%harris comments %hill comments %bernstein comments %tigner comments %rigolin except table check \section{Beam properties} Consider an ensemble of polarized negatively-charged muons. When the muons decay they produce muon neutrinos with a distribution of energies and angles in the muon rest--frame described by~\cite{gaisser}: % \begin{eqnarray} \frac{d^2N_{\nu_\mu}}{dxd\Omega_{cm}} &\propto& {2x^2\over4\pi} \left[ (3-2x) + (1-2x) P_\mu \cos\theta_{cm} \right] \, , \label{eq:n_numu} \end{eqnarray} where $x\equiv 2E_\nu/m_\mu$, $\theta_{cm}$ is the angle between the neutrino momentum vector and the muon spin direction, and $P_\mu$ is the average muon polarization along the beam direction. The electron antineutrino distribution is given by: \begin{eqnarray} \frac{d^2N_{\bar\nu_e}}{dxd\Omega_{cm}} &\propto& {12x^2\over4\pi} \left[ (1-x) + (1-x) P_\mu\cos\theta_{cm} \right] \, , \label{eq:n_nue} \end{eqnarray} % and the corresponding distributions for $\bar\nu_\mu$ and $\nu_e$ from $\mu^+$ decay are obtained by the replacement $P_{\mu} \to -P_{\mu}$. Only neutrinos and antineutrinos emitted in the forward direction ($\cos\theta_{lab}\simeq1$) are relevant to the neutrino flux for long-baseline experiments; in this limit $E_\nu = x E_{max}$ and at high energies the maximum $E_\nu$ in the laboratory frame is given by $E_{max} = \gamma (1 + \beta \cos\theta_{cm})m_{\mu}/2 $. The $\nu_\mu$ and $\overline{\nu}_{e}$ distributions as a function of the laboratory frame variables are then given by: \begin{eqnarray} \frac{d^2N_{\nu_{\mu}}}{dxd\Omega_{lab}} &\propto& {1\over \gamma^2 (1- \beta\cos\theta_{lab})^2}\frac{2x^2}{4\pi} \left[ (3-2x) + (1-2x)P_{\mu}\cos\theta_{cm} \right] , \label{eq:numu} \end{eqnarray} and \begin{eqnarray} \frac{d^2N_{\overline{\nu}_{e}}}{dxd\Omega_{lab}} &\propto& {1\over \gamma^2 (1- \beta\cos\theta_{lab})^2}\frac{12x^2}{4\pi} \left[ (1-x) + (1-x)P_{\mu}\cos\theta_{cm} \right] \; . \label{eq:nue} \end{eqnarray} Thus, for a high energy muon beam with no beam divergence, the neutrino and antineutrino energy-- and angular-- distributions depend upon the parent muon energy, the decay angle, and the direction of the muon spin vector. With the muon beam intensities that could be provided by a muon--collider type muon source~\cite{status_report} the resulting neutrino fluxes at a distant site would be large. For example, Fig.~\ref{fluxes} shows as a function of muon energy and polarization, the computed fluxes per $2\times 10^{20}$ muon decays at a site on the other side of the Earth ($L = 10000$~km). Note that the $\nu_e$ and $\overline{\nu}_e$ fluxes are suppressed when the muons have $P = +1$ (-1). This can be understood by examining Eq.~\ref{eq:nue} and noting that for $P = -1$ the two terms cancel in the forward direction for all $x$. \begin{figure} \epsfxsize3.5in \centerline{\epsffile{s2_fluxes_fig.eps}} %\centerline{\epsffile{cc_rates_fig.ps}} %\renewcommand{\baselinestretch}{2} \caption{Calculated $\nu$ and $\overline{\nu}$ fluxes at a far site located 10000 km from a neutrino factory in which $2 \times 10^{20}$ muons have decayed in the beam--forming straight section. The fluxes are shown as a function of the energy of the stored muons for negative muons (top two plots) and positive muons (bottom two plots), and for three muon polarizations as indicated. The calculated fluxes are averaged over a circular area of radius 1~km at the far site. Calculation from Ref.~\ref{geer98}.} \label{fluxes} \end{figure} \subsection{Interaction rates} % \begin{figure} \epsfxsize3.in \centerline{\epsffile{s2_tau_ratio.eps}} \caption{The total cross section for charged current neutrino scattering by muon and tau neutrinos (top plot), and the ratio of tau to muon neutrino cross sections as a function of neutrino energy (bottom plot).} \label{tau_fig} \end{figure} % Neutrino charged current (CC) scattering cross-sections are shown as a function of energy in Fig.~\ref{tau_fig}. At low energies the neutrino scattering cross section is dominated by quasi-elastic scattering and resonance production. However, if $E_\nu$ is greater than $\sim10$~GeV, the total cross section is dominated by deep inelastic scattering and is approximately~\cite{CCFRsigma}: % \begin{eqnarray} \sigma(\nu +N \gt \lepton^- + X) &\approx& 0.67\times 10^{-38} \; \centi^2\times (E_{\nu}, \GeV) \, , \\ %\end{eqnarray} %\begin{eqnarray} \sigma(\antinu +N \gt \lepton^+ + X) &\approx& 0.34\times 10^{-38} \; \centi^2\times (E_{\antinu}, \GeV) \; . \end{eqnarray} % The number of $\nu_e$ and $\nu_\mu$ CC events observed per incident neutrino on an isoscalar target is given by: \begin{eqnarray} N(\nu +N \gt \lepton^- + X) &=& 4.0 \times 10^{-15} (E_{\nu}, \GeV) \; \hbox{events per gr/cm$^2$} \; , \\ N(\antinu +N \gt \lepton^+ + X) &=& 2.0 \times 10^{-15}( E_{\antinu}, \GeV) \; \hbox{events per gr/cm$^2$} \; . \end{eqnarray} Using this simple form for the energy dependence of the cross section, the predicted energy distributions for $\nu_e$ and $\nu_\mu$ interacting in a far detector ($\cos\theta = 1$) at a neutrino factory are shown in Fig.~\ref{polarization}. The interacting $\nu_\mu$ energy distribution is compared in Fig.~\ref{minos_wbb} with the corresponding distribution arising from the high--energy NUMI wide band beam. Note that neutrino beams from a neutrino factory can be considered narrow band beams. % \begin{figure} \epsfxsize3.in \centerline{\epsffile{s2_polarization.eps}} %\renewcommand{\baselinestretch}{2} \caption{Charged current event spectra at a far detector. %Energies up to 250 GeV are shown to illustrate the scaling. The solid lines indicate zero polarization, the dotted lines indicate polarization of $\pm 0.3$ and the dashed lines indicate full polarization. The $P=1$ case for electron neutrinos results in no events and is hidden by the $x$ axis.} \label{polarization} \end{figure} %% \begin{figure} \begin{center} \mbox{ \epsfxsize=6.5truecm \epsfysize=5.6truecm \epsffile{s2_minos_wbb.ps}} \mbox{ \epsfxsize=6.5truecm \epsfysize=5.6truecm \epsffile{s2_minos_wbb_2900km.ps}} \end{center} %\epsfxsize3.in %\centerline{\epsffile{s2_minos_wbb.ps}} \caption{Comparison of interacting $\nu_\mu$ energy distributions for the NUMI high energy wide band beam (Ref.~\ref{numi}) with a 20~GeV neutrino factory beam (Ref.~\ref{geer98}) at $L = 730$~km and a 30~GeV neutrino factory beam at $L = 2900$~km. The neutrino factory distributions have been calculated based on Eq.~\ref{eq:n_numu} (no approximations), and include realistic muon beam divergences and energy spreads. } \label{minos_wbb} \end{figure} %\begin{figure} %\epsfxsize3.in %\centerline{\epsffile{s2_minos_wbb.ps}} %\caption{Comparison of interacting $\nu_\mu$ energy distributions for %a 20~GeV neutrino factory beam (Ref.~\ref{geer98}) and the %NUMI high energy wide band beam (Ref.~\ref{numi}). %%The neutrino factory distribution has been calculated based on %Eq.~\ref{eq:n_numu} (no approximations), and includes a realistic %muon beam divergence and energy spread. %} %\label{minos_wbb} %\end{figure} % \begin{figure} \epsfxsize3.5in \centerline{\epsffile{s2_elept.ps}} \caption{Lepton energy spectra for CC $\overline{\nu}_\mu$ (top left), $\nu_\mu$ (top right), $\nu_e$ (bottom left), and $\overline{\nu}_e$ (bottom right) interactions. Note that $z$ is the energy normalized to the primary muon energy $z = E_{\lepton}/E_\mu$. Calculation from Ref.~\ref{bgw99}.} \label{fig:elept} \end{figure} In practice, CC interactions can only be cleanly identified when the final state lepton exceeds a threshold energy. The calculated final state lepton distributions are shown in Fig.~\ref{fig:elept}. Integrating over the energy distribution, the total $\nu$ and $\overline{\nu}$ interaction rates per muon decay are given by: % \begin{eqnarray} N_\nu &=& 1.2 \times 10^{-14} \; \biggr[{(E_{\mu}, \GeV)^3\over (L, km)^2}\biggl] \times C(\nu) \;\; \hbox {events per kt} \end{eqnarray} and \begin{eqnarray} N_{\overline{\nu}}&=&0.6\times10^{-14} \; \biggr[{(E_{\mu}, \GeV)^3\over (L, km)^2}\biggl] \times C(\nu) \;\; \hbox{events per kt} \, , \end{eqnarray} % where % \begin{eqnarray} C(\nu_{\mu})&=& {7\over 10} + P_{\mu} {3\over 10}, \ \ \ \ C(\nu_{e}) ={6\over 10} - P_{\mu} {6\over 10}\\ \end{eqnarray} The calculated $\nu_e$ and $\nu_\mu$ CC interaction rates resulting from $10^{20}$ muon decays in the beam--forming straight--section of a neutrino factory are compared in Table~\ref{table:rates} with expectations for the corresponding rates at the next generation of accelerator--based neutrino experiments. Note that event rates at a neutrino factory increase as $E_\mu^3$, and are significantly larger than expected for the next generation of approved experiments if $E_\mu > 20$~GeV. The radial dependence of the event rate is shown in Fig.~\ref{fig:radial} for a 20~GeV neutrino factory and three baselines. \begin{table} %\renewcommand{\baselinestretch}{2} \caption{\label{table:rates} Muon neutrino and electron antineutrino CC interaction rates in the absence of oscillations, calculated for baseline length $L = 732$~km (FNAL $\rightarrow$ Soudan), for MINOS using the wide band beam and a muon storage ring delivering $10^{20}$ decays with $E_\mu=10, 20$, and $50$~GeV at 3 baselines. The neutrino factory calculation includes a realistic muon beam divergence and energy spread.} \begin{center} %\vspace{0.6 cm} \begin{tabular}{|ccc|cc|cc|cc|cc|cc} %\hline \hline \hline & &Baseline & $\langle E_{\nu_\mu} \rangle$ & $\langle E_{\bar \nu_e} \rangle$ & N($\nu_\mu$ CC) & N($\bar\nu_e$ CC) \\ Experiment & &(km) & (GeV) & (GeV) & (per kt--yr) & (per kt--yr) \\ \hline MINOS& Low energy &732& 3 & -- & 458 & 1.3 \\ & Medium energy &732& 6 & -- & 1439 & 0.9 \\ & High energy &732& 12 & -- & 3207 & 0.9 \\ \hline Muon ring & $E_\mu$ (GeV) & & & & & \\ \hline & 10 &732& 7.5 & 6.6 & 1400 & 620 \\ & 20 &732& 15 & 13 & 12000 & 5000\\ & 50 &732& 38 & 33 & 1.8$\times$10$^5$ & 7.7$\times$10$^4$ \\ \hline Muon ring& $E_\mu$ (GeV)& & & & & \\ \hline & 10 &2900& 7.6 & 6.5 & 91 &41\\ & 20 &2900& 15 & 13 & 740 & 330\\ & 50 &2900& 38 & 33 & 11000& 4900 \\ \hline Muon ring& $E_\mu$ (GeV)& & & & & \\ \hline & 10 &7300& 7.5 & 6.4 & 14 & 6 \\ & 20 &7300& 15 & 13 & 110 & 51 \\ & 50 &7300& 38 & 33 & 1900 & 770 \\ \hline %\hline\hline \end{tabular} \end{center} \end{table} % \begin{figure} \epsfxsize 3.in \centerline{\epsffile{s2_20gev.eps}} \caption{Events/kT of detector as a function of distance from the beam center for a 20 GeV muon beam.} \label{fig:radial} \end{figure} Finally, for an isoscalar target the neutral current (NC) cross sections are approximately 0.4 of the CC cross sections\cite{neutralcurrent}, and are given by: \begin{eqnarray} \sigma(\nu +N \gt \nu + X) &\approx& 0.3\times 10^{-38} \; \centi^2\times (E_{\nu}, \GeV) \, , \\ %\end{eqnarray} %\begin{eqnarray} \sigma(\antinu +N \gt \overline{\nu} + X) &\approx& 0.15\times 10^{-38} \; \centi^2\times (E_{\antinu},\GeV) \; . \end{eqnarray} \subsection{Tau neutrino interactions} % Tau neutrino CC interaction rates are substantially less than the corresponding $\nu_e$ and $\nu_\mu$ rates, especially near the tau production threshold of $\sim 3.3$~GeV. %Above threshold the $\tau$--lepton mass terms in the leptonic %current cannot be ignored, allowing the axial vector structure %functions $W_4$ and $W_5$ to play a non-negligible role. The NC rates should be the same as those for electron and muon neutrinos. Figure~\ref{tau_fig} shows the calculated~\cite{goodman} ratio of $\nu_\tau / \nu_\mu$ CC interaction rates as a function of the neutrino energy. Near threshold, contributions from quasi--elastic and resonance production dominate. If the $\nu_\tau$ cross sections from Ref.~\cite{casper} are used, the predicted event rates are 20--30\% higher. \begin{table} \renewcommand{\baselinestretch}{1} \begin{center} \label{tab:com} \vspace{0.6 cm} \caption{Dependence of predicted charged current event rates on muon beam properties at a neutrino factory. The last column lists the required precisions with which each beam property must be determined if the uncertainty on the neutrino flux at the far site is to be less than $\sim1$\%. Here $\Delta$ denotes uncertainty while $\sigma$ denotes the spread in a variable. Table from Ref.~\ref{cg00}. } \vspace{0.2cm} \begin{tabular}{c|c|cc} \hline Muon Beam & Beam & Rate & Target\\ property & Type & Dependence & Precision \\ \hline Energy ($E_\mu$) & $\nu$ (no osc) & $\Delta N / N = 3 \; \Delta E_\mu/E_\mu$ & $\Delta(E_\mu)/E_\mu < 0.003$ \\ & $\nu_{e} \rightarrow \nu_{\mu}$ &$\Delta N / N = 2 \; \Delta E_\mu/E_\mu$ & $\Delta(E_\mu)/E_\mu < 0.005$ \\ \hline Direction ($\Delta\theta$) & $\nu$ (no osc) & $\Delta N/N \leq 0.01$ & $\Delta\theta < 0.6 \; \sigma_\theta$ \\ & & (for $\Delta\theta < 0.6\; \sigma_\theta$) & \\ \hline Divergence ($\sigma_\theta$) & $\nu$ (no osc) & $\Delta N / N \sim 0.03 \; \Delta\sigma_\theta / \sigma_\theta$ & $\Delta\sigma_\theta / \sigma_\theta < 0.2$ \\ & & (for $\sigma_\theta \sim 0.1/\gamma$) & (for $\sigma_\theta \sim 0.1/\gamma$)\\ \hline Momentum spread ($\sigma_p$) & $\nu$ (no osc) & $\Delta N / N \sim 0.06 \; \Delta\sigma_p / \sigma_p$ & $\Delta\sigma_p / \sigma_p < 0.17$ \\ \hline Polarization ($P_\mu$) & $\nu_e$ (no osc) & $\Delta N_{\nu_e} / N_{\nu_e} = \Delta P_\mu$ & $\Delta P_\mu < 0.01$ \\ & $\nu_{\mu}$ (no osc) & $\Delta N_{\nu_\mu} / N_{\nu_\mu} = 0.4 \; \Delta P_\mu$ & $\Delta P_\mu < 0.025$ \\ \hline \end{tabular} \label{tab:flux} \end{center} \end{table} \subsection{Systematic uncertainties on the muon beam and neutrino flux} In the neutrino beam--forming straight section the muon beam is expected to have an average divergence given by $\sigma_\theta =$~O($0.1/\gamma$). The neutrino beam divergence will therefore be dominated by muon decay kinematics, and uncertainties on the beam direction and divergence will yield only small uncertainties in the neutrino flux at a far site. However, if very precise knowledge of the flux is required, the uncertainties on $\theta$ and $\sigma_\theta$ must be taken into account, along with uncertainties on the flux arising from uncertainties on the muon energy distribution and polarization. The relationships between the uncertainties on the muon beam properties and the resulting uncertainties on the neutrino flux are summarized in Table~\ref{tab:flux}. If, for example, we wanted to know the $\nu_e$ and $\nu_{\mu}$ fluxes at a far site with a precision of 1\%, we would need to know the beam divergence $\sigma_\theta$ to 20\% (Fig.~\ref{fig:flux_xy}), and ensure that the beam direction was within $0.6\;\sigma_\theta$ of the nominal direction~\cite{cg00} (Fig.~\ref{fig:flux_d}). % \begin{figure} \epsfxsize3.5in \centerline{\epsffile{s2_flux_xy.ps}} \vspace{-2.3cm} \caption{ Dependence of CC interaction rates on the muon beam divergence for a detector located at $L = 2800$~km from a muon storage ring containing 30~GeV unpolarized muons. Rates are shown for $\nu_e$ (boxes) and $\nu_\mu$ (circles) beams in the absence of oscillations, and for $\nu_e \rightarrow \nu_\mu$ oscillations (triangles) with the three--flavor oscillation parameters IA1. The calculation is from Ref.~\ref{cg00}. } \label{fig:flux_xy} \end{figure} % \begin{figure} \epsfxsize3.5in \centerline{\epsffile{s2_flux_d.ps}} \vspace{-2.0cm} \caption{ Dependence of CC interaction rates on the neutrino beam direction. Relative rates are shown for a detector at a far site located downstream of a storage ring containing 30~GeV unpolarized muons, and a muon beam divergence of 0.33~mr. Rates are shown for $\nu_e$ (triangles) and $\nu_\mu$ (circles) beams in the absence of oscillations, and for $\nu_e \rightarrow \nu_\mu$ oscillations (boxes) with the three--flavor oscillation parameters IA1. The calculation is from Ref.~\ref{cg00}. } \label{fig:flux_d} \end{figure} % \subsection{Event distributions at a near site} The event distributions measured in a detector close to the neutrino factory will be quite different from the corresponding distributions at a far site. There are two main reasons for this difference. First, the near detector accepts neutrinos over a large range of muon decay angles $\theta$, not just those neutrinos traveling in the extreme forward direction. This results in a broader neutrino energy distribution that is sensitive to the radial size of the detector (Fig.~\ref{nearspectra}). Second, if the distance of the detector from the end of the beam forming straight section is of the order of the straight section length, then the $\theta$ acceptance of the detector varies with the position of the muon decay along the straight section. This results in a more complicated radial flux distribution than expected for a far detector. However, since the dominant effects are decay length and muon decay kinematics, it should be modeled quite accurately. (Fig.~\ref{xplot}). \begin{figure} \epsfxsize 3.in \centerline{\epsffile{s2_nearspectra.eps}} \caption{Events per gr/cm$^2$ per GeV for a detector 40~m from a muon storage ring with a 600 m straight section. The 3 curves show all events and those falling within 50 and 20~cm of the beam center. } \label{nearspectra} \end{figure} % \begin{figure} \epsfxsize 3.in \centerline{\epsffile{s2_x.eps}} \caption{Events per gr/cm$^2$ as a function of the transverse coordinate x 50~m downstream of a 50~GeV neutrino factory providing $10^{20}$ muon decays. The central peak is mainly due to decays in the last hundred meters of the decay pipe while the large tails are due to upstream decays.} \label{xplot} \end{figure} Note that, even in a limited angular range, the event rates in a near detector are very high. Figure~\ref{eventrates} illustrates the event rates per gram/cm$^2$ as a function of energy. Because most of neutrinos produced forward in the center of mass reach the detector fiducial volume, the factor of $\gamma^2$ present in the flux for $\theta \sim 0$ is lost and the event rate increases linearly with $E_{\mu}$. For a 50~GeV muon storage ring, the interaction rate per 10$^{20}$ muon decays is 7~million events/gram/cm$^2$. Rate calculations are discussed further in the context of specific experiments in the section on non--oscillation experiments. Finally, in the absence of special magnetized shielding, the high neutrino event rates in any material upstream of the detector will cause substantial backgrounds. The event rate in the last 3 interaction lengths (300~gr/cm$^2$) of the shielding between the detector and the storage ring would be 30 interactions per beam spill at a 15 Hz machine delivering $2\times 10^{20}$ muon decays per year. These high background rates will require clever magnetized shielding designs and fast detector readout to avoid overly high accidental rates in low mass experiments. \begin{figure} \epsfxsize 3.in \centerline{\epsffile{s2_eventrates.eps}} \caption{Events per gr/cm$^2$ at a near detector as a function of muon beam energy. The curves indicate (solid) all events, the dashed and dotted curves show the effects of radial position cuts.} \label{eventrates} \end{figure}