CMSPASSMP23002  
Measurement of the W boson mass in protonproton collisions at $ \sqrt{s} = $ 13 TeV  
CMS Collaboration  
17 September 2024  
Abstract: In the standard model of particle physics, the masses of the carriers of the weak interaction, the W and Z bosons, are uniquely related. Physics beyond the standard model can change this relationship through the effects of virtual particle quantum loops, thus making it of paramount importance to measure these masses with the highest possible precision. While the mass of the Z boson is known to the remarkable precision of nearly 20 parts per million (2 MeV), thanks to the CERN LEP experimental program, the W boson mass is known much less precisely. The current precision of a global fit to electroweak data, used to predict the W boson mass in the standard model, yields an uncertainty of 6 MeV, so that reaching a comparable experimental precision would be a sensitive and fundamental test of the standard model. We report the first W boson mass measurement by the CMS Collaboration at the CERN LHC, based on a data sample collected in 2016 at the protonproton collision energy of 13 TeV. The W boson mass is measured using a sample of $ \mathrm{W}\to\mu\nu $ events via a highly granular maximum likelihood fit to the kinematical distributions of the daughter muons, separated by electric charge. The significant insitu constraints of theoretical inputs and their corresponding uncertainties provided by this novel approach, together with an accurate determination of the experimental effects, lead to a very precise W boson mass measurement, 80 360.2 $ \pm $ 9.9 MeV, in agreement with the standard model prediction.  
Links: CDS record (PDF) ; CADI line (restricted) ; 
Figures & Tables  Summary  Additional Figures  References  CMS Publications 

Figures  
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Figure 1:
Measured and simulated $ \mathrm{Z}\to\mu\mu $ dimuon mass distributions. The prediction reflects the best fit parameter values and uncertainties resulting from the maximum likelihood fit. The total uncertainty after the systematic uncertainty profiling procedure (gray band) and the 4.8 MeV uncertainty in the extracted $ m_{\mathrm{Z}} $ value (magenta lines) are also shown. 
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Figure 2:
The generatorlevel $ p_{\mathrm{T}}^{\mathrm{Z}} $ distribution, compared to the unfolded data, with the prediction and uncertainty before the maximum likelihood fit and with the distribution modified according to the postfit values and uncertainties of the relevant nuisance parameters. The distribution and uncertainties obtained from the Wlike $ m_{\mathrm{Z}} $ measurement are shown in purple, whereas the blue band shows the distribution obtained from the direct fit to the $ p_{\mathrm{T}}^{\mu\mu} $ distribution. The generatorlevel distribution predicted by SCETLIB+DYTURBO (before incorporating the constraints obtained from fits to the data) is shown in black. The ratio of the predictions and unfolded data to the prefit prediction (shown in gray), as well as their uncertainties, are shown by the shaded bands in the bottom panel. 
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Figure 3:
Measured and simulated $ p_{\mathrm{T}}^{\mu} $ distributions, with the prediction adjusted according to the best fit values of the nuisance parameters obtained from the maximum likelihood fit. The solid and dashed pink lines represent, respectively, an increase and decrease of $ m_{\mathrm{W}} $ by 9.9 MeV. The uncertainties in the predictions, after the systematic uncertainty profiling in the maximum likelihood fit, are shown by the shaded band. 
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Figure 4:
The $ m_{\mathrm{W}} $ measurement from this analysis (in red) is compared with those of LEP [9], D0 [14], CDF [17], LHCb [19], and ATLAS [20]. The global EW fit prediction [1] is represented by the gray band. 
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Figure A1:
Measured scale factors (SFs) for muons with positive charge as a function of $ p_{\mathrm{T}}^{\mu} $ in two representative $ \eta^{\mu} $ bins, for the reconstruction (left) and identification (right) selection efficiency. The black (gray) circles are the binned SFs measured with the T&P technique using the nominal (alternative) signal model, with the error bars representing their statistical uncertainty. The solid red (blue) line is the result of the smoothing fit with a third order polynomial for the nominal (alternative) model. The gray band represents the statistical uncertainty from the smoothing of the nominal SFs. The yellow bands represent the regions outside the $ p_{\mathrm{T}}^{\mu} $ range used in the $ m_{\mathrm{W}} $ measurement. 
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Figure A1a:
Measured scale factors (SFs) for muons with positive charge as a function of $ p_{\mathrm{T}}^{\mu} $ in two representative $ \eta^{\mu} $ bins, for the reconstruction (left) and identification (right) selection efficiency. The black (gray) circles are the binned SFs measured with the T&P technique using the nominal (alternative) signal model, with the error bars representing their statistical uncertainty. The solid red (blue) line is the result of the smoothing fit with a third order polynomial for the nominal (alternative) model. The gray band represents the statistical uncertainty from the smoothing of the nominal SFs. The yellow bands represent the regions outside the $ p_{\mathrm{T}}^{\mu} $ range used in the $ m_{\mathrm{W}} $ measurement. 
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Figure A1b:
Measured scale factors (SFs) for muons with positive charge as a function of $ p_{\mathrm{T}}^{\mu} $ in two representative $ \eta^{\mu} $ bins, for the reconstruction (left) and identification (right) selection efficiency. The black (gray) circles are the binned SFs measured with the T&P technique using the nominal (alternative) signal model, with the error bars representing their statistical uncertainty. The solid red (blue) line is the result of the smoothing fit with a third order polynomial for the nominal (alternative) model. The gray band represents the statistical uncertainty from the smoothing of the nominal SFs. The yellow bands represent the regions outside the $ p_{\mathrm{T}}^{\mu} $ range used in the $ m_{\mathrm{W}} $ measurement. 
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Figure A2:
Measured and predicted $ \eta^{\mu} $ distributions in $ \mathrm{Z}\to\mu\mu $ events with the Wlike Z selection for positively (left) and negatively (right) charged muons. The total normalized uncertainties (statistical and systematic) are represented by the shaded bands. 
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Figure A2a:
Measured and predicted $ \eta^{\mu} $ distributions in $ \mathrm{Z}\to\mu\mu $ events with the Wlike Z selection for positively (left) and negatively (right) charged muons. The total normalized uncertainties (statistical and systematic) are represented by the shaded bands. 
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Figure A2b:
Measured and predicted $ \eta^{\mu} $ distributions in $ \mathrm{Z}\to\mu\mu $ events with the Wlike Z selection for positively (left) and negatively (right) charged muons. The total normalized uncertainties (statistical and systematic) are represented by the shaded bands. 
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Figure A3:
Measured and predicted $ m_{\mathrm{T}} $ distributions in Z (left) and W (right) events, after calibrating the hadronic recoil. The total normalized uncertainties (statistical and systematic) are represented by the shaded bands. 
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Figure A3a:
Measured and predicted $ m_{\mathrm{T}} $ distributions in Z (left) and W (right) events, after calibrating the hadronic recoil. The total normalized uncertainties (statistical and systematic) are represented by the shaded bands. 
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Figure A3b:
Measured and predicted $ m_{\mathrm{T}} $ distributions in Z (left) and W (right) events, after calibrating the hadronic recoil. The total normalized uncertainties (statistical and systematic) are represented by the shaded bands. 
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Figure A4:
The observed data and the prediction of the extended ABCD method, as described in the text, for the $ p_{\mathrm{T}}^{\mu} $ (left) and $ \eta^{\mu} $ (right) distributions, in a region enriched in events with nonprompt muons obtained by selecting muons compatible with being produced in a secondary vertex. Small contributions from events with a prompt lepton, evaluated using simulated samples, are shown by the red histogram. The total uncertainties (statistical and systematic) are represented by the shaded bands. 
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Figure A4a:
The observed data and the prediction of the extended ABCD method, as described in the text, for the $ p_{\mathrm{T}}^{\mu} $ (left) and $ \eta^{\mu} $ (right) distributions, in a region enriched in events with nonprompt muons obtained by selecting muons compatible with being produced in a secondary vertex. Small contributions from events with a prompt lepton, evaluated using simulated samples, are shown by the red histogram. The total uncertainties (statistical and systematic) are represented by the shaded bands. 
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Figure A4b:
The observed data and the prediction of the extended ABCD method, as described in the text, for the $ p_{\mathrm{T}}^{\mu} $ (left) and $ \eta^{\mu} $ (right) distributions, in a region enriched in events with nonprompt muons obtained by selecting muons compatible with being produced in a secondary vertex. Small contributions from events with a prompt lepton, evaluated using simulated samples, are shown by the red histogram. The total uncertainties (statistical and systematic) are represented by the shaded bands. 
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Figure A5:
The observed data and the prediction of the extended ABCD method after a fit to the $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}, q^{\mu}) $ distribution, as described in the text, for the $ p_{\mathrm{T}}^{\mu} $ (left) and $ \eta^{\mu} $ (right) distributions, in a region enriched in events with nonprompt muons obtained by selecting muons compatible with being produced in a secondary vertex. The upper plots show the prefit prediction and the lower ones show the prediction after a fit to the $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}, q^{\mu}) $ distribution. Small contributions from events with a prompt lepton, evaluated using simulated samples, are shown by the red histogram. The total uncertainties (statistical and systematic) are represented by the shaded bands. 
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Figure A5a:
The observed data and the prediction of the extended ABCD method after a fit to the $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}, q^{\mu}) $ distribution, as described in the text, for the $ p_{\mathrm{T}}^{\mu} $ (left) and $ \eta^{\mu} $ (right) distributions, in a region enriched in events with nonprompt muons obtained by selecting muons compatible with being produced in a secondary vertex. The upper plots show the prefit prediction and the lower ones show the prediction after a fit to the $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}, q^{\mu}) $ distribution. Small contributions from events with a prompt lepton, evaluated using simulated samples, are shown by the red histogram. The total uncertainties (statistical and systematic) are represented by the shaded bands. 
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Figure A5b:
The observed data and the prediction of the extended ABCD method after a fit to the $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}, q^{\mu}) $ distribution, as described in the text, for the $ p_{\mathrm{T}}^{\mu} $ (left) and $ \eta^{\mu} $ (right) distributions, in a region enriched in events with nonprompt muons obtained by selecting muons compatible with being produced in a secondary vertex. The upper plots show the prefit prediction and the lower ones show the prediction after a fit to the $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}, q^{\mu}) $ distribution. Small contributions from events with a prompt lepton, evaluated using simulated samples, are shown by the red histogram. The total uncertainties (statistical and systematic) are represented by the shaded bands. 
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Figure A6:
Chargeindependent (left) and chargedependent (right) closure results from fits using $ \mathrm{J}/\psi $, $ \Upsilon $ (1S), and Z events. The chargeindependent closure plot shows an equivalent magnetic field scale factor and the chargedependent closure plot shows an equivalent misalignment term. The points with error bars represent the scale and statistical uncertainty associated with the closure test, while the band represents the corresponding statistical uncertainty in the calibration parameters themselves, from the $ \mathrm{J}/\psi $ calibration sample. The calibration uncertainties are fully uncorrelated from the Z and $ \Upsilon $ (1S) closure uncertainties, but very strongly correlated with the $ \mathrm{J}/\psi $ closure uncertainties, since they use the same data. 
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Figure A6a:
Chargeindependent (left) and chargedependent (right) closure results from fits using $ \mathrm{J}/\psi $, $ \Upsilon $ (1S), and Z events. The chargeindependent closure plot shows an equivalent magnetic field scale factor and the chargedependent closure plot shows an equivalent misalignment term. The points with error bars represent the scale and statistical uncertainty associated with the closure test, while the band represents the corresponding statistical uncertainty in the calibration parameters themselves, from the $ \mathrm{J}/\psi $ calibration sample. The calibration uncertainties are fully uncorrelated from the Z and $ \Upsilon $ (1S) closure uncertainties, but very strongly correlated with the $ \mathrm{J}/\psi $ closure uncertainties, since they use the same data. 
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Figure A6b:
Chargeindependent (left) and chargedependent (right) closure results from fits using $ \mathrm{J}/\psi $, $ \Upsilon $ (1S), and Z events. The chargeindependent closure plot shows an equivalent magnetic field scale factor and the chargedependent closure plot shows an equivalent misalignment term. The points with error bars represent the scale and statistical uncertainty associated with the closure test, while the band represents the corresponding statistical uncertainty in the calibration parameters themselves, from the $ \mathrm{J}/\psi $ calibration sample. The calibration uncertainties are fully uncorrelated from the Z and $ \Upsilon $ (1S) closure uncertainties, but very strongly correlated with the $ \mathrm{J}/\psi $ closure uncertainties, since they use the same data. 
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Figure A7:
Measured and simulated $ \mathrm{Z}\to\mu\mu $ dimuon mass distributions, after applying the muon momentum scale and resolution corrections. The normalization of the simulated spectrum is scaled to the measured distribution to better illustrate the agreement between the two $ \mathrm{Z}\to\mu\mu $ line shapes. 
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Figure A8:
Measured and simulated $ p_{\mathrm{T}}^{\mu\mu} $ (left) and $ p_{\mathrm{T}}^{\mu} $ (right) distributions in selected $ \mathrm{Z}\to\mu\mu $ events. The standalone uncorrected MINNLO$_{\mathrm{PS}}$ predictions are shown in dashed gray. The nominal predictions (blue) correct the POWHEG MINNLO$_\mathrm{PS}$ $ p_{\mathrm{T}}^{\mathrm{V}} $ with SCETLIB+DYTURBO at N$^{3}$LL+NNLO, as described in the text. Different sources of uncertainty are shown as solid bands in the bottom panel: the fixedorder uncertainty evaluated with $ \mu_{\mathrm{R}} $ and $ \mu_{\mathrm{F}} $ variations and the uncertainty in the resummation and fixed order matching (orange), resummed prediction using theory nuisance parameters (pink), the CollinsSoper (CS) kernel nonperturbative uncertainty (green), and other nonperturbative uncertainties (light blue). These sources of uncertainty impact different ranges of the $ p_{\mathrm{T}}^{\mathrm{V}} $ and $ p_{\mathrm{T}}^{\mu} $ distributions. 
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Figure A8a:
Measured and simulated $ p_{\mathrm{T}}^{\mu\mu} $ distribution in selected $ \mathrm{Z}\to\mu\mu $ events. The standalone uncorrected MINNLO$_{\mathrm{PS}}$ predictions are shown in dashed gray. The nominal predictions (blue) correct the POWHEG MINNLO$_\mathrm{PS}$ $ p_{\mathrm{T}}^{\mathrm{V}} $ with SCETLIB+DYTURBO at N$^{3}$LL+NNLO, as described in the text. Different sources of uncertainty are shown as solid bands in the bottom panel: the fixedorder uncertainty evaluated with $ \mu_{\mathrm{R}} $ and $ \mu_{\mathrm{F}} $ variations and the uncertainty in the resummation and fixed order matching (orange), resummed prediction using theory nuisance parameters (pink), the CollinsSoper (CS) kernel nonperturbative uncertainty (green), and other nonperturbative uncertainties (light blue). These sources of uncertainty impact different ranges of distribution. 
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Figure A8b:
Measured and simulated $ p_{\mathrm{T}}^{\mu} $ distribution in selected $ \mathrm{Z}\to\mu\mu $ events. The standalone uncorrected MINNLO$_{\mathrm{PS}}$ predictions are shown in dashed gray. The nominal predictions (blue) correct the POWHEG MINNLO$_\mathrm{PS}$ $ p_{\mathrm{T}}^{\mathrm{V}} $ with SCETLIB+DYTURBO at N$^{3}$LL+NNLO, as described in the text. Different sources of uncertainty are shown as solid bands in the bottom panel: the fixedorder uncertainty evaluated with $ \mu_{\mathrm{R}} $ and $ \mu_{\mathrm{F}} $ variations and the uncertainty in the resummation and fixed order matching (orange), resummed prediction using theory nuisance parameters (pink), the CollinsSoper (CS) kernel nonperturbative uncertainty (green), and other nonperturbative uncertainties (light blue). These sources of uncertainty impact different ranges of distribution. 
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Figure A9:
The predicted $ p_{\mathrm{T}}^{\mathrm{Z}} $ (left) and $ p_{\mathrm{T}}^{\mathrm{W}} $ (right) distributions at generatorlevel with no selection applied to the muons (or muon and neutrino, in the W case) from the decay. The bottom panel shows the impact in the distribution of the ten theory nuisance parameters described in the text. The label ``BF" refers to the proton beam functions. 
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Figure A9a:
The predicted $ p_{\mathrm{T}}^{\mathrm{Z}} $ (left) and $ p_{\mathrm{T}}^{\mathrm{W}} $ (right) distributions at generatorlevel with no selection applied to the muons (or muon and neutrino, in the W case) from the decay. The bottom panel shows the impact in the distribution of the ten theory nuisance parameters described in the text. The label ``BF" refers to the proton beam functions. 
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Figure A9b:
The predicted $ p_{\mathrm{T}}^{\mathrm{Z}} $ (left) and $ p_{\mathrm{T}}^{\mathrm{W}} $ (right) distributions at generatorlevel with no selection applied to the muons (or muon and neutrino, in the W case) from the decay. The bottom panel shows the impact in the distribution of the ten theory nuisance parameters described in the text. The label ``BF" refers to the proton beam functions. 
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Figure A10:
The predicted $ p_{\mathrm{T}}^{\mu} $ distribution for Z (left) and W (right) events, normalized to the number of data events, compared to the observed data. The bottom panel shows the impact on the distribution of the ten theory nuisance parameters described in the text. The label ``BF" refers to the proton beam functions. 
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Figure A10a:
The predicted $ p_{\mathrm{T}}^{\mu} $ distribution for Z (left) and W (right) events, normalized to the number of data events, compared to the observed data. The bottom panel shows the impact on the distribution of the ten theory nuisance parameters described in the text. The label ``BF" refers to the proton beam functions. 
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Figure A10b:
The predicted $ p_{\mathrm{T}}^{\mu} $ distribution for Z (left) and W (right) events, normalized to the number of data events, compared to the observed data. The bottom panel shows the impact on the distribution of the ten theory nuisance parameters described in the text. The label ``BF" refers to the proton beam functions. 
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Figure A11:
Measured and predicted dimuon rapidity distributions for $ \mathrm{Z}\to\mu\mu $ events. The nominal prediction, obtained with the CT18Z PDF set, is shown in filled light red. The uncertainty, evaluated as the sum of the eigenvector variation sets, is represented by the filled band in the lower panel. The predictions using the PDF4LHC21, MSHT20, NNPDF4.0, and CT18 sets are also shown (without uncertainty bands). 
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Figure A12:
Measured and predicted $ \eta^{\mu} $ distributions for selected $ \mathrm{W^+} $ (left) and $ \mathrm{W^} $ (right) events. The nominal prediction, obtained with the CT18Z PDF set, is shown in filled light red. The uncertainty, evaluated as the sum of the eigenvector variation sets, is represented by the filled band in the lower panel. The predictions using the PDF4LHC21, MSHT20, NNPDF4.0, and CT18 sets are also shown (without uncertainty bands). 
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Figure A12a:
Measured and predicted $ \eta^{\mu} $ distributions for selected $ \mathrm{W^+} $ (left) and $ \mathrm{W^} $ (right) events. The nominal prediction, obtained with the CT18Z PDF set, is shown in filled light red. The uncertainty, evaluated as the sum of the eigenvector variation sets, is represented by the filled band in the lower panel. The predictions using the PDF4LHC21, MSHT20, NNPDF4.0, and CT18 sets are also shown (without uncertainty bands). 
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Figure A12b:
Measured and predicted $ \eta^{\mu} $ distributions for selected $ \mathrm{W^+} $ (left) and $ \mathrm{W^} $ (right) events. The nominal prediction, obtained with the CT18Z PDF set, is shown in filled light red. The uncertainty, evaluated as the sum of the eigenvector variation sets, is represented by the filled band in the lower panel. The predictions using the PDF4LHC21, MSHT20, NNPDF4.0, and CT18 sets are also shown (without uncertainty bands). 
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Figure A13:
Unfolded differential cross sections as function of $ p_{\mathrm{T}}^{\mathrm{Z}} $ (left) and $ y^{\mathrm{Z}} $ (right) compared to the prefit prediction. 
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Figure A13a:
Unfolded differential cross sections as function of $ p_{\mathrm{T}}^{\mathrm{Z}} $ (left) and $ y^{\mathrm{Z}} $ (right) compared to the prefit prediction. 
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Figure A13b:
Unfolded differential cross sections as function of $ p_{\mathrm{T}}^{\mathrm{Z}} $ (left) and $ y^{\mathrm{Z}} $ (right) compared to the prefit prediction. 
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Figure A14:
Measured and simulated $ p_{\mathrm{T}}^{\mu\mu} $ distributions in the Z boson events, with the normalization and uncertainties of the prediction set to the postfit values. The gray band represents the total systematic uncertainty. 
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Figure A15:
Comparison of the nominal result and its theory uncertainty, using SCETLIB+DYTURBO at N$^{3}$LL+NNLO, with the value of $ m_{\mathrm{W}} $ measured when using alternative approaches to the $ p_{\mathrm{T}}^{\mathrm{Z}} $ modeling and its uncertainty. The impact of correcting the $ p_{\mathrm{T}}^{\mathrm{Z}} $ distribution directly to the $ p_{\mathrm{T}}^{\mu\mu} $ data, via binbybin reweighting, is also shown. The dashdotted black line represents the nominal result, while the shaded gray band shows the $ p_{\mathrm{T}}^{\mathrm{Z}} $modeling uncertainty. The results from alternative approaches to the $ p_{\mathrm{T}}^{\mathrm{Z}} $modeling and uncertainty are shown as points. The $ p_{\mathrm{T}}^{\mathrm{Z}} $modeling uncertainties are shown as the inner bars, while the outer bars denote the total uncertainty. 
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Figure A16:
Comparison of the nominal result and its theory uncertainty, using SCETLIB+DYTURBO at N$^{3}$LL+NNLO, with the value of $ m_{\mathrm{W}} $ measured when using alternative approaches to the $ p_{\mathrm{T}}^{\mathrm{W}} $ modeling and its uncertainty. The impact of correcting the $ p_{\mathrm{T}}^{\mathrm{W}} $ distribution with the $ p_{\mathrm{T}}^{\mu\mu} $ data, both via binbybin reweighting corrections and via a simultaneous maximum likelihood fit, is also shown. The dashdotted black line represents the nominal result, while the shaded gray band shows the $ p_{\mathrm{T}}^{\mathrm{W}} $modeling uncertainty. The results from alternative approaches to the $ p_{\mathrm{T}}^{\mathrm{W}} $modeling and uncertainty are shown as points. The $ p_{\mathrm{T}}^{\mathrm{W}} $modeling uncertainties are shown as the inner bars, while the outer bars denote the total uncertainty. 
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Figure A16a:
Comparison of the nominal result and its theory uncertainty, using SCETLIB+DYTURBO at N$^{3}$LL+NNLO, with the value of $ m_{\mathrm{W}} $ measured when using alternative approaches to the $ p_{\mathrm{T}}^{\mathrm{W}} $ modeling and its uncertainty. The impact of correcting the $ p_{\mathrm{T}}^{\mathrm{W}} $ distribution with the $ p_{\mathrm{T}}^{\mu\mu} $ data, both via binbybin reweighting corrections and via a simultaneous maximum likelihood fit, is also shown. The dashdotted black line represents the nominal result, while the shaded gray band shows the $ p_{\mathrm{T}}^{\mathrm{W}} $modeling uncertainty. The results from alternative approaches to the $ p_{\mathrm{T}}^{\mathrm{W}} $modeling and uncertainty are shown as points. The $ p_{\mathrm{T}}^{\mathrm{W}} $modeling uncertainties are shown as the inner bars, while the outer bars denote the total uncertainty. 
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Figure A16b:
Comparison of the nominal result and its theory uncertainty, using SCETLIB+DYTURBO at N$^{3}$LL+NNLO, with the value of $ m_{\mathrm{W}} $ measured when using alternative approaches to the $ p_{\mathrm{T}}^{\mathrm{W}} $ modeling and its uncertainty. The impact of correcting the $ p_{\mathrm{T}}^{\mathrm{W}} $ distribution with the $ p_{\mathrm{T}}^{\mu\mu} $ data, both via binbybin reweighting corrections and via a simultaneous maximum likelihood fit, is also shown. The dashdotted black line represents the nominal result, while the shaded gray band shows the $ p_{\mathrm{T}}^{\mathrm{W}} $modeling uncertainty. The results from alternative approaches to the $ p_{\mathrm{T}}^{\mathrm{W}} $modeling and uncertainty are shown as points. The $ p_{\mathrm{T}}^{\mathrm{W}} $modeling uncertainties are shown as the inner bars, while the outer bars denote the total uncertainty. 
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Figure A17:
Measured W boson mass with the helicity fit for different scaling scenarios of the prefit helicity cross section uncertainties, compared with the main result. The initial uncertainties of the $ \sigma_3 $ component and of the other components are denoted as $ \Delta_{\sigma_3} $ and $ \Delta_{\sigma_{\text{others}}} $, respectively. 
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Figure A18:
Extraction of the differential cross section as a function of the W boson transverse momentum (left) and rapidity (right) from the muon $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}, q^{\mu}) $ distributions in data, using the helicity fit approach. The generatorlevel distributions predicted by SCETLIB+DYTURBO before incorporating insitu constraints are shown in gray. The ratio of the postfit predictions (in red) to the prefit prediction (in gray), as well as their uncertainties, are shown by the shaded bands in the bottom panel. 
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Figure A18a:
Extraction of the differential cross section as a function of the W boson transverse momentum (left) and rapidity (right) from the muon $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}, q^{\mu}) $ distributions in data, using the helicity fit approach. The generatorlevel distributions predicted by SCETLIB+DYTURBO before incorporating insitu constraints are shown in gray. The ratio of the postfit predictions (in red) to the prefit prediction (in gray), as well as their uncertainties, are shown by the shaded bands in the bottom panel. 
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Figure A18b:
Extraction of the differential cross section as a function of the W boson transverse momentum (left) and rapidity (right) from the muon $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}, q^{\mu}) $ distributions in data, using the helicity fit approach. The generatorlevel distributions predicted by SCETLIB+DYTURBO before incorporating insitu constraints are shown in gray. The ratio of the postfit predictions (in red) to the prefit prediction (in gray), as well as their uncertainties, are shown by the shaded bands in the bottom panel. 
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Figure A19:
Measured and simulated $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ distributions used in the Wlike $ m_{\mathrm{Z}} $ measurement, for positively (upper) and negatively (lower) charged muons. The twodimensional distribution is ``unrolled" such that each bin on the $ x $axis represents one $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ cell. The gray band represents the uncertainty in the prediction, before the fit to the data. 
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Figure A19a:
Measured and simulated $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ distributions used in the Wlike $ m_{\mathrm{Z}} $ measurement, for positively (upper) and negatively (lower) charged muons. The twodimensional distribution is ``unrolled" such that each bin on the $ x $axis represents one $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ cell. The gray band represents the uncertainty in the prediction, before the fit to the data. 
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Figure A19b:
Measured and simulated $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ distributions used in the Wlike $ m_{\mathrm{Z}} $ measurement, for positively (upper) and negatively (lower) charged muons. The twodimensional distribution is ``unrolled" such that each bin on the $ x $axis represents one $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ cell. The gray band represents the uncertainty in the prediction, before the fit to the data. 
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Figure A20:
Measured and simulated $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ distributions for positively (upper) and negatively (lower) charged muons. The twodimensional distribution is ``unrolled" such that each bin on the $ x $axis represents one $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ cell. The gray band represents the uncertainty in the prediction, before the fit to the data. 
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Figure A20a:
Measured and simulated $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ distributions for positively (upper) and negatively (lower) charged muons. The twodimensional distribution is ``unrolled" such that each bin on the $ x $axis represents one $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ cell. The gray band represents the uncertainty in the prediction, before the fit to the data. 
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Figure A20b:
Measured and simulated $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ distributions for positively (upper) and negatively (lower) charged muons. The twodimensional distribution is ``unrolled" such that each bin on the $ x $axis represents one $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ cell. The gray band represents the uncertainty in the prediction, before the fit to the data. 
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Figure A21:
Measured and simulated $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ distributions used in the Wlike $ m_{\mathrm{Z}} $ measurement, for positively (upper) and negatively (lower) charged muons. The predictions and their uncertainties are adjusted to the best fit values obtained from the maximum likelihood fit. The twodimensional distribution is ``unrolled" such that each bin on the $ x $axis represents one $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ cell. The gray band represents the full uncertainty in the prediction, after the nuisances parameters are adjusted to the best fit values. 
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Figure A21a:
Measured and simulated $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ distributions used in the Wlike $ m_{\mathrm{Z}} $ measurement, for positively (upper) and negatively (lower) charged muons. The predictions and their uncertainties are adjusted to the best fit values obtained from the maximum likelihood fit. The twodimensional distribution is ``unrolled" such that each bin on the $ x $axis represents one $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ cell. The gray band represents the full uncertainty in the prediction, after the nuisances parameters are adjusted to the best fit values. 
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Figure A21b:
Measured and simulated $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ distributions used in the Wlike $ m_{\mathrm{Z}} $ measurement, for positively (upper) and negatively (lower) charged muons. The predictions and their uncertainties are adjusted to the best fit values obtained from the maximum likelihood fit. The twodimensional distribution is ``unrolled" such that each bin on the $ x $axis represents one $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ cell. The gray band represents the full uncertainty in the prediction, after the nuisances parameters are adjusted to the best fit values. 
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Figure A22:
Measured and simulated $ p_{\mathrm{T}}^{\mu} $ distributions, with the prediction adjusted according to the best fit values of nuisance parameters obtained from the maximum likelihood fit of the Wlike $ m_{\mathrm{Z}} $ analysis. The solid and dashed purple lines represent, respectively, an increase and decrease of $ m_{\mathrm{Z}} $ by 9.9 MeV. The uncertainties in the predictions, after the systematic uncertainty profiling in the maximum likelihood fit, are shown by the shaded band. 
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Figure A23:
Measured and simulated $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ distributions used in the $ m_{\mathrm{W}} $ measurement, for positively (upper) and negatively (lower) charged muons. The predictions and their uncertainties are adjusted to the best fit values obtained from the maximum likelihood fit. The twodimensional distribution is ``unrolled" such that each bin on the $ x $axis represents one $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ cell. The gray band represents the full uncertainty in the prediction, after the nuisances parameters are adjusted to the best fit values. 
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Figure A23a:
Measured and simulated $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ distributions used in the $ m_{\mathrm{W}} $ measurement, for positively (upper) and negatively (lower) charged muons. The predictions and their uncertainties are adjusted to the best fit values obtained from the maximum likelihood fit. The twodimensional distribution is ``unrolled" such that each bin on the $ x $axis represents one $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ cell. The gray band represents the full uncertainty in the prediction, after the nuisances parameters are adjusted to the best fit values. 
png pdf 
Figure A23b:
Measured and simulated $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ distributions used in the $ m_{\mathrm{W}} $ measurement, for positively (upper) and negatively (lower) charged muons. The predictions and their uncertainties are adjusted to the best fit values obtained from the maximum likelihood fit. The twodimensional distribution is ``unrolled" such that each bin on the $ x $axis represents one $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ cell. The gray band represents the full uncertainty in the prediction, after the nuisances parameters are adjusted to the best fit values. 
png pdf 
Figure A24:
Measured $ m_{\mathrm{W}} $ values, for seven recent PDF sets, when using the original uncertainty for the given set (left) and when the uncertainties are scaled to accommodate the central prediction of the other sets (right). Each point corresponds to the result obtained when using the indicated PDF set and its uncertainty for the simulated predictions. The inner bar shows the uncertainty from the PDF, and the outer bar shows the total uncertainty. The nominal result, using CT18Z, is shown by the red line, with the CT18Z PDF uncertainty shown in light gray. The scaling procedure improves the consistency of the $ m_{\mathrm{W}} $ values across the PDF sets and with the nominal result. 
png pdf 
Figure A24a:
Measured $ m_{\mathrm{W}} $ values, for seven recent PDF sets, when using the original uncertainty for the given set (left) and when the uncertainties are scaled to accommodate the central prediction of the other sets (right). Each point corresponds to the result obtained when using the indicated PDF set and its uncertainty for the simulated predictions. The inner bar shows the uncertainty from the PDF, and the outer bar shows the total uncertainty. The nominal result, using CT18Z, is shown by the red line, with the CT18Z PDF uncertainty shown in light gray. The scaling procedure improves the consistency of the $ m_{\mathrm{W}} $ values across the PDF sets and with the nominal result. 
png pdf 
Figure A24b:
Measured $ m_{\mathrm{W}} $ values, for seven recent PDF sets, when using the original uncertainty for the given set (left) and when the uncertainties are scaled to accommodate the central prediction of the other sets (right). Each point corresponds to the result obtained when using the indicated PDF set and its uncertainty for the simulated predictions. The inner bar shows the uncertainty from the PDF, and the outer bar shows the total uncertainty. The nominal result, using CT18Z, is shown by the red line, with the CT18Z PDF uncertainty shown in light gray. The scaling procedure improves the consistency of the $ m_{\mathrm{W}} $ values across the PDF sets and with the nominal result. 
png pdf 
Figure A25:
The $ m_{\mathrm{Z}} $ fit split in three bins of kinematics of the two muons: $ \eta^{\mu} $ (both central, one central and one forward, and both forward) on the left, and $ \eta^{\mu} $ (both negative, one positive and one negative, and both positive) on the right. The result of a fit with three $ m_{\mathrm{Z}} $ parameters is compared the nominal $ m_{\mathrm{Z}} $ fit result and the $ \chi^2 $like compatibility of the two fits is also shown, as assessed via the saturated goodnessoffit test. The results show the uncertainty in $ m_{\mathrm{Z}} $, separating the calibration and statistical uncertainty contributions. 
png pdf 
Figure A25a:
The $ m_{\mathrm{Z}} $ fit split in three bins of kinematics of the two muons: $ \eta^{\mu} $ (both central, one central and one forward, and both forward) on the left, and $ \eta^{\mu} $ (both negative, one positive and one negative, and both positive) on the right. The result of a fit with three $ m_{\mathrm{Z}} $ parameters is compared the nominal $ m_{\mathrm{Z}} $ fit result and the $ \chi^2 $like compatibility of the two fits is also shown, as assessed via the saturated goodnessoffit test. The results show the uncertainty in $ m_{\mathrm{Z}} $, separating the calibration and statistical uncertainty contributions. 
png pdf 
Figure A25b:
The $ m_{\mathrm{Z}} $ fit split in three bins of kinematics of the two muons: $ \eta^{\mu} $ (both central, one central and one forward, and both forward) on the left, and $ \eta^{\mu} $ (both negative, one positive and one negative, and both positive) on the right. The result of a fit with three $ m_{\mathrm{Z}} $ parameters is compared the nominal $ m_{\mathrm{Z}} $ fit result and the $ \chi^2 $like compatibility of the two fits is also shown, as assessed via the saturated goodnessoffit test. The results show the uncertainty in $ m_{\mathrm{Z}} $, separating the calibration and statistical uncertainty contributions. 
png pdf 
Figure A26:
For the Wlike $ m_{\mathrm{Z}} $ analysis (left) and the $ m_{\mathrm{W}} $ measurement (right) the result of a fit with 24 $ m_{\mathrm{V}} $ parameters corresponding to different $ \eta^{\mu} $ ranges is compared with the nominal $ m_{\mathrm{V}} $ fit result. The $ \chi^2 $like compatibility of the two fits is also shown, assessed via the saturated goodnessoffit test. The results show the uncertainty in $ m_{\mathrm{V}} $, separating the calibration and statistical uncertainty contributions. 
png pdf 
Figure A26a:
For the Wlike $ m_{\mathrm{Z}} $ analysis (left) and the $ m_{\mathrm{W}} $ measurement (right) the result of a fit with 24 $ m_{\mathrm{V}} $ parameters corresponding to different $ \eta^{\mu} $ ranges is compared with the nominal $ m_{\mathrm{V}} $ fit result. The $ \chi^2 $like compatibility of the two fits is also shown, assessed via the saturated goodnessoffit test. The results show the uncertainty in $ m_{\mathrm{V}} $, separating the calibration and statistical uncertainty contributions. 
png pdf 
Figure A26b:
For the Wlike $ m_{\mathrm{Z}} $ analysis (left) and the $ m_{\mathrm{W}} $ measurement (right) the result of a fit with 24 $ m_{\mathrm{V}} $ parameters corresponding to different $ \eta^{\mu} $ ranges is compared with the nominal $ m_{\mathrm{V}} $ fit result. The $ \chi^2 $like compatibility of the two fits is also shown, assessed via the saturated goodnessoffit test. The results show the uncertainty in $ m_{\mathrm{V}} $, separating the calibration and statistical uncertainty contributions. 
Tables  
png pdf 
Table A1:
Breakdown of muon calibration uncertainties. 
png pdf 
Table A2:
Goodnessoffit test statistics for different PDF sets when fitting simultaneously the $ \eta^{\mu} $ distributions for selected $ \mathrm{W^+} $ ($ \mathrm{W^} $) events and the $ y^{\mu\mu} $ distribution for $ \mathrm{Z}\to\mu\mu $ events. The fit is performed in the nominal configuration with all uncertainties (left column), nominal configuration without PDF and $ \alpha_{s} $ uncertainties (middle column), and nominal configuration without theory uncertainties (right column). The $ p $value denotes the probability for the observed data to agree with a given configuration as well as, or worse than, it does. 
png pdf 
Table A3:
Prefit uncertainty scaling factors required to cover the central predictions of the considered PDF sets and postfit impact in $ m_{\mathrm{W}} $, with and without scaled PDF uncertainties. 
png pdf 
Table A4:
Dominant systematic uncertainties in the Wlike $ m_{\mathrm{Z}} $ and $ m_{\mathrm{W}} $ measurements, using the ``nominal'' [nonenonenone] and ``global'' [101] definition of the impacts. 
png pdf 
Table A5:
Number of nuisance parameters for the main groups of systematic uncertainties, for the Wlike Z and W fits. The measured mass parameter is also included, albeit it is treated in a special way in each fit. The number of parameters is displayed only once when it is the same for both fits, while ``$$'' means that this source is not relevant. For completeness, subgroups of parameters are also reported as indented labels for a few groups. 
png pdf 
Table A6:
Dominant systematic uncertainties in the Wlike $ m_{\mathrm{Z}} $ and $ m_{\mathrm{W}} $ measurements, comparing the mass difference between charges and the nominal charge combination, using nominal impacts. 
png pdf 
Table A7:
Dominant systematic uncertainties in the Wlike $ m_{\mathrm{Z}} $ and $ m_{\mathrm{W}} $ measurements, comparing the mass difference between charges and the nominal charge combination, using global impacts. 
png pdf 
Table A8:
The $ m_{\mathrm{W}} $ values measured for different PDF sets, with uncertainties scaled following the procedure described in Section 11.10 and with the default unscaled uncertainties. 
Summary 
In this paper we report the first W mass measurement by the CMS Collaboration at the CERN LHC, with a precision very similar to that of the recent CDF measurement and better than that of all other results. The W mass is extracted from a sample of $ \mathrm{W}\to\mu\nu $ decays, collected in 2016 at the protonproton collision energy of 13 TeV, via a highly granular maximum likelihood fit to the threedimensional distribution of the muon $ p_{\mathrm{T}}^{\mu} $, $ \eta^{\mu} $, and electric charge. A number of novel experimental techniques have been used, together with stateoftheart theoretical models, to improve the measurement accuracy. Both the data analysis methods and the treatment of the theory calculations used in the $ m_{\mathrm{W}} $ measurement have been validated in multiple ways, including a muon momentum calibration using only $ {\mathrm{J}/\psi} \to\mu\mu $ events and the extraction of $ m_{\mathrm{Z}} $ from a Wlike analysis of Z boson dimuon decays. The measured value, $ m_{\mathrm{W}} = $ 80 360.2 $ \pm $ 9.9 MeV, agrees with the expectation from the standard model electroweak fit and is consistent with the present world average (excluding CDF), as shown in Fig. 4. This measurement constitutes a significant step towards reaching an experimental value with a precision approaching that of the standard model prediction. 
Additional Figures  
png pdf 
Additional Figure 1:
Comparison between a fit of the calibration model and simulated data after Kalman Filter track fit (top), CVH refit (middle), and generalized global corrections (bottom). The comparison is performed in two muon pseudorapidity bins. 
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Additional Figure 1a:
Comparison between a fit of the calibration model and simulated data after Kalman Filter track fit (top), CVH refit (middle), and generalized global corrections (bottom). The comparison is performed in two muon pseudorapidity bins. 
png pdf 
Additional Figure 1b:
Comparison between a fit of the calibration model and simulated data after Kalman Filter track fit (top), CVH refit (middle), and generalized global corrections (bottom). The comparison is performed in two muon pseudorapidity bins. 
png pdf 
Additional Figure 1c:
Comparison between a fit of the calibration model and simulated data after Kalman Filter track fit (top), CVH refit (middle), and generalized global corrections (bottom). The comparison is performed in two muon pseudorapidity bins. 
png pdf 
Additional Figure 1d:
Comparison between a fit of the calibration model and simulated data after Kalman Filter track fit (top), CVH refit (middle), and generalized global corrections (bottom). The comparison is performed in two muon pseudorapidity bins. 
png pdf 
Additional Figure 1e:
Comparison between a fit of the calibration model and simulated data after Kalman Filter track fit (top), CVH refit (middle), and generalized global corrections (bottom). The comparison is performed in two muon pseudorapidity bins. 
png pdf 
Additional Figure 1f:
Comparison between a fit of the calibration model and simulated data after Kalman Filter track fit (top), CVH refit (middle), and generalized global corrections (bottom). The comparison is performed in two muon pseudorapidity bins. 
png pdf 
Additional Figure 2:
Example mass fits of $ {\mathrm{J}/\psi} \to\mu\mu $ events in the central (left) and forward (right) region of the detector. 
png pdf 
Additional Figure 2a:
Example mass fits of $ {\mathrm{J}/\psi} \to\mu\mu $ events in the central (left) and forward (right) region of the detector. 
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Additional Figure 2b:
Example mass fits of $ {\mathrm{J}/\psi} \to\mu\mu $ events in the central (left) and forward (right) region of the detector. 
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Additional Figure 3:
Extracted muon calibration factors. 
png pdf 
Additional Figure 4:
Measured and predicted $ p_{\mathrm{T}}^{\mu\mu} $ distribution with the prediction adjusted according to the best fit values of the nuisance parameters obtained from the maximum likelihood fit to the $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}, q^{\mu}) $ distribution with the Wlike selection using the helicity fit. The gray band represents the uncertainty in the prediction, after the fit to the $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}, q^{\mu}) $ data. 
png pdf 
Additional Figure 9:
Measurement of the charge difference between positive and negative W bosons using the Helicity Fit for the nominal configuration and alternative scenarios with modifications to the prefit uncertainty bands. 
png pdf 
Additional Figure 10:
Summary of W boson mass measurements including the Helicity Fit. 
png pdf 
Additional Figure 11:
The generatorlevel $ y^{\mathrm{Z}} $ distribution, compared to the unfolded data, with the prediction and uncertainty before the maximum likelihood fit and with the distribution modified according to the postfit values and uncertainties of the relevant nuisance parameters. The distribution and uncertainties obtained from the Wlike $ m_{\mathrm{Z}} $ measurement are shown in purple, whereas the blue band shows the distribution obtained from the direct fit to the $ p_{\mathrm{T}}^{\mu\mu} $ distribution. The generatorlevel distribution predicted by SCETLIB+DYTURBO (before incorporating the constraints obtained from fits to the data) is shown in black. The ratio of the predictions and unfolded data to the prefit prediction (shown in gray), as well as their uncertainties, are shown by the shaded bands in the bottom panel. 
png pdf 
Additional Figure 12:
Transverse mass distribution for W events in simulation, reconstructed with DeepMET and with Particle Flow MET. 
png pdf 
Additional Figure 13:
Measured and predicted transverse mass distribution in $ \mathrm{Z}\to\mu\mu $ events with the Wlike Z selection with the prediction adjusted according to the best fit values of the nuisance parameters obtained from the maximum likelihood fit to the $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}, q^{\mu}) $ distribution. The gray band represents the uncertainty in the prediction, after the fit to the $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}, q^{\mu}) $ data. 
png pdf 
Additional Figure 14:
For the Wlike $ m_{\mathrm{Z}} $ analysis (left) and the $ m_{\mathrm{W}} $ measurement (right) the result of a fit with 48 $ m_{\mathrm{V}} $ parameters corresponding to different $ \eta^{\mu} $ ranges for the two charges is compared with the nominal $ m_{\mathrm{V}} $ fit result. The $ \chi^2 $like compatibility of the two fits is also shown, assessed via the saturated goodnessoffit test. The results show the uncertainty in $ m_{\mathrm{V}} $, separating the calibration and statistical uncertainty contributions. 
png pdf 
Additional Figure 14a:
For the Wlike $ m_{\mathrm{Z}} $ analysis (left) and the $ m_{\mathrm{W}} $ measurement (right) the result of a fit with 48 $ m_{\mathrm{V}} $ parameters corresponding to different $ \eta^{\mu} $ ranges for the two charges is compared with the nominal $ m_{\mathrm{V}} $ fit result. The $ \chi^2 $like compatibility of the two fits is also shown, assessed via the saturated goodnessoffit test. The results show the uncertainty in $ m_{\mathrm{V}} $, separating the calibration and statistical uncertainty contributions. 
png pdf 
Additional Figure 14b:
For the Wlike $ m_{\mathrm{Z}} $ analysis (left) and the $ m_{\mathrm{W}} $ measurement (right) the result of a fit with 48 $ m_{\mathrm{V}} $ parameters corresponding to different $ \eta^{\mu} $ ranges for the two charges is compared with the nominal $ m_{\mathrm{V}} $ fit result. The $ \chi^2 $like compatibility of the two fits is also shown, assessed via the saturated goodnessoffit test. The results show the uncertainty in $ m_{\mathrm{V}} $, separating the calibration and statistical uncertainty contributions. 
png pdf 
Additional Figure 15:
The $ p_{\mathrm{T}}^{\mu} $ spectrum in simulation, showing variations of one of the eigenvectors of the CT18Z PDF set, and of the $ A_0 $ angular coefficient corresponding to the missing higher order uncertainties as compared to a 100 MeV variation in $ m_{\mathrm{W}} $. 
png pdf 
Additional Figure 16:
The $ p_{\mathrm{T}}^{\mu} $ spectrum in simulation, showing variations of two parameters of the nonperturbative model in the $ \mathrm{W} p_{\mathrm{T}} $ modeling as compared to a 100 MeV variation in $ m_{\mathrm{W}} $. 
png pdf 
Additional Figure 17:
The $ p_{\mathrm{T}}^{\mu} $ spectrum in simulation for $ \mu^+ $ (left) and $ \mu^ $ (right), showing a variation of the $ A_3 $ angular coefficient corresponding to the missing higher order uncertainties, as well as the variation associated with the effect of the Pythia intrinsic $ k_T $ treatment on the angular coefficients. 
png pdf 
Additional Figure 17a:
The $ p_{\mathrm{T}}^{\mu} $ spectrum in simulation for $ \mu^+ $ (left) and $ \mu^ $ (right), showing a variation of the $ A_3 $ angular coefficient corresponding to the missing higher order uncertainties, as well as the variation associated with the effect of the Pythia intrinsic $ k_T $ treatment on the angular coefficients. 
png pdf 
Additional Figure 17b:
The $ p_{\mathrm{T}}^{\mu} $ spectrum in simulation for $ \mu^+ $ (left) and $ \mu^ $ (right), showing a variation of the $ A_3 $ angular coefficient corresponding to the missing higher order uncertainties, as well as the variation associated with the effect of the Pythia intrinsic $ k_T $ treatment on the angular coefficients. 
png pdf 
Additional Figure 18:
The $ m_{\mu\mu} $ distribution in simulation showing variations of the electroweak uncertainties as compared to a 25 MeV variation in $ m_{\mathrm{Z}} $. 
png pdf 
Additional Figure 19:
Measured and predicted $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ distributions for positively (upper) and negatively (lower) charged muons, with the prediction adjusted according to the best fit values of the nuisance parameters obtained from the maximum likelihood fit. The twodimensional distribution is ``unrolled" such that each bin on the $ x $axis represents one $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ cell. The gray band represents the uncertainty in the prediction, after the fit to the data. 
png pdf 
Additional Figure 19a:
Measured and predicted $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ distributions for positively (upper) and negatively (lower) charged muons, with the prediction adjusted according to the best fit values of the nuisance parameters obtained from the maximum likelihood fit. The twodimensional distribution is ``unrolled" such that each bin on the $ x $axis represents one $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ cell. The gray band represents the uncertainty in the prediction, after the fit to the data. 
png pdf 
Additional Figure 19b:
Measured and predicted $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ distributions for positively (upper) and negatively (lower) charged muons, with the prediction adjusted according to the best fit values of the nuisance parameters obtained from the maximum likelihood fit. The twodimensional distribution is ``unrolled" such that each bin on the $ x $axis represents one $ (p_{\mathrm{T}}^{\mu}, \eta^{\mu}) $ cell. The gray band represents the uncertainty in the prediction, after the fit to the data. 
png pdf 
Additional Figure 20:
Measured and simulated $ p_{\mathrm{T}}^{\mu} $ distributions, The solid and dashed pink lines represent, respectively, an increase and decrease of $ m_{\mathrm{W}} $ by 100 MeV. The uncertainties in the predictions, before the systematic uncertainty profiling in the maximum likelihood fit, are shown by the shaded band. 
png pdf 
Additional Figure 21:
The $ p_{\mathrm{T}}^{\mu} $ spectrum in simulation for $ \mu^+ $ events, showing the effect of neglecting the hadronic recoil dependence of the isolation and trigger efficiency scale factors, as compared to a 100 MeV variation in $ m_{\mathrm{W}} $. 
png pdf 
Additional Figure 22:
Variations of quark masses in the variable flavor scheme MSHT PDF set for the muon from the Z (left) and W (right) compared to a 25 MeV variation of the masses of the bosons. 
png pdf 
Additional Figure 22a:
Variations of quark masses in the variable flavor scheme MSHT PDF set for the muon from the Z (left) and W (right) compared to a 25 MeV variation of the masses of the bosons. 
png pdf 
Additional Figure 22b:
Variations of quark masses in the variable flavor scheme MSHT PDF set for the muon from the Z (left) and W (right) compared to a 25 MeV variation of the masses of the bosons. 
png pdf 
Additional Figure 23:
Variations of $ m_{\mathrm{W}} $ by $ \pm $ 10 MeV for selected W events compared to the nominal prediction. 
png pdf 
Additional Figure 24:
Summary of extracted $ m_{\mathrm{Z}} $ values as compared to the PDG value. 
png pdf 
Additional Figure 25:
The $ m_{\mathrm{W}} $ measurement from this analaysis, and CMS measurements of $ m_\mathrm{t} $ and $ \sin^2\theta_{\mathrm{W}} $ are compared with the EW fit prediction in the $ m_\mathrm{t} \mbox{\textsl{vs.}} m_{\mathrm{W}} $ plane (left) and the $ \sin^2\theta_{\mathrm{W}} $ vs $ m_{\mathrm{W}} $ plane (right). 
png pdf 
Additional Figure 25a:
The $ m_{\mathrm{W}} $ measurement from this analaysis, and CMS measurements of $ m_\mathrm{t} $ and $ \sin^2\theta_{\mathrm{W}} $ are compared with the EW fit prediction in the $ m_\mathrm{t} \mbox{\textsl{vs.}} m_{\mathrm{W}} $ plane (left) and the $ \sin^2\theta_{\mathrm{W}} $ vs $ m_{\mathrm{W}} $ plane (right). 
png pdf 
Additional Figure 25b:
The $ m_{\mathrm{W}} $ measurement from this analaysis, and CMS measurements of $ m_\mathrm{t} $ and $ \sin^2\theta_{\mathrm{W}} $ are compared with the EW fit prediction in the $ m_\mathrm{t} \mbox{\textsl{vs.}} m_{\mathrm{W}} $ plane (left) and the $ \sin^2\theta_{\mathrm{W}} $ vs $ m_{\mathrm{W}} $ plane (right). 
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