\subsection{Far Detectors}

\subsubsection{Magnetised Iron Neutrino Detector Performance}
\label{sec:MIND}

\paragraph{Introduction}
\label{par:MindIntro}

Early papers on the physics outcomes of a Neutrino Factory concentrated on the sub-dominant $\nu_e \rightarrow \nu_\mu$ oscillation \cite{DeRujula:1998hd} in which a muon of opposite charge to that stored in the storage ring (wrong-sign muon) would be produced in a far detector by the charge current (CC) interactions of the oscillated $\nu_\mu$. The first analysis of the capabilities of a large Magnetised Iron Neutrino Detector to detect the wrong-sign muon signature (the golden channel) was discussed in~\cite{Cervera:2000kp}, where it was demonstrated that this combination was capable of extracting the remaining unknown parameters in the three-by-three neutrino mixing matrix. This analysis was carried out assuming a Neutrino Factory with 50~GeV muons, and was optimised for high energy using a detector with 4~cm thick iron plates and 1~cm scintillator planes. Hence, the ability to reconstruct low energy muons was not part of the optimisation.

The International Scoping Study for a future Neutrino Factory and super-beam facility (the ISS) \cite{Bandyopadhyay:2007kx,Apollonio:2009,CerveraVillanueva:2008zz,Abe:2007bi} adopted a muon energy of 25~GeV, so the analysis was re-optimised for low-energy neutrino interactions. This study focused on the topology and kinematics of neutrino events in the detector, assuming perfect pattern recognition and smearing of the kinematic variables of the scattered muon and hadronic shower. Using a combination of cuts on the relative length of the two longest particles in the event and the momentum and isolation of this candidate, it was demonstrated that high signal-identification efficiency and background suppression could be achieved. 

The work that has been carried out since the ISS, and is included in this Interim Design Report, has focused on testing these assumptions, by performing a full simulation and reconstruction of the candidate neutrino events, allowing a full characterisation of the detector response. Incorrect charge assignment (charge mis-identification) of non-oscillated $\overline{\nu}_\mu$ CC interactions, background from meson decays in the hadronic shower, and misidentification of neutral current (NC) and $\nu_e$ CC events present the most significant background for the Neutrino Factory beam.

The present study first focused on demonstrating the performance of reconstruction algorithms using deep inelastic scattering (DIS) neutrino events generated using LEPTO61~\cite{Ingelman:1997Cp} and simulated using GEANT3~\cite{geant3wu}, as in the study performed for the ISS. Pattern recognition and reconstruction of candidate muons were carried out using a Kalman filter (RecPack \cite{CerveraVillanueva:2004kt}) and a Cellular Automaton algorithm, particularly useful in extracting low energy neutrino events \cite{Emeliyanov_otr/itr-cats:tracking}. Preliminary results were shown in~\cite{Cervera:2008nf} and were further developed in \cite{Cervera:2010rz}. 

In conjunction with the development of reconstruction algorithms for MIND, an entirely new simulation was developed based on the NUANCE event generator \cite{Casper:2002sd} and GEANT4~\cite{Apostolakis:2007zz}. This simulation, named G4MIND, will ultimately allow for a full optimisation of MIND in terms of segmentation, technology options and analysis algorithms. The work undertaken to develop the simulation and digitisation as well as the application and re-optimisation of the reconstruction algorithm introduced in \cite{Cervera:2010rz} is described in this report. These results were used to determine the response matrices for this detector system (presented in Appendix~\ref{app:response}), which has been used to deduce the expected sensitivity of the combination of the MIND and a 25~GeV Neutrino Factory to key oscillation parameters.

\paragraph{Simulation using NUANCE and GEANT4}
\label{sec:evGen}
In previous studies, only deep inelastic scattering (DIS) events generated with LEPTO61 were considered. However, at energies below 5~GeV there are large contributions from quasi-elastic (QE), Single Pion production (1$\pi$) and other resonant production (DIS) events (see figure \ref{fig:intmuCC}). A QE event may be accurately described by considering the interaction of the neutrino to take place with a nucleon as a whole ($\nu_\mu+n\rightarrow\mu^-+p$). The energy of such an event may readily be reconstructed from the momentum of the muon and the angle it makes with the beam direction. In addition, the low multiplicity of the event makes muon reconstruction simpler. 1$\pi$ events should also improve the purity of the low-energy event sample due to their low multiplicity, although there may be increased background from pion decay. Other nuclear resonant events, producing 2 or 3 pions, as well as diffractive and coherent production, have much smaller contributions.

\begin{figure}
  \begin{center}
    \includegraphics[width=0.6\textwidth]{03-DetWG/figures/xsecZeller.eps}
  \end{center}
  \caption{Experimentally measured cross-sections for quasi-elastic, single pion and deep inelastic $\nu_\mu$ CC interactions with the predictions made by NUANCE~\cite{Zeller:2008zz}.}
  \label{fig:intmuCC}
\end{figure}

Generation of all types of interaction was performed using the NUANCE framework~\cite{Casper:2002sd}. The relative proportions of the interaction types generated by NUANCE are shown in figure \ref{fig:intprops} where `other' interactions include the resonant, coherent and diffractive processes other than single pion production. NUANCE also includes a treatment to simulate the effect of re-interaction within the participant nucleon, which is particularly important for low energy interactions in high-$Z$ targets such as iron. 
\begin{figure}
  \begin{center}$
    \begin{array}{cc}
      \includegraphics[width=0.45\textwidth]{03-DetWG/figures/numubarCCInttype.eps} &
      \includegraphics[width=0.45\textwidth]{03-DetWG/figures/nueCCInttype.eps}\\
      \includegraphics[width=0.45\textwidth]{03-DetWG/figures/nuMuCCInt.eps} &
      \includegraphics[width=0.45\textwidth]{03-DetWG/figures/nuebarCCInt.eps}\\
      \includegraphics[width=0.45\textwidth]{03-DetWG/figures/numubarNCInttype.eps} &
      \includegraphics[width=0.45\textwidth]{03-DetWG/figures/muNCInter.eps}
    \end{array}$
  \end{center}
  \caption{Proportion of total number of interactions of different $\nu$ interaction processes for events generated using NUANCE and passed to the G4MIND simulation. (top) $\overline{\nu}_\mu$ (left) and $\nu_e$ CC (right), (middle) $\nu_\mu$ (left) and $\overline{\nu}_e$ CC (right) and (bottom) NC from anti-neutrinos (left) and neutrinos (right).}
\label{fig:intprops}
\end{figure}

From the flux spectrum used to generate the events and the resultant interaction spectrum, the cross section as a function of energy can be estimated by normalising the highest energy bin to the high-energy limit: $0.67\times 10^{-38}$~cm$^2$~GeV$^{-1}$ for $\nu_\mu$ and $0.34\times 10^{-38}$~cm$^2$~GeV$^{-1}$ for $\overline{\nu}_\mu$~\cite{Amsler20081}. 

A new simulation of MIND using the GEANT4 toolkit G4MIND was developed to give as much flexibility to the geometry as possible so that an optimisation of all aspects of the detector could be carried out. The dimensions and spacing of all scintillator and iron pieces, as well as all external dimensions of the detector, can be controlled. This will allow for easy comparison of the simulation itself and the subsequent analyses to other potential Neutrino Factory detectors.

The detector transverse dimensions ($x$ and $y$ axes) and length in the beam direction ($z$ axis), transverse to the detector face, are controlled from a parameter file. A fiducial cross section of 14~m$\times$14~m, including 3~cm of iron for every 2~cm of polystyrene extruded plastic scintillator (1~cm of scintillator per view), was assumed. A constant magnetic field of 1~T is oriented in the positive $y$ direction throughout the detector volume. Events generated using NUANCE in iron nuclei and scintillator (polystyrene) are selected in the relative proportion of the two materials. An event vertex is then generated within one of the slabs of the appropriate material randomly positioned in three dimensions. Physics processes are modelled using the QGSP\_BERT physics lists provided by GEANT4 \cite{geant4phys}. 

Secondary particles are required to travel at least 30~mm from their production point or to cross a material boundary between the detector sub-volumes to have their trajectory fully tracked. Generally, particles are only tracked down to a kinetic energy of 100~MeV. However, gammas and muons are excluded from this cut. The end-point of a muon track is important for muon pattern recognition. 

A simplified digitisation model was considered for this simulation. Two-dimensional boxes -- termed voxels -- represent view-matched $x$ and $y$ readout positions. Any deposit which falls within a voxel has its energy deposit added to the voxel total raw energy deposit. The thickness of two centimetres of scintillator per plane assumes 1~cm per view. 

Voxels with edge lengths of 3.5~cm were chosen, which results in a position resolution of $3.5/\sqrt{12} \cong 1$~cm. The response of the scintillator bars is derived from the raw energy deposit in each voxel, read out using wavelength shifting (WLS) fibres with attenuation length $\lambda = 5$~m, as reported by the MINERvA collaboration~\cite{PlaDalmau:2005dp}. Assuming that approximately half of the energy will come from each view, the deposit is halved and the remaining energy at each edge in $x$ and $y$ is calculated. This energy is then smeared according to a Gaussian with $\sigma/E = 6\%$ to represent the response of the electronics and then recombined into $x$, $y$ and total = $x+y$ energy deposit per voxel. Assuming an output wavelength of 525~nm, a photo-detector quantum efficiency of  $\sim$30\% can be achieved (see figure \ref{fig:pde_mppc} in section \ref{sec:MIND_conceptual_design}). A threshold of 4.7 photo electrons (pe) per view, as in MINOS~\cite{Michael:2008bc}, was assumed. Any voxel in which the two views do not make this threshold is cut. If only one view is above threshold, then only the view below the cut is excluded (see section~\ref{Sec:RecG4}). The digitisation of an example event is shown in figure \ref{fig:voxclust}.
\begin{figure}
  \begin{center}$
    \begin{array}{cc}
      \includegraphics[width=0.45\textwidth]{03-DetWG/figures/xz.eps} &
      \includegraphics[width=0.45\textwidth]{03-DetWG/figures/yz.eps} \\
    \end{array}$
    \includegraphics[width=0.45\textwidth]{03-DetWG/figures/yx.eps}
  \end{center}
  \caption{The digitisation and voxel clustering of an example event: (top left) bending plane view, (top right) non-bending plane, (bottom) an individual scintillator plane. The individual hits are small dots (in red), the blue squares are the voxels and the black asterisks represent the centroid positions of the clusters.}
  \label{fig:voxclust}
\end{figure}

\paragraph{Reconstruction of events}
\label{Sec:RecG4}

The reconstruction and analysis packages are described in detail in a previous publication \cite{Cervera:2010rz}. We present here an update of the analysis based on the MIND simulation generated using NUANCE and GEANT4. The optimisation of cuts and the extraction of efficiencies were performed using a flux profile designed to maximise statistics in all energy bins of interest. 

Many traversing particles, particularly hadrons, deposit energy in more than one voxel. Forming clusters of adjacent voxels reduces event complexity and can improve pattern recognition in the region of the hadron shower. The clustering algorithm is invoked at the start of each event. The voxels of every plane in which energy has been deposited are considered in sequence. Where an active voxel is in contact with no other active voxel, this voxel becomes a measurement point. If there are adjacent voxels, the voxel with the largest total deposit (at scintillator edge) is sought and all active voxels in the surrounding 3$\times$3 area are considered to be part of the cluster. Adjacent deposits which do not fall into this area are considered separate.
The cluster position is calculated independently in the $x$ and $y$ views as the energy-weighted sum of the individual voxels.

The clusters formed from the hit voxels of an event are then passed to the reconstruction algorithm. 
The separation of candidate muons from hadronic activity is achieved using two methods: a Kalman filter algorithm provided by RecPack~\cite{CerveraVillanueva:2004kt} and a cellular automaton method 
(based on \cite{Emeliyanov_otr/itr-cats:tracking}).

The Kalman filter propagates the track parameters back through the planes using a helix model which takes into account multiple scattering and energy loss.  Since, in general, a muon is a minimum ionising particle (MIP) and will travel further in the detector than hadronic particles, those hits furthest downstream are assumed to be muon hits and used as a seed for the Kalman filter. The seed ``state vector'' is then propagated back to each plane with multiple hits and the matching $\chi^2$ to each of the hits is computed. 
Hits with matching $\chi^2$ below 20 are considered and in each plane the one with the best matching among these is added to the trajectory and filtered (i.e. the track parameters are updated with the new information). All accepted hits constitute the candidate muon and are presented for fitting 
with the remaining hits being considered as hadronic activity. 

Events with high $Q^2$ or low neutrino energy can be rejected by the first method, since in general the muon will not escape the region of hadronic activity.   
The cellular automaton method uses a neighbourhood function first to rank all the hits and then to form all viable combinations into possible trajectories. A ``neighbour'' is defined as a hit in an adjacent plane within a pre-defined transverse distance of the projection into that plane of the straight line connecting hits in the previous two planes. Starting from the plane with lowest $z$ position,
 hits are given incremental ranks higher than neighbours in the previous plane. Trajectories are then formed 
from every possible combination of one hit per plane starting with those of highest rank using the neighbourhood function. 

Those trajectories formed using this method are then categorised according to length and $\chi^2$. The candidate muon is then selected as the longest remaining trajectory with the lowest $\chi^2$. All other hits in the event are considered to be from hadronic activity. Even if three planes in a row have no associated candidate hits, the event can still be accepted if 70\% of the planes are considered to belong to the candidate-muon trajectory. This algorithm improves the efficiency and purity of muon candidates close the hadronic jet, since it ranks the likelihood that a given hit belongs to the muon candidate or the hadronic jet.

An additional step was added to take into account that fully-contained muons (particularly $\mu^-$) can have additional deposits at their endpoint due to captures on nuclei or due to decays. Long, well defined tracks can be rejected if there is additional energy deposited at the muon end point, since the cellular automaton interprets this as hadronic activity or a decay.  Once the celular automaton has completed the pattern recognition, a Kalman fit is used to estimate the track parameters.  Additional energy deposits at the end of the muon track may cause confusion in determining the seed of the Kalman filter. Therefore, after sorting clusters into increasing \emph{z} position, a pattern-recognition algorithm is used to identify the section of the track associated with the long muon section, used for momentum measurement via the Kalman filter, and the endpoint.

The complete pattern-recognition chain using these algorithms leads to candidate purity (fraction of candidate hits of muon origin) for $\nu_\mu~(\overline{\nu}_\mu)$ CC events as shown in figure \ref{fig:G4purity}. A cluster is considered to be of muon origin if greater than 80\% of the raw deposits contained within the cluster were recorded as muon deposits.  Overall, the candidate purity is in excess of 90\% for true muon energies larger than $\sim 2$\,GeV.
\begin{figure}
  \begin{center}$
    \begin{array}{cc}
      \includegraphics[width=8cm, height=6cm]{03-DetWG/figures/kalpurnu.eps} &
      \includegraphics[width=8cm, height=6cm]{03-DetWG/figures/cellpurnu.eps}\\
      \includegraphics[width=8cm, height=6cm]{03-DetWG/figures/kalpurbar.eps} &
      \includegraphics[width=8cm, height=6cm]{03-DetWG/figures/cellpurbar.eps}
    \end{array}$
  \end{center}
  \caption{Muon candidate hit purity for $\nu_\mu$ CC (top) and $\overline{\nu}_\mu$ CC (bottom) interactions extracted using (left) Kalman filter method and (right) cellular automaton method.}
  \label{fig:G4purity}
\end{figure}

Fitting of the candidates proceeds using a Kalman filter to fit a helix to the candidate, using an initial seed estimated by a quartic fit, and then refitting any successes. Projecting successful trajectories back to the true vertex \emph{z} position, the quality of the fitter can be estimated using the pulls of the different parameters in the fit vector (see figure \ref{fig:G4pulls}).
\begin{figure}
  \begin{center}$
    \begin{array}{cc}
      \includegraphics[width=8cm, height=6cm]{03-DetWG/figures/Xpull.eps} &
      \includegraphics[width=8cm, height=6cm]{03-DetWG/figures/Ypull.eps} \\
      \includegraphics[width=8cm, height=6cm]{03-DetWG/figures/dXpull.eps} &
      \includegraphics[width=8cm, height=6cm]{03-DetWG/figures/dYpull.eps} \\
      \includegraphics[width=8cm, height=6cm]{03-DetWG/figures/mompull.eps} &
      \includegraphics[width=8cm, height=6cm]{03-DetWG/figures/G4resolution.eps}
    \end{array}$
  \end{center}
  \caption{Pulls on the five Kalman filter vector parameters and the reconstructed momentum resolution: (top) X (left) and Y (right) position, (middle) X (left) and Y (right) slopes and (bottom) momentum pull (left) and momentum resolution (right).}
  \label{fig:G4pulls}
\end{figure}

The pulls on the position and direction exhibit a larger sigma than expected indicating that the position resolution is underestimated (i.e. that the resolution assumed in the fit is a little too small). This is attributed to clusters being made up of many particles in the high-occupancy vertex region. This effect will be better understood as the digitisation algorithms are improved. The momentum pull, however, behaves as expected but for a slight bias due to the fact that the energy loss is not updated along the track in the version of RecPack used for the study. In this version the energy loss is not changed as the filter progresses along the track and can result in a bias. The muon-momentum resolution is slightly improved from previous studies, particularly at low energies, and its resolution is parametrised as follows:
\begin{equation}
  \label{eq:G4resol}
  \displaystyle\frac{\sigma_{1/p}}{1/p} = 0.18 + \frac{0.28}{p} - 1.17\times 10^{-3}p.
\end{equation}

Hadronic reconstruction is predominantly performed using a smear on the true quantities as described in \cite{Cervera:2010rz}. However, the presence of QE interactions in the sample allows for the reconstruction of certain events using the formula:
\begin{equation}
  \label{eq:quasiEng}
  E_\nu = \displaystyle\frac{m_NE_\mu + \frac{m_{N_X}^2 - m_\mu^2 - m_N^2}{2}}{m_N - E_\mu + |p_\mu|\cos\vartheta} \, ;
\end{equation}
where $\vartheta$ is the angle between the muon momentum vector and the beam direction, $m_N$ is the mass of the initial state nucleon, and $m_{N_X}$ is the mass of the outgoing nucleon for the interactions $\nu_\mu + n \rightarrow \mu^- + p$ and $\overline{\nu}_\mu + p \rightarrow \mu^+ + n$~\cite{Blondel:2004cx}. QE interactions can be selected using their distribution in $\vartheta$ and their event-plane occupancy among other parameters. Should the use of equation~\ref{eq:quasiEng} result in a negative value for the energy, it is recalculated as the total energy of a muon with its reconstructed momentum.

\paragraph{Analysis of potential signal and background}
\label{par:G4ana}
There are four principal sources of background to the wrong sign muon search: incorrect charge assignment (charge misidentification), wrong sign muons from hadron decay in $\overline{\nu}_\mu$ charged current (CC), neutral current (NC) and $\nu_e$ CC events wrongly identified as $\nu_\mu$ CC. In order to reduce these backgrounds while maintaining good efficiency a number of 
offline cuts were employed. They can be organised in four categories: 1) $\nu_\mu$ CC selection cuts; 2) NC rejection cuts; 3) kinematic cuts; and 4) muon quality cuts.
Cut levels and likelihood distributions have been defined using a test statistic of $\nu_\mu$ and $\overline{\nu}_\mu$ CC and NC events.

\subparagraph*{q/p relative error \\}
\label{subpar:qP}
The error on the momentum parameter (charge divided by momentum, q/p) of the Kalman filter is a powerful handle in the rejection of background. The likelihood distributions for the combination of the two signals and of the neutral current and charged-current backgrounds (re-normalising in the latter case to take account of the different data-set sizes) are used to assess events as signal or background (shown in figure \ref{fig:qPlike}-left).
Signal events are selected as those with a log-likelihood parameter $\mathcal{L}_{q/p} > -0.5$. As can be seen from figure \ref{fig:qPlike}-right this cut-level effectively rejects much of the background.
\begin{figure}
  \begin{center}$
    \begin{array}{cc}
      \includegraphics[width=7.5cm, height=5.5cm]{03-DetWG/figures/sigma_p.eps} &
      \includegraphics[width=7.5cm, height=5.5cm]{03-DetWG/figures/errlog.eps}
    \end{array}$
  \end{center}
  \caption{PDF of $\displaystyle\frac{\sigma_{q/p}}{q/p}$ (left) and the resulting log likelihood distribution ($\mathcal{L}_{q/p}$) for a test statistic (right).}
  \label{fig:qPlike}
\end{figure}

\subparagraph*{Neutral current rejection \\}
\label{subpar:NCrej}
Rejection of neutral current events is most efficiently performed using a likelihood analysis of trajectory quantities, as was carried out in MINOS. The three parameters used by MINOS were: number of hits in the candidate, fraction of the total visible energy in the candidate and the mean deposit per plane of the candidate. No improved background rejection was observed by using the latter two parameters, probably due to correlations between them. Only the number of hits, $l_{hit}$, was used to generate particle distribution functions (PDFs) for charged and neutral current events (see figure \ref{fig:NCpdfs}). Candidates with greater than 150 clusters are considered signal and for those with less than or equal 150 clusters, a log likelihood rejection parameter:
\begin{equation}
  \mathcal{L}_1 = \log \left( \frac{l_{hit}^{CC}}{l_{hit}^{NC}} \right) \, ;
\end{equation}
is used, which is also shown in figure \ref{fig:NCpdfs}. Allowing only those candidates where the log parameter is $\mathcal{L}_1 > 1.0$ to remain in the sample ensures that the sample is pure.
\begin{figure}
  \begin{center}$
    \begin{array}{cc}
      \includegraphics[width=8cm, height=6cm]{03-DetWG/figures/hitpdf.eps} &
      \includegraphics[width=8cm, height=6cm]{03-DetWG/figures/an1hitlog.eps}
    \end{array}$
  \end{center} 
  \caption{Number of clusters likelihood distribution function (left) and log-likelihood ratio (right) used to reject NC background. CC in black and NC in red.}
  \label{fig:NCpdfs}
\end{figure}

\subparagraph*{Kinematic cuts \\}
\label{subpar:G4kinCut}
Kinematic cuts based on the momentum and isolation of the candidate, as related to the reconstructed energy of the event $E_{rec}$, can be used to reduce backgrounds from decays. The main isolation variable is defined as $Q_t=p_\mu \sin^2\theta$, where $\theta$ is the angle between the muon candidate and the hadronic-jet vector. Cuts based on these variables are an effective way to reduce all of the relevant beam related backgrounds:
\begin{eqnarray}
  \label{eq:G4kin1}
  E_{rec}~\leq~5\mbox{~GeV or } &Q_t~>~0.25\mbox{~GeV/c} \, {\rm ; and}\\
  \label{eq:G4kin2}
  E_{rec}~\leq~7\mbox{~GeV or } &p_\mu~\geq~0.3\cdot E_{rec} \, .
\end{eqnarray}
The distributions after the application of the preceding cuts are those shown in figure \ref{fig:G4kin}. 

The acceptance level is described in equations~\ref{eq:G4kin1} and~\ref{eq:G4kin2}. With the exception of those events for which the energy is reconstructed using the quasi-elastic formula and hence do not have a hadronic shower the separation of which can be calculated, events are subject to the $Q_t$ cut if their reconstructed energy is greater than 5~GeV. Those events fulfilling the condition of equation~\ref{eq:G4kin1} and those reconstructed using the quasi-elastic formula are then subject to the second cut if they have $E_{rec} > 7$~GeV. All remaining events which fulfil the conditions of equation~\ref{eq:G4kin2} (those above and to the left of the red lines in figure \ref{fig:G4kin}) are kept in the data set for the next series of cuts.
\begin{figure}
  \begin{center}$
    \begin{array}{cc}
      \includegraphics[width=7.5cm, height=4.75cm]{03-DetWG/figures/momEngMuCC.eps} &
      \includegraphics[width=7.5cm, height=4.75cm]{03-DetWG/figures/qtEngMuCC.eps} \\
      \includegraphics[width=7.5cm, height=4.75cm]{03-DetWG/figures/momEngMisIDCC.eps} &
      \includegraphics[width=7.5cm, height=4.75cm]{03-DetWG/figures/qtEngMisIDCC.eps} \\
      \includegraphics[width=7.5cm, height=4.75cm]{03-DetWG/figures/momEngNC.eps} &
      \includegraphics[width=7.5cm, height=4.75cm]{03-DetWG/figures/qtEngNC.eps} \\
      \includegraphics[width=7.5cm, height=4.75cm]{03-DetWG/figures/momEngE.eps} &
      \includegraphics[width=7.5cm, height=4.75cm]{03-DetWG/figures/qtEngE.eps} \\
    \end{array}$
  \end{center}
  \caption{Distributions of kinematic variables: (left) Reconstructed muon momentum with reconstructed neutrino energy for (top$\rightarrow$bottom) $\nu_\mu~(\overline{\nu}_\mu)$ signal, $\nu_\mu~(\overline{\nu}_\mu)$ CC background, NC background, $\nu_e~(\overline{\nu}_e)$ CC background and (right) $Q_t$ variable (in the same order).}
  \label{fig:G4kin}
\end{figure}

\subparagraph*{Additional quality cuts \\}
\label{subpar:addition}
A number of additional cuts have been developed to reduce the background from opposite sign $\nu_\mu~(\overline{\nu}_\mu)$ CC events at high neutrino energy. A set of new cuts in two categories are used: 1) Those related to the range in \emph{z}, displacement in the bending plane, and reconstructed momentum of the candidate; and 2) a cut using the results of a re-fit of the remaining events to a quadratic.

Additionally, a fiducial cut requiring that the first cluster in a candidate be at least 2~m from the end of the detector is employed to reduce the misidentification of candidates originating at high \emph{z}. Moreover, a high proportion of the misidentified candidates are not fully fitted. The distribution of the ratio of the candidate clusters which are fitted with respect to the total number of candidate clusters for signal and background is shown in figure \ref{fig:fitnode}. Accepting only those events with a candidate with 60\% of its clusters fitted reduces further the background levels.
\begin{figure}
  \begin{center}
    \includegraphics[width=8cm, height=6cm]{03-DetWG/figures/fitnodes.eps}
  \end{center}
  \caption{Distribution of the proportion of clusters fitted in the trajectory.}
  \label{fig:fitnode}
\end{figure}

The momentum of the muon candidate can be badly reconstructed due to multiple scattering, badly assigned hits, and other effects. These badly reconstructed events contribute significantly to the overall background. In order to reduce this effect, three separate, but related cuts are employed. Firstly, a maximum reconstructed momentum, due to the finite resolution of the detector, is enforced at 40~GeV. Plotting the reconstructed momentum against the displacement in the longitudinal direction $dispZ$ of the candidate allows the candidate to be separated from the background (see figure \ref{fig:dispMom}). Additionally, the ratio between the lateral displacement in the bending plane $dispX$ and $dispZ$ against the number of hits in the candidate (see figure \ref{fig:dispMom}) provides further separation. The background events tend to be concentrated at low relative displacement and low number of hits. Events are accepted if they meet the conditions described in equation~\ref{eq:dispMom} and illustrated by the red lines in figure \ref{fig:dispMom}:
\begin{eqnarray}
  \label{eq:dispMom}
  \displaystyle \frac{dispX}{dispZ} &>& 0.18 - 0.0026\cdot N_h \, ; {\rm and} \\
  \label{eq:dispMom2}
  dispZ~&>&~6000\mbox{ mm} \mbox{~~~or~~~} p_\mu~\leq~3\cdot~dispZ \, ;
\end{eqnarray}
where $N_h$ is the number of clusters in the candidate, $dispZ$ is in units of mm, and $p_\mu$ in units of MeV/c.

\begin{figure}
  \begin{center}$
    \begin{array}{cc}
      \includegraphics[width=8cm, height=6cm]{03-DetWG/figures/dispHitSig.eps} &
      \includegraphics[width=8cm, height=6cm]{03-DetWG/figures/momDispSig.eps} \\
      \includegraphics[width=8cm, height=6cm]{03-DetWG/figures/dispHitback.eps} &
      \includegraphics[width=8cm, height=6cm]{03-DetWG/figures/momDispBack.eps} \\
    \end{array}$
  \end{center}
  \caption{Distributions of momentum and displacement with cut levels: (top left) relative displacement in the bending plane to the $z$ direction against candidate hits for signal events, (top right) reconstructed momentum against displacement in $z$ for signal events and (bottom) as top for $\nu_\mu~(\overline{\nu}_\mu)$ CC backgrounds.}
  \label{fig:dispMom}
\end{figure}

The final cut involves fitting the candidate's projection onto the bending plane to a parabola. If the charge fitted is opposite to that found by the Kalman filter (in the current simulation a negatively-charged muon bends upwards so that for a parabola defined as $a + bz + cz^2$ the parameter $c$ would be positive and the charge of the muon is $Q_{par} = -sign(c)$) the quality of the fit is assessed using the variable:
\begin{equation}
  \label{eq:qCharge}
  \begin{array}{ll}
  qp_{par} &= \left\{ \begin{array}{c} \left|\displaystyle\frac{\sigma_c}{c}\right|\mbox{, if } Q_{par} = Q_{kal} \, ;\\ \\
      -\left|\displaystyle\frac{\sigma_c}{c}\right|\mbox{, if } Q_{par} = -Q_{kal} \, ;\end{array}\right.
  \end{array}
\end{equation}
where $Q_{kal}$ is the charge fitted by the Kalman filter fit. Defining the parameter in this way ensures that the cut is independent of the initial fitted charge. There are two types of events that are accepted: those with $qp_{par} > 0.0$, in which there is no change in charge, and those candidates with $qp_{par} < -1.0$, in which there is a change in charge but the quality of the fit is poor, so the fit cannot be trusted. This $qp_{par}$ cut effectively reduces the background level from CC mis-identification, as can be seen in figure \ref{fig:qcharge}.
\begin{figure}
  \begin{center}
    \includegraphics[scale=0.6]{03-DetWG/figures/quadQ.eps}
  \end{center}
  \caption{Distribution of the $qp_{par}$ variable with the region where the parameter is $<$ 0 representing those candidates fitted with charge opposite to the initial Kalman filter (region of highest occupancy shown).}
  \label{fig:qcharge}
\end{figure}

\subparagraph*{Cut summary \\}
\label{subpar:Csumm}
In  summary, the test statistic leads to the chain of cuts described above and in table~\ref{tab:G4cutSum}. Application of this analysis and the resulting efficiencies and background suppressions are described in the following section.
\begin{table}
\caption{Summary of cuts applied to select the golden channel appearance signals. The level of absolute efficiency and the main background level (with the dominant neutrino species) after each cut are also shown.}
  \label{tab:G4cutSum}
  \begin{center}
    \begin{tabular}{|c|c|c|c|c|c|}
      \hline
      {\bf Cut} & {\bf Acceptance level} & \multicolumn{2}{c|}{{\bf Eff. after cut}}& \multicolumn{2}{c|}{{\bf Main back ($\times 10^{-3}$)}} \\
                &                        & {\bf $\nu_\mu$}                           & {\bf $\overline{\nu}_\mu$} & {\bf $\nu_\mu$} & {\bf $\overline{\nu}_\mu$} \\
      \hline
      \hline
      Fiducial & \emph{z1} $\leq$ 18000~mm & 0.85 & 0.91 & 120($\nu_e$) & 100($\overline{\nu}_e$)\\
      & {\tiny where \emph{z1} is the lowest \emph{z} cluster in the candidate} & & & &  \\
      \hline
      Track quality & $\mathcal{L}_{q/p} > -0.5$ & 0.76 & 0.85 & 20($\nu_e$) & 20($\overline{\nu}_e$)\\
      \hline
      Max. momentum & $P_\mu \leq 40$~GeV & 0.76 & 0.84 & 20($\nu_e$) & 20($\overline{\nu}_e$)\\
      \hline
      CC selection & $\mathcal{L}_1 > 1.0$ & 0.74 & 0.83 & 0.49($\nu_e$) & 1.6($\nu_\mu$)\\
      \hline
      Fitted proportion & $N_{fit}/N_h \geq 0.6$& 0.73 & 0.83 & 0.46($\nu_e$) & 1.2($\nu_\mu$)\\
      \hline
      Kinematic & $E_{rec} \leq 5~GeV \mbox{ or } Q_t > 0.25$ & 0.63 & 0.77 & 0.65($\overline{\nu}_\mu$) & 0.59($\nu_\mu$)\\
      & $E_{rec} \leq 7~GeV \mbox{ or } P_\mu \geq 0.3E_{rec}$&  &  &  &  \\
      \hline
      Displacement & $dispX/dispZ > 0.18 - 0.0026N_h$ & 0.59 & 0.72 & 0.38($\overline{\nu}_\mu$) & 0.38($\nu_\mu$) \\
      & $dispZ > 6000~mm \mbox{ or } P_\mu \leq 3dispZ$ &  &  &  & \\
      \hline
      Quadratic fit & $qp_{par} < -1.0$ or $qp_{par} > 0.0$ & 0.58 & 0.71 & 0.07($\overline{\nu}_\mu$) & 0.07($\nu_\mu$) \\
      \hline
    \end{tabular}
  \end{center}
\end{table}

\paragraph{MIND response to the golden channel}
\label{par:responseMat}
Using a large data set of $3\times 10^6$ events each of $\nu_\mu$ CC, $\overline{\nu}_\mu$ CC, $\nu_e$ CC, $\overline{\nu}_e$ CC and $7\times 10^6$ NC interactions from neutrinos and anti-neutrinos generated using NUANCE and tracked through the GEANT4 representation of MIND, the expected efficiency and background suppression for the reconstruction and analysis of the golden-channel appearance for both polarities of stored muons has been carried out. In addition, an estimate of the likely systematic error of the analysis has been made.

\subparagraph*{Analysis efficiency \\}
\label{subpar:baseline}
The resultant efficiencies for both polarities and the corresponding background levels expected for the appearance channels are summarised in Figs.~\ref{fig:CCback}~--~\ref{fig:G4Eff}. Numeric response matrices for each of the channels may be found in Appendix~\ref{app:response}.
As can be seen in figure \ref{fig:CCback}, the expected level of background from CC misidentification is significantly below $10^{-3}$ at all energies for the new simulation and re-optimised analysis.
\begin{figure}
  \begin{center}$
    \begin{array}{cc}
      \includegraphics[width=8cm, height=6cm]{03-DetWG/figures/mubarAsmu.eps} &
      \includegraphics[width=8cm, height=6cm]{03-DetWG/figures/muAsmubar.eps} \\
    \end{array}$
  \end{center}
  \caption{Background from mis-identification of $\nu_\mu~(\overline{\nu}_\mu)$ CC interactions as the opposite polarity. (left) $\overline{\nu}_\mu$ CC reconstructed as $\nu_\mu$ CC, (right) $\nu_\mu$ CC reconstructed as $\overline{\nu}_\mu$ CC as a function of true energy.}
  \label{fig:CCback}
\end{figure}

The background from neutral current interactions also lies at or below the $10^{-4}$ level, with the high-energy region exhibiting a higher level than the low-energy region due to the dominance of DIS interactions because of increased visible energy and particle multiplicity. As expected, the NC background tends to be reconstructed at low energy due to the missing energy.
\begin{figure}
  \begin{center}$
    \begin{array}{cc}
      \includegraphics[width=7.5cm, height=5.5cm]{03-DetWG/figures/NCAsMu.eps} &
      \includegraphics[width=7.5cm, height=5.5cm]{03-DetWG/figures/NCAsmubar.eps} \\
    \end{array}$
  \end{center}
  \caption{Background from mis-identification of NC interactions as $\nu_\mu~(\overline{\nu}_\mu)$ CC interactions. (left) NC reconstructed as $\nu_\mu$ CC, (right) NC reconstructed as $\overline{\nu}_\mu$ CC. (top) as a function of true energy.}
  \label{fig:NCback}
\end{figure}

The background from $\nu_e~(\overline{\nu}_e)$ CC interactions is once again expected to constitute a very low level addition to the observed signal. This background is particularly well suppressed due to the thickness of the iron plates and the tendency for the electron shower to overlap with any hadronic activity.
\begin{figure}
  \begin{center}$
    \begin{array}{cc}
      \includegraphics[width=7.5cm, height=5.5cm]{03-DetWG/figures/eAsMu.eps} &
      \includegraphics[width=7.5cm, height=5.5cm]{03-DetWG/figures/ebarAsmubar.eps} \\
    \end{array}$
  \end{center}
  \caption{Background from mis-identification of $\nu_e~(\overline{\nu}_e)$ CC interactions as $\nu_\mu~(\overline{\nu}_\mu)$ CC interactions. (left) $\nu_e$ CC reconstructed as $\nu_\mu$ CC, (right) $\overline{\nu}_e$ CC reconstructed as $\overline{\nu}_\mu$ CC as a function of true energy.}
  \label{fig:eCback}
\end{figure}

The efficiency of detection of the two $\nu_\mu$ polarities was expected to have a threshold lower than that seen in previous studies due to the presence of non-DIS interactions in the data sample. The efficiencies expected for the current analysis are shown in figure \ref{fig:G4Eff}.
\begin{figure}
  \begin{center}$
    \begin{array}{cc}
      \includegraphics[width=7.5cm, height=5.5cm]{03-DetWG/figures/muEff.eps} &
      \includegraphics[width=7.5cm, height=5.5cm]{03-DetWG/figures/mubarEff.eps} \\
    \end{array}$
  \end{center}
  \caption{Efficiency of reconstruction of $\nu_\mu~(\overline{\nu}_\mu)$ CC interactions. (left) $\nu_\mu$ CC efficiency, (right) $\overline{\nu}_\mu$ CC efficiency as a function of true energy.}
  \label{fig:G4Eff}
\end{figure}

Figure \ref{fig:EffEvolv} shows a comparison of the resultant $\overline{\nu}_\mu$ efficiency with that extracted in previous studies.  In the studies described in~\cite{Cervera:2000kp,CerveraVillanueva:2008zz,CerveraVillanueva:2005ym}, the improvement in threshold with the development of the analysis and the introduction of the full spectrum of interactions is clear. The thresholds of the newest results, between 2--3~GeV, correspond well with the expected saturation point of sensitivity to the extraction of the oscillation parameters. The inclusion of non-DIS events is responsible for reducing the threshold, manifested by comparing the black and blue curves (G4 efficiency) with the green curve (G3 efficiency) in figure \ref{fig:EffEvolv} that only included DIS events. 
\begin{figure}
  \begin{center}
    \includegraphics[width=9.5cm, height=7.5cm]{03-DetWG/figures/effcomp.eps}
  \end{center}
  \caption{Evolution of the MIND $\overline{\nu}_\mu$ CC detection efficiency. For the 2000 analysis see~\cite{Cervera:2000kp}, for 2005 see~\cite{CerveraVillanueva:2005ym} and for 2007 see~\cite{CerveraVillanueva:2008zz}.}
  \label{fig:EffEvolv}
\end{figure}

The difference in efficiency between the two appearance channels is effectively described by the difference in the inelasticity of neutrino and anti-neutrino CC interactions. Neutrino DIS interactions with quarks have a  flat distribution in the Bjorken variable:
\begin{equation}
  y = \displaystyle\frac{E_\nu - E_l}{E_\nu} \, ;
\end{equation}
with $E_l$ being the scattered-lepton energy. However, anti-neutrinos interacting with quarks follow a  distribution $\propto~(1-y)^2$ ~\cite{Zuber:2004nz}. For this reason, neutrino interactions generally involve a greater energy transfer to the target. As can be seen from figure \ref{fig:Effbjorken}-(top left), the efficiencies for the two species, as a function of $y$ after all cuts, are the same to within the uncertainties over the full $y$ range. Hence, the difference in neutrino and anti-neutrino efficiencies, when translated into true neutrino-energy, can be explained by the greater abundance of neutrino events at high $y$. However, since the cross section for the interaction of neutrinos is approximately twice that for anti-neutrinos it is not expected that this reduced efficiency will affect the fit to the observed spectrum significantly.
\begin{figure}
  \begin{center}$
    \begin{array}{cc}
      \includegraphics[width=7.5cm, height=5.5cm]{03-DetWG/figures/bjorkenyEffs.eps} &
      \includegraphics[width=7.5cm, height=5.5cm]{03-DetWG/figures/bjorkeny.eps}\\
    \end{array}$
  \end{center}
  \caption{$\nu_\mu$ CC and $\overline{\nu}_\mu$ CC signal detection efficiency as a function of $y$ (left) and the normalised distribution of all events considered in each polarity as a function of $y$ (right).}
  \label{fig:Effbjorken}
\end{figure}
 

\subparagraph*{Study of systematic uncertainties \\}
\label{subSec:syst}
The efficiencies described above will be affected by several systematic effects. There will be many contributing factors including uncertainty in the determination of the parameters used to form the cuts in the analysis, uncertainty in the determination of the hadronic shower energy and direction resolution, uncertainty in the relative proportions of the different interaction types and any assumptions in the representation of the detector and electronics. While exact determination of the overall systematic error in the efficiencies is complicated, an estimate of the contribution of different factors can be obtained by setting certain variables to the extremes of their errors.

Since the neutrino energy is reconstructed generally from the sum of the hadron shower and primary muon energies, and this reconstructed quantity is used in the kinematic cuts, uncertainty in the energy and direction resolution of the detector could contribute significantly to the systematic error in the efficiencies. Taking a 6\% error as quoted for the energy scale uncertainty assumed by the MINOS collaboration~\cite{Michael:2008bc} and varying the constants of the energy and direction smears by this amount, it can be seen (figure \ref{fig:hadresSyst}) that, to this level, the hadronic resolutions have little effect on the true neutrino-energy efficiencies. However, the hadronic direction resolution is likely to have far greater uncertainty and would be very sensitive to noise in the electronics. Also shown in figure \ref{fig:hadresSyst} are the efficiencies when the hadronic energy resolution parameters are 6\% larger but with a 50\% increase in the angular resolution parameters. Even at this level the observed difference in efficiency is only at the level of 1\%.
\begin{figure}
  \begin{center}$
    \begin{array}{cc}
      \includegraphics[width=8cm, height=6cm]{03-DetWG/figures/resVarMu.eps} &
      \includegraphics[width=8cm, height=6cm]{03-DetWG/figures/resVarMubar.eps}\\
      \includegraphics[width=8cm, height=6cm]{03-DetWG/figures/resVarNCneg.eps} &
      \includegraphics[width=8cm, height=6cm]{03-DetWG/figures/resVarNCpos.eps}
    \end{array}$
  \end{center}
  \caption{Variation of signal efficiency (top) and NC backgrounds (bottom) due to a 6\% variation in the hadron shower energy and direction resolution and a more pessimistic 50\% reduction in angular resolution (focused on region of greatest variation).}
  \label{fig:hadresSyst}
\end{figure}

The relative proportions of QE, DIS and other types of interaction in the data sample could have a significant effect on the signal efficiencies and backgrounds. The change in efficiency if the data sample were purely DIS is shown in figure \ref{fig:DISonly}.
\begin{figure}
  \begin{center}$
    \begin{array}{cc}
      \includegraphics[width=7.5cm, height=4.5cm]{03-DetWG/figures/DISEffMu.eps} &
      \includegraphics[width=7.5cm, height=4.5cm]{03-DetWG/figures/DISEffMubar.eps}\\
      \includegraphics[width=7.5cm, height=4.5cm]{03-DetWG/figures/DISmubarAsmu.eps} &
      \includegraphics[width=7.5cm, height=4.5cm]{03-DetWG/figures/DISmuAsmubar.eps}\\
      \includegraphics[width=7.5cm, height=4.5cm]{03-DetWG/figures/DISNCasMu.eps} &
      \includegraphics[width=7.5cm, height=4.5cm]{03-DetWG/figures/DISNCasMubar.eps}\\
      \includegraphics[width=7.5cm, height=4.5cm]{03-DetWG/figures/DISeAsMu.eps} &
      \includegraphics[width=7.5cm, height=4.5cm]{03-DetWG/figures/DISebarAsMubar.eps}
    \end{array}$
  \end{center}
  \caption{Efficiencies for a pure DIS sample compared to the nominal case. (top) Signal efficiency, (second line) $\nu_\mu~(\overline{\nu}_\mu)$ CC background, (third line) NC background and (bottom) $\nu_e~(\overline{\nu}_e)$ CC background. $\nu_\mu$ appearance on the left and $\overline{\nu}_\mu$ appearance on the right.}
  \label{fig:DISonly}
\end{figure}
Although experimental data are available confirming the presence of non-DIS interactions in the energy region of interest, there are significant errors in the transition regions (see for example~\cite{Lyubushkin:2008pe,AguilarArevalo:2010zc}). These errors lead to an uncertainty in the proportion of the different types of interaction that can affect the efficiencies. In order to study the systematic error associated with this effect, many runs over the data-set were performed using different random seeds to exclude events of a particular type from the data-set or alternatively to exclude an equivalent proportion of the ``rest''. As an illustration of the method, consider the contribution from QE interactions. Taking the binned errors on the cross-section measurements from~\cite{Lyubushkin:2008pe,AguilarArevalo:2010zc}, a run to reduce the QE content would exclude a proportion of events in a bin so that instead of contributing a proportion:
\begin{equation}
  \label{eq:nomQE}
  \displaystyle\frac{N_{QE}}{N_{tot}} \, ;
\end{equation}
where $N_{QE}$ is the total number of QE interactions in the bin of interest and $N_{tot}$ is the total number of interactions in the bin, it would instead contribute:
\begin{equation}
  \label{eq:redQE}
  \displaystyle\frac{N_{QE} - \sigma_{QE}N_{QE}}{N_{tot} - \sigma_{QE}N_{QE}} \, ;
\end{equation}
where $\sigma_{QE}$ is the proportional error on the QE cross section for the bin. The equivalent run to increase the QE contribution reduces the contribution of the ``rest'' by an amount calculated to give the corresponding proportional increase in QE interactions:
\begin{equation}
  \label{eq:incQE}
  \displaystyle\frac{N_{QE} + \sigma_{QE}N_{QE}}{N_{tot} + \sigma_{QE}N_{QE}} = \displaystyle\frac{N_{QE}}{N_{tot} - \epsilon N_{rest}} \, ;
\end{equation}
where $N_{rest}$ is the total number of non-QE interactions in the bin and $\epsilon$ is the required proportional reduction in the `rest'. Solving for $\epsilon$ yields the required reduction:
\begin{equation}
  \label{eq:redRest}
  \epsilon = \displaystyle\frac{\sigma_{QE}}{1 + \sigma_{QE}} \, .
\end{equation}
Sampling randomly and repeating runs ensures that any observed change in efficiency is not solely due to the particular events excluded. The 1$\sigma$ systematic error can be estimated as the mean difference between the nominal efficiency and the increase due to a higher QE proportion or decrease due to exclusion. The errors in the true $\nu_\mu$ and $\overline{\nu}_\mu$ efficiencies extracted using this method to vary the contribution of QE, single pion (1$\pi$) and other non-DIS interactions are shown in figure \ref{fig:intvar}. Errors for 1$\pi$ resonant reactions are estimated to be $\sim$20\% below 5~GeV (as measured by the K2K near detector~\cite{:2008eaa}) and at 30\% above. Due to the large uncertainty, both theoretically and experimentally, on the models describing other resonances, coherent, diffractive and elastic processes, a very conservative error of 50\% is taken when varying the contribution of the ``others''. As can be seen, the systematic effect is at the level of 1\% in the efficiency threshold region with increased QE and 1$\pi$ interactions generally increasing the efficiency and increased contribution of the ``other'' interactions having the possibility to decrease efficiency. This last effect is likely to be predominantly due to resonances producing multiple tracks. The effect on backgrounds, while expected to be minimal, has not yet been studied due to the statistical limitations in the data sample, making any calculation using this method unreliable.
\begin{figure}
  \begin{center}$
    \begin{array}{cc}
      \includegraphics[width=7.5cm, height=5.5cm]{03-DetWG/figures/muUpSyst.eps} &
      \includegraphics[width=7.5cm, height=5.5cm]{03-DetWG/figures/mubarUpSyst.eps}\\
      \includegraphics[width=7.5cm, height=5.5cm]{03-DetWG/figures/muDownSyst.eps} &
      \includegraphics[width=7.5cm, height=5.5cm]{03-DetWG/figures/mubarDownSyst.eps}
    \end{array}$
  \end{center}
  \caption{Calculated error on signal efficiencies on increasing (top) and decreasing (bottom) the proportion of non-DIS interactions in the data-set. (left) Errors on true energy $\nu_\mu$ CC efficiency and (right) errors on true energy $\overline{\nu}_\mu$ CC efficiency}
  \label{fig:intvar}
\end{figure}

