\section{Introduction}

This note presents the reconstruction of the radial distribution of the muon beam for the \mbox{\runo} \sixty\ with the Cornell fast rotation Fourier analysis.
The details of the Cornell fast rotation Fourier analysis are presented in~\cite{cornell} and the study of its performance with toy Monte Carlo simulations
presented in~\cite{tmcperf}. More details about the Cornell fast rotation Fourier method can be found in~\cite{daniel, background}. The analysis code's
user guide can be found in~\cite{userguide}.
The fast rotation Fourier method aims at reconstructing the radial distribution of the stored
muon beam, via reconstructing the frequency distribution, in order to estimate
the electric field correction $C_E$ to the anomalous
spin precession frequency of the muon \wa. The electric field correction can
be estimated in first approximation with the following formula:

\begin{equation}
C_E = \frac{\Delta \omega_a}{\omega_a} = -2n(1-n) \beta^2 \frac{\langle x_{e}^{2} \rangle}{R_{0}^{2}},
\label{eq:efield}
\end{equation}

where

\begin{equation}
\langle x_{e}^{2} \rangle = x_{e}^2 + \sigma^2,
\label{eq:xe2}
\end{equation}

where $x_{e}$ is the equilibrium radius (average radial position) and $\sigma$ the radial width of the beam,
$R_{0}$ is the magic radius of 7112 mm, $\beta$ the relativistic speed, and $n$ the field index that relates to the electrostatic quadrupole electric field gradient like:

\begin{equation}
n = \frac{m \gamma r}{p B_0} \frac{\partial E_r}{\partial r},
\end{equation}

where $m$ is the mass, $\gamma$ the Lorentz factor, $r$ the radial distance from the center of the storage ring, $p$ the momentum, and $E_r$ the radial component
of the quadrupole electric field.

The ultimate goal of the \fnal\ experiment is an uncertainty budget on the electric field correction of 20 ppb.
This uncertainty translates into knowing both the average and the with of the cyclotron revolution frequency distribution 
to couple 0.1 kHz, which corresponds to knowing both the equilibrium radius and the width of the radial distribution to couple \mbox{0.1 mm}.
The electric field correction uncertainty budget for the \runo\ data set is not as stringent given the anticipated statistical and systematic
uncertainties on \wa\ of hundreds of ppb. A total uncertainty of 50 ppb on the electric field correction for the \runo\ data set would reach enough precision.

The reader is expected to be familiar with~\cite{cornell} and~\cite{tmcperf} before reading further this analysis note.
