\subsection{Decay ring}

In the Neutrino Factory \cite{Apollonio:2009}  neutrinos are generated
from muon decays according to:
\begin{align}
  \mu^- &\rightarrow e^- \bar{\nu}_e \nu_{\mu} \, ; \text{ and}&
  \mu^+ &\rightarrow e^+ \nu_e \bar{\nu}_{\mu} \, .
\end{align}
Neutrinos are aimed at far detectors by collecting muons in storage rings
with long straight sections pointing at the distant experimental
facilities.
The storage rings dip into the ground with angles of 18$^{\circ}$ and
36$^{\circ}$, for baselines of 4000 and 7500~km, respectively.
 
Two geometries of ring have been considered.
The geometry shown in figure \ref{fig:acc:ring:RaceTrack} is
designed to store either $\mu^+$ or $\mu^-$ with a single straight
pointing into the ground. 
The return straight is used for collimation, RF and tune control. 
A development of this concept can accommodate counter-rotating muons
of both signs.  
An alternative is a triangular lattice (figure
\ref{fig:acc:ring:Triangle}) with two production straights pointing in
different directions sending neutrinos to combination of detectors
dictated by the apex angle \cite{Prior:2009zz}.
Since the beam must travel in a unique direction,
two triangular rings would be built side by side in the same tunnel,
one containing $\mu^+$ and the other $\mu^-$.
\begin{figure}
  \begin{center}
    \includegraphics*[width=0.95\textwidth]%
      {02-AccWG/02-G-DecayRing/RaceTrack-line.eps}
  \end{center}
  \caption{Racetrack design for the Neutrino Factory storage rings.}
  \label{fig:acc:ring:RaceTrack}
\end{figure}
\begin{figure}
  \centering
    \includegraphics*[width=0.5\textwidth]%
      {02-AccWG/02-G-DecayRing/Triangle-line.eps}
  \caption{
     A triangular muon storage ring able to send neutrinos to
     detectors in two different directions.
  }
  \label{fig:acc:ring:Triangle}
\end{figure}

Some parameters driving the choice of the muon storage-ring geometry
are the efficiency, defined as the ratio of the total length of
neutrino production straights to the total circumference, and the
depth of the tunnel, which implies geological and cost considerations.
There will be at most three $\mu^+$ and three $\mu^-$ bunch trains
in the ring.
Based on the proton driver described in section
\ref{Sect:ProtonDriver}, a decay ring of 1.6~km in circumference can
accommodate the equally-spaced, 250~ns long bunch trains and can allow
time intervals of at least 100~ns between the neutrino bursts arising
from the $\mu^+$ and the $\mu^-$ bunches.
The maximal tunnel depths for rings of this size are 444~m for the
racetrack and 493~m for the triangle. 

The optical properties of these rings are challenging. 
In order to keep the divergence of the neutrino beam sufficiently
small, the rms divergence of the muon beam
in the production straight section
must not exceed 10\% of the
natural angular divergence of $1/\gamma$ that arises from the
kinematics of muon decay.
This requirement translates into high values for the $\beta$-functions
in the production straights ($\sim 150$~m) which need to be matched into the low
values of $\sim 14$~m in the arcs. 
At the ends of each straight, small bending magnets are included to
ensure that neutrinos created in the matching section
(where the muon beam still has a large divergence angle)
cannot be seen by the detector.
The production straights for the racetrack design are 599.4~m long,
providing an efficiency of 37.2\% for single-sign muon beams.
The corresponding figures for the triangle lattice is
398.5~m (2 straights, 24.8\% efficiency each).
 
The racetrack design has been chosen as the baseline.
It is the most flexible possibility for directing neutrinos to any
combination of far detector locations, and can direct neutrinos
resulting from the decays of both muon charges to the same far
detector if desired.
In the following paragraphs we focus on the performances of the
race-track rings and on the issue of muon-beam monitoring. 
 
\subsubsection{Performance}

Starting from the initial design described in \cite{Prior:2009zz},
MAD-X~\cite{mad-x} has been used to determine the basic machine
properties: Twiss parameters, working point, and dynamic aperture
(DA).
The structure  of the ring is illustrated in figure
\ref{fig:acc:ring:ring}, which shows the three main elements of the
optics: straight sections; arcs; and matching sections. 
The central momentum is 25~GeV/c and the total length of this machine
is 1608.8~m. 
Figure \ref{fig:acc:ring:twiss} (left) shows the optics of the
decay-ring lattice, while the lattice, subdivided into its
three main parts (straights, arcs, and matching sections), is described
in tables \ref{tab:acc:ring:seq}--\ref{tab:acc:ring:arc}.
\begin{figure}
  \centering
    \includegraphics*[width=0.95\textwidth]%
      {02-AccWG/02-G-DecayRing/Apollonio-4-101211.eps}
  \caption{
    Schematic diagram of the racetrack decay ring illustrating its
    linear optics elements: (black) decay straights, (green/blue/cyan)
    arc sections, (red/brown) matching sections.
  }
  \label{fig:acc:ring:ring}
\end{figure}
\begin{table*}
  \caption{
    Sequence of regions and their lengths for the decay ring.
  }
  \label{tab:acc:ring:seq}
  \centering
  \begin{tabular}{|l|r|}
    \hline
    {\bf Section} & {\bf Length} (m)  \\
    \hline
    1/2 straight  & 299.7             \\
    Matching cell &  36.5             \\
    Arc           & 132.0             \\
    Matching cell &  36.5             \\
    1/2 straight  & 299.7             \\
    1/2 straight  & 299.7             \\
    Matching cell &  36.5             \\
    Arc           & 132.0             \\
    Matching cell &  36.5             \\
    1/2 straight  & 299.7             \\
    \hline
    Total length  & 1608.8            \\
    \hline
  \end{tabular}
\end{table*}
\begin{table*}
  \caption{
    Lattice for straight cell.  
    There are 12 such cells in each of the two straights.
    The magnet type (superconducting, SC, or normal, NC) is given in
    the column headed ``Type''.
  }
  \label{tab:acc:ring:str}
  \centering
  \begin{tabular}{|l|r|l|l|}
    \hline
    {\bf Element}   & {\bf Length} (m)  & {\bf Field/Gradient} (T/Tm$^{-1}$) & {\bf Type} \\
    \hline
    QF              &    1.500          & 0.464                             & NC         \\
    Drift           &   21.975          &                                   &            \\
    QD              &    1.500          & $-0.464$                            & NC         \\
    QD              &    1.500          & $-0.464$                            & NC         \\
    Drift           &   21.975          &                                   &            \\
    QF              &    1.500          & 0.464                             & NC         \\
    \hline
    Cell length     &   49.950          &                                   &            \\
    Straight length &  599.400          &                                   &            \\
    \hline
    Total length    & 1198.800          &                                   &            \\
    \hline
  \end{tabular}
\end{table*}
\begin{table*}
  \caption{Lattice for matching cell.  There is one such cell in each of the
    four matching sections.
    The magnet type (superconducting, SC, or normal, NC) is given in
    the column headed ``Type''.
  }
  \label{tab:acc:ring:match}
  \centering
  \begin{tabular}{|l|r|l|l|}
    \hline
    {\bf Element} & {\bf Length} (m) & {\bf Field/Gradient} (T/Tm$^{-1}$) & {\bf Type} \\
    \hline
    Drift         &   0.41           &                                   &            \\
    Dipole        &   4.00           & $-0.64$                             & NC         \\
    Drift         &   0.41           &                                   &            \\
    QD            &   0.80           & $-9.2$                              & SC         \\
    Drift         &   0.50           &                                   &            \\
    QF            &   1.60           & 11.6                              & SC         \\
    Drift         &   0.45           &                                   &            \\
    Dipole        &   0.60           & $-1.9$                              & SC         \\
    Drift         &  14.25           &                                   &            \\
    QF            &   0.80           & 4.1                               & SC         \\
    Drift         &   7.28           &                                   &            \\
    Dipole        &   2.40           & 0.35                              & NC         \\
    Drift         &   0.90           &                                   &            \\
    \hline 
    Cell length   &  36.50           &                                   &            \\
    \hline
    Total length  & 146.00           &                                   &            \\
    \hline
  \end{tabular}
\end{table*}
\begin{table*}
  \caption{
    Lattice for arc cell.  
    There are 15 such cells in each of two arcs.
    The magnet type (superconducting, SC, or normal, NC) is given in
    the column headed ``Type''.
  }
  \label{tab:acc:ring:arc}
  \centering
  \begin{tabular}{|l|r|l|l|}
    \hline
    {\bf Element} & {\bf Length} (mm) & {\bf Field/Gradient} (T/Tm$^{-1}$) & {\bf Type} \\
    \hline
    QD            &   0.5 & $-23.77$ & SC\\
    Drift         &   0.7 & &\\
    Dipole        &   2.0 & $-4.27$ & SC\\
    Drift         &   0.7 & &\\
    QF            &   0.5 & 24.18 & SC\\
    QF            &   0.5 & 24.18 & SC\\
    Drift         &   0.7 & &\\
    Dipole        &   2.0 & $-4.27$ & SC\\
    Drift         &   0.7 & &\\
    QD            &   0.5 & $-23.77$ & SC\\
    \hline 
    Cell length   &   8.8 & &\\
    Arc length    & 132.0 & &\\
    \hline
    Total length &  264.0 & & \\
    \hline
  \end{tabular}
\end{table*}

Beta functions are kept low in the arcs to reduce the size of the beam and
maximise transmission. Conversely the beam envelope is increased in
the straight sections, where keeping a small divergence
is the primary requirement.
The merging of these two opposite criteria is accomplished by the
matching sections, which also eliminate the high dispersion in
the arcs. 
%A detailed description of the lattice parameters is reported in
%\cite{Apollonio:ShortReportIDS}. 

\begin{figure}
  \begin{center}
    \includegraphics[angle=-90,width=0.48\linewidth]{02-AccWG/02-G-DecayRing/dkr_nosex.eps}%
    \hspace{0.04\linewidth}%
    \includegraphics[angle=-90,width=0.48\linewidth]{02-AccWG/02-G-DecayRing/dkr_phadvance.epsi}
  \end{center}
\caption{(Left): $\beta_x$ (black) and $\beta_y$ (red) functions for the decay rings. Their values 
  are deliberately chosen high in the straight sections to reduce the divergence of the beam. Dispersion in the arcs is also shown (green).
  (Right): phase advance over one period in the decay ring with a fast increase in the arcs due the small beta functions and a slow increase in the straights where $\beta_{x,y}$ are large.}
\label{fig:acc:ring:twiss}
\end{figure}

Figure \ref{fig:acc:ring:twiss} (right) illustrates the phase advance along
the ring: the working point is ($Q_x = 8.5229$, $Q_y = 8.2127$). 
Without sextupoles, a $\pm 10$\% change in momentum causes the working point to 
move substantially in the ($Q_x$, $Q_y$) plane, crossing resonances that could
be detrimental for the stability of the beam. To mitigate this effect,
sextupoles can be introduced in the dispersive arcs. 
Chromaticity correction could be motivated by the desire for large
momentum acceptance, the need for which is not yet fully known due to
concerns about 
longitudinal emittance growth in the FFAG, which is presently under
study~\cite{Berg:2009zz}. 
A visual summary of these results is shown in figure
\ref{fig:acc:ring:reso} where resonance diagrams and chromaticity
plots are displayed.  
\begin{figure}
  \centering
    \includegraphics*[width=0.95\textwidth]%
      {02-AccWG/02-G-DecayRing/reso_plot.eps}%{reso_pic.eps}
  \caption{
    Upper section: linear optics configuration, showing the crossing
    of resonances (left) as a function of momentum spread (green
    line): the red dot shows the working point. 
    The natural chromaticity of the beam is shown on the right for
    both the transverse planes. 
    Lower section: the effect of sextupoles is shown. 
    Chromaticity is strongly reduced (right) and so is the crossing of
    potentially dangerous resonances.
  }
  \label{fig:acc:ring:reso}
\end{figure}

\paragraph{Particle tracking and dynamic aperture}

The introduction of non-linear elements along the ring is advocated to
mitigate resonance-crossing effects, potentially catastrophic for a
storage ring. 
However, two things should be stressed in the case of a
muon ring: (a) at 25~GeV/c the average muon-lifetime corresponds to
fewer than 100 turns in the 1608.8~m long ring, 
(b) the sextupoles introduced in the optics create coupling between
the transverse planes and reduce the dynamic aperture of the
beam. 
Concerning point (a), the question naturally arises whether it is
important to avoid resonance crossing, given the relatively short 
beam-lifetime required.
The second aspect has been investigated to produce a more quantitative
answer and a summary of the dynamic aperture  calculations is shown in
figure \ref{fig:acc:ring:da}.
The dynamic aperture comfortably accommodates the nominal 30~mm~rad
acceptance, at least when errors are not included.
\begin{figure}
  \centering
    \includegraphics*[width=0.94\textwidth]%
    {02-AccWG/02-G-DecayRing/da.eps}
  \caption{
    Dynamic aperture for the decay ring: (left column) linear optics,
    (right column) sextupoles introduced in the lattice. 
    (Top) contour plots defining the maximal radius for a particle
    injected at the midpoint of an arc before being lost in the
    lattice. 
    In off-momentum cases this radius is generally reduced. 
    The green area represents the beam size corresponding to the
    nominal acceptance (30 mm rad). 
    Points on the contour plots $(R_x,R_y)$ are used to define the
    amplitudes in the two transverse planes ($A^N_x$,$A^N_y$), where
    $(A^N_i=\frac{p}{m_{\mu}}R^2_i/\beta_{i})_{i=x,y}$ (bottom). 
    In this case the green area marks the 30 mm rad fiducial
    acceptance.
  }
  \label{fig:acc:ring:da}
\end{figure}

\subsubsection{Decay ring instrumentation}

In order to ensure the precise determination of the neutrino flux and
momentum spectrum, continuous measurements of a number of the muon
beam parameters are required.
The critical parameters are: the muon energy (and polarisation), the
beam divergence, and the beam current. 
Efforts are presently focused on the measurement of the first two
parameters which are considered the more challenging.
The beam current can be measured with sufficient precision using
standard beam-current monitors.
The following sections outline the principles of operation of the
devices being considered to measure the beam energy (and polarisation)
and the beam divergence.

\paragraph{Muon beam energy determination using spin polarisation
           measurements} 
\label{Sect:Polar}

Muons from pion decay are produced with a longitudinal polarisation
of $-100$\% in the decay rest frame. 
In the laboratory frame this number is expected to become $-18$\% for
pions between 200~MeV/c and 300~MeV/c \cite{Blondel:2000vz}.
This residual beam-polarisation can be used to determine the central
energy of the beam \cite{Raja:1997ep}. 
Electrons from muon decays can be collected at the end of a straight
section, exploiting the bending power of the dipoles (e.g., in the
matching sections or in the arcs) and channelling the spectral
components to counting devices \cite{Autin:1999ci}. 
Only part of the electron spectrum can be sampled (since electrons at high
energy travel nearly parallel to the muon beam), however 
a general expression for the expected signal can be
written as \cite{Autin:1999ci}:
\begin{equation}
 \mathcal{E}(t) = \mathcal{E}_0 e^{-\alpha t} \left[ 1 + \frac{\beta}{7} e^{-\frac{1}{2}
     \left( \omega \frac{\Delta E}{E} t \right)^2} \cdot \mathcal{P}
   cos(\phi+\omega t)  \right] \, ;
\label{eq:acc:ring:fit}
\end{equation}
where $\mathcal{E}_0$ is the total energy of the electrons at turn 0 and
the Gaussian term parametrises the beam energy spread. 
If it is assumed that all the decay electrons can be collected, then
$\mathcal{P}$ is the polarisation of the beam.
If this is not the case, the $\mathcal{P}$ is a free parameter that
can be related to the beam polarisation with further study. 
A fit to $\mathcal{E}(t)$ allows both the beam energy the energy
spread to be determined. 
An example of such a fit is shown in figure
\ref{fig:acc:ring:Ee_vs_turn} for two energy bins, [0,5]~GeV and
[15,18]~GeV.
The precision with which $E_{\mu}$ and $\Delta E_{\mu}/E_\mu$ can be
measured is a function of the number of captured electrons and of
the number of sampled turns. 
With $\sim 3\times 10^5$ electrons per turn reaching the device and
50 turns sampled, we expect a statistical precision 0.2\% on the central
muon energy and 4\% on the energy spread for each machine fill. Since 
the physics requires only that the average energy distribution be known over
significant periods of running, with 50 fills of the ring per second,
the statistical uncertainties on the flux will be negligible.
The systematic uncertainties of this method are also expected to be small.
The central energy measurement affects the flux as $E^3$, but the
precision achievable will be sufficient to make this source of flux uncertainty
negligible. The measurement of the polarisation itself is more difficult,
because it relies on knowing the collection efficiency of the decay
electrons, which in turn requires detailed knowledge of the geometry
of both the ring and the polarimeter. In a ring geometry, however,
the polarisation precesses and thus its effect on the flux averages
to zero with sufficient precision~\cite{Blondel:2000vz}.
These preliminary numbers need a more realistic study of the
systematics of the measurement, but they give an idea of the
limits of the method.
The relation between the oscillation excursions in the total recorded
energy and the polarisation provides a measure of the latter.  
\begin{figure}
  \centering
  \includegraphics*[width=0.9\textwidth]%
    {02-AccWG/02-G-DecayRing/WEPE053f3.eps}
  \caption{
     Total energy from electrons belonging to the [0,5]~GeV (top) and
     [15,18]~GeV (bottom) GeV intervals of the spectrum as a function
     of the number of turns. 
     The red dots are samples taken at every turn, the superimposed
     curve is a fit to the function defined in equation
     \ref{eq:acc:ring:fit}.
  }
\label{fig:acc:ring:Ee_vs_turn}
\end{figure} 

Once the storage-ring lattice has been designed, a suitable location
for the electron calorimeter can be determined.
In the studies presented here, the electron calorimeter is taken to
refer to a ``generic'' device that will eventually be designed to
measure the electron energy.
Figure \ref{fig:acc:ring:DR-monitors} shows some of the possible
locations for electron monitors, together with the main features of the
dipole magnets used as spectrometers to separate electrons of
different energies. 
When choosing a particular location, one must consider both the spectral
power and the purity of the signal. Ideally, the device should collect
electrons produced in the proximity of the spectrometer in order to
avoid spurious effects from other magnets. 
With this in mind, we focused on two main cases:
\begin{itemize}
\item (A): a monitor placed downstream of the second bending magnet in
  the matching section, just before a quadrupole and orthogonal to the
  direction of the beam; and
\item (B): a monitor placed in the third bending element of the arc
  section, parallel to the beam direction.
\end{itemize}
In case (A) the device sits transversely to the nominal
beam orbit at a distance of about 13 m from a dipole
($B=1.9\text{ T}$, $L_{\text{eff}}=0.6$ m) with the long drift allowing a good spectral power
despite the relatively low magnetic field. The advantage of this
location is that no special magnet is needed, which simplifies
the engineering. Figure~\ref{fig:acc:ring:DR-monitors} (Case A)
shows that at this
location a device placed at $>$10 cm from the beam axis can
intercept a fraction of the spectrum between 0~GeV and 18~GeV
without disrupting the muon beam. In this case we simulated muon
decays along the whole 13 m of drift to check the uniformity of the signal.
Case (B) is similar to the idea for the polarimeter proposed in
\cite{Autin:1999ci}. 
The advantage here is the good spectral power of the
bending magnet due to its effective length and high field 
($B=4.3$~T, $L_{\text{eff}}=2$~m). 
However, the requirement to modify the shape of the magnet,
introducing apertures through the superconducting element, would pose 
engineering challenges. 
The graph shown in figure~\ref{fig:acc:ring:DR-monitors} is a superposition of
decays that occur in the 2.4~m path between dipole B2 and dipole B3
(where the electron monitor is located). As in case (A), a good
signal uniformity is found.
\begin{figure}
  \begin{center}
    \includegraphics[width=0.5\textwidth]{02-AccWG/02-G-DecayRing/WEPE053f4.eps}%
    \includegraphics[width=0.5\textwidth]{02-AccWG/02-G-DecayRing/WEPE053f5.eps}
  \end{center}
\caption{(Left): G4Beamline rendering of the region at the end of a straight section 
and beginning of the arc-section. In the middle the magnets of a
  matching section are visible. Locations for electron monitors are
  suggested. Blue tracks represent muons, green tracks the electrons from decays.
(Right) Momentum as a function of the impact point position for: (A)
  a monitor placed transversely to the beam 
  after a drift of 13 m past a dipole in the matching section. (B) a
  monitor parallel to the beam trajectory. In case (A) the
  position of the beam and its maximum width (for a normalised
  emittance of 30 mm rad) are shown.}
\label{fig:acc:ring:DR-monitors}
\end{figure}

\paragraph{Muon beam divergence}

The decay rings have been designed in order to keep a low rms
divergence in the straight sections. This should be of the order of
10\% of the natural divergence originating from muon decays into
neutrinos ($1/\gamma=4.0$ mrad). 
In order to measure this quantity, a device has
to be inserted in the beam.  The material budget of the device needs to be
small so as to reduce the effects of multiple scattering and energy
straggling. 
The original idea of a Cherenkov detector has been re-analysed in a
critical way, while a new concept based on optical transition
radiation is under study. 

\subparagraph*{Cherenkov Radiator}

This concept dates back to the first proposals for a Neutrino Factory
and consists of a tank with thin walls containing He at high pressure
($>10$~atm).  
Photons from Cherenkov radiation reach a mirror telescope located
at about 20 m from the source and bounce back to a collector with
sensors. The device is sensitive to the beam divergence, such that an
inclination of the beam translates into a change in the image
point. 
According to \cite{Piteira:2001} this device should be very precise
(0.04~mrad resolution), however a review of this proposal makes it
impractical for a number of reasons.
The geometry and size of the device is related to the Cherenkov
angle, which is a function of the helium-gas pressure. A pressure
of the order of 15~atm is needed in order to fulfil the geometrical constraints
of the telescope. 
This implies the use of a thick transparent window to contain the 
gas and to let the light travel towards the optical system. 
A thickness of 200~$\mu$m, which may be optimistically thin, causes
the initial divergence of the beam of $0.4$~mrad to grow to
0.46~mrad after 200 turns.  
The Cherenkov telescope is not particularly compact, due to the
requirement to exploit a long lever arm.
It should be noted that any layer of material placed across the beam
will heat up due to energy loss, reaching more than 100~$^\circ$C in 
$\sim 10$ seconds.
A heat-dissipation system is therefore needed to maintain an
appropriate working temperature. 

\subparagraph*{Optical transition radiation device}

In order to overcome some of the aforementioned issues, alternative
solutions have been examined that rely on the principle of optical transition
radiation (OTR)~\cite{Qiu:1994}.
When a charged particle crosses the boundary between vacuum and
conductor (dielectric), electromagnetic radiation is produced both in
the forward and backward directions (generated by the image charge
within the conductor). 
A particular configuration is obtained when the 
conductor forms an angle of 45$^{\circ}$ with respect to the incident beam. In this
case the ``backward'' radiation is orthogonal to the incident beam and
can be collected out of the beam pipe. The shape of this radiation
shows a typical two-lobed distribution with an opening angle of
$1/\gamma$. 
Depending on the optics chosen, one can infer the beam
divergence in two ways:
a) by collecting the OTR pattern in the focal plane with one foil; or
b) by reconstructing the beam in the image plane with three stations
and then infer the divergence of the beam by means of the usual
transport-matrix techniques.
In case (a) the OTR pattern definition is a function of the beam
divergence, so the divergence can be found by studying the ratio
between minimum and maximum intensities of the OTR figure. 
From the literature~\cite{Qiu:1995xz,Fiorito:1993zu},
we learn that in our case a resolution of 0.5~mrad
should be achievable. This is not ideal, though further studies
should be done in order to understand the real potential of this
technique with a realistic muon beam, an optical system and the best
sensors available today.
Case (b) relies on the reconstruction of beam size at three different
points. The spatial resolution is dominated by the optical system and
the sensors used. At CEBAF a beam spot of 100 $\mu$m can be resolved
\cite{Denard:1997zz}, which makes this method interesting for our
case, even if the uncertainty of the divergence measurement still has
to be evaluated.
The proposed OTR devices all rely on very thin (order of 50~$\mu$m)
metal foils, which makes the technique interesting as it has the
potential to offer a low material budget and has the capability of
dissipating heat.  

In conclusion, we identified three possible ways of measuring
the divergence, one based on Cher\-en\-kov radiation and the other two
relying on OTR techniques.  
The first one seems impractical, while the OTR techniques necessitate
further studies to understand the level of precision reachable in our case.
