The E-field systematic is associated with the third term on the right hand size of Eq.~(\ref{eq:au}) and depends on momentum and
radial electric field. The contribution from the electric field is minimized by operating near the magic momentum, namely where $a_\mu = \frac{1}{\gamma^2-1}$
The field of the ring magnet is chosen so that the closed orbit of the magic momentum muon corresponds to the design trajectory.
Then the contribution to $\omega_a$ is
\begin{equation}
C_e\sim-2\frac{\Delta p}{p_m}\frac{\vec\beta\times\vec E}{Bc}
\end{equation}
where $p_m$ is the magic momentum $\gamma$ and $\beta$ are evaluated at $p_m$.

The electric field depends on transverse displacement in the quadrupoles. Insofar as the field is
with displacement, the average electric field along the trajectory is proportional to the average radial displacement $x_e$, namely the closed orbit.
$$\langle E_r\rangle = n\left(\frac{v_s B}{R_0}\right)x_e$$
where $n$ is the focusing index.
The average radial displacement is $$x_e = \eta\frac{\Delta p}{p}.$$ Then
the contribution to $\omega_a$ due to electric field is
$$C_e \sim -2\beta^2n(1-n)\frac{x_e^2}{R_0^2}$$ where $\eta =\sim\frac{1-n}{R_0}$. 
Determination of the E-field correction thus depends on measurement of
the equilibrium radial distribution. (The magnitude of the E-field correction is reduced by a few percent due to
the nonlinearity of the quadrupole field.)

The radial distribution is determined by a ``fast rotation analysis'', that exploits the
connection between the revolution frequency and the radial displacement. A technique~\cite{Orlov:2002ag} based on Fourier transform
yields a frequency spectrum that can be correlated with radius (circumference) and momentum. An alternative method extracts the
radial distribution from the measured debunching of the muon beam. The fast rotation data is provided by the calorimeters which
measure the time dependence of the intensity of the distribution. An example of radial (closed orbit) distribution extracted by both the Fourier method and
the debunching analysis are shown in Fig.~\ref{fig:fourierdebunching}.
The dependence of the E-field contribution to $\omega_a$, on the radial offset of the closed orbit,
superimposed on the the measured distribution is
shown in Fig. ~\ref{fig:efield_cor}.


\begin{figure}[]
	\centering
	\begin{tikzpicture}
		\draw (0, 0) node[inner sep=0]{\includegraphics[width=0.95\columnwidth]{fig-dlr/fr-x.pdf}};
		\draw (0, 0) node[rotate=45,opacity=0.4]{{\fontsize{45}{54}\selectfont\textcolor{red}{\textbf{Preliminary}}}}; % rule of thumb: \fontsize{size}{1.2*size}
	\end{tikzpicture}
	%\includegraphics[width=0.95\columnwidth]{fig24.pdf}
	\caption{\textcolor{orange}{David R.} Fourier analysis of the fast rotation signal and parametererized fit give the muon equilibrium radial (closed orbit) distributionrevolution.}
     \label{fig:fourierdebunching}
\end{figure}

\begin{figure}[]
	\centering
	\begin{tikzpicture}
		\draw (0, 0) node[inner sep=0]{\includegraphics[width=0.95\columnwidth]{fig-dlr/efield_cor.pdf}};
		\draw (0, 0) node[rotate=45,opacity=0.4]{{\fontsize{45}{54}\selectfont\textcolor{red}{\textbf{Preliminary}}}}; % rule of thumb: \fontsize{size}{1.2*size}
	\end{tikzpicture}
	%\includegraphics[width=0.95\columnwidth]{fig-dlr/fr-x.pdf}
	\caption{\textcolor{orange}{David R.} The contribution to $\omega_a$ ($C_e$) due to the electric field is computed by spin tracking as a function of muon momentum and plotted in terms of the closed orbit. The measured radial distribution is superimposed.
The average correction is the convolution of the two.} 
\label{fig:efield_cor}
\end{figure}
