\subsection{Measurement of $A_b$}

The dependence of the $b$-tagging efficiency
upon the secondary vertex mass is taken from
the simulation, with the overall tagging efficiency
derived from the single- and double-tagging rates \cite{RB} observed in the data.
Tagging efficiencies for charm and $uds$ events are estimated using
the MC simulation, as is the
charm correct-signing probability $p_{c}$.  The value of
$A_c$ is set to its Standard Model value of 0.67, and the value
of $A_{bckg}$ is set to zero.  
After a small (+0.2\%) correction \cite{ZFITTER} for initial state
radiation and $Z$-$\gamma$ interference,
the value of $A_b$ extracted from the fit
is $A_b=0.897\pm0.027\,\,(stat)$.
This result is found to be insensitive to the value of the 
$b$-tag mass cut. 


  We have investigated a number of systematic effects which can change the
measured value of $A_b$; these are summarized in Table~\ref{SYSERR}.
The uncertainty in $p_b$ due to the
statistical uncertainties in the data self-calibration technique
corresponds to a $+3.4/-3.2\%$ uncertainty in $A_b$\footnote{The error in the self-calibrated $p_b$ is symmetric, but the 
corresponding error in the event weight (the ``analyzing power'', 
$= 2p_b - 1$) is asymmetric.}.
We have estimated the effects of possible biases in the
self-calibration technique by comparing the MC
true value of $p_b(M_{vtx})$ with the 
self-calibrated value of the same quantity determined using
the same MC as a trial dataset. We observe no bias, and
assign a  $1.0$\% systematic uncertainty
in $A_b$ due to our limited MC statistics. 
The uncertainty in the MC modelling of the 
$M_{vtx}$ dependence of $p_b$ is included in the 
tracking efficiency corrections (see below).
In addition, while the mean value of the self-calibration
parameter $p_b$ is constrained by the data, it 
has a $\cos\theta$ dependence due to the fall-off of the
tracking efficiency at high $\cos\theta$ which must be
estimated using the simulation, leading to a $0.6$\% uncertainty
in $A_b$. We also rely on the MC to correctly model the
vertex charge distribution of the light-flavor background
(dominantly $Z^0\rightarrow c \overline{c}$\ ) which is subtracted from the raw 
counts $N_{++}$ and $N_{+-}$; we conservatively take
a $\pm 50\%$ relative uncertainty on this subtraction,
which results in a $0.1$\% uncertainty
in $A_b$. 


The extracted value of $A_b$ is sensitive to our estimate
of the  
$Z^0 \rightarrow c {\overline c}$ background, which tends to reduce
the observed asymmetry due to the positive charge of the underlying
$c$ quark. 
The uncertainty in the purity estimate of $\Pi_b =  (98.6 \pm 0.6)\%$
is dominated by the uncertainties in the charm tagging efficiency
($\epsilon_c = 0.009 \pm 0.001$) 
and leads to
a $0.9\%$ uncertainty in $A_b$. Details of the estimate of the
light and charmed quark efficiencies can be found in Ref.~\cite{RB}.

In addition, agreement between the data and MC simulation charged 
track multiplicity distributions is obtained only after the inclusion 
of additional ad-hoc tracking inefficiency. This random inefficiency was
parametrized as a function of total track momentum, and averages 0.5 
charged tracks per event.
However, without this correction applied, the MC correct-sign probability
is $<p_b(M_{vtx}>2.0)> = (74.4 \pm 0.3)\%$, in better agreement with the data than
the lower value obtained with the correction turned on.
Moreover, the agreement in the data/MC $M_{vtx}$ spectra is somewhat
better without the ad-hoc correction applied. 
Completely removing this
additional correction from the MC results in a $1.0\%$ change
in $A_b$, which is also included as a systematic error.
The uncertainty in the beam polarization $P_e$ is taken
from a preliminary estimate performed for the SLD $A_{LR}$
analysis\cite{PETER}. 

Combining all systematic uncertainties in quadrature yields
a total relative systematic uncertainty of $+4.0/-3.8\%$.


In conclusion, we have exploited the highly polarized SLC electron
beam to perform a direct measurement of
\begin{equation}
  A_b=0.897\pm 0.027(\mbox{stat})^{+0.036}_{-0.034}(\mbox{syst}).
\end{equation}

We thank the personnel of the SLAC accelerator department and the
technical staffs of our collaborating institutions for their
outstanding efforts on our behalf.
This work was supported by the U.S. Department of Energy 
and National Science Foundation,
the UK Particle Physics and Astronomy Research Council,
the Istituto Nazionale di Fisica Nucleare of Italy,
the Japan-US Cooperative Research Project on High Energy Physics,
and the Korea Science and Engineering Foundation.

