\documentstyle[aps,preprint,tighten,rotating]{revtex}
\input hep_macro
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\begin{document}
\title{Non-Oscillation Physics at a Neutrino Factory}
%\author{Your name here}
\maketitle
%\tableofcontents
\section*{Introduction}
Aside from neutrino oscillation physics, a neutrino factory will also be
a formidable tool for other physics such as deep inelastic scattering
experiments and searches for
exotic neutrino interactions. The neutrino factory parameters considered in
this study represent improvements of 3-4 {\bf orders of magnitude }\rm over
existing neutrino beams. Table \ref{otherexpts} compares the numbers
of neutrinos produced by present beamlines. Besides the high intensity, a
muon storage ring neutrino source has two other important aspects not found
in current neutrino beams. First, since the kinematics of muon decay are
well understood and because of the degree of control we will have over the
muons in the storage ring, the neutrino flux will be known to a much higher
accuracy than in any other high energy neutrino beam. Second, the neutrino
factory will not only be a source of high energy muon neutrinos, but also
electron neutrinos as well. The rest of this section
is a short introduction to the phenomenology of deep inelastic scattering
followed descriptions of possible experimental measurents. Later
sections are devoted to searches for novel effects which may appear in
neutrino beams with 10$^4$ time the intensity of present beams.
\it table equivalent to Kevin's comparing NuTeV, MINOS, CERN, Nufac \rm
Possible subsection on statistics.....table and/or figures from Heidi...
\section*{Structure Function Measurements}
In considering measurements of structure functions in neutrino interactions
it is important to remember that almost all present-day experiments have
relied on heavy (isoscalar) targets with an
approximately equal mix of protons and anti-protons. Such measurements
have essentially been the average of neutron and proton structure functions
with additional unknown $A$ dependent effects. In addition, most
measurements have been of the average of neutrino and anti-neutrino
structure functions. We have only really measured one fourth
of the possible combinations of $\nu/\antinu$ neutron/proton.
In addition, no measurements of the spin dependent structure functions
have been made in neutrino scattering.
At a neutrino factory, we can expect to perform measurements with
excellent statistics and good flux estimates on polarized and unpolarized
hydrogen and deuterium
targets. With these data, we should be able to determine the
real structure functions instead of averages.
\subsection*{Statistics}
Overview of statistics
\subsection*{Kinematics}
In neutrino scattering, the incoming beam can assumed to be massless
($m_{\nu}\simeq 0$) but the outgoing lepton mass may be important. If
$\Elab^{\prime}$ denotes the laboratory frame energy of
the scattered lepton $ \lepton$,
\newcommand{\Elep}[0]{E_{\lepton}}
\newcommand{\Enu}[0]{E_{\nu}}
\newcommand{\mlep}[0]{m_{\lepton}}
\begin{eqnarray}
\fourv{q} &=&{\fvk{\nu} - \fvk{\lepton}}, \hskip .4 in Q^2 = -\fourv{q}^2 = 2
\Elep\Enu -\mlep^2 - 2 \Enu p_{\lepton} \cos\theta_{lab}\\
\nu &=&(\fv{\quark}\fourv{q})/M \Elab - \Elab^{\prime}\\
x &=&Q^2/2 \mtarget \nu\\
y &=& \mtarget \nu/(\fvk{\nu} \fv{\quark}) = (1 + cos\theta_{CM})/2 \approx
\nu/\Elab \\
W^2 &=& m_4^2 c^4 = 2 \mtarget \nu + \mtarget^2 -Q^2
\end{eqnarray}
where, $M$ is the target nucleon mass, $\Enu$ is the incoming neutrino
energy $\Elep, p_{\lepton}$ are the outgoing lepton energy and momentum
$\theta_{lab}$ is
the lepton angle with respect to the incoming beam. $\fourv{q}$ is the
four-momentum transfer to the target, $\nu$
is the energy transfer, $x$ is the Bjorken $x$ variable, $y$ is the scaled
energy transfer and $W^2$ is the invariant mass of the final state hadronic
system
squared.
\subsection*{Neutrino Scattering}\label{DIScross}
% The kinematic variables are defined in section \ref{DIS}.
\subsubsection*{Un-polarized}
For an unpolarized target the neutrino(anti-neutrino) scattering cross section
is:
\begin{eqnarray}
{d\sigma^{\nu(\antinu)}\over dx dy} &=&
{M_W^4 G_F^2 M E_{\nu} \over \pi (Q^2 + M_W^2)^2 }
\biggr[
y^2 xF_1(x,Q^2) + (1-y -{M^2 xy\over s-M^2}) F_2(x,Q^2)
\pm (y-{y^2\over 2}) x F_3(x,Q^2)\biggl]
\end{eqnarray}
where $M$ is the mass of the target and the $F_i$ are
Structure Functions. $2xF_1$ is purely transverse while
$F_2 = 2xF_1 + F_L$ where $F_L$ is a purely longitudinal
structure function.
$F_3$ enters with a '-' sign for anti-neutrino scattering.
In the limit that $Q < 0.7$.
\end{enumerate}
\subsection*{Outline of a Muon Storage Ring Neutrino Nuclear Effects
Experiment}
As we discuss possible physics topics we will refer to an experiment
which assumes a 50 GeV storage ring with a detector 50 m away from an 800 m
straight section. With $10^{20}$ mu decays per year per straight section,
we would expect 4 x $10^{6}$ $\nu_{mu}$ and 1.7 x $10^{6}$
$\overline{\nu_{e}}$ interactions per kg-year in
a fiducial volume of radius 10 cm. The detector would consist of a liquid
$D_{2}$ target followed by a rotating support of different A targets -
similar to the geometry of the Fermilab E-665 Tevatron muon experiment. To
acquire $10^{7}$ events per year in all targets we would need targets with
fiducial volume r = 10 cm and length of 50 cm for $D_{2}$ 3.7 cm for C and
0.45 cm for W. The liquid $H_{2}$ or $D_{2}$ targets could be CCD coupled
active liquid targets (ie Bubble Chambers!) which could provide tracking
within the target, followed by the higher A targets interspersed with
tracking chambers, followed by appropriate calorimetry and muon
identification/measurement techniques.
The challenge for the experiment is finding tracking and calorimetry which
can accurately measure the hadron energy without being flooded
with neutrino interactions themselves.
\subsection*{Shadowing Region}
As mentioned these effects have been studied with electromagnetic
probes and a pertinent question is to ask what additional things can we
learn when using a weak probe. In the shadowing region there are several
effects that should yield a different ratio $R_{A}$ when using neutrinos as
the probe. In the limit $Q^{2} --> 0$, the vector current is conserved and
goes to 0, however the axial-vector part of the weak current is only
partially conserved (PCAC) and $F_{2}(x,Q^{2}) -->$ a non-zero constant as
$Q^{2} --> 0$. According to the Adler theorem \cite{Adler} the cross
section of $\nu_{\mu}$ - N can be related to the cross section for $\pi$ - N
at $Q^{2}$ = 0. What effect does a nuclear environment have on the Adler
theorem?
As we increase $Q^{2}$ from 0 but keep it under 10 $GeV^{2}$ in the
shadowing region we enter the region of vector meson dominance (VMD) in
$\mu$/e - A scattering. The physics concept of VMD is the dissociation of
the virtual photon into a $q$ - $\overline{q}$ pair one of which interacts
strongly with the "surface" nucleons of the target nucleus (thus the
"surface" nucleons "shadow" interior nucleons). In $\nu$ - A scattering
there is an additional contribution from axial-vector mesons that is not
present in $\mu$/e - A scattering. Boros et al \cite{Boros} predict
that the resulting shadowing effects in $\nu$ - A scattering will be
roughly 1/2 that measured in $\mu$/e - A scattering.
In a more quantitative analysis, Kulagin \cite{Kulagin} used a
non-perturbative parton model to predict shadowing effects in $\nu$ - A
scattering. At 5 $GeV^{2}$ he predicts equal or slightly more shadowing in
$\nu$ - A scattering than in $\mu$/e - A scattering. He also attempts to
determine quark flavor dependence of shadowing effects by separately
predicting the shadowing observed in $F_{2}(x,Q^{2})$ (sum of all quarks)
and $xF_{3}(x,Q^{2})$ (valance quarks only). Fig.\ \ref{fig:shadow} shows
the results of a 2 year run using predictions of Kulagin's model for
$F_{2}$ and $xF_{3}$. The proposed comparison of W:$D_{2}$ would exhibit a
much sharper decrease in the ratio as $x_{Bj}$ decreases. As can be seen,
the predicted difference between the shadowing on sea and valence quarks is
clearly visible down to $x_{Bj}$ = 0.02 - 0.03.
\begin{figure}
\epsfysize=2.5in
\epsfxsize=5.0in
\centerline{
\epsffile{Kulagin.eps}}
\caption{$R_{Ca:D_{2}}$ for both $F_{2}$ and $xF_{3}$ as measured in a 2
year exposure at the facility described in the text.}
\label{fig:shadow}
\end{figure}
\subsection*{Anti-shadowing Region}
Drell-Yan experiments have also measured nuclear effects. Their results
are quite similar to DIS experiments in the shadowing region. However, in
the anti-shadowing region where $R_{A}$ makes a brief but statistically
significant excursion above 1.0 in DIS, Drell-Yan experiments see no
effect. This could be an indication of difference in nuclear effects
between valence and sea quarks. Eskola et al \cite{Eskola} quantified this
difference by using a leading order/leading twist DGLAP model which used
initial nuclear parton distributions from CTEQ4L and GRV-LO and assumed
scale evolution of nuclear parton distribution is perturbative. The model
predicts that the difference between nuclear effects in $xF_{3}(x,Q^{2})$
and $F_{2}(x,Q^{2})$ persist through the anti-shadowing region as well.
Taking the work of Kulagin and Eskola together implies that nuclear effects
in $xF_{3}(x,Q^{2})$ should be quite dramatic with more shadowing than
$F_{2}(x,Q^{2})$ at lower $x$ and then $R_{A}$ rising fairly rapidly to yield
significant antishadowing around $x = 0.1$. In our standard 2 year
experiment we should be able to measure antishadowing effects and the
difference between shadowing effects in $F_{2}(x,Q^{2})$ and
$xF_{3}(x,Q^{2})$ to the 6 $\sigma$ statistical level
\subsection*{EMC-effect Region}
To determine individual quark contributions to the EMC-effect will be
challenging since the participation of sea quarks, and thus the difference
between $F_{2}(x,Q^{2})$ and $xF_{3}(x,Q^{2})$, shrinks rapidly with
increasing $x_{Bj}$. However, Eskola's predictions for this region indicate
that the contribution of $\overline{u}$ and $\overline{d}$ to $R_{A}$ in the
$Q^{2}$ range of this experiment remains well below 1 so that the quantity
$R(2)_{A}$ - $R(3)_{A}$ should remain negative well into the EMC-effect
region.
\subsection*{Behavior of $F_{2}(x,Q^{2})$ as $x_{Bj} --> 1$ in a Nuclear
Environment}
When working in the "fermi-motion" region it has been shown that we need to
add more than the Fermi gas model to a simple nucleon to reproduce the
behavior of $F_{2}(x,Q^{2})$ at high $x_{Bj}$. Few-nucleon-correlation
models and multi-quark cluster models allow quarks to have a higher
momentum which translates into a high-$x_{Bj}$ tail. In this region
$F_{2}(x,Q^{2})$ should behave as $e^{-ax_{Bj}}$. There have been analysis
of this behavior in similar kinematic domains using $\mu$ + C and $\nu$ +
Fe interactions. The muon experiment finds a = 16.5 +- 0.5 while the neutrino
experiment finds a = 8.3 +- 0.7 +- 0.7 (systematic). Is the value of a
dependent on the nucleus? One would expect any few nucleon correlation or
multi-quark effects to have already saturated by Carbon. Is a dependent
on the probe?
\subsection*{Summary}
There is a very rich program of studying nuclear effects with a neutrino
probe in a high statistics neutrino factory experiment. The effects could
be measured to statistically significant accuracy in a 2 year exposure to
the beam in the near-detector experiment described above. The data
gathered would allow separate measurements of the effects on valence quarks
and sea quarks across much of the $x_{Bj}$ range.
The nuclear community should be informed of what a valuable tool for
nuclear research awaits them in the neutrino factory facility.
\newpage
% Jae Yu's section
\section*{Precision Electroweak Measurements at a Neutrino Factory}
\subsection*{Introduction}
Precision measurements of electroweak parameters from neutrino experiments
have played an important role in testing the Standard Model
and in probing new physics.
However, previous neutrino deep inelstic scttering (DIS) experiments
have been limited by a lack of statistics
and by the coarse granularity of their detectors.
%A neutrino factory would provide a high intensity neutrino flux
% produced from the decay of muons
% decay process, by far one of the best understood.
The intense flux of neutrinos from a neutrino factory opens up a whole
new era of precision electroweak measurements that were previously
limited by statistics.
A precision measurement of \stw or $M_{W}$ could very well still be
important 10 years from now.
At that time the best measurement of $M_{W}$($\pm 30$MeV) would have come from
Fermilab Tevatron collider experiments and LEP-II, and the
LHC experiments will be vigorously looking for the Higgs, if it had not
already been found.
Therefore, complementary mesurements of electroweak parameters from an
independent experiment could play crucial role in LHC experments'
advancements.
In this section, we discuss the prospects of a precision measurement
of \stw and the mass of the W boson ($M_{W}$)
at a neutrino factory facility.
\subsection*{Previous Measurements}
It is always a good idea to learn from past experiments to better
understand what can be done in a future facility.
The CCFR neutrino experiment extracted
$\stw$~\cite{ccfr:bjking,ccfr:mcfarland} through a measurement of
the ratio of the NC to CC cross sections,
as expressed in the following Llewellyn-Smith
formula~\cite{th:llsmith} :
\begin{equation}\label{eq:llsmith}
R^{\nu({\overline{\nu}})}=\frac{\sigma_{NC}^{\nu({\overline{\nu}})}}
{\sigma_{CC}^{\nu({\overline{\nu}})}} =
\rho^{2}\left( \frac{1}{2}-\sin^2\theta_W+
\frac{5}{9}\sin^4\theta_W
\left( 1+\frac{\sigma^{\nub(\nu)}_{CC}}{\sigma^{\nu(\nub)}_{CC}}
\right)\right),
\end{equation}
where the value of $\rho$ depends on the nature of the Higgs sector
and has the value $\rho=1$ in the SM.
This method, although it removed much of the uncertainty due to QCD
effects in the target, does leave some rather large uncertainties associated
with heavy quark production from the quark sea of the nucleon target.
The successor experiment, NuTeV (FNAL-E815) experiment, has improved
upon this measurement by using separate neutrino and anti-neutrino
beams~\cite{ex:ssqt}.
Separation of neutrino and anti-neutrino neutral current
events allows the utilization of the
Paschos-Wolfenstein relationship~\cite{th:paschos}:
\begin{eqnarray}
r=\frac{\sigma({\overline{\nu}},CC)}{\sigma({\nu},CC)} \simeq 0.5.
\end{eqnarray}
NuTeV has presented a preliminary result\cite{NuTeV:prelim} of
\begin{equation}\label{eq:nutev-stw}
\stw=0.2253\pm0.0019{\rmt (stat)}\pm0.0010{\rmt (syst)}.
\end{equation}
There were two dominant systematic uncertainties in the CCFR experiment:
1) $\nu_{e}$ flux and 2) CC charm production.
These two major systematic uncertainties
in CCFR were been dramatically reduced in the NuTeV experiment,
by utilizing sign selected neutrino beam,
so that the remaining dominant uncertainty was the statistical
uncertainty.
Since neutrinos interact weakly, the past neutrino fixed target experiments
have used dense material as neutrino targets in order to increase the
interaction rate.
The calorimetric nature of the targets does not allow one to
distinguish electron neutrino induced charged current interactions (CC)
from neutral current (NC) interactions.
Thus, these old-style heavy target experimental set-ups (such as NuTev
or CCFR)
will be fatal
for neutrino factory analyses because there always will be both
$\nu_{\mu}$ ($\overline{\nu}_{\mu}$) and
$\overline{\nu}_{e}$ ($\nu_{e}$) in the beam.
This condition necessitates the need for light target detector which
is sufficiently different than the normal detectors for
neutrino oscillation measurements.
\subsection*{Experimental Technique}
As in previous neutrino experiments, the measured quantity
used to determine $\stw$ at a neutrino factory will
be the ratio of NC to CC DIS events.
However, the NuTeV-style measurement is not
accessible at the neutrino factory beam because
$\nu_{e}(\overline{\nu}_{e})$ and $\overline{\nu}_{\mu}(\nu_{\mu})$
come in pairs simultaneously and so NC events from neutrinos
and those from anti-neutrinos are not
distinguishable on an event-by-event basis.
This condition restricts the numerator of the ratios to be
sum of NC events.
In addition, it is necessary to minimize uncertainties in sea
quark induced CC production of charm.
Large portion of this uncertainty can be canceled out by taking
a difference between the neutrino and antineutrino induced
CC events.
Therefore the two accessible ratios that satisfy both the above condistions
are the hybrids of the Llewellyn-Smith ratio in
Eq.~\ref{eq:llsmith} and Paschos-Wolfenstein parameter in
Eq.~\ref{eq:r-minus}, as follows:
\begin{eqnarray}\label{eq:Rmuebar}
R_{\nu}^{\mu^{-}}=\frac{\sigma(\nu_{\mu},NC)+\sigma({\overline{\nu}}_{e},NC)}
{\sigma(\nu_{\mu},CC)-\sigma({\overline{\nu}}_{e},CC)}
=\frac{R^{\nu_{\mu}}+grR^{\overline{\nu}_{\mu}}} {1-gr}
\end{eqnarray}
for the $\muebar$ beam, and
\begin{eqnarray}\label{eq:Rmubare}
R_{\overline{\nu}}^{\mu^{+}}=\frac{\sigma({\overline{\nu}}_{\mu},NC)
+\sigma(\nu_{e},NC)}
{\sigma(\nu_{e},CC)-\sigma({\overline{\nu}}_{\mu},CC)}
=\frac{R^{{\nu}_{\mu}}+g^{-1}rR^{\overline{\nu}_{\mu}}}{1-g^{-1}r}
\end{eqnarray}
for the $\mubare$ beam.
The expression of
$R_{\nu}^{\mu^{-}}$ and $R_{\overline{\nu}}^{\mu^{+}}$
in terms of $r$, $R^{\nu_{\mu}}$ and $R^{\overline{\nu}_{\mu}}$
assumes lepton universality.
The second of the two equations differs from the first only in
the replacement of $g$ by $g^{-1}$, where
$g$ is the energy-weighted flux ratio between
$\nu_{\mu}$ and $\overline{\nu}_{e}$ or, equivalently,
between $\overline{\nu}_{\mu}$ and $\nu_{e}$:
\begin{eqnarray}\label{eq:little-g}
g\equiv\frac{\int \Phi(E_{\overline{\nu}_{e}})E_{\overline{\nu}_{e}}dE_{\overline{\nu}_{e}}}
{\int\Phi(E_{{\nu}_{\mu}})E_{{\nu}_{\mu}}dE_{{\nu}_{\mu}}}
=\frac{\int\Phi(E_{{\nu}_{e}})E_{{\nu}_{e}}dE_{{\nu}_{e}}}
{\int \Phi(E_{\overline{\nu}_{\mu}})E_{\overline{\nu}_{\mu}}
dE_{\overline{\nu}_{\mu}}} .
\end{eqnarray}
\subsection*{Experimental and Theoretical Issues}
While the prospects in greatly improving $\stw$ measurements in a neutrino
factory are good, there are both experimental and theoretical
concerns that we need to consider in order to achieve meaningful
precision ($\delta\stw \sim 10^{-3}$).
\subsubsection*{Experimental Issues}
Regarding the beam, a crucial requirement is the ability to reverse
the polarity of the ring so that one can choose between
$\muebar$ or $\mubare$ beams at any given time.
This capability provides the opportunity for studying possible
systematics due to the beam of a given flavor.
Probably the most demanding requirement on the detector for this
analysis is the ability to distinguish events with primary
electrons or muons from neutrino CC interactions from the purely
hadronic showers produced in NC interactions.
Traditional heavy target neutrino detectors can not distinguish
$\nu_{e}$ induced CC interactions from NC interactions.
Other particularly relevant capabilities are the use of a light
isoscalar target, with a trigger capability, that can provide the
charm and strange sea measurements from the same experiment, as
this will be very useful in reducing the remaining systematic
uncertainties.
In order to satisfy the above requirements, the detector needs to have:
\begin{itemize}
\item Good EM and hadronic shower identification.
\item High electron detection efficiency along with high efficiency
particle identification.
\item Good charged particle momentum measurement.
\item Good EM and hadronic shower energy
containment and measurement.
\item Muon identification and momentum measurement.
\end{itemize}
High-performance low density target detectors for neutrino factory
experiments will greatly suppress the expeimental uncertainties in
the $\stw$ measurement relative to today's analyses conducted using
coarse-granularity calorimetric neutrino target.
Probably the most important remaining experimental systematic in
this measurement will be the $e/\mu$ identification efficiency.
Since the expected and meaningful total error in this measurement is
on the order of $10^{-3}$, it is extremely vital to control
the overall efficiency of the particle identification
system to better than $2\times 10^{-4}$.
Otherwise every $\nu_{e}$ CC event where the electron escapes
identification would be a direct background
to NC interactions for both $\nu_{\mu}$ and $\nu_{e}$.
%
\subsubsection*{Theoretical Issues}\label{ss:stw_theo_issues}
%
Theoretical uncertainties will be expected to dominate
the $\stw$ measurement at a neutrino factory, and an
improved understanding of several potential theoretical
uncertainties will be required for a meaningful interpretation
of a sub$-30$ MeV equivalent $W$ mass error.
The CC-charm cross section error needs to be minimized.
In principle, this error can be reduced by measuring
the CC production cross section of charm directly from the
same experiment using vertex-tagged charm events.
Further effects which require investigation include:
a) nucleon isospin-violating effects\cite{sather};
b) strange quark mass effects in $\nu_{\ell}\bar{u}\rightarrow\ell^{-}%
\bar{s}$ and $\nu_{\ell}s\rightarrow\nu_{\ell}s$;
c) Effects from ``radiative higher twist processes'' such as $\nu_{\ell
}n\rightarrow\gamma\ell^{-}p$;
d) other higher twist effects, which are assessed as
one of the dominant errors in today's measurements (
however, a recent study~\cite{ph:arie-unki-ht} indicates that most the
higher twist effects might be higher order QCD corrections in GLS
measurements)
%
\subsection*{Prospects in $\stw$ Measurement}
Using the neutrino factory beam parameter discussed in
Section~\ref{ss:beam} for mean muon energy of
50GeV and scaling the muon decay rate to 40~m long straight section,
one expects approximately $1.5\times10^{7}$ $\nu_{\mu}$
events per year per 0.5~m long CCD target with a radius of 10~cm.
In addition, with a detector that can distinguish electrons and muons
resulting from CC interactions and assuming a 100\%
fiducial acceptance for events occuring in the CCD target,
the effective neutrino interaction statistics double, resulting
in a total of $3\times 10^{7}$ events per year.
Since the number of neutrino events from the NuTeV experiment (with
$3\times 10^{18}$ protons-on-target) was on the order of
1 million events for $\nu_{\mu}$, the expected $3\times 10^{7}$ events
per year from the neutrino factory would cause a reduction in the
statistical uncertainty by a factor of about five within a year
of data taking.
This enormous increase in statistical power also helps to minimize
many of the systematic uncertainties dramatically.
The remaining error of 0.0004 in NuTeV, due to $\nu_{e}$ flux, no
longer exists for a detector that is capable of distinguishing
CC interactions of $\nu_{e}$ from hadronic showers resulting
from NC interactions by identifying the outgoing electrons.
The uncertainties resulting from energy scale are also irrelevant
because they are associated with the muon range through the muon
length cut, which will not be relevant for neutrino factory analysis.
The errors in event length, in principle, also no longer
contribute, because with the
detector described in the following section, one can distinguish CC
from NC interactions on an event-by-event basis.
The above expectations will enable the experiment to reduce the statistical
and experimental systematic uncertainties on $\stw$ to the
equivalent $M_{W}$ uncertainty of $\sim 30$ MeV.
%
\subsection*{Conclusions}
This work has demonstrated that precision test of the electroweak
Standard Model can be expected to play an important part in the
physics program of a neutrino experiment at a neutrino factory.
The \stw measurement at the neutrino factory facility, using the
the variables in Eq.~\ref{eq:Rmuebar}
and Eq.~\ref{eq:Rmubare} will allow tests of unparalleled
statistical accuracy and control over
a potentially large theoretical uncertainty in CC production
of charm.
However, controlling particle ID uncertainty from experimental
systematics and understadning higher twist and radiative
correction theoretical systematics play crucial role in
obtaining meaningful measurements.
\newpage
\def\pl#1#2#3 {{ Phys. Lett.} {\bf#1}, #2 (#3). }
\def\prev#1#2#3 {{ Phys. Rev. } {\bf#1}, #2 (#3). }
\section*{Heavy Lepton Mixing}
A muon storage ring offers ample opportunities to search for new
phenomena in yet unexplored physical regions. One such opportunity
is the ability to search for the
existence of neutral heavy leptons. Several models describe heavy isosinglets
that interact and decay by
mixing with their lighter neutrino counterparts \cite{GLR,Shrock}. The
high intensity neutrino beam created by the muon storage ring provides an
ideal setting to search for neutral heavy leptons with a mass below
the muon mass, 105.6 MeV$/c^2$.
It is postulated that neutral heavy leptons ($L_0$) could be produced
from muon decay when one of the neutrinos mixes with its heavy,
isosinglet partner. Neutral heavy leptons can be produced via one of
two channels:
\begin{equation}
\mu^- \rightarrow L_0 + \overline{\nu}_{e} + e^-
\end{equation}
\begin{equation}
\mu^- \rightarrow \nu_{\mu} + L_0 + e^-
\end{equation}
The branching ratio for each of these reactions is given by:
\begin{equation}
BR(\mu\rightarrow L_0 \mu e) = |U_i|^2 (1 - 8x_m^2 + 8x_m^6 - x_m^8 +
12x_m^4\ln{x_m^2})
\end{equation}
\noindent Here $x_m \equiv m_{L_0}/m_{\mu}$ and $|U_i|^2$ is
the mixing constant between the specific type of neutrino
and the neutral heavy lepton: $U_i \equiv \langle L_0 | \nu_i \rangle$.
Note that $|U_{\mu}|^2$ and $|U_{e}|^2$ need not be identical.
Once produced, a neutral heavy lepton of such low mass will either
decay via $L_0 \rightarrow \nu \nu \nu$, $L_0 \rightarrow \nu e e$, or
$L_0 \rightarrow \gamma \nu$. The most viable mode for detection is
the two-electron channel. For this particular decay mode, the
$L_0$ can decay either via charged current or charged and neutral
current interactions. The branching ratio for this decay process has
been previously calculated \cite{Bolton}. Since the decay
width is proportional to $U_j^2$, the number of $L_0$'s detectable is
proportional to $U_i^2*U_j^2$ in the limit where the
distance from the source to the detector is short compared to the
lifetime of the $L_0$.
Using the above model, one can estimate the number of neutral heavy
leptons produced at the muon storage ring which later decay within a
given detector. The number of detected neutral heavy leptons is given
by the equation below:
\begin{equation}
N_{L_0} = N_{\nu}*
BR(\mu\rightarrow L_0 \nu e)*\epsilon
*e^{-L / \gamma c \tau}*
BR(L_0 \rightarrow detectable)*
(1 - e^{- \delta l / \gamma c \tau})
\end{equation}
\noindent Here $N_{\nu}$ is the number of neutrinos produced from muon
decay, $BR(\mu\rightarrow L_0 \nu e)$ is
the branching ratio of muons decaying into neutral heavy leptons
versus ordinary muon decay, $L$ is the distance from the beamline to
the detector, $\delta l$ is the length of the detector, $\epsilon$ is
the combined detector and geometric efficiency, $\tau$ is the $L_0$
lifetime, and $BR(L_0 \rightarrow detectable)$
is the branching ratio for the neutral heavy lepton decaying via a
detectable channel (presumably $L_0 \rightarrow \nu e e$).
In testing the sensitivity to $L_0$ production at the muon storage
ring, we make a few underlying assumptions. We assume that the
storage ring utilizes a pure, unpolarized muon beam with straight
sections such that 25 percent of the muons will decay to neutrinos
pointing towards the detector. We assume that the fiducial volume is 3 meters
in diameter and 30 meters in length, and that the detector has sufficient
tracking resolution to detect
the $e^+e^-$ vertex from the $L_0$ decay. We assume for now that the
background is negligible. These parameters correspond to the
fiducial volume of the decay channel used for the $L_0$ search at
E815 (NuTeV) \cite{NuTeVNHL,NuTeVQ0}.
The sensitivity of the detector has been calculated for a number of
different muon energies and beam intensities. Figure \ref{nhl_mustore} shows
limits on the $L_0$-$\nu_\mu$ mixing as a function of $L_0$ mass. One
achieves the best limits from using relatively low energy/high
intensity muon beams. This is a major improvement over previous
neutral heavy lepton searches, where limits do not reach below
$6.0*10^{-6}$ in the low mass region \cite{PDB,Shrock}.
The single event sensitivity quoted here depends greatly on having
minimal background levels in the signal region. Part of this can be
achieved by kinematic cuts which discriminate against neutrino interactions
in the detector material. However, it will probably be necessary to
reduce the amount of material in the fiducial region greatly compared
to NuTeV. We estimate that even if the
decay region is composed only of helium gas, the number of
neutrino interactions will approach a few thousand. The ideal
detector, therefore, would consist of a long vacuum or
quasi-vacuum pipe with appropriate segmentation for tracking. The
decay pipe could be used in conjunction with larger neutrino detectors
adapted for the muon storage ring.
The muon storage ring may prove to be an ideal location to continue
the search for neutral heavy leptons. The high intensity neutrino
beam allows for a neutral heavy lepton search to be sensitive to
the 10 -- 100~MeV/$c^2$ mass range. In addition, such a neutral
heavy lepton program could easily interface with a
neutrino detector which uses the same neutrino beam. It is also clear,
however, that a neutral heavy lepton search would receive the most
benefit at lower muon energies, and thus would yield best results at
the earlier stages of the muon storage ring program.
\begin{figure}
\begin{center}
\begin{turn}{90}
\mbox{\epsfysize=6.0in\epsffile{mulimits.ps}}
\end{turn}
\caption{Limits on $|U_{\mu L}|^2$ as a function of $L_0$ mass for one
year of running. The curves show sensitivities for 20 GeV and 50 GeV
muon energies. Sensitivities assume no background events in signal
region.}
\label{nhl_mustore}
\end{center}
\end{figure}
\newpage
\section*{Neutrino Magnetic Moments}
Although neutrino oscillation searches are the primary focus of the neutrino
factory study, these searches are sensitive to the mass differences between
neutrino eigenstates and not to the absolute scale of the neutrino masses.
If discovered, the magnitude of the neutrino magnetic moment (NMM) can
provide the actual neutrino eigenstate mass through a direct relationship
(Equation \ref{eq_mass}). A measurement of the NMM would have great
importance for the development of stellar models, and knowledge of the
neutrino mass would have even greater impact in the field of cosmology. Even
failing the goal of a measurement, an improved limit could constrain several
Standard Model extensions. As shall be seen, a NMM search can run
parasitically as the front--end to a typical long baseline detector.
The following subsections present the current predictions for the magnitude
of the NMM, the state of NMM searches today, and the results of a
sensitivity study for a novel phase rotation scheme that may improve
existing laboratory limits by several orders of magnitude. The final figure
provides the means to estimate the statistical power of the result for many
different neutrino factory scenarios.
\subsection*{Current Predictions and Limits}
Despite their lack of charge, a neutrino can possess a non-zero magnetic
moment that can arise through loop diagrams such as Figure \ref
{fig_feynman_nmm}. In the Standard Model, extended only insofar as the
presence of a right--handed neutrino, or such that left--handed neutrinos
are allowed to have mass, the expected \cite{shrock} neutrino magnetic
moment magnitude is given by
\begin{equation}
\mu _{\nu }\simeq 3\times 10^{-19}\;\mu _{Bohr}\cdot \left( \frac{m_{\nu }}{%
1\;\text{eV}}\right) , \label{eq_mass}
\end{equation}
where $\mu _{Bohr}$ is the Bohr magneton. Several extensions to the SM would
result in a much larger NMM; supersymmeteric models can produce $10^{-14}\mu
_{Bohr}$ to $10^{-12}\mu _{Bohr}$ \cite{frank} and calculations that invoke
large extra dimensions easily yield $10^{-11}\mu _{Bohr}$ or larger \cite{ng}%
.
Relative to the SM expectation, the excluded values of NMM\ are not at all
stringent, being seven to nine orders of magnitude larger. The current
limits on neutrino magnetic moment from laboratory experiments are $\mu
_{\nu }\leq 1.5$ to $1.8\times 10^{-10}\mu _{Bohr}$ for electron neutrinos
\cite{beacom}\cite{mu_e_limit} and $\mu _{\nu }\leq 7.4\times 10^{-10}\mu
_{Bohr}$ for muon neutrinos \cite{mu_mu_limit}. Astrophysical limits are
stronger:\ the slow rate of plasmon decay in horizontal branch stars \cite
{star1} implies $\mu _{\nu }\lesssim 10^{-11}\mu _{Bohr}$, while the
neutrino energy loss rate from supernova 1987a \cite{star2} yields $\mu
_{\nu }\lesssim $ $10^{-12}\mu _{Bohr}$. Note, however, that several
assumptions are implicit to the astrophysics limits, including the core
temperature of the stars; if stellar models omit important processes, these
limits may be overestimated. Also, the supernova limit applies only to Dirac
neutrinos and not to the Majorana case.
From the weakness of the current limits, one can infer the difficulty of
NMM\ experiments. Because it would provide a highly--collimated and
remarkably intense neutrino beam, the neutrino factory promises to improve
the reach of traditional searches as well as to provide a unique opportunity
to explore unconventional NMM search methods. The next section describes a
suggested search method that could provide this improved reach.
\subsection*{The Phase Rotation Scheme}
Existing search schemes possess a weakness that sharply limits their
ultimate sensitivity: the formulae for the hypothesized effect are quadratic
in $\mu _{\nu }$. In contrast, the following scheme is linear in terms of $%
\mu _{\nu }$ and most other parameters, hence it has a good chance of
exceeding the limits from current experiments and possibly the limits from
astrophysics calculations. The experiment relies on the principle that a
neutrino's helicity state (and therefore its magnetic moment) can be
reexpressed in terms of an orthogonal basis. Because the two new components
behave differently when exposed to a magnetic field they will no longer sum
to the original state. The result is a phase rotation from left--handed
helicity to right--handed helicity.
Consider a neutrino of energy $E$ in the presence of a magnetic field B, the
particle's energy gains a new term $\mu _{\nu }\cdot $B. Defining the B
field along $x$, with a neutrino travelling (and having helicity) along the $%
z$-direction, the helicity and magnetic moment of the neutrino may be
decomposed into $+x$ and $-x$ components, yielding a non-zero dot product
for the magnetic potential energy for each state:
\begin{equation}
\left| \uparrow \right\rangle =\frac{e^{-iEt}}{\sqrt{2}}\left| \leftarrow
\right\rangle +\frac{e^{+iEt}}{\sqrt{2}}\left| \rightarrow \right\rangle
\text{ becomes }\frac{e^{-i\left( E+V\right) t}}{\sqrt{2}}\left| \leftarrow
\right\rangle +\frac{e^{+i\left( E-V\right) t}}{\sqrt{2}}\left| \rightarrow
\right\rangle . \label{eq_split}
\end{equation}
Because the magnetic potential has opposite sign for the two energy states,
the two states can be split within the magnetic field. The splitting of the $%
+x$ and $-x$ component states is equivalent to a phase rotation of the
neutrino helicity from $\left| \uparrow \right\rangle $ toward $\left|
\downarrow \right\rangle $, i.e. the neutrino can ``rotate'' to a sterile
state in the case of a Dirac neutrino or to an antineutrino in the Majorana
case. At a far detector, the signal would be a deficit in the number of
neutrinos detected with the B field in place compared the number detected
without the B field.
If there is a force on the neutrino, then the magnetic potential energy
comes from the B field itself and the energy states are indeed split. If
there is no force on the neutrino, then there is no energy exchange with the
external field and the magnetic potential energy comes from other
contributions to the neutrino energy such that the total neutrino energy
remains unchanged. Because the magnetic force on a dipole takes the form $%
F=\nabla (\mu _{\nu }\cdot $B$)$, the presence or absence of a gradient in
the B field will determine whether the total energy changes.
In this phase rotation scheme, the neutrino is allowed to see a gradient in
the magnetic field which causes the energy states to split. Before the
neutrino leaves the influence of the field, the magnetic field energy must
be brought to zero homogeneously in space, such that there is no gradient,
no magnetic force, and the energy is therefore not returned to the field.
The energy difference between the two states then persists indefinitely as
do the resulting phase rotations. Ideally, the magnetic field would be
placed near the neutrino source for a long baseline experiment; that way,
the phase difference between energy states has time to grow and the neutrino
will rotate further toward its alternate state.
\subsubsection*{The Magnetic Field}
There are two basic requirements for the magnetic field:
\begin{itemize}
\item[1) ] The magnetic field must oscillate one quarter cycle while the
neutrinos are within the homogeneous part of the field. If the field covers
too small an area for a given oscillation frequency, the neutrino will pass
through two spatial gradients and the energy states will not remain split.
(Because timing is crucial, this study assumes bunched muons in the ring
providing bunched neutrinos with negligible longitudinal spread.)
\item[2) ] The magnetic field must be as strong as possible while still
satisfying (1). A strong B-field will maximize the resulting phase rotation
angle that occurs in the distance between the cavity and the far detector.
\end{itemize}
We have explored the possibility \cite{norbert} of using two existing
technologies for the B field, resonant cavities and kicker magnets. In both
cases the maximum magnetic field is too small to yield improved magnetic
moment limits given realistic equipment.
As an alternative, we envision many superconducting wires comprising a large
current sheet, not unlike one wall of existing ($\sim 2$ T) collider
detector solenoids. Consider a neutrino passing through an infinite current
sheet from left to right. While the neutrino is on the left, the current
ramps up from zero to maximum, resulting in a B field that is homogeneous in
all space. As the neutrino passes through the sheet, it experiences a
magnetic gradient of $2\cdot $B. As the neutrino continues to the right, the
current drops again to zero, bringing B to zero also. For an actual device,
the sheet must be large in comparison to the width of the neutrino beam and
also in comparison to the distance travelled by the neutrino in one half
cycle of the current.
Superconducting current sheets can provide much larger magnetic fields than
any other method explored to date. In addition, given shielding walls placed
at appropriate distances, several sheets in series linearly increase the
rate of phase rotation.
\subsubsection*{Sensitivity}
The maximum parameter space is explored with the magnetic field(s) in place
for approximately 50\% of the data collection period. Assuming a small
effect, this will maximize the statistical power of the live and null data
sets and result in the best chance of resolving any differences that might
result. Worth noting: the long baseline detector and intervening material
remains the same for both data sets, so most systematic uncertainties cancel
exactly in a ratio.
Assuming for simplicity the absence of neutrino oscillations except those
that arise from the current sheets, the formula for the number of events
lost to sterile states may be expressed very simply as:
\begin{equation}
N_{lost}=N*\sin \left( \mu _{\nu }\text{B}t\right) \label{eq_nlost}
\end{equation}
Figure \ref{mag_mom_signif} compares this result to the expected statistical
fluctuations in the CC event rate at a far detector. The y--axis is the
significance, simply defined as $N_{lost}/\sqrt{N}$. The x--axis displays
the total number of neutrino decays required. The lines assume 5000 CC
events expected in the long baseline detector per $10^{19}$ muon decays \cite
{geer_2day}. The diagonal lines represent contours of constant ``C'', where $%
C=\mu _{\nu }$B$t$ with the given units. For instance, with magnetic
gradients totaling 1 Tesla, long baseline distance of 3000 km, and the
entry--level neutrino factory ($10^{19}$ muon decays), a NMM of magnitude $%
10^{-11}\mu _{Bohr}$ would only appear at the one--sigma level of
significance.
Because the diagonal lines turn out to be logarithmically--spaced, this
formulation allows quick estimation of sensitivity for different parameter
scenarios. Consider the example again but with magnetic gradients totalling
3 Tesla; this change essentially results in a new line with $C=3.16$ and a
3--sigma limit. As can be seen from the figure, the expected sensitivity may
be improved by a rough factor of ten for every: factor of ten in magnetic
field strength, factor of ten in number of current sheets, factor of ten in
distance to the detector, and factor of 100 in muon decays.
\subsection*{Summary}
This section has compared the current best neutrino magnetic moment limits
to what might be accomplished at a neutrino factory. This study has
identified a novel neutrino magnetic moment search technique that is both
feasible and has the potential to provide higher sensitivity than prior
methods. The method relies on the interaction of the neutrino magnetic
moment and an oscillating magnetic field to rotate the neutrino helicity.
The strength of the proposed apparatus is that it could run completely
transparently as the front--end to a long baseline neutrino oscillation
detector. Although the SM prediction for the neutrino magnetic moment lies
out of reach even for this type of experiment, such a search at a neutrino
factory could constrain several Standard Model extensions that predict
larger values for the magnetic moment.
\begin{figure}
\begin{center}
\mbox{\epsfysize=2.0in\epsffile{mag_mom2.eps}}
\caption{One possible loop diagram that results in a non--zero neutrino
magnetic moment.}
\label{fig_feynamnn_nmm}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\mbox{\epsfysize=2.0in\epsfxsize=6.0in\epsfbox{mag_mom_signif.eps}}
\caption{Significance for several scenarios.}
\label{mag_mom_signif}
\end{center}
\end{figure}
\newpage
%Fritz's section 2
\section*{Anomolous Lepton Production}
The neutrino beam from a muon storage ring would consist of a virtually pure
combination of $\bar{\nu}_e$ and $\nu_\mu$ (or charge-conjugate). Also, at the
source of the neutrino beam, the $\bar{\nu}_e$ and $\nu_\mu$ will not have
had time to oscillate into other flavors: For a
20 GeV muon storage ring with a
700 m straight section, and neutrino oscillations with
$\Delta m^2 \ = \ 3.5 \times 10^{-3} \ \rm eV^2$, the oscillation probability
is $\approx 5 \times 10^{-9}$. Furthermore, the neutrino flux is highest
at the source. Thus, an experiment at the neutrino source could be highly
sensitive to exotic processes resulting in production of $e^-$, $\mu^+$,
or $\tau$ of either charge.
While such a search is interesting in its own right, it is also useful
to rule out exotic contributions to long-baseline neutrino oscillation
signals.
These exotic processes would probably have a flat or rising dependence on
the neutrino energy $E_\nu$. In contrast, a neutrino oscillation would
have a $1/E_\nu^2$ dependence. Also, if the distance $L$ of the experiment
changes, the rate of exotic events would decrease with the flux as $1/L^2$.
In contrast, the neutrino oscillation probability would increase as $L^2$
(for $L$ small compared to the oscillation period),
and so the rate of oscillated events would be independent of $L$. Thus,
one could distinguish between exotic processes and the beginning of a
neutrino oscillation.
Anomalous lepton production could occur if muons decay to neutrino flavors
other than those in the usual decay $\mu \to e \bar{\nu}_e \nu_\mu$, and the
anomalous neutrinos then interact in the target. Alternatively, they could
be produced if a $\bar{\nu}_e$ or $\nu_\mu$ interacts with the target
via an exotic process.
The only direct experimental limit on exotic
$\mu \to e \bar{\nu}_x \nu_y$ decays is
$BR(\mu \to e \bar{\nu}_\mu \nu_e) < 1.3\%$\cite{PDG}. Indirect limits are
also very weak. The contribution of non- $V-A$ interactions to the muon
decay rate has been limited to 8\%\cite{PDG}.
Also, the total muon decay rate is one of the main measurements used to
constrain electroweak parameters\cite{PDG}. To first order,
\begin{equation}
\frac{1}{\tau_\mu} = \frac{G_F m_\mu^5}{192\pi^3} .
\end{equation}
Assuming the standard model, $G_F$ is determined to 1 part in $10^5$
from muon lifetime measurements. If there are exotic contributions to
the muon lifetime, the measured value of $G_F$ would be shifted from
the true value.
Since
\begin{equation}
m_W \propto G_F^{-1/2} ,
\end{equation}
the 0.1\%
uncertainty on $m_W$ corresponds to a 0.4\%
shift in the muon lifetime.
Finally, the CKM matrix element $V_{ud}$ is determined from the rate of
nuclear $\beta$-decays relative to the muon lifetime. The assumption
of unitarity on the CKM matrix gives us the following constraint on
the first row:
\begin{equation}
|V_{ud}|^2 + |V_{us}|^2 + |V_{ud}|^2 = 1 .
\end{equation}
The experimental determination is\cite{PDG}:
\begin{equation}
|V_{ud}|^2 + |V_{us}|^2 + |V_{ud}|^2 = 0.991 \pm 0.005 .
\end{equation}
The uncertainty on this constraint corresponds to a 0.5\%
shift in the muon lifetime. Additional contributions to the muon
decay rate would lead to a downward shift in
the determined value of $|V_{ud}|^2$ from the true
value. We conclude that exotic decay modes of the muon with branching
ratios totaling 0.5\%
are possible without contradicting current measurements or tests of the
standard model.
As a concrete example of such an exotic process we consider
R-parity-violating supersymmetric models. These models lead to
lepton-number-violating vertices with couplings $\lambda$, and muon
decay processes such as $\mu \to e \bar{\nu_\tau} \nu_\tau$ as shown
in Fig.~1. The matrix element for these decays turns out to have
the same form as for the standard W-exchange. The current constraints
on the couplings $\lambda$ are reviewed in Ref.~\cite{dreiner}.
These constraints allow a branching ratio of 0.4\%
for the process in Fig.~\ref{exotics:decay}.
Similar processes are allowed for anomalous
lepton production as shown for example in
Fig.~\ref{exotics:decay}.
Estimates for allowed rates are in progress~\cite{quigg}.
These diagrams
involve the $\lambda '$ couplings. Currently, the best limit on one of
these couplings,
$\lambda'_{231}$, is from $\nu_\mu$ deep-inelastic scattering, so existing
neutrino data is already providing constraints!
The search for these types of effects at the muon storage ring could
be input into a decision on whether to build a muon-proton collider
where they could be studied in more detail.
As a start on estimating the capabilities of an experiment at the
neutrino source, we consider the detector concept illustrated in
Fig.~\ref{exotics:detector}.
This concept consists of a repeating sequence of 1.5 mm-thick Tungsten
sheets with Silicon tracking, separated by
4 mm. Tungsten, being dense, provides a high
target mass while being thin enough for a
produced $\tau$ to have a high probability
of hitting the Silicon. The impact parameter of the $\tau$ decay products
is typically 90 microns with a broad distribution, so we would like a
hit resolution of 5 microns or better. Although there is a lot of
multiple scattering in the tungsten, the short extrapolation distance
provides for a good impact parameter resolution on the $\tau$ decay
products. This configuration has been optimized for a 50 GeV muon beam.
For lower energy beams, the planes should be spaced more closely, and the
Tungsten thickness perhaps reduced.
Studies of detectors with passive target mass and tracking with
emulsion sheets~\cite{emulsion} suggest that we can expect $\tau$
reconstruction efficiencies as high as 30\%.
We would propose placing such a detector in a magnetic field, and
measuring the momentum of muons and hadrons should be straightforward.
However, each Tungsten sheet is 0.4 radiations lengths thick, and while
we should obtain good energy resolution for electromagnetic showers,
determining the electron charge will be difficult. We estimate it would
take roughly a 4 Tesla field transverse to the beam direction to have
a good chance of measuring the charge of a 50 GeV electron. Even then,
detailed studies are needed to determine if this can be done reliably.
A total mass of 6 tons of Tungsten corresponds to 200 $\rm m^2$ of Silicon
tracking. For $5\times 10^{20}$ muon decays at 50 GeV, we expect a total
of 35 billion neutrino interactions. Obviously there is much potential for
sensitivity to very rare exotic processes, but detailed simulations and
studies of possible Silicon tracking technologies are needed to quantify
this.
\begin{figure}[h]
\begin{center}
\mbox{\epsfxsize=2.8in\epsffile{exotic_decay.eps}}
\mbox{\epsfxsize=2.8in\epsffile{exotic_interaction.eps}}
\end{center}
\caption{Example of exotic muon decay in R-parity-violating SUSY (Left),
and an example of an exotic neutrino interaction in R-parity-violating
SUSY (Right).}
\label{exotics:decay}
\end{figure}
%\begin{figure}
%\epsfysize=1.5in
%\centerline{
%\epsffile{exotic_decay.eps}}
%\caption{Example of exotic muon decay in R-parity-violating SUSY.}
%\label{exotics:decay}
%\end{figure}
%\begin{figure}
%\epsfysize=1.5in
%\centerline{
%\epsffile{exotic_interaction.eps}}
%\caption{Example of an exotic neutrino interaction in R-parity-violating SUSY.}
%\label{exotics:interaction}
%\end{figure}
\begin{figure}
\epsfysize=1.5in
\centerline{
\epsffile{exotic_detector.eps}}
\caption{One plane of a detector for $\tau$ production.}
\label{exotics:detector}
\end{figure}
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\end{document}