\subsection{\label{sec:pitch}Pitch Correction}
\textcolor{orange}{David R.}
\medskip

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 component of the spin that is
precessing in the $\rho,\phi$ plane  
${\bf s} = s(\hat \phi \cos\omega_a t + \hat\rho\sin\omega_a t)$. In the case of the 'ideal' trajectory, Equation \ref{eq:betadots} becomes
$$\frac{d}{dt}s\cos\omega_a t=-\frac{e}{mc}s\sin\omega_a t  (a_\mu B)$$ and 
$$\omega_a = -\frac{e}{mc}a_\mu B.$$
The dependence of $\omega_a$ on the amplitude of vertical betatron oscillations appears in the term ${\bf\hat\beta\times B}$ in Eq.~(\ref{eq:betadots}). 
Introduce the vertical component of velocity as a small perturbation, so that $\hat\beta = \cos\psi_{y\phi}\hat\rho + \sin\psi_{y\phi}\hat y$, where $\psi_{y\phi}$ is the angle of the trajectory
of the muon in the $y-\phi$ plane.
%We will treat the effect of the vertical oscillations as a perturbation to the ideal muon with magic momentum ($\gamma^2=1+1/a_\mu$) and $\hat\beta = \hat\phi$, where
$\hat\phi$ is the unit vector in the azimuthal direction.
%Then $\Delta \omega_a/\omega_a = (1-\cos\psi_{y\phi})$.   
%If ${\bf B} = B{\bf\hat y}$ and ${\bf\hat \beta} = \sin\psi_{y\phi} {\bf\hat y}+\cos\psi_{y\phi}{\bf\hat z}$ 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. 
Then the contribution to $\omega_a$ due to the pitch for a particular trajectory is 
$\frac{\Delta\omega_a}{\omega_a}=\langle (1-\cos\psi_{y\phi})\rangle$,
where the $\langle \ \rangle$ indicate the average of $(1-\cos\psi_{y\phi})$  along the trajectory. 
For small pitch angle, $\psi_{y\phi}\ll 1$, $$\frac{\Delta\omega_a}{\omega_a}\sim \frac{1}{2}\langle\psi^2_{y\phi}\rangle.$$ 
In the limit of strictly linear betatron motion, the angle of the trajectory in the $y-\phi$ plane is modulated 
according to $$\psi_{y\phi}(s)=\psi_0\sin\phi_y(s)=\sqrt{\frac{a}{\beta_y}}\sin\phi_y(s),$$ where $\phi_y(s)$ is the vertical betatron phase advance
as a function of longitudinal position $s$ and $$\langle \psi_{y\phi}^2\rangle = \frac{1}{2}\psi_0^2=\frac{1}{2}\frac{a}{\beta}$$
and the average is taken over many betatron periods. 
We do not measure the distribution of amplitudes directly but rather the distribution of vertical displacements.
Insofar as $y=\sqrt{a\beta}\cos\phi(s)$, then $\langle y^2\rangle = \frac{1}{2}\beta\langle a\rangle$ and $\langle a \rangle=2\langle y^2\rangle/\beta$. 
Averaged over betatron phase and amplitude  $$\langle \psi^2_{y\phi}\rangle = \frac{\langle y^2\rangle}{\beta^2}$$
Finally the average correction to $\omega_a$ due to the vertical betatron motion
is $$\langle\frac{\Delta\omega_a}{\omega_a}\rangle_a \sim \frac{1}{2}\frac{\langle y^2\rangle}{\langle\beta\rangle^2}.$$
 $\langle\beta\rangle = (\int_{\rm accept}\beta(s)ds)/s_{\rm accept}$ is the average over the azimuthal acceptance of the tracking chambers that measure the vertical distribution.
In the continuous quad limit, where
$1/\beta_y = \frac{\sqrt{n}}{R_0}$.
\begin{equation}
C_p=-\frac{n \langle y^2\rangle}{2R_0^2}\label{eq:pitchlinear}
\end{equation}
\noindent
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:verticaldist}.
\begin{figure}[bt]
\centering
%\subfigure[]{\includegraphics[width=0.49\textwidth]{figures/intensity_allcalo_4us_5us}}
%\subfigure[]{\includegraphics[width=0.49\textwidth]{figures/intensity_allcalo_4us_14us}}
%\subfigure[]{\includegraphics[width=0.49\textwidth]{figures/intensity_allcalo_4us_104us}}
%\subfigure[]{\includegraphics[width=0.49\textwidth]{figures/intensity_allcalo_4us_500us}}
\includegraphics*[width=0.8\textwidth, angle=0, trim = 40 120 40 120, clip = .true.]{figures/TrackerVertical-Mott-2020_01_16.pdf}
\caption{Vertical projection of decay muons from straw trackers}
\label{fig:verticaldist}
\end{figure}

