent with the calculations of Drell-Yan
measurements. The ATLAS and LHCb measurements are compatible and have a total
uncertainty ranging from 14.2 MeV to 16.6 MeV.

| ATLAS (27 d.o.f) |  | LHCb |  | LHC (1 d.o.f) |
---|---|---|---|---|---|---
PDF set | $m_{W}$ | $\chi^{2}$ |  | $m_{W}$ | $\chi^{2}$ |  | $m_{W}$ | $\sigma_{\scriptsize\textrm{PDF}}$ | $\chi^{2}$ | p($\chi^{2},n$)
ABMP16 | $80352.8\pm 16.1$ | 31 |  | $80361.0\pm 30.4$ | – |  | $80354.6\pm 14.2$ | 2.9 | 0.1 | 75%
CT14 | $80363.1\pm 20.4$ | 30 |  | $80354.4\pm 32.2$ | – |  | $80360.4\pm 16.4$ | 6.5 | 0.0 | 100%
CT18 | $80374.5\pm 20.3$ | 30 |  | $80347.3\pm 32.7$ | – |  | $80366.5\pm 16.6$ | 6.3 | 0.5 | 48%
MMHT2014 | $80372.8\pm 18.6$ | 30 |  | $80342.5\pm 31.3$ | – |  | $80364.4\pm 15.4$ | 5.1 | 0.6 | 44%
MSHT20 | $80368.9\pm 17.9$ | 45 |  | $80351.3\pm 31.0$ | – |  | $80364.3\pm 15.0$ | 4.5 | 0.2 | 65%
NNPDF3.1 | $80358.4\pm 17.6$ | 29 |  | $80359.3\pm 31.1$ | – |  | $80358.6\pm 15.0$ | 5.0 | 0.0 | 100%
NNPDF4.0 | $80353.5\pm 16.6$ | 35 |  | $80361.6\pm 30.6$ | – |  | $80355.4\pm 14.5$ | 3.8 | 0.1 | 75%
Table 14: The ATLAS and LHCb $m_{W}$ values obtained from a combination of the
individual measurement distributions and decay channels, along with the
combined LHC $m_{W}$, PDF uncertainty, and $\chi^{2}$, and probability of
obtaining this $\chi^{2}$ or larger. The $\chi^{2}$ of the combination of fit
distributions and decay channels is shown for ATLAS; no $\chi^{2}$ is shown
for LHCb as the measurement is performed using one distribution in one
channel. Mass units are in MeV.

The individual experimental results are shown in Figure 8 for all considered
PDF sets. The combination of ATLAS and LHCb measurements benefits from anti-
correlated PDF uncertainties Bozzi:2015zja . Therefore the combined PDF
uncertainties and the variation of the combined central values are smaller
than for the individual experiments. The ATLAS $m_{W}$ value ranges from
80352.8 MeV for ABMP16 to 80374.5 MeV for CT18. This range is comparable to
that of the Tevatron experiments. A similar spread but opposite trends are
observed for LHCb, and the spread of $m_{W}$ values is reduced from $\approx
20$ MeV to 14.1 MeV in the combination. The PDF uncertainties range from 4.0
MeV to 11.4 MeV for ATLAS and 3.0 to 12.2 MeV for LHCb, but are reduced to
2.9–6.5 MeV for the combined result.

Figure 8: The D0, ATLAS, LHCb, and CDF $m_{W}$ values and uncertainties using
the ABMP16, CT14, CT18, MMHT2014, MSHT20, NNPDF3.1, and NNPDF4.0 PDF sets.

####  5.2.2 All measurements

Tables 15–17 provide the results for various combinations including LEP, whose
uncertainties are treated as uncorrelated with the others. A combination of
all measurements yields a total uncertainty ranging between 7.5 and 11.5 MeV,
though the $\chi^{2}$ probabilities are low, ranging from $8\times 10^{-6}$ to
$5\times 10^{-3}$. The low probabilities reflect the discrepancy between the
CDF measurement and the other measurements. The combined value of $m_{W}$ for
the CT18 PDF set, which gives the largest compatibility with the broader
Drell-Yan measurements, is $m_{W}=80394.6\pm 11.5$ MeV with a probability of
0.5%. The relative weights of the CDF, ATLAS, LHCb, LEP, and D0 measurements
are 41%, 28%, 13%, 12%, and 5%, respectively. Weights for other PDF sets are
given in the Appendix. The largest difference in $m_{W}$ between PDF sets is
10.4 MeV.

All experiments (4 d.o.f.)
---
PDF set | $m_{W}$ | $\sigma_{\scriptsize\textrm{PDF}}$ | $\chi^{2}$ | p($\chi^{2},n$)
ABMP16 | $80392.7\pm 7.5$ | 3.2 | 29 | 0.0008%
CT14 | $80393.0\pm 10.9$ | 7.1 | 16 | 0.3%
CT18 | $80394.6\pm 11.5$ | 7.7 | 15 | 0.5%
MMHT2014 | $80398.0\pm 9.2$ | 5.8 | 17 | 0.2%
MSHT20 | $80395.1\pm 9.3$ | 5.8 | 16 | 0.3%
NNPDF3.1 | $80403.0\pm 8.7$ | 5.3 | 23 | 0.1%
NNPDF4.0 | $80403.1\pm 8.9$ | 5.3 | 28 | 0.001%
Table 15: Combination of $m_{W}$ measurements from the individual experiments.
Shown for each PDF are the PDF uncertainty, $\chi^{2}$, and probability of
obtaining this $\chi^{2}$ or larger. Mass units are in MeV.  All except LEP (3
d.o.f.)
---
PDF set | $m_{W}$ | $\sigma_{\scriptsize\textrm{PDF}}$ | $\chi^{2}$ | p($\chi^{2},n$)
ABMP16 | $80393.6\pm 7.8$ | 3.4 | 19 | 0.03%
CT14 | $80395.1\pm 11.6$ | 8.0 | 16 | 0.1%
CT18 | $80397.1\pm 12.3$ | 8.8 | 15 | 0.2%
MMHT2014 | $80399.9\pm 9.6$ | 6.2 | 17 | 0.7%
MSHT20 | $80396.8\pm 9.7$ | 6.3 | 16 | 0.1%
NNPDF3.1 | $80405.0\pm 9.0$ | 5.6 | 22 | 0.007%
NNPDF4.0 | $80405.3\pm 9.2$ | 5.7 | 27 | 0.0006%
Table 16: Combination of all $m_{W}$ measurements except the LEP average.
Shown for each PDF are the PDF uncertainty, $\chi^{2}$, and probability of
obtaining this $\chi^{2}$ or larger. Mass units are in MeV.  All except CDF (3
d.o.f.)
---
PDF set | $m_{W}$ | $\sigma_{\scriptsize\textrm{PDF}}$ | $\chi^{2}$ | p($\chi^{2},n$)
ABMP16 | $80357.3\pm 11.2$ | 2.6 | 0.4 | 94%
CT14 | $80365.4\pm 12.9$ | 5.8 | 0.3 | 96%
CT18 | $80369.2\pm 13.3$ | 6.2 | 0.5 | 92%
MMHT2014 | $80365.8\pm 12.1$ | 4.7 | 0.8 | 85%
MSHT20 | $80365.1\pm 12.0$ | 4.4 | 0.4 | 94%
NNPDF3.1 | $80364.7\pm 11.9$ | 4.5 | 0.4 | 94%
NNPDF4.0 | $80364.5\pm 11.6$ | 3.9 | 1.2 | 75%
Table 17: Combination of $m_{W}$ measurements from all individual experiments
except CDF. Shown for each PDF are the PDF uncertainty, $\chi^{2}$, and
probability of obtaining this $\chi^{2}$ or larger. Mass units are in MeV.
All except D0 (3 d.o.f.)
---
PDF set | $m_{W}$ | $\sigma_{\scriptsize\textrm{PDF}}$ | $\chi^{2}$ | p($\chi^{2},n$)
ABMP16 | $80397.2\pm 7.8$ | 3.1 | 15 | 0.2%
CT14 | $80395.9\pm 11.0$ | 7.0 | 11 | 1.2%
CT18 | $80397.1\pm 11.6$ | 7.6 | 10 | 1.9%
MMHT2014 | $80401.9\pm 9.4$ | 5.6 | 13 | 0.5%
MSHT20 | $80399.0\pm 9.5$ | 5.7 | 12 | 0.7%
NNPDF3.1 | $80406.6\pm 8.9$ | 5.1 | 19 | 0.03%
NNPDF4.0 | $80406.4\pm 9.0$ | 5.1 | 25 | 0.002%
Table 18: Combination of $m_{W}$ measurements from all individual experiments
except D0. Shown for each PDF are the PDF uncertainty, $\chi^{2}$, and
probability of obtaining this $\chi^{2}$ or larger. Mass units are in MeV.
All except ATLAS (3 d.o.f.)
---
PDF set | $m_{W}$ | $\sigma_{\scriptsize\textrm{PDF}}$ | $\chi^{2}$ | p($\chi^{2},n$)
ABMP16 | $80402.8\pm 8.3$ | 3.5 | 11 | 1.2%
CT14 | $80406.5\pm 12.8$ | 9.1 | 12 | 0.7%
CT18 | $80405.0\pm 13.2$ | 9.6 | 13 | 0.5%
MMHT2014 | $80405.8\pm 10.0$ | 6.3 | 14 | 0.3%
MSHT20 | $80404.2\pm 10.3$ | 6.6 | 12 | 0.7%
NNPDF3.1 | $80414.7\pm 9.5$ | 5.6 | 13 | 0.5%
NNPDF4.0 | $80420.2\pm 10.0$ | 6.2 | 14 | 0.3%
Table 19: Combination of $m_{W}$ measurements from the individual experiments
except for ATLAS. Shown for each PDF are the PDF uncertainty, $\chi^{2}$, and
probability of obtaining this $\chi^{2}$ or larger. Mass units are in MeV.
All except LHCb (3 d.o.f.)
---
PDF set | $m_{W}$ | $\sigma_{\scriptsize\textrm{PDF}}$ | $\chi^{2}$ | p($\chi^{2},n$)
ABMP16 | $80394.8\pm 7.7$ | 3.4 | 18 | 0.04%
CT14 | $80399.8\pm 11.7$ | 8.4 | 14 | 0.3%
CT18 | $80402.6\pm 12.4$ | 9.0 | 12 | 0.7%
MMHT2014 | $80404.4\pm 9.7$ | 6.5 | 13 | 0.5%
MSHT20 | $80400.7\pm 10.0$ | 6.8 | 14 | 0.3%
NNPDF3.1 | $80407.4\pm 9.1$ | 5.8 | 20 | 0.02%
NNPDF4.0 | $80407.3\pm 9.3$ | 5.9 | 26 | 0.001%
Table 20: Combination of $m_{W}$ measurements from the individual experiments
except for LHCb. Shown for each PDF are the PDF uncertainty, $\chi^{2}$, and
probability of obtaining this $\chi^{2}$ or larger. Mass units are in MeV.

A possible procedure for combining measurements with low compatibility is to
scale all uncertainties by the square root of the ratio of the $\chi^{2}$ to
the number of degrees of freedom. This procedure effectively assumes a common
underestimated uncertainty, which is an unlikely scenario for these
measurements. The PDF uncertainty is only partially correlated, and the
uncertainty from the CT18 set is the most conservative. Other measurement
uncertainties are smaller or are statistically constrained and therefore
uncorrelated. Further measurements or studies are required to obtain more
consistent results.

To evaluate the significance of differences between individual measurements
and the others, separate combinations are performed excluding, in turn, each
individual result from the average. Removing LEP, D0, or LHCb from the
combination increases the uncertainty by up to 0.9 MeV and affects the central
value by up to 8 MeV. When removing ATLAS the $\chi^{2}$ probability ranges
from 0.3% to 1.2%, and the uncertainty ranges from 8.3 to 13.2 MeV. The
combinations with CDF excluded have good compatibility and the total
uncertainty increases to 11.2–13.3 MeV, or 2–4 MeV more than the full
combination. The variation of this combination with PDF set is 11.9 MeV, with
the value for the ABMP16 PDF set considerably lower than the others (the
variation is 4.5 MeV without this set). The combination of all measurements
except CDF is $m_{W}=80369.2\pm 13.3$ MeV for the CT18 PDF set, with a 91%
probability of consistency. The relative weights for the ATLAS, D0, LHCb, and
LEP measurements are 42%, 23%, 18%, and 16%, respectively.

The partial combinations are also used to evaluate the difference between each
experimental result and the combination of the others. Considering all PDF
sets, the LEP result is compatible with the average of the others to better
than one standard deviation. The compatibility of D0 or LHCb with the rest
ranges from 1–1.8 standard deviations. The ATLAS result differs from the
others by 1.6–3.6 standard deviations, where the largest difference is
obtained with the NNPDF4.0 PDF set. Finally, the CDF measurement differs from
the others by 3.6–5.2 standard deviations, depending on the choice of the PDF
set. The smallest significance corresponds to the CT18 set and the largest
significance corresponds to the NNPDF4.0 set.

The $m_{W}$ combinations from LEP, the Tevatron, the LHC, and all experiments
are presented in Figure 9 for all PDF sets, along with the corresponding
$\chi^{2}$ probabilities. The same information is also shown for the
combinations removing one experimental result at a time.

Figure 9: Top left: The combined $m_{W}$ values and uncertainties from LEP,
the Tevatron, LHC, and all experiments, using the ABMP16, CT14, CT18,
MMHT2014, MMHT20, NNPDF3.1, and NNPDF4.0 PDF sets. Right: The corresponding
probability of consistency determined using the $\chi^{2}$ per degrees of
freedom. Bottom left: The combined $m_{W}$ values and uncertainties for all
experiments except one using the ABMP16, CT14, CT18, MMHT2014, MMHT20,
NNPDF3.1, and NNPDF4.0 PDF sets. Right: The corresponding probability of
consistency determined using the $\chi^{2}$ per degrees of freedom.

##  6 Conclusion

A combination of $m_{W}$ measurements from the CDF, D0, ATLAS, LHCb, and
combined LEP experiments has been performed. Where necessary, measurement
results have been updated to incorporate an improved theoretical description
of the final state distributions. Experimental resolution effects, which are
required to propagate the impact of variations in the theoretical description
of $W$-boson production and decay, are accounted for using a realistic
emulation of the ATLAS, CDF, and D0 measurement procedures. Results for LHCb
are produced using the published analysis procedures.

The largest theoretical uncertainty arises from the parton distribution
functions. Results are presented for the two most recent PDF sets from the
NNPDF, CTEQ, and M(M/S)HT collaborations, as well as the most recent set from
the ABMP collaboration. Partial or negative correlations of PDF uncertainties
between the Tevatron, ATLAS, and LHCb experiments reduce the dependence of the
combined result on the PDF set. This dependence is nonetheless significant, as
the differences between individual sets is of the same order as the associated
uncertainty. The dependence of the measurements on PDF set are due to
differences in the input data sets and to the modelling assumptions in the
PDFs, and could ultimately limit the precision of future $m_{W}$ measurements
and combinations. Improving the experimental precision on $m_{W}$ requires a
better understanding of PDF model dependence, and of uncertainty correlations
between PDF sets.

The consistency of Drell-Yan cross-section measurements, as well as the
$m_{W}$ combination, is highest for the CT18 PDF set due to its large
uncertainties. With this PDF set the combination of LEP, LHC, and Tevatron Run
2 measurements gives a value $m_{W}=80394.6\pm 11.5$ MeV. This value has a
$\chi^{2}$ probability of $0.5\%$ and is therefore disfavoured. Other PDF sets
give probabilities of consistency between $2\times 10^{-5}$ and $3\times
10^{-3}$.

Good consistency is observed when all experiments other than CDF are combined,
with a resulting $W$-boson mass of $80369.2\pm 13.3$ MeV and a 91% probability
of consistency for the CT18 PDF set. When using this set and uncertainty for
the CDF measurement and for the combination of the others, the values differ
by 3.6 standard deviations. Further measurements or studies of procedures and
uncertainties are required to improve the understanding and consistency of a
world-average value of the $W$ boson mass.

## Acknowledgements

We thank Joshua Isaacson, Pavel Nadolsky, Frank Tackmann, and C.-P. Yuan for
fruitful discussions. We are grateful to Joshua Isaacson and Yao Fu for
providing us with the necessary inputs for generating events with Resbos2. We
thank the ATLAS, CDF, D0 and LHCb collaborations for providing inputs to this
analysis, and the CMS collaboration for participating in the working group. We
thank Raymond Brock, Paul Grannis, Rick Van Kooten, Hugh Montgomery, Pierre
Petroff, Heidi Schellman, Bill Ashmanskas, Bo Jayatilaka, Mark Lancaster,
Stefano Camarda, Aleksandra Dimitrievska, Monica Dunford, Pamela Ferrari,
Oldrich Kepka, Bogdan Malaescu, Philip Sommer, Guillaume Unal, Hengne Li,
Katharina Mueller, Matthew Needham, Monica Pepe-Altarelli, Federico Redi,
Lorenzo Sestini, Frederic Teubert, and Hang Yin for useful comments and
suggestions. We thank the LHC Electroweak Working Group for facilitating
useful discussions and providing computing resources for this work. MV is
supported by the ERC-CoG-865469 SPEAR grant, WB is supported by the UKRI
Future Leaders Fellowships grant MR/W009048/1, and SA is supported by the
Helmholtz Association contract W2/W3-123.

##  Appendix A Further information

We provide here additional information on the combination inputs and results.
Figure 10 shows the correlation matrices for the hadron-collider measurements
for the CT14, MMHT2014, NNPDF3.1 PDF sets. Tables 21–23 give the relative
weight of each measurement to various combinations of measurements.

Figure 10: PDF uncertainty correlation matrices for the CT14, MMHT2014, and NNPDF3.1 PDF sets, shown from top to bottom.  Measurement |  | ABMP16 | CT14 | CT18 | MMHT2014 | MSHT20 | NNPDF3.1 | NNPDF4.0
---|---|---|---|---|---|---|---|---
CDF |  | 85.3 | 86.8 | 88.1 | 86.4 | 86.3 | 86.2 | 86.1
D0 |  | 14.7 | 13.2 | 11.9 | 13.6 | 13.7 | 13.8 | 13.9
Table 21: Relative weights (in percent) of the CDF and D0 measurements for the Tevatron combination. Measurement |  | ABMP16 | CT14 | CT18 | MMHT2014 | MSHT20 | NNPDF3.1 | NNPDF4.0
---|---|---|---|---|---|---|---|---
ATLAS |  | 77.8 | 69.3 | 70.5 | 72.4 | 73.7 | 74.9 | 77.0
LHCb |  | 22.2 | 30.7 | 29.5 | 27.6 | 26.3 | 25.1 | 23.0
Table 22: Relative weights (in percent) of the ATLAS and LHCb measurements for the LHC combination. Measurement |  | ABMP16 | CT14 | CT18 | MMHT2014 | MSHT20 | NNPDF3.1 | NNPDF4.0
---|---|---|---|---|---|---|---|---
ATLAS |  | 19.9 | 28.2 | 28.2 | 20.2 | 22.8 | 19.7 | 24.8
LHCb |  | 6.0 | 12.7 | 13.4 | 9.7 | 10.7 | 8.5 | 8.9
CDF |  | 58.8 | 42.2 | 41.1 | 54.0 | 50.6 | 56.1 | 51.0
D0 |  | 10.1 | 6.0 | 5.1 | 8.3 | 7.9 | 8.7 | 8.1
LEP |  | 5.2 | 10.9 | 12.2 | 7.7 | 8.0 | 7.0 | 7.2
Table 23: Relative weights (in percent) of individual measurements for the combination of all available measurements. Measurement |  | ABMP16 | CT14 | CT18 | MMHT2014 | MSHT20 | NNPDF3.1 | NNPDF4.0
---|---|---|---|---|---|---|---|---
ATLAS |  | 21.0 | 31.6 | 32.1 | 21.9 | 24.7 | 21.2 | 26.7
LHCb |  | 6.3 | 14.3 | 15.3 | 10.5 | 11.7 | 9.1 | 9.5
CDF |  | 62.1 | 47.3 | 46.8 | 58.6 | 55.0 | 60.4 | 55.0
D0 |  | 10.6 | 6.7 | 5.8 | 9.0 | 8.6 | 9.3 | 8.7
Table 24: Relative weights (in percent) of individual measurements for the combination of all except the LEP measurement. Measurement |  | ABMP16 | CT14 | CT18 | MMHT2014 | MSHT20 | NNPDF3.1 | NNPDF4.0
---|---|---|---|---|---|---|---|---
ATLAS |  | 47.4 | 41.5 | 42.2 | 42.5 | 44.2 | 44.4 | 47.5
LHCb |  | 13.7 | 18.6 | 18.4 | 17.1 | 17.0 | 15.8 | 14.9
D0 |  | 27.5 | 24.6 | 23.1 | 26.9 | 25.7 | 26.8 | 25.3
LEP |  | 11.5 | 15.3 | 16.2 | 13.5 | 13.1 | 13.0 | 12.4
Table 25: Relative weights (in percent) of individual measurements for the combination of all except the CDF measurement. Measurement |  | ABMP16 | CT14 | CT18 | MMHT2014 | MSHT20 | NNPDF31 | NNPDF40
---|---|---|---|---|---|---|---|---
ATLAS |  | 21.9 | 28.8 | 28.7 | 21.5 | 24.0 | 21.3 | 26.2
LHCb |  | 6.5 | 13.0 | 13.6 | 10.1 | 11.1 | 8.9 | 9.2
CDF |  | 66.0 | 47.2 | 45.4 | 60.4 | 56.7 | 62.5 | 57.1
LEP |  | 5.6 | 11.1 | 12.3 | 8.0 | 8.2 | 7.3 | 7.5
Table 26: Relative weights (in percent) of individual measurements for the combination of all except the D0 measurement. Measurement |  | ABMP16 | CT14 | CT18 | MMHT2014 | MSHT20 | NNPDF3.1 | NNPDF4.0
---|---|---|---|---|---|---|---|---
LHCb |  | 7.2 | 14.8 | 15.7 | 10.4 | 12.1 | 9.4 | 11.0
CDF |  | 73.8 | 61.0 | 60.4 | 69.5 | 67.4 | 71.0 | 68.7
D0 |  | 12.7 | 9.2 | 7.9 | 10.9 | 10.7 | 11.3 | 11.1
LEP |  | 6.3 | 15.0 | 16.0 | 9.2 | 9.8 | 8.3 | 9.2
Table 27: Relative weights (in percent) of individual measurements for the combination of all except the ATLAS measurement. Measurement |  | ABMP16 | CT14 | CT18 | MMHT2014 | MSHT20 | NNPDF3.1 | NNPDF4.0
---|---|---|---|---|---|---|---|---
ATLAS |  | 21.0 | 30.1 | 30.7 | 21.1 | 24.6 | 21.0 | 27.1
CDF |  | 62.7 | 50.0 | 48.8 | 60.8 | 57.3 | 61.7 | 56.1
D0 |  | 10.8 | 7.2 | 6.2 | 9.4 | 9.0 | 9.6 | 8.9
LEP |  | 5.5 | 12.7 | 14.2 | 8.7 | 9.1 | 7.7 | 8.0
Table 28: Relative weights (in percent) of individual measurements for the
combination of all except the LHCb measurement.

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