\clearpage \subsection{The neutrino factory oscillation physics program} \label{program} We now consider the program of neutrino oscillation measurements at a neutrino factory in the era beyond the next generation of long baseline experiments. The main goals in this era are likely to be to precisely establish the oscillation framework, determine the pattern of neutrino masses, measure matter effects to confirm the MSW phenomenon, make precise measurements or place stringent limits on all of the mixing--matrix elements (and hence mixing--angles), and observe or place stringent limits on CP violation in the lepton sector. A neutrino factory can address each of these goals: \begin{description} \item{(i)} Establishing the oscillation framework. This requires measuring as a function of $L/E$, or putting stringent limits on, all of the oscillation probabilities $P(\nu_e\to\nu_x)$ and $P(\nu_\mu\to\nu_x)$. The oscillation framework can be established by summing the probabilities (a) $P(\nu_e\to\nu_e) + P(\nu_e\to\nu_\mu) + P(\nu_e\to\nu_\tau)$, and (b) $P(\nu_\mu\to\nu_e) + P(\nu_\mu\to\nu_\mu) + P(\nu_\mu\to\nu_\tau)$. In a three--flavor mixing framework, both sums should be unity for all $L/E$. If there are sterile neutrinos participating in the oscillations one or both of the sums will be less than unity. Part (b) of the test will almost certainly be made with conventional neutrino beams, although with a precision that will be limited by the $\nu_\mu\to\nu_\tau$ statistics and by the uncertainty on the $P(\nu_\mu\to\nu_e)$ measurement arising from the O(1\%) $\nu_e$ contamination in the beam. Part (a) of the test, which includes the first observation of (or stringent limits on) $\nu_e\to\nu_\tau$ oscillations, can only be made with an energetic ($E_\nu > 10$~GeV) $\nu_e$ (or $\overline{\nu}_e$) beam, and will therefore be a unique part of the neutrino factory physics program. \item{(ii)} Determining the pattern of neutrino masses. The present experimental data suggests that, within a three--flavor mixing framework, there are two neutrino mass eigenstates separated by a small mass difference, and a third state separated from the pair by a large" mass difference $\delta M^2$. What is unknown is whether there is one low state plus two high states, or two low states plus one high state. This can be determined by measuring the sign of $\delta M^2$. The only way we know of making this measurement is to exploit matter effects which, in a very long baseline experiment, alter the probabilities for oscillations that involve electron neutrinos; the modification being dependent on the sign of $\delta M^2$. In principle the measurement could be made using a conventional neutrino beam and measuring $\nu_\mu\to\nu_e$ and $\overline{\nu}_\mu\to\overline{\nu}_e$ transitions over a baseline of several thousand km. However, the O(1\%) $\nu_e$ ($\overline{\nu}_e$) contamination in the beam will introduce an irreducible background that is comparable to, or larger than, the $\nu_e$ signal. In contrast, at a neutrino factory it appears that the measurement can be done with great precision. Hence, determining the sign of $\delta M^2$ and the pattern of neutrino masses would be a key measurement at a neutrino factory. \item{(iii)} Measuring matter effects to confirm the MSW phenomenon. The same technique used to determine the sign of $\delta m^2_{32}$ can, with sufficient statistics, provide a precise quantitative confirmation of the MSW effect for neutrinos passing through the Earth. The modification to $P(\nu_e\to\nu_\mu)$, for example, depends upon the matter parameter $A$ (Eq.~(\ref{eq:defnA})). Global fits to appearance and disappearance spectra that are used to determine the oscillation parameters can include $A$ as a free parameter. The quantitative MSW test would be to recover the expected value for $A$. This measurement exploits the clean $\nu_e\to\nu_\mu$ signal at a neutrino factory, and would be a unique part of the neutrino factory physics program. \item{(iv)} Making precise measurements or placing stringent limits on all of the mixing--matrix elements. In practice the measured oscillation probability amplitudes are used to determine the mixing angles. If any of the angles are unmeasured or poorly constrained the relevant entries in the mixing matrix will also be poorly determined. At present there is only an upper limit on $\theta_{13}$, the angle that essentially determines the $\nu_e\to\nu_\mu$ oscillation amplitude. A neutrino factory would provide a precise measurement of, or stringent limit on, this difficult angle. In fact, because all of the $\nu_\mu\to\nu_x$ and $\nu_e\to\nu_x$ oscillation amplitudes can be measured at a neutrino factory, global fits can be made to the measured spectra to provide a very precise determination of the mixing angles. This exploits the $\nu_e$ component in the beam. Finally, it should be noted that it is important to test the overall consistency of the oscillation framework by determining the mixing angles in more than one way, i.e. by using more than one independent set of measurements. Clearly the $\nu_e$ beam is an asset for this check. \item{(v)} Placing stringent limits on, or observing, CP violation in the lepton sector. Most of the oscillation scenarios defined for the study predict very small CP violating amplitudes. An important test of these scenarios would be to place stringent experimental limits on CP violation in the lepton sector. The LMA scenario IA1 might result in sufficiently large CP violating effects to be observable at a neutrino factory. The CP test involves comparing $\nu_e\to\nu_\mu$ with $\overline{\nu}_e\to\overline{\nu}_\mu$ oscillation rates, possible at a neutrino factory because backgrounds are very small. A search for CP violation in the lepton sector with the required precision cannot be done with a conventional neutrino beam, and is therefore a unique part of the neutrino factory physics program. \end{description} Note that it is the $\nu_e$ ($\overline{\nu}_e$) component in the neutrino factory beam that drives the oscillation physics program. A $\nu_e$ beam would (a) enable a basic test of the oscillation framework that cannot be made with a $\nu_\mu$ beam, (b) enable the first observation of (or stringent limits on) $\nu_e\to\nu_\tau$ oscillations, (c) make a convincing determination of the pattern of neutrino masses that would be difficult or impossible with a conventional neutrino beam, (d) make a quantitative check of the MSW effect only possible with a neutrino factory beam, (e) enable measurements or stringent limits on all of the (three--flavor) mixing angles with a precision that requires both $\nu_e$ and $\nu_\mu$ beams, and (f) measure or put meaningful limits on CP violation in the lepton sector, which requires a signal purity only available at a neutrino factory. A neutrino factory operating in the next decade, after the next generation of long baseline experiments, would appear to be the right tool at the right time. However, before we can quantitatively assess how well a neutrino factory might realize the physics program we have listed, we must first understand the capabilities of neutrino detectors in the neutrino factory era. %\clearpage \subsection{Detector considerations} \label{detectors} We would like to measure the oscillation probabilities $P(\nu_\alpha \rightarrow \nu_\beta)$ as a function of the baseline $L$ and neutrino energy $E$ (and hence $L/E$) for all possible initial and final flavors $\alpha$ and $\beta$. This requires a beam with a well known initial flavor content, and a detector that can identify the flavor of the interacting neutrino. The neutrinos interact in the detector via charged current (CC) and neutral current (NC) interactions to produce a lepton accompanied by a hadronic shower arising from the remnants of the struck nucleon. In CC interactions the final state lepton tags the flavor ($\beta$) of the interacting neutrino. At a neutrino factory in which, for example, positive muons are stored, the initial beam consists of 50\% $\nu_e$ and 50\% $\overline{\nu}_\mu$. In the absence of oscillations, the $\nu_e$ CC interactions produce electrons and the $\overline{\nu}_\mu$ CC interactions produce positive muons. Note that the charge of the final state lepton tags the flavor ($\alpha$) of the initial neutrino or antineutrino. In the presence of $\nu_e \rightarrow \nu_\mu$ oscillations the $\nu_\mu$ CC interactions produce negative muons (i.e. wrong--sign muons). Similarly, $\overline{\nu}_\mu \rightarrow \overline{\nu}_e$ oscillations produce wrong--sign electrons, $\overline{\nu}_\mu \rightarrow \overline{\nu}_\tau$ oscillations produce events tagged by a $\tau^+$ and $\nu_e \rightarrow \nu_\tau$ oscillations produce events tagged by a $\tau^-$. Hence, there is a variety of information that can be used to measure or constrain neutrino oscillations at a neutrino factory, namely the rates and energy distributions of events tagged by (a) right--sign muons, (b) wrong--sign muons, (c) electrons or positrons (their charge is difficult to determine in a massive detector), (d) positive $\tau$--leptons, (e) negative $\tau$--leptons, and (f) no charged lepton. If these measurements are made when there are alternately positive and negative muons decaying in the storage ring, there are a total of 12 spectra that can be used to extract information about the oscillations. Some examples of the predicted measured spectra are shown as a function of the oscillation parameters in Figs.~\ref{fig:m1} and \ref{fig:m2} for a 10~kt detector sited 7400~km downstream of a 30~GeV neutrino factory. Clearly, the high intensity $\nu_e$, $\overline{\nu}_e$, $\nu_\mu$, and $\overline{\nu}_\mu$ beams at a neutrino factory would provide a wealth of precision oscillation data. \begin{figure} \epsfxsize3.4in \centerline{\epsffile{mum.eps}} %\vspace{-1.0cm} \caption{Visible energy spectra for four event classes when $10^{21} \mu^-$ decay in a 30~GeV neutrino factory at $L = 7400$~km. Black histogram: no oscillations. Blue dotted histogram: $\delta m^2_{32}=3.5\times 10^{-3}$~eV$^2$/c$^4$, $\sin^2\theta_{23}=1$. Red dashed histogram: $\delta m^2_{32}=7\times 10^{-3}$~eV$^2$/c$^4$, $\sin^2\theta_{23}=1$. The distributions in this figure and the following figure are for an ICANOE-type detector, and are from Ref.~\ref{camp00}.} \label{fig:m1} \end{figure} % \begin{figure} \epsfxsize3.4in \centerline{\epsffile{mup.eps}} %\vspace{-1.0cm} \caption{Same as previous figure, but with positive muons circulating in the storage ring. The difference between the two figures is due to the different cross section for neutrinos and antineutrinos, and to matter effects.} \label{fig:m2} \end{figure} The detectors required at a neutrino factory will have many similarities to the detectors that have been designed for the next generation of experiments at conventional neutrino beams. However, there are some important differences. First, we can anticipate more massive detectors. The sensitivity of a neutrino factory oscillation experiment is proportional to the product of the detector mass and beam intensity. It is likely that the cost of increasing the MINOS detector fiducial mass (for example) by a factor of a few is smaller than the cost of increasing the neutrino factory beam intensity by a factor of a few. Therefore, we believe that it is reasonable to assume that detectors at a neutrino factory would be a factor of a few to a factor of 10 more massive than the generation of neutrino detectors presently under construction. Second, the presence of both neutrinos and antineutrinos in the same beam at a neutrino factory places a premium on measuring the sign of the charge of the lepton produced in CC interactions. Charge--sign determination may not be practical for electrons, but is mandatory for muons and highly desirable for $\tau$--leptons. Finally, a relatively low energy threshold for the detection and measurement of wrong--sign muons is very desirable. This is because high muon detection thresholds require high energy interacting neutrinos, and hence a high energy neutrino factory. Since the muon acceleration system at a neutrino factory is likely to be expensive, low energies are preferable. In the following sections we begin by considering general detector issues for the measurement of final state muons and $\tau$--leptons, and then consider some specific candidate detectors for a neutrino factory. Some of these detector types are quite new and are just beginning to be studied; for the more mature detectors the neutrino'' energy resolution, the signal efficiency, background rejection, and fiducial mass are discussed. %\clearpage \subsubsection{Muon identification and measurement} \label{bkgds} The detection and measurement of muons (especially those of opposite sign to the muons in the storage ring) is crucial for many of the key oscillation physics measurements at a neutrino factory. Before considering some specific neutrino factory detectors it is useful to consider more generally muon backgrounds and related issues. Background muons can be produced in NC and CC interactions by: \begin{description} \item{(i)} Pions or kaons from the hadronic shower that decay to produce a muon. \item{(ii)} Non-interacting pions which fake a muon signature (punch-through). \item{(iii)} Charm meson production and muonic decay. \end{description} A background muon event can be produced when a background muon" of the appropriate sign is recorded in (a) a NC event or (b) a CC event in which the primary lepton has been lost. If the background muon has the same charge sign as that in the storage ring the resulting event will be a background for disappearance measurements, but more importantly, if it has the opposite sign then the event will be a background for wrong--sign muon appearance measurements. The integrated wrong-sign background fraction from the hadronic shower is shown in Fig.~\ref{punch-charm} as a function of the minimum muon energy accepted for Steel/Scintillator and water detectors downstream of 20~GeV and 50~GeV neutrino factories. The charm background comes from $\nu_\mu$ CC events where the primary muon was less than 2~GeV. The peak at low muon energies is from the hadron shower itself and from punch through, while the long tail is from shower particles decaying to muons. \begin{figure}[h] \begin{center} \epsfxsize6.0in \centerline{\epsffile{new-punch-charm.eps}} % \epsfxsize=0.75\textwidth % \epsfbox[0 0 570 280]{punch-charm-20.ps} % \epsfxsize=0.75\textwidth % \epsfbox[0 0 570 280]{punch-charm-50.ps} \end{center} \vspace{-0.5cm} \caption{Background levels from punch through, pion/kaon decay, and charm backgrounds for 20~GeV (left) and 50~GeV (right) neutrino factories. The fraction of neutrino interactions that produce a wrong--sign muon background event is shown as a function of the minimum muon energy accepted.} \label{punch-charm} \end{figure} In general there are two different standards for background levels which are relevant: that of a disappearance experiment and that of an appearance experiment. Background estimates are not trivial, but if the backgrounds for a disappearance measurement are at the one per cent level, then the uncertainties on those backgrounds can be expected to be small compared to the flux uncertainty. On the other hand, wrong-sign muon appearance measurement uncertainties are expected to be dominated by the statistics. An extremely aggressive background level requirement would be to have less than of the order of one background event. If there are several thousand CC events expected, then this would require a minimum background rejection factor of $10^{4}$. Backgrounds can be suppressed by imposing a minimum energy requirement on the measured muon. Figure ~\ref{enucut} shows the effect of several different minimum muon energy cuts on a simulated oscillation signal observed in a steel-scintillator type detector at a 20~GeV muon storage ring, at a baseline length of 2800km \cite{bgrw00}. A muon threshold energy of 4~GeV for example depletes the low energy part of observed measured neutrino energy" distribution, degrading but not completely removing the information about the neutrino oscillation parameters that is encoded in the shape of the distribution. A 4~GeV threshold at a 20~GeV neutrino factory is probably tolerable. If higher thresholds are needed to reject backgrounds, then a higher energy neutrino factory is desirable. If a lower energy neutrino factory is to be viable, then lower muon thresholds are desirable. \begin{figure} %\epsfig{file=steveplot.ps,width=8.cm} \epsfxsize3.0in \centerline{\epsffile{steveplot.ps}} \bigskip \caption{Reconstructed neutrino energy distribution for several different minimum muon energy cuts for a 20~GeV ring. Result is from Ref.~\ref{bgrw00}. } \label{enucut} \end{figure} As is shown in Fig.~\ref{punch-charm}, to get to a background level of $10^{-4}$ one would need a 5 (6.5) GeV muon momentum cut in Steel/scintillator (Water) for a 20 GeV muon storage ring, and a 10 (12) GeV muon momentum cut in Steel/Scintillator (Water) for a 50 GeV muon storage ring. Clearly more background rejection is desirable. Fortunately muons from hadron decay in the hadronic shower are likely to be more aligned with the shower direction than muons from the leptonic vertex of the CC interaction. This provides another handle on the background. A useful variable to cut on is the momentum of the muon in the direction transverse to the hadronic shower ($p_t$). Figure~\ref{fig:pt2gen} shows the generated $p_t^2$ distribution for background and signal events, with no cut on the final state muon momentum. Note that requiring $p_t^2 > 1$ the background is extremely low, while the signal efficiency is high. The resolution with which $p_t^2$ is determined is detector dependent, and for detectors with reasonable transverse and longitudinal segmentation is dominated by the hadronic energy resolution. \begin{figure}[tb] \centering \epsfig{file=pt2_gen.eps,width=\textwidth} \caption{Distributions of the square of the muon momentum component transverse to the hadronic shower direction for $\nu_\mu$ charged current events compared to background muons for a 20 and 50~GeV muon storage ring. } \label{fig:pt2gen} \end{figure} %\clearpage \subsubsection{$\tau$--lepton identification and measurement} The detection and measurement of $\tau$--leptons is crucial for $\nu_\mu \rightarrow \nu_\tau$ and $\nu_e \rightarrow \nu_\tau$ measurements at a neutrino factory. Note that $\nu_e \rightarrow \nu_\tau$ oscillations will be of special interest since they will not have been previously observed. The $\nu_e \rightarrow \nu_\tau$ signal can be separated from $\nu_\mu \rightarrow \nu_\tau$ background" if the sign of the $\tau$--lepton charge is measured. The majority of $\tau$--lepton decays produce either one charged track (electron, muon, of hadron) or three charged tracks (hadrons). There are two general techniques that can be used to identify $\tau$--leptons. The first technique exploits the one-prong and three-prong topologies, and uses kinematic cuts to suppress backgrounds. The second technique uses a detector with a high spatial resolution to look for the displaced vertex or kink resulting from $\tau$--lepton decay. The advantage of the displaced vertex or kink detection $\tau$--lepton technique is that the detailed $\tau$--lepton decay is measured and background suppression is therefore large. The disadvantage is that detectors that have sufficient spatial resolution are necessarily less massive than coarse--grained detectors. The advantage of the kinematic technique is that a very massive detector can be used. If the $\tau$--leptons decay muonically (BR = 17\%) a measurement of the muon charge--sign determines the sign of the $\tau$ charge. However, there are substantial backgrounds that must be reduced. In the case of muonic $\tau$ decays, the backgrounds are from (a) $\nu_\mu$ (or $\bar\nu_\mu$) CC interactions which typically produce muons at high momentum and high $p_t^2$, and (b) meson decays (discussed earlier) which are at low momentum and low $p_t^2$. For $\tau \to e$ decays, the main background comes from $\nu_e$ and $\bar\nu_e$ CC interactions. Fortunately the undetected neutrinos from $\tau$ decays result in a larger missing transverse momentum than expected for background events. Exploiting these kinematic characteristics the backgrounds can be reduced by a large factor. For example, for an ICANOE--type detector a background rejection factor of 200 has been estimated, with a corresponding signal efficiency of 30\%. In the electron channel background can also come from NC interactions which produce photon conversions or Dalitz $\pi^0$ decays. These backgrounds can be suppressed in detectors with good pattern recognition allowing conversions, for example, to be identified and rejected. The analysis of hadronic $\tau$ decays requires the identification of the $\tau$ decay product inside a jet. This can only be done with a detector having good pattern recognition. It has been demonstrated that with an ICANOE--type detector a background rejection factor of 200 can be expected for $\tau\to$ 1 prong, $\tau\to\rho$, and $\tau\to 3\pi$ decays, with a signal efficiency of 8\%. %\begin{figure}[tb] %\centering %%\epsfig{file=CIN_HB_pr1.eps,width=5.5cm,angle=-90} %\epsfig{file=Rubbia-Plenary.2.1-fig1.eps,width=7.5cm,angle=-90} %\caption{Perspective view of the Liquid Argon TPC detector with %4 supermodules. HOW ABOUT REPLACING THIS WITH AN ELECTRON EVENT PIC} %\label{icanoe_pr} %%\epsfig{file=overfig/NICm_HB.eps,width=12cm,angle=90} %%\caption{Perspective view of the baseline detector with %%8 supermodules. \ednote{SIZE NEED TO BE FIXED!}} %\end{figure} \begin{figure}[tb] \centering %\epsfig{file=icanoe_event.ps,width=7.5cm,angle=-90} %\rotate[r]{ \epsfxsize=\textwidth \epsfbox[60 150 800 490]{icanoe_event.ps} %} \label{fig:icanoe} \caption{Example of a $\nu_e$ Charged current event from the full simulation of the ICANOE detector.} \end{figure} %\clearpage \subsubsection{A Liquid Argon neutrino detector} We have studied the performance of a large Liquid Argon neutrino detector at a neutrino factory using the ICANOE monte carlo program. One ICANOE detector unit consists of a liquid argon TPC followed by a magnetic spectrometer. %The layout is shown in Fig.~\ref{icanoe_pr}. The Liquid Argon TPC has extremely fine granularity, producing bubble chamber like event images. %Figure~\ref{fig:icanoe} Figure~21 shows an example of an electron neutrino charged current event--note the separation between the electromagnetic shower and the hadronic shower of the nucleon remnant. The TPC is instrumented with 3~mm pitch wires which allow tracking, $dE/dx$ measurements, electromagnetic, and hadronic calorimetry. Electrons and photons can be identified and their energies are measured with a resolution given by $\sigma_E/E = 0.03/\sqrt{E} \oplus 0.01$. The hadron energy resolution is given by $\sigma_E/E = 0.2/\sqrt{E} \oplus 0.05$. The magnetic spectrometer is primarily needed to measure muon energy and charge, but it is assumed that it will also be instrumented as a calorimeter to allow the hadron energy of showers which leak into the spectrometer to be correctly measured (albeit with worse resolution). The muon momentum resolution is expected to be $20\%$. In the design we have simulated, the liquid Argon module is 18~m deep with a cross section of $11.3m\times11.3m$. The active (total) mass of one Liquid Argon module is 1.4~kt (1.9~kt). The magnetized calorimeter module is 2.6~m deep with a cross section of $9m\times 9m$, and has a mass of 0.8~kt. It consists of 2~m of steel, corresponding to $7.4\ \lambda_{int}$ and $59\ X_0$, interleaved with tracking chambers. Four Super-Modules are assumed, yielding a total detector length of $82.5$~m and a total active mass of 9.3~kt that is fully instrumented. ICANOE can reconstruct neutrino (and antineutrino) events of all active flavors, and with an energy ranging from tens of MeV to tens of GeV, for the relevant physics analyses. The unique imaging capabilities of the liquid argon TPC allow one to cleanly determine whether a given event is a $\nu_\mu$ CC event, a $\nu_e$ CC event, or a NC event. For our studies the ICANOE fast simulation was used. Neutrino interaction events are generated, with a proper treatment of quasi-elastic interactions, resonance and deep-inelastic processes. The 4-vectors for all the particles generated are smeared, according to the resolutions derived from the full simulation. Muonic decays of pions and kaons are also considered, for a proper wrong- and right-sign muon background treatment. Once a 2-GeV cut is placed on the outgoing muon momentum, the background levels tend to be about $10^{-5}$ times the actual charged current event rate, and are dominated by meson decay in the hadronic shower. Examples of simulated oscillation signals in an ICANOE--type detector at a neutrino factory are shown in Figs.~\ref{fig:m1} and \ref{fig:m2}. More detailed results from a study of the sensitivity that might be achieved using an ICANOE--type detector are discussed in the oscillation measurements section of this report. %\clearpage \subsubsection{A magnetized Steel/Scintillator neutrino detector} Steel/Scintillator calorimeters have been used extensively in past neutrino experiments. Their performance is well understood and well simulated. Typically a magnetized Steel/Scintillator (MINOS--like) neutrino detector consists of iron plates interspersed with scintillator planes. To obtain transverse position information the scintillator can be segmented transversely, or a separate detector system (e.g. drift chambers) used. Penetrating charged particles (muon candidates) can then be reconstructed. With a reasonable transverse segmentation, the transverse position resolution is dominated by multiple coulomb scattering. The detector performance depends primarily on its longitudinal segmentation. The segmentation needs to be fine enough to determine whether a charged track has penetrated beyond the region of the accompanying hadronic shower. If it has, then the penetrating track is a muon candidate. The muon momentum resolution is determined by the magnetic field and the thickness of the steel plates. Neutrino CC and NC interactions have well defined signatures. In a MINOS--like detector NC interactions produce a hadronic shower reconstructed as a large energy deposition in a small number of scintillator units. A $\nu_\mu$ or $\overline{\nu}_\mu$ CC interaction will produce a muon in the final state, characterized by a long track in addition to the hadronic shower. These events can be identified provided the muon penetrates well beyond the hadronic shower. This imposes a minimum track-length, and hence minimum energy, requirement on muons that can be identified. If the muon is not identified the CC interaction will look like a NC event. A $\nu_e$ or $\bar{\nu_e}$ CC interaction, will produce an electron in the final state which cannot be resolved, so these events look similar to NC interactions. A $\nu_\tau$ or $\bar{\nu_\tau}$ CC interaction will also look like a NC interaction unless the $\tau$--lepton decays muonically. %The second consideration comes from the %possibility of reconstructing as primary muons non-prompt muons or %punchthrough pions in NC-like events ($\nu_mu$ NC, all $\nu_e$, %and most $\nu_\tau$). The sources of these muons were listed in %section \ref{bkgds}. % %The experimental technique proposed for this study is the observation of %wrong-sign'' muons. Given the above discussion, the issues which we need %to address in order to understand the sensitivity of the magnetized Fe/Sci %detector are: the efficiency to identify and measure the charge of the primary %muon (from the leptonic vertex) and the efficiency to remove secondary %muons (from the hadronic vertex) without eliminating the signal. To study the performance of a magnetized Steel/Scintillator detector at a neutrino factory we have considered a detector geometry similar to the CCFR/NuTeV calorimeter~\cite{nutevdet}, but with the addition of a toroidal magnetic field of 1T. The detector is constructed from $3\times 3 \times 0.3$~m$^3$ modules (see Fig.~22). %(see Fig.~\ref{fig:ccfr}). The 0.7~kt CCFR detector consists of 42 modules. A neutrino factory detector with a mass of 50~kt (10 $\times$ the MINOS detector) would require 3000 of these modules. The ultimate transverse size (and hence module mass) that is practical is probably determined by the largest size over which a large magnetic field can be generated. \begin{figure} [h] \begin{center} % \epsfxsize=0.6\textwidth % \epsfysize=0.3\textheight % \epsfbox{unit-target-metric.eps} \epsfxsize3.0in \centerline{\epsffile{unit-target-metric.eps}} \caption{Schematic of a CCFR/NuTeV calorimeter module.} \end{center} \label{fig:ccfr} \end{figure} In the following we consider how well a magnetized Steel/Scintillator detector can identify and measure wrong--sign muon events at a neutrino factory. For our simulations, we used the parameterized Monte Carlo developed by the NuTeV collaboration~\cite{nutevpub}, modified to include particle tracking in the magnetic field. The hadron energy resolution of this detector is described in detail in ~\cite{nutevdet}, and is approximately given by $\sigma_E / E = 0.85 / \sqrt{E}$. The muon momentum resolution depends on the track length in the steel, and whether the muon is contained within the detector. For muons which range out in the detector the effective momentum resolution is $\sigma_P/P = 0.05$, while for tracks which leave the fiducial volume of the detector the resolution is described by $\sigma_P/P \sim \theta_{MCS}\theta_{BdL}$, where the angles $\theta_{MCS}$ and $\theta_{BdL}$ describe respectively the change in direction due to multiple scattering and curvature in the magnetic field. The simulation includes a detailed parameterization of the hadron-shower development, with the inclusion of charm production and $\pi$, $K$ decays (the data set on which the decay probability parameterization was tuned contained only muons with momentum higher than 4~GeV/c). Note that $\pi$ punchthrough was not included in the parameterization, but is expected to make only a small contribution to background muons above 4~GeV. \begin{figure} [h] %\epsfig{file=pttalk20.eps,width=7.cm} %\epsfig{file=pttalk50.eps,width=7.cm} \epsfxsize=7.cm \epsfbox[40 10 520 520]{pttalk20a.eps} \epsfxsize=7.cm \epsfbox[40 10 520 520]{pttalk50a.eps} \caption{Reconstructed $\mu^-$ $P_t^2$ with respect to the shower direction for 20~GeV and 50~GeV $\mu^+$ decaying in a neutrino factory. The muons are required to have energies exceeding 4~GeV.} \label{pt} \end{figure} To be conservative, and reduce the dependence of our study on low energy processes that may not be adequately described by the Monte Carlo program, in our analysis all muons with generated energy below 4~GeV are considered lost. Muons with track length in steel less than 50~cm past the hadronic shower are also considered lost. All other muons are assumed to be identified with 100\% efficiency, and measured sufficiently well to determine their charge sign. For the background events we considered (i) all the $\pi$, $K$ decay events producing wrong--sign'' muons in NC interactions, and (ii) all the charm production and $\pi$, $K$ decay events producing wrong--sign" muons in CC events where the primary muon was considered lost. To reduce the backgrounds, we cut on $P_t^2$. The reconstructed $P_t^2$ distribution is shown in Fig.~\ref{pt} for signal and background muons in a 10~kt detector 2800~km downstream of 20~GeV and 50~GeV neutrino factories which provide $10^{20}$ $\mu^+$ decays. The oscillation parameters corresponding to the LMA scenario IA1. As expected, background wrong--sign muons, tend to have smaller $P_t^2$ than genuine wrong--sign muons from the leptonic vertex. The reconstructed wrong-sign muon spectrum is shown in Fig.~\ref{data} for a 20~GeV storage ring before (top plot) and after (bottom plot) muon energy, track length and $P_t^2 > 2$~GeV$^2$ cut were applied. Signal and background rates are summarized in Table~\ref{thetable}. After the cuts the signal/background ratio is above 10 to 1 in scenario IA1 for a detector 2800~km away, while $40-50\%$ of the $\nu_e \rightarrow \nu_\mu$ signal events are retained. \begin{figure} [h] \begin{center} \epsfxsize=0.75\textwidth \epsfbox{{signalwriteplus20.eps}} \caption{Reconstructed wrong-sign muons as a function of the muon energy for a $\mu+$ 20 GeV ring. Top plot accepted events for the signal ($\nu_e \rightarrow \nu_\mu$--stars-- and $\nu_e \rightarrow \nu_\tau \rightarrow \mu + X$--crosses) and the potential backgrounds (x). The bottom plot shows the signal and the background after cuts.} \label{data} \end{center} \end{figure} \begin{table} \caption{Wrong-sign muon rates after all cuts for a 10~kt steel-scintillator detector downstream of a neutrino factory providing $10^{20}$ muon decays. The oscillation parameters correspond to scenario IA1. The loss of signal acceptance and the background rejection are due solely to the kinematic and reconstruction cuts.} \bigskip \begin{tabular}{cc|ccccc} \hline \multicolumn{2}{c}{$\mu$ Ring}&$\nu_e \rightarrow \nu_\mu$ & $\nu_e \rightarrow \nu_\tau$ & & & \\ Energy & Charge & events &$\rightarrow \mu + X$ & background & signal & background/CC \\ GeV & &accepted& events & events & acceptance & rate \\ \hline 50 & + & 268.5 & 15.4 & 21.6 & 0.50$\pm$0.02 &$4.5\times10^{-4}$ \\ 50 & $-$ & 55.2 & 4.7 & 3.5 &0.48$\pm$0.02&$4.0\times10^{-5}$ \\ 20 & + & 85.7 & 3.5 & 0.7 & 0.41$\pm$0.02 & $2.0\times10^{-4}$ \\ \hline \end{tabular} \label{thetable} \end{table} %\clearpage \subsubsection{A Water Cerenkov detector} Preliminary studies have explored the possibility of using a large water Cerenkov detector as a distant target for a neutrino factory beam. Traditionally this type of detector has been used for measuring much lower energy neutrinos than expected at a muon storage ring, but to date water Cerenkov neutrino detectors are the only existing neutrino detectors with masses already in the 50~kt range. Water is of course the lightest target material under consideration in this report, but this type of detector has several advantages when extrapolating to large masses, namely (i) low cost target material, (ii) only the surface of a very large volume needs to be instrumented, and (iii) good calorimetry. A large volume guarantees containment of hadronic and electromagnetic showers (as well as muons up to a certain energy). The low density of the target and good angular resolution from the Cerenkov cone might yield an overall hadron angle resolution that is as good as or better than the corresponding resolution obtained with steel-scintillator calorimeters. Water Cerenkov devices as large as 50~kt (SuperK) are already in operation and is expected to continue data-taking for ten years or more. Therefore, the response of the existing SuperK detector at a baseline distance of 9100~km has been studied as a test case. Next generation detectors, up to 1~Mton in mass, are technically feasible and are currently under consideration for proton decay and neutrino measurements, sited perhaps at the Kamioka mine or elsewhere. For this initial study, the primary question is the suitability of a water Cerenkov detector for the higher energy neutrino beam produced by a 10-50 GeV muon storage ring. At these energies, the multiplicity of hadrons is greater than for typical atmospheric neutrino interactions, and event topologies are correspondingly more complex. Figure~\ref{fig:f2kevt} shows the Cerenkov light produced in a typical neutrino event from a 50 GeV muon storage ring at the SuperK detector: the circles in the display are estimates of the outgoing angles of different charged particles produced in the hadronic shower. Some particle identification is possible from the pulse-height information. Reconstruction software from the SuperK experiment must be further optimized to study the detector response to neutrinos from 10~GeV and 50~GeV muon storage rings. It is worth noting that neutrinos produced by a 50~GeV muon beam induce a large number of events in the material (rock) surrounding any detector (producing an entering muon), and for a SuperK sized device these events outnumber the those produced in the detector's water volume. Both contained and entering events have therefore been studied. \begin{figure} [h] \begin{center} \epsfxsize=0.75\textwidth \epsfbox[0 310 550 530]{f2kevt.ps} \caption{Simulated neutrino event from a 50~GeV muon storage ring in the SuperKamiokande detector. The rings indicate where the reconstruction software found charged particles in the hadronic shower, as well as the exiting muon.} \label{fig:f2kevt} \end{center} \end{figure} The response of a detector the size of SuperK changes drastically as a function of beam energy. At a 10~GeV neutrino factory, 57\% of the muon CC events are fully contained in the inner water volume, whereas only 11\% are fully contained at a 50~GeV neutrino factory. This large difference only exists for events containing penetrating muons; at 10~GeV (50~GeV) both ${\bar \nu_e}$ and NC events are contained greater than 98\% (90\%) of the time. The existing $\mu$-like particle identification algorithm works to produce a reasonably pure (89\%) $\nu_\mu$-CC sample for fully contained events in the 10~GeV beam, but $e$-like events are a mixture of of ${\bar \nu_e}$-CC, NC and $\nu_\mu$-CC contamination. Exiting and entering events are pure samples of $\nu_\mu$-CC simply because of their penetrating nature. The muon angular resolution (3${}^{\circ}$) is much less than the muon-neutrino angular correlation (15${}^{\circ}$). With $2 \times 10^{20}$ decays at a 50~GeV neutrino factory and a baseline of 9100~km, approximately $200,000$ $\nu_\mu$ CC events would be observed entering or exiting the current SuperK detector. Combined with muon charge identification this sample should be able to provide good oscillation measurements. Implementing charge identification in a water Cerenkov detector is not trivial. Two possibilities have been proposed: (i) several large water targets, each one followed by a thin external muon spectrometer, and (ii) a magnetic field introduced into the water volume itself. Although the first design would have lower geometrical acceptance and a higher muon energy threshold, it would pose much less of a problem for the phototubes since the magnetic field would presumably be well-contained in the spectrometer. The second proposal could in principle have good low energy muon momentum acceptance, but the resolution on the muon and hadron shower angles might be compromised. For a magnetic field internal to the target, 0.5-1~kG is sufficient to visually determine the charge of a several meter-long ($>1$~GeV) muon, but no automated algorithms have yet been developed. A number of conceptual magnet designs have been studied: solenoidal, toroidal, and concentric current loops in the center or at the ends of the detector. A detailed study of one particular design has shown that one can immerse the central volume of a SuperK sized detector in a 0.5~kG magnetic field while leaving only a 0.5~G fringe field (which may be acceptable with shielding and/or local compensation) in the region of the PMTs. Many of the difficulties inherent in placing a field inside a water detector would be avoided if an alternative light collector (insensitive to the field) were used. Work on magnet design and alternative light collection is ongoing, but the internal magnetic field option must be considered speculative at this point. The results we will describe in the remainder of this section are for a water Cerenkov detector with an external magnetic field, because neutrino event reconstruction is more straightforward to simulate and the spectrometer technology is well-understood. Although the studies of this detector are very preliminary, they look promising and warrant further investigation. We have used a LUND/GEANT Monte Carlo program which uses as its geometry a $40\times40\times100$~m$^3$ box of water, followed by a 1~m long muon spectrometer. This simulation can be used to study acceptance issues and background contamination for a range of geometries and storage ring energies. %The background levels for this %type of detector compared to that of steel-scintillator detectors %are discussed above. Figure~\ref{fig:wateracc} shows the geometrical acceptance for the box-like water Cerenkov detector as a function of distance of the neutrino interaction vertex from the spectrometer, for CC $\nu_\mu$ events from 20 and 50~GeV storage rings. The loss in acceptance close to the spectrometer is due to rejection of events where there is more than one muon which traverses the spectrometer (where the extra muon comes from background processes). It is clear that for a 20~GeV muon storage ring one would want a muon spectrometer much more frequently than once every 100~m. Of course, noting that steel has a density of 8 times that of water, the smaller the ratio of water thickness to steel thickness the more it approximates a magnetized steel/scintillator target interspersed with water volumes with fine granularity. \begin{figure} [h] \begin{center} \epsfxsize=0.75\textwidth \epsfbox[0 0 520 530]{muon-acc.ps} \caption{Acceptance in a water target for charged current $\nu_\mu$ (solid) and $\overline\nu_\mu$ (dashed) events in a 20 and 50 GeV storage ring, as a function of distance of the neutrino interaction vertex from the muon spectrometer.} \end{center} \label{fig:wateracc} \end{figure} Clearly more work is needed to optimize the design for this kind of detector, but it might be an inexpensive compromise between a coarse-grained sampling calorimeter and a very fine-grained liquid argon TPC. %\clearpage \subsubsection{Specialized $\tau$--lepton detectors} The measurement of $\tau$--lepton appearance in large mass neutrino detectors is challenging. There are several ideas that might lead to viable new $\tau$--appearance detectors within the next 5--10~years, and that might be suitable for use at a neutrino factory. We briefly describe three examples in the following: (i) a perfluorohexane Cerenkov detector, (ii) a hybrid emulsion detector, and (iii) a very fine--grained micro--strip gas chamber target. Consider first a Cerenkov detector filled with perfluorohexane ($C_6F_{14}$), which has a density 1.7 times that of water. This has been proposed~\cite{forty} for use in the CERN to Gran Sasso beamline. The detector geometry consists of several target volumes followed by short muon spectrometer modules. A 1~Ton perfluorohexane detector (with a very different geometry) exists at DELPHI. The $\tau$--lepton signature in this type of detector consists of a sparsely populated Cerenkov ring from the $\tau$ before it decays, together with a more densely populated ring from the daughter muon. The two rings would have offset centers. Figure~\ref{fig:forty} shows a simulated quasi-elastic $\nu_\tau$ event (no hadron energy) from this kind of detector. This technique would probably not work for events with high energy hadron showers because of the large number of charged particles that would result in overlapping rings near the initial $\tau$--lepton ring. \begin{figure}[h] \epsfxsize=\textwidth \epsfbox[50 400 580 690]{forty_fig3.ps} \caption{Quasi-elastic $\nu_\tau$ event in a perfluoroHexane Cerenkov detector: the ring described by about eight hits on the left is from the tau before it decays.} \label{fig:forty} \end{figure} Next consider a hybrid emulsion detector consisting of, for example, thin ($\sim 100\mu$m) sheets of emulsion combined with low-density ($\sim 300\mu$m) spacers. The signature for a $\tau$--lepton 1--prong decay would be a change in direction of the track measured before and after the spacer~\cite{strolin}. For charge identification the detector could be within a large magnetic field volume. With an emulsion track angular resolution of 2~mrad, a 5$\sigma$ charge--sign determination of a 10~GeV/c charged particle could be achieved with a 2~T field and a 1.2~mm thick spacer \cite{para}. An $\sim 20$~kt hybrid emulsion detector of this type might consist of 20~kt of steel segmented into 1~mm thick sheets, and an equal volume of thin emulsion layers plus low density spacers. The resulting detector would fit into the ATLAS barrel toroid magnet, which has a magnetic field ranging from 2 to 5 Tesla~\cite{atlas}. A hybrid emulsion detector with an external downstream muon spectrometer will be used by the OPERA experiment, which is to be put in the CERN to Gran Sasso beam. The muon spectrometer will determine the charge sign for $\tau\to\mu$ decays provided the muon reaches the spectrometer. According to the OPERA studies~\cite{opera}, with an average neutrino energy of 20~GeV the total efficiency for seeing the $\tau$ decays is 29\% (including the branching ratios). %for $\delta m_{23}^2 = 2.5x10^{-3}$ in the CERN to The efficiency is largely geometric and should not be compromised by the addition of a magnetic field, provided the bend in the spacer due to the magnetic field is much less than the "apparent bend" due to the $\tau$--lepton decay. %However, it should be noted that for an average neutrino energy of %20GeV the muon storage ring would have to be roughly 30GeV, and %the efficiency of $\tau$ detection and charge measurement %will deteriorate severely as the energy of the taus decreases. Finally, consider a target consisting of a tracking chamber constructed from micro--strip gas chambers (MSGCs) and a low $Z$ material (for example, nylon) in a large magnetic field volume. This would be a NOMAD~\cite{nomad}--like detector with a much larger O(1~kt) fiducial mass and an improved spatial resolution. Because of the low $Z$ of the material electrons can travel a long distance in the detector before showering, and with a high enough field their charge can therefore be measured. Although a kink is not seen, the tau decay could be distinguished kinematically. For example, nylon has a radiation length of 37 cm. With a $B$ field of 1~Tesla and MSGC's every 10~cm one would have an $8\sigma$ measurement of a 50~GeV electron's charge. This idea is worthy of further consideration, particularly if the LSND signal is confirmed and lower-mass tau detectors are warranted. %\clearpage \subsubsection{Detector summary} In our initial studies we have simulated the performance of steel/scintillator and liquid Argon detectors at a neutrino factory. Results are encouraging. These technologies could provide detectors with masses of order 10~kt (liquid Argon) to a few $\times 10$~kt (steel/scintillator) that yield good wrong--sign muon identification and adequate background rejection. Our simulations of the capabilities of water Cerenkov detectors at a neutrino factory are less advanced, but initial results are encouraging, and this detector technology might permit very large detector masses to be realized. Some relevant characteristics of steel/scintillator, liquid Argon, and water Cerenkov detectors are listed in Table~\ref{dettab}. It is premature to choose between detector types at this early stage. However, some general points are worth noting: \begin{description} \item{(i)} We believe that a cost optimization of detector mass (cost) versus neutrino factory beam intensity (cost) will probably favor detectors that are at least a factor of a few to a factor of 10 more massive than, for example, ICANOE or MINOS. A detector mass in the range 10~kt to 50~kt does not seem unreasonable. \item{(ii)} The minimum energy a muon must have for good identification and measurement may well determine the minimum viable muon storage ring energy. This threshold is a few GeV, and is detector technology dependent. With a steel/scintillator detector and a threshold of 4~GeV, for example, the minimum acceptable neutrino factory energy appears to be in the neighborhood of 20~GeV. \end{description} In this initial study we have not comprehensively considered to what extent massive detectors at a neutrino factory need to be deep underground. It seems very likely that detectors with low detection thresholds (water Cerenkov and liquid argon) will need to be well protected from cosmic ray backgrounds, regardless of the neutrino factory energy. For the steel/scintillator detector, the cosmic ray backgrounds for charged current events with muons in them are likely to be small for a detector at the surface of the earth, but there will be substantial background to neutral current or $\nu_e$ charged current interactions. Finally, we note that the development of a new generation of very massive detectors capable of identifying and measuring the charge--sign of muons and $\tau$--leptons, would be of great benefit to a neutrino factory. There is a possible area of mutual interest with the nucleon decay community in developing the technology for a really massive 1~Mton scale water Cerenkov detector. This possibility deserves further investigation. \begin{table}[h] \caption{Comparison of detector parameters for candidate detectors at a neutrino factory.} \bigskip \begin{center} \begin{tabular}{|l|l|l|l|} \hline Characteristic & \multicolumn{3}{|c|}{Detector Technology } \\ & Steel/Scint & Liquid Argon TPC & Water Cerenkov \\\hline Resolutions of: & & & \\ Electron Energy& $50\%/\sqrt{E}$ & $3\%/\sqrt{E}\oplus 1\%$ & $0.6\oplus 2.6\%/\sqrt{E}$ \\ Hadron Energy & $85\%/\sqrt{E}$ & $20\%/\sqrt{E}\oplus 5\%$ & 20-30\% \\ Muon Energy &$5\%$ & $20\%$ & $20\%^\dagger$ \\ Hadron Shower Angle & &$.13/\sqrt{{\rm p}}$ rad & \\ & & (each hadron)& \\ Muon Angle & 5\% for & & $3^\circ$ \\ & 50cm track & $.02\oplus .21/\sqrt{{\rm p}}$ & \\ Maximum mass &50 kton & 30 kton & 1Mton?\\ What limits size? & & safety, tunnel & tunnel\\ Required Overburden$^{**}$ & 0 m &50 m & 50-100m\\ Analysis Cuts & $P_{\mu}>4$ GeV &$P_{\mu}>2$ GeV& \\ & $P_t^2 > 2$ GeV$^2$ & & \\ %Signal acceptance from these cuts & 50\% &$>99\%$ & 50\% \\ Background level & $10^{-4}$ &$2\times 10^{-5}$ & \\ %Maximum B field& 1T & 2 T & 1T \\ %Transverse segmentation&& 3 mm wire pitch & xxxpmts/4$\pi$\\ \hline\hline \end{tabular} \end{center} $^{**}$ The overburden required for all technologies depends on the neutrino factory duty factor. The overburden required for a steel-scintillator calorimeter also depends on the energy of the muon storage ring; but in the past this type of detector has been used at ground level with minimal contamination in the $\nu_\mu$ charged current sample above a neutrino energy of 5~GeV.$^{\dagger}$ The muon momentum resolution would be comparable to that of an ICANOE detector if the muon spectrometer were separated from the water tank volume. \label{dettab} \end{table} \clearpage \subsection{Oscillation measurements} Using the oscillation scenarios described in section~\ref{theory} as examples, we can now assess how well the neutrino oscillation physics program outlined in section~\ref{program} can be pursued at a neutrino factory with the detectors described in section~\ref{detectors}. In the following sub-sections the oscillation measurements that can be made at a neutrino factory are discussed as a function of baseline, muon beam energy, and muon beam intensity. In particular we consider: % \begin{description} \item{(i)} The first observation of $\nu_e \rightarrow \nu_\mu$ oscillations, the measurement of the sign of $\delta m^2$ and hence the pattern of neutrino masses (section~3.5.1). \item{(ii)} The first observation of $\nu_e \rightarrow \nu_\tau$ oscillations (section~3.5.2). \item{(iii)} The measurement of $\nu_\mu \rightarrow \nu_\tau$ oscillations (section~3.5.3). \item{(iv)} Precision measurements of the oscillation parameters (section~3.5.4). \item{(v)} The search for CP violation in the lepton sector (section~3.5.5). \end{description} % The results are based on the calculations described in more detail in Refs.~\cite{cerv00, bgrw00, bern00, camp00}. The calculations from Ref.~\cite{camp00} are for an ICANOE type detector, and include realistic resolution functions, analysis cuts, and background modeling, but use a constant average matter density to compute matter effects. The calculations from Refs.~\cite{bern00, bgrw00, cerv00} all use resolution functions typical of steel/scintillator detectors and, unless explicitly stated, reasonable thresholds on the detected muon energies. The calculation from Ref.~\cite{bgrw00} does not include backgrounds but covers a broad range of scenarios, and uses the explicit trans--Earth density profile to compute matter effects. In contrast, the calculations from Refs.~\cite{bern00,cerv00} have been used to look at only a few oscillation scenarios, but include backgrounds and use respectively the average Earth density and the explicit density profile in computing matter effects. It should be noted that although there are significant differences in the details implemented in the calculations, in general all the four groups arrive at similar assessments for the measurement sensitivity at a neutrino factory as a function of energy, intensity, and baseline. \subsubsection{Observation of $\nu_e \rightarrow \nu_\mu$ oscillations and the pattern of neutrino masses} At a neutrino factory $\nu_e \rightarrow \nu_\mu$ oscillations would be signaled by the appearance of CC interactions tagged by a wrong--sign muon~\cite{geer98}. Within the framework of three--flavor oscillations the $\nu_e \rightarrow \nu_\mu$ oscillation amplitude is approximately proportional to $\sin^2 2\theta_{13}$. At the present time only an upper limit exists on $\sin^2 2\theta_{13}$. The next generation long-baseline oscillation experiments are expected to be able to improve the sensitivity to $\sin^2\theta_{13}\approx 10^{-2}$, i.e. about one order of magnitude below the present bound. If $\sin^2 2\theta_{13}$ is in this range we would expect to observe $\nu_e \rightarrow \nu_\mu$ oscillations at a relatively low intensity neutrino factory, measure matter effects, and determine the pattern of neutrino masses~\cite{bgrw99}. This is discussed further in the remainder of this sub--section. % \begin{figure}[h] \epsfxsize3.0in \centerline{\epsffile{fig_v1.ps}} \vspace{0.3cm} \caption{Reach in $\sin^22\theta_{13}$ for the observation of 10 $\mu^-$ events from $\nu_e \rightarrow \nu_\mu$ oscillations, shown versus baseline for three $\delta m^2_{32}$ spanning the favored SuperK range. The other oscillation parameters correspond to the LMA scenario IA1. The curves correspond to $10^{19} \mu^+$ decays in a 20~GeV neutrino factory with a 50~kt detector, and a minimum muon detection threshold of 4~GeV. Results are from Ref.~\ref{bgrw00}.} \label{fig:v1} \end{figure} It is useful to define~\cite{bgrw00} the $\sin^22\theta_{13}$ reach" for an experiment as the value of $\sin^22\theta_{13}$ for which a given physics goal would be met. We take as our initial goal the observation of 10 $\nu_e \rightarrow \nu_\mu$ events tagged by a wrong--sign muon. Consider first the $\sin^22\theta_{13}$ reach for a 50~kt detector sited a distance $L$ from a 20~GeV neutrino factory in which there are $10^{19} \mu^+$ decays in the beam--forming straight section. The baseline--dependent $\sin^22\theta_{13}$ reach is shown in Fig.~\ref{fig:v1} for a three-flavor oscillation scenario in which $\delta m^2_{21}, \sin^22\theta_{12}$, and $\sin^22\theta_{23}$ correspond to the LMA scenario IA1, and the value of $\delta m^2_{32}$ is varied over the favored SuperK range. Backgrounds are expected to be less than one event for $L \ge 2800$~km (Table~\ref{dm2table}), and are not included in the calculation shown in the figure. If $\delta m^2_{32}$ is in the center of the SuperK range, the $\sin^22\theta_{13}$ reach is about an order of magnitude below the currently excluded region, improving slowly with decreasing $L$. However, at short baselines ($L < 2800$~km) backgrounds may degrade the $\sin^22\theta_{13}$ reach. The reach improves (degrades) by a about a factor of 2 (3) if $\delta m^2_{32}$ is at the upper (lower) end of the current SuperK range. If the oscillation parameters correspond to the LMA scenario IA1 ($\sin^22\theta_{13} = 0.04$), then only $2 \times 10^{18}$ muon decays are required at a 20~GeV neutrino factory to observe 10~signal events in a 50~kt detector at $L = 2800$~km. The calculation~\cite{bgrw00} assumes that CC events producing muons with energy less than (greater than) 4~GeV are detected with an efficiency of 0 (1). The number of muon decays needed to observe 10 $\nu_e \rightarrow \nu_\mu$ events is shown in Fig.~\ref{fig:v2} as a function of $E_\mu$ for the LMA scenario IA1, the SMA scenario IA2, and the LOW scenario IA3. The required muon beam intensities decrease with increasing $E_\mu$, and are approximately proportional to $E_\mu^{-1.5}$. Compared with the SMA and LOW scenarios, slightly less intensity is needed for the LMA scenario, showing the small but finite contribution to the signal rate from the sub--leading $\delta m^2$ scale. In all three scenarios (LMA, SMA, LOW) a 20~GeV neutrino factory providing $10^{19}$ decays in the beam--forming straight section would enable the first observation of $\nu_e \rightarrow \nu_\mu$ oscillations in a 50~kt detector provided $\sin^22\theta_{13} > 0.01$. It should be noted that although $\sin^22\theta_{13}$ could be very small, there are models~\cite{albright} that predict $\sin^2 2\theta_{13} \simeq 0.01$. % \begin{figure} \epsfxsize5.0in \centerline{\epsffile{fig_v2.ps}} \vspace{0.3cm} \caption{The required number of muon decays needed in the beam--forming straight section of a neutrino factory to achieve the physics goals described in the text, shown as a function of storage ring energy for the LMA scenario IA1, SMA scenario IA2, LOW scenario IA3, and a bimaximal mixing scenario BIMAX. The baseline is taken to be 2800~km, and the detector is assumed to be a 50~kt wrong--sign muon appearance device with a muon detection threshold of 4~GeV or, for $\nu_e \rightarrow \nu_\tau$ appearance, a 5~kt detector. Results are from Ref.~\ref{bgrw00}.} \label{fig:v2} \end{figure} % \begin{figure} \epsfxsize3.5in \centerline{\epsffile{fig_v3.ps}} \caption{Predicted measured energy distributions for CC events tagged by a wrong-sign (negative) muon from $\nu_e \rightarrow\nu_\mu$ oscillations (no cuts or backgrounds), shown for various $\delta m^2_{32}$, as labeled. The predictions correspond to $2 \times 10^{20}$ decays, $E_\mu = 30$~GeV, $L = 2800$~km, with the values for $\delta m^2_{12}$, $\sin^22\theta_{13}$, $\sin^22\theta_{23}$, $\sin^22\theta_{12}$, and $\delta$ corresponding to the LMA scenario IA1. Results are from Ref.~\ref{bgrw99}.} \label{fig:v3} \end{figure} \begin{figure} \epsfxsize3.5in \centerline{\epsffile{fig_v4.ps}} \caption{Same as previous figure, for CC events tagged by a wrong-sign (positive) muon from $\bar{\nu}_e \rightarrow \bar{\nu}_\mu$ oscillations. } \label{fig:v4} \end{figure} Having established $\nu_e \rightarrow \nu_\mu$ oscillations, further data taking would facilitate the measurement of matter effects and the determination of the sign of $\delta m^2$, and hence the pattern of neutrino masses. To illustrate the effect of matter on the $\nu_e \rightarrow \nu_\mu$ oscillation probability, the predicted measured energy distributions 2800~km downstream of a 30~GeV neutrino factory are shown in Figs.~\ref{fig:v3} and \ref{fig:v4} for respectively $\nu_e \rightarrow \nu_\mu$ and $\overline{\nu}_e \rightarrow \overline{\nu}_\mu$ wrong--sign muon events. The distributions are shown for a range of positive and negative values of $\delta m^2_{32}$. Note that for a given $|\delta m^2_{32}|$, if $\delta m^2_{32} < 0$ we would expect to observe a lower wrong--sign muon event rate and a harder associated spectrum when positive muons are stored in the neutrino factory than when negative muons are stored. On the other hand, if $\delta m^2_{32} > 0$ we would expect to observe a higher wrong--sign muon event rate and a softer associated spectrum when positive muons are stored in the neutrino factory than when negative muons are stored. Hence, measuring the differential spectra when positive and negative muons are alternately stored in the neutrino factory can enable the sign of $\delta m^2_{32}$ to be unambiguously determined~\cite{bgrw99}. \begin{table}[t] \caption{Wrong-sign muon rates for a 50~kt detector (with a muon threshold of 4~GeV) a distance $L$ downstream of a neutrino factory (energy $E_\mu$) providing $10^{19}$ muon decays. Rates are shown for LMA scenario IA1 with both signs of $\delta m^2_{32}$ considered separately. The background rates listed correspond to an assumed background level of $10^{-4}$ times the total CC rates (see section~\ref{detectors}) with no energy dependence. Energy dependent cuts might suppress backgrounds further. Results are from Ref.~\ref{bgrw99}. } \bigskip \begin{tabular}{cc|ccc|ccc} \hline $E_\mu$&$L$&\multicolumn{3}{c}{$\mu^+$ stored}&\multicolumn{3}{c}{$\mu^-$ stored}\\ GeV & km & $\delta m^2_{32} > 0$ & $\delta m^2_{32} < 0$ & Backg & $\delta m^2_{32} > 0$ &$\delta m^2_{32} < 0$ & Backg \\ \hline 20 & 732 & 52. & 36. & 7.3 & 32. & 26. & 6.5 \\ &2800 & 46. & 9.2 &0.43 & 7.1 & 26. & 0.36\\ &7332 & 33. & 0.97&0.063&0.55 & 19. & 0.05\\ \hline 30 & 732 &100. & 72. & 25. & 58. & 45. & 24.\\ &2800 & 90. & 26. & 1.6 & 19. & 43. & 1.4\\ &7332 & 43. & 3.3 & 0.19& 2.1 & 33. & 0.17\\ \hline 40 & 732 &150. &110. & 60. & 83. & 65. & 58.\\ &2800 &140. & 48. & 4.0 & 36. & 64. & 3.8\\ &7332 & 54. & 5.6 & 0.49& 3.1 & 28. & 0.43\\ \hline 50 & 732 &200. & 140.& 120.& 110.& 84. & 120.\\ &2800 &180. & 71. & 7.9& 53.& 82. & 7.7\\ &7332 & 56. & 8.0 & 1.1& 5.0 & 34. & 1.0\\ \hline \end{tabular} \label{dm2table} \end{table} The expected number of wrong--sign muon events are listed in Table~\ref{dm2table} for the LMA scenario IA1, and a 50~kt detector downstream of a neutrino factory providing $10^{19} \mu^+$ decays and the same number of $\mu^-$ decays. The event rates are shown for both signs of $\delta m^2_{32}$, and for various storage ring energies and baselines. Even at a 20~GeV neutrino factory the signal rates at $L = 7332$ and 2800~km are large enough to permit the sign of $\delta m^2_{32}$ to be determined with a few years of data taking. We conclude that for the LMA, SMA, and LOW three--flavor mixing scenarios we have considered, a 20~GeV neutrino factory providing $10^{19}$ decays in the beam--forming straight section would be a viable entry--level facility. In particular, with a 50~kt detector and a few years of data taking either $\nu_e \rightarrow \nu_\mu$ oscillations would be observed and the sign of $\delta m^2_{32}$ determined or a very stringent upper limit on $\sin^2 2\theta_{13}$ will have been obtained (discussed later). Long baselines ($>2000$~km) are preferred. The longest baseline we have considered (7332~km) has the advantage of lower total event rates and hence lower background rates. %\clearpage \subsubsection{Observation of $\nu_e \rightarrow \nu_\tau$ oscillations} We begin by considering the LMA scenario IA1, and ask: What beam intensity is needed to make the first observation of $\nu_e \rightarrow \nu_\tau$ oscillations in a detector that is 2800~km downstream of a 20~GeV neutrino factory ? The $\nu_e \rightarrow \nu_\tau$ and the accompanying $\overline{\nu}_\mu \rightarrow \overline{\nu}_\tau$ event rates are shown in Fig.~\ref{fig:t1} as a function of the oscillation parameters $\sin^2 2\theta_{13}$ and $\delta m^2_{32}$. The $\nu_e \rightarrow \nu_\tau$ signal rate is sensitive to both of these parameters, and hence provides an important consistency check for three-flavor mixing: the observation or non--observation of a $\nu_e \rightarrow \nu_\tau$ signal must be consistent with the oscillation parameters measured from, for example, $\nu_e \rightarrow \nu_\mu$, $\nu_\mu \rightarrow \nu_\tau$, and $\nu_\mu$ disappearance measurements. For the LMA scenario IA1 the observation of 10 signal events in a 5~kt detector (with 30\% $\tau$--lepton efficiency) would require 3 years with $7 \times 10^{19} \mu^+$ decays per year in the beam forming straight section. Very similar beam intensities are required for the SMA and LOW scenarios (IA2 and IA3). Note that, over the $\sin^22\theta_{13}$ range shown in Fig.~\ref{fig:t1}, the $\overline{\nu}_\mu \rightarrow \overline{\nu}_\tau$ rates are one to two orders of magnitude higher than the $\nu_e \rightarrow \nu_\tau$ rates. Hence, we will need a detector that can determine the sign of the tau--lepton charge at the $2\sigma-3\sigma$ level, or better. \begin{figure} \epsfxsize3.5in \centerline{\epsffile{fig_t1.ps}} \caption{$\nu_\tau$ CC appearance rates in a 5~kt detector 2800~km downstream of a 20~GeV neutrino factory in which there are $10^{20} \mu^+$ decays in the beam--forming straight section. The rates are shown as a function of $\sin^22\theta_{13}$ and $\delta m_{32}^2$ with the other oscillation parameters corresponding to the LMA scenario IA1. The top 3 curves are the predictions for $\overline{\nu}_\mu \rightarrow \overline{\nu}_\tau$ events and the lower curves are for $\nu_e \rightarrow \nu_\tau$ events. Results are from Ref.~\ref{bgrw00}. } \label{fig:t1} \end{figure} Let us define the $\sin^22\theta_{13}$ reach" for an experiment as the value of $\sin^22\theta_{13}$ for which we would observe 10 $\nu_e \rightarrow \nu_\tau$ events when there are $10^{20}$ muon decays in the beam--forming straight section of a neutrino factory. The $\sin^22\theta_{13}$ reach is shown as a function of the baseline and storage ring energy in Fig.~\ref{fig:t2} for a 5~kt detector and an oscillation scenario in which all of the parameters except $\sin^22\theta_{13}$ correspond to scenario IA1. The reach improves with energy (approximately $\sim E^{-1.5}$~\cite{bgrw00}) and is almost independent of baseline except for the highest energies and baselines considered. However, backgrounds to a $\nu_e \rightarrow \nu_\tau$ oscillation search have not been studied in detail, and are not included in the calculation. Background considerations will favor longer baselines. The number of muon decays needed to observe 10 $\nu_e \rightarrow \nu_\tau$ events is shown as a function of muon beam energy in Fig.~\ref{fig:v2} for the LMA scenario IA1, the SMA scenario IA2 , and the LOW scenario IA3. We conclude that within these three--flavor mixing scenarios, a 20~GeV storage ring in which there are O($10^{20}$) muon decays per year would begin to permit an observation of, or meaningful limits on, $\nu_e \rightarrow \nu_\tau$ oscillations provided a multi-kt detector with good tau--lepton identification and charge discrimination is practical. \begin{figure} \epsfxsize3.3in \centerline{\epsffile{fig_t2.ps}} \caption{Reach in $\sin^22\theta_{13}$ for the observation of 10 $\nu_e \rightarrow \nu_\tau$ oscillation events, shown as a function of baseline for four storage ring energies. The oscillation parameters correspond to the LMA scenario IA1. The curves correspond to $10^{20}$ $\mu^+$ decays in a 20~GeV neutrino factory with a 5~kt detector. Results are from Ref.~\ref{bgrw00}. } \label{fig:t2} \end{figure} Next, consider the oscillation scenarios IB1 (atmospheric + LSND scales) and IC1 (three--flavor with atmospheric, solar, and LSND data stretched). In these cases the leading $\delta m^2$ is large (0.3~eV$^2$/c$^4$) and medium baseline experiments ($L =$ 10-100~km) become interesting. As an example, consider a medium baseline experiment a few $\times 10$~km downstream of a 20~GeV neutrino factory in which there are $10^{20} \mu^+$ decays. The $\nu_e \rightarrow \nu_\tau$ and accompanying $\overline{\nu}_\mu \rightarrow \overline{\nu}_\tau$ event rates are shown in Fig.~\ref{fig:t3} as a function of the baseline and the phase $\delta$ with the other oscillation parameters corresponding to scenario IB1. In contrast to the $\overline{\nu}_\mu \rightarrow \overline{\nu}_\tau$ rates, the $\nu_e \rightarrow \nu_\tau$ rates are very sensitive to $\delta$, and for $|\delta| > 20^\circ$ can be very large, yielding thousands of events per year in a 1~kt detector at $L = 60$~km, for example. Note that the corresponding $\overline{\nu}_\mu \rightarrow \overline{\nu}_\tau$ rate is of order 100~events. For small $|\delta|$ the $\overline{\nu}_\mu \rightarrow \overline{\nu}_\tau$ rate will dominate the $\tau$ appearance event sample. For larger $|\delta|$ the $\nu_e \rightarrow \nu_\tau$ rate dominates. Good $\tau$ charge determination will therefore be important to measure both $\nu_e \rightarrow \nu_\tau$ and $\overline{\nu}_\mu \rightarrow \overline{\nu}_\tau$ oscillations. \begin{figure} \epsfxsize3.3in \centerline{\epsffile{fig_t3.ps}} \caption{$\nu_\tau$ CC appearance rates in a 1~kt detector downstream of a 20~GeV neutrino factory in which there are $10^{20} \mu^+$ decays. Rates are shown as a function of the baseline $L$ and phase $\delta$, with the other oscillation parameters corresponding to the LSND + Atmospheric scenario IB1. Predictions for $\nu_e \rightarrow \nu_\tau$ and $\overline{\nu}_\mu \rightarrow \overline{\nu}_\tau$ are shown separately, as labeled. Results are from Ref.~\ref{bgrw_prep}. } \label{fig:t3} \end{figure} Now consider the $\tau$ appearance rates in scenario IC1. In this case the rates are not sensitive to $\delta$ and, for a 1~kt detector at $L = 60$~km, there are about 8000 $\nu_e \rightarrow \nu_\tau$ events and 93000 $\overline{\nu}_\mu \rightarrow \overline{\nu}_\tau$ events~\cite{bgrw_prep}. A detector with $3\sigma$ (or better) $\tau$-lepton charge discrimination would enable these two rates to be separately measured. We conclude that measurements of the $\nu_e \rightarrow \nu_\tau$ oscillation rate at a neutrino factory would provide an important test of the oscillation scenario. In LMA, SMA, and LOW three-flavor oscillation scenarios, a 20~GeV neutrino factory providing O($10^{20}$) muon decays could permit an observation of, or meaningful limits on, $\nu_e \rightarrow \nu_\tau$ oscillations. In LSND-type scenarios where the leading $\delta m^2$ scale is large, a 20~GeV neutrino factory providing O($10^{19}$) muon decays might already permit hundreds of $\nu_e \rightarrow \nu_\tau$ events to be measured. It should be noted that the feasibility of a multi-kt detector with good $\tau$ identification efficiency (for example 30\%) and good charge sign determination has not been explored in detail at this stage, and further work is required to identify the best detector technology for this, and determine the expected resolutions and efficiencies. %\clearpage \subsubsection{Measurement of $\nu_\mu \rightarrow \nu_\tau$ oscillations} The present SuperK data suggests that the atmospheric neutrino deficit is due to $\nu_\mu \rightarrow \nu_\tau$ oscillations. If this is correct the next generation of accelerator based long baseline experiments are expected to measure these oscillations. %Hence %$\nu_\mu \rightarrow \nu_\tau$ appearance measurements at a neutrino %factory are likely to be less interesting than $\nu_e \rightarrow \nu_\mu$ %or $\nu_e \rightarrow \nu_\tau$ appearance measurements. Nevertheless, for a fixed neutrino factory energy and baseline, it is important to measure or put stringent constraints on all of the appearance channels so that the sum of the appearance modes can be compared with the disappearance measurements. Hence, we briefly consider $\nu_\mu \rightarrow \nu_\tau$ rates at a neutrino factory. Note that at a 20~GeV neutrino factory the average interacting neutrino energy is of order 15~GeV, and for $\delta m^2$ within the favored SuperK range, the first oscillation maximum occurs at baselines of $7000 \pm 3000$~km. At shorter baselines the oscillation probabilities are lower and hence the signal/background ratio is lower, although the signal rate can be higher. Consider first a 5~kt detector 2800~km downstream of a 20~GeV neutrino factory in which there are $10^{20}$ muon decays. The $\overline{\nu}_\mu \rightarrow \overline{\nu}_\tau$ event rates are shown in Fig.~\ref{fig:t1} as a function of $\sin^2 2\theta_{13}$ and $\delta m^2_{32}$, with the other oscillation parameters corresponding to the LMA scenario IA1. If negative muons are stored in the neutrino factory, the resulting $\nu_\mu \rightarrow \nu_\tau$ event rates would be about a factor of two higher than the $\overline{\nu}_\mu \rightarrow \overline{\nu}_\tau$ rates shown in the figure. A neutrino factory providing O($10^{20}$) muon decays would enable $\nu_\mu \rightarrow \nu_\tau$ appearance data samples of a few hundred to a few thousand events to be obtained. Similar rates are expected in SMA and LOW three-flavor mixing scenarios. Next consider a longer baseline example in which a 10~kt ICANOE--type detector is 7400~km downstream of a 30~GeV neutrino factory which provides $10^{20}$ muon decays in the beam forming straight section~\cite{camp00}. The main advantage of a longer baseline is that the total interaction rate, and hence the $\tau$--lepton background, is reduced. The energy distribution for events in which there is no charged lepton can directly reflect the presence of a $\nu_\tau$ signal (see Fig.~\ref{fig:m1}). The non--$\tau$ events in this event sample can be suppressed using topology--dependent kinematic cuts. It is desirable that the $\tau$ charge--sign also be determined which, with an external muon spectrometer, will be possible for the $\tau \rightarrow \mu$ subsample. We conclude that the measurement of $\nu_\mu \rightarrow \nu_\tau$ oscillations with high statistical precision will be possible at a neutrino factory in long and very long baseline experiments. A more complete study is warranted. %\clearpage \subsubsection{Determination of $\sin^2 2\theta_{13}$, $\sin^2 2\theta_{23}$, and $\delta m^2_{32}$} Consider first the determination of $\sin^2 2\theta_{13}$. The most sensitive way to measure $\sin^2 2\theta_{13}$ at a neutrino factory is to measure the $\nu_e \rightarrow \nu_\mu$ oscillation amplitude, which is approximately proportional to $\sin^2 2\theta_{13}$. More explicitly, the value of $\sin^2 2\theta_{13}$ is extracted from a fit to the spectrum of CC interactions tagged by a wrong--sign muon. Background contributions from, for example, muonic decays of charged mesons must be kept small, which favors small total event samples and hence long baselines. \begin{figure}[th] \epsfxsize3.3in \centerline{\epsffile{s2_contrib_to_ws_muons.eps}} \caption{Visible energy spectrum for events tagged by wrong-sign muons in an ICANOE--type detector (full histogram). The oscillation parameters are $\delta m^2_{32}=3.5\times 10^{-3}$~eV$^2$/c$^4$, $\sin^2\theta_{23}=1$, and $\sin^2 2\theta_{13}=0.05$. Also shown are the contributions from $\nu_e\to\nu_\mu$ oscillations (black dashed curve), $\nu_e\to\nu_\tau$, with a subsequent muonic decay of the $\tau$ lepton (red curve), and background from muonic decays of pions or kaons in neutral current or charged current events (blue dot-dashed curve). Results are from Ref.~\ref{camp00}. } \label{fig:m3} \end{figure} % As an example, consider a 10~kt ICANOE--like detector that is 7400~km downstream of a 30~GeV neutrino factory~\cite{camp00}. The simulated energy spectrum of wrong-sign muon events is shown in Fig.~\ref{fig:m3} for three--flavor oscillations with the parameters $\delta m^2_{32}=3.5\times 10^{-3}$~eV$^2$/c$^4$, $\sin^2 2\theta_{23}=1$, and $\sin^2 2\theta_{13}=0.05$. Note that the backgrounds predominantly contribute to the low energy part of the spectrum. To fit the observed spectrum and extract $\sin^2 2\theta_{13}$ matter effects must be taken into account. The modification of the oscillation probability due to matter effects is a function of the profile of the matter density $\rho$ between the neutrino source and the detector. The density profile is known from geophysical measurements, and this knowledge can either be used in the fit, or alternatively $\rho$ can be left as a free parameter. It has been shown that both methods give consistent results~\cite{camp00}, and that the uncertainties on the fitted values of $\rho$ and $\sin^2 2\theta_{13}$ are not strongly correlated. However, the fitted value for $\sin^2 2\theta_{13}$ does depend on the assumed values for $\sin^2 \theta_{23}$ and $\delta m^2_{32}$. The measured right--sign muon ($\nu_\mu$ disappearance) distribution, together with the distributions of events tagged by electrons, $\tau$--leptons, or the absence of a lepton, can be used to constrain these additional oscillation parameters. Hence, the best way to extract $\sin^2 2\theta_{13}$ is from a global fit to all of the observed event distributions, with the oscillation parameters (and optionally $\rho$) left as free parameters. If the density profile is left as a free parameter, the fit determines its value with an uncertainty of about 10\%~\cite{camp00,cerv00}. This provides a quantitative test of the MSW effect ! Examples of fit results in the ($\sin^2 2\theta_{13}$,~$\sin^2 \theta_{23}$)--plane are shown in Fig.~\ref{fig:m7} for $10^{19}$, $10^{20}$, and $10^{21}$ muon decays in the neutrino factory. As the beam intensity increases the measurements become more precise. With $10^{19} \mu^+$ and $\mu^-$ decays $\sin^2 2\theta_{13}$ and $\sin^2 2\theta_{23}$ are determined with precisions of 40\% and 20\% respectively. With $10^{21}$ decays these precisions have improved to $\sim5$\%. If the baseline is decreased from 7400~km to 2900~km the oscillation parameters are determined with comparable (although slightly worse) precisions (Fig.~\ref{fig:m7a}). We conclude that within the framework of three--flavor mixing, provided $\sin^2 2\theta_{13}$ is not too small, a global fit to the observed oscillation distributions would enable $\sin^2 2\theta_{13}$, $\sin^2 \theta_{23}$, and $\delta m^2_{32}$ to be simultaneously determined, and the MSW effect to be measured. \begin{figure}[t] \epsfxsize3.3in \centerline{\epsffile{contgeers.eps}} %\centerline{\epsffile{contdgeers2.eps}} \caption{Results from a global fit to the visible energy distributions for various event classes recorded in a 10~kt ICANOE--type detector 7400~km downstream of a 30~GeV neutrino factory. The 68\% CL contours correspond to experiments in which there are $10^{19}$, $10^{20}$, and $10^{21} \mu^+$ decays in the neutrino factory (as labeled) followed by the same number of $\mu^-$ decays. Upper plot: density fixed to its true value. Lower plot: density is a free parameter of the fit. Results are from Ref.~\ref{camp00}. } \label{fig:m7} \end{figure} \begin{figure}[t] \epsfxsize3.3in \centerline{\epsffile{contdgeers2.eps}} %\centerline{\epsffile{contgeers2.eps}} \caption{Results from a global fit to the visible energy distributions for various event classes recorded in a 10~kt ICANOE--type detector downstream of a 30~GeV neutrino factory in which there are $10^{20} \mu^+$ decays in the neutrino factory followed by the same number of $\mu^-$ decays. The 68\% CL contours correspond to baselines of 7400~km and 2900~km, as labeled. Results are from Ref.~\ref{camp00}. } \label{fig:m7a} \end{figure} \begin{figure} \epsfxsize3.3in \centerline{\epsffile{s15.ps}} \caption{Fit results in the ($A$, $\theta_{13}$)--plane for a simulated experiment in which a 40~kt Fe-scintillator detector is a distance $L$~km downstream of a 50~GeV neutrino factory in which there are $10^{21} \mu$ decays. The density parameter $A$ is defined in Eq.~(\ref{eq:defnA}). The curves are 68.5, 90, and 99\% CL contours. Results are from Ref.~\ref{cerv00}.} \label{fig:s15} \end{figure} \begin{figure} \epsfxsize3.3in \centerline{\epsffile{s2_l7400_sensi_t13.eps}} %\centerline{\epsffile{s2_l7400_exclu_t13.eps}} \caption{Allowed regions in oscillation parameter space calculated for a simulated experiment in which there are $N$~$\mu^+$ decays followed by $N$~$\mu^-$ decays in a 30~GeV neutrino factory that is 7400~km from a 10~kt ICANOE--type detector. The contours correspond to $N = 10^{20}$ and $10^{21}$ with and without backgrounds included in the calculation. Results are from Ref.~\ref{camp00}.} \label{fig:m4} \end{figure} \begin{figure} \epsfxsize3.3in \centerline{\epsffile{s13.ps}} \caption{Allowed regions in oscillation parameter space calculated for a simulated experiment in which a 40~kt Fe-scintillator detector is a distance $L$~km downstream of a 50~GeV neutrino factory in which there are $10^{21} \mu$ decays. The curves are 90\% CL contours for $L = 732$~km (dashed), 3500~km (solid), and 7332~km (dotted). Results are from Ref.~\ref{cerv00}.} \label{fig:s13} \end{figure} \begin{figure} \epsfxsize3.0in \centerline{\epsffile{mu3-30gev_2800km_1234.eps}} \caption{Visible energy distributions for events tagged by a right--sign muon in a MINOS--type detector 2800~km downstream of a 20~GeV neutrino factory in which there are $2 \times 10^{20} \mu^-$ decays. Predicted distributions are shown for four values of $\delta m^2_{32}$, with the other parameters corresponding to the LMA scenario IA1. For each panel, the points with statistical error bars show an example of a simulated experiment. The light shaded histograms show the predicted distributions in the absence of oscillations. Results are from Ref.~\ref{bgrw00}. } \label{fig:v5} \end{figure} \begin{figure} \epsfxsize2.8in \centerline{\epsffile{30gev_disap_fit.eps}} \caption{Fit results for simulated $\nu_\mu$ disappearance measurements with a 10~kt MINOS-type detector 2800~km downstream of a 30~GeV neutrino factory in which there are $2 \times 10^{20} \mu^-$ decays. For each trial point the $1\sigma$, $2\sigma$, and $3\sigma$ contours are shown for a perfect detector (no backgrounds) and no systematic uncertainty on the beam flux. The 68\%, 90\% and 95\% SuperK regions are indicated. Results are from Ref.~\ref{bgrw00}. } \label{fig:v6} \end{figure} %\begin{figure} %\epsfxsize3.0in %\centerline{\epsffile{bob2.ps}} %\caption{Bobs backgrounds %Results are from Ref.~\ref{bern00}. %} %\label{fig:b2} %\end{figure} \begin{figure} \epsfxsize4.8in \centerline{\epsffile{newbob1.ps}} %\centerline{\epsffile{disappearance_with_no_flux_error.ps}} \caption{Fit results (1~$\sigma$ contours) for (a) simulated $\nu_\mu$ disappearance measurements with a 10~kt MINOS-type detector 2800~km downstream of a 30~GeV neutrino factory in which there are $2 \times 10^{20} \mu^-$ decays, with and without an ad hoc 2\% systematic uncertainty on the neutrino flux, and (b) wrong--sign muon appearance measurements including an ad hoc 2\% systematic uncertainty on the flux. The acceptance for a muon is zero for $p_{\mu} < 4$ GeV and unity for $p_{\mu} \geq 4$ GeV. Backgrounds are included but no $p_{\perp}^2$ cut has been used. Results are from Ref.~\ref{bern00}. } \label{fig:b1} \end{figure} Consider as a second example a 20~kt MINOS--type detector 2800~km downstream of a 30~GeV neutrino factory providing $10^{20}$ muon decays. Some preliminary fit results~\cite{bern00} for a LMA type scenario are shown in Fig.~\ref{fig:b1}b. For this example the analysis required $p_{\mu} < 4$~GeV/c, but did not use the full set of cuts described in section~\ref{detectors}, and therefore tolerated a background level a factor of a few greater than shown in Fig.~\ref{data} and Table~\ref{thetable}. Nevertheless, the fits to the measured distribution of energies for events tagged by wrong--sign muons were able to determine $\sin^2 2\theta_{13}$ and $\delta m^2_{32}$ with precisions of 14\% and 10\% respectively for scenario IA1. To illustrate the ultimate sensitivity to the oscillation parameters that might be achievable at a high intensity neutrino factory, consider next a 40~kt Fe-scintillator detector downstream of a 50~GeV neutrino factory in which there are $10^{21} \mu^+$ decays followed by $10^{21} \mu^-$ decays~\cite{cerv00}. Fit results in the (matter density, $\sin^2 2\theta_{13}$)--plane are shown in Fig.~\ref{fig:s15} for three baselines. The precision on the $\sin^2 2\theta_{13}$ determination is a few percent. Note that the analysis described in Ref.~\cite{cerv00} suggests that backgrounds can be suppressed to less than $10^{-5}$ of the total CC rate in the detector. This impressive level of background rejection deserves further study. At the shortest baselines (732~km) matter effects are too small to obtain a good determination of the matter density parameter. Consider next the precision with which the oscillation parameters can be determined if $\sin^2 2\theta_{13}$ is very small, and hence no $\nu_e \rightarrow \nu_\mu$ oscillation signal is observed. The resulting limits on $\sin^2 2\theta_{13}$ are shown as a function of $\delta m^2_{32}$ in Fig.~\ref{fig:m4} for a 10~kt ICANOE type detector 7400~km downstream of a 30~GeV neutrino factory in which there are $10^{20} \mu^+$ decays followed by $10^{20} \mu^-$ decays~\cite{camp00}. The resulting upper limit on $\sin^2 2\theta_{13}$ would be O($10^{-3}-10^{-4}$), about three orders of magnitude below the present experimental bound, and one to two orders of magnitude below the bound that would be expected at the next generation of long--baseline experiments. The limit would become even more stringent at a higher intensity neutrino factory. As an example of the ultimate sensitivity that might be achievable, in Fig.~\ref{fig:s13} the limits on $\sin^2 2\theta_{13}$ are shown as a function of $\delta m^2_{32}$ and baseline for a 40~kt Fe-scintillator detector downstream of a 50~GeV neutrino factory in which there are $10^{21} \mu$ decays~\cite{cerv00}. The non--observation of $\nu_e \rightarrow \nu_\mu$ oscillations could result in an upper limit on $\sin^2 2\theta_{13}$ below $10^{-5}$ ! With this level of sensitivity $\nu_e \rightarrow \nu_\mu$ oscillations driven by the sub--leading $\delta m^2$ scale might be observed~\cite{bgrw00}. For example, the number of muon decays required to produce 10 $\nu_e \rightarrow \nu_\mu$ events in a 50~kt detector 2800~km downstream of a neutrino factory is shown for a bimaximal mixing scenario ($\sin^22\theta_{13} = 0$) in Fig.~\ref{fig:v2} as a function of the stored muon energy. Approaching $10^{21}$ muon decays might be sufficient to observe oscillations driven by the sub--leading scale, but would require background levels of the order of $10^{-5}$ of the total CC rate, or better. With a vanishing or very small $\sin^2 2\theta_{13}$ only the $\nu_\mu\to\nu_\tau$ oscillations will have a significant rate, and the oscillation parameters $\sin^2 2\theta_{23}$ and $\delta m^2_{32}$ can be determined by fitting the right--sign muon ($\nu_\mu$ disappearance) spectrum. Good sensitivity can be obtained provided the baseline is chosen such that the first oscillation maximum occurs in the middle of the visible energy spectrum. As a first example, spectra of events tagged by right--sign muons are shown in Fig.~\ref{fig:v5} as a function of $\delta m^2_{32}$ for a 10~kt MINOS--type detector 2800~km downstream of a 30~GeV neutrino factory in which there are $2 \times 10^{20} \mu^-$ decays in the beam--forming straight section~\cite{bgrw00}. The position of the oscillation maximum (resulting in a dip in the observed distributions) is clearly sensitive to $\delta m^2_{32}$. The depth of the observed dip is sensitive to the oscillation amplitude, and hence to $\sin^2 2\theta_{23}$. The visible energy spectrum of right--sign muon events can be fit to obtain $\sin^2 2\theta_{23}$ and $\delta m^2_{32}$. We begin by considering the statistical precision that could be obtained with a perfect detector having MINOS--type resolution functions, no backgrounds, no selection requirements, and no systematic uncertainty on the neutrino flux. Fit results are shown in Fig.~\ref{fig:v6}. For $\delta m^2_{32}=3.5\times 10^{-3}$~eV$^2$/c$^4$ the fit yields statistical precisions of a few percent on the the values of the oscillation parameters. If $L$ is increased to 7332~km, the statistical precision improves to about 1\%. With this level of precision it is likely that systematic uncertainties will be significant~\cite{bern00}. To illustrate this in Fig.~\ref{fig:b1}a the $1\sigma$ contours are shown in the ($\delta m^2_{32}$, $\sin^2 2\theta_{23}$) from fits which include backgrounds together with 0\% and 2\% systematic uncertainties on the beam flux. With a 2\% flux uncertainty the precision on $\delta m^2_{32}$ and $\sin^2 2\theta_{23}$ are respectively 11\% and 14\%. \begin{figure}[t] \epsfxsize3.5in \centerline{\epsffile{s1_syst_err_732.eps}} \caption{Fit results for simulated $\nu_\mu$ disappearance measurements with a 10~kt ICANOE type detector 732~km downstream of a 30~GeV neutrino factory in which there are $10^{20} \mu$ decays. The effect of a systematic uncertainty on the neutrino flux is shown. Results are from Ref.~\ref{camp00}. } \label{fig:m8} \end{figure} \begin{figure} \epsfxsize3.5in \centerline{\epsffile{s1_comp_fit_mix23_diff_munorm.eps}} \caption{Fit results for simulated $\nu_\mu$ disappearance measurements with a 10~kt ICANOE type detector 2900~km (top plot) and 7400~km (bottom plot) downstream of a 30~GeV neutrino factory in which there are (a) $10^{20} \mu$ decays and (b) $10^{21} \mu$ decays. Results are from Ref.~\ref{camp00}. } \label{fig:m9} \end{figure} \begin{figure}[t] \epsfxsize3.5in \centerline{\epsffile{s1_diff_dm_3500.eps}} \caption{Fit results for simulated $\nu_\mu$ disappearance measurements with a 10~kt ICANOE type detector 2900~km downstream of a 30~GeV neutrino factory in which there are $10^{20} \mu^-$ decays followed by $10^{20} \mu^+$ decays. Results are shown for 3 values of $\delta m^2_{32}$, and are from Ref.~\ref{camp00}. } \label{fig:m10} \end{figure} \begin{figure} \epsfxsize3.5in \centerline{\epsffile{s1_diff_dm_7344.eps}} \caption{Fit results for simulated $\nu_\mu$ disappearance measurements with a 10~kt ICANOE type detector 7400~km downstream of a 30~GeV neutrino factory in which there are $10^{20} \mu^+$ decays followed by $10^{20} \mu^+$ decays. Results are shown for 3 values of $\delta m^2_{32}$, and are from Ref.~\ref{camp00}. } \label{fig:m11} \end{figure} As a second example, consider a 10~kt ICANOE--type detector that is downstream of a 30~GeV neutrino factory in which there are $10^{20} \mu^+$ decays in the beam--forming straight section followed by $10^{20} \mu^-$ decays~\cite{camp00}. The sensitivity to the oscillation parameters has been studied by fitting simulated visible energy distributions for events tagged by a right--sign muon. The analysis includes a 2\% bin-to-bin uncorrelated systematic error on the number of neutrino interactions which takes into account the uncorrelated uncertainties on neutrino flux, the cross section, and the selection efficiency. To reduce background from charged meson decays, the events entering the fit are those with muons having momenta $>2$~GeV/c. Figures~\ref{fig:m8}-\ref{fig:m11} show fit results in the ($\sin^2 2\theta_{23}$,~$\delta m^2_{32}$)--plane as a function of the oscillation parameters and baseline. Note that for the short'' baseline ($L =732$~km) the first oscillation maximum for the reference value of $\delta m^2_{32}=3.5\times 10^{-3}$~eV$^2$/c$^4$ occurs at a neutrino energy of about 2~GeV. This is too low to produce a clear dip in the visible energy spectrum, and as a result $\sin^2 2\theta_{23}$ and $\delta m^2_{32}$ can only be determined with relatively low precision (Fig.~\ref{fig:m8}), and the fit results are sensitive to systematic uncertainties on the neutrino flux. At the longer baselines ($L = 2900$~km and 7400~km) the oscillation dip is visible, and the oscillation parameters can be measured with a precision that is mostly determined by the statistical uncertainty (Fig.~\ref{fig:m9}). For a 30~GeV neutrino factory and $\delta m^2_{32} = 3.5 \times 10^{-3}$~eV$^2$/c$^4$ the longer baseline (7400~km) yields the most precise result. Specifically, for $10^{20} \mu$ decays the statistical precisions on $\sin^2 2\theta_{23}$ and $\delta m^2_{32}$ are respectively about 10\% and 1\%. With $10^{21} \mu$ decays the $\sin^2 2\theta_{23}$ precision improves by about a factor of 2. It should be noted that the best baseline choice depends on $\delta m^2_{32}$ (Figs.~\ref{fig:m10}-\ref{fig:m11}), or more specifically $\delta m^2_{32}/E$. We conclude that, within the framework of three--flavor mixing, the oscillation parameters $\sin^2 2\theta_{13}$, $\sin^2 2\theta_{23}$, and $\delta m^2_{32}$ can be determined at a neutrino factory by fitting the observed visible energy distributions for various event types. A comprehensive study of the expected precisions of the measurements as a function of the oscillation parameters, baseline, and neutrino factory parameters has not yet been undertaken. However, detailed studies have been made for some examples in which there are $10^{20} \mu^+$ decays followed by $10^{20} \mu^-$ decays in a 30~GeV neutrino factory. For these examples we find that (i) if $\sin^2 2\theta_{13} >$~O($10^{-2}$) global fits can be used to determine its value, (ii) if $\sin^2 2\theta_{13}$ is too small to observe $\nu_e \rightarrow \nu_\mu$ oscillations then we would expect to place the very stringent upper limit on its value of $10^{-3}$ or better, and (iii) the values of $\sin^2 2\theta_{23}$, and $\delta m^2_{32}$ could be determined with precisions of respectively better than or of order 10\% and of order 1\%, provided the baseline is chosen so that the dip corresponding to the first oscillation maximum is in the middle of the visible energy distribution. At a high--intensity neutrino factory (for example with $10^{21}$ decays of 50~GeV muons) the mixing angles could be measured with a precision of a few percent, and if $\sin^2 2\theta_{13}$ is vanishingly small, the resulting upper limit could be at the O($10^{-5}$)--level. %\clearpage \subsubsection{Search for CP violation} \begin{figure} \epsfxsize2.8in \centerline{\epsffile{kerry.eps}} \caption{The ratio $R$ of $\bar\nu_e \to \bar\nu_\mu$ to $\nu_e \to \nu_\mu$ event rates at a 20~GeV neutrino factory for $\delta = 0$ and $\pm\pi/2$. The upper group of curves is for $\delta m^2_{32} < 0$, the lower group is for $\delta m^2_{32} > 0$. The statistical errors correspond to $10^{21}$ muon decays of each sign and a 50~kt detector. The oscillation parameters correspond to the LMA scenario IA1. With no matter or CP effects $R\sim0.5$ for all baselines. Results are from Ref.~\ref{bgrw00}. } \label{fig:cp1} \end{figure} \begin{figure} \epsfxsize3.2in \centerline{\epsffile{cp_sensitivity_6c.ps}} \caption{Reach in $\sin^22\theta_{13}$ that yields a $3\sigma$ discrimination between (a) $\delta = 0$ and $\pi/2$ with $\delta m^2_{32} > 0$, (b) $\delta = 0$ and $\pi/2$ with $\delta m^2_{32} < 0$, (c) $\delta = 0$ and $-\pi/2$ with $\delta m^2_{32} > 0$, and (d) $\delta = 0$ and $-\pi/2$ with $\delta m^2_{32} < 0$. The discrimination is based on a comparison of wrong--sign muon CC event rates in a 50~kt detector when $10^{21}$ positive and negative muons alternately decay in the neutrino factory. The reach is shown versus baseline for four storage ring energies. The oscillation parameters correspond to the LMA scenario IA1. Results are from Ref.~\ref{bgrw00}. } \label{fig:cp2} \end{figure} \begin{figure} \epsfxsize3.2in \centerline{\epsffile{s18.ps}} \caption{Fit results in the CP phase $\delta$ versus $\theta_{13}$ plane for a LMA scenario with $\delta m^2_{21} = 1 \times 10^{-4}$~eV$^2$/c$^4$. The 68.5, 90, and 99\% CL contours are shown for a 40~kt detector a distance $L$~km downstream of a 50~GeV neutrino factory in which there are $10^{21} \mu^+$ and $10^{21} \mu^-$ decays. Results are from Ref.~\ref{cerv00}. } \label{fig:cp3} \end{figure} \begin{figure} \epsfxsize3.2in \centerline{\epsffile{s22.ps}} \caption{The lowest value of $\delta m^2_{21}$, shown as a function of $\theta_{13}$, for which the maximal CP phase $\delta = \pi/2$ can be distinguished from a vanishing phase in a LMA oscillation scenario. The curve corresponds to a 40~kt detector 3500~km downstream of a 50~GeV neutrino factory in which there are $10^{21} \mu^+$ and $10^{21} \mu^-$ decays. Results are from Ref.~\ref{cerv00}. } \label{fig:cp4} \end{figure} In the majority of the three--flavor oscillation scenarios described in section~\ref{theory} the CP violating amplitude is too small to produce an observable effect. Nevertheless, in these cases stringent limits on CP violation would provide an important check of the overall interpretation of the oscillation data. If however the LMA scenario provides the correct description of neutrino oscillations, CP violating effects might be sufficiently large to be observable at a high--intensity neutrino factory~\cite{cerv00,bgrw00}. This is illustrated in Fig.~\ref{fig:cp1} which shows, as a function of baseline at a 20~GeV neutrino factory, the ratio $R$ for $\delta = 0$ and $\pm\pi/2$, where $R$ is defined as the $\overline{\nu}_e \rightarrow \overline{\nu}_\mu$ event rate divided by the $\nu_e \rightarrow \nu_\mu$ event rate. The upper group of curves is for $\delta m^2_{32} < 0$, the lower group is for $\delta m^2_{32} > 0$, and the statistical errors correspond to $10^{21}$ muon decays of each sign with a 50~kt detector. If $L$ is a few thousand~km a non--zero $\delta$ can produce a modification to $R$ that is sufficiently large to be measured ! Since the $\nu_e \rightarrow \nu_\mu$ oscillation rates are to a good approximation proportional to $\sin^22\theta_{13}$, it is useful to define the $\sin^22\theta_{13}$ reach as that value of $\sin^22\theta_{13}$ that will produce a 3$\sigma$ change in the predicted ratio $R$ when $\delta$ is changed from $0$ to $\pm\pi/2$. The $\sin^22\theta_{13}$ reach is shown as a function of baseline and stored muon energy in Fig.~\ref{fig:cp2} for a 50~kt detector at a neutrino factory in which there are $10^{21} \mu^+$ decays followed by $10^{21} \mu^-$ decays. With an optimum baseline of about 3000~km (for $\delta m^2_{32} = 3.5 \times 10^{-3}$~eV$^2$/c$^4$) the $\sin^22\theta_{13}$ reach is approximately $10^{-2}$, an order of magnitude below the current experimental bound. Thus, in a LMA scenario, CP violation in the lepton sector might be measurable at a neutrino factory providing O($10^{21}$) muon decays. As an example, consider a 40~kt Fe-scintillator detector downstream of a 50~GeV neutrino factory providing $10^{21} \mu^+$ decays followed by $10^{21} \mu^-$ decays in the beam--forming straight section~\cite{cerv00}. The results of fits to the simulated wrong--sign muon event distributions, with $\delta$ and $\sin^22\theta_{13}$ left as free parameters, are shown in Fig.~\ref{fig:cp3} for various baselines, with the sub--leading scale $\delta m^2_{21} = 1 \times 10^{-4}$~eV$^2$/c$^4$. The analysis includes the detector resolutions, reasonable event selection criteria, and backgrounds. As might be expected from Fig.~\ref{fig:cp1} at $L = 7332$~km there is little sensitivity to $\delta$, and at the short" baseline $L = 732$~km the fit has difficulty untangling $\delta$ from $\sin^22\theta_{13}$. However, at a baseline of $L = 3500$~km for the example shown $\delta$ and $\sin^22\theta_{13}$ can be determined with precisions of respectively about $15^\circ$ and a few percent. Note that a combination of baselines can yield a modest improvement in the precision of the measurement. The sensitivity to CP violation decreases with decreasing $\delta m^2_{21}$. Figure~\ref{fig:cp4} shows as a function of $\sin^22\theta_{13}$ the lowest value of $\delta m^2_{21}$ for which the maximal CP phase $\delta = \pi/2$ can be distinguished from a vanishing phase at $L = 3500$~km. This limiting $\delta m^2_{21}$ is below the current central value for the LMA parameter space suggested by solar neutrino deficit, and is about $2 \times 10^{-5}$~eV$^2$/c$^4$, independent of $\sin^22\theta_{13}$. Finally we note that the sensitivity of short and medium baseline experiments to CP violation in a three--active plus one sterile neutrino scenario has been considered in Ref.~\cite{doninietal}. They concluded that a 1~kt detector and a 100~km baseline could provide a clean test of CP violation, particularly in the $\tau--lepton$ appearance channel. %\clearpage \subsection{Summary} The oscillation physics that could be pursued at a neutrino factory is compelling. In particular, experiments at a neutrino factory would be able to simultaneously measure, or put stringent limits on, all of the appearance modes $\nu_e \rightarrow \nu_\tau$, $\nu_e \rightarrow \nu_\mu$, and $\nu_\mu \rightarrow \nu_\tau$. Comparing the sum of the appearance modes with the disappearance measurements would provide a unique basic check of candidate oscillation scenarios that cannot be made with a conventional neutrino beam. In addition, for all of the specific oscillation scenarios we have studied, the $\nu_e$ component in the beam can be exploited to enable crucial physics questions to be addressed. These include (i) the pattern of neutrino masses (sign of $\delta m^2$) and a quantitative test of the MSW effect, (ii) the precise determination of (or stringent limits on) all of the leading oscillation parameters, which in a three--flavor mixing scenario would be $\sin^22\theta_{13}$, $\sin^22\theta_{23}$, and $\delta m^2_{32}$, and (iii) the observation of, or stringent limits on, CP violation in the lepton sector. To be more quantitative in assessing the beam energy, intensity, and baseline required to accomplish a given set of physics goals it is necessary to consider two very different experimental possibilities: (a) the LSND oscillation results are not confirmed by the MiniBooNE experiment, or (b) the LSND results are confirmed. \begin{description} \item{(a) LSND not confirmed.} Fairly extensive neutrino factory studies have been made within the framework of three--flavor oscillation scenarios in which there is one large" $\delta m^2$ scale identified with the atmospheric neutrino deficit results, and one small $\delta m^2$ identified with the solar neutrino deficit results. A summary of the energy dependent beam intensities required to cross a variety of thresholds of interest" is provided by Fig.~\ref{fig:v2}. A 20~GeV neutrino factory providing $10^{19}$ muon decays per year is a good candidate entry--level" facility which would enable either (i) the first observation of $\nu_e \rightarrow \nu_\mu$ oscillations, the first direct measurement of matter effects, and a determination of the sign of $\delta m^2_{32}$ and hence the pattern of neutrino masses, or (ii) a very stringent limit on $\sin^22\theta_{13}$ and a first comparison of the sum of all appearance modes with the disappearance measurements. The optimum baselines for this entry--level physics program appears to be of the order of 3000~km or greater, for which matter effects are substantial. Longer baselines also favor the precise determination of $\sin^22\theta_{13}$. A 20~GeV neutrino factory providing $10^{20}$ muon decays per year is a good candidate upgraded neutrino factory (or alternatively a higher energy facility providing a few $\times 10^{19}$ decays per year). This would enable the first observation of, or meaningful limits on, $\nu_e \rightarrow \nu_\tau$ oscillations, and precision measurements of the leading oscillation parameters. In the more distant future, a candidate for a second (third ?) generation neutrino factory might be a facility that provides O($10^{21}$) decays per year and enables the measurement of, or stringent limits on, CP violation in the lepton sector. \item{(b) LSND confirmed.} Less extensive studies have been made for the class of scenarios that become of interest if the LSND oscillation results are confirmed. However, in the scenarios we have looked at (IB1 and IC1) we find that the $\nu_e \rightarrow \nu_\tau$ rate is sensitive to the oscillation parameters and can be substantial. With a large leading $\delta m^2$ scale medium baselines (for example a few $\times 10$~km) are of interest, and the neutrino factory intensity required to effectively exploit the $\nu_e$ beam component might be quite modest ($< 10^{19}$ decays per year). \end{description} The neutrino factory oscillation physics study we have pursued goes beyond previous studies. In particular we have explored the physics capabilities as a function of the muon beam energy and intensity, and the baseline. Based on the representative oscillation scenarios and parameter sets defined for the study, it would appear that a 20~GeV neutrino factory providing O($10^{19}$) decays per year would be a viable entry--level facility for experiments at baselines of $\sim3000$~km or greater. There are still some basic open questions that deserve further study: (1) We have sampled, but not fully explored, the beam energy and intensity required to explore the scenarios that become relevant if the LSND oscillation results are confirmed. (2) Possible technologies for a very massive neutrino factory detector have been considered, but these considerations deserve to be pursued further. The chosen detector technology will determine whether it is necessary to go deep underground. (3) We have developed tools that can explore the utility of having polarized muon beams. The physics payoff with polarization is a detailed issue. It deserves to be studied in the coming months. Based on our study, we believe that a neutrino factory in 5--10~years from now would be the right tool at the right time.