\documentclass{article}

\title{Contributions to measured Asymmetry}
\author{R.J. Feuerbach \\ {\it College of William and Mary, Williamsburg, VA}}
\begin{document}

\maketitle

The asymmetry measured during the experiment ($A$) is a combination of the
desired physics asymmetry ($A_\mathit{phys}$), as well as instrumental
effects.  The observed asymmetry is assembled directly through the yields of
events ($N$) with helicity $+$ or $-$.
\begin{eqnarray*}
A &= &\frac{N_+ - N_-}{N_+ + N_-} \\
N_\pm & = & \mathcal{L}_\pm \sigma_\pm \eta_\pm d\Omega \\
\mathcal{L}_\pm & = & Q_\pm T \\
\eta_\pm & = & D_\pm \epsilon \\
\sigma_\pm & = & \Sigma \pm P_B P_T \Delta \\
\Delta' & = & P_B P_T \Delta \\
\end{eqnarray*}
where $\mathcal{L_\pm}$ is the helicity-dependent integrated luminosity,
$Q_\pm$ is the accumulated charge per helicity state, $T$ is the target
density in nucleons per $\mathrm{cm^2}$, $\eta_\pm$ is the helicity-dependent
livetime constructed of the DAQ ($D_\pm$) and electronic ($\epsilon_\pm$)
livetime fractions, and $\sigma_\pm$ is the helicity dependent cross-section
that contains the actual asymmetry we wish to extract.  The beam and target
polarizations are given as $P_B$ and $P_T$, respectively.

\begin{eqnarray*}
A & = & \frac{\mathcal{L}_+ \sigma_+ \eta_+ - \mathcal{L}_- \sigma_- \eta_-}
{\mathcal{L}_+ \sigma_+ \eta_+ + \mathcal{L}_- \sigma_- \eta_-} \\
  & = & \frac{Q_+ \eta_+ (\Sigma + \Delta') - Q_- \eta_-
  (\Sigma - \Delta')}
{Q_+ \eta_+ (\Sigma + \Delta') + Q_- \eta_-
  (\Sigma - \Delta')} \\
 & = & \frac{\Sigma(Q_+ \eta_+ - Q_- \eta_-) 
  + \Delta'(Q_+ \eta_+ + Q_- \eta_-)}
{\Sigma(Q_+ \eta_+ + Q_- \eta_-) 
  + \Delta'(Q_+ \eta_+ - Q_- \eta_-)} \\
 & & \mathrm{for\; clarity,\; assign}\; f = 1 - \frac{Q_- \eta_-}{Q_+ \eta_+}  \\
A & = & \frac{\Sigma f + \Delta' ( 2 - f )}{\Sigma (2-f) + \Delta' f} \\
  & = & \frac{ f + \frac{\Delta'}{\Sigma} (2-f) }
 {(2-f) + \frac{\Delta'}{\Sigma} f} \\
  & \approx & \frac{\Delta'}{\Sigma} + f, \;  \mathrm{since}\; f\approx.01\; \mathrm{and}\; \frac{\Delta'}{\Sigma}<.1. \\
\end{eqnarray*}

So the false asymmetry contributions, encapulased in $f$, contribute directly
to the observed asymmetry and must be removed.  Rewriting, to first order then
in the corrections

\begin{eqnarray*}
f & = & 1 - \frac{Q_- \eta_-}{Q_+ \eta_+} \\
  & = & 1 - ( 1 + (\frac{Q_-}{Q_+} - 1) ) 
            ( 1 + (\frac{\eta_-}{\eta_+} - 1) ) \\
  & \approx & (1 - (\frac{Q_-}{Q_+}) + (1 - \frac{\eta_-}{\eta_+}) \\
\frac{\Delta'}{\Sigma} & = & A - f \\
& = & A + (\frac{Q_-}{Q_+}-1) + (\frac{\eta_-}{\eta_+}-1) \\
\end{eqnarray*}

\begin{table}[h]
\centering
\caption{Contributions to the observed asymmetry and their approximate
  uncertainties (when known).}
\begin{tabular}{|c r r|}
\hline
term  & value & uncertainty \\ \hline
$1 - Q_-/Q_+$ & $<0.0002$ & \\ \hline
$1 - D_-/D_+$ & $0.001$ & \\ \hline
$1 - \epsilon_-/\epsilon_+$ & $0.01$? & \\ \hline
$A$ & $0.04$ & $0.005$ \\ \hline
\end{tabular}
\end{table}
\end{document}