\section{Physics Motivation} \label{physics} In this Section we cover the physics potential of the Neutrino Factory accelerator complex, which includes superbeams of conventional neutrinos that are possible using the proton driver needed for the factory, and intense beams of cold muons that become available once the muon cooling and collection systems for the factory are in place. Once the cold muons are accelerated and stored in the muon storage ring, we realize the full potential of the factory in both neutrino oscillation and non-oscillation physics. Cooling muons will be a learning experience. We hope that the knowledge gained in constructing a Neutrino Factory can be used to cool muons sufficiently to produce the first muon collider operating as a Higgs factory. We examine the physics capabilities of such a collider, which if realized, will invariably lead to higher energy muon colliders with exciting physics opportunities. \subsection{Neutrino Oscillation Physics} Here we discuss~\cite{study2} the current evidence for neutrino oscillations, and hence neutrino masses and lepton mixing, from solar and atmospheric data. A review is given of some theoretical background including models for neutrino masses and relevant formulas for neutrino oscillation transitions. We next mention the near-term and mid-term experiments in this area and comment on what they hope to measure. We then discuss the physics potential of a muon storage ring as a Neutrino Factory in the long term. \subsubsection{Evidence for Neutrino Oscillations} In a modern theoretical context, one generally expects nonzero neutrino masses and associated lepton mixing. Experimentally, there has been accumulating evidence for such masses and mixing. All solar neutrino experiments (Homestake, Kamiokande, SuperKamiokande, SAGE, GALLEX and SNO) show a significant deficit in the neutrino fluxes coming from the Sun~\cite{sol}. This deficit can be explained by oscillations of the $\nu_e$'s into other weak eigenstate(s), with $\Delta m^2_{\rm sol}$ of the order $10^{-5}$ eV$^2$ for solutions involving the Mikheyev-Smirnov-Wolfenstein (MSW) resonant matter oscillations~\cite{wolf}--\cite{ms} or of the order of $10^{-10}\,$eV$^2$ for vacuum oscillations~\cite{just-so}. Accounting for the data with vacuum oscillations (VO) requires almost maximal mixing. The MSW solutions include one for small mixing angle (SMA) and one for large mixing angle (LMA). Another piece of evidence for neutrino oscillations is the atmospheric neutrino anomaly, observed by Kamiokande~\cite{kam}, IMB~\cite{imb}, SuperKamiokande~\cite{sk} with the highest statistics, and by Soudan~\cite{soudan2} and MACRO~\cite{macro}. These data can be fit by the inference of $\nu_{\mu} \rightarrow \nu_x$ oscillations with $\Delta m^2_{\rm atm}\sim 3 \times 10^{-3}\rm\,eV^2$~\cite{sk} and maximal mixing $\sin^2 2 \theta_{\rm atm} = 1$. The identification $\nu_x = \nu_\tau$ is preferred over $\nu_x=\nu_{sterile}$, and the identification $\nu_x=\nu_e$ is excluded by both the Superkamiokande data and the Chooz experiment~\cite{chooz}. In addition, the LSND experiment~\cite{lsnd} has reported $\bar\nu_\mu \to \bar \nu_e$ and $\nu_{\mu} \to \nu_e$ oscillations with $\Delta m^2_{\rm LSND} \sim 0.1\mbox{--}1\rm~eV^2$ and a range of possible mixing angles. This result is not confirmed, but also not completely ruled out, by a similar experiment, KARMEN~\cite{karmen}. The miniBOONE experiment at Fermilab is designed to resolve this issue, as discussed below. If one were to try to fit all of these experiments, then, since they involve three quite different values of $\Delta m^2_{ij}=m(\nu_i)^2-m(\nu_j)^2$, which could not satisfy the identity for three neutrino species, % \begin{equation} \Delta m^2_{32} + \Delta m^2_{21} + \Delta m^2_{13}=0 \,, \label{mident} \end{equation} % it would follow that one would have to introduce at least one further neutrino. Since it is known from the measurement of the $Z$ width that there are only three leptonic weak doublets with associated light neutrinos, it follows that such further neutrino weak eigenstate(s) would have to be electroweak singlet(s) (``sterile'' neutrinos). Because the LSND experiment has not been confirmed by the KARMEN experiment, we choose here to use only the (confirmed) solar and atmospheric neutrino data in our analysis, and hence to work in the context of three active neutrino weak eigenstates. \subsubsection{Neutrino Oscillation Formalism} In this theoretical context, consistent with solar and atmospheric data, there are three electroweak-doublet neutrinos and the neutrino mixing matrix is described by % \begin{equation} U=\left( \begin{array}{ccc} c_{12} c_{13}&c_{13} s_{12}&s_{13} e^{-i\delta}\\ -c_{23}s_{12}-s_{13}s_{23}c_{12}e^{i\delta} &c_{12}c_{23}-s_{12}s_{13}s_{23}e^{i\delta}&c_{13}s_{23}\\ s_{12}s_{23}-s_{13}c_{12}c_{23}e^{i\delta} &-s_{23}c_{12}-s_{12}c_{23}s_{13}e^{i\delta}&c_{13}c_{23} \end{array} \right)K^\prime \,, \end{equation} % where $c_{ij}=\cos\theta_{ij}$, $s_{ij}=\sin\theta_{ij}$, and $K^\prime = {\rm diag}(1,e^{i\phi_1},e^{i\phi_2})$. The phases $\phi_1$ and $\phi_2$ do not affect neutrino oscillations. Thus, in this framework, the neutrino mixing relevant for neutrino oscillations depends on the four angles $\theta_{12}$, $\theta_{13}$, $\theta_{23}$, and $\delta$, and on two independent differences of squared masses, $\Delta m^2_{\rm atm}$, which is $\Delta m^2_{32} = m(\nu_3)^2-m(\nu_2)^2$ in the favored fit, and $\Delta m^2_{\rm sol}$, which may be taken to be $\Delta m^2_{21}=m(\nu_2)^2- m(\nu_1)^2$. Note that these $\Delta m^2$ quantities involve both magnitude and sign; although in a two-species neutrino oscillation in vacuum the sign does not enter, in the three-species-oscillation, which includes both matter effects and $CP$ violation, the signs of the $\Delta m^2$ quantities enter and can, in principle, be measured. For our later discussion it will be useful to record the formulas for the various neutrino-oscillation transitions. In the absence of any matter effect, the probability that a (relativistic) weak neutrino eigenstate $\nu_a$ becomes $\nu_b$ after propagating a distance $L$ is % \begin{eqnarray} P(\nu_a \to \nu_b) &=& \delta_{ab} - 4 \sum_{i>j=1}^3 Re(K_{ab,ij}) \sin^2 \Bigl ( \frac{\Delta m_{ij}^2 L}{4E} \Bigr ) + \nonumber\\ && {}+ 4 \sum_{i>j=1}^3 Im(K_{ab,ij}) \sin \Bigl ( \frac{\Delta m_{ij}^2 L}{4E} \Bigr ) \cos \Bigl ( \frac{\Delta m_{ij}^2 L}{4E} \Bigr ) \label{pab} \end{eqnarray} where \begin{equation} K_{ab,ij} = U_{ai}U^*_{bi}U^*_{aj} U_{bj} \label{k} \end{equation} and \begin{equation} \Delta m_{ij}^2 = m(\nu_i)^2-m(\nu_j)^2 \,. \label{delta} \end{equation} Recall that in vacuum, $CPT$ invariance implies $P(\bar\nu_b \to \bar\nu_a)=P(\nu_a \to \nu_b)$ and hence, for $b=a$, $P(\bar\nu_a \to \bar\nu_a) = P(\nu_a \to \nu_a)$. For the CP-transformed reaction $\bar\nu_a \to \bar\nu_b$ and the T-reversed reaction $\nu_b \to \nu_a$, the transition probabilities are given by the right-hand side of (\ref{pab}) with the sign of the imaginary term reversed. (Below we shall assume $CPT$ invariance, so that $CP$ violation is equivalent to $T$ violation.) In most cases there is only one mass scale relevant for long-baseline neutrino oscillations, $\Delta m^2_{\rm atm} \sim {\rm few} \times 10^{-3}\rm\,eV^2$, and one possible neutrino mass spectrum is the hierarchical one \begin{equation} \Delta m^2_{21} = \Delta m^2_{\rm sol} \ll \Delta m^2_{31} \approx \Delta m^2_{32}=\Delta m^2_{\rm atm} \,. \label{hierarchy} \end{equation} In this case, $CP$ $(T)$ violation effects may be negligibly small, so that in vacuum \begin{equation} P(\bar\nu_a \to \bar\nu_b) = P(\nu_a \to \nu_b) \label{pcp} \end{equation} and \begin{equation} P(\nu_b \to \nu_a) = P(\nu_a \to \nu_b) \,. \label{pt} \end{equation} In the absence of $T$ violation, the second equality (\ref{pt}) would still hold in uniform matter, but even in the absence of $CP$ violation, the first equality (\ref{pcp}) would not hold. With the hierarchy (\ref{hierarchy}), the expressions for the specific oscillation transitions are \begin{eqnarray} P(\nu_\mu \to \nu_\tau) & = & 4|U_{33}|^2|U_{23}|^2 \sin^2 \Bigl ( \frac{\Delta m^2_{\rm atm}L}{4E} \Bigr ) \nonumber\\ & = & \sin^2(2\theta_{23})\cos^4(\theta_{13}) \sin^2 \Bigl (\frac{\Delta m^2_{\rm atm}L}{4E} \Bigr ) \,, \label{pnumunutau} \end{eqnarray} % \begin{eqnarray} P(\nu_e \to \nu_\mu) & = & 4|U_{13}|^2 |U_{23}|^2 \sin^2 \Bigl ( \frac{\Delta m^2_{\rm atm}L}{4E} \Bigr ) \nonumber\\ & = & \sin^2(2\theta_{13})\sin^2(\theta_{23}) \sin^2 \Bigl (\frac{\Delta m^2_{\rm atm}L}{4E} \Bigr ) \,, \label{pnuenumu} \end{eqnarray} % \begin{eqnarray} P(\nu_e \to \nu_\tau) & = & 4|U_{33}|^2 |U_{13}|^2 \sin^2 \Bigl ( \frac{\Delta m^2_{\rm atm}L}{4E} \Bigr ) \nonumber\\ & = & \sin^2(2\theta_{13})\cos^2(\theta_{23}) \sin^2 \Bigl (\frac{\Delta m^2_{\rm atm}L}{4E} \Bigr ) \,. \label{pnuenutau} \end{eqnarray} In neutrino oscillation searches using reactor antineutrinos, i.e,\ tests of $\bar\nu_e \to \bar\nu_e$, the two-species mixing hypothesis used to fit the data is % \begin{eqnarray} P(\nu_e \to \nu_e) & = & 1 - \sum_x P(\nu_e \to \nu_x) \nonumber\\ & = & 1 - \sin^2(2\theta_{\rm reactor}) \sin^2 \Bigl (\frac{\Delta m^2_{\rm reactor}L}{4E} \Bigr ) \,, \label{preactor} \end{eqnarray} % where $\Delta m^2_{\rm reactor}$ is the squared mass difference relevant for $\bar\nu_e \to \bar\nu_x$. In particular, in the upper range of values of $\Delta m^2_{\rm atm}$, since the transitions $\bar\nu_e \to \bar\nu_\mu$ and $\bar\nu_e \to \bar\nu_\tau$ contribute to $\bar\nu_e$ disappearance, one has \begin{equation} P(\nu_e \to \nu_e) = 1 - \sin^2(2\theta_{13})\sin^2 \Bigl (\frac{\Delta m^2_{\rm atm}L}{4E} \Bigr ) \,, \label{preactoratm} \end{equation} % i.e., $\theta_{\rm reactor}=\theta_{13}$, and, for the value $|\Delta m^2_{32}| = 3 \times 10^{-3}\rm\,eV^2$ from SuperK, the CHOOZ experiment on $\bar\nu_e$ disappearance yields the upper limit~\cite{chooz} % \begin{equation} \sin^2(2\theta_{13}) < 0.1 \,, \label{chooz} \end{equation} which is also consistent with conclusions from the SuperK data analysis~\cite{sk}. Further, the quantity ``$\sin^2(2\theta_{\rm atm})$'' often used to fit the data on atmospheric neutrinos with a simplified two-species mixing hypothesis, is, in the three-generation case, % \begin{equation} \sin^2(2\theta_{\rm atm}) \equiv \sin^2(2\theta_{23})\cos^4(\theta_{13}) \,. \label{thetaatm} \end{equation} % The SuperK experiment finds that the best fit to their data is $\nu_\mu \to \nu_\tau$ oscillations with maximal mixing, and hence $\sin^2(2\theta_{23})=1$ and $|\theta_{13}| \ll 1$. The various solutions of the solar neutrino problem involve quite different values of $\Delta m^2_{21}$ and $\sin^2(2\theta_{12})$: (i)~large mixing angle solution, LMA: $\Delta m^2_{21} \simeq {\rm few} \times 10^{-5}\rm\,eV^2$ and $\sin^2(2\theta_{12}) \simeq 0.8$; (ii)~small mixing angle solution, SMA: $\Delta m^2_{21} \sim 10^{-5}\rm\,eV^2$ and $\sin^2(2\theta_{12}) \sim 10^{-2}$, (iii)~LOW: $\Delta m^2_{21} \sim 10^{-7}\rm\,eV^2$, $\sin^2(2\theta_{12}) \sim 1$, and (iv)~``just-so'': $\Delta m^2_{21} \sim 10^{-10}\rm\,eV^2$, $\sin^2(2\theta_{12}) \sim 1$. The SuperK experiment favors the LMA solutions~\cite{sol}; for other global fits, see, e.g., Ref.~\cite{sol}. We have reviewed the three neutrino oscillation phenomenology that is consistent with solar and atmospheric neutrino oscillations. In what follows, we will examine the neutrino experiments planned for the immediate future that will address some of the relevant physics. We will then review the physics potential of the Neutrino Factory. \subsubsection{Relevant Near- and Mid-Term Experiments} There are currently intense efforts to confirm and extend the evidence for neutrino oscillations in all of the various sectors --- solar, atmospheric, and accelerator. Some of these experiments are running; in addition to SuperKamiokande and Soudan-2, these include the Sudbury Neutrino Observatory, SNO, and the K2K long baseline experiment between KEK and Kamioka. Others are in development and testing phases, such as miniBOONE, MINOS, the CERN--Gran Sasso program, KamLAND, Borexino, and MONOLITH~\cite{anl}. Among the long baseline neutrino oscillation experiments, the approximate distances are $L \simeq 250$~km for K2K, 730~km for both MINOS (from Fermilab to Soudan) and the proposed CERN--Gran Sasso experiments. K2K is a $\nu_\mu$ disappearence experiment with a conventional neutrino beam having a mean energy of about 1.4~GeV, going from KEK 250~km to the SuperK detector. It has a near detector for beam calibration. It has obtained results consistent with the SuperK experiment, and has reported that its data disagree by $2\sigma$ with the no-oscillation hypothesis~\cite{k2k}. MINOS is another conventional neutrino beam experiment that takes a beam from Fermilab 730~km to a detector in the Soudan mine in Minnesota. It again uses a near detector for beam flux measurements and has opted for a low-energy configuration, with the flux peaking at about 3~GeV. This experiment is scheduled to start taking data in 2005 and, after some years of running, to obtain higher statistics than the K2K experiment and to achieve a sensitivity down to the level $|\Delta m^2_{32}| \sim 10^{-3}\rm\,eV^2$. The CERN--Gran Sasso program will also come on in 2005. It will use a higher-energy neutrino beam, $E_\nu\sim17$~GeV, from CERN to the Gran Sasso deep underground laboratory in Italy. This program will emphasize detection of the $\tau$'s produced by the $\nu_\tau$'s that result from the inferred neutrino oscillation transition $\nu_\mu \to \nu_\tau$. The OPERA experiment will do this using emulsions~\cite{opera}, while the ICARUS proposal uses a liquid argon chamber~\cite{icanoe}. For the joint capabilities of MINOS, ICARUS and OPERA experiments see Ref.~\cite{minicop}. Plans for the Japan Hadron Facility (JHF), also called the High Intensity Proton Accelerator (HIPA), include the use of a 0.77~MW proton driver to produce a high-intensity conventional neutrino beam with a path length of 300~km to the SuperK detector~\cite{jhf}. Moreover, at Fermilab, the miniBOONE experiment is scheduled to start data taking in the near future and to confirm or refute the LSND claim after a few years of running. There are several neutrino experiments relevant to the solar neutrino anomaly. The SNO experiment is currently running and has recently reported their first results that confirm solar neutrino oscillations~\cite{snolatest}. These involve measurement of the solar neutrino flux and energy distribution using the charged current reaction on heavy water, $\nu_e + d \to e + p + p$. They are expected to report on the neutral current reaction $\nu_e + d \to \nu_e + n + p$ shortly. The neutral current rate is unchanged in the presence of oscillations that involve standard model neutrinos, since the neutral current channel is equally sensitive to all the three neutrino species. If however, sterile neutrinos are involved, one expects to see a depletion in the neutral current channel also. However, the uncertain normalization of the $^8$B flux makes it difficult to constrain a possible sterile neutrino component in the oscillations~\cite{unknowns}. The KamLAND experiment~\cite{kamland} in Japan started taking data in January 2002. This is a reactor antineutrino experiment using baselines of 100--250 km. It will search for $\bar\nu_e$ disappearance and is sensitive to the solar neutrino oscillation scale. KamLAND can provide precise measurements of the LMA solar parameters~\cite{bmw-kamland}. On a similar time scale, the Borexino experiment in Gran Sasso is scheduled to turn on and measure the $^7$Be neutrinos from the sun. These experiments should help us determine which of the various solutions to the solar neutrino problem is preferred, and hence the corresponding values of $\Delta m^2_{21}$ and $\sin^2(2\theta_{12})$. This, then, is the program of relevant experiments during the period 2000--2010. By the end of this period, we may expect that much will be learned about neutrino masses and mixing. However, there will remain several quantities that will not be well measured and which can be measured by a Neutrino Factory. \subsubsection{Oscillation Experiments at a Neutrino Factory } \label{neuf} Although a Neutrino Factory based on a muon storage ring will turn on several years after this near-term period in which K2K, MINOS, and the CERN-Gran Sasso experiments will run, it has a valuable role to play, given the very high-intensity neutrino beams of fixed flavor-pure content, including, uniquely, $\nu_e$ and $\bar\nu_e$ beams in addition to $\nu_\mu$ and $\bar\nu_\mu$ beams. A conventional positive charge selected neutrino beam is primarily $\nu_\mu$ with some admixture of $\nu_e$'s and other flavors from $K$ decays ({\cal O}(1\%) of the total charged current rate) and the fluxes of these neutrinos can only be fully understood after measuring the charged particle spectra from the target with high accuracy. In contrast, the potential of the neutrino beams from a muon storage ring is that the neutrino beams would be of extremely high purity: $\mu^-$ beams would yield 50\% $\nu_\mu$ and 50\% $\bar\nu_e$, and $\mu^+$ beams, the charge conjugate neutrino beams. Furthermore, these could be produced with high intensities and low divergence that make it possible to go to longer baselines. In what follows, we shall take the design values from Study-II of $10^{20}$ $\mu$ decays per ``Snowmass year'' ($10^7$ sec) as being typical. The types of neutrino oscillations that can be searched for with the Neutrino Factory based on the muon storage ring are listed in Table~\ref{tab:nu-osc-ratings} for the case of $\mu^-$ which decays to $ \nu_\mu e^- \bar\nu_e$: \begin{table} \caption[Neutrino Oscillation Modes]{Neutrino-oscillation modes that can be studied with conventional neutrino beams or with beams from a Neutrino Factory, with ratings as to degree of difficulty in each case; * = well or easily measured, $\surd$ = measured poorly or with difficulty, --- = not measured.\label{tab:nu-osc-ratings}} \begin{center} \begin{tabular}{|llcc|} \hline & & Conventional & Neutrino \\[-2ex] \raisebox{1ex}[0pt]{Measurement } & \raisebox{1ex}[0pt]{Type} & beam & Factory \\ \hline $\nu_\mu\to\nu_\mu,\,\nu_\mu\to\mu^-$ & survival & $\surd$ & *\\ $\nu_\mu\to\nu_e,\,\nu_e\to e^-$ & appearance & $\surd$ & $\surd$\\ $\nu_\mu\to\nu_\tau,\,\nu_\tau\to\tau^-,\,\tau^-\to(e^-,\mu^-)...$ & appearance & $\surd$ & $\surd$ \\ \hline $\bar \nu_e\to\bar \nu_e,\,\bar\nu_e\to e^+$ & survival & --- & $*$\\ $\bar\nu_e\to\bar\nu_\mu,\,\bar\nu_\mu\to\mu^+$ & appearance & --- & $*$ \\ $\bar\nu_e\to\bar\nu_\tau,\,\bar\nu_\tau\to\tau^+,\,\tau^+\to(e^+,\mu^+)...$ & appearance & ---& $\surd$ \\ \hline \end{tabular} \end{center} \end{table} It is clear from the processes listed that since the beam contains both neutrinos and antineutrinos, the only way to determine the flavor of the parent neutrino is to determine the identity of the final state charged lepton and measure its charge. A capability unique to the Neutrino Factory will be the measurement of the oscillation $\bar\nu_e \to \bar\nu_\mu$, giving a wrong-sign $\mu^+$. Of greater difficulty would be the measurement of the transition $\bar\nu_e \to \bar\nu_\tau$, giving a $\tau^+$ which will decay part of the time to $\mu^+$. These physics goals mean that a detector must have excellent capability to identify muons and measure their charges. Especially in a steel-scintillator detector, the oscillation $\nu_\mu \to \nu_e$ would be difficult to observe, since it would be difficult to distinguish an electron shower from a hadron shower. From the above formulas for oscillations, one can see that, given the knowledge of $|\Delta m^2_{32}|$ and $\sin^2(2\theta_{23})$ that will be available by the time a Neutrino Factory is built, the measurement of the $\bar\nu_e \to \bar\nu_\mu$ transition yields the value of $\theta_{13}$. To get a rough idea of how the sensitivity of an oscillation experiment would scale with energy and baseline length, recall that the event rate in the absence of oscillations is simply the neutrino flux times the cross section. First of all, neutrino cross sections in the region above about 10 GeV (and slightly higher for $\tau$ production) grow linearly with the neutrino energy. Secondly, the beam divergence is a function of the initial muon storage ring energy; this divergence yields a flux, as a function of $\theta_d$, the angle of deviation from the forward direction, that goes like $1/\theta_d^2 \sim E^2$. Combining this with the linear $E$ dependence of the neutrino cross section and the overall $1/L^2$ dependence of the flux far from the production region, one finds that the event rate goes like \begin{equation} \frac{dN}{dt} \sim \frac{E^3}{L^2} \,. \label{eventrate} \end{equation} We base our discussion on the event rates given in the Fermilab Neutrino Factory study~\cite{INTRO:ref9}. For a stored muon energy of 20~GeV, and a distance of $L=2900$ to the WIPP Carlsbad site in New Mexico, these event rates amount to several thousand events per kton of detector per year, i.e,\ they are satisfactory for the physics program. This is also true for the other path lengths under consideration, namely $L=2500$~km from BNL to Homestake and $L=1700$~km to Soudan. A usual racetrack design would only allow a single pathlength $L$, but a bowtie design could allow two different path lengths (e.g.,~\cite{zp}). We anticipate that at a time when the Neutrino Factory turns on, $|\Delta m^2_{32}|$ and $\sin^2(2\theta_{23})$ would be known at perhaps the 10\% level (while recognizing that future projections such as this are obviously uncertain). The Neutrino Factory will significantly improve precision in these parameters, as can be seen from Fig.~\ref{fig:30gev_disap_fit} which shows the error ellipses possible for a 30~GeV muon storage ring. \begin{figure}[tbh!] \centerline{\includegraphics[width=4.0in]{30gev_disap_fit.eps}} \bigskip \caption[Error ellipses in $\delta m^2$ sin$^2 2\theta$ space for a Neutrino Factory] { \label{fig:30gev_disap_fit} Fit to muon neutrino survival distribution for $E_\mu=30$ GeV and $L=2800$~km for 10 pairs of sin$^2 2\theta$, $\delta m^2$ values. For each fit, the 1$\sigma$,\ 2$\sigma$ and 3$\sigma$ contours are shown. The generated points are indicated by the dark rectangles and the fitted values by stars. The SuperK 68\%, 90\%, and 99\% confidence levels are superimposed. Each point is labelled by the predicted number of signal events for that point.} \end{figure} In addition, the Neutrino Factory can contribute to the measurement of: (i) $\theta_{13}$, as discussed above; (ii) measurement of the sign of $\Delta m^2_{32}$ using matter effects; and (iii) possibly a measurement of $CP$ violation in the leptonic sector, if $\sin^2(2\theta_{13})$, $\sin^2(2\theta_{21})$, and $\Delta m^2_{21}$ are sufficiently large. To measure the sign of $\Delta m^2_{32}$, one uses the fact that matter effects reverse sign when one switches from neutrinos to antineutrinos, and carries out this switch in the charges of the stored $\mu^\pm$. We elaborate on this next. \subsubsection{Matter Effects} With the advent of the muon storage ring, the distances at which one can place detectors are large enough so that for the first time matter effects can be exploited in accelerator-based oscillation experiments. Simply put, matter effects are the matter-induced oscillations that neutrinos undergo along their flight path through the Earth from the source to the detector. Given the typical density of the earth, matter effects are important for the neutrino energy range $E \sim {\cal O}(10)$ GeV and $\Delta m^2_{32} \sim 10^{-3}$~eV$^2$, values relevant for the long baseline experiments. Matter effects in neutrino propagation were first pointed out by Wolfenstein~\cite{wolf} and Barger, Pakvasa, Phillips and Whisnant~\cite{bppw-1980}. (See the papers~\cite{dgh}--\cite{cpv} for details of the matter effects and their relevance to neutrino factories.) In brief, assuming a normal hierarchy, the transition probabilities for propagation through matter of constant density are~\cite{golden,formcon} % \begin{eqnarray} P(\nu_e \to \nu_\mu) &=& x^2 f^2 + 2 x y f g (\cos\delta\cos\Delta + \sin\delta\sin\Delta) + y^2 g^2\,,\\ P(\nu_e \to \nu_\tau) &=& {\rm cot}^2 \theta_{23} x^2 f^2 - 2 x y f g (\cos\delta\cos\Delta + \sin\delta\sin\Delta) + {\rm tan}^2 \theta_{23} y^2 g^2\,,\\ P(\nu_\mu \to \nu_\tau) &=& \sin^2 2\theta_{23} \sin^2\Delta \\ & + &\alpha \sin 2\theta_{23} \sin 2\Delta \bigg({\hat A \over 1-\hat A} \sin \theta_{13} \sin 2\theta_{12} \cos 2\theta_{23} \sin\Delta-\Delta \cos^2 \theta_{12} \sin 2\theta_{23}\bigg)\,, \nonumber \end{eqnarray} where % \begin{eqnarray} \Delta &\equiv& |\delta m_{31}^2| L/4E_\nu = 1.27 |\delta m_{31}^2/{\rm eV^2}| (L/{\rm km})/ (E_\nu/{\rm GeV}) \,, \label{eq:D}\\ \hat A &\equiv& |A/\delta m_{31}^2| \,, \label{eq:Ahat}\\ \alpha &\equiv& |\delta m^2_{21}/\delta m^2_{31}| \,,\\ x &\equiv& \sin\theta_{23} \sin 2\theta_{13} \,, \label{eq:x}\\ y &\equiv& \alpha \cos\theta_{23} \sin 2\theta_{12} \,, \label{eq:y}\\ f &\equiv& \sin((1\mp\hat A)\Delta)/(1\mp\hat A) \,, \label{eq:f}\\ g &\equiv& \sin(\hat A\Delta)/\hat A \,. \label{eq:alpha} \end{eqnarray} % The amplitude $A$ for $\nu_e e$ forward scattering in matter is given by % \begin{equation} A = 2\sqrt2 G_F N_e E_\nu = 1.52 \times 10^{-4}{\rm\,eV^2} Y_e \rho({\rm\,g/cm^3}) E({\rm\,GeV}) \,. \label{eq:A} \end{equation} % Here $Y_e$ is the electron fraction and $\rho(x)$ is the matter density. For neutrino trajectories that pass through the earth's crust, the average density is typically of order 3~gm/cm$^3$ and $Y_e \simeq 0.5$. For neutrinos with $\delta m^2_{31} > 0$ or anti-neutrinos with $\delta m^2_{31} < 0$, $\hat A = 1$ corresponds to a matter resonance. Thus, for a Neutrino Factory operating with positive stored muons (producing a $\nu_e$ beam) one expects an enhanced production of opposite sign ($\mu^-$) charged-current events as a result of the oscillation $\nu_e\to \nu_\mu$ if $\delta m^2_{32}$ is positive and vice versa for stored negative beams. Figure~\ref{fig:hists}~\cite{barger-raja} shows the wrong-sign muon appearance spectra as function of $\delta m^2_{32}$ for both $\mu^+$ and $\mu^-$ beams for both signs of $\delta m^2_{32}$ at a baseline of 2800~km. The resonance enhancement in wrong sign muon production is clearly seen in Fig.~\ref{fig:hists}(b) and (c). % \begin{figure}[tbh!] \centerline{\includegraphics[width=4.0in]{plt_paper_503.eps}} \caption[Wrong sign muon appearance rates and sign of $\delta m^2_{32}$] {The wrong sign muon appearance rates for a 20 GeV muon storage ring at a baseline of 2800~km with 10$^{20}$ decays and a 50 kiloton detector for (a)~$\mu^+$ stored and negative $\delta m^2_{32}$\,, (b)~$\mu^-$ stored and negative $\delta m^2_{32}$\,, (c)~$\mu^+$ stored and positive $\delta m^2_{32}$\,, (d)~$\mu^-$ stored and positive $\delta m^2_{32}$. The values of $|\delta m^2_{32}|$ range from 0.0005 to 0.0050 eV$^2$ in steps of 0.0005~eV$^2$. Matter enhancements are evident in (b) and (c). \label{fig:hists}} \end{figure} By comparing these (using first a stored $\mu^+$ beam and then a stored $\mu^-$ beam) one can thus determine the sign of $\Delta m^2_{32}$ as well as the value of $\sin^2(2\theta_{13})$. Figure~\ref{fig:sigmas}~\cite{barger-raja} shows the difference in negative log-likelihood between a correct and wrong-sign mass hypothesis expressed as a number of equivalent Gaussian standard deviations versus baseline length for muon storage ring energies of 20, 30, 40 and 50~GeV. The values of the oscillation parameters are for the LMA scenario with $\sin^22\theta_{13}=0.04$. Figure~\ref{fig:sigmas}(a) is for 10$^{20}$ decays for each sign of stored energy and a 50 kiloton detector and positive $\delta m^2_{32}$ , (b) is for negative $\delta m^2_{32}$ for various values of stored muon energy. Figures~\ref{fig:sigmas} (c) and (d) show the corresponding curves for 10$^{19}$ decays and a 50 kiloton detector. An entry-level machine would permit one to perform a 5$\sigma$ differentiation of the sign of $\delta m^2_{32}$ at a baseline length of $\sim$2800~km. %3 \begin{figure}[tbh!] \centerline{\includegraphics[width=4.0in]{letter_plots.eps}} \caption[$\delta m_{32}^2$ sign determination at a Neutrino Factory] {The statistical significance (number of standard deviations) with which the sign of $\delta m_{32}^2$ can be determined versus baseline length for various muon storage ring energies. The results are shown for a 50~kiloton detector, and (a)~10$^{20}$ $\mu^+$ and $\mu^-$ decays and positive values of $\delta m_{32}^2$; (b)~10$^{20}$ $\mu^+$ and $\mu^-$ decays and negative values of $\delta m_{32}^2$; (c)~10$^{19}$ $\mu^+$ and $\mu^-$ decays and positive values of $\delta m_{32}^2$; (d)~10$^{19}$ $\mu^+$ and $\mu^-$ decays and negative values of $\delta m_{32}^2$. \label{fig:sigmas}} \end{figure} For the Study II design, in accordance with the previous Fermilab study~\cite{INTRO:ref9}, one estimates that it is possible to determine the sign of $\delta m^2_{32}$ even if $\sin^2(2\theta_{13})$ is as small as $\sim 10^{-3}$. \subsubsection{CP Violation} $CP$ violation is measured by the (rephasing-invariant) product % \begin{eqnarray} J & =& Im(U_{ai}U_{bi}^* U_{aj}^* U_{bj}) \cr\cr & = &\frac{1}{8} \sin(2\theta_{12})\sin(2\theta_{13})\cos(\theta_{13})\sin(2\theta_{23})\sin \delta \,. \end{eqnarray} % Leptonic CP violation also requires that each of the leptons in each charge sector be nondegenerate with any other leptons in this sector; this is, course, true of the charged lepton sector and, for the neutrinos, this requires $\Delta m^2_{ij} \ne 0$ for each such pair $ij$. In the quark sector, $J$ is known to be small: $J_{\rm CKM} \sim {\cal O}(10^{-5})$. A promising asymmetry to measure is $P(\nu_e \to \nu_\mu)-P(\bar\nu_e - \bar\nu_\mu)$. As an illustration, in the absence of matter effects, % \begin{eqnarray} P(\nu_e \to \nu_\mu) - P(\bar\nu_e \to \bar\nu_\mu) & = & -4J(\sin 2\phi_{32}+ \sin 2\phi_{21} + \sin 2\phi_{13}) \cr & = & -16J \sin \phi_{32} \sin \phi_{13} \sin \phi_{21} \,, \label{pnuenumudif} \end{eqnarray} % where % \begin{equation} \phi_{ij} = \frac{\Delta m^2_{ij}L}{4E} \,. \label{phiijdef} \end{equation} % In order for the $CP$ violation in Eq.~(\ref{pnuenumudif}) to be large enough to measure, it is necessary that $\theta_{12}$, $\theta_{13}$, and $\Delta m^2_{\rm sol} = \Delta m^2_{21}$ not be too small. From atmospheric neutrino data, we have $\theta_{23}\simeq \pi/4$ and $\theta_{13} \ll 1$. If LMA describes solar neutrino data, then $\sin^2(2\theta_{12}) \simeq 0.8$, so $J \simeq 0.1\sin(2\theta_{13})\sin \delta$. For example, if $\sin^2(2\theta_{13})=0.04$, then $J$ could be $\gg J_{CKM}$. Furthermore, for parts of the LMA phase space where $\Delta m^2_{\rm sol} \sim 4 \times 10^{-5}$ eV$^2$ the CP violating effects might be observable. In the absence of matter, one would measure the asymmetry % \begin{equation} \frac{P(\nu_e \to \nu_\mu) - P(\bar\nu_e \to \bar\nu_\mu)}{ P(\nu_e \to \nu_\mu) + P(\bar\nu_e \to \bar\nu_\mu)} = -\frac{\sin(2\theta_{12})\cot(\theta_{23})\sin\delta \sin \phi_{21}}{ \sin \theta_{13}} \end{equation} % However, in order to optimize this ratio, because of the smallness of $\Delta m^2_{21}$ even for the LMA, one must go to large pathlengths $L$, and here matter effects are important. These make leptonic $CP$ violation challenging to measure, because, even in the absence of any intrinsic $CP$ violation, these matter effects render the rates for $\nu_e \to \nu_\mu$ and $\bar\nu_e \to \bar\nu_\mu$ unequal since the matter interaction is opposite in sign for $\nu$ and $\bar\nu$. One must therefore subtract out the matter effects in order to try to isolate the intrinsic $CP$ violation. Alternatively, one might think of comparing $\nu_e \to \nu_\mu$ with the time-reversed reaction $\nu_\mu \to \nu_e$. Although this would be equivalent if $CPT$ is valid, as we assume, and although uniform matter effects are the same here, the detector response is quite different and, in particular, it is quite difficult to identify $e^\pm$. Results from SNO and KamLAND testing the LMA~\cite{bmw-kamland} will help further planning. The Neutrino Factory provides an ideal set of controls to measure $CP$ violation effects since we can fill the storage ring with either $\mu^+$ or $\mu^-$ particles and measure the ratio of the number of events $\bar\nu_e\rightarrow \bar\nu_\mu$/$\nu_e\rightarrow\nu_\mu$. Figure~\ref{cpfig} shows this ratio for a Neutrino Factory with 10$^{21}$ decays and a 50~kiloton detector as a function of the baseline length. The ratio depends on the sign of $\delta m^2_{32}$. The shaded band around either curve shows the variation of this ratio as a function of the $CP$-violating phase $\delta$. The number of decays needed to produce the error bars shown is directly proportional to $\sin^2\theta_{13}$, which for the present example is set to 0.004. Depending on the magnitude of $J$, one may be driven to build a Neutrino Factory just to understand $CP$ violation in the lepton sector, which could have a significant role in explaining the baryon asymmetry of the Universe~\cite{yanag}. \begin{figure}[tbh!] \centerline{\includegraphics[width=4.0in]{cp_fig.eps}} \bigskip \caption[CP violation effects in a Neutrino Factory] { \label{cpfig} Predicted ratios of wrong-sign muon event rates when positive and negative muons are stored in a 20~GeV Neutrino Factory, shown as a function of baseline. A muon measurement threshold of 4~GeV is assumed. The lower and upper bands correspond, respectively, to negatve and positive $\delta m^2_{32}$. The widths of the bands show how the predictions vary as the $CP$ violating phase $\delta$ is varied from $-\pi$/2 to $\pi$/2, with the thick lines showing the predictions for $\delta=0$. The statistical error bars correspond to a high-performance Neutrino Factory yielding a data sample of 10$^{21}$ decays with a 50~kiloton detector. The curves are based on calculations presented in~\cite{barger-entry}. } \end{figure} \subsection{Physics Potential of Superbeams} It is possible to extend the reach of the current conventional neutrino experiments by enhancing the capabilities of the proton sources that drive them. These enhanced neutrino beams have been termed ``superbeams'' and form an intermediate step on the way to a Neutrino Factory. Their capabilities have been explored in recent papers~\cite{superbeams,bargersuperbeam,superbeam-peak}. These articles consider the capabilities of enhanced proton drivers at (i) the proposed 0.77~MW 50~GeV proton synchrotron at the Japan Hadron Facility (JHF)~\cite{jhf}, (ii) a 4~MW upgraded version of the JHF, (iii) a new $\sim 1$~MW 16~GeV proton driver~\cite{brighter} that would replace the existing 8~GeV Booster at Fermilab, or (iv) a fourfold intensity upgrade of the 120~GeV Fermilab Main Injector (MI) beam (to 1.6~MW) that would become possible once the upgraded (16~GeV) Booster was operational. Note that the 4~MW 50~GeV JHF and the 16~GeV upgraded Fermilab Booster are both suitable proton drivers for a neutrino factory. The conclusions of both reports are that superbeams will extend the reaches in the oscillation parameters of the current neutrino experiments but ``the sensitivity at a Neutrino Factory to $CP$ violation and the neutrino mass hierarchy extends to values of the amplitude parameter $\sin^2 2\theta_{13}$ that are one to two orders of magnitude lower than at a superbeam''~\cite{bargersuperbeam,superbeam-peak}. To illustrate these points, we choose one of the most favorable superbeam scenarios studied: a 1.6~MW NuMI-like high energy beam with $L = 2900$~km, detector parameters corresponding to the liquid argon scenario in~\cite{bargersuperbeam,superbeam-peak}, and oscillation parameters $|\delta m^2_{32}| = 3.5 \times 10^{-3}$~eV$^2$ and $\delta m^2_{21} = 1 \times 10^{-4}$~eV$^2$. The calculated three-sigma error ellipses in the $\left(N(e^+), N(e^-)\right)$ plane are shown in Fig.~\ref{fig:signdm2} for both signs of $\delta m^2_{32}$, with the curves corresponding to various $CP$ phases $\delta$ (as labeled). The magnitude of the $\nu_\mu \to \nu_e$ oscillation amplitude parameter $\sin^2 2\theta_{13}$ varies along each curve, as indicated. The two groups of curves, which correspond to the two signs of $\delta m^2_{32}$, are separated by more than $3\sigma$ provided $\sin^2 2\theta_{13} \gsim 0.01$. Hence the mass heirarchy can be determined provided the $\nu_\mu \to \nu_e$ oscillation amplitude is not more than an order of magnitude below the currently excluded region. Unfortunately, within each group of curves, the $CP$-conserving predictions are separated from the maximal $CP$-violating predictions by at most $3\sigma$. Hence, it will be difficult to conclusively establish $CP$ violation in this scenario. Note for comparison that a very long baseline experiment at a neutrino factory would be able to observe $\nu_e \to \nu_\mu$ oscillations and determine the sign of $\delta m^2_{32}$ for values of $\sin^2 2\theta_{13}$ as small as ${\cal O}(0.0001)$. This is illustrated in Fig.~\ref{fig:nufact}. A Neutrino Factory thus outperforms a conventional superbeam in its ability to determine the sign of $\delta m^2_{32}$. Comparing Fig.~\ref{fig:signdm2} and Fig.~\ref{fig:nufact} one sees that the value of $\sin^2 2\theta_{13}$, which has yet to be measured, will determine the parameters of the first Neutrino Factory. % \begin{figure}[tbh!] \centerline{\includegraphics[width=4.0in]{fig15_superbeams.ps}} \caption[Error ellipses for superbeams for electron appearance] {Three-sigma error ellipses in the $\left(N(e^+), N(e^-)\right)$ plane, shown for $\nu_\mu \to \nu_e$ and $\bar\nu_\mu \to \bar\nu_e$ oscillations in a NuMI-like high energy neutrino beam driven by a 1.6~MW proton driver. The calculation assumes a liquid argon detector with the parameters listed in \cite{superbeams}, a baseline of 2900~km, and 3~years of running with neutrinos, 6~years running with antineutrinos. Curves are shown for different CP phases $\delta$ (as labelled), and for both signs of $\delta m^2_{32}$ with $|\delta m^2_{32}| = 0.0035$~eV$^2$, and the sub-leading scale $\delta m^2_{21} = 10^{-4}$~eV$^2$. Note that $\sin^22\theta_{13}$ varies along the curves from 0.001 to 0.1, as indicated~\cite{bargersuperbeam}. } \label{fig:signdm2} \end{figure} % \begin{figure}[tbh!] %\epsfxsize=3.1in\epsffile{bgrwfig12.eps} \centerline{\includegraphics[width=4.0in]{fig16_superbeams.ps}} \caption[Error ellipses for Neutrino Factory for muon appearance] {Three-sigma error ellipses in the $\left(N(\mu+), N(\mu-)\right)$ plane, shown for a 20~GeV neutrino factory delivering $3.6\times10^{21}$ useful muon decays and $1.8\times10^{21}$ antimuon decays, with a 50~kt detector at $L = 7300$~km, $\delta m^2_{21} = 10^{-4}$~eV$^2$, and $\delta = 0$. Curves are shown for both signs of $\delta m^2_{32}$; $\sin^22\theta_{13}$ varies along the curves from 0.0001 to 0.01, as indicated~\cite{bargersuperbeam}. } \label{fig:nufact} \end{figure} Finally, we compare the superbeam $\nu_\mu \to \nu_e$ reach with the corresponding Neutrino Factory $\nu_e \to \nu_\mu$ reach in Fig.~\ref{fig:reach}, which shows the $3\sigma$ sensitivity contours in the $(\delta m^2_{21}, \sin^2 2\theta_{13})$ plane. The superbeam $\sin^2 2\theta_{13}$ reach of a few $\times 10^{-3}$ is almost independent of the sub-leading scale $\delta m^2_{21}$. However, since the neutrino factory probes oscillation amplitudes $O(10^{-4})$ the sub-leading effects cannot be ignored, and $\nu_e \to \nu_\mu$ events would be observed at a Neutrino Factory over a significant range of $\delta m^2_{21}$ even if $\sin^2 2\theta_{13} = 0$. % \begin{figure}[tbh!] \centerline{\includegraphics[width=4.0in]{fig20_superbeams.ps}} \caption[Comparison of superbeams and Neutrino Factories] {Summary of the $3\sigma$ level sensitivities for the observation of $\nu_\mu \to \nu_e$ at various MW-scale superbeams (as indicated) with liquid argon ``A'' and water cerenkov ``W'' detector parameters, and the observation of $\nu_e \to \nu_\mu$ in a 50~kt detector at 20, 30, 40, and 50~GeV neutrino factories delivering $2 \times 10^{20}$ muon decays in the beam-forming straight section. The limiting $3\sigma$ contours are shown in the ($\delta m^2_{21}, \sin^2 2\theta_{13}$) plane. All curves correspond to 3~years of running. The grey shaded area is already excluded by current experiments. } \label{fig:reach} \end{figure} %% restart here 1.16.02, noon \subsection{Non-oscillation physics at a Neutrino Factory} The study of the utility of intense neutrino beams from a muon storage ring in determining the parameters governing non-oscillation physics was begun in 1997~\cite{rajageer}. More complete studies can be found in~\cite{INTRO:ref9} and recently a European group has brought out an extensive study on this topic~\cite{cern-nonosc}. A Neutrino Factory can measure individual parton distributions within the proton for all light quarks and anti-quarks. It could improve valence distributions by an order of magnitude in the kinematical range $x\gsim 0.1$ in the unpolarized case. The individual components of the sea ($\bar{u}$, $\bar{d}$, ${s}$ and $\bar{s}$), as well as the gluon, would be measured with relative accuracies in the range of 1--10\%, for $0.1\lsim x \lsim 0.6$. A full exploitation of the Neutrino Factory potential for polarized measurements of the shapes of individual partonic densities requires an {\it a priori} knowledge of the polarized gluon density. The forthcoming set of polarized deep inelastic scattering experiments at CERN, DESY and RHIC may provide this information. The situation is also very bright for measurements of $C$-even distributions. Here, the first moments of singlet, triplet and octet axial charges can be measured with accuracies that are up to one order of magnitude better than the current uncertainties. In particular, the improvement in the determination of the singlet axial charge would allow a definitive confirmation or refutation of the anomaly scenario compared to the `instanton' or `skyrmion' scenarios, at least if the theoretical uncertainty originating from the small-$x$ extrapolation can be kept under control. The measurement of the octet axial charge with a few percent uncertainty will allow a determination of the strange contribution to the proton spin better than 10\%, and allow stringent tests of models of $SU(3)$ violation when compared to the direct determination from hyperon decays. A measurement of $\as(M_Z)$ and $\sin^2\theta_W$ will involve different systematics from current measurements and will therefore provide an important consistency check of current data, although the accuracy of these values is not expected to be improved. The weak mixing angle can be measured in both the hadronic and leptonic modes with a precision of approximately $2\times 10^{-4}$, dominated by the statistics and the luminosity measurement. This determination would be sensitive to different classes of new-physics contributions. Neutrino interactions are a very good source of clean, sign-tagged charm particles. A Neutrino Factory can measure charm production with raw event rates up to 100 million charm events per year with $\simeq$ 2 million double-tagged events. (Note that charm production becomes significant for storage ring energies above 20~GeV). Such large samples are suitable for precise extractions of branching ratios and decay constants, the study of spin-transfer phenomena, and the study of nuclear effects in deep inelastic scattering. The ability to run with both hydrogen and heavier targets will provide rich data sets useful for quantitative studies of nuclear models. The study of $\Lambda$ polarization both in the target and in the fragmentation regions will help clarify the intriguing problem of spin transfer. Although the neutrino beam energies are well below any reasonable threshold for new physics, the large statistics makes it possible to search for physics beyond the Standard Model. The high intensity neutrino beam allows a search for the production and decay of neutral heavy leptons with mixing angle sensitivity two orders of magnitude better than present limits in the 30--80 MeV range. The exchange of new gauge bosons decoupled from the first generation of quarks and leptons can be seen via enhancements of the inclusive charm production rate, with a sensitivity well beyond the present limits. A novel neutrino magnetic moment search technique that uses oscillating magnetic fields at the neutrino beam source could discover large neutrino magnetic moments predicted by some theories. Rare lepton-flavor-violating decays of muons in the ring could be tagged in the deep inelastic scattering final states through the detection of wrong-sign electrons and muons, or of prompt taus. \subsection{Physics that can be done with Intense Cold Muon Beams} Experimental studies of muons at low and medium energies have had a long and distinguished history, starting with the first search for muon decay to electron plus gamma-ray~\cite{Hincks-Pontecorvo}, and including along the way the 1957 discovery of the nonconservation of parity, in which the $g$ value and magnetic moment of the muon were first measured~\cite{Garwinetal}. The years since then have brought great progress: limits on the standard-model-forbidden decay $\mu\to e\gamma$ have dropped by nine orders of magnitude, and the muon anomalous magnetic moment $a_\mu=(g_\mu-2)/2$ has yielded one of the more precise tests ($\approx1$ ppm) of physical theory~\cite{BNLg-2}. The front end of a Neutrino Factory has the potential to provide $\sim10^{21}$ muons per year, five orders of magnitude beyond the most intense beam currently available\footnote{The $\pi$E5 beam at PSI, Villigen, providing a maximum rate of $10^9$ muons/s~\cite{Edgecock}.}. Such a facility could enable precision measurements of the muon lifetime $\tau_\mu$ and Michel decay parameters as well as sensitive searches for lepton-flavor nonconservation (LFV), a possible ($P$- and $T$-violating) muon electric dipole moment (EDM) $d_\mu$~\cite{HIMUS99}, and $P$ and $T$ violation in muonic atoms. It could also lead to an improved direct limit on the mass of the muon neutrino~\cite{numass}. Of these possibilities, Marciano~\cite{Marciano97} has suggested that muon LFV (especially coherent muon-to-electron conversion in the field of a nucleus) is the ``best bet" for discovering signatures of new physics using low-energy muons; measurement of $d_\mu$ could prove equally exciting but is not yet as well developed, being only at the Letter of Intent stage at present~\cite{EDMLOI}\footnote{Experimentalists might argue that extending such measurements as $\tau_\mu$ and the Michel parameters is worthwhile whenever the state of the art allows substantial improvement. However, their comparison with theory is dominated by theoretical uncertainties. Thus, compared to Marciano's ``best bets," they represent weaker arguments for building a new facility.}. The search for $\mu\to e \gamma$ is also of great interest. The MEGA experiment recently set an upper limit $B(\mu^+\to e^+\gamma)<1.2\times10^{-11}$~\cite{MEGA}. Ways to extend sensitivity to the $10^{-14}$ level have been discussed~\cite{Cooper97}. Sensitivity greater than this may be possible but will be difficult since at high muon rate there will be background due to accidental coincidences; a possible way around this relies on the correlation between the electron direction and the polarization direction using a polarized muon beam. The $\mu$-to-$e$-conversion approach does not suffer from this drawback and has the additional virtue of sensitivity to possible new physics that does not couple to the photon. In the case of precision measurements ($\tau_\mu$, $a_\mu$, etc.), new-physics effects can appear only as small corrections arising from the virtual exchange of new massive particles in loop diagrams. In contrast, LFV and EDMs are forbidden in the standard model, thus their observation at any level constitutes evidence for new physics. The current status and prospects for advances in these areas are summarized in Table~\ref{tab:LEmuons}. It is worth recalling that LFV as a manifestation of neutrino mixing is suppressed as $(\delta m^2)^2/m_W^4$ and is thus entirely negligible. However, a variety of new-physics scenarios predict observable effects. Table~\ref{tab:newmuphys} lists some examples of limits on new physics that would be implied by nonobservation of $\mu$-to-$e$ conversion ($\mu^-N\to e^-N$) at the $10^{-16}$ level~\cite{Marciano97}. \begin{table} \caption[Current and future tests in low energy muons] {Some current and future tests for new physics with low-energy muons (from~\protect\cite{Marciano97}, \protect\cite{PDG}, and \protect\cite{Aoki01}). Note that the ``Current prospects" column refers to anticipated sensitivity of experiments currently approved or proposed; ``Future" gives estimated sensitivity with the Neutrino Factory front end. (The $d_\mu$ measurement is still at the Letter of Intent stage and the reach of experiments is not yet entirely clear.)\label{tab:LEmuons}} \begin{center} \begin{tabular}{|lccc|} \hline Test & Current bound & Current prospects & Future \\ \hline $B(\mu^+\to e^+\gamma)$ & $<1.2\times10^{-11}$ & $\approx5\times10^{-12}$ & $\sim10^{-14}$\\ $B(\mu^-{\rm Ti}\to e^-{\rm Ti})$ & $<4.3\times10^{-12}$ & $\approx2\times10^{-14}$ & $<10^{-16}$\\ $B(\mu^-{\rm Pb}\to e^-{\rm Pb})$ & $<4.6\times10^{-11}$ & & \\ $B(\mu^-{\rm Ti}\to e^+{\rm Ca})$ & $<1.7\times10^{-12}$ & & \\ $B(\mu^+\to e^+e^-e^+)$ & $<1\times10^{-12}$ & & \\ $d_\mu$ & $(3.7\pm3.4)\times10^{-19}\,e\cdot$cm & $10^{-24}\,e\cdot$cm? & ? \\ \hline \end{tabular} \end{center} \end{table} \begin{table} \caption[New physics probed by $\mu\rightarrow e$ experiments] {Some examples of new physics probed by the nonobservation of $\mu\rightarrow e$ conversion at the $10^{-16}$ level (from~\protect\cite{Marciano97}).\label{tab:newmuphys}} \begin{center} \begin{tabular}{|lc|} \hline New Physics & Limit \\ \hline Heavy neutrino mixing & $|V_{\mu N}^*V_{e N}|^2<10^{-12}$\\ Induced $Z\mu e$ coupling & $g_{Z_{\mu e}}<10^{-8}$\\ Induced $H\mu e$ coupling & $g_{H_{\mu e}}<4\times10^{-8}$\\ Compositeness & $\Lambda_c>3,000\,$TeV\\ \hline \end{tabular} \end{center} \end{table} Precision studies of atomic electrons have provided notable tests of QED ({ e.g,}\ the Lamb shift in hydrogen) and could in principle be used to search for new physics were it not for nuclear corrections. Studies of muonium ($\mu^+e^-$) are free of such corrections since it is a purely leptonic system. Muonic atoms also can yield new information complementary to that obtained from electronic atoms. A number of possibilities have been enumerated by Kawall {\it et al.}~\cite{Kawall97} and Molzon~\cite{Molzon97}. As an example we consider the hyperfine splitting of the muonium ground state, which has been measured to 36 ppb~\cite{Mariam} and currently furnishes the most sensitive test of the relativistic two-body bound state in QED~\cite{Kawall97}. The precision could be further improved with increased statistics. The theoretical error is 0.3 ppm but could be improved by higher-precision measurements in muonium and muon spin resonance, also areas in which the Neutrino Factory front end could contribute. Another interesting test is the search for muonium-antimuonium conversion, possible in new-physics models that allow violation of lepton family number by two units. The current limit is $R_g \equiv G_C / G_F< 0.0030$~\cite{PDG}, where $G_C$ is the new-physics coupling constant and $G_F$ is the Fermi coupling constant. This sets a lower limit of $\approx 1 \,$TeV$/c^2$ on the mass of a grand-unified dileptonic gauge boson and also constrains models with heavy leptons~\cite{Abela}. \subsection[Physics Potential of a Higgs Factory Muon Collider]% {Physics potential of a Low energy Muon Collider operating as a Higgs Factory} Muon colliders~\cite{bargersnow,clinehanson} have a number of unique features that make them attractive candidates for future accelerators~\cite{INTRO:ref5}. The most important and fundamental of these derive from the large mass of the muon in comparison to that of the electron.The synchrotron radiation loss in a circular accelerator goes as the inverse fourth power of the mass and is two billion times less for a muon than for an electron. Direct $s$ channel coupling to the higgs boson goes as the mass squared and is 40,000 greater for the muon than for the electron. This leads to: a)~the possibility of extremely narrow beam energy spreads, especially at beam energies below $100\gev$; b)~the possibility of accelerators with very high energy; c)~the possiblity of employing storage rings at high energy; d)~the possibility of using decays of accelerated muons to provide a high luminosity source of neutrinos as discussed in Section~\ref{neuf}; e)~increased potential for probing physics in which couplings increase with mass (as does the SM $\hsm f\bar f$ coupling) . The relatively large mass of the muon compared to the mass of the electron means that the coupling of Higgs bosons to $\mu^+\mu^-$ is very much larger than to $e^+e^-$, implying much larger $s$-channel Higgs production rates at a muon collider as compared to an electron collider. For Higgs bosons with a very small (MeV-scale) width, such as a light SM Higgs boson, production rates in the $s$-channel are further enhanced by the muon collider's ability to achieve beam energy spreads comparable to the tiny Higgs width. In addition, there is little beamstrahlung, and the beam energy can be tuned to one part in a million through continuous spin-rotation measurements~\cite{Raja:1998ip}. Due to these important qualitative differences between the two types of machines, only muon colliders can be advocated as potential $s$-channel Higgs factories capable of determining the mass and decay width of a Higgs boson to very high precision~\cite{Barger:1997jm,Barger:1995hr}. High rates of Higgs production at $\epem$ colliders rely on substantial $VV$ Higgs coupling for the $Z+$Higgs (Higgstrahlung) or $WW\to$Higgs ($WW$ fusion) reactions. In contrast, a $\mupmum$ collider can provide a factory for producing a Higgs boson with little or no $VV$ coupling so long as it has SM-like (or enhanced) $\mupmum$ couplings. Of course, there is a tradeoff between small beam energy spread, $\delta E/E=R$, and luminosity. Current estimates for yearly integrated luminosities (using $\call=1\times 10^{32}\rm\,cm^{-2}\ s^{-1}$ as implying $ L=1\fbi/{\rm yr}$) are: $\lyear\gsim 0.1,0.22,1 \fbi$ at $\rts\sim 100\gev$ for beam energy resolutions of $R=0.003\%,0.01\%,0.1\%$, respectively; $\lyear\sim 2,6,10 \fbi$ at $\rts\sim 200,350,400\gev$, respectively, for $R\sim 0.1\%$. Despite this, studies show that for small Higgs width the $s$-channel production rate (and statistical significance over background) is maximized by choosing $R$ to be such that $\srts\lsim \gamhtot$. In particular, in the SM context for $\mhsm\sim 110\gev$ this corresponds to $R\sim 0.003\%$. If the $\mh\sim 115\gev$ LEP signal is real, or if the interpretation of the precision electroweak data as an indication of a light Higgs boson (with substantial $VV$ coupling) is valid, then both $\epem$ and $\mupmum$ colliders will be valuable. In this scenario the Higgs boson would have been discovered at a previous higher energy collider (even possibly a muon collider running at high energy), and then the Higgs factory would be built with a center-of-mass energy precisely tuned to the Higgs boson mass. The most likely scenario is that the Higgs boson is discovered at the LHC via gluon fusion ($gg\to H$) or perhaps earlier at the Tevatron via associated production ($q\bar{q}\to WH, t\overline{t}H$), and its mass is determined to an accuracy of about 100~MeV. If a linear collider has also observed the Higgs via the Higgs-strahlung process ($e^+e^-\to ZH$), one might know the Higgs boson mass to better than 50~MeV with an integrated luminosity of $500$~fb$^{-1}$. The muon collider would be optimized to run at $\sqrt{s}\approx m_H$, and this center-of-mass energy would be varied over a narrow range so as to scan over the Higgs resonance (see Fig.~\ref{mhsmscan} below). \subsubsection{Higgs Production} The production of a Higgs boson (generically denoted $\h$) in the $s$-channel with interesting rates is a unique feature of a muon collider~\cite{Barger:1997jm,Barger:1995hr}. The resonance cross section is % \begin{equation} \sigma_h(\sqrt s) = {4\pi \Gamma(h\to\mu\bar\mu) \, \Gamma(h\to X)\over \left( s - m_h^2\right)^2 + m_h^2 \left(\Gamma_{\rm tot}^h \right)^2}\,. \label{rawsigform} \end{equation} In practice, however, there is a Gaussian spread ($\srts$) to the center-of-mass energy and one must compute the effective $s$-channel Higgs cross section after convolution assuming some given central value of $\rts$: % \begin{eqnarray} \bar\sigma_h(\sqrt s) & =& {1\over \sqrt{2\pi}\,\srts} \; \int \sigma_h (\sqrt{\what s}) \; \exp\left[ -\left( \sqrt{\what s} - \sqrt s\right)^2 \over 2\sigma_{\sqrt s}^2 \right] d \sqrt{\what s}\\ &&\stackrel{\rts=\mh}{\simeq} {4\pi\over m_h^2} \; {\br(h\to\mu\bar\mu) \, \br(h\to X) \over \left[ 1 + {8\over\pi} \left(\srts\over\gamhtot \right)^2 \right]^{1/2}} \ . % %\bar\sigma_h(\sqrt s) & =& {1\over \sqrt{2\pi}\,\srts} \; \int \sigma_h %(\sqrt{\what s}) \; \exp\left[ -\left( \sqrt{\what s} - \sqrt s\right)^2 \over %2\sigma_{\sqrt s}^2 \right] d \sqrt{\what s}\\ %\stackrel{\rts=\mh}{\simeq} {4\pi\over m_h^2} \; {\br(h\to\mu\bar\mu) \, %\br(h\to X) \over \left[ 1 + {8\over\pi} \left(\srts\over\gamhtot %\right)^2 \right]^{1/2}} \,. \label{sigform} \end{eqnarray} % \begin{figure}[tbh!] \centering\leavevmode %\epsfxsize=3.5in\epsffile{singlemhscan_mh110_r003_brem.ps} \centerline{\includegraphics[width=4.0in]{singlemhscan_mh110_r003_brem.ps}} \caption[Scan of the Higgs resonance using a muon collider]{ Number of events and statistical errors in the $b\overline{b}$ final state as a function of $\protect\rts$ in the vicinity of $\mhsm=110\gev$, assuming $R=0.003\%$, and $\epsilon L=0.00125$~fb$^{-1}$ at each data point. %The precise theoretical prediction is given by the solid line. %The dotted (dashed) curve is the theoretical prediction %if $\Gamma _{tot}$ is decreased (increased) by 10\%, {\it keeping %the $\Gamma(h\to\mu^+\mu^-)$ and $\Gamma(h\to b\overline{b})$ %partial widths fixed at the predicted SM value.} \label{mhsmscan}} \end{figure} % It is convenient to express $\srts$ in terms of the root-mean-square (rms) Gaussian spread of the energy of an individual beam, $R$: % \begin{equation} \srts = (2{\rm~MeV}) \left( R\over 0.003\%\right) \left(\sqrt s\over 100\rm~GeV\right) \,. \end{equation} % From Eq.~(\ref{rawsigform}), it is apparent that a resolution $\srts \lsim \gamhtot$ is needed to be sensitive to the Higgs width. Further, Eq.~(\ref{sigform}) implies that $\bar\sigma_h\propto 1/\srts$ for $\srts>\gamhtot$ {\it and} that large event rates are only possible if $\gamhtot$ is not so large that $\br(\h\to \mu\bar\mu)$ is extremely suppressed. The width of a light SM-like Higgs is very small ({ e.g}, a few MeV for $\mhsm\sim 110\gev$), implying the need for $R$ values as small as $\sim 0.003\%$ for studying a light SM-like $\h$. Figure~\ref{mhsmscan} illustrates the result for the SM Higgs boson of an initial centering scan over $\rts$ values in the vicinity of $\mhsm=110\gev$. This figure dramatizes: a)~that the beam energy spread must be very small because of the very small $\gamhsmtot$ (when $\mhsm$ is small enough that the $WW^\star$ decay mode is highly suppressed); b)~that we require the very accurate {\it in situ} determination of the beam energy to one part in a million through the spin precession of the muon noted earlier in order to perform the scan and then center on $\rts=\mhsm$ with a high degree of stability. If the $\h$ has SM-like couplings to $WW$, its width will grow rapidly for $\mh>2m_W$ and its $s$-channel production cross section will be severely suppressed by the resulting decrease of $\br(\h\to\mu\mu)$. More generally, any $\h$ with SM-like or larger $\h\mu\mu$ coupling will retain a large $s$-channel production rate when $\mh>2m_W$ only if the $\h WW$ coupling becomes strongly suppressed relative to the $\hsm WW$ coupling. The general theoretical prediction within supersymmetric models is that the lightest supersymmetric Higgs boson $\hl$ will be very similar to the $\hsm$ when the other Higgs bosons are heavy. This `decoupling limit' is very likely to arise if the masses of the supersymmetric particles are large (since the Higgs masses and the superparticle masses are typically similar in size for most boundary condition choices). Thus, $\hl$ rates will be very similar to $\hsm$ rates. In contrast, the heavier Higgs bosons in a typical supersymmetric model decouple from $VV$ at large mass and remain reasonably narrow. As a result, their $s$-channel production rates remain large. For a SM-like $\h$, at $\sqrt s = \mh \approx 115$~GeV and $R=0.003\%$, the $b\bar b$ rates are \vspace{-.05in} \begin{eqnarray} \rm signal &\approx& 10^4\rm\ events\times L(fb^{-1})\,,\\ \rm background &\approx& 10^4\rm\ events\times L(fb^{-1})\,. \end{eqnarray} \subsubsection{What the Muon Collider Adds to LHC and LC Data} An assessment of the need for a Higgs factory requires that one detail the unique capabilities of a muon collider versus the other possible future accelerators as well as comparing the abilities of all the machines to measure the same Higgs properties. Muon colliders, and a Higgs factory in particular, would only become operational after the LHC physics program is well-developed and, quite possibly, after a linear collider program is mature as well. So one important question is the following: if a SM-like Higgs boson and, possibly, important physics beyond the Standard Model have been discovered at the LHC and perhaps studied at a linear collider, what new information could a Higgs factory provide? The $s$-channel production process allows one to determine the mass, total width, and the cross sections $\overline \sig_h(\mupmum\to\h\to X)$ for several final states $X$ to very high precision. The Higgs mass, total width and the cross sections can be used to constrain the parameters of the Higgs sector. For example, in the MSSM their precise values will constrain the Higgs sector parameters $\mha$ and $\tanb$ (where $\tanb$ is the ratio of the two vacuum expectation values (vevs) of the two Higgs doublets of the MSSM). The main question is whether these constraints will be a valuable addition to LHC and LC constraints. The expectations for the luminosity available at linear colliders has risen steadily. The most recent studies assume an integrated luminosity of some $500$~fb$^{-1}$ corresponding to 1--2 years of running at a few$\times100$~fb$^{-1}$ per year. This luminosity results in the production of greater than $10^4$ Higgs bosons per year through the Bjorken Higgs-strahlung process, $e^+e^-\to Z\h$, provided the Higgs boson is kinematically accessible. This is comparable or even better than can be achieved with the current machine parameters for a muon collider operating at the Higgs resonance; in fact, recent studies have described high-luminosity linear colliders as ``Higgs factories,'' though for the purposes of this report, we will reserve this term for muon colliders operating at the $s$-channel Higgs resonance. A linear collider with such high luminosity can certainly perform quite accurate measurements of certain Higgs parameters, such as the Higgs mass, couplings to gauge bosons and couplings to heavy quarks ~\cite{Battaglia:2000jb}. Precise measurements of the couplings of the Higgs boson to the Standard Model particles is an important test of the mass generation mechanism. In the Standard Model with one Higgs doublet, this coupling is proportional to the particle mass. In the more general case there can be mixing angles present in the couplings. Precision measurements of the couplings can distinguish the Standard Model Higgs boson from that from a more general model and can constrain the parameters of a more general Higgs sector. \begin{table*}[h!] \begin{center} \caption[Comparison of a Higgs factory muon collider with LHC and LC] {Achievable relative uncertainties for a SM-like $\mh=110$~GeV for measuring the Higgs boson mass and total width for the LHC, LC (500~fb$^{-1}$), and the muon collider (0.2~fb$^{-1}$). }\label{unc-table} \protect\protect \begin{tabular}{|cccc|} \hline \ & LHC & LC & $\mu^+\mu^-$\\ \hline $\mh$ & $9\times 10^{-4}$ & $3\times 10^{-4}$ & $1-3\times 10^{-6}$ \\ $\gamhtot$ & $>0.3$ & 0.17 & 0.2 \\ \hline \end{tabular} \end{center} \end{table*} The accuracies possible at different colliders for measuring $\mh$ and $\gamhtot$ of a SM-like $\h$ with $\mh\sim 110\gev$ are given in Table~\ref{unc-table}. Once the mass is determined to about 1~MeV at the LHC and/or LC, the muon collider would employ a three-point fine scan~\cite{Barger:1997jm} near the resonance peak. Since all the couplings of the Standard Model are known, $\gamhsmtot$ is known. Therefore a precise determination of $\gamhtot$ is an important test of the Standard Model, and any deviation would be evidence for a nonstandard Higgs sector. For a SM Higgs boson with a mass sufficiently below the $WW^\star$ threshold, the Higgs total width is very small (of order several MeV), and the only process where it can be measured {\it directly} is in the $s$-channel at a muon collider. Indirect determinations at the LC can have higher accuracy once $\mh$ is large enough that the $WW^\star$ mode rates can be accurately measured, requiring $\mh>120\gev$. This is because at the LC the total width must be determined indirectly by measuring a partial width and a branching fraction, and then computing the total width, \begin{eqnarray} &&\Gamma _{tot}={{\Gamma(h\to X)}\over {BR(h\to X)}}\;, \end{eqnarray} for some final state $X$. For a Higgs boson so light that the $WW^\star$ decay mode is not useful, the total width measurement would probably require use of the $\gamma \gamma $ decays~\cite{Gunion:1996cn}. This would require information from a photon collider as well as the LC and a small error is not possible. The muon collider can measure the total width of the Higgs boson directly, a very valuable input for precision tests of the Higgs sector. To summarize, if a Higgs is discovered at the LHC or possibly earlier at the Fermilab Tevatron, attention will turn to determining whether this Higgs has the properties expected of the Standard Model Higgs. If the Higgs is discovered at the LHC, it is quite possible that supersymmetric states will be discovered concurrently. The next goal for a linear collider or a muon collider will be to better measure the Higgs boson properties to determine if everything is consistent within a supersymmetric framework or consistent with the Standard Model. A Higgs factory of even modest luminosity can provide uniquely powerful constraints on the parameter space of the supersymmetric model via its very precise measurement of the light Higgs mass, the highly accurate determination of the total rate for $\mupmum\to\hl\to b\bar b$ (which has almost zero theoretical systematic uncertainty due to its insensitivity to the unknown $m_b$ value) and the moderately accurate determination of the $\hl$'s total width. In addition, by combining muon collider data with LC data, a completely model-independent and very precise determination of the $h^0\mu^+\mu^-$ coupling is possible. This will provide another strong discriminator between the SM and the MSSM. Further, the $h^0\mu^+\mu^-$ coupling can be compared to the muon collider and LC determinations of the $h^0\tau^+\tau^-$ coupling for a precision test of the expected universality of the fermion mass generation mechanism. \subsection{Physics Potential of a High Energy Muon Collider} Once one learns to cool muons, it becomes possible to build muon colliders with energies of $\approx$ 3 TeV in the center of mass that fit on an existing laboratory site~\cite{INTRO:ref5,rajawitherell}. At intermediate energies, it becomes possible to measure the W mass \cite{bbgh-wtt,bergerw} and the top quark mass~\cite{bbgh-wtt,bergertop} with high accuracy, by scanning the thresholds of these particles and making use of the excellent energy resolution of the beams. We consider further here the ability of a higher energy muon collider to scan higher-lying Higgs like objects such as the H$^0$ and the A$^0$ in the MSSM that may be degenerate with each other. \subsubsection{Heavy Higgs Bosons} As discussed in the previous section, precision measurements of the light Higgs boson properties might make it possible to not only distinguish a supersymmetric boson from a Standard Model one, but also pinpoint a range of allowed masses for the heavier Higgs bosons. This becomes more difficult in the decoupling limit where the differences between a supersymmetric and Standard Model Higgs are smaller. Nevertheless with sufficiently precise measurements of the Higgs branching fractions, it is possible that the heavy Higgs boson masses can be inferred. A muon collider light-Higgs factory might be essential in this process. In the context of the MSSM, $\mha$ can probably be restricted to within $50\gev$ or better if $\mha<500\gev$. This includes the $250-500\gev$ range of heavy Higgs boson masses for which discovery is not possible via $\hh\ha$ pair production at a $\rts=500\gev$ LC. Further, the $\ha$ and $\hh$ cannot be detected in this mass range at either the LHC or LC in $b\bar b \hh,b\bar b\ha$ production for a wedge of moderate $\tanb$ values. (For large enough values of $\tanb$ the heavy Higgs bosons are expected to be observable in $b\bar b \ha,b\bar b \hh$ production at the LHC via their $\tau ^+\tau ^-$ decays and also at the LC.) A muon collider can fill some, perhaps all of this moderate $\tanb$ wedge. If $\tanb$ is large, the $\mupmum \hh$ and $\mupmum\ha$ couplings (proportional to $\tanb$ times a SM-like value) are enhanced, thereby leading to enhanced production rates in $\mupmum$ collisions. The most efficient procedure is to operate the muon collider at maximum energy and produce the $\hh$ and $\ha$ (often as overlapping resonances) via the radiative return mechanism. By looking for a peak in the $b\bar b$ final state, the $\hh$ and $\ha$ can be discovered and, once discovered, the machine $\rts$ can be set to $\mha$ or $\mhh$ and factory-like precision studies pursued. Note that the $\ha$ and $\hh$ are typically broad enough that $R=0.1\%$ would be adequate to maximize their $s$-channel production rates. In particular, $\Gamma\sim 30$~MeV if the $t\overline{t}$ decay channel is not open, and $\Gamma\sim 3$~GeV if it is. Since $R=0.1\%$ is sufficient, much higher luminosity ($L\sim 2-10~{\rm fb}^{-1} /{\rm yr}$) would be possible as compared to that for $R=0.01\%-0.003\%$ required for studying the $\hl$. In short, for these moderate $\tanb$--$\mha\gsim 250\gev$ scenarios that are particularly difficult for both the LHC and the LC, the muon collider would be the only place that these extra Higgs bosons can be discovered and their properties measured very precisely. In the MSSM, the heavy Higgs bosons are largely degenerate, especially in the decoupling limit where they are heavy. Large values of $\tan \beta$ heighten this degeneracy. A muon collider with sufficient energy resolution might be the only possible means for separating out these states. Examples showing the $H$ and $A$ resonances for $\tan \beta =5$ and $10$ are shown in Fig.~\ref{H0-A0-sep}. For the larger value of $\tan \beta$ the resonances are clearly overlapping. For the better energy resolution of $R=0.01\%$, the two distinct resonance peaks are still visible, but become smeared out for $R=0.06\%$. \begin{figure}[tbh!] \centering\leavevmode \centerline{\includegraphics[width=4.0in]{hh_ha_susy_rtsscan.ps}} \caption[Separation of $A$ and $H$ signals for $\tan\beta=5$ and $10$] {Separation of $A$ and $H$ signals for $\tan\beta=5$ and $10$. From Ref.~\cite{Barger:1997jm}. \label{H0-A0-sep}} \end{figure} Once muon colliders of these intermediate energies can be built, higher energies such as 3--4~TeV in the center of mass become feasible. Muon colliders with these energies will be complementary to hadron colliders of the SSC class and above. The background radiation from neutrinos from the muon decay becomes a problem at $\approx$~3~TeV in the CoM. Ideas for ameliorating this problem have been discussed and include optical stochastic cooling to reduce the number of muons needed for a given luminosity, elimination of straight sections via wigglers or undulators, or special sites for the collider such that the neutrinos break ground in uninhabited areas.