\subsection{$\nu_\mu$ disappearance} \begin{figure}[htbp] \begin{center} \includegraphics*[width=\textwidth]{angular.eps} \caption[Neutrino produced muon angle distribution, data and Monte Carlo.] {Angular distribution of muons from the process $\nu_\mu n \rightarrow \mu^- p$ (top curve) and background from $\nu_\mu N \rightarrow \mu^- N' \pi$ (bottom curve). The histogram is data from AGS experiment E734 (year 1986) and the lines are Monte Carlo.} \label{fig:e734mu} \end{center} \end{figure} \begin{figure} \begin{center} \includegraphics*[width=\textwidth]{ness_figs/nodes.eps} \caption[Oscillation nodes {\it vs.} distance.] {Nodes of neutrino oscillations for disappearance (Not affected by matter effects) as a function of oscillation length and energy for $\mdmatm = 0.0025 \meV^2$. The distances from FNAL to Soudan (the distance from BNL to Morton salt works is approximately the same\cite{imb}) and from BNL to Homestake are shown by the vertical lines. } \label{nodes} \end{center} \end{figure} \begin{figure} \begin{center} % \includegraphics*[width=\textwidth]{figs/numu-matter_numu_2540_0_0.8_1.0_0.04_5.0_2.6.eps} % \includegraphics*[width=\textwidth]{numu-2540-003.eps} % \includegraphics*[width=\textwidth]{ness_figs/dm2003.eps} \includegraphics*[width=\textwidth]{ness_figs/dm20025.eps} \caption[Expected $\nu_\mu$ disappearance spectra, $\Delta m^2_{32} = 0.0025$] {Spectrum of detected events in a 0.5 MT detector at 2540 km from BNL including quasielastic signal and CC-single pion background. We have assumed 1.0 MW of beam power and 5 years of running. The top histogram is without oscillations; the middle error bars are with oscillations and the bottom histogram is the contribution of the background to the oscillated signal only. This plot is for $\mdmatm = 0.0025 \meV^2$. The error bars correspond to the statistical error expected in the bin. A 10 \% detector energy resolution is assumed. At low energies the Fermi movement, which is included in simulation, will dominate the resolution.} \label{wcnodesa} \end{center} \end{figure} \begin{figure} \begin{center} % \includegraphics*[width=\textwidth]{figs/numu-matter_numu_2540_0_0.8_1.0_0.04_5.0_1.0.eps} % \includegraphics*[width=\textwidth]{numu-2540-001.eps} \includegraphics*[width=\textwidth]{ness_figs/dm2001.eps} \caption[Expected $\nu_\mu$ disappearance spectra, $\Delta m^2_{32} = 0.001$] {Spectrum of detected events in a 0.5 MT detector at 2540 km from BNL including quasielastic signal and CC-single pion background. We have assumed 1.0 MW of beam power and 5 years of running. The top histogram is without oscillations; the middle error bars are with oscillations and the bottom histogram is the contribution of the background to the oscillated signal only. This plot is for $\mdmatm = 0.001 \meV^2$. The error bars correspond to the statistical error expected in the bin. A 10 \% detector energy resolution is assumed. At low energies the Fermi movement, which is included in simulation, will dominate the resolution.} \label{wcnodesb} \end{center} \end{figure} The angular distribution of the muons from the quasi-elastic process $\nu_{\mu} + n \rightarrow \mu^- + p$ produced by the $0^o$ beam in Figure~\ref{bnlspec} was measured in experiment E734 (1986) at BNL. It is shown again in Figure~\ref{fig:e734mu} along with the principal background, $\nu_{\mu} + N \rightarrow \mu^- + N + \pi$ \cite{e734d}. A variety of strategies is possible to reduce this background further in a water \cerenkov{} detector. Knowing the direction of an incident $\nu_{\mu}$ accurately and measuring the angle of the observed muon allows the energy of the $\nu_{\mu}$ to be calculated, up to Fermi momentum effects. This method is used by the currently running K2K experiment \cite{k2k}. The known capability of large water \cerenkov{} detectors indicates that at energies lower than 1 \GeV{} the $\nu_\mu$ energy resolution will be dominated by Fermi motion and nuclear effects\cite{kasuga}. The contribution to the resolution from water \cerenkov{} track reconstruction depends on the photo-multiplier tube coverage. With coverage greater than $\sim$ 10\%, we expect that the reconstruction resolution should be more than adequate for our purposes \cite{e889}. In the following discussion we assume a 10\% resolution on the $\nu_\mu$ energy. This is consistent with the resolution projected for 10\% coverage from the K2K experience \cite{sharkey}. The range of $\mdmatm \sim 1.24{E_\nu\mbox{[\GeV{}]} \over L\mbox{[km]}}$ covered by the proposed experiment using the beam in Figure \ref{wspec} extends to the low value of about $5 \times 10^{-4} \ \meV^2$. The lower end of this extensive range of values is considerably below the corresponding values for other long baseline terrestrial experiments~\cite{minos,cngs}. If the value of \dmatm{} turns out to be towards the lower end ($\sim 10^{-3}$) of its current range, or if the value of \dmsol{} turns out to be towards its high end ($\sim$ $10^{-4} \meV^2$), then large and very interesting interference effects in the very long baseline experiment will be possible. Extra-long neutrino flight paths open the possibility of observing multiple nodes (minimum intensity points) of the neutrino oscillation probability in the disappearance experiment. Observation of one such pattern will for the first time directly demonstrate the oscillatory nature of the flavor changing phenomenon. The nodes occur at distances $L_n = 1.24 (2n-1) E_{\nu}/\mdmatm$, $n= 1,2,3, $ \ldots. In Figure \ref{nodes}, as an example, we show the flight path $L$ versus $E_{\nu}$ relationship of the nodes for $\Delta m^2 = 0.003 ~eV^2$, a value close to the value measured in atmospheric neutrino experiments \cite{sk}. An advantage of having a very long baseline is that the multiple node pattern is detectable over a broad range of $\Delta m^2$. For \dmatm{} as small as 0.001 $e\mbox{V}^2$, the oscillation effects will be very large. The two single charged pion reactions $\nu_\mu + p \rightarrow \mu^- + p + \pi^+$ and $\nu_{\mu} + n \rightarrow \mu^- + n + \pi^+$ produce a signal which is somewhat larger than the quasi-elastic total in Table \ref{evcount}. For these events, if both the muon and the pion produce more than 50 photoelectrons each, the event can be easily identified as a two ring event in a water \cerenkov{} detector and rejected. 50 photoelectrons corresponds to about 170 MeV/c (250 MeV/c) for muons (pions) for a detector with 10\% photo-multiplier coverage. An additional cut to require the muon to be within 60$^o$ of the neutrino direction reduces the background further. With such a cut, we find that 18\% of the events will show one ring (principally the $\mu^-$). % We see %that 0.65 of these single ring events are above 300 photoelectrons while %93\% of the quasi-elastic muons will be above 300 photoelectrons (Figure \ref{muspec}). %Therefore %about 0.12 ($0.48\times 0.40 \times 0.65$) of the events identified as %quasi-elastic muons could be %from charged current single pion channels. The detection of two muon decays, one from the $\mu^-$ the other from the decay chain $\pi \to \mu \to e$, could be used to further suppress this background by approximately a factor of 2. More importantly, background events can be tagged by the two muon decays to determine the shape of the background from the data itself. This will greatly increase the confidence in the systematic error due to this background. The reaction $\nu_{\mu} + n \rightarrow \mu^- + p + \pi^0$ (the only allowed CC-$\pi^0$ reaction) is $\sim$15\% of the total quasi-elastic rate. The momentum distribution of $\mu^-$ and $\pi^0$ are essentially the same as those for CC-charged pion production. Only 0.5\% of the CC-$\pi^0$ events will look like quasi-elastic muon events because at least one of the gamma rays from the $\pi^0$ decay is usually visible. Thus this background is negligible in the quasi-elastic sample. The expected plot of signal and background is shown in Figures \ref{wcnodesa} and \ref{wcnodesb}. They show the disappearance of muon type neutrinos as a function of neutrino energy measured in quasi-elastic events. The background, which will be mainly charged current, will also oscillate, but the reconstructed neutrino energy will be systematically lower for the background. Nevertheless, the main effect will be to slightly broaden the large dips due to disappearing muon neutrinos. In Figure \ref{cntr1} we show the statistical precision expected on the measurement of $\Delta m^2_{32}$ and $\sin^2 2 \theta_{23}$ for several different points in the parameter space. It is clear that since the signal and the statistics are large, the systematic error in fitting the spectrum will dominate the final error. We have listed various effect that must be considered for the measurement with brief comments about each. \begin{itemize} \item The determination of $\Delta m^2$ has a statistical uncertainty of approximately $\pm 0.7\%$ at $\Delta m^2 = 0.0025 ~eV^2$ with maximum mixing. It is about $\pm 1.0\% $ when $\sin^2 2 \theta_{23} = 0.75$. Clearly the knowledge of the energy scale will be very important in measuring this number. If the energy scale uncertainty is $\delta E/E$ then the final error will be given by $$({\sigma(\Delta m^2)\over \Delta m^2})^2 = ({\sigma_{stat} \over \Delta m^2})^2 + ({\delta E \over E})^2$$ Therefore it will be very important to understand the energy calibration of the detector to about 1 \% for muon energy of $\sim 1$ GeV. One solution could be a magnetic spectrometer to measure the momentum of cosmic ray muons entering the detector. This consideration could affect the depth at which this detector should be mounted. Another option could be a linear accelerator that could provide protons or electrons at a rate of few Hz at $\sim 100$ MeV. \item Even if the overall energy scale is known well, the energy calibration could vary non-linearly over the entire spectrum. The worst effects of these fluctuations will be where the spectrum has the maximum slope. This effect will cause additional smearing of the spectrum and reduce the resolution on $\Delta m^2$. We assume a 5\% uncertainty of the energy calibration over the entire range. It should be pointed out that the oscillation minima should be at energies that are in precisely known ratios of integers: 3, 5, 3/5, etc. This could be used to determine the relative energy scale precisely. On the other hand these ratios could be important to determine the presence of new physics (non-sinusoidal depletion of muon neutrinos) in the oscillations. \item The model of Fermi motion and reconstruction resolution will affect both the shape of the signal and the background used in the fit. The consequences of this effect are probably the same as the previous one in terms of the resolution of fitted parameters. It was pointed out earlier that some of the the CC-$\pi^+$ background could be tagged by two muon decays. This sample of events can be used in separate fits to put more constraints on the detector simulations. The large number of charged current events ($\sim 52000$) that are not quasielastic could also be used in the same manner. \item The statistical uncertainty in the determination of $\sin^2 2 \theta_{23}$ is $\pm 0.016$ at $\sin^2 2 \theta_{23}=0.75$ and $\Delta m^2_{32} = 0.0025 ~eV^2$. This determination is somewhat better at smaller $\Delta m^2$. At maximum mixing, Figure \ref{cntr1} shows that we can determine $\sin^2 2 \theta_{23} > 0.99$ at 90\% confidence level. We expect this error to be even smaller if proper background subtraction is performed on the data. Normally the determination of this quantity depends on the systematic error for the normalization of the flux. However, in the case of very long baseline, the largest part of the sensitivity comes from the shape of the spectrum or how deep the valleys are compared to the peaks (see Figure \ref{wcnodesa}). Therefore this determination is not affected greatly by the systematic error for the overall normalization. This is demonstrated as follows: for $\Delta m^2_{32} = 0.003 ~eV^2$, even without background subtraction, the valleys at $\pi /2$ and $3\pi/2$ have only 2\% and 30\% of the un-oscillated event rate (see Figure \ref{wcnodesa}). If we assume the flux normalization error to be 5\%, which is consistent with what has been achieved by the K2K experiment\cite{sharkey}, then the expected error due to flux normalization on $\sin^2 2 \theta_{23}$ is $0.02\times 0.05 = 0.001$. \item We note that within the parameter region of interest there should be very little correlation in the determination of $\Delta m^2_{32}$ and $\sin^2 2 \theta_{32}$. \end{itemize} \begin{figure} \begin{center} \includegraphics*[width=\textwidth]{dm2res_stat.eps} \caption[Statistical uncertainty for $\Delta m^2_{32}$ and $\sin^22\theta_{23}$] { Statistical resolution at 68\%, 90\% and 99\% confidence level on $\Delta m_{32}^2$ and $\sin^2 2\theta_{23}$ for the 2540 baseline experiment; assuming 1 MW, 0.5 MT, and 5 years of exposure. } \label{cntr1} \end{center} \end{figure} \begin{figure} \begin{center} \includegraphics*[width=\textwidth]{ness_figs/dm2res_sys.eps} \caption[Statistical and systematic uncertainty for $\Delta m^2_{32}$ and $\sin^22\theta_{23}$, includes other's allowed regions.] { Resolution including statistical and systematic effects at 68\%, 90\% and 99\% confidence level on $\Delta m_{32}^2$ and $\sin^2 2\theta_{23}$ for the 2540 baseline experiment; assuming 1 MW, 0.5 MT, and 5 years of exposure. We have included a 5\% bin-to-bin systematic uncertainty in the energy calibration as well as a 5\% systematic uncertainty in the normalization. The expected resolution from the MINOS experiment at Fermilab and the allowed region from SuperK is also indicated. } \label{cntr2} \end{center} \end{figure} %\begin{figure} % \begin{center} % \includegraphics*[width=\textwidth]{ccsig_le_syst.eps} % \caption{ The expected resolution from the MINOS experiment %at Fermilab using a low energy beam from the main injector % superimposed on the allowed region from Super Kamiokande %data. } % \label{ccminos} % \end{center} %\end{figure} \begin{figure} \begin{center} \includegraphics*[width=\textwidth]{k2k-allowed.ps} \caption[The allowed region from the K2K experiment.]{ The allowed region for $\Delta m^2_{32}$ and $\sin^22\theta_{23}$ from the K2K experiment. From thesis by Eric Sharkey, SUNY at Stony Brook.} \label{k2kallowed} \end{center} \end{figure} With the assumption on the systematic errors as above we obtain Figure \ref{cntr2}. The systematic errors introduce a small correlation in the $\Delta m^2_{32}$ vs. $\sin^2 2 \theta_{32}$ measurement. The error on the determination of $\Delta m^2_{32}$ at 0.0025 $eV^2$ increases to about $\pm 1.2\%$ at maximum mixing, but there is only a small effect on the determination of $\sin^2 2 \theta_{23}$. As mentioned before, the energy scale uncertainty must be added in quadrature to the calculated uncertainty on $\Delta m^2_{32}$. The precision of this experiment can be compared with the precision expected from MINOS (Figure \ref{cntr2}) and the precision obtained so far from the K2K experiment (Figure \ref{k2kallowed}). It is expected that K2K will obtain twice as much data; therefore we could naively estimate that the precision on the parameter determination will improve as $1/\sqrt{2}$. Finally, we note that the flux normalization is usually obtained by placing a detector close to the neutrino source. For example, both K2K and MINOS have large near detectors to determine the flux. Since absolute flux determination is not very important for parameter determination in our case, we argue that the requirements on a near detector need not be very severe for this measurement. It may not be necessary to build a near detector until sufficient statistics are obtained in the far detector to demand the required systematic error reduction of a near detector.