\section{Interaction Spectra}

\subsection{Types of Beam Plugs}

Various types of beam plugs were considered.  The ``base line'' beam
plug for this study was taken to be the optimized one\footnote{The
  final radius chosen in the IHEP study was based on available
  graphite stock.  For this study the idealized optimum was taken.}
found in the IHEP study\cite{numi-b-543}.  This plug is a graphite
cylinder of 3.0 cm diameter, 1.5 m long and starting at z = 4~m.
Figure~\ref{fig:baseline-plug} shows a diagram of the placement of
this plug in the target area.  From this starting point different
lengths, materials and placements were considered.
Table~\ref{tab:plug-types} gives a summary of the different plug
parameters.  In this note, if no descriptive qualifier is used then it
is the ``short'' beam which is being discussed.


\begin{figure}[htbp]
  \begin{center}
    \sizedfig{0.9\textwidth}{baseline-plug.eps}
    \caption[Beam plug location.]{Beam plug location.  Note that X and Y scales are vastly different.}
    \label{fig:baseline-plug}
  \end{center}
\end{figure}


\begin{table}[htbp]
  \begin{center}
    \begin{tabular}{c|c|c|c} 
      Name & Material & Length & Location \\ 
      \hline
      ``Base line'' & No plug \\
      ``Short'' & Graphite & 1.5 m & 4.0 m \\
      ``Long'' & Graphite & 2.5 m & 3.5 m \\
      ``Copper'' & Copper & 1.5 m & 4.0 m \\
      ``Composite'' & Graphite/Copper & 1.5 m / 0.3 m & 4.0 m 
    \end{tabular}
    \caption[Summary of beam plug types.]{Summary of the parameters of the various beam plugs studied.  All plugs are coaxial with the beam.  Location refers to the distance from the front face of the first horn to the front face of the beam plug.  The composite plug has a 0.3~m copper cap at the downstream end.  All plugs have a diameter of 3~cm.}
    \label{tab:plug-types}
  \end{center}
\end{table}

\subsection{Effects of Short Graphite Plug and the Hadron Hose}

As stated above, the general effect of a plug is to reduce the high
energy tail of the \numu{} spectrum by a lot and the low energy peak
by a little, while the hadron hose increases both, with a relatively
higher increase in the tail.  Figure~\ref{fig:spect-short-hose} shows
the changes in this spectrum with and without these beam line
elements.


\begin{figure}[htbp]
  \begin{center}
    \overlay{(a) near $\nu_e$}{\sizedfig{\twofigsize}{plots/simple-spect-near-nue.eps}}{0.2}{0.2}
    \overlay{(b) near $\nu_\mu$}{\sizedfig{\twofigsize}{plots/simple-spect-near-numu.eps}}{0.2}{0.2}
    \overlay{(c) far $\nu_e$}{\sizedfig{\twofigsize}{plots/simple-spect-far-nue.eps}}{0.2}{0.2}
    \overlay{(d) far $\nu_\mu$}{\sizedfig{\twofigsize}{plots/simple-spect-far-numu.eps}}{0.2}{0.2}
    \caption[Neutrino interaction spectra, near/far, with/no hose, plug.]{Neutrino, near and far detector interaction spectra with combinations of hadron hose and the short graphite plug.  Data shows: No plug with hose ($\Box$), no plug no hose ($\circ$), with plug with hose ($\Diamond$), and with plug no hose ($\bigtriangleup$).}
    \label{fig:spect-short-hose}
  \end{center}
\end{figure}

It is illustrative to break up the spectrum into various regions.  The
oscillations are expected to have a minimum around 1 to 3 GeV, which
covers about half of the ``peak'' which extends to about 6 GeV.
There is then a ``shoulder'' to 10 GeV.  Above 10 GeV is taken to be
the ``tail''.  Table~\ref{tab:percent-changes} shows the percent
change in these regions between various beam line configurations.
Some of the beam plug configurations will be discussed in more detail
later.

% used, eg: 
% cd ~/work/beamplug && paw
% PAW > numi-note#load-hists
% PAW > hist#percent-change 14001 24001 1 3; message [@]
% PAW > hist#percent-change 14001 24001 4 6; message [@]
% PAW > hist#percent-change 14001 24001 7 10; message [@]
% PAW > hist#percent-change 14001 24001 11 50; message [@]
% to generate these numbers
\begin{table}[htbp]
  \begin{center}
    \begin{tabular}{c|c|c|c|c|c}
      \multicolumn{2}{c|}{Beam line config.} & \multicolumn{4}{c}{Energy range (GeV):} \\
      From: & To:      & 0 - 3 & 3 - 6 & 6 - 10 & 10 - 50 \\ \hline
      BL   & BL+HH     & +20\% & +25\% & +53\%  & +68\%   \\
      BL+HH& BL+HH+SBP & -7.6\%& -2.5\%& -26\%  & -70\%   \\
      BL+HH& BL+HH+LBP & -10\% & -3.4\%& -41\%  & -82\%   \\
      BL+HH& BL+HH+CuBP & -11\%& -4.8\%& -38\%  & -85\%   \\
      BL+HH& BL+HH+CBP & -8.6\%& -3.1\%& -30\%  & -79\%   \\
    \end{tabular}
    \caption[Change in number of interactions with different beam elements.]{Percent changes in number of $\nu_\mu$ interactions in the far detector when going from one beam line configuration to another. BL = Base line beam configuration (no plug, no hose), HH = with hadron hose, SBP = ``short'' graphite beam plug, LBP = ``long'' graphite beam plug, CuBP = ``copper'' beam plug, CBP = ``composite'' graphite+copper beam plug.}
    \label{tab:percent-changes}
  \end{center}
\end{table}

The hose increases the expected number of far detector events in the
low enegy peak by about 20\% to 25\% while increasing this in the
shoulder and tail by 50\% to 70\%.  Adding in the short graphite plug
will knock down the gain in the peak by a few percent.
Unfortunately this reduction is largest at the low end of the peak
where effects of neutrino oscillation are expected to be most prominent.  Here
7.6\% of the 20\% increase the hose gives is taken away by the plug.
With hose and plug in place there is still a net gain in the region of
the peak compared to the case of neither hose nor plug.

\subsection{Other Plug Designs}

Adding more material to the plug will help reduce the high energy tail
even further, but increasing the plug dimensions tends to cause the
useful hadrons to be absorbed as well.  The past
studies~\cite{numi-b-543} found that increasing the radius lead to a
stronger reduction in the low energy part of the low energy peak as
did, to a lesser extent, increasing the plug length.  Increasing the
length and changing the composition is examined in this section.

\subsubsection{Long Graphite Plug}

Adding on 0.5 m to each end of the ``short'' graphite plug gives the
``long'' graphite plug.  The radius is the same.  See
Table~\ref{tab:percent-changes} for percent changes in various energy
ranges.  Figure~\ref{fig:long-plug-relative} shows the far detector
\numu{} interaction spectra for the case of the hadron hose and no
plug, short and long plug relative to the no plug, no hose case.  


\begin{figure}[htbp]
  \begin{center}
    \sizedfig{\textwidth}{plots/relative-normed-graphite.eps}
    \caption[Rel.  $\nu_\mu$ int. spectra, different beam elements, far det.]{Far detector $\nu_\mu$ interactions relative to no plug, no hose (solid line) case for the cases of the hose and no plug (dashed), short plug (dotted) and long plug (dot-dashed).  The error bars show normalized statistical errors.}
    \label{fig:long-plug-relative}
  \end{center}
\end{figure}

The same tradeoff observed in the previous IHEP studies is seen here.
There is a desirable further reduction of 10\% to 20\% in the tail but
it is accompanied by an additional couple of percent suppression of
the low energy peak.

\subsubsection{Copper Plug}

A copper plug was considered as a way of placing more material in the
beam without presenting a larger profile.  However, it was
learned~\cite{hylen-copper-sucks} that copper is a poor choice as it
would be destroyed by beam heating by a direct hit by the full beam.
This being a possibility that must be planned for, the idea of a solid
copper plug was excluded.  In any case, the 1.5 meter copper plug
performs about as well as the ``long'' graphite plug, so this
optimization is not of value.

\subsubsection{Composite Copper/Graphite Plug}
% from plug vs from target - bare target

It was found that the short graphite plug reduces the high energy tail
so well, that a significant part of what is left in the tail is
actually from hadrons produced in the plug by the protons in the
primary beam which survive the passage through the target.  Since the
plug is acting as a bare target, the ``useful'' hadrons tend to be
very forward going and thus exit the plug from the end cap.  These two
results are shown in Fig.~\ref{fig:from-plug-target-ratio}.

\begin{figure}[htbp]
  \begin{center}
    \overlay{(a)}{\sizedfig{\twofigsize}{plots/from-plug-target.eps}}{0.3}{0.15}
    \overlay{(b)}{\sizedfig{\twofigsize}{plots/from-plug-side-end.eps}}{0.3}{0.15}
    \caption[Interaction spectra, parents from target and plug.]{Far detector $\nu_\mu$ interactions due to (a) parents produced in the short graphite plug relative to those produced in the target and (b) broken up by parents which exit the side of the plug ($\bigtriangleup$) or exit the end of the plug ($\bigtriangledown$) and their sum ($\circ$).}
    \label{fig:from-plug-target-ratio}
  \end{center}
\end{figure}


This lead to the idea~\cite{milind-plug-the-plug} of ``plugging the
plug'' by adding a denser end cap.  This was expected to reduce the
high energy tail further while not being vulnerable to a full proton
beam hit.  A simulation with an additional 30 cm of copper just
following the short graphite plug was run.  The results are shown in
Fig.~\ref{fig:composite-plug-relative}.  It shows the far detector
\numu{} interaction spectra with this composite plug in place relative
to that of just the short graphite beam plug.  Also shown is the case
of no plug.  All plots have the hose turned on.  It can be seen that
adding the copper end cap results in an even further reduction beyond
that due to just the short graphite plug.  This further reduction is
about 30\% in the tail while only about 1\% in the peak.

\begin{figure}[htbp]
  \begin{center}
    \sizedfig{\textwidth}{plots/relative-normed-composite.eps}
    \caption[Relative interaction spectra, composite plug.]{Far detector $\nu_\mu$ interaction spectra with (dashed) composite plug and hose in place, relative to (solid) short graphite plug.  Also shown (dotted) case of no plug, but with hose.}
    \label{fig:composite-plug-relative}
  \end{center}
\end{figure}

\subsection{Other Plug Locations}


\begin{figure}[htbp]
  \begin{center}
    \sizedfig{\textwidth}{plots/relative-normed-moved-short.eps}
    \caption[Relative interaction spectra, plug in different locations.]{Far detector $\nu_\mu$ interaction spectra relative to the short graphite beam plug at the nominal position of z = 4~m for the case of (dashed line) the short plug moved to z = 3 m (just past end of horn 1) and the case of (dotted line) the short plug moved to z = 2 m (inside the end of horn 1).  The error bars show the statistical uncertainty of the spectrum with the plug at the nominal position normilized to the strength of this spectrum bin-by-bin.  The statistics of the two cases with relocated plugs are half that of the nominal case.  Note, the hadron hose is not used in this plot.}
    \label{fig:moved-plug-relative}
  \end{center}
\end{figure}

In addition to the different plug designs presented in the previous
section, different locations for the  ``short'' graphite plug
were considered.  Figure~\ref{fig:moved-plug-relative} shows the far
detector \numu{} interaction spectra for the case of the short beam
plug moved to two different locations relative to the nominal
location (z = 4 m). 

Moving the plug closer to the first horn helps to reduce the shoulder
of the spectrum but also reduces the high energy side of the low
energy peak.  This could be advantageous for \numunue{} searches and
not detrimental to disappearance searches depending on where the value
of the oscillation parameters lay.  If the minium oscillation
probability is at low energies ($E < 3$ GeV), the appearance search
would benefit from a reduction of neutral current background from the
lower energy neutrinos in this shoulder, but if the oscillation
probability has a minimum at energies around 7 - 8 GeV, then both
searches will suffer a reduction in statistics of more than a factor
of 2 from the case of the plug being at the nominal location.


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