Our measurement of the anomalous precession frequency of the muon depends on the correlation of muon velocity and muon polarization.
The projection of polarization onto velocity is given by (Jackson 11.171)
\begin{equation}
\frac{d}{dt}({\bf\hat\beta\cdot s})=-\frac{e}{mc}{\bf s_\perp \cdot}\left[a_\mu({\bf\hat\beta\times B})+\left(a_\mu-\frac{1}{\gamma^2-1}\right)\beta{\bf E}\right]\label{eq:betadots}
\end{equation}  
The dependence of $\omega_a$ on the amplitude of vertical betatron oscillations appears in the term ${\bf\hat\beta\times B}$ in Equation ~\ref{eq:betadots}.
If ${\bf B} = B{\bf\hat y}$ and ${\bf\hat \beta} = \sin\psi_{yz} {\bf\hat y}+\cos\psi_{yz}{\bf\hat z}$ then
${\bf\hat\beta\times B} \sim \beta_z B = \beta(\cos\psi_{yz})B$, where $\psi_{yz}$ is the angle of the trajectory in the $y-z$ plane.
The contribution to $\omega_a$ due to the pitch for a particular trajectory is $\frac{\Delta\omega_a}{\omega_a}=(1-\cos\psi_{yz})$ and
for $\psi_{yz}\ll 1$, 
$$\frac{\Delta\omega_a}{\omega_a}\sim \frac{1}{2}\psi^2_{yz},$$
The contribution from the pitch angle averaged over all $\phi$ for each amplitude $a$ and then over all possible amplitudes $a$,
is
$$\langle\langle\frac{\Delta\omega_a}{\omega_a}\rangle_{\phi}\rangle_a \sim \frac{1}{2}\langle\langle\psi^2_{yz}\rangle_\phi\rangle_a,$$
In the limit of strictly linear betatron motion, the displacement $y$ is related to amplitude $a$ according to
$y=\sqrt{a\beta_y}\cos\phi_y$, where $\phi_y$ is the betatron phase advance. Then the angle
$$\psi_{yz}= \psi_0\sin\phi=\sqrt{\frac{a}{\beta_y}}\sin\phi_y.$$
The average of $\psi_{yz}^2$ over all $\phi_y$ for amplitude $a$ is
$$\langle\psi^2_{yz}(a)_\phi =\frac{1}{2}\psi_0^2(a)=\frac{\langle y^2(a)\rangle_\phi}{\beta^2}$$
Next, average over all amplitudes $a$. We suppose that the distribution of amplitudes is given by some function $P(a)$ defined over the physical aperture.
Then
\begin{eqnarray*}
\langle\langle y^2\rangle_\phi\rangle_a &=& \frac{1}{2}\beta\frac{\int_{aper}aP(a)da}{\int_{aper}P(a) da}\\
\langle\langle \psi^2\rangle_\phi\rangle_a &=& \frac{1}{2}\langle\psi_0^2\rangle_a =\frac{a}{\beta^2}\langle\langle y^2\rangle_\phi\rangle_a\\
C_p &\equiv& \frac{\Delta \omega_a}{\omega_a} = \frac{1}{4}\langle\psi_0^2\rangle_a = -\frac{n \langle y^2\rangle}{2R_0^2}
\end{eqnarray*}
where $\beta_y = \frac{\sqrt{n}}{R_0}$.

The vertical distribution of decay positrons is measured by the straw tracking chambers.
Tracks of positrons are extrapolated back to the parent muon decay point to determine 
the vertical distribution of the muons as shown for a subset of the data in Fig. ~\ref{fig:verticaltracker}
\begin{figure}[]
	\centering
	\begin{tikzpicture}
		\draw (0, 0) node[inner sep=0]{\includegraphics[width=0.95\columnwidth]{fig-dlr/tracker_pitch_15922.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]{fig23.pdf}
	\caption{\textcolor{orange}{David R.} Vertical distribution measured by straw tracker for a subset of the data. The data is not corrected for acceptance or resolution.}\label{fig:verticaltracker}
\end{figure}
