
\section{Corrections applied to the F$_{\pi}$-1 and F$_{\pi}$-2 data sets}

As the experimental technique and analysis of the $^1$H data acquired in the 
F$_{\pi}$-1 and F$_{\pi}$-2 experiments have already been presented in detail
in Ref. \cite{Blok08}, this report will concentrate on those areas where the
$^2$H experimental data differ from the $^1$H data.

\subsection{Event Reconstruction and Tracking}

In order to bring the F$_{\pi}$-1 and F$_{\pi}$-2 deuterium experimental data
sets to the same level of reconstruction and analysis, the data from the first
F$_{\pi}$-1 experiment (two channels: $\pi^{-}$ and $\pi^{+}$ electroproduction
from deuterium) were replayed using the new (2003) analysis engine.  It was a
considerable effort to adapt the new engine to accept the format of the older
F$_{\pi}$-1 data, which previously had been replayed with the 1999 engine.  The
reason for this effort is that the 2003 engine incorporates a tracking
algorithm with improved multitrack event capabilities, implicitly extending the
accuracy of the event reconstruction.  The details about the new tracking
algorithm implemented in the ENGINE are presented in Ref.~\cite{Vladasthesis}.
Charged particle trajectories are measured by two drift chambers, each with six
planes.  In the old (1999) Analyzer, the HMS tracking selection criteria was
that 4
out 6 planes should have fired in each drift chamber in order to select a
``good track'', while 5/6 was used for SOS tracking.  The new (2003) Analyzer
uses the 5/6 method (5 out of 6 planes should have fired in each drift chamber)
for both spectrometers, which is much better suited for high rate data (in this
case the $\pi^{-}$ channel data).  The same 5/6 method was used for both the
F$_{\pi}$-1 and F$_{\pi}$-2 data sets.

Using matrix elements to describe the transport through the spectrometer, the
vertex at the target is then reconstructed from the focal plane coordinates
(position; $x_{fp}$ and $y_{fp}$ and direction/slope: $x_{fp}^{\prime}$ and
$y_{fp}^{\prime}$ of the trajectory at the focal plane).  The new tracking
algorithm implemented in the Engine has several requirements:
\begin{itemize}
\item {If the program reconstructed only one track, then that track was used.}
\item {If the calorimeter reconstructed two or more tracks, then the track that
hit the cluster in the calorimeter was used.  In case two or more tracks hit
the cluster in the calorimeter (or neither of them), then in order to select a
correct track the following criteria was used:}
\item {If the program reconstructs two or more tracks, then the nearest track
to the fired bar in 2Y was chosen.}
\item {If two or more reconstructed tracks hit the same fired bar in 2Y, then
the track nearest to the fired bar in 2X was chosen.}
\item {If two or more reconstructed tracks hit the same fired bar in 2Y and 2X,
then the track with the best $\chi^2$ was chosen (this was the only criteria in
the old algorithm).}
\end{itemize}
Use of the new tracking algorithm required several adjustments in the
calculation of the tracking efficiency.  The tracking efficiency analysis
incorporates a cut on good PMT ADC which does not allow for multiple
hits per scintillator plane.  In the case where there are multiple
tracks in the same scintillator plane, this cut places a bias on the
event sample used to calculate the HMS tracking efficiency.  Since
2-track events have a lower efficiency than 1-track events, the
resulting bias causes the tracking efficiency to be over estimated.  

In the case of the F$_{\pi}$-2 data set, the requirement of rejecting
additional PMT hits outside the fiducial region due mostly to multiple track
events was removed.  The effect of this action was checked using carbon
luminosity runs for F$_{\pi}$-2 and the overall results indicated that the high
rates up to 1 MHz were understood within $1\%$, which is a reasonable
uncertainty.  The overall procedure of extracting the final tracking
efficiencies and an additional carbon target check are presented in 
Section \ref{sec:trackeff}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{SOS/HMS Momentum-Angle Correlation Corrections}

As discussed in Ref.~\cite{Volmerthesis}, the iron in the SOS magnets saturates
above a certain magnetic field.  Below this threshold field strength the shape
of the magnetic field does not change.  Beyond this point, the magnetic field
shape changes with momentum, and the effective field length decreases.
Although the actual field changes are not measured, the effect of these changes
upon the particle trajectories are easily observed.

There are two effects, which are corrected for in the reconstruction of the SOS
target variables.  The first effect is that the central momentum $p_0=\Gamma B$
receives an additional $B$ dependence as $\Gamma$ changes with $B$, due to a
decrease of the SOS effective field length.  The central momenta are
accordingly corrected, $p_o^{corr}=p_0(1+\epsilon_p)$.  The second effect is
that the reconstructed $\delta=(p-p_0)/p_0)$ depends on $x'_fp$ (when going
away from the 1.65 GeV setting where the F$_{\pi}$-1 matrix elements were
determined).  This effect arises from changes in the shape of the fringe fields
when saturation sets in, and is corrected for on an event-by-event basis.  Both
of these corrections are explained in further detail in
Ref.~\cite{Volmerthesis}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0,width=0.9\textwidth]{eps/SOS_corr1.ps}
\vskip 0.05cm
\caption[Invariant proton mass M$_{p}$-$W$ peak resolution as a function of
$x_{fp}^{\prime}$.]{\label{fig:SOS_corr1} \small Proton invariant mass $W$ peak
resolution as a function of $x_{fp}^{\prime}$ for three SOS central momentum
settings:1.234 GeV/c (upper panels) and 1.396 GeV/c (middle panels), and 1.564
GeV/c (lower panels) and for SOS Optics Matrix elements extracted from the
F$_{\pi}$-1 (left panels) and from the F$_{\pi}$-2 (right panels) elastic
data.}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\includegraphics[angle=0,width=0.9\textwidth]{eps/SOS_corr2.ps}
\vskip 0.05cm
\caption[Invariant proton mass M$_{p}$-$W$ peak resolution as a function of
$x_{fp}^{\prime}$.]{\label{fig:SOS_corr2} \small Proton invariant mass
M$_{p}$-$W$ peak resolution as a function of $x_{fp}^{\prime}$ for two SOS
central momentum settings:1.67 GeV/c (upper panels) and 1.715 GeV/c (lower
panels), and for SOS Optics Matrix elements extracted from the F$_{\pi}$-1
(left panels) and from the F$_{\pi}$-2 (right panels) elastic data.}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\vskip -1.5cm
\hskip -2cm \includegraphics[angle=0,width=0.5\textwidth]{eps/w_q3_hms.eps}
\caption[HMS Q3 Correction.]{\label{fig:HMSQ3} \small HMS Q3 correction
coefficient (in percent) plotted as a function of HMS elastic electron momentum
$p_{HMS}$.}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

Due to these effects, there is a residual dependence of the reconstructed
SOS momentum, $\delta_{SOS}$, upon the reconstructed particle trajectory,
$x_{tar}^{\prime}$.  To account for this, the reconstructed SOS momentum was
corrected by adding a central momentum dependent quadratic term,
$\epsilon_{\delta}$,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{equation}
\delta_{SOS}^{corr} = \delta_{SOS} + \epsilon_{\delta},
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
where the correction is calculated as
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{equation}
\epsilon_{\delta} = (x_{fp}^{\prime})^2\left[ \sum_{i=0}^3 a_{i}
  p_{0}^{i}\right],
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
with $x_{fp}^{\prime}$ in units of radians, and with $a_{0}=0.47$,
$a_{1}=-1.37$, $a_{2}=1.81$ and $a_{3}=-0.76$ for $p_{0}$ above 1 GeV/c. For
$p_{0}<1$ GeV/c, the correction used was
$\epsilon_{\delta}=0.15(x_{fp}^{\prime})^2$, with $\epsilon_{\delta}=-0.004$.

For the deuterium data sets studied here, elastic runs were used to check the
validity of these corrections for several momentum ranges.  The SOS
optics matrix elements used were those extracted
during the 1998 (F$_{\pi}$-1) and 2003 (F$_{\pi}$-2) experimental runs.  As can
be seen from Fig.~\ref{fig:SOS_corr1}, for lower SOS central momenta
($p_{SOS}<1.7$ GeV/c) the optics matrix elements extracted from F$_{\pi}$-1
(left panels) elastic data seem to work better than those extracted from the
F$_{\pi}$-2 (right panels) elastic data.  For higher central SOS momenta
($>1.7$ GeV/c), the optics matrix elements extracted from F$_{\pi}$-2 (right
panels) work better (Fig.~\ref{fig:SOS_corr2}).  The corrections appropriate to
each SOS central momentum range were thus applied to each F$_{\pi}$-1 data set
($\pi^{-}$ and $\pi^{+}$).  The F$_{\pi}$-2 matrix elements determined in Ref.
\cite{chuncheng} and discussed in Ref. \cite{Tanjathesis} were used in the
analysis of the F$_{\pi}$-2 data, so no further corrections were needed in this
case.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[hb]
\begin{center}
\vskip -1.0cm
\hskip -1.0cm \includegraphics[angle=0,width=0.6\textwidth]{eps/HMS_5.0.eps}
\vskip -1.0cm
\caption[Invariant mass peak plotted as a function of
$x_{fp}^{\prime}$.]{\label{fig:Wvsxpfp} \small Proton invariant mass $W$ peak
resolution plotted as a function of $x_{fp}^{\prime}$ for an HMS delta
correction of $c_{\delta}=5.0\%$ and for a central HMS momentum of 1.559
GeV/c.}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht]
\begin{center}
\vskip -0.5cm
\hskip -2cm \includegraphics[angle=0,width=0.5\textwidth]{eps/resfit.eps}
\vskip 0.5cm
\caption[Invariant mass peak resolution plotted as a function of delta
correction $c_{\delta}$.]{\label{fig:HMSdeltacorr} \small Proton invariant mass
$W$ peak resolution plotted as a function of HMS delta correction $c_{\delta}$
for a central HMS momentum of $1.559$ GeV/c.}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

A separate correction for the HMS momentum dependence on the focal plane
trajectory, $x_{fp}^{\prime}$, arises because the Q3 current offset causes a
mis-setting of the quadrupole current relative to the tune value.  The
resulting HMS correlations between the reconstructed HMS momentum,
$\delta_{HMS}$, and the reconstructed focal plane trajectory,
$x_{fp}^{\prime}$, were addressed in the same manner as for the SOS.  While for
the SOS a quadratic form dependent on central momentum was added to
$\delta_{SOS}$, for the HMS a simple linear function $c_{\delta}$ (in \% units)
of momentum sufficed,
\begin{equation}
c_{\delta}=a + b\delta_{HMS},
\end{equation}
where $a$=-2.795 and $b$=-1.280 (Fig. \ref{fig:HMSQ3}).

The final reconstructed HMS momentum, $\delta_{HMS}^{corr}$, was obtained by
adding a term dependent on both $\delta_{HMS}$ and $x_{fp}^{\prime}$,
\begin{equation}
\delta_{HMS}^{corr}=\delta_{HMS} + c_{\delta}x_{fp}^{\prime}.
\end{equation}
During the F$_{\pi}$-1 experiment, no elastic runs with an electron detected in
the HMS were taken, with the exception of few runs taken at the end of the run
with $p_{HMS}=3.67$ GeV/c.  For this momentum, the correlation between the
invariant mass $W$ and the focal plane trajectory slope $x_{fp}^{\prime}$,
using two sets of HMS optic matrix elements (1999 and 2003 versions) were used
to set the value of correction factor $c_{\delta}$ for the F$_{\pi}$-1 data, as
shown in Figs. \ref{fig:Wvsxpfp},\ref{fig:HMSdeltacorr}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\clearpage
\subsection{Tracking Efficiencies and Carbon Luminosity Scans
\label{sec:trackeff}
}

\subsubsection{Tracking in F$_{\pi}$-1 and F$_{\pi}$-2}

The tracking efficiency is defined~\cite{Tanjathesis} as the ratio of events
that should pass through the drift chambers, to the number of events for which
an actual track is found.  The fraction of events expected to pass through the
drift chambers is extracted from a fiducial area where the scintillator
paddles presented the highest hit efficiency.

The tracking efficiency depends on both the drift chamber hit efficiency and
the tracking algorithm used in finding the track. From previous experiments, we
know that the efficiency for tracking electrons through the HMS falls linearly
with rates, due mainly to the presence of multiple tracks in the drift chambers
at high rates. In order to accurately calculate the tracking efficiency, we
applied tight particle identification (PID) requirements to select a pure data
sample. These requirements are stricter than those used in the regular
analysis. In the HMS, the particle identification requirements used to select
pions in the tracking efficiency calculation consisted of cuts on the gas
Cerenkov and the calorimeter for F$_{\pi}$-1 data, while for F$_{\pi}$-2 an
additional cut on the aerogel Cerenkov was applied.

As already mentioned, one of the objectives of this effort was to align both of
the F$_{\pi}$-1 and F$_{\pi}$-2 experimental data sets to the same level of
reconstruction and analysis.  For the first pass reconstruction of the
F$_{\pi}$-1 data set, a tracking algorithm was used that did not address
multiple track events.  Since the $\pi^-$ data are taken at high rates under
the condition of potentially high electron contamination, this deficiency in
the tracking algorithm could affect the accurate determination of the
$\pi^-/\pi^+$ ratios.  Thus, it was decided to reanalyze the F$_{\pi}$-1 data
using the new tracking algorithm~\cite{Vladasthesis} used for the F$_{\pi}$-2
data set reconstruction. The newly (2003) redesigned tracking algorithm does a
significantly better job in selecting the best one of several tracks (multiple
tracks) per event than the older version.  After adapting the analysis package
to accept the format of the older F$_{\pi}$-1 data, the various analysis
parameters characteristic to F$_{\pi}$-1 were used in the redesigned
reconstruction software, as appropriate.

In the initial tracking efficiency analysis (used in F$_{\pi}$-1), the routine
Analyzer/HTRACKING/h\_track\_tests.f incorporated cuts to
exclude multiple good PMT ADC signals within the fiducial region of any
hodoscope plane.

The specific lines implementing these cuts are:\\
{\tt{
do i=1,hxloscin(1)-1\\
   if (hscinhit(1,i).EQ.1) hhitsweet1x=-1\\
enddo\\
do i=hxhiscin(1)+1,hscin\_1x\_nr\\
   if (hscinhit(1,i).EQ.1) hhitsweet1x=-1\\
enddo\\
\ \\
do i=1,hxloscin(2)-1\\
\ \ \ \ \ \ if (hscinhit(3,i).EQ.1) hhitsweet2x=-1\\
\ \ \ enddo\\
\ \ \ do i=hxhiscin(2)+1,hscin\_2x\_nr\\
\ \ \ \ \ \ if (hscinhit(3,i).EQ.1) hhitsweet2x=-1\\
\ \ \ enddo\\
\ \\
\ \ \ do i=1,hyloscin(1)-1\\
\ \ \ \ \ \ if (hscinhit(2,i).EQ.1) hhitsweet1y=-1\\
\ \ \ enddo\\
\ \ \ do i=hyhiscin(1)+1,hscin\_1y\_nr\\
\ \ \ \ \ \ if (hscinhit(2,i).EQ.1) hhitsweet1y=-1\\
\ \ \ enddo\\
\ \\
\ \ \ do i=1,hyloscin(2)-1\\
\ \ \ \ \ \ if (hscinhit(4,i).EQ.1) hhitsweet2y=-1\\
\ \ \ enddo\\
\ \ \ do i=hyhiscin(2)+1,hscin\_2y\_nr\\
\ \ \ \ \ \ if (hscinhit(4,i).EQ.1) hhitsweet2y=-1\\
\ \ \ enddo\\
\ \\
\ \ \ if (abs(sweet1xscin-sweet2xscin).gt.3) hgoodscinhits=0\\
\ \ \ if (abs(sweet1yscin-sweet2yscin).gt.2) hgoodscinhits=0\\
}}

In the case where there were multiple tracks in the same scintillator plane,
these cuts place a bias on the event sample used to calculate the HMS tracking
efficiency.  Since 2-track events have lower efficiencies than 1-track events,
the resulting bias caused the tracking efficiencies to be over estimated.  We
will call this first method of selecting the data sample the ``old'' tracking
efficiency method.  For the F$_{\pi}$-2 hydrogen analysis, the above lines were
commented out, removing the requirement of rejecting additional PMT hits
outside the fiducial region and resulting in a ``dirty'' data sample which
included multiple track events.  We will call this later method the ``new''
tracking efficiency method.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\vskip -0.05cm
\includegraphics[angle=0,width=12.0cm,totalheight=8.0cm]
{eps/trk_eff_piminus_corr.ps}
\caption[Tracking efficiency extracted using ``new'' and ``old'' methods
plotted as a function of rates.]{\label{fig:Tracking_efficiency} \small
F$_{\pi}$-1 tracking efficiencies extracted using the ``new'' (green squares)
and ``old'' (blue dots) methods, plotted as a function of rates. Also shown are
the over estimated tracking efficiencies obtained during the first F$_{\pi}$-1
analysis in 1998 (so called ``replay 4'' in black triangles), and the tracking
efficiencies obtained with the ``old'' recipe plus a rate dependent correction
which accounts for the fact that the $^{12}$C should present no boiling effect
(red upside down triangles).}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

The ``new'' tracking efficiency method, with the bias against multiple track
events removed, was compared to the ``old'' tracking efficiency method for both
the F$_{\pi}$-1 and F$_{\pi}$-2 $\pi^{-}$ deuterium data sets.  When making
this comparison, it is important to note that the maximum rate encountered in
the F$_{\pi}$-1 experiment was 1.3 MHz, while in F$_{\pi}$-2 (benefiting from
the operational experience of the first experiment) the maximum rate was
limited to 600 kHz.  Thus, the F$_{\pi}$-1 deuterium data are much more
sensitive to high rate tracking issues than the F$_{\pi}$-2 data.

F$_{\pi}$-1 tracking efficiencies were extracted using the ``old'' and ``new''
tracking efficiency methods described above, and they are shown as a function
of rates in Fig.~\ref{fig:Tracking_efficiency}.  The blue dots indicate the
tracking efficiencies obtained using the prescription of the ``old'' tracking
efficiency method plus PID cuts, while the green squares indicate the
efficiencies obtained with the ``new'' recipe plus PID cuts.  Also shown (black
triangles) are the initial tracking efficiencies obtained during the last pass
of analysis back in 1998, so called ``replay 4''.  Clearly, they were over
estimated.  The red upside down triangles show the tracking efficiencies
obtained using the ``old'' method, plus a rate dependent correction which will
be explained later in this section.  For the case of the F$_{\pi}$-1 $\pi^{-}$
data set, it can be seen (Fig.~\ref{fig:Tracking_efficiency}) that the ``new''
tracking efficiency method compares nicely with the ``old'' method up to rates
$\approx 600$ kHz, but fails for high rates ($>600$ kHz).  The ``old'' method
appears to do a fairly good job in describing the expected fall of tracking
efficiencies with rates.

\subsubsection{F$_{\pi}$-2 Carbon Rate Studies
\label{sec:carbon_boiling}}

To obtain a better understanding of the HMS tracking efficiencies, a study of
yields from carbon target versus rate and current was performed.  In
F$_{\pi}$-2, special luminosity scans using different beam currents and
different targets (carbon, hydrogen and deuterium) were used to investigate
rate and current dependent effects.  Due to its very high boiling temperature
(4554 K), the normalized yields from the carbon target should present no
significant current or rate dependence if the various efficiencies are
calculated correctly.  Unfortunately, no $^{12}$C luminosity scans were taken
at different beam currents in the F$_{\pi}$-1 experiment, so any conclusions
obtained from the F$_{\pi}$-2 study will have to be applied also to the
F$_{\pi}$-1 data.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\vskip -0.05cm
\hskip -2.cm
\includegraphics[angle=0,width=8.0cm,totalheight=7.5cm]
{eps/lumi_c12_curr_beam.ps}
\includegraphics[angle=0,width=8.0cm,totalheight=7.5cm]
{eps/lumi_c12_rates_beam.ps}
\caption{\label{fig:Carbon_boiling_initial} Normalized
$C(e,e^{\prime})$ yields versus current and rate taken during the F$_{\pi}$-2
experiment.  Since carbon targets are insensitive to beam heating effects at
high current, the plot of yields versus current should be a horizontal line if
all rate dependent effects are correctly taken into account in the data
analysis.  It is seen that the ``new'' tracking efficiencies do a better job at
high rates than the ``old'' tracking efficiencies.}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

We compared the ``new'' tracking efficiencies to the scaler-based luminosity
scans, which are independent of the tracking algorithm.  Carbon runs were used
to establish if the ``new'' tracking efficiency would have been consistent with
the expected non-boiling effect of a solid target (related to beam current
intensity dependence of normalized electron yields).  Using tight PID cuts for
electrons detected in the HMS, we extracted the normalized yields as a function
of beam current, using the form:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
N_{yields}=\frac{N_{e}*PS1}{Q_{el}*El_{dt}*CPU_{dt}*Trk_{eff}},
\label{eqn:yields}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $N_{e}$ is the number of electrons obtained using PID cuts, $PS1$ is the
prescaling factor applied during the data acquisition, $Q_{el}$ is the
accumulated charge, $El_{dt}$ is the electron dead time, $CPU_{dt}$ is the
computer dead time, and $Trk_{eff}$ is the tracking efficiency obtained using
the scaler information and the prescription given by the ``old'' or ``new''
methods described earlier.  The beam current dependence of normalized yields
(including a 200 nA beam current uncertainty due to an offset in the beam
current calibration) is shown in Fig.~\ref{fig:Carbon_boiling_initial}.

From Fig.~\ref{fig:Tracking_efficiency}, one can see that the ``old'' method of
extracting tracking efficiencies does a better job for rates higher than 1 MHz,
as is the case of the F$_{\pi}$-1 data set, while for rates lower than 600 MHz
the ``new'' method of selecting the data sample for tracking efficiency is
better suited.  However, Fig.~\ref{fig:Carbon_boiling_initial} indicates that
the ``new'' method of extracting tracking efficiencies (which as well includes
the improved tracking algorithm) is better as far as the F$_{\pi}$-2 carbon
luminosity scans are concerned.  Yet, when the tracking efficiency corrections
calculated with the ``old'' method are inserted into Eqn. \ref{eqn:yields}, a
larger effect is observed than when the ``new'' method efficiencies are used.
Since the ``old'' method for computing the tracking efficiencies is needed for
the F$_{\pi}$-1 data set, it is necessary to implement a rate-dependent
correction to the ``old'' method tracking efficiencies.  This correction will
be applied to both the F$_{\pi}$-1 and F$_{\pi}$-2 $\pi^-$ data.

To carefully determine this correction to the ``old'' tracking efficiencies, a
second study was performed incorporating a greater number of carbon runs taken
in two different kinematic settings.  Since the probability of a second
particle traversing the HMS during the event resolving time is greater at high
rates, it is likely that tight electron PID cuts will cause the rate dependence
to be underestimated.  Therefore only the following cuts
\begin{equation}
abs(hsdelta)<8.5\ \ \ \ \ abs(hsytar)<5\ \ \ \ \ 
abs(hsxptar)<0.08\ \ \ \ \ abs(hsyptar)<0.05
\label{eqn:carbon_cuts}
\end{equation}
were applied in the second ntuples study, resulting in the data tabulated in
Table \ref{tab:carbon_lumi}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[h]
%\vskip -1.cm
\begin{center}
\begin{tabular}{|c|cccc|ccccc|c|}\hline
Run  &$Q_{tot}$&  hELLO  &   hS1X   &   BOT &  PS1  & htr1 & htr2 & cpult& hele & SING \\
\hline
\multicolumn{11}{|c|}{$E_e=4.210$ GeV, $\theta_{HMS}=12.00$, $P_{HMS}=-3.000$}\\ 
\hline
47012 &100621 &181614321 &279010493 &1106.5 & 697.9 &0.9775&0.8781&0.9671&0.9879& 168963\\
47017 & 44330 & 80184449 &123248274 & 608.5 & 299.3 &0.9794&0.8822&0.9425&0.9903& 170741\\
47018 & 31249 & 56737629 & 87321171 & 600.5 & 199.7 &0.9802&0.8844&0.9394&0.9931& 180750\\
47023 & 28692 & 52272441 & 80556749 & 923.5 & 199.8 &0.9827&0.8882&0.9607&0.9959& 171278\\
\hline
\multicolumn{11}{|c|}{$E_e=4.702$ GeV, $\theta_{HMS}=10.57$, $P_{HMS}=-4.050$}\\ 
\hline
47757 & 42974 &233316132 &303696380 & 575.5 & 496.4 &0.9687&0.9261&0.6415&0.9705& 223257\\
47758 & 52953 &301225113 &373411559 & 660.5 & 992.4 &0.9687&0.9242&0.7730&0.9693& 165871\\
47759 &124775 &675461151 &880596706 &1590.5 &1985.1 &0.9685&0.9243&0.8738&0.9700& 220334\\
47760 & 19126 &107791771 &138777445 & 710.5 & 249.3 &0.9788&0.9059&0.7171&0.9896& 233050\\
47763 & 56962 &308207310 &402136284 & 724.5 & 744.4 &0.9690&0.8896&0.7225&0.9699& 221788\\
47764 & 29473 &160260551 &209175826 & 376.5 & 248.2 &0.9689&0.9236&0.4685&0.9701& 221964\\
\hline
\end{tabular}
\caption{HMS carbon luminosity study data taken during F$_{\pi}$-2.  No PID
  cuts are applied.  BOT is the average of beam on times 1 \& 2 (Threshold
  cuts: 5 $\mu$A for BCM 1 and 1 $\mu$A for BCM 2).  htr1 is the ``old''
  tracking efficiency (pass 1), while htr2 is the ``new'' tracking efficiency
  (pass 2).  The final column is the number of HMS singles passing the cuts
  listed in Eqn. \ref{eqn:carbon_cuts}.
\label{tab:carbon_lumi}}
\end{center}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
%\vskip -0.05cm
\hskip -2.cm
\includegraphics[angle=89.9,width=10.0cm]{eps/gh_c_boiling_rate_s1x.ps}
\caption{\label{fig:Carbon_boiling_rate}
Normalized yields (no PID cut) from carbon target versus HMS singles rate.  The
``old'' tracking efficiency calculation, using a data sample where multiple
track events are rejected, is better suited for higher rates ($>600$
kHz). Tracking efficiencies for both of the F$_{\pi}$-1 and F$_{\pi}$-2 data
sets were computed using the ``old'' method and corrected with the linear rate
dependent function presented here.
}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}[h]
%\begin{center}
%%\vskip -0.05cm
%\hskip -2.cm
%\includegraphics[angle=0,width=10.0cm]{eps/trk_eff_corr_C12.ps}
%\caption{\label{fig:Carbon_boiling_rate_old} 
%THIS IS CORNEL'S OLD TRACKING EFFICIENCY CORRECTION FOR REFERENCE.
%The correction is determined from the ratio of the two lines.}
%\end{center}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{figure}[ht]
\begin{center}
%\vskip -0.05cm
\hskip -2.cm 
\includegraphics[angle=90,width=10.0cm]{eps/gh_c_boiling_corrected_s1x.ps}
\caption{\label{fig:Carbon_boiling_curr} 
Carbon yields, after the ``old'' tracking efficiencies are corrected
according to Eqn. \ref{eqn:tracking_corr} versus current, confirming
that the applied correction yields consistent results for both kinematic
settings.  The dashed lines were obtained by applying to the data a tracking
efficiency correction raised/lowered by $1\sigma$
(Fig. \ref{fig:Carbon_boiling_rate}), and refitting the current dependence.
}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

Normalized yields from the carbon target were computed using
Eqn. \ref{eqn:yields} and the ``old'' (htr1) tracking efficiencies in Table
\ref{tab:carbon_lumi}.  They are plotted versus rate in
Fig. \ref{fig:Carbon_boiling_rate}.  The error bars include statistical
uncertainties and an estimated systematic uncertainty of 0.3\% added in
quadrature, to take into account beam steering on the target and other
sensitive effects when no PID cut is applied.  Data from the two kinematic
settings were separately fit versus rate (blue and red curves in the figure)
and normalized to unity at zero rate.  The two data sets, thus normalized, were
then fit together, yielding the black curve.

The observed rate dependence suggests that the ``old'' tracking efficiencies
should be corrected in the following manner
\begin{equation}
htr\_corrected=htr\_old\times (1-S1Xrate(kHz)*6.76236\times 10^{-5}).
\label{eqn:tracking_corr}
\end{equation}
This correction nearly accounts for the difference between the ``old'' (htr1)
and ``new'' (htr2) efficiencies listed for the second kinematic setting in
Table \ref{tab:carbon_lumi}, but yields only a small correction to the ``old''
tracking efficiencies listed for the first kinematic setting.

Normalized carbon yields, after the ``old'' tracking efficiencies are corrected
according to Eqn.~\ref{eqn:tracking_corr}, are plotted versus current in
Fig.~\ref{fig:Carbon_boiling_curr}.  This confirms that the applied correction
eliminates spurious rate-dependent effects for both kinematic settings, even
though they are taken at significantly different combinations of rates and
currents.

To summarize, the tracking efficiencies plotted in
Fig.~\ref{fig:Tracking_efficiency} were corrected for the tracking efficiency
dependence upon rate shown in Fig. \ref{fig:Carbon_boiling_rate} and applied to
the normalized $\pi^{-}$ pion yields (for both F$_{\pi}$-1 and F$_{\pi}$-2).

The systematic uncertainties in the HMS tracking efficiencies were estimated as
follows.  In the F$_{\pi}$-2 hydrogen analysis, the tracking efficiencies were
assigned a 1.0\% scale and an 0.4\% $\epsilon$-uncorrelated systematic
uncertainty \cite{Tanjathesis,Blok08}, where the first is the scale uncertainty
primarily at low rate and the second is due to a variety of factors which may
affect high and low $\epsilon$ settings differently, such as the greater
scatter exhibited by tracking efficiencies at high rates.  The same effect is
seen in Fig. \ref{fig:Tracking_efficiency}, where the F$_{\pi}$-1 $\pi^-$
``old'' tracking efficiencies have an increased scatter of approximately $\pm
1.25\%$ at 1.3 MHz.  There is an additional uncertainty of 0.2\%/MHz due to the
tracking efficiency correction discussed above.  Since the maximum rate
variation for the F$_{\pi}$-1 $\pi^+$ settings, and for all F$_{\pi}$-2
$\pi^{\pm}$ settings, is about 400~kHz, this gives an $\epsilon$-uncorrelated
systematic uncertainty of 0.45\%.  However, the F$_{\pi}$-1 $\pi^-$ settings
have a maximum rate variation of 1.2 MHz, corresponding to an
$\epsilon$-uncorrelated systematic uncertainty of 1.3\%.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}[hb]
%\begin{center}
%\vskip -0.05cm
%\hskip -2.cm 
%\includegraphics[angle=0,width=10.0cm,totalheight=8.5cm]
%{eps/trk_eff_fpi1_fpi2_corr2.ps}
%\caption[Tracking efficiency as a function of rates for $\pi^{-}$ F$_{\pi}$-1
%and F$_{\pi}$-2 data sets with and without ``$^{12}$C boiling
%correction''.]{\label{fig:Fpi1_fpi2_track_corr} \small Tracking efficiencies
%as a function of rates for the $\pi^{-}$ F$_{\pi}$-1 and F$_{\pi}$-2 data sets
%with and without the ``$^{12}$C boiling correction''.}
%\end{center}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

\subsection{Cryotarget Boiling Correction
\label{sec:ld2_boiling}
}

When the electron beam hits the liquid target, it produces a large power
deposition per unit target area and as a result induces localized density
fluctuations referred to as ``target boiling''.  In order to reduce these
fluctuations, the beam was rastered over a small area rather than localizing it
at one point on the target.  The target boiling effect can be measured by
comparing the yields at fixed kinematics and varying beam current.  During both
experiments (F$_{\pi}$-1 and F$_{\pi}$-2), dedicated luminosity elastic runs
were taken for both liquid targets (Hydrogen and Deuterium).  The two
experiments used cryotargets with significantly different geometries, leading
to very different boiling effects.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[h]
%\vskip -1.cm
\begin{center}
\begin{tabular}{|c|cccc|ccccc|ccc|}\hline
Run  &$Q_{tot}$&hELLO   &  hS1X  & BOT & PS1 & htr1 & htr2 & cpult& hele & S1.5 & S2 & S99 \\
\hline
\multicolumn{13}{|c|}{$Q^2$=2.45, low $\epsilon$, $E_e=4.21$ GeV, $\theta_{HMS}=10.54$}\\
\hline
47190 &35632& 593645749&759588827&1032.5&1978.1&0.9658&0.8383&0.9047&0.9589&  2834&  2999& 150861\\
47191 &21863& 370444977&477096355&1115.5& 596.3&0.9734&0.8528&0.9170&0.9765&  6577&  6926& 327205\\
47192 &10301& 176604087&228444444&1089.5& 299.1&0.9790&0.8636&0.9347&0.9886&  6620&  6990& 323435\\
47241 &27317& 454374146&581377706& 809.5& 989.2&0.9656&0.8351&0.8788&0.9592&  4215&  4478& 225052\\
47242 &17066& 284001092&363506739& 530.5& 989.4&0.9656&0.8372&0.8800&0.9600&  2633&  2768& 141946\\
47243 &15933& 269682797&347254506& 828.5& 298.2&0.9732&0.8518&0.8671&0.9769&  8872&  9435& 451950\\
47244 &12196& 206390831&265833195& 632.5& 298.2&0.9734&0.8526&0.8671&0.9768&  6898&  7294& 346316\\
47245 & 6118& 104824677&135564178& 661.5&  99.7&0.9792&0.8633&0.8435&0.9890& 10561& 11098& 521854\\
47246 & 5437&  93131752&120435159& 578.5&  99.7&0.9790&0.8621&0.8403&0.9888&  9214&  9722& 460107\\
\hline
\multicolumn{13}{|c|}{$Q^2$=2.45, low $\epsilon$, $E_e=4.21$ GeV, $\theta_{HMS}=12.21$}\\
\hline
47266 &56228& 431953502&559403421&1134.5& 992.9&0.9718&0.8599&0.9006&0.9734&  4611&  4876& 234392\\
47267 &32848& 255082402&331707652&1114.5& 398.3&0.9766&0.8704&0.9163&0.9840&  7211&  7597& 357745\\
47296 &10513&  82800190&108090906&1132.5&  99.9&0.9824&0.8809&0.9234&0.9950&  9913& 10298& 475933\\
47297 &49966& 383049896&494990723& 921.5& 991.4&0.9697&0.8552&0.8743&0.9689&  3976&  4195& 201027\\
\hline
\multicolumn{13}{|c|}{$Q^2$=2.45, high $\epsilon$, $E_e=5.25$ GeV, $\theta_{HMS}=10.61$}\\
\hline
47480 & 9204&  65677147&106143067& 955.8& 149.8&0.9802&0.8615&0.8848&0.9945& 37161& 37467& 256839\\
47481 & 3327&  23751902& 38382157& 349.0&  99.8&0.9802&0.8598&0.8425&0.9945& 19019& 19187& 132571\\
47483 &38898& 272602333&435706188&806.5& 992.1&0.9697&0.8397&0.7671&0.9723& 18521& 18793& 133363\\
\hline
\multicolumn{13}{|c|}{$Q^2$=2.45, high $\epsilon$, $E_e=5.25$ GeV, $\theta_{HMS}=16.61$}\\ 
\hline
47510 &65033&  48592122&139812146& 912.5&  59.9&0.9784&0.8785&0.7819&0.9958& 54175& 54716& 442817\\
47511 &14012&  10469724& 30143430& 194.5&  59.9&0.9751&0.8746&0.7815&0.9958& 11706& 11820&  95212\\
\hline
\end{tabular}
\caption{Singles data taken with SCIN-3/4 HMS trigger during F$_{\pi}$-2
  $\pi^-$ running.  No PID cuts are applied.  BOT is the average of beam on
  times 1 \& 2 (Threshold cuts: 5 $\mu$A for BCM 1 and 1 $\mu$A for BCM 2).
  htr1 is the ``old'' tracking efficiency (pass1), while htr2 is the ``new''
  tracking efficiency (pass2).  The htr1 tracking efficiency is subsequently
  corrected by Eqn.~\ref{eqn:tracking_corr}.  The final column (S99) is the
  number of HMS singles events passing the cuts listed in
  Eqn. \ref{eqn:cc_cuts}, while S1.5 and S2 have in addition cuts of
  $hcer\_{npe}<1.5$ and $hcer\_{npe}<2$ applied.
\label{tab:cerblock}}
\end{center}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The F$_{\pi}$-2 cryotarget uses the ``tuna can'' geometry which is expected to
have boiling corrections $<1\%$ \cite{Tanjathesis}.  However, the correction
determined in Ref. \cite{Tanjathesis} was determined with the ``new'' tracking
efficiencies.  To ensure that the appropriate correction is applied when
``corrected old'' tracking efficiencies are used, this study was repeated,
using the $^2$H data taken with SCIN-3/4 trigger to also determine the
Cerenkov blocking correction.  Dedicated runs with a wide variety of
electron beam currents were taken for all $\pi^-$ kinematic settings except
$Q^2$=2.45, high $\epsilon$, $E_e=5.25$ GeV, $\theta_{HMS}=13.61$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\hskip -2cm
\includegraphics[angle=89.9,width=10cm]{eps/gh_ld2_boiling_corrected.ps}
\caption{\label{fig:Fpi2_LD2_boiling} 
Normalized HMS singles yields from F$_{\pi}$-2 $^2$H $\pi^-$ production data
taken with SCIN-3/4 trigger obtained to also determine the Cerenkov
blocking correction, plotted as a function of beam current.  
The dashed lines were obtained by applying to the data a tracking
efficiency correction raised/lowered by $1\sigma$
n(Fig. \ref{fig:Carbon_boiling_rate}), and refitting the current dependence.
}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\hskip -2cm
\includegraphics[angle=90,,width=10.0cm]{eps/gh_ld2_boiling_fpi1.ps}
\caption{\label{fig:Fpi1_LD2_boiling} 
Normalized HMS yields from F$_{\pi}$-1 $^2$H elastics data taken with electron
trigger plotted as a function of beam current.}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}[h]
%\begin{center}
%\hskip -2cm
%\includegraphics[angle=0,width=0.5\textwidth]{eps/boiling.eps}
%\caption{\label{fig:target_boiling} 
%THIS IS CORNEL'S OLD BOILING CORRECTION FOR REFERENCE, DETERMINED USING THE
%TRACKING EFFICIENCY CORRECTION IN FIG. \ref{fig:Carbon_boiling_rate_old}.  The
%new boiling correction is approximately 1/3 the previous value, but the sum of
%the tracking+boiling corrections is about the same as before.  The difference
%is how the corrections are divided versus rate and current.
%}
%\end{center}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Cuts of
\begin{equation}
abs(hsdelta)<8.0\ \ \ \ \ abs(hsxptar)<0.09\ \ \ \ \    abs(hsyptar)<0.055
\label{eqn:cc_cuts}
\end{equation}
were applied to ntuples from each of these runs, resulting in the data
tabulated in Table \ref{tab:cerblock}.  The ``old'' tracking efficiencies (htr1
in the table) were subsequently corrected via Eqn.~\ref{eqn:cc_cuts}.  The
normalized pion yield was calculated for each run according to 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
N_{yields}=\frac{N_{sing}*PS1+N_{coin}}{Q_{el}*El_{dt}*CPU_{dt}*Trk_{eff}}.
\label{eqn:yieldsc}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$N_{sing}$ is the column labeled S99 (no Cerenkov cut) in the table.
$N_{coin}$ is the corresponding number of coincidences obtained by the same
analysis, corrected for the SOS electronic live time.  [For brevity, the
coincidence data (C99) is not listed in the table.]

The normalized yields are plotted versus current in
Fig.~\ref{fig:Fpi2_LD2_boiling}.  The error bars include statistical
uncertainties and an estimated systematic uncertainty of 0.3\% added in
quadrature.  Data from the three kinematic settings were separately fit versus
current and normalized to unity at zero current.  The three data sets, thus
normalized, were then fit together, yielding the black curve.  The observed
current dependence suggests that no correction should be applied, which is
similar to the conclusion reached in Ref.~\cite{Tanjathesis}, where a
$<$1\%/100$\mu$A correction was determined with ``new'' tracking efficiencies.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[h]
%\vskip -1.cm
\begin{center}
\begin{tabular}{|c|ccc|cccc|c|}\hline
Run  &$Q_{tot}$&hS1X & BOT & PS1 & htr & cpult& hele & SING\\
\hline
\multicolumn{9}{|c|}{$E_e=4.044$ GeV, $\theta_{HMS}=12.50$, $P_{HMS}=-3.670$}\\
\hline
17476 &  4381&  39912098&  387.5&   50& 0.9816& 0.5432& 0.9983& 241548\\
17481 &  7346&  67005769&  680.1&   50& 0.9818& 0.5551& 0.9984& 414269\\
17487 &  6182&  54761791&  241.6&  200& 0.9763& 0.6877& 0.9964& 104189\\
17488 & 23038& 198857869&  912.3&  200& 0.9762& 0.6905& 0.9963& 387456\\
17491 & 69286& 594124739& 1351.4&  500& 0.9658& 0.7335& 0.9934& 477416\\
17511 & 40311& 345624687&  777.5&  300& 0.9662& 0.6196& 0.9937& 390612\\
17513 & 73205& 613217684& 1039.2&  600& 0.9569& 0.6792& 0.9913& 375792\\
\hline
\end{tabular}
\caption{HMS elastics luminosity study data taken during F$_{\pi}$-1.  No PID
  cuts are applied.  htr is the ``old'' tracking efficiency, which is
  subsequently corrected by Eqn.~\ref{eqn:tracking_corr}.  The final column is
  the number of HMS singles events passing the cuts listed in
  Eqn. \ref{eqn:carbon_cuts}.
\label{tab:fpi1_lumi}}
\end{center}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

F$_{\pi}$-1 used the so-called ``soda can'' targets, which require
significantly larger boiling corrections.  The magnitude of this correction is
sensitive to the rate-dependent correction applied to the HMS tracking
efficiencies.  The cuts listed in Eqn.~\ref{eqn:carbon_cuts} were applied to
ntuples from each of these runs, resulting in the data tabulated in Table
\ref{tab:fpi1_lumi}.  The ``old'' tracking efficiencies were corrected via
Eqn.~\ref{eqn:cc_cuts} and normalized yields calculated according to
Eqn.~\ref{eqn:yields}.  They are plotted versus current in
Fig. \ref{fig:Fpi1_LD2_boiling}.  The error bars include statistical
uncertainties and an estimated systematic uncertainty of 0.3\% added in
quadrature.  Data were fit versus current and normalized to unity at zero
current, yielding the black curve.  The observed current dependence yields a
$^2$H target density correction of $(7.11578\times 10^{-4})/\mu$A.  This is
somewhat larger than the $(1.15\pm 0.31\%)/100\mu$A correction determined for a
nearly identical $^2$H cell in Ref. ~\cite{Gaskellthesis}, but is consistent
with the $(6\pm 1\%)/100\mu$A correction determined for the F$_{\pi}$-1 $^1$H
cell in Ref. \cite{Volmerthesis}.

The fact that these target boiling corrections are consistent with previous
studies supports the validity of the tracking efficiency corrections determined
from the carbon luminosity studies without PID cut applied.

%\clearpage
\subsection{HMS Cerenkov Blocking Correction
\label{sec:cerblock}}

The HMS threshold gas Cerenkov detector is used to ensure good $e^{-}/\pi^{-}$
separation. In both F$_{\pi}$-1 and F$_{\pi}$-2, the HMS gas Cerenkov detector
was used as a veto in the trigger for $\pi^{-}$ runs to avoid high DAQ deadtime
due to large $e^{-}$ rates in the HMS.  The effective gas Cerenkov thresholds
used in the hardware veto are similar for both experiments.  Since the actual
veto threshold varies slightly from run to run due to PMT gain variations at
high rates, slightly more restrictive software thresholds of were applied in
the data analysis.  Additionally in the F$_{\pi}$-2 experiment, an Aerogel
Cerenkov detector was used for separating protons and $\pi^{+}$ for central
momenta above 3 GeV/c.

The loss of pions due to Cerenkov blocking is due to electrons passing through
the gas Cerenkov within $\approx 100$ ns after a pion has traversed the
detector, resulting in a mis-identification of the pion event as an electron
and being eliminated by the analysis cuts applied.  Thus, the correct
estimation of the gas Cerenkov blocking correction is essential in extracting
the $\pi^{-}$ cross section and is implicit in the final estimation of
$\pi^{-}/\pi^{+}$ ratios of separated response functions.  A variety of
studies were performed to determine the corrections that should be applied to
both experiments.

\subsubsection{F$_{\pi}$-2 TDC Studies
\label{sec:cer_tdc}}

Fig.~\ref{fig:Cer_tdc_tworuns} shows multi-hit TDC spectra of the Cerenkov signal
into the HMS trigger for two $\pi^-$ SCIN-3/4 runs.  The TDC is started by the
HMS pretrigger signal and can be stopped multiple times by the retimed
(i.e. delayed and discriminated) Cerenkov signal.  The main peak corresponds
corresponds to signals (primarily electrons) that result in the trigger,
starting the TDC.  Events not associated with the original trigger (other
electrons, or pions that are mis-identified as electrons due to Cerenkov
blocking) appear as additional events to the left and right of the main
electron peak.  The second peak to the right is due to a second electron
arriving within the timing window, but after the discriminator ``dead window''
of $\sim$40 ns (caused by the length of the discriminator pulse).  The
backgrounds to the left and right of the two peaks are due to earlier and later
electrons, while the tail extending to 410 ns is due to pedestal noise that
crosses the discriminator threshold.  The peak at channel 4096 is the
accumulation of very late TDC stops, while zeros correspond to electrons (or
pions) that did not give a stop.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\vskip -0.05cm
\hskip -2.cm
\includegraphics[angle=0,width=0.6\textwidth]{eps/tdc_tworuns.eps}
\caption{\label{fig:Cer_tdc_tworuns}
TDC spectra of the Cerenkov signal into the HMS trigger for two $\pi^-$ runs
where the HMS trigger was SCIN-3/4.  A series of cuts are placed on the number
of Cerenkov photoelectrons to select ``electron'' events.  The TDC scale is
100~ps/chan.
}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\vskip -0.05cm
\hskip -2.cm
\includegraphics[angle=0,width=0.6\textwidth]{eps/coin_comparo_47243.eps}
\caption{\label{fig:Cer_comparo_47243}
Cerenkov trigger TDC for $\pi^-$ run 47243, where the HMS trigger was SCIN-3/4.
The mean HMS rate for this run, as computed from the ELLO scaler is 325 kHz.
The top panel is the TDC spectrum for HMS singles events, while the bottom
panel is for coincidence events.  The TDC scale is 100~ps/chan.
}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\vskip -0.05cm
\hskip -2.cm
\includegraphics[angle=0,width=0.6\textwidth]{eps/trig_comparo_1set.eps}
\caption{\label{fig:trig_comparo_1set}
Cerenkov trigger TDC for two $\pi^-$ runs taken under identical beam current
and singles rate conditions, but different HMS trigger conditions.  The only
other differences between these runs are the singles prescale factor and the 
CPU livetime.  The bottom panel is their difference, where the data run has
been normalized by charge, cpult and PS1 to the SCIN-3/4 run and then
subtracted.  The TDC scale is 100~ps/chan.
}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[h]
%\vskip -1.cm
\begin{center}
\begin{tabular}{|c|cccc|ccccc|cccc|}\hline
Run  &$Q_{tot}$&hELLO   &  hS1X    & BOT & PS1 & htr1 & htr2 & cpult& hele & S1.5 & S2 & C1.5& C2 \\
\hline
\multicolumn{14}{|c|}{$Q^2$=2.45, low $\epsilon$, $E_e=4.21$ GeV, $\theta_{HMS}=10.54$}\\
\hline
47190 &356312& 593645749& 759588827&1032.5&1978.1&0.9311&0.3590&0.9047&0.9589&  2046& 2071& 3894& 3941\\
47193 &117729&1951678625&2497749072&3400.5&2994.4&0.9299&0.3585&0.9851&0.9941&  4899& 4963& 13964& 14194\\
\hline
\multicolumn{14}{|c|}{$Q^2$=2.45, low $\epsilon$, $E_e=4.21$ GeV, $\theta_{HMS}=12.21$}\\
\hline
47266 & 56228& 431953502& 559403421&1134.5& 992.9&0.9286&0.4008&0.9006&0.9734&  3314& 3335& 4227& 4264\\
47269 &169246&1301200129&1684928207&3587.5&2996.1&0.9324&0.4043&0.9874&0.9965&  3687& 3720&14577&14746\\
\hline
\multicolumn{14}{|c|}{$Q^2$=2.45, low $\epsilon$, $E_e=4.21$ GeV, $\theta_{HMS}=12.21$}\\
\hline
47297 & 49966& 383049896& 494990723& 921.5& 991.4&0.9326&0.4078&0.8743&0.9689&  3070& 3094& 4600& 4731\\
47298 &166228&1274314538&1646468086&2986.5&3993.8&0.9333&0.4032&0.9840&0.9958&  2776& 2799&16838&17017\\
\hline
\multicolumn{14}{|c|}{$Q^2$=2.45, high $\epsilon$, $E_e=5.25$ GeV, $\theta_{HMS}=10.61$}\\
\hline
47483 & 38898& 272602333& 435706188& 806.5& 992.1&0.9667&0.7294&0.7671&0.9723& 16075&16188&37391&37645\\
47477 &137020& 959986136&1533128246&2872.5&5785.6&0.9657&0.7271&0.9397&0.9927& 11619&11704&156783&158043\\
\hline
\multicolumn{14}{|c|}{$Q^2$=2.45, high $\epsilon$, $E_e=5.25$ GeV, $\theta_{HMS}=16.61$}\\ 
\hline
47510 & 65033&  48592122&139812146& 912.5&  59.9&0.9641&0.7608&0.7819&0.9958& 46298&46446&10983&11020\\
47514 &251361& 186978517&531086990&3510.5& 699.0&0.9642&0.7672&0.9766&0.9990& 18820&18879&52502&52663\\
\hline
\end{tabular}
\caption{Selected F$_{\pi}$-2  $\pi^-$ runs taken under identical beam and rate
  conditions, except that the HMS trigger conditions are different.  The first
  run of each pair is a Cerenkov veto Test Run with SCIN-3/4 HMS trigger, and the
  second is a regular $\pi^-$ Data Run.  The column quantities are as in Table 
  \ref{tab:cerblock}, except that C1.5, C2 indicate coincidence events with
  $hcer\_{npe}<1.5$ and $hcer\_{npe}<2$ cuts applied.
\label{tab:cer_yield_study}}
\end{center}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The features of the TDC spectra were investigated for a variety of SCIN-3/4
$\pi^-$ runs, for HMS singles rates up to $\sim$600 kHz.  As indicated by the
differences between the low rate and high rate runs plotted in
Fig.~\ref{fig:Cer_tdc_tworuns}, the main peak to pedestal ratio degrades with
increasing rate, and the second peak to first peak ratio gets larger.  The
growth of the second peak relative to the first peak was confirmed to have a
$\tau=37\pm 1$ns time constant, which is consistent with the expected TDC
deadtime as indicated by the spacing between the main and second peaks.

We also confirmed that the basic features of the TDC spectra are the same for
HMS singles and HMS+SOS coincidences.  Fig.~\ref{fig:Cer_comparo_47243} shows
this for a representative run.  While the structure of the second peak is a bit
different for reasons which we do not completely understand, the pedestal width
and peak locations are the same in both cases.  This lends confidence that any
Cerenkov blocking correction determined from HMS singles data (due to smaller
statistical uncertainties) should be applicable to the coincidence data as
well.

While the events to the left of the `notch' next to the main peak correspond to
early electrons passing through the detector before the electrons associated
with the trigger (already addressed in the coincidence time blocking
correction), events to the right of the main peak correspond to electrons
traversing the detector after the original trigger electrons, and the pedestal
immediately beneath the main peak is due to concurrent electrons within the TDC
deadtime.  The concurrent and later events contribute to Cerenkov blocking
which must be corrected for.  The time from the notch to the left of the main
peak (Chan 900) to the edge of the plateau (Chan 2050) indicates an effective
Cerenkov TDC gate width of $116.4\pm 6.3$ ns, where the uncertainty is
estimated from the slope of the right falling edge and the width of the notch.

Finally, five pairs of $\pi^-$ runs were identified where for each pair the
beam and rate conditions were identical but the HMS trigger conditions were
different (see Table~\ref{tab:cer_yield_study}).  The TDC spectra for one of
these pairs is shown in Fig.~\ref{fig:trig_comparo_1set}.  While one should
expect the second peak to be much smaller in the $\pi^-$ data run than in the
SCIN-3/4 run, we are a bit surprised that it is completely absent in the data
run.  Perhaps the lack of a TDC deadtime effect when the first electron is
vetoed causes later electron stops to be spread further to the right, instead
of piled up in a second peak.  The regular $\pi^-$ data run (middle panel) also
exhibits some structure to the left of the main peak but otherwise the features
are consistent with our expectations.  The ratio of zeros to main peak counts
is much greater in the $\pi^-$ data run, which is consistent with the zeros
being primarily pions and the main peak being electrons.  The normalized
difference of the two runs (bottom panel) indicates that the Cerenkov trigger
is about 90\% efficient at vetoing electrons.

%\clearpage
\subsubsection{F$_{\pi}$-2 Singles Yield Study}

The same comparison of runs with same rate but different trigger condition can
also be used to determine the effective threshold of the Cerenkov trigger veto.
The sample comparison shown in Fig.~\ref{fig:trig_comparo_adc_1set} indicates
an effective veto threshold of approximately 2.5 npe.  Because PMT gain
variations and pile-up effects will cause the actual veto threshold to vary
with rate, a slightly more restrictive software threshold of $hcer_{npe}<2.0$
was uniformly applied in the F$_{\pi}$-2 data analysis.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\vskip -0.05cm
\hskip -2.cm
\includegraphics[angle=0,width=0.6\textwidth]{eps/trig_comparo_adc_1set.eps}
\caption{\label{fig:trig_comparo_adc_1set}
Cerenkov photoelectron spectra for two $\pi^-$ runs taken under identical beam
current and singles rate conditions, but different HMS trigger conditions.  The
bottom panel is their charge normalized difference, where the normalization
factor has been multiplied by 0.4 to avoid negative counts at low npe.
}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\hskip -2.cm
\includegraphics[angle=90,width=10cm]{eps/gh_hms_cerblock_2p0.ps}
\caption{\label{fig:cerblock_2p0}
Normalized experimental yields as a function of HMS singles rate, for data with
SCIN-3/4 HMS trigger, with a $N_{p.e.}>2.0$ Cerenkov particle identification
cut applied.  The tracking efficiency and cryotarget boiling corrections
discussed in Secs.\ref{sec:carbon_boiling},\ref{sec:ld2_boiling} are applied.
The colored lines are fits of the form $Y=Ae^{-R\tau}$ for each kinematic
setting separately.  The black line is a combined fit to all of the data, as
explained in the text.}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

As already discussed in Sec.~\ref{sec:ld2_boiling}, dedicated runs were taken
with SCIN-3/4 trigger for a variety of electron beam currents for all
F$_{\pi}$-2 $\pi^-$ kinematic settings except $Q^2$=2.45, high $\epsilon$,
$E_e=5.25$ GeV, $\theta_{HMS}=13.61$.  The purpose of these runs was to
determine the Cerenkov blocking correction versus electron rate into the HMS.
The data with $hcer_{npe}<2.0$ software threshold listed in
Table~\ref{tab:cerblock} were used to compute normalized pion yields in a
manner similar to Eqn.~\ref{eqn:yieldsc}, except that the tracking efficiency
correction Eqn.~\ref{eqn:cc_cuts} was applied.  [Note that it might have been
better to use the ELCLEAN rate in this study, however, the object is to obtain
a correction which can be applied to both experiments and ELLO is the closest
equivalent scaler which is available in F$_{\pi}$-1.]

To determine whether the yields for the different sets of kinematics exhibit
similar rate dependencies, the three sets spanning a large range of rates were
fit to functions of the form $Y=A^{-R\tau}$, where both $A$ and $\tau$ were
free parameters.  Although $A$ introduces an additional free parameter to the
fit, this allows $\tau$ to be extracted for each set without being sensitive to
the normalization of the `low rate yield'.  The normalized yields versus rate,
and the fit $\tau$ values are shown in Fig.~\ref{fig:cerblock_2p0}.  The
obtained effective gate widths are reasonably independent of kinematic setting.
Because the runs at $Q^2$=2.45, high $\epsilon$, $E_e=5.25$ GeV,
$\theta_{HMS}=16.61$ are only at low rates, that $\tau$ value is not shown.

Two of the three $\tau_{eff}$ values have nearly same the values, while the
third is $1.5\sigma$ larger.  Additional studies were performed to see if the
larger $\tau$ value could due to a different $\pi/e$ ratio at this setting
compared to the others.  The $\pi/e$ ratio was found to depend upon incident
beam energy, and was nearly insensitive to spectrometer rate and angle, so it
could not be the cause of the discrepancy.  We conclude the three $\tau$ values
are consistent within normal statistical fluctuation, and that other factors
which could affect the conclusions are reasonably well under control.  Since
they are consistent, a combined fit of all the data was made.  In this case,
the three data sets spanning wide ranges of rates were independently normalized
to unity at zero rate, effectively removing the cross section variation with
kinematics.  The $E_e=5.25$ GeV, $\theta_{HMS}=16.61$ setting was then
normalized to the lowest rate run of the $E_e=5.25$ GeV, $\theta_{HMS}=10.61$
setting.  The combined fit to the renormalized data is shown by the black line
in Fig. \ref{fig:cerblock_2p0}.  The combined fit unavoidably depends on the
renormalization constants chosen for the four kinematic settings, but the
methodology employed does not allow too much arbitrary judgment.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
%\vskip -0.5cm
\hskip -2.cm
\includegraphics[angle=90,width=0.5\textwidth]{eps/tau_fit_2p0.ps}
\caption{\label{fig:tau_fit_2p0}
Effective Cerenkov gate widths for the $hcer_{npe}<2.0$ cut used in the
F$_{\pi}$-2 data analysis, color coded according
to the fits versus rate in Fig.~\ref{fig:cerblock_2p0}.
The black line is the error-weighted average, while the dashed lines indicate
the uncertainty.}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[hb]
\begin{center}
%\vskip -0.5cm
\includegraphics[angle=90,width=10.0cm]{eps/fpi2_ccblk_rate.ps}
\caption{\label{fig:fpi2_ccblock} 
F$_{\pi}$-2 HMS Cerenkov blocking correction $\delta_{CCblock}$ plotted as a
function of the HMS electron singles rate for the $Q^2=2.45$ GeV$^2$ setting.}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The Cerenkov blocking $\tau_{eff}$ value and its uncertainty was estimated from
the fits to the three kinematic settings shown in Fig.~\ref{fig:cerblock_2p0}.
Note that these fits depend sensitively upon the tracking efficiency and
cryotarget boiling corrections used.  Taking into account the respective
uncertainties in these fits yields a central value of $139\pm 19$ ns.

Since this $\tau$ value was determined with singles events, it needs to be
adjusted to yield the effective gate width for coincidence events.  As
discussed in Sec.~\ref{sec:cer_tdc}, the region to the left of the main peak in
Figs.\ref{fig:Cer_tdc_tworuns}-\ref{fig:trig_comparo_1set} corresponds to early
electrons passing through the detector before the electrons associated with the
trigger, already addressed in the coincidence time blocking correction.  The
correction factor is the ratio of the time from the notch to the left of
the main peak (Chan 900) to the right edge of the plateau (Chan 2050), divided
by the time of the full distribution (Chan 425 to Chan 2050),
\begin{equation}
\frac{116.4\pm 6.3\ {\rm ns}}{162.9\ {\rm ns}}=0.714\pm 0.039.
\label{eqn:tdc_factor}
\end{equation}
Multiplication by this factor gives $99.2\pm 19$ ns.

The two F$_{\pi}$-2 Cerenkov blocking studies (TDC gate width and singles
yield) are statistically consistent.  It is difficult to tell which one is more
definitive, so the error weighted average $\tau_{eff}=114.7\pm 6.0$ ns, is used
to compute the Cerenkov blocking correction
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\delta_{CCblock}=e^{-ELLOrate*\tau_{eff}}.
\label{eqn:ccblock}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
appropriate for the F$_{\pi}$-2 $\pi^-$ analysis.  This correction is shown as
a function of the HMS electron singles rate in Fig.~\ref{fig:fpi2_ccblock}.

%\clearpage
\subsubsection{Cerenkov Blocking Correction for F$_{\pi}$-1 Data}

Open trigger data at different electron rates were unfortunately not taken
during the F$_{\pi}$-1 experiment, so the Cerenkov blocking correction cannot
be directly determined for those data.  We therefore modify the Cerenkov
blocking correction determined from F$_{\pi}$-2 data for use in the F$_{\pi}$-1
analysis according to the following procedure.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\includegraphics[angle=0,width=0.5\textwidth]{eps/elastic_17462.eps}
\vskip 0.05cm
\caption{A summed photoelectron histogram from the HMS Cerenkov for a
representative F$_{\pi}$-1 carbon elastics run, $^{nat}C(e,e')$.  The
Cerenkov veto is not in the trigger; those events which would have most likely
been vetoed (as determined from a nonzero Cerenkov trigger TDC value) are
indicated in red.  The effective veto threshold appears to be at about 2
photoelectrons.
\label{fig:cc_elastic}}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Fig.~\ref{fig:cc_elastic} shows a HMS Cerenkov photoelectron histogram for a
carbon elastics run taken at the very beginning of F$_{\pi}$-1, immediately
before the first $\pi$ data run.  It indicates that the effective veto
threshold in the F$_{\pi}$-1 experiment is slightly lower than that used in 
F$_{\pi}$-2.  Therefore, a slightly more restrictive software threshold of
$hcer_{npe}<1.5$ was applied in the analysis of the F$_{\pi}$-1 data.  The
figure also indicates that the Cerenkov veto is about 80\% efficient for this
run.  Very few timing checks were performed after the initial experimental
setup, and our investigations indicate that the Cerenkov veto efficiency varied
with kinematic setting during the F$_{\pi}$-1 experiment.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
%\vskip -0.5cm
\hskip -2.cm
\includegraphics[angle=90,width=10cm]{eps/gh_hms_cerblock_1p5.ps}
\caption{\label{fig:cerblock_1p5}
The data are the same as in Fig.~\ref{fig:cerblock_2p0}, except that a
$N_{p.e.}>1.5$ Cerenkov particle identification appropriate to the F$_{\pi}$-1
analysis is applied.}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}[h]
%\begin{center}
%%\vskip -0.5cm
%\hskip -2.cm
%\includegraphics[angle=90,width=10cm]{eps/gh_hms_cerblock_trig_1p5.ps}
%\caption{\label{fig:cerblock_trig_1p5}
%The data are the same as in Fig.~\ref{fig:cerblock_trig_2p0}, except that a
%$N_{p.e.}>1.5$ Cerenkov particle identification appropriate to the F$_{\pi}$-1
%analysis is applied.}
%\end{center}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
%\vskip -0.5cm
\hskip -2.cm
\includegraphics[angle=90,width=0.5\textwidth]{eps/tau_fit_1p5.ps}
\caption{\label{fig:tau_fit_1p5}
Effective Cerenkov gate widths determined from F$_{\pi}$-2 data, but with
a $hcer_{npe}<1.5$ cut appropriate for the F$_{\pi}$-1 data analysis.}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\vskip -0.05cm
\hskip -2.cm
\includegraphics[angle=0,width=0.6\textwidth]{eps/tdc_fpi1_fpi2_comparo.eps}
\caption{\label{fig:Cerenkov TDC}
Cerenkov Trigger TDC histogram for the one F$_{\pi}$-1 $\pi^-$ run with
SCIN-3/4 trigger (blue), compared to a F$_{\pi}$-2 SCIN-3/4 run at similar rate
(red).  HMS singles events, subject to a $N_{p.e.}>2.0$ Cerenkov cut, are used
in both spectra.  The TDC scale is 100~ps/chan.}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\vskip -0.05cm
\includegraphics[angle=90,width=10.0cm,totalheight=8.0cm]
{eps/fpi1_ccblk_rate.ps}
\vskip -0.05cm
\caption{\label{fig:fpi1_ccblock} 
F$_{\pi}$-1 HMS Cerenkov blocking correction $\delta_{CCblock}$ plotted as a
function of the HMS electron singles rate.  Note that the maximum HMS singles
rate was much higher in F$_{\pi}$-1 than F$_{\pi}$-2.  The plotting symbols
indicate the $Q^2$ setting as follows: [red triangle] $Q^2=0.6$; [cyan diamond]
$Q^2=0.75$; [green circle] $Q^2=1.0$; [black square] $Q^2=1.6$ GeV$^2$.
}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We therefore reanalyzed the F$_{\pi}$-2 dedicated $\pi^-$ runs with SCIN-3/4
trigger, except that a $N_{p.e.}>1.5$ Cerenkov particle identification
appropriate to the F$_{\pi}$-1 analysis was applied, and obtained the results
shown in Figs.~\ref{fig:cerblock_1p5},\ref{fig:tau_fit_1p5}.  Taking into
account the respective uncertainty in these three fits yields a central value
of $\tau=162\pm 19$ ns, which is then corrected by Eqn.~\ref{eqn:tdc_factor} to
give $\tau=115.7\pm 20$ ns.

Finally, we used the TDC timing information from the only F$_{\pi}$-1 ``open
trigger'' SCIN-3/4 run taken just before the main data taking to estimate the
scaling with respect to the F$_{\pi}$-2 timing information.  As shown in
Fig. \ref{fig:Cerenkov TDC}, the TDC timing window used during F$_{\pi}$-1 is
wider than in F$_{\pi}$-2.  Comparing the time from the notch to the left of
the main to the edge of the plateau in the F$_{\pi}$-2 spectrum ($116.4\pm 
6.3$~ns) to the equivalent features in the the F$_{\pi}$-1 spectrum ($138.4\pm
6.3$~ns), indicates a scale factor of $1.19\pm 0.084$.  Application of this
scale factor to the $\tau$ value determined from the F$_{\pi}$-2 data yields
$\tau=137.7\pm 26$ ns.

The above singles yield $\tau$ value compares well with the one obtained
from the F$_{\pi}$-1 TDC gate width.  The error-weighted average of the two
gives an effective $\tau$ value for the F$_{\pi}$-1 analysis of
$\tau_{eff}=138.4\pm 6.1$ ns.  This value is used to compute the CC blocking
corrections used for the F$_{\pi}$-1 data normalization, shown as a function of
the electron rate in Fig.~\ref{fig:fpi1_ccblock}.  Evaluating the effect of the
$\pm 6.1$ ns uncertainty at 1.2 MHz gives an uncorrelated systematic
uncertainty of 0.7\%, while the 20 ns difference in $\tau$ values from the two
methods gives a scale uncertainty of 2.0\%.

%\clearpage
%\subsection{Cerenkov Cut Inefficiency}
%
%The $N_{p.e.}<2.0$ used in F$_{\pi}$-2 may exclude some pions.  To determine
%the appropriate correction factor, we investigated Cerenkov photoelectron
%spectra for $\pi^-$ runs with SCIN-3/4 trigger, subject to a $hsshtrk<0.75$
%calorimeter cut which excludes most electrons.  A sample histogram for a high
%rate $\pi^-$ run is given in Fig.~\ref{fig:Cerenkov_npe_47190}.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}[h]
%\begin{center}
%\vskip -0.05cm
%\hskip -2.cm
%\includegraphics[angle=0,width=0.6\textwidth]{eps/hcernpe_sshcut_47190.eps}
%\caption{\label{fig:Cerenkov_npe_47190}
%Cerenkov photoelectron histogram for the one F$_{\pi}$-1 $\pi^-$ run with
%SCIN-3/4 trigger.  In the top panel the red curve indicates all events passing
%the cuts listed in Eqn:\ref{eqn:cc_cuts}, while the blue curve has in addition
%a $hsshtrk<0.75$ cut applied.  A pion peak atop an electron background is
%seen.  The background is fit with a Gaussian+constant and subtracted to yield
%the data in the bottom panel.
%}
%\end{center}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}[h]
%\begin{center}
%%\vskip -0.5cm
%\hskip -2.cm
%\includegraphics[angle=90,width=10cm]{eps/pieff_rate.ps}
%\caption{\label{fig:pieff_rate}
%Yield ratios versus HMS singles rate for $\pi^-$ runs where the Cerenkov veto
%was included in the HMS trigger, divided by the corresponding run at same
%current and rate where the HMS trigger was SCIN-3/4.  A 1\% systematic
%uncertainty has been added in quadrature to all ratio error bars.}
%\end{center}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%After subtracting the background due to the remaining electrons, a clear
%$\pi^-$ peak below $N_{p.e.}<1$ is seen.  The pion distribution extends
%slightly past 2 {\it p.e.} due to knock-on electrons from reactions of the pion
%in the Cerenkov gas and interactions of secondary particles in the Cerenkov at
%high rate.  The ratios of counts surviving the $N_{p.e.}<2.0$ cut, to the
%numbers of counts surviving $N_{p.e.}<4.0$ and $N_{p.e.}<5.0$ cuts, are listed
%in the figure.  The rate dependencies of these ratios are shown in
%Fig.~\ref{fig:pieff_rate}.  As expected, there is a small rate dependence due
%Cerenkov signals moving from $N_{p.e.}<2.0$ to the tail at high rate and the
%rate dependence induced by the $hsshtrk<0.75$ calorimeter cut.  Since the rate
%dependence induced by the Cerenkov cut has already been taken into account by
%the correction factors discussed in Sec.\ref{sec:cerblock}, we extrapolate the
%ratios to zero rate.  This yields a Cerenkov inefficiency of approximately 1\%,
%which is applied to both the F$_{\pi}$-1 and F$_{\pi}$-2 data.

\subsection{F$_{\pi}$-2 Yield Ratio Study}

The five pairs of $\pi^-$ runs identified in Table~\ref{tab:cer_yield_study},
where for each pair the beam and rate conditions were identical but the HMS
trigger conditions different, can be used to probe additional rate dependent
effects.  The ratios of normalized yields
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
R=\frac{N_{\pi^-\_run\_with\_veto}}{N_{\pi^-\_run\_without\_veto}}
\label{eqn:yield_ratio}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
were thus formed, using the $N_{p.e.}<2.0$ data (columns S2 and C2) in
Table~\ref{tab:cer_yield_study}.  As indicated from the values in the table,
the uncertainty in the tracking efficiency and target boiling corrections
cancel in the formation of the ratio, depending only on the knowledge of the
cpu livetimes and prescale factors.  These ratios are plotted versus rate in
Fig.~\ref{fig:cerblock_trig_2p0}.  The effective gate width determined from
this study, determined from a fit which is constrained to unity at zero rate,
is $\tau=91\pm 22$ ns, while a fit without this constraint gives $\tau=37\pm
37$ ns.  This indicates a small residual rate dependence beyond those taken
into account in the analysis.  Since the prescale factors should be known, it
is likely the issue is due to the cpu livetime.  It is sometimes an issue that
heavily prescaled singles events encounter a different cpu livetime than
coincidence events.  It seems likely that the residual rate dependence of the
singles ratios shown in Fig.~\ref{fig:cerblock_trig_2p0} is due to this issue.
Without accurate information on the singles cpu livetime we cannot proceed
further.  We note that studies of this nature could be a valuable tool in our
planned 12 GeV experiments, and we should aim to acquire better data to allow a
more complete study in the future.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
%\vskip -0.5cm
\hskip -2.cm
\includegraphics[angle=90,width=9cm]{eps/gh_hms_cerblock_trig_2p0.ps}
\caption{\label{fig:cerblock_trig_2p0}
Yield ratios versus HMS singles rate for $\pi^-$ runs where the Cerenkov veto
was included in the HMS trigger, divided by the corresponding run at same
current and rate where the HMS trigger was SCIN-3/4.  A 1\% systematic
uncertainty has been added in quadrature to all ratio error bars.}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\clearpage
\subsection{HMS Beta Cut Correction}
In the main analysis package, ${\beta}$ is the velocity of the detected charged
particle determined from the time of flight between the two scintillator
hodoscopes in the detector stack.  The coincidence time spectrum at low
${\beta}$ (Fig. \ref{fig:beta_HMS_for_Fpi-1_data_set}) displays a ``tail'' due
to pions undergoing nuclear interactions in the scintillators, Cerenkov
detector material, and in the case of F$_{\pi}$-2 experiment the Aerogel
Cerenkov detector material which was absent during F$_{\pi}$-1.  These
scattered pions have a larger time of flight.  There are also events with
${\beta}=0$, which means that no hits in the relevant scintillators were found
when projecting the reconstructed track to the hodoscopes.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\vskip -0.2cm 
\begin{center}
\includegraphics[angle=0,width=0.6\textwidth]{eps/betacut-8_fpi1.eps}
\vskip -0.5cm
\caption[$\beta_{HMS}$ spectrum for both channels ($\pi^{-}$ and $\pi^{+}$) for
the F$_{\pi}$-1 data set.]{\label{fig:beta_HMS_for_Fpi-1_data_set} \small
$\beta_{HMS}$ spectrum for the $\pi^{-}$ channel (red thin line) and the
$\pi^{+}$ channel (black thick line), scaled with respect to each other, and
overlapped with the $\beta_{HMS}$ region defined by the $\beta>0.95$ cut (solid
area) for one of the F$_{\pi}$-1 kinematic settings.}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\vskip -2.0cm 
\begin{center}
\includegraphics[angle=0,width=0.6\textwidth]{eps/betacut-7_fpi1.eps}
\vskip -0.5cm
\caption[Coincidence timing plotted as a function of $\beta$ for the
F$_{\pi}$-1data set.]{\label{fig:Betacut_Fpi1} \small Coincidence timing
spectrum plotted versus $\beta$ for the F$_{\pi}$-1 data set.  The solid lines
indicate the cuts applied on both variables in order to select ``good
electrons'' as well as to compute the $\beta$ cut correction.}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[h]
%\vskip -1.cm
\begin{center}
\begin{tabular}{|c|ccr|c|}\hline
{ \small Kin}& { \small $Q^2$} & $\epsilon$ & $\theta_{pq} [deg]$ &
{ \small $\beta_{cut}$ corr. [\%]}\\ \hline \hline
{ \small 1}& 0.6  & 0.37 & 0.5  & { \small 5.46}\\ \hline
{ \small 2}& 0.6  & 0.37 & 4    & { \small 5.42}\\ \hline
{ \small 3}& 0.6  & 0.74 & 0    & { \small 4.81}\\ \hline
{ \small 4}& 0.75 & 0.43 & 0    & { \small 5.11}\\ \hline
{ \small 5}& 0.75 & 0.43 & 4    & { \small 4.69}\\ \hline
{ \small 6}& 1.0  & 0.33 & 0    & { \small 5.02}\\ \hline
{ \small 7}& 1.0  & 0.33 & 4    & { \small 4.57}\\ \hline
{ \small 8}& 1.0  & 0.65 & 0    & { \small 5.23}\\ \hline
{ \small 9}& 1.6  & 0.27 & 0    & { \small 5.05}\\ \hline
{ \small 10}& 1.6 & 0.27 & 4    & { \small 4.09}\\ \hline
{ \small 11}& 1.6 & 0.63 & -4   & { \small 5.04}\\ \hline
{ \small 12}& 1.6 & 0.63 & 0    & { \small 4.80}\\ \hline
{ \small 13}& 1.6 & 0.63 & 4    & { \small 4.27}\\ \hline \hline
{ \small Mean} &  &      &      & { \small $4.89\pm 0.41$}\\ \hline
\end{tabular}
\end{center}
\vskip -.5cm
\caption[F$_{\pi}$-1 data set, $\beta_{cut}$ corrections for the $\pi^{-}$
channel.]{\label{table:Betacut_corr_mn_Fpi1} \small $\beta_{cut}$ corrections
for the $\pi^{-}$ channel for each of the 13 F$_{\pi}$-1 kinematic settings.
Note that the correction was applied as a normalization constant for each
kinematic setting separately.}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

Due to the fact that the Aerogel Cerenkov detector was not yet in existence at
the time of the F$_{\pi}$-1 experiment, it was not possible to cleanly
eliminate the proton background which contaminated the $\pi^{+}$ data set.  In
order to reduce the proton contamination of $\pi^{+}$ data set, we applied a
beta cut ${\beta}>0.95$, as shown in
Fig.~\ref{fig:beta_HMS_for_Fpi-1_data_set}.  The so called ``beta cut
correction'' was defined as the ratio of the "blob'' (defined by the following
cuts: $-1<$ coincidence timing $<1$ and ${\beta}>0.95$), and ``blob'' plus
``tail'' as shown in the Fig.~\ref{fig:Betacut_Fpi1}.  This correction was
extracted from $\pi^{-}$ data and applied to both data sets ($\pi^{+}$ and
$\pi^{-}$) as a normalization constant to account for the pions lost due to
this cut.  The $\beta$ cut correction used in the final F$_{\pi}$-1
normalization was 4.89\% with an uncertainty of 0.41\% determined from the
standard deviation, as indicated in Table~\ref{table:Betacut_corr_mn_Fpi1}.

%\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\vskip -0.5cm 
\begin{center}
\includegraphics[angle=0,width=0.6\textwidth]{eps/betacut-8_fpi2.eps}
\vskip -0.5cm
\caption[$\beta_{HMS}$ spectrum for both channels ($\pi^{-}$ and $\pi^{+}$) for
F$_{\pi}$-2 data set.]{\label{fig: beta for Fpi-2 data set} \small
$\beta_{HMS}$ spectrum for $\pi^{-}$ (red thin line) channel and $\pi^{+}$
(black thick line) channel, scaled with respect to each other, and overlapped
with the $\beta_{HMS}$ region defined by the $\beta >0.95$ cut (solid area) for
one of the F$_{\pi}$-2 kinematic settings.}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\vskip -0.5cm 
\begin{center}
\includegraphics[angle=0,width=0.6\textwidth]{eps/betacut-7_fpi2.eps}
\vskip -0.5cm 
\caption[Coincidence timing plotted as a function of $\beta$ for the
F$_{\pi}$-2 data set.]{\label{fig:Betacut_Fpi2} \small Coincidence timing
plotted versus the $\beta$ of the detected charged particle for the
F$_{\pi}$-2 data set. The solid lines indicate the cuts applied on both
variables in order to select ``good electrons'' as well as to compute the
$\beta$ cut correction.}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[h]
%\vskip -1.cm
\begin{center}
\begin{tabular}{|c|ccr|c|}\hline
{ \small Kin}& { \small $Q^2$} & $\epsilon$ & $\theta_{pq}$ &
{ \small $\beta_{cut}$ corr. [\%]}\\ \hline \hline
\multicolumn{5}{|c|}{$\pi^{+}$ Settings}\\ \hline\hline
{ \small 1}& 2.45 & 0.27 & 1.35 & { \small 2.44}\\ \hline
{ \small 2}& 2.45 & 0.27 & 3    & { \small 2.59}\\ \hline
{ \small 3}& 2.45 & 0.54 & 0    & { \small 2.30}\\ \hline
{ \small 4}& 2.45 & 0.54 & 3    & { \small 2.30}\\ \hline
{ \small 5}& 2.45 & 0.54 & -3   & { \small 2.45}\\ \hline\hline
{ \small Mean} &  &      &      & { \small $2.42\pm 0.122$}\\ \hline\hline
\multicolumn{5}{|c|}{$\pi^{-}$ Settings}\\ \hline\hline
{ \small 1}& 2.45 & 0.27 & 1.35 & { \small 2.61}\\ \hline
{ \small 2}& 2.45 & 0.27 & 3    & { \small 2.60}\\ \hline
{ \small 3}& 2.45 & 0.54 & 0    & { \small 2.71}\\ \hline
{ \small 4}& 2.45 & 0.54 & 3    & { \small 2.33}\\ \hline
{ \small 5}& 2.45 & 0.54 & -3   & { \small 2.31}\\ \hline\hline
{ \small Mean} &  &      &      & { \small $2.51\pm 0.179$}\\ \hline
\end{tabular}
\end{center}
\vskip -.5cm
\caption[$\beta_{cut}$ corrections for the F$_{\pi}$-2 kinematic settings.]
{\label{table:Betacut_correction_Fpi2} \small $\beta_{cut}$ corrections for
each of the F$_{\pi}$-2 kinematic settings.}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

For the F$_{\pi}$-2 data set, the same procedure was used, but adjusted for the
presence of the Aerogel Cerenkov detector.  This permitted the separation
of protons from pions, leading to a much cleaner pion sample (Fig. 
\ref{fig: beta for Fpi-2 data set}).  For each $\pi^{+}$ and $\pi^{-}$
kinematic setting, ``beta cut corrections'' were extracted in the same fashion
as explained earlier (see Fig. \ref{fig:Betacut_Fpi2}).  The corrections are
summarized in Table~\ref{table:Betacut_correction_Fpi2}.  The $\beta$ cut
corrections used in the final F$_{\pi}$-2 normalization were 2.42\% with an
uncertainty of 0.12\% in the $\pi^+$ case, and 2.51\% with an uncertainty of
0.18\% in the $\pi^-$ case.

%\clearpage
\vskip 1.cm
\subsection{Pion Absorption in the HMS} 
A fraction of the produced pions are lost as a result of nuclear interactions
in the materials that they traverse before reaching the HMS detectors.  For the
pion momenta used in F$_{\pi}$-1,2, the dominant pion reaction types are the
inelastic scattering and charge exchange processes, resulting in a deflection
and loss of momentum of the pion, and the pion absorption process, resulting
primarily in the emission of energetic protons and neutrons.  The proportion of
pions lost due to these processes must be correctly accounted for.  The
situation is complicated by the fact that pion absorptions in the
scintillators, aerogel, and Cerenkov detector material leading to the emission
of energetic protons are already included in the $\beta$ efficiency correction.
The pion absorption correction is intended to account for the HMS triggers
which are lost due to pion interactions in materials upstream of the drift
chambers, or interactions in the detector stack such as large angle deflection,
or absorption processes leading to the emission of only low momentum or neutral
particles.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\vskip -0.5cm
\includegraphics[angle=0,width=0.8\textwidth]{eps/test/pionabs-1.eps}
\vskip -0.5cm
\caption[HMS $\beta$ spectra for several F$_{\pi}$-1 settings.]
{\label{fig:Pion_absorption_Fpi1} \small HMS $\beta$ spectrum for a
representative $\pi^+$ run in F$_{\pi}$-1, showing the proportion of HMS events
with $\beta=0$, the proportions in the `low' ($0<\beta<0.7$) and `high tails'
($0.7<\beta<0.9$), and those in the `blob' ($\beta>0.9$).  To avoid proton
contamination due to the lack of an Aerogel Cerenkov detector in F$_{\pi}$-1, 
the higher cut value of $\beta=0.95$ was used to define the `blob' and `tail'
regions in the calculation of the pion absorption correction.}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

To study the interplay between the pion absorption and $\beta$ corrections, the
transmission of pions from the target through to S2X was calculated and used to
estimate which fractions of these end up in the various parts of the $\beta$
versus coincidence time spectrum.  The pion transmission for each material was
calculated by making use of their known areal densities and the nuclear
collision lengths $\lambda$ from Ref. \cite{pdg}.  Since these $\lambda$ values
are only appropriate for high energy particles, where the energy dependence is
small, they were rescaled according to the energy dependence of the $\pi^+p$
interaction.  It was assumed that all pion reactions from the target to
spectrometer exit window resulted in lost triggers.  For the pions reacting
from the drift chambers to S1, it was assumed that a fraction $f_1$ were lost
triggers, while the remaining $1-f_1$ were not lost, but ended up with either
$\beta=0$ (meaning no valid track was found by the tracking algorithm) or in
the `tail' ($\beta$ below the cut value).  Finally, for the interactions from
the Cerenkov through to the first 1/4 of S2, it was assumed that fraction $f_2$
resulted in a low $\beta$ value, while the remaining $1-f_2$ were
indistinguishable from those pions that did not undergo nuclear reactions.

To determine the fractions $f_1$, $f_2$ appropriate for the F$_{\pi}$-1,2
deuterium data, the following procedure was employed.  The fractions of
$\beta=0$, `tail' and `blob' events were determined for each F$_{\pi}$-1
$\pi^-$ setting, and for each F$_{\pi}$-2 $\pi^-$ and $\pi^+$ setting (see
Fig.~\ref{fig:Pion_absorption_Fpi1}).  Since low $\beta$ values could also be
due to instrumental timing effects, the same proportions were also determined
from representative F$_{\pi}$-1,2 runs with electrons in the HMS.  The electron
fractions were then subtracted from the pion fractions, yielding typical
`zero+tail' values of 2.2+4.0=6.2\% (2.6+2.2=4.8\%) for F$_{\pi}$-1
(F$_{\pi}$-2), with the remainder in the `blob'.  $f_1$, $f_2$ were then
inferred by comparison of the observed `zero+tail' values to the calculated
interaction probabilities.  For F$_{\pi}$-1, the calculation predicted too few
zero+tail events and too many in the blob, with the closest agreement being for
the assumption that none of the interacting pions resulted in lost triggers
($f_1=0$), and 100\% of the interacting pions ended up in the `zero+tail'
($f_2=1$).  For F$_{\pi}$-2, values of $f_1$=30-45\% and $f_2=75-100\%$
resulted in good agreement with the data.

Since the calculated interaction probability from the target to exit window was
0.55\% for F$_{\pi}$-1, and 0.70\% for F$_{\pi}$-2, we estimated pion
absorption corrections of $1\%\pm 1\%$ for F$_{\pi}$-1 and $2\%\pm 1\%$ for
F$_{\pi}$-2.  The higher F$_{\pi}$-2 value is mainly due to the thicker
titanium spectrometer exit window and the addition of the aerogel Cerenkov in
the detector stack.

\vskip 1.cm
\subsection{Data Corrections Summary} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[h]
%\vskip -1.cm
\begin{center}
\begin{tabular}{|l|c|}\hline
\multicolumn{2}{|c|}{Summary of F$_{\pi}$-1 Correction Factors}\\
\hline\hline
HMS tracking efficiency correction    & $1-(0.0676\pm 0.002)/$S1Xrate(MHz)\\
LD$_2$ Cryotarget Boiling             & $1-(0.0712\pm 0.01)/100\mu$A\\
HMS Cerenkov blocking                 & $e^{{\rm -ELLOrate}*(138.4\pm 6.1 {\rm ns})}$\\
$\beta_{cut}$ correction ($\pi^{\pm}$)& $4.89\% \pm 0.41\%$\\
Pion Absorption                       & $1\%\pm 1\%$\\
\hline
SOS Cerenkov efficiency               & $99.92\%\pm 0.02\%$ \cite{Tanjathesis}\\
SOS Calorimeter efficiency            & $99.5\%\pm 0.1\%$ \cite{Tanjathesis}\\
HMS Cerenkov efficiency               & $99.6\%\pm 0.05\%$ \cite{Tanjathesis}\\
Coincidence Time Blocking             & $e^{{\rm -Total Pretrig rate}*(140 {\rm ns})}$
\cite{Volmerthesis}\\ 
HMS electronic live time              & $1 - 5/6(hEL60-hEL120)/hELREAL$
\cite{Volmerthesis}\\
SOS electronic live time              & $1 - 5/6(sEL60-sEL120)/sELREAL$
\cite{Volmerthesis}\\

\hline\hline

\multicolumn{2}{|c|}{Summary of F$_{\pi}$-2 Correction Factors}\\
\hline\hline
HMS tracking efficiency correction    & $1-(0.0676\pm 0.002)/$S1Xrate(MHz)\\
LD$_2$ Cryotarget Boiling             & No correction.  $\pm 0.3\%/100 100\mu A$\\
HMS Cerenkov blocking                 & $e^{{\rm -ELLOrate}*(114.7\pm 6.0 {\rm ns})}$\\
$\beta_{cut}$ correction ($\pi^-$)    & $2.51\% \pm 0.18\%$\\
$\beta_{cut}$ correction ($\pi^+$)    & $2.42\% \pm 0.12\%$\\
Pion Absorption                       & $2\%\pm 1\%$\\
\hline
SOS Cerenkov efficiency               & $99.92\%\pm 0.02\%$ \cite{Tanjathesis}\\
SOS Calorimeter efficiency            & $99.5\%\pm 0.1\%$ \cite{Tanjathesis}\\
HMS Cerenkov efficiency               & $99.6\%\pm 0.05\%$ \cite{Tanjathesis}\\
HMS Aerogel efficiency                & $99.5\%\pm 0.02\%$ \cite{Tanjathesis}\\
Coincidence Time Blocking             & $e^{{\rm -SOS Pretrig rate}*(92 {\rm ns})}$
\cite{Tanjathesis}\\
HMS electronic live time              & $1 - 6/5(hPRE100-hPRE150)/hPRE100$
\cite{Tanjathesis}\\
SOS electronic live time              & $1 - 6/5(sPRE100-sPRE150)/sPRE100$
\cite{Tanjathesis}\\

\hline
\end{tabular}
\end{center}
\vskip -.5cm
\caption{Summary of corrections applied to the deuterium data.  In addition, 
  HMS and SOS tracking efficiencies and computer live times are applied on a
  run-by-run basis.
\label{tab:eff_summary}}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
