\section{Scaling FFAG replacing the final RLA stage}
\label{App:FFAG}

The possibility of using the scaling type of FFAG ring for muon
acceleration has already been proposed \cite{Kuno:2001tb}, but it was
done assuming very low RF frequencies, incompatible with a frequency
of the order of 200~MHz or higher. 
Here we present a scheme based on stationary-bucket acceleration
\cite{Mori:2006zn} to accelerate muon beams in a scaling FFAG ring
with an RF frequencey of 200~MHz. 

\subsection{Muon Beam Acceleration}

The example of a 3.6~GeV to 12.6~GeV ring with parameters given in
table \ref{tab:acc:sffag:muonrpara} is considered. 
It is assumed that scaling FFAG magnets with a maximum field of
about 4~T are used. 
This is a reasonable assumption once superconducting magnets with a
left-right asymmtric coil distribution \cite{Nakamoto:2004nj} are
employed to realize the scaling-field law.  
In order to allow the simultaneous acceleration of $\mu^+$ and $\mu^-$
beams, the path length per cell of the synchronous particle is
adjusted to be a multiple of $\frac{1}{2}\beta_s \lambda_{RF}$, with
$\beta_s$ the ratio of the synchronous particle velocity to the speed
of light, and $\lambda_{RF}$ the RF wavelength. 
\begin{table}
    \caption{Ring parameters.}
    \label{tab:acc:sffag:muonrpara}
    \begin{center}
    \begin{tabular}{|l|r|}
\hline
	Lattice type		& scaling FFAG FDF triplet\\
	Injection energy & 3.6 GeV\\
	Extraction energy & 12.6 GeV\\
	RF frequency & 200 MHz\\
	Mean radius		& $\sim$ 160.9 m \\
	Synchronous energy (kinetic) & 8.04 GeV\\
	Hormonic number $h$ & 675\\
	Number of cells 	& 225\\
	Field index $k$		& 1\,390\\
	RF peak voltage (per turn)& 1.8 GV\\
	Number of turns & 6\\
	$B_{max}$ (at 12.6 GeV)	& 3.9~T\\
	Drift length & $\sim$ 1.5~m\\
        	Horizontal phase advance per cell & 85.86 deg.\\
	Vertical phase advance per cell &  33.81 deg.\\
	Excursion & 14.3 cm\\ 
\hline 
    \end{tabular}
    \end{center}
\end{table}

We use stepwise particle-tracking in a geometrical field model with
Enge-type fringe field \cite{Enge:1967} to study the beam dynamics. 
Results of single-particle tracking at fixed energy show a normalized
transverse acceptance larger than 30~$\pi$.mm.rad for both horizontal
and vertical planes.

\subsubsection{6D Simulation of a Whole Acceleration Cycle}
\label{sec:acc:sffag:6Dmu}

At the beginning of 6D tracking, the bunch of particles is prepared as
follows: 1\,000 particles are uniformly distributed inside a
transverse, 4D ellipsoid (Waterbag distribution) and uniformly inside
an ellipse in the longitudinal plane. 
The initial normalised bunch emittances are 30~$\pi$.mm.rad in both
horizontal and vertical planes and 150~mm in the longitudinal plane.

Tracking results show no beam lost during the acceleration cycle.
No significant emittance blow-up is observed in either the
longitudinal (see figure \ref{fig:acc:sffag:longit_mu_6d}) or the
transverse planes (see figure \ref{fig:acc:sffag:trans_mu_6d}).  
\begin{figure}
  \begin{center}
    \includegraphics[width=8cm]{06-Appendix/02-04-ScalingFFAG/longit_mu_6d.eps}
  \end{center}
  \caption{
    6-turn acceleration cycle plotted in the longitudinal phase
    space. Hamiltonian contours are superimposed.
  } 
  \label{fig:acc:sffag:longit_mu_6d}
\end{figure}
\begin{figure}
  \begin{minipage}[b]{.5\linewidth}
    \includegraphics[width=8cm]{06-Appendix/02-04-ScalingFFAG/RRprime_mu_6d.eps}
  \end{minipage} \hfill
  \begin{minipage}[b]{.5\linewidth}
    \includegraphics[width=8cm]{06-Appendix/02-04-ScalingFFAG/ZZprime_mu_6d.eps}
  \end{minipage}
  \caption{
    Initial (blue squares) and final (red dots) particle positions
    plotted in the horizontal (left figure) and vertical (right
    figure) phase spaces.
  } 
  \label{fig:acc:sffag:trans_mu_6d}
\end{figure}

Tracking introducing random alignment errors in the lattice has been
performed. 
No beam loss has been observed with an rms alignment error smaller
than 1~mm. 

\subsection{Summary}

A scheme to accelerate muon beams inside the stationary RF bucket of
scaling FFAG rings using 200~MHz RF cavities has been proposed. 
The ring can accelerate both $\mu^+$ and $\mu^-$ beams simultaneously. 
Acceleration is performed within 6 turns, the RF is used in an efficient
manner. 
A detailed tracking-study of the example of a 3.6~GeV to 12.6~GeV muon
ring has been presented.  
6D particle tracking in a soft-edge-field model shows that the
acceptance of this scheme is larger than 30~$\pi$.mm.rad in both
horizontal and vertical planes, and larger than 150~mm in the
longitudinal plane.  
No beam loss and no significant emittance blow-up was observed in
neither the transverse nor the longitudinal planes. 
This scheme also shows a good tolerance to errors.
