\newcommand{\ignore}[1]{}
\section*{Introduction}
{\it version from Eric Hawker Mar 14, 2000}\\
{\it small mods by Mayda Velasco Mar 15, 2000}\\
{\it rewrite of DIS sections, Heidi Schellman, Mar 16, 2000}\\
{\it rewrite the introduction, structure fuctions, ew and charm, Mar 21, 2000\\}
Due to the theoretically clean nature of weak interactions, conventional
neutrino scattering experiments have always provided very precise
measurements of fundamental parameters. These include
measurements of the Weinberg angle and the strong coupling constant
$\alpha_s$ which are competitive with any other methods. Perhaps because
of this success, we forget how crude existing neutrino experiments are.
The high statistics experiments such as CDHS and CCFR/NuTeV, in order
to obtain statistical samples of more that 10$^5$ events, rely on
coarsely segmented massive iron/scintillator calorimeters weighing
close to 1000 tons. Measurements on proton targets and detailed studies
of the final state have been confined to very low statistics bubble chamber
and electronic detectors. As a result we have virtually no precise measurements
of neutrino-proton scattering and no measurements on polarized targets
which offer totally new insights into the spin structure of the nucleon.
The advent of a high flux neutrino factory, with neutrino fluxes of
10$^{20}$/year instead of the 10$^{15-16}$ at existing facilities would
open a new era in conventional neutrino physics. We would be able
to use low mass targets and high resolution detection technologies and
still achieve better statistical power than present-day experiments.
For example a 50 GeV muon storage ring would produce around 400 M events
neutrino charged current interactions per year in a cubic meter of
water,
5-10 times the statistics of the CCFR/NuTeV experiments with 600 ton detectors
Better understanding of neutrino fluxes from the decay of monochromatic
muons will also reduce many of the dominant systematic errors.
In this study we have concentrated on new measurements only possible
with higher fluxes rather than repeating older measurements with 1000
times the statistics. As a result, the statistical errors shown are
often not negligible, but without the high flux at a neutrino factor the
measurements themselves would
be impossible.
\subsection*{Outline}
Due to the breadth of the field we were unable to give a complete
survey and have instead highlighted a few
of the areas where the high flux beam at a neutrino factory allows
new measurements.
\begin{itemize}
\item A description of a low mass target/detector and typical rates in such a detector.
\item Conventional deep inelastic scattering measurements and a proposed
detector design.
\item Measurements of neutrino cross sections on nuclear targets, a topic
of great interest to the nuclear physics community and also needed to
understand normalization at a far neutrino oscillation detector.
\item Spin structure functions, which have never been measured in neutrino
beams.
\item The potential of the neutrino factory as a clean source
of single tagged charm mesons and baryons.
\item Electroweak measurements in both the hadronic and purely leptonic sectors.
\item Use of the very clean initial state to search for exotic interactions
\item Searches for anomalous neutrino interactions with electromagnetic fields.
\end{itemize}
%\subsection*{Structure Function Measurements}
\subsection*{Possible detector configurations and statistics}
For studies of charged current deep-inelastic scattering on proton
targets, the optimal detector system is probably a target followed by
precision magnetic tracking sytems, an electromagnetic calorimeter
and a muon detection system. Such detectors have been used in
muon scattering experiments at CERN and FNAL and in the new generation
neutrino scattering experiments CHORUS and NOMAD. A low mass target
followed by tracking and electromagnetic calorimetry makes the
electron anti-neutrinos in the beam a source of additional statistics
rather than a backround, except in the case of neutral current studies.
The numerical estimates in this study use, unless otherwise noted,
$10^20$ 50 GeV muon decays in a 600 m straight section.
The target itself should be thin enough that particles produced within
it have a small probability of interacting before they reach the tracking
systems.
In this study we considered liquid hydrogen and deuterium targets -
both polarized and unpolarized - and heavier solid nuclear targets.
The hydrogen and deuterium targets are 1m long while the polarized target
is 50 cm long. All targets
are 20 cm in radius, to fit the central beam spot at 50 GeV.
Nuclear targets are scaled so that the interaction length
in the material is constant at 14\%. The charge current muon neutrino
interaction rates are summarized in table \ref{rates}.
\begin{table}[b]
\caption{\label{rates} Charged current muon-neutrino scattering
rates in a small target located near a muon storage ring. Rates
are per $10^{20}$ muon decays. The detector is located ($E_{\mu}$, GeV)
meters away from the ring to assure that primary muons have ranged out
before the detector.}\begin{center}
\begin{tabular}{|c|c|r|r|}
\hline
Machine& Target & Thickness,cm & Events \\
\hline
50 GeV neutrino factory &Liquid H$_2$& 100 & 12.1M\\
&Liquid D$_2$& 100 & 29.0M\\
&solid HD & 50 &9.3M\\
&C&5.3&20.7M\\
&Si&6.3&25.4M\\
&Fe&2.3&31.6M\\
&Sn&3.1&39.1M\\
&W&1.3&44.3M\\
&Pb&2.4&46.5M\\
\hline
CCFR/NuTeV&Fe& 600& $\sim$ 2M\\
\end{tabular}\end{center}
\end{table}
%
%
\begin{figure}
%\epsfysize=2.5in
\epsfxsize=3.0in
\centerline{
\epsffile{diskin.eps}}
\caption{Kinematics of neutrino scattering in the parton model.
The energy-momentum tranfer from the leptons to the proton
is $\fv{q}$ and the fraction of the proton momentum carried
by the struck quark is approximately $x$.
\label{fig:comparejlab}}
\end{figure}
These are the total event rates for muon-neutrino scattering. The anti-neutrino
rates are half as large. Kinematic cuts reduce the statistics
by less than a factor of two.
We have only considered muon-neutrino charge current scattering for
structure function measurements, although for such thin targets, electron
neutrino scatters should also be reconstructable with high precision.
\subsection*{Neutrino Scattering Kinematics}
\newcommand{\Elep}[0]{E_{\lepton}}
The kinematic variables for neutrino deep inelastic scattering are
illustrated in figure \ref{diskin}:
\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 \Elep - \Elep^{\prime},\\
x &=&Q^2/2 \mtarget \nu,\\
y &=& \mtarget \nu/(\fvk{\nu} \fv{\quark}) = (1 + cos\theta_{CM})/2 \approx
\nu/\Elep,\\
W^2 &=& 2 \mtarget \nu + \mtarget^2 -Q^2,
\end{eqnarray}
where the $\fvk{}$ are the neutrino and final state four vectors,
$\fv{p}$ is the proton four-vector, $M$ 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.
\begin{figure}
%\epsfysize=2.5in
\epsfxsize=5.0in
\centerline{
\epsffile{10_comparejlab.eps}}
\caption{Comparison of kinematic ranges for present DIS experiments
with a 50 GeV Neutrino factory. }
\label{fig:comparejlab}
\end{figure}
Fig. \ref{fig:comparejlab} show the kinematic region for a neutrino
factory as compared to other deep-inelastic scattering experiments.
%\end{document}
%\subsubsection{Unpolarized Neutrino Scattering}\label{DIScross}
% The kinematic variables are defined in section \ref{DIS}.
For $Q << E$ and $s << M_W$ the the unpolarized neutrino(anti-neutrino) scattering cross section
is:
\begin{eqnarray}
{d\sigma^{\nu(\antinu)}\over dx dy} =
{ G_F^2 M E_{\nu} \over 2\pi }\biggr[ [F_2(x,Q^2) \pm xF_3] + [F_2(x,Q^2)\mp xF_3(x,Q^2)] (1-y)^2 \\\nonumber
- 2 y^2 F_L(x,Q^2).
\end{eqnarray}
where the $F_i$ are
Structure Functions. $F_L = F_2 - 2xF_1 $ is a purely longitudinal
structure function.
$F_3$ changes sign for anti-neutrino scattering.
There are additional structure functions $F_4$ and $F_5$ which are
suppressed by factors of the lepton mass squared. For $\nutau$ and $\numu$
scattering at very low energies, these terms can become
quite important.
%\subsection{Polarized Scattering}
If the target is longitudinally polarized with respect to the
neutrino polarization, then the cross section difference:
\begin{eqnarray}
{d^2(\sigma_{\Rightarrow}^{\leftarrow}
-\sigma_{\Leftarrow}^{\leftarrow})^{\nu(\bar\nu) } \over dxdy} =
{G_F^2 M E_{\nu} \over \pi }
\bigl\{\pm
y(1-{y \over 2}-{xyM \over 2E})xg_1
\mp
{x^2yM \over E}g_2
+y^2x(1+{xM \over E})
g_3\\\nonumber
+(1-y-{xyM \over 2E})
[(1+{xM \over E})g_4+g_5]
\bigr\},
\label{pol_lon}
\end{eqnarray}
\noindent
is described by two parity conserving Polarized Structure
Functions $g_1$ and $g_2$, and by three parity
violating Polarized Structure Functions $g_3, g_4$ and $g_5$.
However, if the nucleon is transversely polarized:
\begin{eqnarray}
{d^2(\sigma_{\Uparrow}^{\leftarrow}
-\sigma_{\Downarrow}^{\leftarrow})^{\nu(\bar\nu) } \over dxdy} =
\frac{ G_F^2 M}{ 16 \pi^2 }
\sqrt{xyM \left[ 2(1-y)E-xyM \right] }
\bigr\{\mp2xy({y\over 2}g_1+g_2)\\\nonumber
+xy^2g_3
+ (1-y-{xyM \over 2E})g_4
-{y\over 2}g_5\bigr\} \/.
\label{pol_tra}
\end{eqnarray}
%{\it is there a factor of 2 missing for the subtraction here}
\noindent The transverse cross section is suppressed by
${M/Q}$ with respect to the longitudinal cross section.
\subsection{Structure function measurements at a neutrino factory}
In principle, the structure functions can be extracted by fits to the $y$ dependence of the
cross section. To date this
has proven very difficult as the data must be binned in $x$, $y$ and $Q^2$ and
no experiment has had sufficient statistics to perform such an analysis with high accuracy\cite{VallageThesis}.
Instead, high statistics experiments such as CHARM, CCFR, CDHSW have relied on massive targets (Iron, Calcium) which are
approximately iso-scalar and have combined neutrino and anti-neutrino information in
order to extract average structure functions.
The structure functions averages have leading-order parton model interpretations:
%
\begin{eqnarray}
\overline{F}_2^N(x,Q^2) &\simeq &\sum (x\quark(x,Q^2) + x\antiquark(x,Q^2)),\\
\overline{F}_3^N(x,Q^2) &\simeq &\sum(x\quark(x,Q^2) - x\antiquark(x,Q^2)),\\\nonumber
\end{eqnarray}
%
where $\overline{F}_2(x,Q^2)$ and $\overline{F}_3(x,Q^2) $ are the average of neutrino
and antineutrino structure functions measured on a target which is an average of neutron and proton and $\quark(x,Q^2)$ and $\antiquark(x,Q^2)$ represent the
parton distribution functions or total probability of finding a quark
or antiquark in the proton:
\begin{eqnarray}
\quark(x,Q^2) &=& \uquark(x,Q^2) + \dquark(x,Q^2)+\squark(x,Q^2)+\cquark(x,Q^2)...\\
\antiquark(x,Q^2) &=& \antiuquark(x,Q^2) + \antidquark(x,Q^2)+\antisquark(x,Q^2)+\anticquark(x,Q^2)...\\\nonumber
\end{eqnarray}
%\subsubsection{Measurements with high statistics}
Given the expectation of 12 M events/year in a 1 m hydrogen or deuterium target at a 50 GeV
muon collider we can do a complete analysis of each channel $\nu p, \nu n, \antinu p,
\antinu n$ without averaging. Such an analysis allows a unique extraction of individual quark flavor parton
distribution functions.
Consider $\nu$p scattering, a $W^{+}$ is exchanged and the reaction can only be
sensitive to negatively charged quarks. Due to the helicity dependence of the interaction
only left-handed $\dquark$ type and right handed $\antiuquark$ quarks will be involved.
The leading order parton model cross section is simply
\begin{eqnarray}
{d\sigma^{\nu p}\over dx dy} &\simeq& {4 G_F^2 M E_{\nu}\over \pi }x [(\dquark_L (x,Q^2) + \squark_L(x,Q^2)) +(\antiuquark_R(x,Q^2) +\\
&&\hskip 2 in \anticquark_R(x,Q^2))(1-y)^2],\\\nonumber
\label{partonformula}\end{eqnarray}
%
and the different contributions can be extracted from the $y$ dependence of this
cross section and the corresponding anti-neutrino cross section. The relative
$\squark$ and $\dquark$ contributions can be measured in charm production.
For an unpolarized target
$\quark_L(x) = \quark_R(x) = \onehalf \quark(x)$. For
a polarized quark $\quark_L(x) = \onehalf (\quark(x) + \delta\quark(x))$
and $\quark_R(x) = \onehalf(\quark(x) - \delta\quark(x))$ where
$\delta\quark(x)$ is the degree to which the quark spin
is aligned with the proton spin\footnote{ The traditional $\Delta q$ spin distributions from
electron and muon scattering measure the sum $\Delta \quark = \delta \quark + \delta \antiquark$ as
photon probes cannot tell quarks and anti-quarks apart.}.
Thus a $\sigma_{\nu p}$ measurement on an unpolarized target can determine $\dquark+\squark$ and $\antiuquark +
\anticquark$ by averaging over the proton spin, while by measuring the polarization
asymmetry one can measure $\delta \dquark+\delta\squark$ and $\delta\antiuquark +
\delta\anticquark$.
Scattering on neutrons can be related to scattering on protons by an isospin
transformation which exchanges $\uquark$ and $\dquark$ quarks and anti-quarks.
Differences of neutron and proton cross sections can then be used to cancel the
$\uquark$ and $\dquark$ components leaving observables
sensitive only to $\squark$ and $\cquark$
distributions.
\subsection{Perturbative QCD}
Neutrinos do not couple directly to gluons. As a result, QCD effects
appear in neutrino scattering as well-understood higher order corrections to
the leading order parton model. Measurements
of the $\qsq$ dependence of neutrino cross sections are one of the most
sensitive measurements of the strong coupling constant $\alpha_s$ and
some information on the gluon distribution can be obtained from its coupling
to the structure functions via the DGLAP evolution equations.
The neutrino structure functions can be divided into two types, singlet
and non-singlet, depending on their sensitivity to gluon effects in
their evolution.
The structure functions $2x F_1$, $F_2$ and $g_1$ are singlet functions
and are directly coupled to the gluon distribution via the evolution equations.
The structure functions $x F_3 + x F_3$, $2x g_3,g_4$ and $g_5$
averaged over neutrino and anti-neutrino are non-singlet
functions and their evolution is independent of the gluon distribution.
The combinations $F_2^p - F_2^n$ and $g_1^p - g_1^n$ also cancel the
gluon contributions and are thus non-singlet in nature.
To date, extractions of $\alpha_s$ from non-singlet distributions have
been statistics limited and strongly affected by flux uncertainties.
The additional factor of 10-100 in statistics
and improved flux understanding
at a neutrino factory should allow vastly improved measurements of strong
interaction parameters in this very clean channel.
Once the quark distributions and strong interaction effects have been thoroughly studied in the non-singlet structure function, that knowledge can be used for
improved constraints on the gluon distributions via the evolution of the
singlet structure functions.
%Jorge's section
\subsection{Nuclear Effects}
Experiments at a neutrino factory of nuclear effects in the distribution of partons within nuclei relative
to protons and deuterons are
of interest to both the nuclear and high energy communities.
These nuclear effects have been studied extensively using muon
and electron beams but have only been glanced at in low-statistics bubble
chamber experiments using neutrinos. If we consider the behavior of the
structure functions $F_{2}(x,Q^{2})$ measured on a nucleus (A) to
$F_{2}(x,Q^{2})$ measured on a nucleon as a function of $x$ we pass
through four distinct regions in going from $x$ = 0 to $x$ =
1.0:
\subsubsection*{Shadowing Region $x < 0.1$}
In the shadowing region ($x < 0.1$) there are several
effects that should yield a different ratio $R_{A}\equiv F_{2(A)}/F_{2(N)}$
when using neutrinos as
the probe. In the limit $Q^{2} \gt 0$, the vector current is conserved and
goes to 0, however the axial-vector part of the weak current is only
partially conserved (PCAC) and $F_{2}(x,Q^{2}) \rightarrow$ a non-zero constant as
$Q^{2} \gt 0$. 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 boson into a quark/antiquark 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 $xF_{3}$. As can be seen,
the predicted difference between the shadowing on sea and valence quarks is
clearly visible down to $x$ = 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 with
14 M events on each target. }
\label{fig:shadow}
\end{figure}
\subsection*{Anti-shadowing Region (0.1 < x < 0.2) }
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 $x = 0.1$. With 14 M events on each target
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.
\subsubsection*{EMC-effect Region($ 0.2 < x < 0.7$)}
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 $x$. However, Eskola's predictions for this region indicate
that the contribution of $\overline{u}$ and $\overline{d}$ to $R^{(2)}_{A}$ 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 $x \rightarrow 1$ 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 $x$. Few-nucleon-correlation
models and multi-quark cluster models allow quarks to have a higher
momentum which translates into a high-$x$ tail. In this region
$F_{2}(x,Q^{2})$ should behave as $e^{-ax}$. 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~$\pm$~0.5 while the neutrino
experiment finds a = 8.3~$\pm$~0.7~$\pm$~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?
\subsubsection*{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 $x$ range.
The nuclear community should be informed of what a valuable tool for
nuclear research awaits them in the neutrino factory facility.
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\ba}{\begin{array}}
\newcommand{\ea}{\end{array}}
\newcommand{\bmat}{\left(\ba}
\newcommand{\emat}{\ea\right)}
\def\3{\ss}
\def\p{p\llap{/}}
\def\d{\delta}
\def\ga{\gamma}
\def\Ga{\Gamma}
\def\s{s\llap{/}}
\def\k{k\llap{/}}
\def\g5{\gamma_5}
\def\mn{\mu\nu}
\def\rs{\rho\sigma}
\def\b{\beta}
\def\a{\alpha}
\def\ve{\varepsilon}
\def\r{\rho}
\def\si{\sigma}
\def\as2{\alpha^2_s}
\def\ha{{1\over 2}}
\def\pa{\partial}
\def\du{\delta u}
\def\GeV{{\rm GeV}}
\def\Pon{P^{(0)n}}
\def\hPon{\hat P^{(0)n}}
\def\Q2{(Q^2_0)}
\def\zweib{\frac{2}{\beta_0}}
\def\vph{\varphi}
\def\nspm{NS\pm}
\def\gen{\gamma^{(1)n}}
\def\aspi{\frac{\a_s}{2\pi}}
\def\Pen{P^{(1)n}}
\def\hPen{\hat P^{(1)n}}
\def\tolimit_#1{\mathrel{\mathop{\longrightarrow}\limits_{#1}}}
\def\tosim_#1{\mathrel{\mathop{\thicksim}\limits_{#1}}}
\subsection{Spin Structure}%Mayda's section
An intense neutrino beam at a neutrino factory would create significant
event rates in very compact detectors. This opens the possibility of
using a polarized target, and hence a completely new class of neutrino
measurements becomes possible. At present we know
very little about the spin structure functions $g_1^\nu - g_5^\nu$
introduced in Eq.s \ref{pol_lon} and \ref{pol_tran}. In particular,
the parity violating functions have only been explored via weak-interference
measurements of proton form factors by the SAMPLE collaboration
\cite{SAMPLE} with very low statistics. A neutrino factory
would allow direct high-statistics measurements of all of these
structure functions and should be able to answer many
unresolved questions about the spin structure of the nucleon.
\ignore{
In the naive parton model,
\begin{eqnarray}
g_1^{\nu p}(x,Q^2)&=&\delta \dquark(x,Q^2) +\delta \squark(x,Q^2) + \delta
\antiuquark(x,Q^2) + \delta \anticquark(x,Q^2),\\
g_1^{\bar\nu p}(x,Q^2)&=&\delta \uquark(x,Q^2) +\delta \cquark(x,Q^2) + \delta
\antidquark(x,Q^2) + \delta \antisquark(x,Q^2).
\label{pol_g1}
\end{eqnarray}
\nonindent Note that $g_2$ has a a twist--2
($g_2^{WW}$) and a twist--3 ($\bar g_2$) contribution and has no
simple parton model interpretation,
\begin{eqnarray}
g_2&=&g_2^{WW}+\bar g_2\\
g_2^{WW}(x,Q^2)&=&-g_1(x,Q^2)+ \int_x^1 {dy
\over y}g_1(y,Q^2).
\label{pol_g2}
\end{eqnarray}
For $g_3$ and $g_4+g_5$ the parton model predictions are:
}
The nucleon spin ($\frac{1}{2}$)
can decomposed in terms of quark and gluon contributions:
%
\begin{equation}
\frac{1}{2}= \frac{1}{2}\Delta\Sigma + \Delta g + L_q + L_g,
\end{equation}
%
where $\Delta\Sigma \equiv \Delta u+ \Delta d+ \Delta s+\Delta c $
is the net quark helicity and
$\Delta g$ is the net gluon helicity along the nucleon
spin direction, while $L_i$ are their relative orbital angular
momentum.( We use $\Delta \quark$ as a shorthand for the integral $\int \Delta \quark(x) dx$.)
To date, the only experiments which have studied the
spin structure of the nucleon are low energy charged
lepton polarized deep-inelastic scattering experiments (PDIS) where only
the parity conserving polarized structure functions $g_1^l$ and $g_2^l$
can be measured.
$g_1^\lepton$ can be written in the leading order parton model as a sum of a nonsinglet and singlet part:
\begin{eqnarray}
g_1^\lepton(x,Q^2) &=& g_{1,NS}^\lepton(x,\qsq) + g_{1,S}^\lepton(x,\qsq) \\
&=&\onehalf \sum (e_i^2 - )\Delta \quark_i(x,Q^2) +
\onehalf \sum \Delta \quark_i(x,Q^2)\\
\end{eqnarray}
The first non-singlet term evolves independently of the gluonic spin contribution while the second is coupled to, and thus depends on the gluon spin
contribution $\Delta g$.
The integral structure functions have the following relation
to the parton spin contributions:
\begin{eqnarray}
\Gamma_1^{\lepton}(Q^2)& =& \int dx g_1^{l p/n} (x,Q^2)\\
&=& \Gamma^{\lepton }_{1,NS}(\qsq) + \Gamma^{\lepton }_{1,S}(Q^2)\\
\Gamma_1^{\lepton p}(Q^2)&=& \frac{C_1^{NS}(\qsq)}{6}\biggr [\onehalf a_3 + \frac{1}{6} a_8\biggl ] +
\frac{C_1^S}{9} a_0\\
\Gamma_1^{\lepton n}(Q^2)&=& \frac{C_1^{NS}(\qsq)}{6}\biggr [-\onehalf a_3 + \frac{1}{6} a_8\biggl ] +
\frac{C_1^S}{9} a_0\\\end{eqnarray}
Where the $C_1$ are coefficient functions and the axial charge matrix elements
\begin{eqnarray}
a_3 &\equiv& F+D \simeq \Delta \uquark - \Delta \dquark \\
a_8 &\equiv& 3F-D \simeq \Delta \uquark + \Delta \dquark - 2 \Delta \squark \\
a_0 &\equiv& \Delta \uquark + \Delta \dquark + \Delta \squark = \Delta \Sigma \\
\end{eqnarray}
can be expressed in terms of coupling constants $F$ and $D$ obtained from
neutron and hyperon beta decays \cite{betadecays}.
Because the interaction between $\Delta g $ and $\Delta \Sigma $
in the evolution of the singlet ($a_0$) component, interpretation of
$\Gamma_1^\lepton$ in terms of the quark spin is problematic.
Fig. \label{gluon} shows NLO QCD predictions for $\Delta \Sigma$ as a function of $\Delta g$.
\begin{figure}
\centerline{\epsffile{delta.eps}}
\caption{
Model dependent decomposition of singlet term into quarks and gluon based
on PDIS data,
$a_0 \rightarrow \Delta q - 3{\alpha_s \over 2 \pi} \Delta g$.
The QPM expectation and the results from a NLO fit of the
$Q^2$ evolution of most of the available data on $g_1^l$
are also shown. From the fit it was found that under this singlet
%decomposition, $\Delta g = 1.6 \pm 0.3\pm 1.0$, where
the error is dominated by theoretical
uncertainties.
\label{gluon}}
\end{figure}
Neutrino beams introduce both additional parity violating spin
structure functions $g_3, g_4 $ and $g_5$ and new combinations based
on sums and differences of neutrino and anti-neutrino scattering.
For example the sum
$$\Gamma_1^{\nu } + \Gamma_1^{\nu } = \int dx (g_1^{\nu } + g_1^{\nubar }) $$
for both proton and neutron targets is only sensitive to the singlet $a_0$ term and no input from
beta decay is necessary.
The parton model interpretation of these new structure functions is:
\begin{eqnarray}
g_{4+5}^{\nu p}(x,Q^2)&=&2xg_3^{\nu p}(x,Q^2)\\\nonumber&=&-x[\delta d(x,Q^2)+\delta
s(x,Q^2)-\delta\antiuquark(x,Q^2)-\delta \anticquark(x,Q^2)],\\
g_{4+5}^{\bar\nu p}(x,Q^2)&=&2xg_3^{\bar\nu p}(x,Q^2)\\\nonumber&=&-x[\delta
u(x,Q^2)+\delta c(x,Q^2)-\delta \antidquark(x,Q^2)-\delta \antisquark(x,Q^2)].
\label{pol_g3}
\end{eqnarray}
On a deuterium target, the $\uquark$ and $\dquark$ contributions
to $g_3$ can be cancelled leading to a direct measurement of
the strange sea contribution to the nucleon spin
$$g_3^{\nu (np)} - g_3^{\nubar (np)} = -2 (\delta \squark + \delta \antisquark)+2 (\delta \cquark + \delta \anticquark) $$.
which can also be studied via polarization asymmetries in charm production
from strange quarks.
The structure functions $g_3 -- g_5$, like $F_3$ are non-singlet functions
in which contribution from gluons cancel. Comparison of the non-singlet
functions with the single functions $g_1$ and $F_2$ is an indirect way
of measuring the contribution of gluons $\Delta g$.
%
\subsection{Experimental Setup at the Neutrino Factory}
A very promising target technology is the 'ICE' target \cite{ICE},
a solid HD compound in which the protons or the
deuterons can be polarized independently.
The expected polarization and dilution are
$P_H$=80\%, and $f_H=1/3$ for H, and $P_D$=50\% and $f_D=2/3$ for deuteron.
A 7 kg ($\rho_t$=1.1gr/cm$^2$) polarized target with the qualities mentioned
above would be 20~cm in radius and 50~cm thick, similar to
the other light targets proposed for structure
function studies. Raw event rates in
such a detector would be around 20M per 10$^{20}$ muon decays.
If such a data sample is analyzed in 10 in $x$ bins,
the error
in each $x$ bin would be:
$\delta {g_1}\simeq (fP_T\sqrt{N})^{-1} \sim 1\%$.
%measurement of the strange polarization from final charm state
%(di-lepton events), instead of the evaluation of Eq.~\ref{g3}.
%The total di-lepton cross-section is approximately $2\%$.
If the neutrino beam intensities and polarized target described above are
feasible, the physics motivations would be very strong. We
will be able to do high precision
measurements where we can cleanly separate singlet
(gluon-dependent) from non-singlet (gluon-free) terms. Furthermore,
due to the nature of the neutrino charged interactions it will be
possibility to perform a measurement of the polarization of the
proton's quarks by flavor, with sea and valence contributions separated.
\newcommand{\stw}{\mbox{$\sin^2\theta_W$}}
\newcommand{\nub}{\overline{\nu}}
\newcommand{\qbar}{\overline{q}}
%\newcommand{\nue}{\nu_{e}}
%\newcommand{\numu}{\nu_{\mu}}
\newcommand{\nubmu}{\overline{\nu_{\mu}}}
\newcommand{\nube}{\overline{\nu_{e}}}
\newcommand{\muebar}{\numu\nube}
\newcommand{\mubare}{\nubmu\nue}
\newcommand{\ubar}{\antiuquark}
\newcommand{\dbar}{\antidquark}
\newcommand{\alps}{\mbox{$\alpha_s$}}
\newcommand{\asop}{\mbox{$\frac{\alpha_s}{\pi}$}}
%\newcommand{\qsq}{\mbox{$Q^2$}}
\newcommand{\qnsq}{\mbox{$Q_0^2$}}
\newcommand{\mztwo}{\mbox{$M_Z^2$}}
\newcommand{\mz}{\mbox{$M_Z$}}
\newcommand{\mw}{\mbox{$M_W$}}
\newcommand{\mtop}{\mbox{$M_{\rms top}$}}
\newcommand{\mhiggs}{\mbox{$M_{\rms Higgs}$}}
\newcommand{\lmsb}{\mbox{$\Lambda_{\overline{MS}}$}}
\newcommand{\avgth}{\left< \theta_\nu\right> }
\subsection{Charm Production and $\dzero- \dzerobar$ Mixing}
\begin{figure}[tpb]
\begin{center}
\epsfxsize=5 in
\epsfbox{heavy-flavor-grv.eps}
%\epsfxsize=0.8\textwidth\epsfbox{heavy-flavor-grv.ps}
\end{center}
\caption{Charm and bottom quark production as a fraction of the total
cross-section as a function of $E_\nu$. }
\label{fig:charmrate}
\end{figure}
\begin{figure}[tpb]
\begin{center}
\epsfxsize=5 in
\epsfbox{charm-spect-grv.eps}
\end{center}
\caption{Charmed hadron spectra from neutrino interactions in a near detector
from a $50$~GeV muon storage ring.}
\label{fig:charmspect}
\end{figure}
Neutrino interactions are a very good source of clean, sign-tagged charm
particles. Single charm quarks are produced
via the processes
\begin{eqnarray}
\nu \squark &\gt& \lminus \cquark \hbox{\ \ \ Cabbibo favored} \\
\nu \dquark &\gt& \lminus \cquark \hbox{\ \ \ Cabbibo suppressed} \\
\nu \antisquark &\gt& \lplus \anticquark \hbox{\ \ \ Cabbibo favored} \\
\nu \antidquark &\gt& \lplus \anticquark \hbox{\ \ \ Cabbibo suppressed} \\
\end{eqnarray}
The fraction of heavy flavor produced as
a function of $E_\nu$ is shown in Fig.~\ref{fig:charmrate}.
An experiment
at a 50 GeV muon storage ring with 10$^{20}$ muon decays and a
a one ton (fiducial) target made up of silicon strip detectors
interleaved with heavier material would observe $\approx 3\times10^{9}$ muon-neutrino charged-current
interactions and around $1.2\times 10^8$ charm hadrons with energies
above 4 GeV/year.
All of these charmed
hadrons are flavor tagged at the point of production by the charge of the
outgoing primary lepton ($c$ production with $\ell^-$ and $\overline{c}$
production with $\ell^+$).
There are several interesting physics motivations for charm studies at muon
storage rings, including measurements of the strange contribution
to proton structure and spin; however, the primary motivation for producing
charm by this method is the cleanliness of the final state relative to
hadroproduction and the flavor tagging in production. This experimental fact
compliments the theoretically ``clean laboratory'' of charm in searches for
FCNC, CP asymmetries and ${\rm D^0 \rightarrow \overline{D^0}}$ oscillations,
all of which are very small in the standard model because of the lack of
coupling of charm to the heavy top quark.
As an example of the physics reach of a neutrino charm factory, consider the
example of $D^0-\overline{D^0}$ mixing measurements. The most sensitive
current searches for time-integrated mixing place limits on the process of
$\sim 5\times 10^{-3}$ \cite{E791,CLEO}. BaBar expects to have sensitivity to
mixing at the $\sim 5\times10^{-4}$ level after several years at design
luminosity \cite{BaBar}. These measurements are ultimately limited by
tagging mistakes and backgrounds to final state $D^0$ or $\overline{D^0}$
identification from doubly-Cabibbo suppressed decays, such as
$D^0\to K^+\pi^-$ which occur at the few part per thousand level.
At a $50$~GeV muon storage ring, with a high mass detector,
one could probe $D^0-\overline{D^0}$ mixing
{\em via}
\begin{eqnarray*}
\nu N \to & c \ell^- X\hspace*{8ex} \\
& \hookrightarrow \ell^+ X\hspace*{5ex} \\
& \hookrightarrow \overline{c} \to \ell^- X,
\end{eqnarray*}
and its charge conjugates. The appearance of like-signed leptons would
indicate mixing, where opposite-signed leptons are expected. Assuming $50\%$ of the charm produces hadronizes as a $D^0$ or
$\overline{D^0}$, this would result in the observation of $2\times 10^6$
tagged neutral charm meson semi-leptonic decays in either the muon or electron
channel.% These should allow limits on D meson mixing at the $
\approx\sqrt{\frac{\pi^2}{16\gamma_\mu^2}+\frac{m_eE_\nu}{2}}.
\end{equation}
For a $50$~GeV storage ring, this factor is dominated by the fundamental
$p_t$ of the interaction and is typically $\sim90$~MeV. For a lower
energy storage ring of about $15$~GeV, these factors become equal.
\begin{figure}
\begin{center}
\mbox{\epsfxsize=5.0in\epsffile{nue-sn.eps}}
\caption{Signal to noise in the low $p_t$ region
($p_t^2<\frac{\pi^2}{16\gamma_\mu^2}+\frac{m_eE_\nu}{2}$) as
a function of $E_\nu$.}\label{fig:nue-sn}
\end{center}
\end{figure}
The primary background
to this measurement is from quasi-elastic $\nue-N$ or
$\bar{\nu}_e-N$ scattering events which occur at $p_t$ up to $\sqrt{m_N E_\nu}$.
Fig.~\ref{fig:nue-sn} shows the estimated signal to background
ratios expected in the low $p_t$ region.
Because of the exceptionally low cross section, the target must be
very massive.
The detector must therefore be capable of resolving the
$p_t$ with much better resolution than the background spread.
This favors the use of a fully active, high resolution tracking detector with
sub-radiation length sampling in order to resolve the $p_t$ of the single
electron before it is significantly broadened by shower development. A
liquid Argon TPC, such as the one proposed for the ICANOE
experiment\cite{ICANOE} might be ideal for such a measurement. Another
possibility would be a scintillating fiber/tungsten calorimeter.
% Bruce had L of 10^46 and E_\nu=100 GeV which is out of the energy
% range! So I decreased the beam energy by three and upped the
% luminosity by 10 to keep the same sensitivity.
\begin{figure}
\begin{center}
\mbox{\epsfxsize=5.0in\epsffile{nue-beam-sens.eps}}
\caption{Statistical uncertainty in $\siniiW$ for a luminosity of
$10^{46}$~cm$^{-2}$ as a function of $y_{\rm cut}$ for a $30$~GeV
neutrino beam. Note that the $\mu^-$ produced beam is much less sensitive
to $\siniiW$ due to nearly exactly cancellation in the $\siniiW$ dependence of
the two neutrino species in the beam.}\label{fig:nue-sens}
\end{center}
\end{figure}
The largest experimental challenge
for measurement is likely to be the normalization of the absolute
neutrino flux. Despite the precise knowledge of muon decays, it
would be extremely difficult to predict the precise neutrino flux
at the $10^{-4}$ level merely from monitoring the parent muon beam.
Instead, the signal processes will probably have to be normalized
to the theoretically predictable processes of inverse muon decay,
$\numu e^-\to\nue \mu^-$, and muon production through annihilation,
$\antinu_e e^-\to \antinu_{\mu}\mu^-$, both of which occur only in the
$\muebar$ beam. Normalization of the $\nue$ beam may be possible
through comparison neutrino-nucleon
scattering, $\nu N \to \l^\pm N'$, in the $\mubare$ and $\muebar$
beam.
For a 20 ton detector at a 50 GeV muon storage ring,
with $2\times 10^{20}$ $\muminus$ decays/year there will be approximately
$1.5\times 10^{10}$ DIS charged current events and 8.5M $\numu/antinue$-electron
scatters per year. These lead to an estimated sensitivity of $\delta\siniiW^{\rm (stat)}\sim0.0002$
% Jae Yu's section *******************************************************
\subsection*{Electro-weak measurements in neutrino-nucleon scattering}
There were two dominant systematic uncertainties in present-day meaurements
of the weak mixing angle in neutrino nucleon scatering,
$\nu_{e}$ contamination in the $\nu_{\mu} $ beam and the kinematic
suppression of scattering from strange quarks in the charged current channel.
For an isoscalar target, the neutral current rates can be related to the charged current rates via \cite{LL-SMITH}:
\begin{eqnarray}\label{NCCC}
R_{\nu} - \Delta R_s &=& (\onehalf - \siniiW +{5\over 9} \sin^4\theta_W)[ 1 - \Delta R_c] + \\
&&\hskip 2 in{5\over 9} \sin^4\theta_W [r - r\Delta \overline{R}_c]\\
%
R_{\nubar} - \Delta \overline{R}_s&=& (\onehalf - \siniiW +{5\over 9} \sin^4\theta_W)[ 1 - \Delta\overline{R}_c]+\\
& & \hskip 2 in{5\over 9}\sin^4\theta_W[ {1\over r}- {1\over r} \Delta R_c]
%\nonumber &\simeq& 0.12\times 10^{-38} (E_{\nu}, \GeV), \centii\\
\end{eqnarray}
where $R_{\nu/\nubar}$ is the ratio of neutral to charged current
cross sections, $r \sim 0.5$ is the ratio of charged current anti-neutrino to neutrino cross sections and $\Delta R_s$ and $\Delta R_c$ are small corrections for
the kinematic suppression of $\squark \gt \cquark$ in charged current scattering where the neutral current process $\squark\gt \squark$ has no suppression.
The charm corrections can be eliminated by a judicious subtraction
of neutrino and anti-neutrino rates \cite{ref:paschos} but with a consequent
reduction in statistical power.
Experiments CITE(CDHSW, CCFR, NUTEV) to date have had integrated fluxes of 10$^{15}$-10$^{16}$ neutrinos
and have relied on dense iron targets. In such targets neutral current events are distinguished from charged current events by
the presence or absence of a muon in the final state. But in a
calorimeter, electron neutrino
charged current
induced events look very similar to neutral current events and form a significant background for precision measurements with conventional beams
produced by pion and kaon decay.
The most precise measurement to date is from the NuTeV collaboration \cite{NuTeV:prelim} of
\begin{equation}\label{eq:nutev-stw}
\siniiW=0.2253\pm0.0019{ (stat)}\pm0.0010{ (syst)}.
\end{equation}
At a neutrino factory, the neutrino flux will be several orders of
magnitude higher but the beam will consist of approximately equal numbers
of $\numu$ and $\antinue$. This makes a detector capable of
distinguishing electron charged current events from neutral current
events desirable and implies a low density detector such as those
considered for the deep-inelastic scattering studies.
We have considered several possible observables for a neutrino factory
measurement and propose:
\begin{eqnarray}\label{eq:Rmuebar}
R_e^{\muminus}=\frac{\sigma(\nu_{\mu},NC)+\sigma(\nubar_{e},NC)}
{\sigma(\nu_{\mu},CC)-\sigma(\nubar_{e},CC)}
&=&{R^{\nu}+grR^{\nubar}\over {1-gr}}
\end{eqnarray}
or
\begin{eqnarray}\label{eq:Rmuebarhat}
\hat{R}^{\muminus}=\frac{\sigma(\nu_{\mu},NC)+\sigma(\nubar_{e},NC)+\sigma(\nubar_{e},CC)}
{\sigma(\nu_{\mu},CC)}
&=&{R^{\nu}+grR^{\nubar} + gr}
\end{eqnarray}
for the $\muebar$ beam, and
\begin{eqnarray}\label{eq:Rmubare}
R_e^{\muplus}=\frac{\sigma(\nubar_{\mu},NC)
+\sigma(\nu_{e},NC)}
{\sigma(\nu_{e},CC)-\sigma(\nubar_{\mu},CC)}
={R^{\nu}+g^{-1}rR^{\nubar}\over{1-g^{-1}r}}
\end{eqnarray}
or
\begin{eqnarray}\label{eq:Rmubarehat}
\hat{R}^{\muplus}&=&{\sigma(\numubar,NC)
+\sigma(\nue,NC)+\sigma(\nue),CC)
\over\sigma(\numubar,CC)}\\
&=&{{g\over r}R^{\nu}+R^{\nubar} + {g\over r}}
\end{eqnarray}
for the $\mubare$ beam, where
$R_{\nu/\nubar}$ is the ratio of neutral to charged current cross sections
from Eq. \ref{NCCC}.
The variable $g$ is the energy-weighted flux ratio between $\nu_{\mu}$
and $\overline{\nu}_{e}$ or, equivalently, between $\overline{\nu}_{\mu}$
and $\nu_{e}$:
The observable $R_e^{\mu}$ requires electron identification while
$\hat{R}^{\mu}$ requires only muon identification.
The flux ratio for neutrinos and anti-neutrinos $g$ is:
\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}}} \simeq {6\over7 }.
\end{eqnarray}
Due to the nature of muon decay kinematics, the flux ratio should
be quite well known. However, the relative detection efficiencies for muons
and electrons must be known at the $2\times 10^{-4}$ level in order to
determine $\siniiW$ to 10$^{-3}$ by the first method.
In addition, the charm contributions are not cancelled in this observable
and must be measured directly in the same experiment.
%\end{itemize}
For the $R_e$ measurement, which requires electron identification,
an active target of 20~cm radius, 10 gr/cm$^2$ thick consisting of either CCD's or silicon strip detectors ($\sim$ 140 300-$\mu$m detectors)
spaced over a meter and
followed by the
tracking, electromagnetic and hadron calorimetry and muon identification proposed above
for structure function measurements
would yield 15M muon and 8M electron charged current
deep-inelastic scattering events/10$^{20}$ $\muminus$ decays and would yield a statistical precision
of 0.0004 in $\siniiW$. The charm corrections partially cancel in this
observable and would also be measured directly
via the 2M charm events/year produced in such a detector.
The $\hat{R}$ measurement, which relies only on muon identification
can be done with a much denser target, perhaps an iron/silicon sandwich
calorimeter. Such a calorimeter 200 gr/cm$^2$ thick would have a
statistical sensitivity of $\Delta \siniiW ~ 0.0001$ per year at a 50 GeV machine. This method is quite similar to the method used in the NuTeV \ref{nutev-stw}
measurement and would be dominated by systematic errors.
%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 isospin
singlets
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,
isospin singlet 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 $U_i^2\cdot U_j^2$ 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:
\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 $BR(L_0 \rightarrow \hbox{detectable})$
is the branching ratio for the neutral heavy lepton decaying via a
detectable channel (presumably $L_0 \rightarrow \nu e e$).
In estimating 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. Fig. \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
$6.0\times 10^{-6}$ in the low mass region \cite{PDB,Shrock}.
The single event sensitivity quoted here depends 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 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}
%%% John Krane, March 2000
\subsection{Neutrino Magnetic Moments}
Although neutrino oscillation searches focus on the mass differences between
neutrino eigenstates, neutrinos can possess other observable properties
such as a magnetic moment. A measurement of the neutrino magnetic
moment (NMM) would not only have great impact in the field of cosmology,
particularly the development of stellar models, but would also help
constrain several Standard Model extensions. An important experimental
advantage is that a NMM search can run parasitically as the front--end to a
typical long baseline detector.
Despite their lack of charge, neutrinos can possess a non-zero magnetic
moment that can arise through loop diagrams. In the Standard Model, extended
to include a right--handed neutrino or with left--handed neutrinos which have mass, the expected magnitude of the \cite{shrock} neutrino magnetic moment
is given by
\begin{equation}
\label{krane}
\mu _{\nu }\simeq 3\times 10^{-19}\;\mu _{B}\cdot ( \frac{m_{\nu }}{%
1\;\hbox{eV}}) ,
\end{equation}
where $\mu _{B}$ is the Bohr magneton. Although quite minuscule, several
extensions to the Standard Model could dramatically increase $\mu_\nu$ : supersymmeteric
models can produce $10^{-14}\mu_{B}$ to $10^{-12}\mu _{B}$
\cite{frank} and calculations that invoke large extra dimensions easily
yield $10^{-11}\mu _{B}$ or larger \cite{ng}.
Relative to the Standard Model 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 $1.8\times 10^{-10}\mu _{B}$ for electron
neutrinos \cite{beacom}\cite{mu_e_limit} and $\mu _{\nu }\leq 7.4\times
10^{-10}\mu _{B}$ for muon neutrinos \cite{mu_mu_limit}. Astrophysical
limits are stronger:\ the slow rate of plasmon decay in horizontal branch
stars \cite {star1} implies $\mu _{\nu }\le 10^{-11}\mu _{B}$,
while the neutrino energy loss rate from supernova 1987a \cite{star2}
yields $\mu _{\nu }\le $ $10^{-12}\mu _{B}$. 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 might be overestimates. 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 a carefully designed and executed experiment could improve the limits
from current experiments and possibly the limits from astrophysics
calculations, or actually detect a NMM.
The energy $E$ of a neutrino with a magnetic moment 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
or increase in the number of antineutrinos detected with the B field in
place compared to the sample detected with no B field turned on.
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 this energy difference, which drives the phase
difference in the absence of the B-field, 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.
{\it Editor's note - is this possible or does the neutrino see
an effective spatial gradient in all case}
\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. We are exploring configurations
involving 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 ^{2}\left( \mu _{\nu }\hbox{B}t\right) \label{eq_nlost}
\end{equation}
\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}
\noindent
where $t$ is the neutrino flight time from entering the magnetic field to
detection. We note that, in contrast to an oscillation disappearence
signal, this effect is explicitly independant of the neutrino energy. Fig.
\ref{mag_mom_signif} compares the number of events that vanish
because of phase rotations 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 a 10 T field gradient and $10^{19}$ muon decays
we expect a $> 10 \sigma$ significance for a NMM of
$10^{-11}\mu _{B}$. With a 3T gradient, the limit drops below
two sigma. The sensitivity can be increased very strongly by increasing the
field strength and more weakly by
increasing the number of events in the far detector. Because the detector
distance deterimines both $t$ and $N$ in Eqn. \ref {eq_nlost}, the
``significance'' in the figure is linearly dependent on 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 a controllable parameter, and
thus has the potential for improved sensitivity as we improve our ability
to create oscillating magnetic field gradients.
%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 $\le 5 \times 10^{-9}$. 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, neutrino oscillations
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 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!
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}
\section{Summary}
We have investigated possible conventional neutrino physics studies done
at a detector located near a muon storage ring. We emphasized novel methods
rather than extensions of existing experiments with additional statistics.
For a reference machine with 50 GeV stored muons and 10$^{20}$ muon decays
per year we find that it is possible to:
\begin{itemize}
\item Measure individual parton distributions within the proton for all
light quarks and anti-quarks.
\item Determine the effects of a nuclear environment on individual quark species.
\item Measure the spin contributions of individual quark species, including
strange quarks and do precision studies of the QCD evolution of spin
effects without a need for data from beta decay measurements.
\item Measure charm production with raw event rates of up to 100 million
charm events/year with $\simeq$ 2M double tagged events.
\item Measure the Weinberg angle in both hadronic and purely leptonic
modes with a precision of 0.0001 to 0.0002.
\item Search for the production and decay of neutral heavy leptons
with mixing angle sensitivity 2 orders of magnitude better than
present limits in the 30-80 MeV region.
\item Search for a neutrino magnetic moment which may be much large
than the Standard Model prediction in
some supersymmetric theories.
\item Search for anomolous tau production predicted
by some R-parity violating supersymmetric models.
\end{itemize}
We note that the event rates at a near detector increase
linearly with neutrino energy. In addition, the acceptance
of small detectors is better for the narrower beam produced
by higher energy machines. Almost
all of the above measurements, with the exception of the neutral heavy
lepton search, lose sensitivity if the beam energy is less than 50 GeV
and gain if it is greater.
If the storage ring beam energy is lowered to 20 GeV, the statistical power of almost all of the
measurements considered here would drop a factor of 2.5 or more. The number
of deep-inelastic scattering events with $\qsq$ high enough for perturbative
QCD to be meaningful drops even further and the minimum $x$ rises to 0.05.
Measurements involving charm or tau production in the final
state would be severely limited, as would the inverse muon
decay normalization for $\nu-e$ scattering, which has a threshold of $\sim$ 11 GeV.