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\input hep_macro
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\begin{document}
%Joe and Eric's section ***************************************************
\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}
\mbox{\epsfxsize=6.0in\epsffile{mulimits.eps}}
\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
%John's section ***********************************************************
\section*{Neutrino Magnetic Moments}
Although neutrino oscillation searches focus on the mass differences between
neutrino eigenstates, neutrinos can possess other observable properties. If
discovered, the magnitude of the neutrino magnetic moment (NMM) would
provide the Standard Model 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.
Despite their lack of charge, a neutrino can possess a non-zero magnetic
moment that can arise through loop diagrams. 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) ,
\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.
Existing search schemes possess a weakness that sharply limits their
ultimate sensitivity: the formulae for the hypothesized effect are quadratic
in $\mu _{\nu }$ but linear in terms of the experimenter--controlled
parameters. In contrast, the following scheme is quadratic in terms of the
product of the NMM$\;$and a magnetic field strength, $\mu _{\nu }\cdot $B;
hence it has a chance of exceeding the limits from current experiments and
possibly the limits from astrophysics calculations.
The energy $E$ of a neutrino in a magnetic field B gains a new term $\mu
_{\nu }\cdot $B. Consider a B field along the $x$-axis, and a neutrino with
momentum and helicity along the $z$-axis at $t=0$. The eigenstates of the
neutrino are projections along the $x$-axis, and the state of the neutrino
is expressed as:
\begin{equation}
\left| \uparrow \right\rangle =\frac{e^{-i\left( E+\mu _{\nu }B\right) t}}{%
\sqrt{2}}\left| \leftarrow \right\rangle +\frac{e^{-i\left( E-\mu _{\nu
}B\right) t}}{\sqrt{2}}\left| \rightarrow \right\rangle . \label{eq_split}
\end{equation}
As the neutrino propagates, the relative phase of the two components
changes, corresponding to a rotation 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.
In this phase rotation scheme, the energy splitting occurs as the neutrino
passes through a field gradient and experiences a force $F=\nabla (\mu _{\nu
}\cdot $B$)$. To preserve the energy difference, so that the phase
difference can continue to accumulate on the way to the far detector, the
field must be turned off instead of allowing the neutrino to experience the
reverse gradient as it exits the field region.
There are thus two basic requirements for the magnetic field:
\begin{itemize}
\item[1) ] The magnetic field must oscillate such that the neutrino
experiences only one sign of the gradient. This study assumes that the
neutrino bunch length is small compared to the oscillation length. If this
assumption is not true, the effects discussed here will be diluted but the
basic conclusions will still apply.
\item[2) ] The magnetic field must be as strong as possible.
\end{itemize}
We have explored the possibility \cite{norbert} of using two existing
technologies for the B field: resonant cavities and kicker magnets. In both
cases the maximum magnetic field is too small to yield improved magnetic
moment limits given realistic equipment.
As an alternative, we envision many superconducting wires comprising a large
current sheet, not unlike one wall of existing ($\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
space. As the neutrino passes through the sheet, it experiences a magnetic
gradient of $2\cdot $B. As the neutrino continues to the right, the current
drops again to zero, bringing B to zero also. Placing sheets in series
linearly increases the rate of phase rotation.
The formula for the number of events lost to sterile states may be expressed
very simply as:
\begin{equation}
N_{lost}=N*\sin ^{2}\left( \mu _{\nu }\text{B}t\right) \label{eq_nlost}
\end{equation}
We note that, in contrast to an oscillation disappearence signal, this
effect is independant of energy. Figure \ref{mag_mom_signif} compares this
result to the expected statistical fluctuations in the number of CC events ($%
N$) observed in a 50 kton \cite{geer_2day} far detector. We see that for ten
1 T field gradients and $10^{19}$ muon decays we expect a
greater--than--ten--sigma sensitivity to a NMM of $10^{-11}\mu _{Bohr}$.
With only three 1 T gradients, the limit drops below two sigma. The
sensitivity can be increased very strongly by increasing the field strength
and having many gradient stages, and also by increasing the number of events
in the far detector by increasing the number of muon decays or the detector
mass. Because the detector distance deterimines both $t$ and $N$ in Eqn. \ref
{eq_nlost}, the ``significance'' in the figure is linear in terms of
distance.
To conclude, we have discussed a novel neutrino magnetic moment search
technique that uses oscillating magnetic fields at the source of a long
baseline detector's neutrino beam. This is the only technique we know of
that is quadratic in both $\mu _{\nu }$ and another parameter, and thus has
the potential for arbitrary sensitivity as we improve our ability to create
oscillating magnetic field gradients.
\begin{figure}
\begin{center}
\mbox{\epsfxsize=4.5in\epsfbox{mag_mom_signif.eps}}
\caption{Significance for several scenarios.}
\label{mag_mom_signif}
\end{center}
\end{figure}
\newpage
%Fritz's section *********************************************************
\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}
\newpage
\section*{References}
\begin{references}
%NHL ref
\bibitem{GLR}
M.Gronau, C.N. Leung, and J.L. Rosner, \prev{D29}{2539}{1984}
%
\bibitem{Shrock}
R. E. Shrock, \prev{D24}{1232}{1981}
%
\bibitem{Bolton}
T. Bolton, L. Johnson, and D. McKay, Phys. Rev. D {\bf 56} (1997) 2970.
%
\bibitem{NuTeVNHL}
A. Vaitaitis {\em et al.}, Phys. Rev. Lett. {\bf 83} (1999) 4943.
%
\bibitem{NuTeVQ0}
J. Formaggio {\em et al.}, Phys. Rev. Lett. {\bf XXX} (2000) XXX.
%
\bibitem{PDB}
{\it Review of Particle Physics} Eur. Phys. J. C. 3,320 (1998).
%
\bibitem{Shrock}
R. Shrock, Phys. Rev. D {\bf 24} (1981) 1232.
%John Krane's ref
\bibitem{shrock} B.W.Lee and R.E.Shrock, Phys. Rev. {\bf D16} 1444 (1977).
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\bibitem{norbert} Private discussions with Norbert Holtkamp, Fermilab.
\bibitem{geer_2day} See Steve Geer's talk at the 2-day meeting. Number of
events (5000) was quoted for a 50 kTon detector at L=2800 km and a 20 GeV
muon storage ring.
% Fritz's ref
\bibitem{PDG} Review of Particle Properties, C.~Caso {\it et. al.},
Euro. Phys. J. {\bf C}3, 1 (1998).
\bibitem{dreiner} H. Dreiner, hep-ph/9707435
\bibitem{quigg} Chris Quigg, private communication
\bibitem{emulsion}
K. Kodama {\it et. al.} (OPERA Collaboration), CERN/SPSC 98-25. \\
A. E. Asratyan {\it et. al.}, hep-ex/0002019
\end{references}
\end{document}