\section{Introduction}

The angular distribution of electrons from the
Drell-Yan~\cite{DrellYan} process is used to measure the
electroweak-mixing parameter $\sin^2\theta_W$ \cite{EWKsw2Ref}.
At the Tevatron, Drell-Yan pairs are produced by the process
$p\bar{p} \rightarrow e^+e^-  + X$, where the $e^+e^-$ pair is
produced through an intermediate $\gamma^*/Z$ boson, and $X$ is
the hadronic final state associated with the production of
the boson. In the standard model, the Drell-Yan process at the
Born level is described by two parton-level amplitudes:
\begin{eqnarray*}
  q\bar{q} & \rightarrow & \gamma^* \rightarrow e^+e^-, \; {\rm and} \\
  q\bar{q} & \rightarrow & Z \rightarrow e^+e^-.
\end{eqnarray*}
The fermions $(f)$ couple to the virtual photon via a vector
coupling, $Q_f \gamma_\mu$, where $Q_f$ is the fermion charge
(in units of $e$). The fermion coupling to $Z$ bosons consists
of both vector $(V)$ and axial-vector $(A)$ couplings:
$g_V^f \gamma_\mu + g_A^f \gamma_\mu\gamma_5$.
The Born-level couplings are
\begin{eqnarray*}
  g_V^f & = & T_3^f - 2Q_f \: \sin^2\theta_W  \\
  g_A^f & = & T_3^f ,
\end{eqnarray*}
where $T_3^f$ is the third component of the fermion weak isospin.
The $\sin^2\theta_W$ parameter is related to the $W$-boson mass
$M_W$, and the $Z$-boson mass $M_Z$, by the relationship
$\sin^2\theta_W =  1 - M_W^2/M_Z^2$ which holds to all orders
in the on-shell scheme. These couplings have been investigated
both at the Tevatron~\cite{CDFIIsw2e,D0sw2e}, and at LEP-1 and
SLD \cite{LEPfinalZ}.
\par
In this article, the parameter $\sin^2\theta_W$ is inferred
from a previous measurement~\cite{CSangcoef21} of the angular
distribution of Drell-Yan $e^+e^-$ pairs produced at the
Tevatron. The measurement investigates higher-order quantum
chromodynamic (QCD) corrections to the angular distribution,
using electron pairs in the $Z$-boson region 66--116~GeV/$c^2$
from 2.1~fb$^{-1}$ of collisions. This analysis utilizes the
results of that measurement to test a new method to obtain
$\sin^2\theta_W$. One of the measurements, the $A_4$ angular
coefficient, is sensitive to $\sin^2\theta_W$ and is compared
with QCD predictions for various values of $\sin^2\theta_W$.
The predictions also include electroweak-radiative corrections
comparable to those utilized at LEP-1 and SLD \cite{LEPfinalZ}.

\par
Section~\ref{ElecAngDistr} provides an overview of both
the electron angular distributions and the
method used to obtain $\sin^2 \theta_W$.
Section~\ref{QCDEWKpred} discusses QCD calculations required
by the new method. A technique to use and incorporate electroweak
radiative-correction form factors for high-energy $e^+e^-$
collisions into the Drell-Yan process is presented.
Section~\ref{A4expInput} reviews and documents the event sample,
simulation of the data, and methods used in the previous
measurement, and describes how the measurement is used in
this analysis.
Section~\ref{systUncerts} describes the systematic uncertainties.
Finally, Sec.~\ref{finalResults} gives the results, and
Sec.~\ref{theEndSummary} the summary.
The units $\hbar = c =1$ are used for equations
and symbols, but standard units are used for numerical values.

\section{\label{ElecAngDistr}
Electron Angular Distributions}

The angular distribution of electrons in the boson rest frame
is governed by the polarization state of the $\gamma^*/Z$ boson.
In amplitudes at higher order than tree level, initial-state
QCD interactions of the colliding partons impart transverse momentum,
relative to the collision axis, to the $\gamma^*/Z$ boson.
This affects the polarization states.
\par
The polar and azimuthal angles of the $e^-$ in the rest frame of the boson
are denoted as $\vartheta$ and $\varphi$, respectively. For this analysis,
the ideal positive-$z$ axis coincides with the direction of the incoming
quark so that $\vartheta$ parallels the definition used in $e^+e^-$
collisions at LEP~\cite{LEPfinalZ}.
This frame is approximated by the Collins-Soper
(CS) rest frame~\cite{CollinsSoperFrame} for $p\bar{p}$ collisions.
The CS frame is reached from the laboratory frame via a Lorentz boost
along the laboratory $z$ axis into a frame where the $z$ component of
the lepton-pair momentum is zero, followed by a boost along the
transverse momentum of the pair. The transverse momentum ($P_{\rm T}$)
in a reference frame is the magnitude of momentum transverse to the
$z$ axis. Within the CS frame, the $z$ axis for the polar
angle is the angular bisector between the proton direction and the
negative of the anti-proton direction. The $x$ axis for the azimuthal
angle is the direction of the lepton-pair $P_{\rm T}$.
At $P_{\rm T} = 0$, the CS and laboratory coordinate systems are the
same, and if the incoming quark of the Drell-Yan parton amplitude
is from the proton, the $z$ axis and quark directions coincide.
\par
The general structure of the Drell-Yan lepton angular
distribution in the boson rest frame consists of nine helicity
cross sections~\cite{MirkesA0to7a, *MirkesA0to7b},
\begin{eqnarray*}
\frac{dN}{d\Omega}
  & \propto &
        \: (1 + \cos^2 \vartheta) +  \nonumber \\
  &   & A_0 \:\frac{1}{2} \:
             (1 -3\cos^2 \vartheta) + \nonumber \\
  &   & A_1 \: \sin 2\vartheta
               \cos \varphi +   \nonumber \\
  &   & A_2 \: \frac{1}{2} \:
               \sin^2 \vartheta
               \cos 2\varphi +  \nonumber \\
  &   & A_3 \: \sin \vartheta
               \cos \varphi +   \nonumber \\
  &   & A_4 \: \cos \vartheta + \nonumber \\
  &   & A_5 \: \sin^2 \vartheta
               \sin 2\varphi +  \nonumber \\
  &   & A_6 \: \sin 2\vartheta
               \sin \varphi +   \nonumber \\
  &   & A_7 \: \sin \vartheta
               \sin \varphi \: .
\end{eqnarray*}
The $A_{0-7}$ coefficients are cross section ratios, and are
functions of the boson kinematic variables. They vanish at
$P_{\rm T} = 0$, except for the electroweak part of $A_4$
responsible for the forward-backward $e^-$ asymmetry in
$\cos \vartheta$.
The $A_4$ coefficient is relatively uniform across the
range of transverse momentum where the cross section is large,
but slowly drops for larger values of $P_{\rm T}$ where the
cross section is very small.
The $A_{5-7}$ coefficients appear at second
order in the QCD strong coupling, $\alpha_s$, and are
small in the CS frame \cite{MirkesA0to7a, *MirkesA0to7b}.
Hereafter, the angles $(\vartheta, \: \varphi)$ and the
angular coefficients $A_{0-7}$ are specific to the CS rest
frame.
\par
The $A_4 \cos\vartheta$ term is parity violating, and is due to
vector and axial-vector current amplitude interference. Its
presence adds an asymmetry to the $\varphi$-integrated
$\cos \vartheta$ cross section.
Two sources contribute: the interference between
the $Z$-boson vector and axial-vector amplitudes, and the
interference between the photon vector and $Z$-boson axial-vector
amplitudes. The asymmetric component from the $\gamma$-$Z$
interference cross section is proportional to $g_A^f$. The asymmetric
component from $Z$ boson self-interference has a coupling
factor that is a product of $g_V^f/g_A^f$ from the electron and quark
vertices, and thus is related to $\sin^2 \theta_W$.
At the Born level, this product is
\begin{displaymath}
   (1 - 4|Q_e|\sin^2\theta_W) \: (1 - 4|Q_q|\sin^2\theta_W) ,
\end{displaymath}
where $e$ and $q$ denote the electron and quark, respectively.
For the Drell-Yan process, the quarks are predominantly light
quarks: $u$, $d$, or $s$. As $\sin^2\theta_W \approx 0.223$,
the coupling factor has an enhanced sensitivity to
$\sin^2\theta_W$ at the electron-$Z$ vertex. A 1\% variation
in $\sin^2\theta_W$ changes the electron factor (containing
$Q_e$) by $\approx 8$\%, while the quark factor (containing
$Q_q$) changes by $\approx 1.5$\% for the $u$ quark, and
$\approx 0.4$\% for the $d$ and $s$ quarks.
Loop and vertex electroweak-radiative corrections are
multiplicative form-factor corrections to the couplings that
change their value by a few percent.
\par
Traditionally, $\sin^2 \theta_W$ is inferred from the
forward-backward asymmetry of the $e^-$ $\cos \vartheta$
distribution as a function of the dielectron-pair mass. The new
method for the inference  of  $\sin^2 \theta_W$ has two inputs: an
experimental measurement of the  $A_4$ angular-distribution
coefficient, and predictions of the $A_4$  coefficient for various
input values of $\sin^2 \theta_W$. Electroweak and QCD radiative
corrections are included in the predictions of the $A_4$ coefficient.
\par
The new method to infer $\sin^2 \theta_W$ utilizes the value of the
cross-section weighted average, $\bar{A}_4$, for both the experimental
input and predictions. The average is 
\begin{displaymath}
  \bar{A}_4  = 
	\frac{1}{\sigma}
	\int_{-\infty}^{\infty} dy
        \int_0^{\infty} dP_{\rm T}^2
        \int dM
        \: A_4
	\frac{d^3\sigma}{dy dP_{\rm T}^2 dM} ,
\end{displaymath}
where $\sigma$ is the  integrated cross-section, and
$y$, $P_{\rm T}$, and $M$ are the lepton-pair rapidity,
transverse momentum, and mass, respectively. The energy
and momentum of particles are denoted as $E$ and $P$,
respectively. For a given coordinate frame, the rapidity is
$y = \frac{1}{2} \, \ln[\,(E + P_{\rm z})/(E - P_{\rm z})\,]$,
where $P_{\rm z}$ is the component of momentum along the
$z$ axis of the coordinate frame. The mass integration is
limited to the $Z$-boson region 66--116~GeV/$c^2$.
\par
The experimental input for the $\bar{A}_4$ coefficient
is derived from
a previous measurement of the angular-distribution
coefficients $A_0$, $A_2$, $A_3$, and $A_4$,
in independent ranges of the dielectron-pair
$P_{\rm T}$ \cite{CSangcoef21}. In this analysis,
the individual measurements for the $A_4$ coefficient are
combined into an average.
The predictions provide the relationship between $\sin^2 \theta_W$
and $\bar{A}_4$. The QCD predictions of $\bar{A}_4$
include an implementation of electroweak radiative
corrections derived from an approach adopted at LEP
\cite{Dizet,*zfitter621,*zfitter642}.


\section{\label{QCDEWKpred}
Enhanced QCD Predictions}

Drell-Yan process calculations with QCD radiation do not
typically include the full electroweak-radiative corrections.
However, the QCD, quantum electrodynamic (QED), and weak
corrections can be organized to be individually gauge invariant
so that they can be applied separately and independently.
\par
QED radiative corrections with photons in the final state are not
included in the calculation of the $\bar{A}_4$ coefficient. Instead,
they are applied in the physics and detector simulation of the
Drell-Yan process used in the measurement of the $A_4$ coefficients.
For the process
$q\bar{q} \rightarrow e^+e^-$, QED final-state radiation is most
important, and is included. The effects of QED radiative corrections
are removed from the measurement of the $A_4$ coefficients.
\par
The Drell-Yan process and the production of quark pairs in high
energy $e^+e^-$ collisions are analog processes:
   $q\bar{q} \rightarrow e^-e^+$ and
   $e^-e^+   \rightarrow q\bar{q}$.
At the Born level,
the process amplitudes are of the same form except for the
interchange of the electron and quark labels. Electroweak radiative
corrections, calculated and extensively used for precision
fits of LEP-1 and SLD measurements to the standard model \cite{LEPfinalZ},
can be applied to the Drell-Yan process.
\par
In the remainder of this section, the technique used to incorporate
independently
calculated electroweak radiative corrections for $e^+e^-$ collisions
into existing QCD calculations for the Drell-Yan process is
presented. The results of the QCD calculations for the value of
the $\bar{A}_4$ coefficient are also presented.

\subsection{\label{EWKradcor}
Electroweak radiative corrections}

The effects of electroweak radiative corrections are incorporated
into Drell-Yan QCD calculations via form factors for fermion-pair
production in $e^+e^-$ collisions,
$e^+e^- \rightarrow Z \rightarrow f\bar{f}$. 
The form factors are calculated by
\textsc{zfitter 6.43}~\cite{Dizet,*zfitter621,*zfitter642},
which is used with LEP-1 and SLD measurement inputs for standard-model
tests \cite{LEPfinalZ}. It is a semi-analytical calculation for
fermion-pair production and radiative corrections for high-energy
$e^+e^-$ collisions. The set of radiative corrections in each form
factor is gauge invariant. Thus it includes
$W$-boson loops in the photon propagator and $Z$ propagators at
fermion-photon vertices. Consequently, the weak and QED corrections
are separately gauge invariant. 
The renormalization scheme used by \textsc{zfitter} is the on-shell
scheme~\cite{OnShellScheme}, where particle masses are on-shell,
and
\begin{equation}
  \sin^2 \theta_W = 1 - M_W^2/M_Z^2
\label{baseSW2}
\end{equation}
holds to all orders of perturbation theory by definition.
Since the $Z$-boson mass is accurately known
(to $\pm 0.0021$ GeV/$c^2$ \cite{LEPfinalZ}),
the inference of $\sin^2 \theta_W$ is equivalent to an indirect
$W$-boson mass measurement.
\par
Form factors calculated by \textsc{zfitter} are stored
for later use in QCD calculations.
Details of the form-factor calculation with its specific
standard-model assumptions and parameters are presented in
Appendix~\ref{appendixZFITTER}. The calculated form factors are
$\rho_{eq}$, $\kappa_e$, $\kappa_q$, and $\kappa_{eq}$, where
the label $e$ denotes an electron, and $q$ a quark. As the
calculations use the massless-fermion approximation, 
the form factors only depend on the charge and weak isospin
of the fermions. Consequently, the stored form factors are
distinguished by three labels: $e$ (electron type),
$u$ (up-quark type), and $d$ (down-quark type). The form factors
are complex valued, and functions of the
$\sin^2\theta_W$ parameter and the Mandelstam $s$ variable of the
$e^+e^- \rightarrow Z \rightarrow f\bar{f}$ process.
\par
The first three form factors can be trivially incorporated into
the $q\bar{q} \rightarrow Z \rightarrow e^+e^-$ interaction
currents. The Born-level $g_A^f$ and $g_V^f$ couplings within the
currents are replaced with
\begin{eqnarray*}
  g_V^f & \rightarrow & \sqrt{\rho_{eq}}\,
                       ( T_3^f - 2Q_f \kappa_f \: \sin^2\theta_W ) ,
		       \:\: {\rm and} 
                       \nonumber \\
  g_A^f & \rightarrow & \sqrt{\rho_{eq}} \, T_3^f ,
\end{eqnarray*}
where $f = e$ or $q$. The resulting electron-quark
current-current interaction amplitude contains a term proportional
to $\kappa_e \kappa_q \sin^4 \theta_W$. However, as this is an
approximation of the desired coefficient,
$\kappa_{eq} \sin^4\theta_W$,
a further correction to the amplitude (which is discussed in
Sec.~\ref{QCDcalcs}) is required.
\par
The combination $\kappa_f \sin^2 \theta_W$, called an
effective-mixing parameter, is directly accessible from measurements
of the asymmetry in the $\cos \vartheta$ distribution.
However, neither the $\sin^2 \theta_W$ parameter nor the
form factors can be inferrred from experimental measurements
without the standard model. The effective-mixing parameters
are denoted as $\sin^2 \theta_{\rm eff}$ to distinguish them from
the on-shell definition of $\sin^2 \theta_W$ (Eq.~(\ref{baseSW2})).
The Drell-Yan process is most sensitive to the parameter
$\sin^2 \theta_{\rm eff}$ of the lepton vertex, or
$\kappa_e \sin^2 \theta_W$, which is commonly
denoted as $\sin^2 \theta^{\rm lept}_{\rm eff}$.
At the $Z$ pole, $\kappa_e$ is independent of the quark type.
For comparisons with other measurements, the value of
$\sin^2 \theta^{\rm lept}_{\rm eff}$ at the $Z$ pole
$\operatorname{Re}\kappa_e(s_Z) \sin^2 \theta_W$ ($s_Z = M_Z^2$),
is used.
\par
Only the photon self-energy correction from fermion loops
is used with the \textsc{zfitter} $Z$-amplitude form factors.
The self-energy correction is a complex-valued form factor
of the photon propagator, and its effect is often described as
the running of the electromagnetic interaction coupling.
The corrections from $W$-boson loops in the photon propagator
and $Z$ propagators at the fermion-photon vertices have been
combined with their gauge-dependent counter terms in the
$Z$-amplitude form factors. With this reorganization of terms,
all form factors are gauge invariant.



\subsection{\label{QCDcalcs}
QCD calculations}

The Drell-Yan QCD calculations are improved by incorporating
the \textsc{zfitter} form factors into the process amplitude. This
provides an enhanced Born approximation (EBA) to the electroweak
terms of the amplitude. The QED photon self-energy correction
is included as part of the EBA. The photon amplitude influences
the shape of $A_4$ away from the $Z$ pole via its interference with
the axial-vector part of the $Z$ amplitude. The $\gamma$-$Z$
interference, whose cross section is proportional to $(s-M_Z^2)$,
begins to dominate the total-interference cross section away from
the $Z$ pole. As it dilutes measurements of
$\sin^2 \theta_{\rm eff}$, photonic corrections also need
to be included.
\par
The \textsc{zfitter} form factors, $\rho_{eq}$, $\kappa_e$, and
$\kappa_q$ are inserted into the Born $g_A^f$ and $g_V^f$ couplings
for the  Drell-Yan process.
To accomodate the $\kappa_{eq}$ form factor,
a correction term proportional to the
$(\kappa_{eq} - \kappa_e\kappa_q)$ form factor is added
to the Born amplitude. The photon self-energy correction
is incorporated with
the photon propagator in the amplitude. Complex-valued form
factors are used in the amplitude. Operationally, only the
electroweak-coupling factors in the QCD cross sections are
affected. To be consistent with the standard LEP
$Z$-boson resonant line shape, the $Z$-boson propagator is
defined as in $A_q$ (Eq.~(\ref{eqn_Aq})). The total-decay
width $\Gamma_Z$, calculated with \textsc{zfitter}
is also used.
\par
A leading-order (LO) QCD or {\it tree} calculation
of $\bar{A}_4$ for the process,
$p\bar{p} \rightarrow \gamma^*/Z \rightarrow e^+e^-$, is
used as the baseline EBA calculation with \textsc{zfitter}
form factors. It is used to provide a reference for the 
sensitivity of  $\bar{A}_4$ to QCD radiation.
The CT10~\cite{ct10pdfs} next-to-leading-order (NLO) parton
distribution functions (PDF) provide the incoming parton
flux used in all QCD calculations discussed in this section
except where specified otherwise.
The EBA calculation using \textsc{zfitter} form-factor tables
is developed for this analysis. The EBA implementation of
the form factors in the tree calculation is tested against
\textsc{zgrad2}, a LO QCD calculation with electroweak radiative
corrections. Only expected differences are found.
The details of the tests are in Appendix~\ref{ebaXchecks}.
\par
Two NLO calculations,
\textsc{resbos}~\cite{ResBos1,*ResBos2,*ResBos3,*ResBosc221}
and the \textsc{powheg-box} framework \cite{Powheg-Box},
are modified to be EBA-based QCD calculations.
For both calculations, the boson $P_{\rm T}^2$ distribution
is finite as $P_{\rm T}^2$ vanishes. The \textsc{resbos}
calculation combines a NLO fixed-order calculation at high
boson-$P_{\rm T}$ with the Collins-Soper-Sterman
resummation formalism
\cite{methodCSS,*wfactorCSS1,*wfactorCSS2,*wfactorCSS3}
at low boson-$P_{\rm T}$,
which is an all-orders summation of large terms
from gluon emission. The \textsc{resbos} calculation uses
CTEQ6.6~\cite{cteq66pdf} NLO PDFs.
The \textsc{powheg-box} is a fully unweighted partonic-event
generator that implements Drell-Yan production of $ee$-pairs
at LO and NLO. The NLO
production implements a Sudakov form factor that controls the
infrared diverence at low $P_{\rm T}$, and is constructed to be
interfaced with parton showering to avoid double counting.
The \textsc{pythia}~6.41~\cite{pythia64} parton-showering
algorithm is used to produce the final hadron-level event.
\par
At tree level, the
electron angular-distribution coefficient $A_4$ is a function
of the $ee$-pair rapidity $(y)$ and mass $(M)$:
$A_4(y,M)$. The mass dependence is significant, and typically
represented as the forward-backward asymmetry in
$\cos \vartheta$,
\begin{displaymath}
  A_{\rm fb}(M) = \frac{\sigma^+(M) - \sigma^-(M)}
                   {\sigma^+(M) + \sigma^-(M)} 
                = \frac{3}{8}A_4(M) \:,
\end{displaymath}
where $\sigma^+(M)$ is the total cross section for
$\cos \vartheta > 0$, and $\sigma^-(M)$ is the cross section for
$\cos \vartheta < 0$. Figure~\ref{fig_treeAfb} shows
\begin{figure}
\includegraphics
   [width=85mm]
   {fig01_treeAfb.eps}
\caption{\label{fig_treeAfb}
Value of $A_{\rm fb}$ as a function of mass as resulting from a
tree-level calculation with 
$\sin^2 \theta_W = 0.223$. The horizontal line corresponds to
$A_{\rm fb} = 0$ and the vertical line corresponds to $M = M_Z$.
}
\end{figure}
the typical behavior of $A_{\rm fb}(M)$. At $M=M_Z$, the asymmetry
$A_{\rm fb}$ originates purely from $Z$ bosons, and
is sensitive to $\sin^2 \theta_{\rm eff}$.
\par
Beyond leading order, the angular coefficients begin to depend on
the boson $P_{\rm T}$, {\it i.e.}, $A_4(y,M,P_{\rm T})$. The projections
$A_4(y)$ and  $A_4(P_{\rm T})$ for $66 < M < 116$~GeV/$c^2$ are
approximately constant except
at the extremes of large $|y|$ or $P_{\rm T}$. The \textsc{powheg-box}
events are post-processed by the \textsc{pythia} parton showering,
which adds additional boson $P_{\rm T}$, i.e., higher-order QCD
corrections. While the angular-distribution coefficients of the
\textsc{powheg-box} LO events with \textsc{pythia} parton showering
and the NLO-based coefficients are similar at low $P_{\rm T}$,
they can differ at large $P_{\rm T}$.
\par
The tree and NLO calculations of the $\bar{A}_4$ coefficient
for various input values of  $\sin^2 \theta_W$ are shown in
\begin{figure}
\includegraphics
   [width=85mm]
   {fig02_A4vsSW2.eps}
\caption{\label{fig_a4vssw2}
Dependence of $\sin^2 \theta_W$ on $\bar{A}_4$ 
for various $\sin^2 \theta_W$ values from different QCD
calculations.
The tree calculation is represented by the solid (black) curve,
the \textsc{resbos} calculation is represented by the dashed
(blue) curve, and the \textsc{powheg-box} NLO calculation is
represented by the dots-dashed (red) curve.
}
\end{figure}
Fig.~\ref{fig_a4vssw2}.
To quantify the effects of higher-order QCD corrections on
$\bar{A}_4$, the ratio
$R_4 = \bar{A}_4({\rm NLO}) / \bar{A}_4({\rm tree})$
is used, where NLO and tree denote $\bar{A}_4$ evaluated at
NLO and at the tree level, respectively.
Figure~\ref{fig_R4ratio} shows the fractional difference
$1 - R_4$
\begin{figure}
\includegraphics
   [width=80mm]
   {fig03_R4ratio.eps}
\caption{\label{fig_R4ratio}
$1 - R_4$ as a function of  $\sin^2 \theta_W$.
The open squares, circles, and diamonds correspond to 
the \textsc{resbos}, \textsc{powheg-box} NLO, and
\textsc{powheg-box} LO calculations, respectively. The
\textsc{powheg-box} LO prediction includes higher-order
QCD corrections from the parton-showering algorithm of
\textsc{pythia}.
}
\end{figure}
for the \textsc{resbos} and \textsc{powheg-box} calculations
with various values of $\sin^2 \theta_W$.
Higher-order QCD corrections do not significantly alter
$\bar{A}_A$ with respect to its value from tree-level
amplitudes.
\par
The \textsc{resbos} and \textsc{powheg-box} NLO calculations
are similar and consistent. The \textsc{resbos}
calculation is chosen as the default EBA-based QCD calculation of
$\bar{A}_4$ with various input values of $\sin^2 \theta_W$.
As the \textsc{powheg-box} NLO program has a diverse
and useful set of calculation options, it is used to estimate
QCD systematic uncertainties. 



\section{\label{A4expInput}\boldmath
Experimental input to $\bar{A}_4$}\unboldmath

The value of the $\bar{A}_4$ angular-distribution coefficient
is derived from the previous measurement of
electron angular-distribution coefficients \cite{CSangcoef21}.
Elements of the  measurement are summarized in this section
for completeness and supplemental documentation.
\par
The coefficients $A_0$, $A_2$, $A_3$, and $A_4$ are measured in
the CS rest frame and in independent ranges of the
dielectron-pair $P_{\rm T}$. These measurements are reproduced in
Table~\ref{tblAngCoeff21},
\begin{table}
\caption{\label{tblAngCoeff21}
Measured angular coefficients \cite{CSangcoef21}.
The first contribution to the uncertainty is statistical,
and the second systematic. The lepton-pair mass range is
restricted to 66--116~GeV/$c^2$, and the mean lepton-pair
$P_{\rm T}$ values of the events in the five bins are
4.8, 14.1, 26.0, 42.9, and 73.7 GeV/$c$, respectively. }
\begin{ruledtabular}
\begin{tabular}{ccc} 
$P_{\rm T}$ bin (GeV/$c$) &  $A_0$ ($\times 10^{-1}$)  &  $A_2$ ($\times 10^{-1}$)  \\ \hline
0--10
    &   $0.17 \pm 0.14 \pm 0.07$   &  $0.16 \pm 0.26 \pm 0.06$    \\ 
10--2
    &   $0.42 \pm 0.25 \pm 0.07$   &  $-0.01 \pm 0.35 \pm 0.16$   \\ 
20--3
    &   $0.86 \pm 0.39 \pm 0.08$   &  $0.52 \pm 0.51 \pm 0.29$   \\ 
35--5
    &   $3.11 \pm 0.59 \pm 0.10$   &  $2.88 \pm 0.84 \pm 0.19$  \\ 
$>$ 5
    &  $4.97 \pm 0.61 \pm 0.10$    &  $4.83 \pm 1.24 \pm 0.02$   \\
\end{tabular}
\end{ruledtabular}
\begin{ruledtabular}
\begin{tabular}{ccc} 
$P_{\rm T}$ bin (GeV/$c$)&  $A_3$ ($\times 10^{-1}$)  &  $A_4$ ($\times 10^{-1}$) \\ \hline 
0--10
     & $-0.04 \pm 0.12 \pm 0.01$  &   $1.10 \pm 0.10 \pm 0.01$    \\ 
10--2
     & $0.18 \pm 0.16 \pm 0.01$   &  $1.01 \pm 0.17 \pm 0.01$ \\ 
20--3
     & $0.14 \pm 0.24 \pm 0.01$   &  $1.56 \pm 0.26 \pm 0.01$ \\ 
35--5
     & $-0.19 \pm 0.41 \pm 0.04$   &  $0.52 \pm 0.42 \pm 0.03$ \\ 
$>$ 5
     & $-0.47 \pm 0.56 \pm 0.02$   & $0.85 \pm 0.50 \pm 0.05$   \\
\end{tabular}
\end{ruledtabular}
\end{table}
and are derived from a $p\bar{p}$ collision sample corresponding
to an integrated luminosity  is 2.1~fb$^{-1}$. The data and
simulation are understood,
and the modeling of the data in the simulation is accurate.
The measurement of the angular coefficients is data driven,
and fully corrected for acceptance and detector resolution.
\par
The description of the data simulation, Sec.~\ref{A4expDatSim},
is presented before the description of the event sample,
Sec.~\ref{A4expSample}, to aid in the discussion of the
data-driven corrections to the simulation.
Section~\ref{A4expAngCoeff} describes the method used to
measure the angular coefficients, $A_0$, $A_2$, $A_3$,
and $A_4$ in independent ranges of the dielectron-pair
$P_{\rm T}$. Finally, Sec.~\ref{A4expAverage} describes the
method used to average the previous independent measurements
of $A_4$, and to estimate the uncertainties on the combination.



\subsection{\label{A4expDatSim}
Data simulation}
Drell-Yan pair production is simulated using the Monte Carlo
event generator, \textsc{pythia}~\cite{Pythia621}, and CDF~II
detector-simulation programs. This simulation is only used for
the measurement of the angular coefficients.
\textsc{pythia} generates the hard,
leading-order QCD interaction, $q+\bar{q} \rightarrow \gamma^*/Z$,
simulates initial-state QCD radiation via its parton-shower algorithms,
and generates the decay $\gamma^*/Z \rightarrow l^+l^-$.
The CTEQ5L~\cite{Cteq5pdf} nucleon parton-distribution functions are
used in the QCD calculations.
The underlying event and boson $P_{\rm T}$ parameters are from
\textsc{pythia} tune \textsc{aw} (i.e., \textsc{pytune} 101, which is a
tuning to previous CDF data)~\cite{Pythia621,run1CDF-Z,PyTuneAW}.
In addition, \textsc{photos} 2.0~\cite{Photos20a,*Photos20b,Photos20c},
adds final-state QED radiation to decay vertices with charged particles
(e.g. $\gamma^*/Z \rightarrow ee$).
The parton-shower simulation of \textsc{pythia} uses a QCD resummation
calculation. The resulting physics model is adequate to allow 
data-driven adjustments to the underlying angular-distribution coefficients
and other physics distributions.
\par
The measurement of the electron angular coefficient depends on the
correct modeling of the physics and both the detector acceptance
and efficiency.
All data efficiencies, global and particle-trajectory dependent,
as well as time-dependent, are measured in the data and
incorporated into the simulation. The simulation also uses the
calorimeter energy scales and resolutions measured in the data.
The data-driven approach is iterative with simultaneous
tuning of both the generator physics-model distributions and the
detector-modeling parameters that make the distributions of
reconstructed quantities of simulated events match the data
precisely.
The tuning of the generator physics-model distributions include
adjustments to both the boson production kinematics ($y$, $M$, and
$P_{\rm T}$), and the lepton angular distributions ($A_0$, $A_2$,
$A_3$, and $A_4$).
\par
The \textsc{photos} program generates multiple photons at the
$\gamma^*/Z \rightarrow ee$ vertex via a form factor to the production
cross section. Soft and collinear photons are simulated to
$\alpha_{em}^2$ leading-logarithmic accuracy, where $\alpha_{em}$ is
the fine-structure constant. The simulation of hard, non-collinear
photon emission is a full $\alpha_{em}$ matrix-element algorithm,
except that the interference terms are removed to make the algorithm
process-independent \cite{Photos20c}. For the $\gamma^*/Z \rightarrow ee$
process, the interference terms are restored in an approximate way.
The real and virtual photon-emission cross-section infrared
divergences at each order are regularized and analytically
combined to cancel the divergences. Photons with energies smaller
than the default regularization energy are not generated.
\par
In addition to QCD initial-state radiation, \textsc{pythia}
adds initial- and final-state QED
radiation via its parton-showering algorithm. The
regularization-energy threshold is very low, and most of
the photons are very soft. This threshold is lower than the
one in \textsc{photos}, so the soft-photon emission of
\textsc{pythia} is complementary to the hard-photon emission
of \textsc{photos}.
\par
The default implementation of \textsc{pythia} plus
\textsc{photos} (\textsc{pythia+photos}) QED radiation
in the CDF data-simulation infrastructure is validated
with \textsc{zgrad2}~\cite{ZGrad2},
a leading-order QCD Drell-Yan calculation with a
${\cal O}(\alpha_{em})$ matrix-element calculation for the
emission of zero or one real photon. Both initial-state
and final-state radiation are included. As \textsc{zgrad2}
has soft and collinear photon-regularization regions for
the cancellation of divergences, these regions are
excluded from comparisons with \textsc{pythia+photos}.
\par
The $e^+e^- + {\rm n}\gamma$ systems are first boosted
to their center-of-momentum frames to minimize
distortions to the electron and photon kinematic
distributions from QCD (QED) initial-state radiation.
To simplify the comparison of the multi-photon system
of \textsc{pythia+photos} to the single photon of
\textsc{zgrad2}, the multi-photon system is
clustered by adding up the photon momentum vectors.
Events with cluster energies under 0.5~GeV,
the \textsc{zgrad2} regularization energy,
are classified as events without photons.
The photon (cluster) energy distributions are shown in
Fig.~\ref{fig_egph2zg2}.
\begin{figure}
\includegraphics
   [width=85mm]
   {fig04_egamPh2Zg2.eps}
\caption{\label{fig_egph2zg2}
Photon (cluster) energy: $E_{n\gamma}$. Events without
photons are included in the lowest energy bin. The
bold histogram is \textsc{pythia+photos}. The lighter
histogram is \textsc{zgrad2}. The integral of the
\textsc{pythia+photos} distribution is normalized to the
\textsc{zgrad2} total cross section.
}
\end{figure}
For events with photons, the smallest angle between 
the photon (cluster) and either lepton is denoted as
$\beta$. The $\cos \beta$
distribution is shown in Fig.~\ref{fig_cbph2zg2}.
\begin{figure}
\includegraphics
   [width=85mm]
   {fig05_cosbPh2Zg2.eps}
\caption{\label{fig_cbph2zg2}
Separation between the photon (cluster) and the nearest
lepton: $\cos \beta$. The bold histogram is
\textsc{pythia+photos}. The lighter histogram is \textsc{zgrad2}.
The integral of the \textsc{pythia+photos} distribution within
$0 < \cos\beta < 0.8$ is normalized to the corresponding
\textsc{zgrad2} cross section.
}
\end{figure}
The overall consistency is good. Differences are expected as the
\textsc{pythia+photos} correction is ${\cal O}(\alpha_{em}^2)$
or larger, while the \textsc{zgrad2} correction is
${\cal O}(\alpha_{em})$.

\subsection{\label{A4expSample}
Measurement event sample}

The CDF experimental apparatus is a general-purpose
detector~\cite{refCDFII} at the Fermilab Tevatron $p\bar{p}$ collider
whose center-of-momentum (cm)
energy is 1.96~TeV. The positive $z$-axis is directed along the proton
direction. For particle trajectories, the polar angle $\theta_{\rm cm}$ is
relative to the proton direction and the azimuthal angle $\phi_{\rm cm}$ is
oriented about the beamline axis with $\pi/2$ being vertically upwards.
The component of the particle energy transverse to the beamline
is defined as $E_{\rm T} = E \sin \theta_{\rm cm}$. 
The pseudorapidity of a particle trajectory is
$\eta = -\ln \tan (\theta_{\rm cm}/2)$. Detector coordinates are
specified as $(\eta_{\rm det}, \phi_{\rm cm})$, where $\eta_{\rm det}$ is
the pseudorapidity relative to the detector center ($z=0$).
\par
The central charged-particle tracking-detector (tracker)
is a 3.1~m long, open-cell drift chamber~\cite{refCOT}
that radially extends from 0.4 to 1.4~m. Between the
Tevatron beam pipe and the central tracker is a 2~m long silicon
vertex-tracker~\cite{refSVXII}. 
Both trackers are immersed in a 1.4~T axial
magnetic field. Outside the central tracker is a central barrel
calorimeter~\cite{refCEM,refChad}
that covers the region $|\eta_{\rm det}|<1.1$.  The forward
end-cap regions are covered by the end-plug (``plug'')
calorimeters~\cite{refPEM,refPES,refPHA} that
cover the regions $1.1<|\eta_{\rm det}|<3.5$. Both the central
and plug calorimeters are segmented into electromagnetic
and hadronic sections. The electromagnetic sections of both
calorimeters have preshower and shower-maximum detectors for
electron identification. The silicon tracker, in conjunction with
the plug shower-maximum detector, provides tracking coverage
in the plug region to $|\eta_{\rm det}|$ of about 2.8. As
$|\eta_{\rm det}|$ increases for plug-region tracks, the
transverse track length within the magnetic field decreases,
resulting in increasingly poorer track-curvature resolutions.

\par
Events are required to contain two electron candidates having a
pair mass in the $Z$-boson region of 66--116 GeV/$c^2$.
Electrons in both the central and plug calorimeters are used.
The events are classified into three dielectron topologies:
CC, CP, and PP, where C (P) denotes that the electron is detected
in the central (plug) calorimeter. Electrons are required to
have an associated track, pass standard selection and fiducial
requirements \cite{refCDFII}, and be isolated from other
calorimeter activity. The electron
kinematic variables are based on the electron energy measured in the
calorimeters and the track direction. The kinematic and fiducial
regions of acceptance for electrons in the three topologies are
summarized below.
\begin{enumerate}
 \item Central--Central (CC)
    \begin{itemize}
      \item $E_{\rm T} > 25$ (15)~GeV for electron 1 (2)
      \item $0.05 < |\eta_{\rm det}| < 1.05$
    \end{itemize}
 \item Central--Plug (CP)
    \begin{itemize}
      \item $E_{\rm T} > 20$~GeV for both electrons
      \item Central electron: $0.05 < |\eta_{\rm det}| < 1.05$
      \item Plug electron: $1.2 < |\eta_{\rm det}| < 2.8$
    \end{itemize}
 \item Plug--Plug (PP)
    \begin{itemize}
      \item $E_{\rm T} > 25$~GeV for both electrons
      \item $1.2 < |\eta_{\rm det}| < 2.8$
    \end{itemize}
\end{enumerate}
The CC-electron $E_{\rm T}$ selection is asymmetric, with electron 1
having the highest $E_{\rm T}$. The asymmetric selection, an
optimization from the previous measurement of electron
angular-distribution coefficients, improves the acceptance in the
electron phase space \cite{CSangcoef21}.  The PP-electron candidates,
required to be in the same end of the CDF~II detector,
extend the rapidity coverage to $|y| \approx 2.9$.
The kinematic limit of $|y|$ for the production of $ee$-pairs 
at the $Z$-boson mass is 3.1. The acceptance is limited for
PP-topology Drell-Yan electrons on opposite ends of the CDF~II detector;
the dielectrons tend to be at low $ee$-pair
rapidities, and are overwhelmed by the QCD di-jet backgrounds.
\par
The numbers of events passing all requirements in the CC, CP, and PP topologies
are 51~951, 63~752, and 22~469, respectively. All requirements in
the association of charged-particle tracks to both final-state electrons
significantly reduces the backgrounds coming from QCD, the electroweak
(EWK) processes of $WW$, $WZ$, $ZZ$, $t\bar{t}$, $W+$jets, and also
$Z \rightarrow \tau^+\tau^-$.
The QCD background is primarily from dijets where a particle in a jet
is misidentified as an electron or is an electron from a photon conversion.
The high-$E_{\rm T}$ electron sources have at least one real electron.
The second electron is either a real second electron or a fake one.
The backgrounds and the methods used to determine them are
described further in previous measurements \cite{CSangcoef21,zpt21}.
The QCD backgrounds, determined from the same dielectron sample used
for the measurement, constitute 0.3\% of the sample.
The EWK
backgrounds are derived from \textsc{pythia}~\cite{Pythia621} samples
with detector simulation, and amount to 0.2\%. The fraction of QCD plus
EWK backgrounds is approximately constant over $\cos \vartheta$ for each
topology. Background-subtracted distributions are used in measurements.
\par
The online-event selection and electron-identification efficiencies
are measured as functions of 
$\eta_{\rm det}$ for both central and plug electrons. The measured
efficiencies are incorporated in the simulation as scale factors
(event weights). Plug-electron efficiencies are separately measured
for the CP and PP electrons. A significant fraction of the PP-toplology
electrons are in more forward regions of the calorimeter relative
to those of the CP topology. The efficiencies for electrons to be
identified in the plug calorimeter particularly in
the very forward regions, have a
significant time-dependence (due to increasing instantaneous
luminosities) which are measured and incorporated into the
simulation.
\par
Corrections to the simulated-event electron energy-scales and
resolutions are determined using both the $ee$-pair mass
and electron-$E_{\rm T}$ distributions. The energy scales and
resolutions of the simulation are adjusted so that both the
simulated-electron $E_{\rm T}$ distributions and the $ee$-pair
mass distributions are matched to the observed
distributions \cite{zpt21}.
The central- and plug-electron energy scales are accurately
constrained by the three independent $ee$-pair topologies.
Figures \ref{zptmeecc}, \ref{zptmeecp}, and \ref{zptmeepp}
\begin{figure}
\includegraphics
   [width=85mm]
   {fig06_meecc.eps}
\caption{\label{zptmeecc}
The CC-topology $ee$-pair mass distribution. The crosses are the
data and the histogram is the simulation.
}
\end{figure}
\begin{figure}
\includegraphics
   [width=85mm]
   {fig07_meecp.eps}
\caption{\label{zptmeecp}
The CP-topology $ee$-pair mass distribution. The crosses are the
data and the histogram is the simulation.
}
\end{figure}
\begin{figure}
\includegraphics
   [width=85mm]
   {fig08_meepp.eps}
\caption{\label{zptmeepp}
The PP-topology $ee$-pair mass distribution. The crosses are the
data and the histogram is the simulation.
}
\end{figure}
show the $ee$-pair mass distributions for the CC, CP, and PP
topologies, respectively. The simulated-data to data $\chi^2$
for the CC-, CP-, and PP-topology $ee$-pairs are
117, 126, and 127, respectively, for 100 bins. The event count
of the simulated data is normalized to that of the data, and
only statistical uncertainties are used in the calculation.


\par
The Collins-Soper frame angle,
$\cos \vartheta$~\cite{CollinsSoperFrame}, is reconstructed
using these laboratory-frame quantities: the lepton energies
$(E)$, the lepton momenta along the beam line $(P_z)$, the
dilepton mass $(M)$, and the dilepton transverse momentum
$(P_{\rm T})$. The angle of the negatively-charged lepton is
\begin{displaymath}
  \cos \vartheta = \frac{ l^-_+l^+_- - l^-_-l^+_+ }
                 { M \sqrt{M^2 + P_{\rm T}^2} }  \; ,
\end{displaymath}
where $l_\pm = (E \pm P_z)$ and
the $+$ $(-)$ superscript specifies that $l_\pm$ is for
the positively- (negatively-)charged lepton. A similar expression
is used for $\varphi$. For plug electrons,
charge identification is not used because of significant
charge misidentification probability at large $|\eta_{\rm det}|$.
As an interchange
of the $e^-$ with the $e^+$ changes the sign of $\cos \vartheta$,
$|\!\cos \vartheta|$ is used for the PP-topology dielectrons. For
CP-topology dielectrons, the central-electron charge determines
whether the $e^-$ is the central or plug electron. For the CC- and
CP-topology dielectrons, the charge-misidentification probabilities
are 0.3\% and 0.4\% respectively.
\par
The $\cos \vartheta$ bias and resolution of the observed
events are estimated using the simulation. The bias
$\Delta \cos \vartheta$, is the difference between the true
$\cos \vartheta$ before final-state QED radiation and the
measurement. The $\Delta \cos \vartheta$ distribution is
affected by the electron-energy resolution of the calorimeters and
electron-charge misidentification. The effect of 
calorimeter energy-resolution smearing is small for all dielectron
topologies. The bias distribution has a narrow non-Gaussian central
core centered at zero with less than 1\% rms deviation.
The calorimeters have a negligible effect on the mean of the bias
but dominate the resolution. Charge misidentification
in the CC- and CP-dielectron topologies contributes a relatively flat
background with a negligible bias.

\subsection{\label{A4expAngCoeff}
Angular coefficient measurement}

The angular distribution integrated over $\varphi$ is
\begin{equation}
   N(\vartheta,A_0,A_4) \propto
   1 + \cos^2 \vartheta + A_0 \, \frac{1}{2}(1 - 3\cos^2 \vartheta)
                        + A_4\cos \vartheta  .
\label{eqnA0A4cos}
\end{equation}
In each $P_{\rm T}$ bin, this distribution is modified by the
acceptance and resolution of the detector into the observed
$\cos \vartheta$ distribution.
The simulated events used to model the $\cos \vartheta$
distribution are selected as data.
The underlying $A_0$ and $A_4$ values in the simulation physics
model are simultaneously varied until the simulated
$\cos \vartheta$ distributions match the corresponding data
distributions. The variation is accomplished with an event weight
\begin{displaymath}
   w = \frac{N(\vartheta, A_0^\prime, A_4^\prime)}
            {N(\vartheta, A_0,        A_4 )} \; .
\end{displaymath}
The base physics-model angular coefficients are denoted as $A_0$
and $A_4$, and variations to them are denoted as $A_0^\prime$ and
$A_4^\prime$. The best-fit values for $A_0^\prime$ and $A_4^\prime$
are determined using a binned log-likelihood fit between the
data and simulation. The event normalization of the simulation
sample relative to the data is a parameter in the log-likelihood
fit as the detector acceptance depends on $A_0$ and $A_4$.
The log-likelihood of each dielectron topology is separately
evaluated and then combined into a joint probability-density
function.
\par
The  best-fit values of $A_0^\prime$ and $A_4^\prime$ for each
$P_{\rm T}$ bin are incorporated into the physics model prior to
the determination of $\varphi$-based angular coefficients. The
angular distribution integrated over $\cos \vartheta$ is
\begin{displaymath}
   N(\varphi,A_2,A_3) \propto
   \frac{8}{3} +
   \frac{2}{3} A_2 \: \cos 2\varphi +
   \frac{\pi}{2} A_3 \: \cos \varphi  .
\end{displaymath}
The $A_5$ and $A_7$ terms, expected to be relatively
small \cite{MirkesA0to7a,MirkesA0to7a}, are dropped.
The best-fit values to $A_2$ and $A_3$, denoted as $A_2^\prime$
and $A_3^\prime$ respectively, are also obtained using the same
method as for $A_0^\prime$ and $A_4^\prime$.
The fits to the observed $\cos \vartheta$
and $\varphi$ distributions are iterated to obtain the
final values of $A_0^\prime$, $A_1^\prime$,  $A_3^\prime$, and
$A_4^\prime$ for each $P_{\rm T}$ bin. The measurements are
fully corrected for detector acceptance and resolution.

\subsection{\label{A4expAverage}\boldmath
${A_4}$ average}\unboldmath

The measured values of $A_0$, $A_2$, $A_3$, and $A_4$
(Table~\ref{tblAngCoeff21}) are incorporated into the
physics model. The one-dimensional $\cos\vartheta$
distribution of events with $ee$-pair masses in the range
66--116~GeV/$c^2$ has the functional
form, $N(\vartheta,\bar{A}_0,\bar{A}_4)$
(Eq.~(\ref{eqnA0A4cos})). The best fit to the distribution
for the functional form yields the parameters
$\bar{A}_0 = 0.0514 \pm 0.0010$ and
$\bar{A}_4 = 0.1100 \pm 0.0008$,
where the uncertainties are due to the limited size of the
simulated sample. The parameters are uncorrelated because
their angular functions are orthogonal Legendre polynomials.
These angular-coefficient parameters are the cross-section
weighted averages based on the measurements. Without the
data-driven corrections, the baseline (\textsc{pythia})
model gives $\bar{A}_4 = 0.1128$ and $\bar{A}_0 = 0.0304$.
\par
Experimental uncertainties of $\bar{A}_4$ are evaluated
directly from the observed and simulated $\cos \vartheta$
distributions of events selected for the angular-distribution
measurement.
As the simulated distributions include the measured values
of $A_0$, $A_2$, $A_3$, and $A_4$, the variations of
$A_0^\prime$ and $A_4^\prime$ are via scale factors to
$\bar{A}_0$ and $\bar{A}_4$ of the simulation physics model.
The best-fit values from the log-likelihood fits are
$\bar{A}_0 = 0.0497 \pm 0.0073$ and
$\bar{A}_4 = 0.1095 \pm 0.0079$,
and the central values are consistent with the cross-section
weighted averages. The uncertainties reflect the limited
data-sample size.
The correlation coefficient between the uncertainty of
$\bar{A}_4$ and $\bar{A}_0$ or the simulation normalization
is under 0.01. The angular function of the $A_4$
coefficient is an odd function in $\cos \vartheta$, and
it is also orthogonal to $1+\cos^2\vartheta$.
\par
The experimental value of $\bar{A}_4$ used to infer
$\sin^2 \theta_W$ is
\begin{displaymath}
  \bar{A}_4  =  0.1100 \pm  0.0079 ,
\end{displaymath}
where the central value is the cross-section weighted average,
and its uncertainty is the statistical
uncertainty from the log-likelihood fit.
\par
The $\cos \vartheta$ distribution for the combined CC- and
CP-topology dielectrons is shown in Fig.~\ref{fig_coscsCCCP}.
The comparison of the simulation with the data yields a
$\chi^2$ of 44.8 for 50 bins.
\begin{figure}
\includegraphics
   [width=85mm]
   {fig09_coscsCCCP.eps}
\caption{\label{fig_coscsCCCP}
The observed $\cos \vartheta$ distribution for the
combined CC and CP topologies. The crosses
are the background-subtracted data, and the solid
histogram is the simulation.
}
\end{figure}
The yield of simulated events is determined by the fit.
For the separate
CC- and CP-topology $\cos \vartheta$ distributions, the
comparison between the simulation and the data yields
a CC-topology $\chi^2$ of 49.0 for 50 bins, and 
a CP-topology $\chi^2$ of 46.9 for 46 bins.
\begin{figure}
\includegraphics
   [width=85mm]
   {fig10_coscsPP.eps}
\caption{\label{fig_coscsPP}
The observed $|\cos \vartheta|$ distribution for the
PP topology. The crosses are the background-subtracted
data, and the solid histogram is the simulation.
}
\end{figure}
Figure~\ref{fig_coscsPP} shows the $\cos \vartheta$ distribution
of the PP topology. The comparison of the simulation with the data
yields a $\chi^2$ of 31.7 for 35 bins.
The CC and CP topologies are the ones that 
mainly constrain the fit for $\bar{A}_0$ and $\bar{A}_4$.
The PP topology helps to constrain the simulation event
normalization.
\par
The observed $\varphi$ distributions are also well described
by the simulation. Figure~\ref{fig_phicsCCCP} shows the
\begin{figure}
\includegraphics
   [width=85mm]
   {fig11_phicsCCCP.eps}
\caption{\label{fig_phicsCCCP}
The observed $\varphi$ distribution for the 
combined CC and CP topologies. The crosses
are the background-subtracted data, and the solid
histogram is the simulation.
}
\end{figure}
distribution for the combined CC and CP $ee$-pair topologies.
The comparison of the simulation with the data yields a
$\chi^2$ of 51.5 for 50 bins. For the separate
CC- and CP-topology $\varphi$ distributions, the
$\chi^2$ between the simulation and the data are
56.1 and 46.9, respectively, for 50 bins.
Figure~\ref{fig_phicsPP} shows the $\varphi$ distribution
\begin{figure}
\includegraphics
   [width=85mm]
   {fig12_phicsPP.eps}
\caption{\label{fig_phicsPP}
The observed $\varphi$ distribution of electrons for the PP topology.
The crosses are the background-subtracted
data, and the solid histogram is the simulation.
}
\end{figure}
for events in the PP topology. The comparison of simulation with
the data yields a $\chi^2$ of 47.4 for 50 bins.

\section{\label{systUncerts}
Systematic Uncertainties}

The systematic uncertainties on the inference of $\sin^2\theta_W$
(or $M_W$) contain contributions from both the experimental input 
for $\bar{A}_4$
and the predictions of $\bar{A}_4$ for various input values of
$\sin^2\theta_W$. The prediction uncertainties dominate.
Both the experimental and prediction systematic uncertainties are
small compared to the experimental statistical uncertainty.

\subsection{Experimental input}

The $\bar{A}_4$ angular-coefficient uncertainties considered
include the simulation energy scale, the
background estimates, and the single-electron selection and
tracking-efficiency measurements.
\par
The central- and plug-electron energy scales for the simulation
are accurately constrained by the data. Their residual
uncertainties correspond to an estimated uncertainty for the
$\bar{A}_4$ coefficient of $\pm 0.0003$. This is not
completely independent of the experimental statistical
uncertainty, but is included in quadrature with the
other experimental systematic uncertainties.
\par
The largest independent uncertainty is from the
background subtraction. It is estimated by varying the fraction
of the default background that is subtracted, then re-fitting
the observed $\cos\vartheta$ distribution for a modified best-fit
value of $\bar{A}_4$. The level of background subtracted from the
data is varied so that the change in the corresponding
likelihood value corresponds to the nominal one-standard-deviation 
change of the results with respect to the central value.
The result is $\Delta \bar{A}_4 = \pm 0.0003$.
\par 
The measured single-electron efficiences incorporated in the
simulation have uncertainties. When propagated to the 
$\cos \vartheta$ bins, the fractional uncertainties of the CC, CP,
and PP topologies are relatively constant. The levels of uncertainty
for the CC, CP, and PP topology yields are 0.9\%, 0.6\%, and 4\%,
respectively. The PP-topology electron acceptance extends into
very forward regions of the plug calorimeter, and signficantly
beyond that for CP-topology electrons. As measurements are
difficult in this far forward region, the PP uncertainty is larger.
Since the same single-electron measurements are
used in each bin, they are treated as 100\% correlated across
the $\cos \vartheta$ bins. To estimate uncertainties, the overall
dielectron-topology efficiency is rescaled within its uncertainty
prior to log-likelihood fits of the observed $\cos\vartheta$
distribution. This is equivalent to a systematic offset in its
event normalization relative to the other topologies. The
uncertainty on the $\bar{A}_4$ coefficient from this source is
found to be negligible. Because the angular function
of the $\bar{A}_4$ coefficient $\cos\vartheta$, is odd, the
normalization of the simulated events and $\bar{A}_4$ are nearly
uncorrelated in all fits.

\subsection{Predictions}

The QCD mass-factorization and renormalization scales and
uncertainties in the CT10 PDFs affect the calculated value of
$\bar{A}_4$. The corresponding systematic uncertainties on
$\bar{A}_4$ are evaluated using \textsc{powheg-box} NLO. As the
\textsc{resbos} calculation is chosen as the default
for $\bar{A}_4$, the associated uncertainty is
also included in the overall systematic uncertainty.
\par
In all QCD calculations, the mass-factorization and
renormalization scales are both set to the $ee$-pair mass.
To evaluate the effect on $\bar{A}_4$ from different scales,
the running scales are varied independently by a factor
ranging from $0.5$ to $2$ in the calculations.
The largest observed deviation in $\bar{A}_4$ from the default
value is the QCD-scale uncertainty. This uncertainty
is $\Delta\bar{A}_4({\rm QCD \; scale)} = \pm 0.0004$.
\par
The CT10 set of 26 eigenvector pairs of 90\% C.L. uncertainty
PDFs are used to evaluate the effect of PDF uncertainties on
$\bar{A}_4$: The quadrature sum of the PDF uncertainties
to $\bar{A}_4$ from each pair gives the total PDF uncertainty.
The 68\% C.L. uncertainty to $\bar{A}_4$ is obtained by
rescaling the 90\% C.L. uncertainty down by a factor of 1.645
to give
$\Delta\bar{A}_4({\rm PDF}) = \pm 0.0026$.
\par
The default \textsc{resbos} calculation of the $\bar{A}_4$
coefficent for various input values of $\sin^2\theta_W$ yields
coefficent values 0.5--0.8\% larger than the baseline
tree calculation. The \textsc{powheg-box} calculations
are slightly different. A conservative systematic uncertainty
of $\pm 1$\% is assigned for differences, and this is
denoted as the EBA uncertainty.
\par
In summary, the total systematic uncertainty from the QCD
mass-factorization and renormalization scales, and uncertainties
in the CT10 PDFs is $\Delta\bar{A}_4({\rm QCD}) = \pm 0.0026$. The
EBA uncertainty is
$\Delta\bar{A}_4({\rm EBA}) = \pm 0.01 \bar{A}_4$.
These prediction uncertainties are combined in quadrature.
At the measured value of $\bar{A}_4$ (0.1100), the total
prediction uncertainty is $\pm 0.0029$.

\section{\label{finalResults}
Results}

The fully-corrected value of the $\bar{A}_4$ coefficient
for this analysis is
\begin{displaymath}
  \bar{A}_4 = 0.1100 \pm 0.0079 
                     \pm 0.0004  ,
\end{displaymath}
where the first contribution to the uncertainty is statistical
and the second systematic. Prediction uncertainties are separated
from experimental uncertainties, but for the total uncertainties of
derived results  presented in this section, all uncertainties
are combined in quadrature.
\par
The $A_4$ angular coefficient is directly sensitive to the
$\sin^2\theta_{\rm eff}$ parameter at the lepton and quark vertices of
the Drell-Yan amplitude. However, it is most sensitive to
the effective-mixing parameter at the lepton vertex,
and consequently, the $A_4$ coefficient is primarily a measure of
$\sin^2\theta^{\rm lept}_{\rm eff}$.
The standard model (SM) provides the means to express the
effective-mixing parameters in terms of its static parameters and the
collision dynamics, to map the correspondence between the
effective-mixing parameters and the angular coefficient $A_4$,
\begin{displaymath}
  {\rm SM}(\sin^2\theta_W) 
	   \stackrel{{\rm EWK}}{\longmapsto}
		\sin^2\theta_{\rm eff}(s)
           \stackrel{{\rm QCD}}{\longleftrightarrow} A_4(s)  ,
\end{displaymath}
and to interpret measurements of this coefficient in terms
of the fundamental $W$-boson mass, $M_W$, or the
$\sin^2\theta_W$ parameter. The symbol EWK denotes electroweak
radiative corrections, and the symbol QCD denotes EBA-based
QCD calculations. For the $\bar{A}_4$ coefficient, the
kinematic dependencies of the $\sin^2\theta_{\rm eff}(s)$
parameters are averaged by the integration over the $\sqrt{s}$
range of 66--116~GeV. Over this range, the predicted differences
between the effective-leptonic and effective-quark mixing
parameters
are under $0.0005$ in magnitude. The interpretation of the
measured $\bar{A}_4$ coefficient in terms of the $\sin^2\theta_W$
or $M_W$ parameter is interesting, but model dependent.
Under different standard-model contexts, the same value
of an effective-mixing parameter can be associated with different
values of the $\sin^2\theta_W$ parameter.
\par
The \textsc{resbos}
predictions of $\bar{A}_4$ for various values of the $M_W$
(or $\sin^2\theta_W$) parameter are shown in
Fig.~\ref{fig_A4vsMW}
\begin{figure}
\includegraphics
   [width=85mm]
   {fig13_A4vsMW.eps}
\caption{\label{fig_A4vsMW}
Distribution of $M_W$ as a function of the $\bar{A}_4$ value
as predicted by \textsc{resbos}.
The prediction is the solid (blue) diagonal line and its
one standard-deviation limits are the bands.
The $\bar{A}_4$ measurement is the bold vertical line, and its
one standard-deviation limits are the lighter vertical lines.
The hatched horizontal bands are uncertainty limits from other
$W$-mass measurements (see text).
}
\end{figure}
along with the observed value. The intersection of the measured
value with the prediction can be interpreted as the
indirect measurement of
$M_W$ or $\sin^2 \theta_W$ within the context of standard-model
assumptions specified in Appendix~\ref{appendixZFITTER}:
\begin{eqnarray*}
  \sin^2 \theta_W & = & 0.2246 \pm 0.0009 \\
  M_W ({\rm indirect}) & = & 80.297 \pm 0.048 \;{\rm GeV}/c^2,
\end{eqnarray*}
where the uncertainty includes both measurement and prediction
uncertainties. The experimental statistical uncertainty for
the value of $M_W$ is $\pm 0.045$~GeV/$c^2$.
The systematic uncertainty,
predominantly from the prediction, is $\pm 0.017$~GeV/$c^2$.
The corresponding statistical and systematic uncertainties
for the value of $ \sin^2 \theta_W$ are $\pm 0.0009$ and
$\pm 0.0003$, respectively. The other $W$-mass
measurements shown in Fig.~\ref{fig_A4vsMW} are from combinations
of the Tevatron, and the LEP-1 and SLD measurements~\cite{EWKsw2Ref}:
\begin{eqnarray*}
  M_W & = & 80.385 \pm 0.015 \; {\rm GeV}/c^2, \;\; {\rm direct}\\
      & = & 80.365 \pm 0.020 \; {\rm GeV}/c^2, \;\; Z\;{\rm pole},
\end{eqnarray*}
where direct refers to the combination of LEP-2 and Tevatron
$W$-mass measurements, and $Z$ pole is an indirect measurement
from electroweak standard-model fits to LEP-1 and SLD $Z$-pole
measurements with the top-quark mass measurement. 
Figure~\ref{fig_compareMW} shows the comparison of
\begin{figure}
\includegraphics
   [width=85mm]
   {fig14_cmpWmass.eps}
\caption{\label{fig_compareMW}
Comparisons of experimental measurements of the $W$-boson mass:
``TeV and LEP-2'' represents direct measurements of the $W$-boson mass;
``LEP-1 and SLD (m$_{\rm t}$)'' represents the standard-model analysis
of $Z$-pole measurements;
and ``CDF ee 2~fb$^{-1}$'' represents this analysis.
The horizontal bars represent total uncertainties. For this analysis,
the inner uncertainty bar is the measurement uncertainty.
}
\end{figure}
these $W$-boson mass results.
\par
The $\sin^2\theta_W$ parameter also specifies the correspondence
between the $A_4$ angular coefficient and the effective-mixing
parameters. As the  parameters are averaged in the $\bar{A}_4$
angular coefficient, a reference value of the effective-leptonic
mixing parameter at the $Z$ pole,
\begin{displaymath}
    \sin^2\theta^{\rm lept}_{\rm eff}   =  
         \operatorname{Re}\kappa_e(s_Z,\sin^2\theta_W) 
                 \sin^2\theta_W ,
\end{displaymath}
is provided for comparisons.
Although the $\bar{A}_4$ coefficient is integrated across the
$\sqrt{s}$ range of 66--116~GeV, the bulk of the integrated cross
section is near the vicinity of the $Z$ pole ($s_Z=M_Z^2$).
Therefore, it is an effective probe of the leptonic
$\sin^2\theta_{\rm eff}$ at the reference $s_Z$ value.
The reference value of $\sin^2\theta^{\rm lept}_{\rm eff}$
corresponding to the $\bar{A}_4$ angular-coefficient
measurement is
\begin{displaymath}
     \sin^2\theta^{\rm lept}_{\rm eff} = 0.2328 \pm 0.0010 ,
\end{displaymath}
where both statistical and systematic uncertanities are included.
The experimental statistical uncertainty is $\pm 0.0009$.
The systematic uncertainty, predominantly from the prediction,
is $\pm 0.0003$. Relative to $\sin^2\theta^{\rm lept}_{\rm eff}$,
the effective-mixing parameters of the $u$- and $d$-type quarks
$\operatorname{Re}\kappa_{u,d}\sin^2\theta_W$ (at $s_Z$),
are lower by 0.0001 and 0.0002, respectively. The corresponding
$\sin^2\theta^{\rm lept}_{\rm eff}$ measurements
from LEP-1 and SLD are
\begin{eqnarray*}
   & & 0.23153 \pm 0.00016 \;\;(Z{\rm\mbox{-}pole}) \:\: {\rm and} \\
   & & 0.2320  \pm 0.0021  \;\;({\rm light \; quarks}) ,
\end{eqnarray*}
where the ``$Z$-pole'' measurement is from the standard-model
analysis of the combined $Z$-pole results, and the ``light quarks''
measurement is from the light-quark ($u$, $d$, and $s$)
asymmetries~\cite{LEPfinalZ}. The previous corresponding Tevatron
value from D0 derived from a measurement of $A_{\rm fb}(M)$ is
$\sin^2\theta^{\rm lept}_{\rm eff} = 0.2309 \pm 0.0008  \pm 0.0006$,
where the first contribution to the uncertainty is statistical and
second systematic \cite{D0sw2e}.
Figure~\ref{fig_compareSW2leff} shows a comparison of
\begin{figure}
\includegraphics
   [width=85mm]
   {fig15_cmpSw2leff.eps}
\caption{\label{fig_compareSW2leff}
Comparisons of experimental measurements of 
$\sin^2\theta^{\rm lept}_{\rm eff}$:
``All $Z$ pole'' represents the LEP-1 and SLD standard-model
analysis of $Z$-pole measurements and ``light quarks'' represents
the LEP-1 and SLD results from the light-quark asymmetries;
``D0 ee 5~fb$^{-1}$'' represents the D0 $A_{\rm fb}(M)$ analysis;
and ``CDF ee 2~fb$^{-1}$'' represents this analysis.
The horizontal bars represent total uncertainties. For this analysis,
the inner uncertainty bar is the measurement uncertainty.
}
\end{figure}
these $\sin^2\theta^{\rm lept}_{\rm eff}$ measurements.
\par
The admixture of light quarks in the Drell-Yan production and
$e^+e^-$ collisions is somewhat different. The contributions of
the various quarks to the incoming parton flux in Tevatron
$p\bar{p}$ collisions are evaluated with the CT10 PDFs at a
virtuality scale of $Q = M_Z$ and at a momentum fraction of
$x=0.047$ (corresponding to $\sqrt{s} = M_Z$). The $q\bar{q}$
fluxes of the $d$, $s$, $c$, and $b$ quarks relative to the
$u$-quark flux are 0.51, 0.06, 0.02, and 0.01, respectively.
\par
The EBA-based QCD calculations include the full electroweak
radiative correction formalism of \textsc{zfitter}.
Without this formalism, the extracted values of
$\sin^2\theta^{\rm lept}_{\rm eff}$ tend to be slightly
lower. For the value $\bar{A}_4 = 0.1100$, the difference
between the derived value of $\sin^2\theta^{\rm lept}_{\rm eff}$
with and without the \textsc{zfitter} formalism for the
\textsc{resbos} calculation is 0.0002. The corresponding value
for the \textsc{powheg-box} calculation is 0.0003. The
difference between the EBA-based \textsc{resbos} value and
the non-EBA \textsc{pythia}~6.41 value obtained with
CTEQ5L PDFs is 0.0005.
These differences are not negligible for precision measurements.


\section{\label{theEndSummary}
Summary}

The angular distribution of Drell-Yan $e^+e^-$ pairs provides
information on the electroweak-mixing parameter $\sin^2\theta_W$.
The electron forward-backward asymmetry in the polar-angle
distribution $\cos\vartheta$ is governed by the
$A_4\cos\vartheta$ term, whose $A_4$ coefficient is directly
related to the $\sin^2 \theta^{\rm lept}_{\rm eff}$ mixing
parameter
at the lepton vertex, and indirectly to $\sin^2\theta_W$.
A new method for the determination of
$\sin^2 \theta^{\rm lept}_{\rm eff}$
using the average value of $A_4$ $(\bar{A}_4)$ for $ee$-pairs
in the $Z$-boson mass region of 66--116~GeV/$c^2$ is tested.
The method utilizes standard-model calculations of
$\bar{A}_4$ for different input values of $\sin^2 \theta_W$,
or equivalently, $\sin^2 \theta^{\rm lept}_{\rm eff}$,
for comparison with the measured value of $\bar{A}_4$.
These calculations include both
quantum chromodynamic and electroweak radiative corrections. 
The result for $\sin^2 \theta_W$ is equivalent to
an indirect determination of the $W$-boson mass.
However, unlike $\sin^2 \theta^{\rm lept}_{\rm eff}$, the
interpretation of $\sin^2 \theta_W$ or the $W$-boson mass
is dependent on the standard-model context.
Using the value $\bar{A}_4 = 0.1100 \pm 0.0079$ observed in
a sample corresponding to 2.1~fb$^{-1}$ of integrated luminosity
from $p\bar{p}$ collisions at a center-of-momentum
energy of 1.96~TeV,
\begin{eqnarray*}
  \sin^2 \theta^{\rm lept}_{\rm eff} & = & 0.2328 \pm 0.0010 , \\
  \sin^2 \theta_W & = & 0.2246 \pm 0.0009 ,
				      \: {\rm and}\\   
  M_W ({\rm indirect}) & = & 80.297 \pm 0.048 \;{\rm GeV}/c^2 \: .
\end{eqnarray*}
Each uncertainty includes statistical and systematic
contributions. Both results are consistent with LEP-1 and SLD
$Z$-pole measurements. The value of
$\sin^2 \theta^{\rm lept}_{\rm eff}$
is also consistent with the previous Tevatron value from D0.
The results of the test for the new method are promising.
As the uncertainties are predominantly statistical, the
measurement will improve with the analysis of the full
Tevatron sample corresponding to 9~fb$^{-1}$ of
integrated luminosity.


\begin{acknowledgments}

We thank T.~Riemann for useful discussions
and help on \textsc{zfitter}.
We thank D.~Wackeroth for useful discussions
and help on the \textsc{zgrad2} calculation.
\input{cdf_ack_021111.itex}

\end{acknowledgments}


