CMSSMP18012 ; CERNEP2020116  
Measurements of the W boson rapidity, helicity, doubledifferential cross sections, and charge asymmetry in pp collisions at 13 TeV  
CMS Collaboration  
10 August 2020  
Phys. Rev. D 102 (2020) 092012  
Abstract: The differential cross section and charge asymmetry for inclusive W boson production at $\sqrt{s} = $ 13 TeV is measured for the two transverse polarization states as a function of the W boson absolute rapidity. The measurement uses events in which a W boson decays to a neutrino and either a muon or an electron. The data sample of protonproton collisions recorded with the CMS detector at the LHC in 2016 corresponds to an integrated luminosity of 35.9 fb$^{1}$. The differential cross section and its value normalized to the total inclusive W boson production cross section are measured over the rapidity range ${y_{\mathrm{W}}} < $ 2.5. In addition to the total fiducial cross section, the W boson doubledifferential cross section, ${\mathrm{d}}^2\sigma/{\mathrm{d}}{p_{\mathrm{T}}}^{\ell}{\mathrm{d}}{\eta^{\ell}} $, and the charge asymmetry are measured as functions of the charged lepton transverse momentum and pseudorapidity. The precision of these measurements is used to constrain the parton distribution functions of the proton using the nexttoleading order NNPDF3.0 set.  
Links: eprint arXiv:2008.04174 [hepex] (PDF) ; CDS record ; inSPIRE record ; HepData record ; CADI line (restricted) ; 
Figures & Tables  Summary  Additional Figures & Material  References  CMS Publications 

Figures  
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Figure 1:
Generatorlevel distributions of the W boson ${{p_{\mathrm {T}}} ^{\mathrm{W}}}$ (top), ${{ y_{\mathrm{W}} }}$ (center), and the resulting $\eta $ distribution of the charged lepton (bottom) after reweighting each of the helicity components for positively (left) and negatively (right) charged W bosons. 
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Figure 1a:
Generatorlevel distributions of the W boson ${{p_{\mathrm {T}}} ^{\mathrm{W}}}$ after reweighting each of the helicity components for positively charged W bosons. 
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Figure 1b:
Generatorlevel distributions of the W boson ${{p_{\mathrm {T}}} ^{\mathrm{W}}}$ after reweighting each of the helicity components for negatively charged W bosons. 
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Figure 1c:
Generatorlevel distributions of the W boson ${{ y_{\mathrm{W}} }}$ after reweighting each of the helicity components for positively charged W bosons. 
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Figure 1d:
Generatorlevel distributions of the W boson ${{ y_{\mathrm{W}} }}$ after reweighting each of the helicity components for negatively charged W bosons. 
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Figure 1e:
$\eta $ distributions of the charged lepton after reweighting each of the helicity components for positively charged W bosons. 
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Figure 1f:
$\eta $ distributions of the charged lepton after reweighting each of the helicity components for negatively charged W bosons. 
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Figure 2:
Distributions of 2D templates of ${{p_{\mathrm {T}}} ^{\ell}}$ versus ${\eta ^{\ell}}$, for simulated positively charged W bosons events in different helicity or rapidity bins. Templates for the muon channel are shown. Blue: ${\mathrm{W} ^+_\mathrm {R}}$ with $ {{ y_{\mathrm{W}} }} < 0.25$, red: ${\mathrm{W} ^+_\mathrm {R}}$ with 0.50 $ < {{ y_{\mathrm{W}} }} < $ 0.75, green: ${\mathrm{W} ^+_\mathrm {L}}$ with 2.00 $ < {{ y_{\mathrm{W}} }} < $ 2.25. 
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Figure 3:
Upper: relative impact of groups of uncertainties (as defined in the text) on the normalized signal cross sections as functions of the W boson rapidity for the ${\mathrm{W} ^_\mathrm {L}}$ case. Lower: absolute impact of uncertainties on the charge asymmetry of the ${\mathrm{W} _\mathrm {L}}$ boson. All impacts are shown for the combination of the muon and electron channels in the helicity fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure 3a:
Relative impact of groups of uncertainties (as defined in the text) on the normalized signal cross sections as functions of the W boson rapidity for the ${\mathrm{W} ^_\mathrm {L}}$ case. All impacts are shown for the combination of the muon and electron channels in the helicity fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure 3b:
Absolute impact of uncertainties on the charge asymmetry of the ${\mathrm{W} _\mathrm {L}}$ boson. All impacts are shown for the combination of the muon and electron channels in the helicity fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure 4:
Upper: relative impact of groups of systematic uncertainties (as defined in the text) on the normalized cross sections for the ${\mathrm{W^{+}}}$ case as functions of ${{ \eta ^{\ell} }}$. Lower: relative impact of uncertainties on charge asymmetry. All impacts are shown for the combination of the muon and electron channels in the doubledifferential cross section fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure 4a:
Relative impact of groups of systematic uncertainties (as defined in the text) on the normalized cross sections for the ${\mathrm{W^{+}}}$ case as functions of ${{ \eta ^{\ell} }}$. All impacts are shown for the combination of the muon and electron channels in the doubledifferential cross section fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure 4b:
Relative impact of uncertainties on charge asymmetry. All impacts are shown for the combination of the muon and electron channels in the doubledifferential cross section fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure 5:
Distributions of ${\eta ^{\mu}}$ (upper left), ${{p_{\mathrm {T}}} ^{\mu}}$ (upper right) and $\mathrm {bin_{unrolled}}$ (lower) for $ {\mathrm{W^{+}}} \to \mu ^+\nu $ events for observed data superimposed on signal plus background events. The signal and background processes are normalized to the result of the template fit. The cyan band over the datatoprediction ratio represents the uncertainty in the total yield in each bin after the profiling process. 
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Figure 5a:
Distribution of ${\eta ^{\mu}}$ for $ {\mathrm{W^{+}}} \to \mu ^+\nu $ events for observed data superimposed on signal plus background events. The signal and background processes are normalized to the result of the template fit. The cyan band over the datatoprediction ratio represents the uncertainty in the total yield in each bin after the profiling process. 
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Figure 5b:
Distribution of ${{p_{\mathrm {T}}} ^{\mu}}$ for $ {\mathrm{W^{+}}} \to \mu ^+\nu $ events for observed data superimposed on signal plus background events. The signal and background processes are normalized to the result of the template fit. The cyan band over the datatoprediction ratio represents the uncertainty in the total yield in each bin after the profiling process. 
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Figure 5c:
Distribution of $\mathrm {bin_{unrolled}}$ for $ {\mathrm{W^{+}}} \to \mu ^+\nu $ events for observed data superimposed on signal plus background events. The signal and background processes are normalized to the result of the template fit. The cyan band over the datatoprediction ratio represents the uncertainty in the total yield in each bin after the profiling process. 
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Figure 6:
Distributions of ${\eta ^{\mu}}$ (upper left), ${{p_{\mathrm {T}}} ^{\mu}}$ (upper right) and $\mathrm {bin_{unrolled}}$ (lower) for $ {\mathrm{W^{}}} \to \mu ^\bar{\nu} $ events for observed data superimposed on signal plus background events. The signal and background processes are normalized to the result of the template fit. The cyan band over the datatoprediction ratio represents the uncertainty in the total yield in each bin after the profiling process. 
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Figure 6a:
Distribution of ${\eta ^{\mu}}$ for $ {\mathrm{W^{}}} \to \mu ^\bar{\nu} $ events for observed data superimposed on signal plus background events. The signal and background processes are normalized to the result of the template fit. The cyan band over the datatoprediction ratio represents the uncertainty in the total yield in each bin after the profiling process. 
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Figure 6b:
Distribution of ${{p_{\mathrm {T}}} ^{\mu}}$ for $ {\mathrm{W^{}}} \to \mu ^\bar{\nu} $ events for observed data superimposed on signal plus background events. The signal and background processes are normalized to the result of the template fit. The cyan band over the datatoprediction ratio represents the uncertainty in the total yield in each bin after the profiling process. 
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Figure 6c:
Distribution of $\mathrm {bin_{unrolled}}$ for $ {\mathrm{W^{}}} \to \mu ^\bar{\nu} $ events for observed data superimposed on signal plus background events. The signal and background processes are normalized to the result of the template fit. The cyan band over the datatoprediction ratio represents the uncertainty in the total yield in each bin after the profiling process. 
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Figure 7:
Distributions of ${\eta ^{\mathrm{e}}}$ (upper left), ${{p_{\mathrm {T}}} ^{\mathrm{e}}}$ (upper right) and $\mathrm {bin_{unrolled}}$ (lower) for $ {\mathrm{W^{+}}} \to \mathrm{e} ^+\nu $ events for observed data superimposed on signal plus background events. The signal and background processes are normalized to the result of the template fit. The cyan band over the datatoprediction ratio represents the uncertainty in the total yield in each bin after the profiling process. 
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Figure 7a:
Distribution of $\mathrm {bin_{unrolled}}$ for $ {\mathrm{W^{+}}} \to \mathrm{e} ^+\nu $ events for observed data superimposed on signal plus background events. The signal and background processes are normalized to the result of the template fit. The cyan band over the datatoprediction ratio represents the uncertainty in the total yield in each bin after the profiling process. 
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Figure 7b:
Distributions of ${\eta ^{\mathrm{e}}}$ (upper left), ${{p_{\mathrm {T}}} ^{\mathrm{e}}}$ (upper right) and $\mathrm {bin_{unrolled}}$ (lower) for $ {\mathrm{W^{+}}} \to \mathrm{e} ^+\nu $ events for observed data superimposed on signal plus background events. The signal and background processes are normalized to the result of the template fit. The cyan band over the datatoprediction ratio represents the uncertainty in the total yield in each bin after the profiling process. 
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Figure 7c:
Distributions of ${\eta ^{\mathrm{e}}}$ (upper left), ${{p_{\mathrm {T}}} ^{\mathrm{e}}}$ (upper right) and $\mathrm {bin_{unrolled}}$ (lower) for $ {\mathrm{W^{+}}} \to \mathrm{e} ^+\nu $ events for observed data superimposed on signal plus background events. The signal and background processes are normalized to the result of the template fit. The cyan band over the datatoprediction ratio represents the uncertainty in the total yield in each bin after the profiling process. 
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Figure 8:
Distributions of ${\eta ^{\mathrm{e}}}$ (upper left), ${{p_{\mathrm {T}}} ^{\mathrm{e}}}$ (upper right) and $\mathrm {bin_{unrolled}}$ (lower) for $ {\mathrm{W^{}}} \to \mathrm{e} ^\bar{\nu} $ events for observed data superimposed on signal plus background events. The signal and background processes are normalized to the result of the template fit. The cyan band over the datatoprediction ratio represents the uncertainty in the total yield in each bin after the profiling process. 
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Figure 8a:
Distribution of ${\eta ^{\mathrm{e}}}$ for $ {\mathrm{W^{}}} \to \mathrm{e} ^\bar{\nu} $ events for observed data superimposed on signal plus background events. The signal and background processes are normalized to the result of the template fit. The cyan band over the datatoprediction ratio represents the uncertainty in the total yield in each bin after the profiling process. 
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Figure 8b:
Distribution of ${{p_{\mathrm {T}}} ^{\mathrm{e}}}$ for $ {\mathrm{W^{}}} \to \mathrm{e} ^\bar{\nu} $ events for observed data superimposed on signal plus background events. The signal and background processes are normalized to the result of the template fit. The cyan band over the datatoprediction ratio represents the uncertainty in the total yield in each bin after the profiling process. 
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Figure 8c:
Distribution of $\mathrm {bin_{unrolled}}$ for $ {\mathrm{W^{}}} \to \mathrm{e} ^\bar{\nu} $ events for observed data superimposed on signal plus background events. The signal and background processes are normalized to the result of the template fit. The cyan band over the datatoprediction ratio represents the uncertainty in the total yield in each bin after the profiling process. 
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Figure 9:
Measured normalized $ {\mathrm{W^{+}}} \to \ell ^+\nu $ (left plot) or $ {\mathrm{W^{}}} \to \ell ^\bar{\nu} $ (right plot) cross section as functions of ${{ y_{\mathrm{W}} }}$ for the lefthanded and righthanded helicity states from the combination of the muon and electron channels, normalized to the total cross section. Also shown is the ratio of the prediction from MadGraph5_aMC@NLO to the data. The MadGraph5_aMC@NLO $^{*}$ spectrum stands for the prediction with the ${{p_{\mathrm {T}}} ^{\mathrm{W}}}$ weighting applied. The lightlyfilled band corresponds to the expected uncertainty from the PDF variations, ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales, and ${\alpha _S}$. 
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Figure 9a:
Measured normalized $ {\mathrm{W^{+}}} \to \ell ^+\nu $ cross section as functions of ${{ y_{\mathrm{W}} }}$ for the lefthanded and righthanded helicity states from the combination of the muon and electron channels, normalized to the total cross section. Also shown is the ratio of the prediction from MadGraph5_aMC@NLO to the data. The MadGraph5_aMC@NLO $^{*}$ spectrum stands for the prediction with the ${{p_{\mathrm {T}}} ^{\mathrm{W}}}$ weighting applied. The lightlyfilled band corresponds to the expected uncertainty from the PDF variations, ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales, and ${\alpha _S}$. 
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Figure 9b:
Measured normalized $ {\mathrm{W^{}}} \to \ell ^\bar{\nu} $ cross section as functions of ${{ y_{\mathrm{W}} }}$ for the lefthanded and righthanded helicity states from the combination of the muon and electron channels, normalized to the total cross section. Also shown is the ratio of the prediction from MadGraph5_aMC@NLO to the data. The MadGraph5_aMC@NLO $^{*}$ spectrum stands for the prediction with the ${{p_{\mathrm {T}}} ^{\mathrm{W}}}$ weighting applied. The lightlyfilled band corresponds to the expected uncertainty from the PDF variations, ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales, and ${\alpha _S}$. 
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Figure 10:
Measured W boson charge asymmetry as functions of ${{ y_{\mathrm{W}} }}$ for the lefthanded and righthanded helicity states from the combination of the muon and electron channels. Also shown is the ratio of the prediction from MadGraph5_aMC@NLO to the data. The MadGraph5_aMC@NLO $^{*}$ spectrum stands for the prediction with the ${{p_{\mathrm {T}}} ^{\mathrm{W}}}$ weighting applied. The lightlyfilled band corresponds to the expected uncertainty from the PDF variations, ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales, and ${\alpha _S}$. 
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Figure 11:
Measured normalized $ {\mathrm{W^{+}}} \to \ell ^+\nu $ (left plot) and $ {\mathrm{W^{}}} \to \ell ^\bar{\nu} $ (right plot) cross section as a function of ${{ y_{\mathrm{W}} }}$ from the combination of the muon and electron channels, normalized to the total cross section, integrated over the W polarization states. Also shown is the ratio of the prediction from MadGraph5_aMC@NLO to the data. The MadGraph5_aMC@NLO $^{*}$ spectrum stands for the prediction with the ${{p_{\mathrm {T}}} ^{\mathrm{W}}}$ weighting applied. The lightlyfilled band corresponds to the expected uncertainty from the PDF variations, ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales, and ${\alpha _S}$. 
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Figure 11a:
Measured normalized $ {\mathrm{W^{+}}} \to \ell ^+\nu $ cross section as a function of ${{ y_{\mathrm{W}} }}$ from the combination of the muon and electron channels, normalized to the total cross section, integrated over the W polarization states. Also shown is the ratio of the prediction from MadGraph5_aMC@NLO to the data. The MadGraph5_aMC@NLO $^{*}$ spectrum stands for the prediction with the ${{p_{\mathrm {T}}} ^{\mathrm{W}}}$ weighting applied. The lightlyfilled band corresponds to the expected uncertainty from the PDF variations, ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales, and ${\alpha _S}$. 
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Figure 11b:
Measured normalized $ {\mathrm{W^{}}} \to \ell ^\bar{\nu} $ cross section as a function of ${{ y_{\mathrm{W}} }}$ from the combination of the muon and electron channels, normalized to the total cross section, integrated over the W polarization states. Also shown is the ratio of the prediction from MadGraph5_aMC@NLO to the data. The MadGraph5_aMC@NLO $^{*}$ spectrum stands for the prediction with the ${{p_{\mathrm {T}}} ^{\mathrm{W}}}$ weighting applied. The lightlyfilled band corresponds to the expected uncertainty from the PDF variations, ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales, and ${\alpha _S}$. 
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Figure 12:
Measured W charge asymmetry as a function of ${{ y_{\mathrm{W}} }}$ from the combination of the muon and electron channels, integrated over the W polarization states. Also shown is the ratio of the prediction from MadGraph5_aMC@NLO to the data. The MadGraph5_aMC@NLO $^{*}$ spectrum stands for the prediction with the ${{p_{\mathrm {T}}} ^{\mathrm{W}}}$ weighting applied. The lightlyfilled band corresponds to the expected uncertainty from the PDF variations, ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales, and ${\alpha _S}$. 
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Figure 13:
Measured absolute $ {\mathrm{W^{+}}} \to \ell ^+\nu $ (left plot) or $ {\mathrm{W^{}}} \to \ell ^\bar{\nu} $ (right plot) cross section as functions of ${{ y_{\mathrm{W}} }}$ from the combined flavor fit. The ratio of the prediction from MadGraph5_aMC@NLO to the data is also shown. The MadGraph5_aMC@NLO $^{*}$ spectrum stands for the prediction with the ${{p_{\mathrm {T}}} ^{\mathrm{W}}}$ weighting applied. The lightlyfilled band corresponds to the expected uncertainty from the PDF variations, ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales, and ${\alpha _S}$. 
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Figure 13a:
Measured absolute $ {\mathrm{W^{+}}} \to \ell ^+\nu $ cross section as functions of ${{ y_{\mathrm{W}} }}$ from the combined flavor fit. The ratio of the prediction from MadGraph5_aMC@NLO to the data is also shown. The MadGraph5_aMC@NLO $^{*}$ spectrum stands for the prediction with the ${{p_{\mathrm {T}}} ^{\mathrm{W}}}$ weighting applied. The lightlyfilled band corresponds to the expected uncertainty from the PDF variations, ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales, and ${\alpha _S}$. 
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Figure 13b:
Measured absolute $ {\mathrm{W^{}}} \to \ell ^\bar{\nu} $ cross section as functions of ${{ y_{\mathrm{W}} }}$ from the combined flavor fit. The ratio of the prediction from MadGraph5_aMC@NLO to the data is also shown. The MadGraph5_aMC@NLO $^{*}$ spectrum stands for the prediction with the ${{p_{\mathrm {T}}} ^{\mathrm{W}}}$ weighting applied. The lightlyfilled band corresponds to the expected uncertainty from the PDF variations, ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales, and ${\alpha _S}$. 
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Figure 14:
Normalized doubledifferential cross section as a function of ${{p_{\mathrm {T}}} ^{\ell}}$ and ${{ \eta ^{\ell} }}$, unrolled in a 1D histogram along ${{ \eta ^{\ell} }}$ for the positive (negative) charge on the upper (lower) plot. The lower panel in each plot shows the ratio of the observed and expected cross sections. The colored bands represent the prediction from MadGraph5_aMC@NLO with the expected uncertainty from the quadrature sum of the PDF$\oplus {\alpha _S} $ variations (blue) and the ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales (bordeaux). 
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Figure 14a:
Normalized doubledifferential cross section as a function of ${{p_{\mathrm {T}}} ^{\ell}}$ and ${{ \eta ^{\ell} }}$, unrolled in a 1D histogram along ${{ \eta ^{\ell} }}$ for the positive charge. The lower panelshows the ratio of the observed and expected cross sections. The colored bands represent the prediction from MadGraph5_aMC@NLO with the expected uncertainty from the quadrature sum of the PDF$\oplus {\alpha _S} $ variations (blue) and the ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales (bordeaux). 
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Figure 14b:
Normalized doubledifferential cross section as a function of ${{p_{\mathrm {T}}} ^{\ell}}$ and ${{ \eta ^{\ell} }}$, unrolled in a 1D histogram along ${{ \eta ^{\ell} }}$ for the negative charge. The lower panelshows the ratio of the observed and expected cross sections. The colored bands represent the prediction from MadGraph5_aMC@NLO with the expected uncertainty from the quadrature sum of the PDF$\oplus {\alpha _S} $ variations (blue) and the ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales (bordeaux). 
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Figure 15:
Doubledifferential W boson charge asymmetry as a function of ${{p_{\mathrm {T}}} ^{\ell}}$ and ${{ \eta ^{\ell} }}$, unrolled in a 1D histogram along ${{ \eta ^{\ell} }}$. The lower panel shows the difference of the observed and expected charge asymmetry. The colored bands represent the prediction from MadGraph5_aMC@NLO with the expected uncertainty from the quadrature sum of the PDF$\oplus {\alpha _S} $ variations (blue) and the ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales (bordeaux). 
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Figure 16:
Absolute doubledifferential cross section as a function of ${{p_{\mathrm {T}}} ^{\ell}}$ and ${{ \eta ^{\ell} }}$, unrolled in a 1D histogram along ${{p_{\mathrm {T}}} ^{\ell}}$ in bins of ${{ \eta ^{\ell} }}$ for the positive (negative) charge on upper (lower) panel. The combined muon and electron fit is shown in green markers, the muononly fit in blue markers, and the electrononly fit in red markers. The error bars correspond to the total uncertainty from the respective fits. The filled gray band in the lower panel represents the total uncertainty from the combined fit. 
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Figure 16a:
Absolute doubledifferential cross section as a function of ${{p_{\mathrm {T}}} ^{\ell}}$ and ${{ \eta ^{\ell} }}$, unrolled in a 1D histogram along ${{p_{\mathrm {T}}} ^{\ell}}$ in bins of ${{ \eta ^{\ell} }}$ for the positive charge. The combined muon and electron fit is shown in green markers, the muononly fit in blue markers, and the electrononly fit in red markers. The error bars correspond to the total uncertainty from the respective fits. The filled gray band in the lower panel represents the total uncertainty from the combined fit. 
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Figure 16b:
Absolute doubledifferential cross section as a function of ${{p_{\mathrm {T}}} ^{\ell}}$ and ${{ \eta ^{\ell} }}$, unrolled in a 1D histogram along ${{p_{\mathrm {T}}} ^{\ell}}$ in bins of ${{ \eta ^{\ell} }}$ for the negative charge. The combined muon and electron fit is shown in green markers, the muononly fit in blue markers, and the electrononly fit in red markers. The error bars correspond to the total uncertainty from the respective fits. The filled gray band in the lower panel represents the total uncertainty from the combined fit. 
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Figure 17:
Absolute differential cross section as a function of ${{ \eta ^{\ell} }}$ for the ${\mathrm{W^{+}} \to \ell ^+\nu}$ (left) and ${\mathrm{W^{}} \to \ell ^\bar{\nu}}$ channel (right). The measurement is the result of the combination of the muon and electron channels. The lower panel in each plot shows the ratio of observed and expected cross section. The colored bands represent the prediction from MadGraph5_aMC@NLO with the expected uncertainty from the quadrature sum of the PDF$\oplus {\alpha _S} $ variations (blue) and the ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales (bordeaux). 
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Figure 17a:
Absolute differential cross section as a function of ${{ \eta ^{\ell} }}$ for the ${\mathrm{W^{+}} \to \ell ^+\nu}$ channel. The measurement is the result of the combination of the muon and electron channels. The lower panel shows the ratio of observed and expected cross section. The colored bands represent the prediction from MadGraph5_aMC@NLO with the expected uncertainty from the quadrature sum of the PDF$\oplus {\alpha _S} $ variations (blue) and the ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales (bordeaux). 
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Figure 17b:
Absolute differential cross section as a function of ${{ \eta ^{\ell} }}$ for the ${\mathrm{W^{}} \to \ell ^\bar{\nu}}$ channel. The measurement is the result of the combination of the muon and electron channels. The lower panel shows the ratio of observed and expected cross section. The colored bands represent the prediction from MadGraph5_aMC@NLO with the expected uncertainty from the quadrature sum of the PDF$\oplus {\alpha _S} $ variations (blue) and the ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales (bordeaux). 
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Figure 18:
Absolute differential W boson charge asymmetry as a function of ${{ \eta ^{\ell} }}$. The measurement is the result of the combination of the muon and electron channels. The lower panel shows the difference of observed and expected charge asymmetry. The colored bands represent the prediction from MadGraph5_aMC@NLO with the expected uncertainty from the quadrature sum of the PDF$\oplus {\alpha _S} $ variations (blue) and the ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales (bordeaux). 
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Figure 19:
Ratio of the measured over predicted absolute inclusive cross section in the fiducial region 26 $ < {{p_{\mathrm {T}}} ^{\ell}} < $ 56 GeV and $ {{ \eta ^{\ell} }} < $ 2.5, chargeintegrated, chargedependent, and the ratio for ${\mathrm{W^{+}}}$ and ${\mathrm{W^{}}}$. The measurement is the result of the combination of the muon and electron channels. The colored bands represent the prediction from MadGraph5_aMC@NLO with the expected uncertainty from the quadrature sum of the PDF$\oplus {\alpha _S} $ variations (blue) and the ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales (bordeaux). 
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Figure 20:
Pulls and constraints of the 60 Hessian variations of the NNPDF3.0 PDF set, and of the ${\alpha _S}$ parameter, from the combined fit of muon and electron channels. The underlying fit is performed by fixing the W boson cross sections to their expectation in all helicity and charge processes. The cyan band represents the input values (which all have zero mean and width one), the orange bands show the postfit expected values, and black points represent the observed pulls and constraint values. 
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Figure A1:
Measured absolute $ {\mathrm{W^{+}}} \to \ell ^+\nu $ (left) and $ {\mathrm{W^{}}} \to \ell ^\bar{\nu} $ (right) cross section as a function of ${{ y_{\mathrm{W}} }}$ for the lefthanded and righthanded helicity states from the combination of muon and electron channels. The ratio of the prediction from MadGraph5_aMC@NLO to the data is also shown. The lightlyfilled band corresponds to the expected uncertainty from the PDF variations, ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales, and ${\alpha _S}$. 
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Figure A1a:
Measured absolute $ {\mathrm{W^{+}}} \to \ell ^+\nu $ (left) and $ {\mathrm{W^{}}} \to \ell ^\bar{\nu} $ (right) cross section as a function of ${{ y_{\mathrm{W}} }}$ for the lefthanded and righthanded helicity states from the combination of muon and electron channels. The ratio of the prediction from MadGraph5_aMC@NLO to the data is also shown. The lightlyfilled band corresponds to the expected uncertainty from the PDF variations, ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales, and ${\alpha _S}$. 
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Figure A1b:
Measured absolute $ {\mathrm{W^{+}}} \to \ell ^+\nu $ (left) and $ {\mathrm{W^{}}} \to \ell ^\bar{\nu} $ (right) cross section as a function of ${{ y_{\mathrm{W}} }}$ for the lefthanded and righthanded helicity states from the combination of muon and electron channels. The ratio of the prediction from MadGraph5_aMC@NLO to the data is also shown. The lightlyfilled band corresponds to the expected uncertainty from the PDF variations, ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales, and ${\alpha _S}$. 
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Figure A2:
Measured absolute $ {\mathrm{W^{+}}} \to \ell ^+\nu $ cross section as a function of ${{ y_{\mathrm{W}} }}$ from three distinct fits: the combination of muon and electron channels (green), the muononly fit (blue), and the electrononly fit (red). The ratio of the prediction from MadGraph5_aMC@NLO to the data is also shown. The lightlyfilled band corresponds to the expected uncertainty from the PDF variations, ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales, and ${\alpha _S}$. 
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Figure A3:
Measured absolute $ {\mathrm{W^{}}} \to \ell ^\bar{\nu} $ cross section as a function of ${{ y_{\mathrm{W}} }}$ from three distinct fits: the combination of muon and electron channels (green), the muononly fit (blue), and the electrononly fit (red). The ratio of the prediction from MadGraph5_aMC@NLO to the data is also shown. The lightlyfilled band corresponds to the expected uncertainty from the PDF variations, ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales, and ${\alpha _S}$. 
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Figure A4:
Measured W boson charge asymmetry as a function of ${{ y_{\mathrm{W}} }}$ from the combination of the muon and electron channels (black dots), compared with different theoretical predictions. The yellow band represents the default generator used in this analysis, MadGraph5_aMC@NLO with NNPDF3.0 PDF set, the pink band represents the {fewz} generator with NNPDF3.1 PDF set, and the cyan band represents the {fewz} generator with CT18 PDF set. The uncertainty bands of the prediction include PDF uncertainties only, which are dominant with respect to ${\alpha _S}$ or QCD scale variations for this quantity. 
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Figure A5:
Measured $A_4$ coefficient for $ {\mathrm{W^{+}}} \to \ell ^+\nu $ extracted from the fit of the polarized cross sections to the combined muon and electron channel fit. Note that $A_4$ is negative in this case, and the plotted quantity is $A_4$. The difference between the prediction from MadGraph5_aMC@NLO and the measured values is also shown. The lightlyfilled band corresponds to the expected uncertainty from the PDF variations, ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales, and ${\alpha _S}$. 
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Figure A6:
Measured $A_4$ coefficient for $ {\mathrm{W^{}}} \to \ell ^\nu $ extracted from the fit of the polarized cross sections to the combined muon and electron channel fit. The difference between the prediction from MadGraph5_aMC@NLO and the measured values is also shown. The lightlyfilled band corresponds to the expected uncertainty from the PDF variations, ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales, and ${\alpha _S}$. 
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Figure A7:
Correlation coefficients between the helicitydependent signal cross sections for $ {\mathrm{W^{+}}} \to \ell ^+\nu $ (left) and $ {\mathrm{W^{}}} \to \ell ^\bar{\nu} $ (right) extracted from the fit to the combined muon and electron channel fit. 
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Figure A7a:
Correlation coefficients between the helicitydependent signal cross sections for $ {\mathrm{W^{+}}} \to \ell ^+\nu $ (left) and $ {\mathrm{W^{}}} \to \ell ^\bar{\nu} $ (right) extracted from the fit to the combined muon and electron channel fit. 
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Figure A7b:
Correlation coefficients between the helicitydependent signal cross sections for $ {\mathrm{W^{+}}} \to \ell ^+\nu $ (left) and $ {\mathrm{W^{}}} \to \ell ^\bar{\nu} $ (right) extracted from the fit to the combined muon and electron channel fit. 
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Figure A8:
Correlation coefficients between the 60 PDF nuisance parameters extracted from the fit to the combined muon and electron channel fit. The underlying fit is performed by fixing the W boson cross sections to their expectation in all helicity and charge processes. 
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Figure A9:
Postfit pulls and constraints of the nuisance parameters associated with the ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scale systematic uncertainties. The numbering refers to bins in the ${{p_{\mathrm {T}}} ^{\mathrm{W}}}$ spectrum in increasing order. The nuisance parameters applied to the "left'' polarization are shown on the upper panel while the ones associated with the "right'' polarization are shown on the lower panel. 
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Figure A9a:
Postfit pulls and constraints of the nuisance parameters associated with the ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scale systematic uncertainties. The numbering refers to bins in the ${{p_{\mathrm {T}}} ^{\mathrm{W}}}$ spectrum in increasing order. The nuisance parameters applied to the "left'' polarization are shown on the upper panel while the ones associated with the "right'' polarization are shown on the lower panel. 
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Figure A9b:
Postfit pulls and constraints of the nuisance parameters associated with the ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scale systematic uncertainties. The numbering refers to bins in the ${{p_{\mathrm {T}}} ^{\mathrm{W}}}$ spectrum in increasing order. The nuisance parameters applied to the "left'' polarization are shown on the upper panel while the ones associated with the "right'' polarization are shown on the lower panel. 
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Figure A10:
Remaining impacts on the normalized polarized cross sections as functions of the W boson rapidity. Shown are the impacts of the nuisance groups for ${\mathrm{W} ^+_\mathrm {R}}$ (upper), ${\mathrm{W} ^+_\mathrm {L}}$ (middle), and ${\mathrm{W} ^_\mathrm {R}}$ (lower) bosons in the helicity fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A10a:
Remaining impacts on the normalized polarized cross sections as functions of the W boson rapidity. Shown are the impacts of the nuisance groups for ${\mathrm{W} ^+_\mathrm {R}}$ (upper), ${\mathrm{W} ^+_\mathrm {L}}$ (middle), and ${\mathrm{W} ^_\mathrm {R}}$ (lower) bosons in the helicity fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A10b:
Remaining impacts on the normalized polarized cross sections as functions of the W boson rapidity. Shown are the impacts of the nuisance groups for ${\mathrm{W} ^+_\mathrm {R}}$ (upper), ${\mathrm{W} ^+_\mathrm {L}}$ (middle), and ${\mathrm{W} ^_\mathrm {R}}$ (lower) bosons in the helicity fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A10c:
Remaining impacts on the normalized polarized cross sections as functions of the W boson rapidity. Shown are the impacts of the nuisance groups for ${\mathrm{W} ^+_\mathrm {R}}$ (upper), ${\mathrm{W} ^+_\mathrm {L}}$ (middle), and ${\mathrm{W} ^_\mathrm {R}}$ (lower) bosons in the helicity fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A11:
Impacts on the absolute polarized cross sections as functions of the W boson rapidity. Shown are the impacts of the nuisance groups for ${\mathrm{W} ^+_\mathrm {L}}$ (upper) and ${\mathrm{W} ^+_\mathrm {R}}$ (lower) in the helicity fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A11a:
Impacts on the absolute polarized cross sections as functions of the W boson rapidity. Shown are the impacts of the nuisance groups for ${\mathrm{W} ^+_\mathrm {L}}$ (upper) and ${\mathrm{W} ^+_\mathrm {R}}$ (lower) in the helicity fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A11b:
Impacts on the absolute polarized cross sections as functions of the W boson rapidity. Shown are the impacts of the nuisance groups for ${\mathrm{W} ^+_\mathrm {L}}$ (upper) and ${\mathrm{W} ^+_\mathrm {R}}$ (lower) in the helicity fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A12:
Impacts on the absolute polarized cross sections as functions of the W boson rapidity. Shown are the impacts of the nuisance groups for ${\mathrm{W} ^_\mathrm {L}}$ (upper) and ${\mathrm{W} ^_\mathrm {R}}$ (lower) bosons in the helicity fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A12a:
Impacts on the absolute polarized cross sections as functions of the W boson rapidity. Shown are the impacts of the nuisance groups for ${\mathrm{W} ^_\mathrm {L}}$ (upper) and ${\mathrm{W} ^_\mathrm {R}}$ (lower) bosons in the helicity fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A12b:
Impacts on the absolute polarized cross sections as functions of the W boson rapidity. Shown are the impacts of the nuisance groups for ${\mathrm{W} ^_\mathrm {L}}$ (upper) and ${\mathrm{W} ^_\mathrm {R}}$ (lower) bosons in the helicity fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A13:
Impacts on the charge asymmetry as functions of the W boson rapidity for ${\mathrm{W} _\mathrm {R}}$ bosons in the helicity fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A14:
Impacts on the unpolarized absolute cross sections as functions of the W boson rapidity for ${\mathrm{W^{+}}}$ (upper), ${\mathrm{W^{}}}$ (middle), and the unpolarized charge asymmetry (lower) bosons in the helicity fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A14a:
Impacts on the unpolarized absolute cross sections as functions of the W boson rapidity for ${\mathrm{W^{+}}}$ (upper), ${\mathrm{W^{}}}$ (middle), and the unpolarized charge asymmetry (lower) bosons in the helicity fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A14b:
Impacts on the unpolarized absolute cross sections as functions of the W boson rapidity for ${\mathrm{W^{+}}}$ (upper), ${\mathrm{W^{}}}$ (middle), and the unpolarized charge asymmetry (lower) bosons in the helicity fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A14c:
Impacts on the unpolarized absolute cross sections as functions of the W boson rapidity for ${\mathrm{W^{+}}}$ (upper), ${\mathrm{W^{}}}$ (middle), and the unpolarized charge asymmetry (lower) bosons in the helicity fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A15:
Impacts on the unpolarized normalized cross sections as functions of the W boson rapidity for ${\mathrm{W^{+}}}$ (upper) and ${\mathrm{W^{}}}$ (lower) bosons in the helicity fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A15a:
Impacts on the unpolarized normalized cross sections as functions of the W boson rapidity for ${\mathrm{W^{+}}}$ (upper) and ${\mathrm{W^{}}}$ (lower) bosons in the helicity fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A15b:
Impacts on the unpolarized normalized cross sections as functions of the W boson rapidity for ${\mathrm{W^{+}}}$ (upper) and ${\mathrm{W^{}}}$ (lower) bosons in the helicity fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A16:
Impacts on the A$_4$ coefficient as functions of the W boson rapidity for ${\mathrm{W^{+}}}$ (upper) and ${\mathrm{W^{}}}$ (lower) bosons in the helicity fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A16a:
Impacts on the A$_4$ coefficient as functions of the W boson rapidity for ${\mathrm{W^{+}}}$ (upper) and ${\mathrm{W^{}}}$ (lower) bosons in the helicity fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A16b:
Impacts on the A$_4$ coefficient as functions of the W boson rapidity for ${\mathrm{W^{+}}}$ (upper) and ${\mathrm{W^{}}}$ (lower) bosons in the helicity fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A17:
Unrolled cross sections for the combined muon and electron channel fit unrolled along ${{p_{\mathrm {T}}} ^{\ell}}$ in bins of ${{ \eta ^{\ell} }}$ for ${\mathrm{W^{+}}}$ (upper) and ${\mathrm{W^{}}}$ (lower) bosons. The colored bands represent the prediction from MadGraph5_aMC@NLO with the expected uncertainty from the quadrature sum of the PDF$\oplus {\alpha _S} $ variations (blue) and the ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales (bordeaux). 
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Figure A17a:
Unrolled cross sections for the combined muon and electron channel fit unrolled along ${{p_{\mathrm {T}}} ^{\ell}}$ in bins of ${{ \eta ^{\ell} }}$ for ${\mathrm{W^{+}}}$ (upper) and ${\mathrm{W^{}}}$ (lower) bosons. The colored bands represent the prediction from MadGraph5_aMC@NLO with the expected uncertainty from the quadrature sum of the PDF$\oplus {\alpha _S} $ variations (blue) and the ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales (bordeaux). 
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Figure A17b:
Unrolled cross sections for the combined muon and electron channel fit unrolled along ${{p_{\mathrm {T}}} ^{\ell}}$ in bins of ${{ \eta ^{\ell} }}$ for ${\mathrm{W^{+}}}$ (upper) and ${\mathrm{W^{}}}$ (lower) bosons. The colored bands represent the prediction from MadGraph5_aMC@NLO with the expected uncertainty from the quadrature sum of the PDF$\oplus {\alpha _S} $ variations (blue) and the ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales (bordeaux). 
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Figure A18:
Absolute cross sections as functions of ${{p_{\mathrm {T}}} ^{\ell}}$, integrated over ${{ \eta ^{\ell} }}$ for ${\mathrm{W^{+}}}$ (left) and ${\mathrm{W^{}}}$ (right) bosons. The colored bands represent the prediction from MadGraph5_aMC@NLO with the expected uncertainty from the quadrature sum of the PDF$\oplus {\alpha _S} $ variations (blue) and the ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales (bordeaux). 
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Figure A18a:
Absolute cross sections as functions of ${{p_{\mathrm {T}}} ^{\ell}}$, integrated over ${{ \eta ^{\ell} }}$ for ${\mathrm{W^{+}}}$ (left) and ${\mathrm{W^{}}}$ (right) bosons. The colored bands represent the prediction from MadGraph5_aMC@NLO with the expected uncertainty from the quadrature sum of the PDF$\oplus {\alpha _S} $ variations (blue) and the ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales (bordeaux). 
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Figure A18b:
Absolute cross sections as functions of ${{p_{\mathrm {T}}} ^{\ell}}$, integrated over ${{ \eta ^{\ell} }}$ for ${\mathrm{W^{+}}}$ (left) and ${\mathrm{W^{}}}$ (right) bosons. The colored bands represent the prediction from MadGraph5_aMC@NLO with the expected uncertainty from the quadrature sum of the PDF$\oplus {\alpha _S} $ variations (blue) and the ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales (bordeaux). 
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Figure A19:
Normalized cross sections as functions of ${{p_{\mathrm {T}}} ^{\ell}}$, integrated over ${{ \eta ^{\ell} }}$ for ${\mathrm{W^{+}}}$ (left) and ${\mathrm{W^{}}}$ (right) bosons. The colored bands represent the prediction from MadGraph5_aMC@NLO with the expected uncertainty from the quadrature sum of the PDF$\oplus {\alpha _S} $ variations (blue) and the ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales (bordeaux). 
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Figure A19a:
Normalized cross sections as functions of ${{p_{\mathrm {T}}} ^{\ell}}$, integrated over ${{ \eta ^{\ell} }}$ for ${\mathrm{W^{+}}}$ (left) and ${\mathrm{W^{}}}$ (right) bosons. The colored bands represent the prediction from MadGraph5_aMC@NLO with the expected uncertainty from the quadrature sum of the PDF$\oplus {\alpha _S} $ variations (blue) and the ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales (bordeaux). 
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Figure A19b:
Normalized cross sections as functions of ${{p_{\mathrm {T}}} ^{\ell}}$, integrated over ${{ \eta ^{\ell} }}$ for ${\mathrm{W^{+}}}$ (left) and ${\mathrm{W^{}}}$ (right) bosons. The colored bands represent the prediction from MadGraph5_aMC@NLO with the expected uncertainty from the quadrature sum of the PDF$\oplus {\alpha _S} $ variations (blue) and the ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales (bordeaux). 
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Figure A20:
W charge asymmetry as functions of ${{p_{\mathrm {T}}} ^{\ell}}$, integrated over ${{ \eta ^{\ell} }}$. The colored bands represent the prediction from MadGraph5_aMC@NLO with the expected uncertainty from the quadrature sum of the PDF$\oplus {\alpha _S} $ variations (blue) and the ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales (bordeaux). 
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Figure A21:
Normalized cross sections as functions of ${{ \eta ^{\ell} }}$, integrated over ${{p_{\mathrm {T}}} ^{\ell}}$ for ${\mathrm{W^{+}}}$ (left) and ${\mathrm{W^{}}}$ (right) bosons. The colored bands represent the prediction from MadGraph5_aMC@NLO with the expected uncertainty from the quadrature sum of the PDF$\oplus {\alpha _S} $ variations (blue) and the ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales (bordeaux). The uncertainty band is almost entirely dominated by the PDF$\oplus {\alpha _S} $ variations, while the missing higher order QCD uncertainties almost perfectly cancel and are therefore invisible. 
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Figure A21a:
Normalized cross sections as functions of ${{ \eta ^{\ell} }}$, integrated over ${{p_{\mathrm {T}}} ^{\ell}}$ for ${\mathrm{W^{+}}}$ (left) and ${\mathrm{W^{}}}$ (right) bosons. The colored bands represent the prediction from MadGraph5_aMC@NLO with the expected uncertainty from the quadrature sum of the PDF$\oplus {\alpha _S} $ variations (blue) and the ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales (bordeaux). The uncertainty band is almost entirely dominated by the PDF$\oplus {\alpha _S} $ variations, while the missing higher order QCD uncertainties almost perfectly cancel and are therefore invisible. 
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Figure A21b:
Normalized cross sections as functions of ${{ \eta ^{\ell} }}$, integrated over ${{p_{\mathrm {T}}} ^{\ell}}$ for ${\mathrm{W^{+}}}$ (left) and ${\mathrm{W^{}}}$ (right) bosons. The colored bands represent the prediction from MadGraph5_aMC@NLO with the expected uncertainty from the quadrature sum of the PDF$\oplus {\alpha _S} $ variations (blue) and the ${\mu _\mathrm {F}}$ and ${\mu _\mathrm {R}}$ scales (bordeaux). The uncertainty band is almost entirely dominated by the PDF$\oplus {\alpha _S} $ variations, while the missing higher order QCD uncertainties almost perfectly cancel and are therefore invisible. 
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Figure A22:
Remaining impacts of the nuisance groups on the normalized cross sections as functions of ${{ \eta ^{\ell} }}$, integrated in ${{p_{\mathrm {T}}} ^{\ell}}$, for ${\mathrm{W^{}}}$ bosons in the doubledifferential cross section fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A23:
Remaining impacts of the nuisance groups on the absolute cross sections as functions of ${{ \eta ^{\ell} }}$, integrated in ${{p_{\mathrm {T}}} ^{\ell}}$, for ${\mathrm{W^{+}}}$ (upper) and ${\mathrm{W^{}}}$ (lower) bosons in the doubledifferential cross section fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A23a:
Remaining impacts of the nuisance groups on the absolute cross sections as functions of ${{ \eta ^{\ell} }}$, integrated in ${{p_{\mathrm {T}}} ^{\ell}}$, for ${\mathrm{W^{+}}}$ (upper) and ${\mathrm{W^{}}}$ (lower) bosons in the doubledifferential cross section fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A23b:
Remaining impacts of the nuisance groups on the absolute cross sections as functions of ${{ \eta ^{\ell} }}$, integrated in ${{p_{\mathrm {T}}} ^{\ell}}$, for ${\mathrm{W^{+}}}$ (upper) and ${\mathrm{W^{}}}$ (lower) bosons in the doubledifferential cross section fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A24:
Remaining impacts of the nuisance groups on the normalized cross sections as functions of ${{p_{\mathrm {T}}} ^{\ell}}$, integrated over ${{ \eta ^{\ell} }}$, for ${\mathrm{W^{+}}}$ (upper), ${\mathrm{W^{}}}$ (middle) bosons, and the resulting charge asymmetry (lower) in the doubledifferential cross section fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A24a:
Remaining impacts of the nuisance groups on the normalized cross sections as functions of ${{p_{\mathrm {T}}} ^{\ell}}$, integrated over ${{ \eta ^{\ell} }}$, for ${\mathrm{W^{+}}}$ (upper), ${\mathrm{W^{}}}$ (middle) bosons, and the resulting charge asymmetry (lower) in the doubledifferential cross section fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A24b:
Remaining impacts of the nuisance groups on the normalized cross sections as functions of ${{p_{\mathrm {T}}} ^{\ell}}$, integrated over ${{ \eta ^{\ell} }}$, for ${\mathrm{W^{+}}}$ (upper), ${\mathrm{W^{}}}$ (middle) bosons, and the resulting charge asymmetry (lower) in the doubledifferential cross section fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A24c:
Remaining impacts of the nuisance groups on the normalized cross sections as functions of ${{p_{\mathrm {T}}} ^{\ell}}$, integrated over ${{ \eta ^{\ell} }}$, for ${\mathrm{W^{+}}}$ (upper), ${\mathrm{W^{}}}$ (middle) bosons, and the resulting charge asymmetry (lower) in the doubledifferential cross section fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A25:
Remaining impacts of the nuisance groups on the absolute cross sections as functions of ${{p_{\mathrm {T}}} ^{\ell}}$, integrated over ${{ \eta ^{\ell} }}$, for ${\mathrm{W^{+}}}$ (upper) and ${\mathrm{W^{}}}$ (lower) bosons in the doubledifferential cross section fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A25a:
Remaining impacts of the nuisance groups on the absolute cross sections as functions of ${{p_{\mathrm {T}}} ^{\ell}}$, integrated over ${{ \eta ^{\ell} }}$, for ${\mathrm{W^{+}}}$ (upper) and ${\mathrm{W^{}}}$ (lower) bosons in the doubledifferential cross section fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
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Figure A25b:
Remaining impacts of the nuisance groups on the absolute cross sections as functions of ${{p_{\mathrm {T}}} ^{\ell}}$, integrated over ${{ \eta ^{\ell} }}$, for ${\mathrm{W^{+}}}$ (upper) and ${\mathrm{W^{}}}$ (lower) bosons in the doubledifferential cross section fit. The groups of uncertainties subleading to the ones shown are suppressed for simplicity. 
Tables  
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Table 1:
Estimated backgroundtosignal ratios in the $\mathrm{W} \to \mu \nu $ and $\mathrm{W} \to \mathrm{e} \nu $ channels. The DY simulation includes $\ell =$ e, $\mu$, $\tau $. 
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Table 2:
Systematic uncertainties for each source and process. Quoted numbers correspond to the size of lognormal nuisance parameters applied in the fit, while a "yes'' in a given cell corresponds to the given systematic uncertainty being applied as a shape variation over the full 2D template space. 
Summary 
The differential W boson cross sections as functions of the W boson rapidity, ${y_{\mathrm{W}}} $, and for the two charges separately, ${\mathrm{W^{+}}} \to \ell^+\nu$ and ${\mathrm{W^{}}} \to \ell^\bar{\nu}$, are measured in the W boson helicity states. Doubledifferential cross sections of the W boson are measured as a function of the chargedlepton transverse momentum ${p_{\mathrm{T}}}^{\ell}$ and absolute pseudorapidity ${\eta^{\ell}}$. For both ${\mathrm{W^{+}}}$ and ${\mathrm{W^{}}}$ bosons, the differential charge asymmetry is also extracted. The measurement is based on data taken in protonproton collisions at the LHC at a centerofmass energy of $\sqrt{s} = $ 13 TeV, corresponding to an integrated luminosity of 35.9 fb$^{1}$. Differential cross sections are presented, both absolute and normalized to the total production cross section within a given acceptance. For the helicity measurement, the range ${y_{\mathrm{W}}} < $ 2.5 is presented, whereas for the doubledifferential cross section the range ${\eta^{\ell}} < $ 2.4 and 26 $ < {p_{\mathrm{T}}}^{\ell} < $ 56 GeV is used. The measurement is performed using both the muon and electron channels, combined together considering all sources of correlated and uncorrelated uncertainties. The precision in the measurement as a function of ${y_{\mathrm{W}}}$, using a combination of the two channels, is about 2% in central ${y_{\mathrm{W}}}$ bins, and 5 to 20%, depending on the chargepolarization combination, in the outermost acceptance bins. The precision of the doubledifferential cross section, relative to the total, is about 1% in the central part of the detector, ${\eta^{\ell}} < $ 1, and better than 2.5% up to ${\eta^{\ell}} < $ 2 for each of the two W boson charges. Charge asymmetries are also measured, differentially in ${y_{\mathrm{W}}}$ and polarization, as well as in ${p_{\mathrm{T}}}^{\ell}$ and ${\eta^{\ell}}$. The uncertainties in these asymmetries range from 0.1% in highacceptance bins to roughly 2.5% in regions of phase space with lower detector acceptance. Furthermore, fiducial cross sections are presented by integrating the twodimensional differential cross sections over the full acceptance of the analysis. The measurement of the W boson polarized cross sections as functions of ${y_{\mathrm{W}}}$ is used to constrain the parameters related to parton distribution functions in a simultaneous fit of the two channels and the two W boson charges. The constraints are derived at the detector level on 60 uncorrelated eigenvalues of the NNPDF3.0 set of PDFs within the MadGraph5+MCatNLO event generator, and show a total constraint down to $\simeq$70% of the prefit uncertainties for certain variations of the PDF nuisance parameters. 
Additional Figures  
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Additional Figure 1:
Distributions of ${\eta ^{\ell}}$ and ${{p_{\mathrm {T}}} ^{\ell}}$ of the reconstructed muon in W boson decays, split into W boson charge and polarization (left, longitudinal, right). For a given charge and helicity state, the particular shape of each distribution is determined by the underlying W rapidity distribution, which in turn depends on the proton PDFs. These distributions are obtained from simulated $\mathrm{W} \rightarrow \mu \nu $+jets events at $\sqrt {s} = $ 13 TeV using MadGraph5_aMC@NLO. Event yields are normalized to an integrated luminosity of 35.9 fb$^{1}$. No event selection is applied, except for the requirement that a muon is identified and reconstructed within the CMS detector and the plot boundaries in ${{p_{\mathrm {T}}} ^{\ell}}$ from 25 to 50 GeV and $ \eta ^{\ell}  < $ 2.4. 
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Additional Figure 1a:
Distribution of ${\eta ^{\ell}}$ and ${{p_{\mathrm {T}}} ^{\ell}}$ of the reconstructed muon in W boson decays, for positive charge and left polarization. For a given charge and helicity state, the particular shape of each distribution is determined by the underlying W rapidity distribution, which in turn depends on the proton PDFs. This distribution is obtained from simulated $\mathrm{W} \rightarrow \mu \nu $+jets events at $\sqrt {s} = $ 13 TeV using MadGraph5_aMC@NLO. Event yields are normalized to an integrated luminosity of 35.9 fb$^{1}$. No event selection is applied, except for the requirement that a muon is identified and reconstructed within the CMS detector and the plot boundaries in ${{p_{\mathrm {T}}} ^{\ell}}$ from 25 to 50 GeV and $ \eta ^{\ell}  < $ 2.4. 
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Additional Figure 1b:
Distribution of ${\eta ^{\ell}}$ and ${{p_{\mathrm {T}}} ^{\ell}}$ of the reconstructed muon in W boson decays, for positive charge and longitudinal polarization. For a given charge and helicity state, the particular shape of each distribution is determined by the underlying W rapidity distribution, which in turn depends on the proton PDFs. This distribution is obtained from simulated $\mathrm{W} \rightarrow \mu \nu $+jets events at $\sqrt {s} = $ 13 TeV using MadGraph5_aMC@NLO. Event yields are normalized to an integrated luminosity of 35.9 fb$^{1}$. No event selection is applied, except for the requirement that a muon is identified and reconstructed within the CMS detector and the plot boundaries in ${{p_{\mathrm {T}}} ^{\ell}}$ from 25 to 50 GeV and $ \eta ^{\ell}  < $ 2.4. 
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Additional Figure 1c:
Distribution of ${\eta ^{\ell}}$ and ${{p_{\mathrm {T}}} ^{\ell}}$ of the reconstructed muon in W boson decays, for positive charge and right polarization. For a given charge and helicity state, the particular shape of each distribution is determined by the underlying W rapidity distribution, which in turn depends on the proton PDFs. This distribution is obtained from simulated $\mathrm{W} \rightarrow \mu \nu $+jets events at $\sqrt {s} = $ 13 TeV using MadGraph5_aMC@NLO. Event yields are normalized to an integrated luminosity of 35.9 fb$^{1}$. No event selection is applied, except for the requirement that a muon is identified and reconstructed within the CMS detector and the plot boundaries in ${{p_{\mathrm {T}}} ^{\ell}}$ from 25 to 50 GeV and $ \eta ^{\ell}  < $ 2.4. 
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Additional Figure 1d:
Distribution of ${\eta ^{\ell}}$ and ${{p_{\mathrm {T}}} ^{\ell}}$ of the reconstructed muon in W boson decays, for negative charge and left polarization. For a given charge and helicity state, the particular shape of each distribution is determined by the underlying W rapidity distribution, which in turn depends on the proton PDFs. This distribution is obtained from simulated $\mathrm{W} \rightarrow \mu \nu $+jets events at $\sqrt {s} = $ 13 TeV using MadGraph5_aMC@NLO. Event yields are normalized to an integrated luminosity of 35.9 fb$^{1}$. No event selection is applied, except for the requirement that a muon is identified and reconstructed within the CMS detector and the plot boundaries in ${{p_{\mathrm {T}}} ^{\ell}}$ from 25 to 50 GeV and $ \eta ^{\ell}  < $ 2.4. 
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Additional Figure 1e:
Distribution of ${\eta ^{\ell}}$ and ${{p_{\mathrm {T}}} ^{\ell}}$ of the reconstructed muon in W boson decays, for negative charge and longitudinal polarization. For a given charge and helicity state, the particular shape of each distribution is determined by the underlying W rapidity distribution, which in turn depends on the proton PDFs. This distribution is obtained from simulated $\mathrm{W} \rightarrow \mu \nu $+jets events at $\sqrt {s} = $ 13 TeV using MadGraph5_aMC@NLO. Event yields are normalized to an integrated luminosity of 35.9 fb$^{1}$. No event selection is applied, except for the requirement that a muon is identified and reconstructed within the CMS detector and the plot boundaries in ${{p_{\mathrm {T}}} ^{\ell}}$ from 25 to 50 GeV and $ \eta ^{\ell}  < $ 2.4. 
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Additional Figure 1f:
Distribution of ${\eta ^{\ell}}$ and ${{p_{\mathrm {T}}} ^{\ell}}$ of the reconstructed muon in W boson decays, for negative charge and right polarization. For a given charge and helicity state, the particular shape of each distribution is determined by the underlying W rapidity distribution, which in turn depends on the proton PDFs. This distribution is obtained from simulated $\mathrm{W} \rightarrow \mu \nu $+jets events at $\sqrt {s} = $ 13 TeV using MadGraph5_aMC@NLO. Event yields are normalized to an integrated luminosity of 35.9 fb$^{1}$. No event selection is applied, except for the requirement that a muon is identified and reconstructed within the CMS detector and the plot boundaries in ${{p_{\mathrm {T}}} ^{\ell}}$ from 25 to 50 GeV and $ \eta ^{\ell}  < $ 2.4. 
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Additional Figure 2:
The postfit nuisance parameter values and covariance matrix for the PDF and $\alpha _S$ uncertainties from the fixedparameterofinterest fit described in Section 8.2 are propagated to the corresponding change in central value and uncertainty with respect to the input NNPDF3.0 NLO PDF set as a function of x for fixed $Q^2=M_\mathrm {W}^2$. The profiling procedure employed here has the caveat that the results cannot be interpreted as a rigorous PDF determination in case the results are far from the input PDF, and that the measurement under consideration tends to have a larger weight than if it were included in a global PDF determination. Nevertheless the profiling procedure is useful to assess the sensitivity of the different parton distributions to this measurement, and its compatibility with the predictions of the input PDF set. Particular to this result is the fact that constraints are derived only at NLO + parton shower accuracy in QCD, and that there are possible limitations of the PDF set used with respect to newer sets. The presence of oscillatory behaviour in the valence quark distributions is in part related to anticorrelations between different x values arising from the statistical uncertainties in the underlying measurement. 
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Additional Figure 2a:
The postfit nuisance parameter values and covariance matrix for the PDF and $\alpha _S$ uncertainties from the fixedparameterofinterest fit described in Section 8.2 are propagated to the corresponding change in central value and uncertainty with respect to the input NNPDF3.0 NLO PDF set as a function of x for fixed $Q^2=M_\mathrm {W}^2$. The profiling procedure employed here has the caveat that the results cannot be interpreted as a rigorous PDF determination in case the results are far from the input PDF, and that the measurement under consideration tends to have a larger weight than if it were included in a global PDF determination. Nevertheless the profiling procedure is useful to assess the sensitivity of the different parton distributions to this measurement, and its compatibility with the predictions of the input PDF set. Particular to this result is the fact that constraints are derived only at NLO + parton shower accuracy in QCD, and that there are possible limitations of the PDF set used with respect to newer sets. The presence of oscillatory behaviour in the valence quark distributions is in part related to anticorrelations between different x values arising from the statistical uncertainties in the underlying measurement. 
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Additional Figure 2b:
The postfit nuisance parameter values and covariance matrix for the PDF and $\alpha _S$ uncertainties from the fixedparameterofinterest fit described in Section 8.2 are propagated to the corresponding change in central value and uncertainty with respect to the input NNPDF3.0 NLO PDF set as a function of x for fixed $Q^2=M_\mathrm {W}^2$. The profiling procedure employed here has the caveat that the results cannot be interpreted as a rigorous PDF determination in case the results are far from the input PDF, and that the measurement under consideration tends to have a larger weight than if it were included in a global PDF determination. Nevertheless the profiling procedure is useful to assess the sensitivity of the different parton distributions to this measurement, and its compatibility with the predictions of the input PDF set. Particular to this result is the fact that constraints are derived only at NLO + parton shower accuracy in QCD, and that there are possible limitations of the PDF set used with respect to newer sets. The presence of oscillatory behaviour in the valence quark distributions is in part related to anticorrelations between different x values arising from the statistical uncertainties in the underlying measurement. 
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Additional Figure 3:
The postfit nuisance parameter values and covariance matrix for the PDF and $\alpha _S$ uncertainties from the fixedparameterofinterest fit described in Section 8.2 are propagated to the corresponding change in central value and uncertainty with respect to the input NNPDF3.0 NLO PDF set as a function of x for fixed $Q^2=M_\mathrm {W}^2$. The profiling procedure employed here has the caveat that the results cannot be interpreted as a rigorous PDF determination in case the results are far from the input PDF, and that the measurement under consideration tends to have a larger weight than if it were included in a global PDF determination. Nevertheless the profiling procedure is useful to assess the sensitivity of the different parton distributions to this measurement, and its compatibility with the predictions of the input PDF set. Particular to this result is the fact that constraints are derived only at NLO + parton shower accuracy in QCD, and that there are possible limitations of the PDF set used with respect to newer sets. The presence of oscillatory behaviour in the valence quark distributions is in part related to anticorrelations between different x values arising from the statistical uncertainties in the underlying measurement. 
png pdf 
Additional Figure 3a:
The postfit nuisance parameter values and covariance matrix for the PDF and $\alpha _S$ uncertainties from the fixedparameterofinterest fit described in Section 8.2 are propagated to the corresponding change in central value and uncertainty with respect to the input NNPDF3.0 NLO PDF set as a function of x for fixed $Q^2=M_\mathrm {W}^2$. The profiling procedure employed here has the caveat that the results cannot be interpreted as a rigorous PDF determination in case the results are far from the input PDF, and that the measurement under consideration tends to have a larger weight than if it were included in a global PDF determination. Nevertheless the profiling procedure is useful to assess the sensitivity of the different parton distributions to this measurement, and its compatibility with the predictions of the input PDF set. Particular to this result is the fact that constraints are derived only at NLO + parton shower accuracy in QCD, and that there are possible limitations of the PDF set used with respect to newer sets. The presence of oscillatory behaviour in the valence quark distributions is in part related to anticorrelations between different x values arising from the statistical uncertainties in the underlying measurement. 
png pdf 
Additional Figure 3b:
The postfit nuisance parameter values and covariance matrix for the PDF and $\alpha _S$ uncertainties from the fixedparameterofinterest fit described in Section 8.2 are propagated to the corresponding change in central value and uncertainty with respect to the input NNPDF3.0 NLO PDF set as a function of x for fixed $Q^2=M_\mathrm {W}^2$. The profiling procedure employed here has the caveat that the results cannot be interpreted as a rigorous PDF determination in case the results are far from the input PDF, and that the measurement under consideration tends to have a larger weight than if it were included in a global PDF determination. Nevertheless the profiling procedure is useful to assess the sensitivity of the different parton distributions to this measurement, and its compatibility with the predictions of the input PDF set. Particular to this result is the fact that constraints are derived only at NLO + parton shower accuracy in QCD, and that there are possible limitations of the PDF set used with respect to newer sets. The presence of oscillatory behaviour in the valence quark distributions is in part related to anticorrelations between different x values arising from the statistical uncertainties in the underlying measurement. 
png pdf 
Additional Figure 4:
The postfit nuisance parameter values and covariance matrix for the PDF and $\alpha _S$ uncertainties from the fixedparameterofinterest fit described in Section 8.2 are propagated to the corresponding change in central value and uncertainty with respect to the input NNPDF3.0 NLO PDF set as a function of x for fixed $Q^2=M_\mathrm {W}^2$. The profiling procedure employed here has the caveat that the results cannot be interpreted as a rigorous PDF determination in case the results are far from the input PDF, and that the measurement under consideration tends to have a larger weight than if it were included in a global PDF determination. Nevertheless the profiling procedure is useful to assess the sensitivity of the different parton distributions to this measurement, and its compatibility with the predictions of the input PDF set. Particular to this result is the fact that constraints are derived only at NLO + parton shower accuracy in QCD, and that there are possible limitations of the PDF set used with respect to newer sets. The presence of oscillatory behaviour in the valence quark distributions is in part related to anticorrelations between different x values arising from the statistical uncertainties in the underlying measurement. 
png pdf 
Additional Figure 4a:
The postfit nuisance parameter values and covariance matrix for the PDF and $\alpha _S$ uncertainties from the fixedparameterofinterest fit described in Section 8.2 are propagated to the corresponding change in central value and uncertainty with respect to the input NNPDF3.0 NLO PDF set as a function of x for fixed $Q^2=M_\mathrm {W}^2$. The profiling procedure employed here has the caveat that the results cannot be interpreted as a rigorous PDF determination in case the results are far from the input PDF, and that the measurement under consideration tends to have a larger weight than if it were included in a global PDF determination. Nevertheless the profiling procedure is useful to assess the sensitivity of the different parton distributions to this measurement, and its compatibility with the predictions of the input PDF set. Particular to this result is the fact that constraints are derived only at NLO + parton shower accuracy in QCD, and that there are possible limitations of the PDF set used with respect to newer sets. The presence of oscillatory behaviour in the valence quark distributions is in part related to anticorrelations between different x values arising from the statistical uncertainties in the underlying measurement. 
png pdf 
Additional Figure 4b:
The postfit nuisance parameter values and covariance matrix for the PDF and $\alpha _S$ uncertainties from the fixedparameterofinterest fit described in Section 8.2 are propagated to the corresponding change in central value and uncertainty with respect to the input NNPDF3.0 NLO PDF set as a function of x for fixed $Q^2=M_\mathrm {W}^2$. The profiling procedure employed here has the caveat that the results cannot be interpreted as a rigorous PDF determination in case the results are far from the input PDF, and that the measurement under consideration tends to have a larger weight than if it were included in a global PDF determination. Nevertheless the profiling procedure is useful to assess the sensitivity of the different parton distributions to this measurement, and its compatibility with the predictions of the input PDF set. Particular to this result is the fact that constraints are derived only at NLO + parton shower accuracy in QCD, and that there are possible limitations of the PDF set used with respect to newer sets. The presence of oscillatory behaviour in the valence quark distributions is in part related to anticorrelations between different x values arising from the statistical uncertainties in the underlying measurement. 
png pdf 
Additional Figure 5:
The postfit nuisance parameter values and covariance matrix for the PDF and $\alpha _S$ uncertainties from the fixedparameterofinterest fit described in Section 8.2 are propagated to the corresponding change in central value and uncertainty with respect to the input NNPDF3.0 NLO PDF set as a function of x for fixed $Q^2=M_\mathrm {W}^2$. The profiling procedure employed here has the caveat that the results cannot be interpreted as a rigorous PDF determination in case the results are far from the input PDF, and that the measurement under consideration tends to have a larger weight than if it were included in a global PDF determination. Nevertheless the profiling procedure is useful to assess the sensitivity of the different parton distributions to this measurement, and its compatibility with the predictions of the input PDF set. Particular to this result is the fact that constraints are derived only at NLO + parton shower accuracy in QCD, and that there are possible limitations of the PDF set used with respect to newer sets. The presence of oscillatory behaviour in the valence quark distributions is in part related to anticorrelations between different x values arising from the statistical uncertainties in the underlying measurement. 
png pdf 
Additional Figure 5a:
The postfit nuisance parameter values and covariance matrix for the PDF and $\alpha _S$ uncertainties from the fixedparameterofinterest fit described in Section 8.2 are propagated to the corresponding change in central value and uncertainty with respect to the input NNPDF3.0 NLO PDF set as a function of x for fixed $Q^2=M_\mathrm {W}^2$. The profiling procedure employed here has the caveat that the results cannot be interpreted as a rigorous PDF determination in case the results are far from the input PDF, and that the measurement under consideration tends to have a larger weight than if it were included in a global PDF determination. Nevertheless the profiling procedure is useful to assess the sensitivity of the different parton distributions to this measurement, and its compatibility with the predictions of the input PDF set. Particular to this result is the fact that constraints are derived only at NLO + parton shower accuracy in QCD, and that there are possible limitations of the PDF set used with respect to newer sets. The presence of oscillatory behaviour in the valence quark distributions is in part related to anticorrelations between different x values arising from the statistical uncertainties in the underlying measurement. 
png pdf 
Additional Figure 5b:
The postfit nuisance parameter values and covariance matrix for the PDF and $\alpha _S$ uncertainties from the fixedparameterofinterest fit described in Section 8.2 are propagated to the corresponding change in central value and uncertainty with respect to the input NNPDF3.0 NLO PDF set as a function of x for fixed $Q^2=M_\mathrm {W}^2$. The profiling procedure employed here has the caveat that the results cannot be interpreted as a rigorous PDF determination in case the results are far from the input PDF, and that the measurement under consideration tends to have a larger weight than if it were included in a global PDF determination. Nevertheless the profiling procedure is useful to assess the sensitivity of the different parton distributions to this measurement, and its compatibility with the predictions of the input PDF set. Particular to this result is the fact that constraints are derived only at NLO + parton shower accuracy in QCD, and that there are possible limitations of the PDF set used with respect to newer sets. The presence of oscillatory behaviour in the valence quark distributions is in part related to anticorrelations between different x values arising from the statistical uncertainties in the underlying measurement. 
png pdf 
Additional Figure 6:
The postfit nuisance parameter values and covariance matrix for the PDF and $\alpha _S$ uncertainties from the fixedparameterofinterest fit described in Section 8.2 are propagated to the corresponding change in central value and uncertainty with respect to the input NNPDF3.0 NLO PDF set as a function of x for fixed $Q^2=M_\mathrm {W}^2$. The profiling procedure employed here has the caveat that the results cannot be interpreted as a rigorous PDF determination in case the results are far from the input PDF, and that the measurement under consideration tends to have a larger weight than if it were included in a global PDF determination. Nevertheless the profiling procedure is useful to assess the sensitivity of the different parton distributions to this measurement, and its compatibility with the predictions of the input PDF set. Particular to this result is the fact that constraints are derived only at NLO + parton shower accuracy in QCD, and that there are possible limitations of the PDF set used with respect to newer sets. The presence of oscillatory behaviour in the valence quark distributions is in part related to anticorrelations between different x values arising from the statistical uncertainties in the underlying measurement. 
png pdf 
Additional Figure 6a:
The postfit nuisance parameter values and covariance matrix for the PDF and $\alpha _S$ uncertainties from the fixedparameterofinterest fit described in Section 8.2 are propagated to the corresponding change in central value and uncertainty with respect to the input NNPDF3.0 NLO PDF set as a function of x for fixed $Q^2=M_\mathrm {W}^2$. The profiling procedure employed here has the caveat that the results cannot be interpreted as a rigorous PDF determination in case the results are far from the input PDF, and that the measurement under consideration tends to have a larger weight than if it were included in a global PDF determination. Nevertheless the profiling procedure is useful to assess the sensitivity of the different parton distributions to this measurement, and its compatibility with the predictions of the input PDF set. Particular to this result is the fact that constraints are derived only at NLO + parton shower accuracy in QCD, and that there are possible limitations of the PDF set used with respect to newer sets. The presence of oscillatory behaviour in the valence quark distributions is in part related to anticorrelations between different x values arising from the statistical uncertainties in the underlying measurement. 
png pdf 
Additional Figure 6b:
The postfit nuisance parameter values and covariance matrix for the PDF and $\alpha _S$ uncertainties from the fixedparameterofinterest fit described in Section 8.2 are propagated to the corresponding change in central value and uncertainty with respect to the input NNPDF3.0 NLO PDF set as a function of x for fixed $Q^2=M_\mathrm {W}^2$. The profiling procedure employed here has the caveat that the results cannot be interpreted as a rigorous PDF determination in case the results are far from the input PDF, and that the measurement under consideration tends to have a larger weight than if it were included in a global PDF determination. Nevertheless the profiling procedure is useful to assess the sensitivity of the different parton distributions to this measurement, and its compatibility with the predictions of the input PDF set. Particular to this result is the fact that constraints are derived only at NLO + parton shower accuracy in QCD, and that there are possible limitations of the PDF set used with respect to newer sets. The presence of oscillatory behaviour in the valence quark distributions is in part related to anticorrelations between different x values arising from the statistical uncertainties in the underlying measurement. 
png pdf 
Additional Figure 7:
The postfit nuisance parameter values and covariance matrix for the PDF and $\alpha _S$ uncertainties from the fixedparameterofinterest fit described in Section 8.2 are propagated to the corresponding change in central value and uncertainty with respect to the input NNPDF3.0 NLO PDF set as a function of x for fixed $Q^2=M_\mathrm {W}^2$. The profiling procedure employed here has the caveat that the results cannot be interpreted as a rigorous PDF determination in case the results are far from the input PDF, and that the measurement under consideration tends to have a larger weight than if it were included in a global PDF determination. Nevertheless the profiling procedure is useful to assess the sensitivity of the different parton distributions to this measurement, and its compatibility with the predictions of the input PDF set. Particular to this result is the fact that constraints are derived only at NLO + parton shower accuracy in QCD, and that there are possible limitations of the PDF set used with respect to newer sets. The presence of oscillatory behaviour in the valence quark distributions is in part related to anticorrelations between different x values arising from the statistical uncertainties in the underlying measurement. 
png pdf 
Additional Figure 7a:
The postfit nuisance parameter values and covariance matrix for the PDF and $\alpha _S$ uncertainties from the fixedparameterofinterest fit described in Section 8.2 are propagated to the corresponding change in central value and uncertainty with respect to the input NNPDF3.0 NLO PDF set as a function of x for fixed $Q^2=M_\mathrm {W}^2$. The profiling procedure employed here has the caveat that the results cannot be interpreted as a rigorous PDF determination in case the results are far from the input PDF, and that the measurement under consideration tends to have a larger weight than if it were included in a global PDF determination. Nevertheless the profiling procedure is useful to assess the sensitivity of the different parton distributions to this measurement, and its compatibility with the predictions of the input PDF set. Particular to this result is the fact that constraints are derived only at NLO + parton shower accuracy in QCD, and that there are possible limitations of the PDF set used with respect to newer sets. The presence of oscillatory behaviour in the valence quark distributions is in part related to anticorrelations between different x values arising from the statistical uncertainties in the underlying measurement. 
png pdf 
Additional Figure 7b:
The postfit nuisance parameter values and covariance matrix for the PDF and $\alpha _S$ uncertainties from the fixedparameterofinterest fit described in Section 8.2 are propagated to the corresponding change in central value and uncertainty with respect to the input NNPDF3.0 NLO PDF set as a function of x for fixed $Q^2=M_\mathrm {W}^2$. The profiling procedure employed here has the caveat that the results cannot be interpreted as a rigorous PDF determination in case the results are far from the input PDF, and that the measurement under consideration tends to have a larger weight than if it were included in a global PDF determination. Nevertheless the profiling procedure is useful to assess the sensitivity of the different parton distributions to this measurement, and its compatibility with the predictions of the input PDF set. Particular to this result is the fact that constraints are derived only at NLO + parton shower accuracy in QCD, and that there are possible limitations of the PDF set used with respect to newer sets. The presence of oscillatory behaviour in the valence quark distributions is in part related to anticorrelations between different x values arising from the statistical uncertainties in the underlying measurement. 
png pdf 
Additional Figure 8:
The postfit nuisance parameter values and covariance matrix for the PDF and $\alpha _S$ uncertainties from the fixedparameterofinterest fit described in Section 8.2 are propagated to the corresponding change in central value and uncertainty with respect to the input NNPDF3.0 NLO PDF set as a function of x for fixed $Q^2=M_\mathrm {W}^2$. The profiling procedure employed here has the caveat that the results cannot be interpreted as a rigorous PDF determination in case the results are far from the input PDF, and that the measurement under consideration tends to have a larger weight than if it were included in a global PDF determination. Nevertheless the profiling procedure is useful to assess the sensitivity of the different parton distributions to this measurement, and its compatibility with the predictions of the input PDF set. Particular to this result is the fact that constraints are derived only at NLO + parton shower accuracy in QCD, and that there are possible limitations of the PDF set used with respect to newer sets. The presence of oscillatory behaviour in the valence quark distributions is in part related to anticorrelations between different x values arising from the statistical uncertainties in the underlying measurement. 
png pdf 
Additional Figure 8a:
The postfit nuisance parameter values and covariance matrix for the PDF and $\alpha _S$ uncertainties from the fixedparameterofinterest fit described in Section 8.2 are propagated to the corresponding change in central value and uncertainty with respect to the input NNPDF3.0 NLO PDF set as a function of x for fixed $Q^2=M_\mathrm {W}^2$. The profiling procedure employed here has the caveat that the results cannot be interpreted as a rigorous PDF determination in case the results are far from the input PDF, and that the measurement under consideration tends to have a larger weight than if it were included in a global PDF determination. Nevertheless the profiling procedure is useful to assess the sensitivity of the different parton distributions to this measurement, and its compatibility with the predictions of the input PDF set. Particular to this result is the fact that constraints are derived only at NLO + parton shower accuracy in QCD, and that there are possible limitations of the PDF set used with respect to newer sets. The presence of oscillatory behaviour in the valence quark distributions is in part related to anticorrelations between different x values arising from the statistical uncertainties in the underlying measurement. 
png pdf 
Additional Figure 8b:
The postfit nuisance parameter values and covariance matrix for the PDF and $\alpha _S$ uncertainties from the fixedparameterofinterest fit described in Section 8.2 are propagated to the corresponding change in central value and uncertainty with respect to the input NNPDF3.0 NLO PDF set as a function of x for fixed $Q^2=M_\mathrm {W}^2$. The profiling procedure employed here has the caveat that the results cannot be interpreted as a rigorous PDF determination in case the results are far from the input PDF, and that the measurement under consideration tends to have a larger weight than if it were included in a global PDF determination. Nevertheless the profiling procedure is useful to assess the sensitivity of the different parton distributions to this measurement, and its compatibility with the predictions of the input PDF set. Particular to this result is the fact that constraints are derived only at NLO + parton shower accuracy in QCD, and that there are possible limitations of the PDF set used with respect to newer sets. The presence of oscillatory behaviour in the valence quark distributions is in part related to anticorrelations between different x values arising from the statistical uncertainties in the underlying measurement. 
Covariance Matrices among Parameters of Interest and Nuisance Parameters 
Compressed tarball (202M) including covariance matrices among parameters of interest (POIs) and nuisance parameters (NPs) in yaml format. The lists of postfit values and uncertainties for POIs and NPs are also included for each matrix. Note that for each of the 3 fits detailed in the paper (doubledifferential cross section, helicity with floating POIs, and helicity with fixed POIs), the postfit values of NPs and their mutual correlations do not change across different tables.  
License ccby4.0. The content of this file can be shared and adapted but you must give appropriate credit and cannot restrict access to others.  
The list of tables is available in yaml format. The convention adopted to name POIs and NPs in the tables is reported in a glossary of parameters of interest and nuisance parameters.  
Below are root files containing the covariance matrices and postfit values for nuisance parameters and parameters of interest. They are equivalent to the yaml tables in the compressed tarball.  
chargepois helicity
Covariance matrix among parameters of interest and nuisance parameters (root file, 14M) Postfit values and uncertainties for parameters of interest and nuisance parameters (root file, 33k) for polarized charge asymmetry versus $y_{\mathrm{W}}$, for combination of muon and electron channel. The relevant POIs are shown in Fig.10 in paper. 

chargemetapois helicity
Covariance matrix among parameters of interest and nuisance parameters (root file, 14M) Postfit values and uncertainties for parameters of interest and nuisance parameters (root file, 33k) for unpolarized charge asymmetry versus $y_{\mathrm{W}}$, for combination of muon and electron channel. The relevant POIs are shown in Fig.12 in paper. 

polpois helicity
Covariance matrix among parameters of interest and nuisance parameters (root file, 14M) Postfit values and uncertainties for parameters of interest and nuisance parameters (root file, 33k) for A4 angular coefficient versus $y_{\mathrm{W}}$, for combination of muon and electron channel. The relevant POIs are shown in Figs.A5 and A6 in paper. 

fixedpoi helicity
Covariance matrix among parameters of interest and nuisance parameters (root file, 13M) in the helicity fit with fixed signal cross sections, for combination of muon and electron channel. Postfit values and uncertainties for parameters of interest and nuisance parameters (root file, 32k) for nuisance parameters, for combination of muon and electron channel. 

pmaskedexpnorm helicity
Covariance matrix among parameters of interest and nuisance parameters (root file, 14M) Postfit values and uncertainties for parameters of interest and nuisance parameters (root file, 33k) for polarized normalized cross section versus $y_{\mathrm{W}}$, for combination of muon and electron channel. The relevant POIs are shown in Fig.9 in paper. 

sumpoisnorm helicity
Covariance matrix among parameters of interest and nuisance parameters (root file, 14M) Postfit values and uncertainties for parameters of interest and nuisance parameters (root file, 33k) for unpolarized normalized cross section versus $y_{\mathrm{W}}$, for combination of muon and electron channel. The relevant POIs are shown in Fig.11 in paper. 

sumpois helicity
Covariance matrix among parameters of interest and nuisance parameters (root file, 14M) Postfit values and uncertainties for parameters of interest and nuisance parameters (root file, 33k) for unpolarized absolute cross section versus $y_{\mathrm{W}}$, for combination of muon and electron channel. The relevant POIs are shown in Fig.13 in paper 

pmaskedexp helicity
Covariance matrix among parameters of interest and nuisance parameters (root file, 14M) Postfit values and uncertainties for parameters of interest and nuisance parameters (root file, 33k) for polarized absolute cross section versus $y_{\mathrm{W}}$, for combination of muon and electron channel. The relevant POIs are shown in Fig.A1 in paper. 

chargepois 2dxsec
Covariance matrix among parameters of interest and nuisance parameters (root file, 14M) Postfit values and uncertainties for parameters of interest and nuisance parameters (root file, 33k) for double differential charge asymmetry versus lepton $\eta$$p_{\mathrm{T}}$, for combination of muon and electron channel. The relevant POIs are shown in Fig.15 in paper. 

ratiometapois 2dxsec
Covariance matrix among parameters of interest and nuisance parameters (root file, 8.2M) Postfit values and uncertainties for parameters of interest and nuisance parameters (root file, 27k) for ratio ($W^{+}$/$W{}$) of total fiducial cross sections, for combination of muon and electron channel. The relevant POIs are shown in Fig.19 in paper. 

pmaskedexp 2dxsec
Covariance matrix among parameters of interest and nuisance parameters (root file, 22M) Postfit values and uncertainties for parameters of interest and nuisance parameters (root file, 39k) for absolute double differential cross section versus lepton $\eta$$p_{\mathrm{T}}$, for combination of muon and electron channel. The relevant POIs are shown in Fig.A17 in paper. 

chargemetapois 2dxsec
Covariance matrix among parameters of interest and nuisance parameters (root file, 8.7M) Postfit values and uncertainties for parameters of interest and nuisance parameters (root file, 33k) for single differential charge asymmetry versus lepton $\eta$ or $p_{\mathrm{T}}$, for combination of muon and electron channel. The relevant POIs are shown in Figs.18 and A20 in paper. 

sumpoisnorm 2dxsec
Covariance matrix among parameters of interest and nuisance parameters (root file, 9.4M) Postfit values and uncertainties for parameters of interest and nuisance parameters (root file, 29k) for normalized single differential cross section versus lepton $\eta$ or $p_{\mathrm{T}}$, for combination of muon and electron channel. The relevant POIs are shown in Figs.A19 and A21 in paper. 

sumpois 2dxsec
Covariance matrix among parameters of interest and nuisance parameters (root file, 9.3M) Postfit values and uncertainties for parameters of interest and nuisance parameters (root file, 29k) for absolute single differential cross section versus lepton $\eta$ or $p_{\mathrm{T}}$, and total fiducial cross section ($\mathrm{W}^{+}$, $\mathrm{W}^{}$, and charge inclusive), for combination of muon and electron channel. The relevant POIs are shown in Figs.17, 19, and A18 in paper. 
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