The numbers of events with correctly and incorrectly reconstructed charges for
the simulation are given by $N_{\rm t}^{\pm}(1-m^{\pm})$ and $N_{\rm
t}^{\pm}m^{\pm}$, respectively, where $N_{\rm t}^{\pm}$ is the number of
events with the truth-level charge specified in the superscript. Charge-
misidentification corrections based on the measured rates from $ee$ pairs are
applied as event weights to the simulated events.

For $ee$ pairs, the weight for events with incorrectly reconstructed charges
is the data-to-simulation ratio $m_{ee\text{-}d}/m_{ee\text{-}s}$, where the
$\pm$ superscripts for the truth-level charge are suppressed for clarity, and
the $m_{ee\text{-}d}$ and $m_{ee\text{-}s}$ symbols are the misidentification
rates observed in the data and simulation, respectively. For $e\nu$ pairs,
only $m_{e\nu\text{-}s}$ is measured, and it differs from $m_{ee\text{-}s}$.
In addition to the alignment of the PES wedge, there is an effect from the
differences in the $P_{\rm T}$ distributions of the electrons (positrons) from
$\gamma^{*}/Z$\- and $W$-boson decays.

The corrected misidentification rate for $e\nu$ pairs,
$m_{e\nu\text{-}s}^{\prime}$, is given by the product
$m_{ee\text{-}d}\>(m_{e\nu\text{-}s}/m_{ee\text{-}s})$. The leading term is
the measured rate from $ee$ pairs, and the following term in parentheses is a
relative correction that accounts for the additional effect of
electron-$P_{\rm T}$ differences between $W$\- and $\gamma^{*}/Z$-boson
decays. The event-weight correction for charge-misidentified events is
$m_{e\nu\text{-}s}^{\prime}/m_{e\nu\text{-}s}$, and
$(1-m_{e\nu\text{-}s}^{\prime})/(1-m_{e\nu\text{-}s})$ for events with the
correct charge.

To confirm the efficacy of the charge-misidentification corrections to the
simulated rate of positrons and electrons for the $W\rightarrow e\nu$ process,
the following asymmetry $A_{\pm}=(n^{+}-n^{-})/(n^{+}+n^{-})$, where
$n^{+(-)}$ is the number of positrons (electrons) as a function of $\eta_{\rm
det}$, is used. All of the charge-independent corrections discussed in this
section are applied. Charge-misidentification corrections affect the numerator
difference, but leave the denominator sum unchanged. Prior to the application
of the charge-misidentification corrections, the $A_{\pm}$ asymmetries of the
simulation differed from those of the data for $|\eta_{\rm det}|>1.5$.
Disagreements increased with increasing values of $|\eta_{\rm det}|$ and the
amount of disagreement varied with the $\phi$-sector locations of the
positrons and electrons. The amounts of disagreement are large, significant,
and different for positive and negative $\eta_{\rm det}$ regions within a
$\phi$-sector. After the application of the charge-misidentification
corrections, the differences of the $\phi$-sector asymmetries of the
simulation relative to the data are significantly decreased.

##  IX The $A_{\ell}$ Measurement

Equation (4) is the basis of the asymmetry measurement. The corrections
discussed in Sec. VIII are incorporated into the evaluation of the $N^{+}$,
$N^{-}$, $(\epsilon A)^{+}$, and $(\epsilon A)^{-}$ quantities. Using the
simulation, the product of the efficiency and acceptance is derived bin-by-bin
with the formula $(\epsilon A)^{\pm}=N_{\rm r}^{\pm}/N_{\rm g}^{\pm}$, where
$N_{\rm r}^{\pm}$ is the number of reconstructed and selected events in a bin
of the reconstructed pseudorapidity, and $N_{\rm g}^{\pm}$ is the number of
accepted events at the event-generation level in the corresponding bin of
generated pseudorapidity. In the determination of the generated-level
acceptance, the kinematic restrictions on the reconstructed quantities $E_{\rm
T}^{e}>25$ GeV, $E_{\rm T}^{\nu}>25$ GeV, and $M_{\rm T}>45$ GeV/$c^{2}$ are
applied to the corresponding generator-level quantities.

Alternatively, the asymmetry can be measured using $(N^{+}\\!-\rho
N^{-})/(N^{+}\\!+\rho N^{-})$ or
$(N^{+}\\!/\rho\,-N^{-})/(N^{+}\\!/\rho\,+N^{-})$, where $\rho=(\epsilon
A)^{+}\\!/(\epsilon A)^{-}$. Figure 15 shows the function
$\rho^{\prime}(\eta)$ defined as $1/\rho$ for the $\eta<0$ region and

Figure 15:  Dependence of $\rho^{\prime}(\eta)$ on $\eta$. The bold (black)
histogram includes the accounting of charge misidentification while the
lighter (blue) histogram does not. The uncertainties shown are statistical and
are evaluated bin-by-bin. The horizontal (gray) line is a unit-value
reference.

$\rho$ for the $\eta\geq 0$ region. The effect of the $\rho^{\prime}(\eta)$
correction on the measurement can be gauged using $D(\eta)=1-2m(\eta)$, where
$D(\eta)$ approximates the ratio of the uncorrected to corrected asymmetry,
and $m(\eta)$ is the mean charge-misidentification rate of both charges.

As the $A_{\ell}$ distribution is an antisymmetric function of the
pseudorapidity, events from the $\eta\geq 0$ and $\eta<0$ regions provide
independent measurements of the asymmetry distribution. They are combined to
improve the statistical precision. Prior to the combination, the measurement
over the $\eta<0$ region is transformed via the “CP-folding” operation,
$A_{\ell}(\eta)\rightarrow-A_{\ell}(-\eta)$. The measurement over the $\eta<0$
region, the measurement over the $\eta\geq 0$ region, and the combined
measurement are generically denoted by the symbol $A_{\ell}(|\eta|)$.

The fully corrected measurements of the $A_{\ell}(|\eta|)$ from the $\eta\geq
0$ and $\eta<0$ regions are shown in Fig. 16.

Figure 16:  Observed lepton asymmetry $A_{\ell}(|\eta|)$ as a function of
$|\eta|$. The bold (black) histogram is the measurement in the $\eta\geq 0$
region, and the lighter (blue) histogram is in the $\eta<0$ region. The
measurements include all corrections, and the uncertainties are statistical
only, and evaluated bin-by-bin. Bin-centering corrections are not a part of
the measurements.

The uncertainties shown in Fig. 16 are the diagonal elements of the covariance
matrix for the data, which is discussed in the next paragraph. The $\chi^{2}$
comparison between the $A_{\ell}(|\eta|)$ measurements of the $\eta\geq 0$ and
$\eta<0$ regions yields a value of 11 over the 15 bins. As the measurements
are consistent, they are combined into the CP-folded asymmetry.

Measurement uncertainties require a covariance matrix since the uncertainty of
an $\eta$ bin $i$ and of another bin $j$ are correlated from the measurement
uncertainties of the charge-misidentification rates in conjunction with the 30
cm spread of the $p\bar{p}$-collision vertex along the beam line. The charge-
misidentification rate for the tracks traversing a detector region $(\eta_{\rm
det})$ affects the asymmetry uncertainties of multiple $\eta$ bins because
those tracks are from electrons produced over a wide range of $\eta$. The
covariance matrix is calculated using the simulation, which provides the
distributions for $N_{\rm r}^{\pm}$, $N_{\rm g}^{\pm}$, and $N_{\rm t}^{\pm}$.
Recall that $N_{\rm r}^{\pm}$ is the number of reconstructed and selected
events in a bin of the reconstructed pseudorapidity, $N_{\rm g}^{\pm}$ is the
number of accepted events at the event-generation level in the corresponding
bin of the generated pseudorapidity, and $N_{\rm t}^{\pm}$ is the analog of
$N_{\rm r}^{\pm}$ but with the true charges of the generator level.

In Eq. (4), the $N_{\rm r}^{\pm}/(\epsilon A)^{\pm}$ terms are the corrected
event counts $N_{\rm c}^{\pm}$ in a data $\eta$ bin. The $N_{\rm r}^{\pm}$ and
$(\epsilon A)^{\pm}$ components contain sums of events from different detector
regions. Expressions for the first-order fluctuations of $N_{\rm c}^{\pm}$ due
to input uncertainties are derived in terms of the fluctuations from its
component $N_{\rm r}^{\pm}$ and $(\epsilon A)^{\pm}$ sums. Fluctuation
distributions for the $N_{\rm r}^{\pm}$ terms are based on the statistical
precision of the data, while those for $(\epsilon A)^{\pm}$ are based on that
of the simulation. For both terms, the uncertainties of their $N_{\rm
r}^{\pm}$ values are estimated using $N_{\rm r}^{\pm}=N_{\rm
t}^{\pm}(1-m^{\pm})+N_{\rm t}^{\mp}m^{\mp}$. Charge-misidentification related
uncertainties consist of two components, those from the binomial distribution
among the number of events with correctly and incorrectly reconstructed
charges, and those from the measured values of $m^{\pm}$. Uncertainties for
the measured values of $m^{\pm}$ are systematic uncertainties of $(\epsilon
A)^{\pm}$, but they are accounted for here with the statistical uncertainties
of $(\epsilon A)^{\pm}$.

To obtain the final expression for the fluctuations of the asymmetry
measurement, the expressions derived for the fluctuations of $N_{\rm c}^{\pm}$
are incorporated into the asymmetry, Eq. (4). Then, the covariances of
fluctuations between the $\eta$ bins of the measurement are calculated. Two
covariance matrices are calculated as there are two asymmetry measurements,
the base measurement with 30 bins covering the range $-3<\eta<3$ and the
combination of the $\eta\geq 0$ and $\eta<0$ measurements with half the number
of bins.

The covariance matrix of uncertainties for the combined measurement, denoted
by $V$, is expanded and inverted to the error matrix using singular-value
decomposition methods. As this is a real-valued symmetric $15\times 15$
matrix, its 15 eigenvalues and eigenvectors are the rank-1 matrix components
in the decomposition of the covariance matrix and of the error matrix

$V$ $\displaystyle=$ $\displaystyle\sum_{n}\lambda_{n}\>|v_{n}\rangle\langle
v_{n}|\;{\rm and}$ $\displaystyle\mbox{\boldmath$V$\unboldmath}^{-1}$
$\displaystyle=$ $\displaystyle\sum_{n}\lambda_{n}^{-1}\>|v_{n}\rangle\langle
v_{n}|\,,$ (5)

where $\lambda_{n}$ and $|v_{n}\rangle$ are the eigenvalues and eigenvectors
of $V$, respectively, and $|v_{n}\rangle\langle v_{n}|$ represents a vector
projection operator in the notation of Dirac bra-kets. In the basis space of
the eigenvectors where the error matrix is diagonal, the $\chi^{2}$ comparison
of a calculation to the data is $\sum_{n}\Delta_{n}^{2}/\lambda_{n}$, where
$\Delta_{n}$ is the difference between a calculation and the data along the
$n{\rm th}$ eigenvector, and $\lambda_{n}$ represents the squared uncertainty
of the difference. Figure 17 shows the eigenvalues.

Figure 17:  Eigenvalues $\lambda_{n}$ as a function of the eigenvector number
$n$ of the covariance matrix for data uncertainties. The eigenvalues, ordered
from the largest to the smallest, are numbered from 0 to 14, respectively. The
bold (black) histogram shows the eigenvalues, and the lighter (blue)
horizontal line corresponds to the smallest value of the diagonal terms of the
covariance matrix.

Also shown is the smallest value of the diagonal terms of the covariance
matrix. As this value is on par with the smallest eigenvalue, the error matrix
does not have any anomalously small eigenvalues that need regulation.

##  X  Systematic uncertainties

Systematic uncertainties are evaluated for the electron-energy scales and
resolutions, the recoil-system energy scale of the simulation relative to the
data, the $|\eta_{\rm det}|$-dependent scale factors, the backgrounds, and the
PDFs. Of these, the PDF uncertainties are the largest.

###  X.1  Non-PDF uncertainties

For plug-region events, additional uncertainties are evaluated for the
correction of the QCD-background shape at large $E_{\rm T}$ values of the
electron, the correction of the simulation efficiency relative to that for
data at large $E\\!\\!\\!/_{\rm T}$ values, and the charge-dependent bias of
the track reconstruction. Except for the backgrounds, the uncertainties affect
the $(\epsilon A)^{\pm}$ components of the asymmetry measurement. All
systematic uncertainties, except those from the PDFs, are small in relation to
the statistical uncertainties on the data. The various categories of
systematic uncertainties are treated as uncorrelated.

To obtain most of the systematic uncertainties, the corresponding measurement
uncertainties are propagated to the asymmetry. The systematic uncertainties of
the $|\eta_{\rm det}|$-dependent scale factors are derived from their
statistical uncertainties. One standard-deviation shifts of all $|\eta_{\rm
det}|$ bins are coherently propagated to the asymmetry measurement to obtain
upper-limit estimates of the systematic uncertainties. Systematic
uncertainties of plug region events due to the QCD-background shape and
simulation-efficiency corrections at large $E\\!\\!\\!/_{\rm T}$ values are
taken to be half the difference of the asymmetries observed with and without
the correction. For the uncertainty from the plug track-finding bias, the
uncertainty of the integrated value of the bias $A_{\rm b}$ is propagated to
the asymmetry measurement.

###  X.2  PDF uncertainties

Systematic uncertainties due to the PDFs used in the simulation enter through
the $(\epsilon A)^{\pm}$ corrections, which depend on the asymmetry. For
$(\epsilon A)^{+}$, the explicit expression is

$\frac{N_{\rm r}^{+}}{N_{\rm g}^{+}}=(1-m^{+})\frac{N_{\rm t}^{+}}{N_{\rm
g}^{+}}+m^{-}\frac{N_{\rm t}^{-}}{N_{\rm g}^{-}}\frac{N_{\rm g}^{-}}{N_{\rm
g}^{+}}\;,$

where $N_{\rm g}^{-}/N_{\rm g}^{+}$ equals $(1-A_{\ell})/(1+A_{\ell})$. The
expression for $(\epsilon A)^{-}$ is obtained by interchanging the $+$ and $-$
charge superscripts. The $N_{\rm t}^{\pm}/N_{\rm g}^{\pm}$ ratios are the
combined efficiencies and acceptances when $m^{\pm}=0$.

The implementation of PDFs from the NNPDF collaboration that is used in this
paper is the ensemble set of 100 equally probable PDFs based on the fit to the
input data, along with a default or best-fit PDF. For such probabilistic PDFs,
the prediction is the average value of $A_{\ell}(|\eta|)$ calculated over the
ensemble, and the dispersion rms about the average is the PDF uncertainty of
the prediction. These uncertainties are correlated across $|\eta|$ bins.

The simulation is used to calculate the covariance matrix of uncertainties due
to PDF effects. For the calculation of the corrected number of events $N_{\rm
r}^{\pm}/(\epsilon A)^{\pm}$ in the expression for the asymmetry $A_{\ell}$,
the data term $N_{\rm r}^{\pm}$ of the numerator is fixed to its default value
from the simulation. The denominator term $(\epsilon A)^{\pm}$ is modified for
each ensemble PDF to include its effect on its asymmetry relative to the
default. Covariance sums are evaluated using the differences of asymmetries
calculated with the modified values of $(\epsilon A)^{\pm}$ relative to the
default asymmetries. The eigenvalues and eigenvectors of the covariance matrix
are determined with the method used for the covariance matrix of data
uncertainties described at the end of Sec. IX. The three largest eigenvalues
are comparable to or larger in value than those of the covariance matrix for
data uncertainties, but the others are smaller.

###  X.3  Total systematic uncertainties

A summary of the minimum and maximum values of the systematic uncertainties
from each non-PDF source across the $|\eta|$ bins of the measurement is shown
in Table 2.

Table 2:  Minimum and maximum values of the systematic uncertainties from each non-PDF source for $A_{\ell}$ over the $|\eta|$ bins of the measurement. Except for the specific sources of the plug region, the minimum values correspond to $|\eta|\leq 1$ electrons and the maximum values to $|\eta|>1$. In general, uncertainty values increase with increasing $|\eta|$ values.  Source | Minimum value | Maximum value
---|---|---
Electron-energy scale | $3.1\times 10^{-6}$ | $2.4\times 10^{-4}$
Electron-energy resolution | $5.2\times 10^{-6}$ | $5.9\times 10^{-5}$
Recoil-energy scale | $6.5\times 10^{-6}$ | $3.6\times 10^{-4}$
Efficiency-scale factor | $1.2\times 10^{-6}$ | $6.8\times 10^{-4}$
Backgrounds | $7.6\times 10^{-6}$ | $5.4\times 10^{-4}$
Plug-QCD shape | $8.1\times 10^{-6}$ | $1.5\times 10^{-4}$
Plug high-$E\\!\\!\\!/_{\rm T}$ efficiency | $1.7\times 10^{-6}$ | $3.2\times 10^{-5}$
Plug track-finding bias | $6.0\times 10^{-7}$ | $6.1\times 10^{-5}$

Table 3 shows the data, non-PDF, and PDF uncertainties across the $|\eta|$
bins of the measurement,

Table 3:  Data, non-PDF, and PDF uncertainties of each $|\eta|$ bin. The data and PDF uncertainties are taken from the diagonal terms of their respective covariance matrices. The non-PDF entry is the quadrature sum of the non-PDF uncertainties specified in Table 2. All uncertainties are bin-by-bin.  $|\eta|$ bin | Data | Non-PDF | PDF
---|---|---|---
0.0–0.2 | $2.0\times 10^{-3}$ | $6.8\times 10^{-5}$ | $4.0\times 10^{-6}$
0.2–0.4 | $1.7\times 10^{-3}$ | $3.2\times 10^{-5}$ | $2.0\times 10^{-5}$
0.4–0.6 | $1.6\times 10^{-3}$ | $1.1\times 10^{-4}$ | $3.3\times 10^{-5}$
0.6–0.8 | $1.6\times 10^{-3}$ | $8.0\times 10^{-5}$ | $4.9\times 10^{-5}$
0.8–1.0 | $1.7\times 10^{-3}$ | $1.1\times 10^{-4}$ | $6.7\times 10^{-5}$
1.0–1.2 | $2.8\times 10^{-3}$ | $1.8\times 10^{-4}$ | $2.3\times 10^{-4}$
1.2–1.4 | $2.3\times 10^{-3}$ | $1.3\times 10^{-4}$ | $6.2\times 10^{-4}$
1.4–1.6 | $2.5\times 10^{-3}$ | $1.5\times 10^{-4}$ | $1.1\times 10^{-3}$
1.6–1.8 | $3.3\times 10^{-3}$ | $1.5\times 10^{-4}$ | $2.0\times 10^{-3}$
1.8–2.0 | $4.2\times 10^{-3}$ | $1.8\times 10^{-4}$ | $3.0\times 10^{-3}$
2.0–2.2 | $5.8\times 10^{-3}$ | $2.1\times 10^{-4}$ | $4.5\times 10^{-3}$
2.2–2.4 | $8.4\times 10^{-3}$ | $5.9\times 10^{-4}$ | $6.9\times 10^{-3}$
2.4–2.6 | $1.1\times 10^{-2}$ | $3.4\times 10^{-4}$ | $1.0\times 10^{-2}$
2.6–2.8 | $1.5\times 10^{-2}$ | $6.3\times 10^{-4}$ | $1.7\times 10^{-2}$
2.8–3.0 | $2.3\times 10^{-2}$ | $8.3\times 10^{-4}$ | $2.7\times 10^{-2}$

and Fig. 18 shows the corresponding plot of the data, PDF, and non-PDF
uncertainties presented in Table 3.

Figure 18:  $\delta A_{\ell}(|\eta|)$ as a function of $|\eta|$, where $\delta
A_{\ell}$ denotes the data, PDF, or non-PDF uncertainty. The bold (black)
histogram shows the statistical uncertainty of the data. The lowest (red)
histogram is the non-PDF systematic uncertainty. The remaining lighter-shaded
(blue) histogram is the uncertainty due to PDFs.

For the total systematic uncertainty, the non-PDF and PDF components are
combined. The non-PDF components are negligible, but are added in quadrature
with the diagonal elements of the covariance matrix of PDF uncertainties.

##  XI  Results

The final CDF measurement of the charge asymmetry $A_{\ell}$ using the
electrons from the production and decay of $W$ bosons is presented in Table 4,

Table 4:  Electron-asymmetry results compared with predictions of the NLO calculations. The NNPDF 3.0 and NNPDF 3.1 columns respectively show the predictions of the NLO calculations with the NNPDF 3.0 and NNPDF 3.1 PDFs, and the PDF uncertainties shown are evaluated bin-by-bin. The measurement uncertainties shown are from the diagonal elements of the covariance matrix for the measurement.  $|\eta|$ bin | Measurement | NNPDF 3.0 | NNPDF 3.1
---|---|---|---
| $(\times 10)$ | $(\times 10)$ | $(\times 10)$
0.0–0.2 | $0.164\pm 0.020$ | $0.185\pm 0.011$ | $0.184\pm 0.004$
0.2–0.4 | $0.549\pm 0.017$ | $0.549\pm 0.032$ | $0.545\pm 0.011$
0.4–0.6 | $0.921\pm 0.016$ | $0.893\pm 0.049$ | $0.887\pm 0.016$
0.6–0.8 | $1.246\pm 0.016$ | $1.198\pm 0.063$ | $1.190\pm 0.019$
0.8–1.0 | $1.479\pm 0.018$ | $1.442\pm 0.073$ | $1.433\pm 0.023$
1.0–1.2 | $1.634\pm 0.029$ | $1.600\pm 0.081$ | $1.588\pm 0.025$
1.2–1.4 | $1.647\pm 0.024$ | $1.640\pm 0.086$ | $1.621\pm 0.027$
1.4–1.6 | $1.487\pm 0.027$ | $1.525\pm 0.090$ | $1.496\pm 0.030$
1.6–1.8 | $1.178\pm 0.038$ | $1.214\pm 0.094$ | $1.182\pm 0.035$
1.8–2.0 | $0.688\pm 0.052$ | $0.679\pm 0.100$ | $0.656\pm 0.039$
2.0–2.2 | $-0.009\pm 0.073$ | $-0.082\pm 0.109$ | $-0.072\pm 0.046$
2.2–2.4 | $-1.149\pm 0.109$ | $-1.036\pm 0.124$ | $-0.958\pm 0.058$
2.4–2.6 | $-1.976\pm 0.150$ | $-2.134\pm 0.153$ | $-1.930\pm 0.083$
2.6–2.8 | $-3.115\pm 0.225$ | $-3.302\pm 0.210$ | $-2.889\pm 0.137$
2.8–3.0 | $-4.605\pm 0.356$ | $-4.428\pm 0.324$ | $-3.680\pm 0.248$

along with the default NLO prediction using the NNPDF 3.0 R. D. Ball _et al._
(2015); *nnpdf302; *nnpdf303; *nnpdf304; *nnpdf305; *nnpdf306; *nnpdf306e;
*nnpdf307. Uncertainties of the measurement are represented by the sum of the
covariance matrices for the statistical uncertainties and the systematic
uncertainties. Table 4 also includes a calculation using the NNPDF 3.1 PDFs
derived with the value of $\alpha_{s}=0.118$ at the $Z$-pole mass R. D. Ball
_et al._ (2017). For consistency with the measurement, the predictions using
the NNPDF 3.0 and 3.1 PDFs are restricted to the kinematic region of $P_{\rm
T}^{e}>25$ GeV/$c$, $P_{\rm T}^{\nu}>25$ GeV/$c$, and $M_{\rm T}>45$
GeV/$c^{2}$.

The input data used in the global fits for the NNPDF 3.0 PDFs do not include
any Tevatron measurements of the charge asymmetry in the production of $W$
bosons but do include the lepton-charge asymmetry measurements in the electron
and muon channels from CMS using $pp$ collisions at $\sqrt{s}=7$ TeV S.
Chatrchyan _et al._ (2012, 2014). The input data for the NNPDF 3.1 PDFs
include the final Tevatron measurements of the lepton-charge asymmetry in the
electron and muon channels from D0 V. M. Abazov _et al._ (2013); V. M. Abazov
_et al._ (2015a); *D0196Ae2E. This inclusion significantly reduces the
uncertainty relative to NNPDF 3.0. In addition, the ensemble methodology for
NNPDF 3.1 is more robust in that the ensemble represents a better sampling of
the probability distribution of the fit to input data.

Figures 19 and 20 show the final results for the charge asymmetry using the
electrons at the Tevatron from this measurement and D0 V. M. Abazov _et al._
(2015a); *D0196Ae2E.

Figure 19:  CP-folded distributions of $A_{\ell}(|\eta|)$ as a function of
$|\eta|$. The bold (black) histogram is the result of this measurement, and
the uncertainties shown are evaluated bin-by-bin and include both statistical
and systematic contributions. The (blue) crosses represent the measurement
from D0 V. M. Abazov _et al._ (2015a); *D0196Ae2E. For D0, the bin size is
also 0.2 $|\eta|$-units wide, except for these regions: 1.2–1.6, 2.4–2.7, and
2.7–3.2. Both the CDF and D0 measurements use all the data from Run II of the
Tevatron Collider. The thin (green) histogram is the NLO prediction using the
NNPDF 3.0 PDFs.  Figure 20:  Distributions of $A_{\ell}(|\eta|)$ as a function
of $|\eta|$ for the Tevatron measurements previously shown in Fig. 19, along
with the NLO prediction with NNPDF 3.1 PDFs. The bold (black) histogram is the
result of this measurement and the (blue) crosses the D0 measurement V. M.
Abazov _et al._ (2015a); *D0196Ae2E. The thin (green) histogram is the NLO
prediction using the NNPDF 3.1 PDFs.

All uncertainties presented in Table 4 and shown in Figs. 19 and 20 are bin-
by-bin, and do not reflect their correlations with the uncertainties of
neighboring bins. For the data, interbin correlations increase from about 0.03
to about 0.80 as $|\eta|$ increases from 0 to 3.0.

To compare the CDF measurement with predictions, the $\chi^{2}$ statistic is
evaluated over all bins using the error matrix of the measurement. The
eigenvalues of the corresponding covariance matrix are shown in Fig. 21.

Figure 21:  Eigenvalues $\lambda_{n}$ as a function of the eigenvector number
$n$ from the covariance matrix of the asymmetry measurement. The bold (black)
histogram shows the eigenvalues, and the lighter (blue) horizontal line
corresponds to the smallest value of the diagonal terms of the covariance
matrix.

The comparison of the asymmetry measurement with the prediction derived from
the NNPDF 3.0 (3.1) ensemble yields the $\chi^{2}$ value 32.6 (44.9) for the
15 bins of the measurement. Calculation of the corresponding $\chi^{2}$ value
with the bin-by-bin uncertainties of the measurement shown in Table 4 instead
of the error matrix yields 26.4 (41.2).

The cumulative-$\chi^{2}$ distribution versus $|\eta|$ as a function of
$|\eta|$ is used to assess how the goodness-of-fit varies across the $|\eta|$
bins. For the $\chi^{2}$ evaluated with the error matrix, the $\chi^{2}$
increment per eigenvector covers several $|\eta|$ bins. Consequently, the
increment for each eigenvector is associated with its expectation value of the
$|\eta|$-bin centers, $0.1+0.2j$, where $j$ is the number of the $|\eta|$ bin,
which ranges from 0 to 14. The expectation value is denoted by
$|\eta|^{\prime}$. Figure 22 shows the cumulative-$\chi^{2}$ distributions

Figure 22:  Cumulative $\chi^{2}$ as a function of $|\eta|^{\prime}$. The bold
(black) [light (blue)] histogram is for NNPDF 3.0 [3.1]. The lowermost (gray)
histogram is for an ideal 15-bin measurement and a prediction whose underlying
physics matches that of the measurement. Differences between the underlying
PDFs of the data and that of the calculation result in differences between the
ideal and observed $\chi^{2}$ distributions.

versus $|\eta|^{\prime}$ for the NNPDF 3.0 and 3.1 predictions. These
distributions show that the measurement can tighten the constraints to the
PDFs over a broad region, $|\eta|>0.5$.

Since each of the ensemble PDFs is equally probable, the distribution of
$\chi^{2}$ values from the comparisons between the measurement and the
individual predictions from each of the ensemble PDFs quantifies the
consistency between the ensemble and the underlying PDFs of the measurement.

Figure 23:  Distribution of $\chi^{2}$ values from the comparisons between the
measurement and the individual predictions over 15 bins for the NNPDF 3.0 and
NNPDF 3.1 ensembles of PDFs. The light (blue) vertical line of each panel show
the $\chi^{2}$ value of the ensemble-averaged prediction for NNPDF 3.0 (32.6)
and NNPDF 3.1 (44.9). For NNPDF 3.0, one of the ensemble PDFs gives a
$\chi^{2}$ value of 1548, and is not shown. For NNPDF 3.1, all values are
contained within the panel.

Figure 23 shows the $\chi^{2}$ distributions for the NNPDF 3.0 and NNPDF 3.1
ensemble of PDFs. As the mean and rms of the NNPDF 3.1 ensemble distribution
are much smaller than those of NNPDF 3.0, the NNPDF 3.1 ensemble is thus found
to be a more robust representation of the Tevatron PDFs and their
uncertainties.

The inclusion of this measurement in global PDF fits will improve the
precision of the PDFs over the kinematic region for the Tevatron. Numerical
tables for the measurement and its covariance matrix of uncertainties are
provided as supplemental materials to this paper. Also included are numerical
tables of the $\chi^{2}$ values for each PDF of the NNPDF 3.0 and NNPDF 3.1
ensembles.

##  XII  Summary

The yield asymmetry between positrons and electrons from the decays of
$W^{\pm}$ bosons produced in $p\bar{p}$ collisions at the center-of-momentum
energy of 1.96 TeV is measured as a function of the electron pseudorapidity
using the full Run II data set of CDF, corresponding to 9.1 fb-1 of integrated
luminosity. Results are in Table 4, and Figs. 19 and 20. The uncertainties in
the results are dominated approximately equally by the statistical precision
of the data and the effect of PDF uncertainties on the modeling of acceptance
and efficiencies.

At the Tevatron collider, the asymmetry is sensitive to the slope of the ratio
of $d$\- to $u$-quark parton-distribution functions of the proton versus the
Bjorken-$x$ parameter. Inclusion of this asymmetry measurement in global fits
to PDFs will reduce the overall uncertainties of the PDFs within the kinematic
region of Tevatron collisions.

###### Acknowledgements.

This document was prepared by the CDF collaboration using the resources of the
Fermi National Accelerator Laboratory (Fermilab), a U.S. Department of Energy,
Office of Science, HEP User Facility. Fermilab is managed by Fermi Research
Alliance, LLC (FRA), acting under Contract No. DE-AC02-07CH11359. We thank the
Fermilab staff and the technical staffs of the participating institutions for
their vital contributions. This work was supported by the U.S. Department of
Energy and National Science Foundation; the Italian Istituto Nazionale di
Fisica Nucleare; the U.S. Department of Energy and the Italian Istituto
Nazionale di Fisica Nucleare summer-student exchange program for the years
2014 and 2015; the Ministry of Education, Culture, Sports, Science and
Technology of Japan; the Natural Sciences and Engineering Research Council of
Canada; the National Science Council of the Republic of China; the Swiss
National Science Foundation; the A.P. Sloan Foundation; the Bundesministerium
für Bildung und Forschung, Germany; the National Research Foundation of Korea
(Grants No. 2018R1A6A1A06024970); the Science and Technology Facilities
Council and the Royal Society, United Kingdom; the Russian Foundation for
Basic Research; the Ministerio de Ciencia e Innovación, and Programa
Consolider-Ingenio 2010, Spain; the Slovak R&D Agency; the Academy of Finland;
and the Australian Research Council (ARC).

##  Appendix A  Plug electron selection

The background in the plug-electron sample varies significantly with the
topology of the reconstructed track in the silicon detector. The purity is
adjusted as a function of the quality parameters of the electron candidate,
which are the goodness-of-fit between the measured and expected transverse-
shower shapes $\chi^{2}_{3\times 3}$, and the goodness-of-fit between the
track helix and the hits attached to the helix $\chi^{2}_{\rm trk}$.

Tracks in the plug region are reconstructed with the calorimetry-seeded
tracking algorithm (“Phoenix”), which searches for hits in seven layers of the
silicon detector. The tracks are characterized by two parameters,
$(n_{l},\>n_{h})$, where $n_{l}$ is the number of fiducial layers of the
silicon tracker traversed by the particle, and $n_{h}$ is the number of hits
detected in those layers that are associated with the track by the algorithm.
Multiple hits per layer can be attached to the track by the algorithm. The
number of layers is a prediction based on the track-helix parameters and a
simplified model of the silicon-detector geometry. It restricts the electron
candidate to a region of $|\eta_{\rm det}|$.

The maximum allowed $\chi^{2}_{\rm trk}/n_{h}$ and $\chi^{2}_{3\times 3}$
values are both 10 in the default selection. For electron candidates with
lower quality tracks, the maximum $\chi^{2}$ values are reduced to improve the
signal purity. For events with high-$E_{\rm T}$ electrons, the distributions
of these quantities are peaked at values of about 1.0 and decrease
exponentially beyond the peak. For background events, the distributions are
broad and relatively uniform across the $\chi^{2}$ values in relation to those
for the electrons. The events used to adjust the maximum values must pass the
asymmetry-measurement criteria of Sec. V.3, except for the $E\\!\\!\\!/_{\rm
T}$ criterion. After an adjustment, the $E\\!\\!\\!/_{\rm T}$ distribution of
the event is used to evaluate independently the purity of the electron sample
from $W$-boson decays. The results are shown in Table 5.

Table 5:  Selection criteria matrix for plug-region electrons. The index $n_{l}$ is the number of fiducial layers in the silicon tracker traversed by the particle, and $n_{h}$ is the number of hits detected in those layers that are associated with the track. In each table-entry pair, the first value is the maximum value for the $\chi^{2}_{\rm trk}/n_{h}$ quantity, and the second is the maximum value for the $\chi^{2}_{3\times 3}$ quantity.  | $n_{h}=3$ | $n_{h}=4$ | $n_{h}=5$ | $n_{h}=6$ | $n_{h}=7$
---|---|---|---|---|---
$n_{l}=7$ | $2.5/3.0$ | $2.5/4.0$ | $10/10$ | $10/10$ | $10/10$
$n_{l}=6$ | $2.5/4.0$ | $2.5/5.0$ | $10/10$ | $10/10$ | $10/10$
$n_{l}=5$ | $3.0/3.0$ | $10/3.0$ | $10/3.0$ | $10/3.0$ | $10/3.0$
$n_{l}=4$ | $10/3.0$ | $10/3.0$ | $10/3.0$ | $10/3.0$ | $10/3.0$
$n_{l}=3$ | $10/3.0$ | $10/3.0$ | $10/3.0$ | $10/3.0$ | $10/3.0$

In addition, the lateral-shower profile measured in the PES detector, which
consists of 5 mm wide scintillator strips, is required to be consistent with
that of an electron. The profile is measured with the ratio of the shower
energy observed in five strips relative to nine strips, $R_{5/9}$. For EM
showers, the $R_{5/9}$ distribution is peaked near the value of 0.9. The
consistency criterion is $R_{5/9}>0.75$.

The $E\\!\\!\\!/_{\rm T}$ distribution after the application of the additional
selection criteria is shown in Fig. 24.

Figure 24:  $E\\!\\!\\!/_{\rm T}$ distribution for plug-region electrons
accepted by the additional selection criteria. The (black) crosses are the
data, and the (blue) histogram is the simulation. The online-trigger selection
allows the peak at low values of $E\\!\\!\\!/_{\rm T}$.

The $E\\!\\!\\!/_{\rm T}$ distribution for events passing the default
selection criteria but failing the additional criteria is shown in Fig. 25.

Figure 25:  $E\\!\\!\\!/_{\rm T}$ distribution for plug-region electrons
rejected by the additional selection criteria. The (black) crosses are the
data, and the (blue) histogram is the simulation, which contains about 18% of
the events without the additional requirements. The online-trigger selection
allows the peak at low values of $E\\!\\!\\!/_{\rm T}$.

The simulation is described in Secs. VI and VIII.

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