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+all: sddm.pdf
+
+sddm.pdf: sddm.tex
+	pdflatex $^
+
+clean:
+	rm sddm.pdf
+
+.PHONY: clean sddm.pdf
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+\documentclass{article}
+\usepackage{amsmath} % for \text command
+\usepackage{fullpage}
+\usepackage{tikz}
+\usepackage{hyperref}
+\usepackage{amsfonts}
+\usepackage{algorithmic}
+\renewcommand{\algorithmiccomment}[1]{\# #1}
+\usepackage{algorithm}
+\newcommand*\diff{\mathrm{d}}
+\usetikzlibrary{shapes}
+\author{Anthony LaTorre}
+\date{\today}
+\title{Searching for Dark Matter with the Sudbury Neutrino Observatory}
+\begin{document}
+\maketitle
+\section{Introduction}
+\section{Estimating the Event rate in the SNO detector}
+The event rate of self destructing dark matter events, $R$, in the SNO detector is given by first integrating over the detector.
+\begin{equation}
+R = \int_\mathrm{SNO} \mathrm{d}^3r \, R(r) 
+\end{equation}
+Next, we integrate over the earth where the dark matter annihilates:
+\begin{equation}
+R = \int_\mathrm{SNO} \mathrm{d}^3r \, \int_{r'} \mathrm{d}^3r' R(r') \mathrm{P}(\text{detect at r} | \text{DM scatters at r'})
+\end{equation}
+where we have assumed above that the dark matter annihilates immediately after
+scattering. The event rate for scattering at a position $r'$ in the earth is:
+\begin{equation}
+R(r') = \Phi(r') \eta(r') \sigma(r')
+\end{equation}
+where $\Phi(r')$ is the flux at $r'$, $\eta(r')$ is the number density of
+scatterers at $r'$, and $\sigma$ is the cross section for the dark matter to
+scatter and annihilate. In general this will be a sum over the elemental
+composition of the earth at $r'$, but for notational simplicity we will assume
+a single cross section. We will also assume that the cross section is small
+enough that the flux is essentially constant over the whole earth so that the
+rate may be written as:
+\begin{equation}
+R(r') = \Phi \eta(r') \sigma(r').
+\end{equation}
+The rate may then be written as:
+\begin{equation}
+R = \Phi \int_\mathrm{SNO} \mathrm{d}^3r \, \int_{r'} \mathrm{d}^3r' \, \eta(r') \sigma(r') \mathrm{P}(\text{detect at r} | \text{DM scatters at r'})
+\end{equation}
+If we assume that the probability of detecting the dark matter is uniform
+across the SNO detector we may write it as:
+\begin{equation}
+R = \Phi \int_\mathrm{SNO} \mathrm{d}^3r \, \int_{r'} \mathrm{d}^3r' \, \eta(r') \sigma(r') \mathrm{P}(\text{detect at SNO} | \text{DM scatters at r'})
+\end{equation}
+This assumption is pretty well motivated since for most values of the mediator
+decay length the probability will be uniform across the detector. The only
+value for which it might not be a good approximation is if the mediator decay
+length is on the order of the detector radius in which case DM scattering in
+the rock of the cavity walls might have a higher event rate at the edge of the
+detector. Since the integrand no longer depends on $r$, we may write it as:
+\begin{equation}
+R = \Phi V_\text{fiducial} \int_{r'} \mathrm{d}^3r' \, \eta(r') \sigma(r') \mathrm{P}(\text{detect at SNO} | \text{DM scatters at r'})
+\end{equation}
+where $V_\text{fiducial}$ is the fiducial volume of the detector. The
+probability that the mediator $V$ is emitted in a direction $\theta$ and
+travels a distance $r$ in a spherical coordinate system centered on $r'$ may be
+written as:
+\begin{equation}
+f(r,\theta) = \frac{\sin\theta}{4\pi}\frac{e^{-r/L_V}}{L_V}
+\end{equation}
+To transform this probability distribution to the coordinate system centered on the SNO detector we first transform it to a cartesian coordinate system:
+\begin{align}
+f(r,\theta) &= \frac{\sin\theta}{4\pi}\frac{e^{-r/L_V}}{L_V}\frac{1}{r^2sin\theta} \\
+&= \frac{1}{4\pi r^2}\frac{e^{-r/L_V}}{L_V}
+\end{align}
+Then, the distribution is translated to the center of the detector, which
+doesn't change the form since the radial coordinate $r$ is the same in both
+coordinate systems. Finally, we switch back into spherical coordinates:
+\begin{align}
+f(r,\theta') &= \frac{1}{4\pi r^2}\frac{e^{-r/L_V}}{L_V}r^2\sin\theta' \\
+&= \frac{1}{4\pi}\frac{e^{-r/L_V}}{L_V}\sin\theta'
+\end{align}
+where $\theta'$ is the polar angle in the SNO coordinate system. We can now write the rate as:
+\begin{align}
+R &= \Phi V_\text{fiducial} \int_r \mathrm{d}r \, \int_\theta \mathrm{d}\theta \, \int_\phi \mathrm{d}\phi \, \eta(r,\theta,\phi) \sigma(r,\theta,\phi) \frac{1}{4\pi}\frac{e^{-r/L_V}}{L_V}\sin\theta
+\end{align}
+We now assume that the number density of scatterers $\eta$ and the cross
+section $\sigma$ are independent of the position in the earth. This is a good
+approximation for certain values of $L_V$ since the integral will be dominated
+by a single material. For example, if the mediator decay length $L_V$ is
+approximately 1 meter, then the vast majority of the events in the detector
+will be caused by dark matter scattering off of water. Similarly if the
+mediator decay length is approximately 1 km then the majority of the events in
+the detector will be caused by the dark matter scattering off of the norite
+rock surrounding the detector. With this approximation, the rate may be
+written:
+\begin{align}
+R &= \Phi V_\text{fiducial} \eta \sigma \int_r \mathrm{d}r \, \int_\theta \mathrm{d}\theta \, \int_\phi \mathrm{d}\phi \, \frac{1}{4\pi}\frac{e^{-r/L_V}}{L_V}\sin\theta \\
+&= \Phi V_\text{fiducial} \eta \sigma \int_r \mathrm{d}r \, \frac{1}{2}\frac{e^{-r/L_V}}{L_V} \int_\theta \mathrm{d}\theta \, \sin\theta
+\end{align}
+The $\theta$ integral goes from $\theta_\text{min}$ to $\pi$:
+\begin{align}
+R &= \Phi V_\text{fiducial} \eta \sigma \int_r \mathrm{d}r \, \frac{1}{2}\frac{e^{-r/L_V}}{L_V} \int_{\theta_\text{min}}^\pi \mathrm{d}\theta \, \sin\theta
+\end{align}
+where $\theta_\text{min}$ is equal to:
+\begin{equation}
+\theta_\text{min} =%
+\begin{cases}
+0 & \text{if } r < \text{depth} \\
+\pi - \arccos\left(\frac{\text{depth}^2 + r^2 - 2R\text{depth}}{2r(R-\text{depth})}\right) & \text{if } \text{depth} < r < 2R-\text{depth} \\
+\end{cases}
+\end{equation}
+where $R$ is the radius of the earth and $\text{depth}$ is the distance from
+the surface of the earth to the SNO detector.
+
+\section{Cross Section}
+In \cite{grossman2017} the differential scattering cross section for dark
+matter off a nucleus is calculated as
+\begin{equation}
+\frac{\diff \sigma_\text{scatter}}{\diff q^2} = \frac{g_V^2 \epsilon^2 e^2}{4\pi v^2 (q^2 + m_V^2)^2} |F_D(q^2)|^2 Z^2 F^2(q),
+\end{equation}
+where $q$ is the momentum transferred, $g_V$ and $\epsilon$ are coupling
+constants (FIXME: is this true?), $v$ is the velocity of the dark matter
+particle, $m_V$ is the mass of the mediator, $F_D(q^2)$ is a form factor for
+the dark matter to transition from a high angular momentum state to a lower
+angular momentum state, $Z$ is the atomic number of the nucleus, and $F^2(q)$
+is a nuclear form factor.
+
+In the limit of low momentum transfer, the cross section is approximately
+\begin{equation}
+\frac{\diff \sigma_\text{scatter}}{\diff q^2} \simeq \frac{g_V^2 \epsilon^2 e^2}{4\pi v^2 m_V^4} |F_D(q^2)|^2 Z^2 F^2(q).
+\end{equation}
+
+For existing direct detection dark matter experiments, the relevant cross
+section is (FIXME: is this true?)
+\begin{equation}
+\frac{\diff \sigma_\text{scatter}}{\diff q^2} \simeq \frac{g_V^2 \epsilon^2 e^2}{4\pi v^2 m_V^4} Z^2 F^2(q).
+\end{equation}
+
+A standard cross section can be defined as the total cross section in the zero
+momentum limit\cite{pepin2016}
+\begin{align}
+\sigma_0 &= \int_0^{4\mu_T^2 v^2} \frac{\diff \sigma_\text{scatter}}{\diff q^2}\bigg\rvert_{q \rightarrow 0} \diff q^2 \\
+&= \frac{\mu_T^2 g_V^2 \epsilon^2 e^2}{\pi m_V^4} Z^2,
+\end{align}
+where $\mu_T$ is the reduced mass of the WIMP and target nucleus.
+
+Since different experiments use different detector targets, it is also useful
+to define a standard cross section, $\sigma_p$ which is independent of the
+nuclear target:
+\begin{equation}
+\sigma_p = \left(\frac{\mu_p}{\mu_T}\frac{1}{Z}\right)^2 \sigma_0.
+\end{equation}
+
+The direct detection cross section is then:
+\begin{equation}
+\frac{\diff \sigma_\text{scatter}}{\diff q^2} \simeq \frac{1}{4 \mu_p^2 v^2} \sigma_p Z^2 F^2(q).
+\end{equation}
+and the cross section for the dark matter to annihilate is:
+\begin{equation}
+\frac{\diff \sigma_\text{scatter}}{\diff q^2} \simeq \frac{1}{4 \mu_p^2 v^2} \sigma_p |F_D(q)|^2 Z^2 F^2(q).
+\end{equation}
+
+\subsection{Nuclear Form Factor}
+The nuclear form factor, $F(q)$, characterizes the loss of coherence as the de
+Broglie wavelength of the WIMP approaches the radius of the
+nucleus\cite{caldwell2015}. The most commonly used form factor calculation used
+in the direct detection community is that of Helm which is given by:
+\begin{equation}
+F(q) = 3\frac{j_1(q r_1)}{q r_1} e^{-\frac{(q s)^2}{2}},
+\end{equation}
+where $j_1$ is the spherical bessel function of the first order, $s$ is a
+measure of the nuclear skin thickness, and $r_1$ is a measure of the nuclear
+radius\cite{pepin2016}. The values used for these constants were
+\begin{align}
+s &= 0.9 \text{ fm} \\
+a &= 0.52 \text{ fm} \\
+c &= 1.23 A^\frac{1}{3} - 0.60 \text{ fm} \\
+r_1 &= \sqrt{c^2 + \frac{7}{3}\pi^2 a^2 - 5 s^2}
+\end{align}
+
+\begin{figure}
+\centering
+\begin{tikzpicture}[scale=0.1]
+% earth
+\draw [thick,domain=120:150] plot[smooth] ({200*cos(\x)},{200*sin(\x)});
+\begin{scope}[shift={(-100,100)},rotate=45]
+% interaction
+\node[star,star points=9,draw] at (20,20){};
+% acrylic vessel
+\draw [thick,domain={90+asin(75/600)}:{360+90-asin(75/600)}] plot[smooth] ({6*cos(\x)},{6*sin(\x)});
+\draw [thick] ({6*cos(90+asin(75/600))},{6*sin(90+asin(75/600))}) -- ({6*cos(90+asin(75/600))},{6*sin(90+asin(75/600))+7.5}) --
+      ({6*cos(90+asin(75/600))+2*0.75},{6*sin(90+asin(75/600))+7.5}) --
+      ({6*cos(90+asin(75/600))+2*0.75},{6*sin(90+asin(75/600))});
+% PSUP
+\draw [domain=0:360] plot ({8.89*cos(\x)},{8.89*sin(\x)});
+% cavity
+\draw (-9.5,-10.5) --
+      (-10.6,-10.5+5.6) --
+      (-10.6,-10.5+14.93) --
+      (-9.5,-10.5+30) --
+      (9.5,-10.5+30) --
+      (10.6,-10.5+14.93) --
+      (10.6,-10.5+5.6) --
+      (9.5,-10.5) --
+      (-9.5,-10.5);
+\draw[->,ultra thick] (-25,0) -- (25,0) node[right]{$x$};
+\draw[->,ultra thick] (0,-25) -- (0,25) node[right]{$y$};
+\end{scope}
+\end{tikzpicture}
+\end{figure}
+
+\section{Event Reconstruction}
+In order to reconstruct the physical parameters associated with an event we
+compute a likelihood for that event given a proposed energy, position,
+direction, and initial time. The likelihood may be written as:
+\begin{equation}
+\label{likelihood}
+\mathcal{L}(E, \vec{x}, \vec{v}, t_0) = P(\vec{q}, \vec{t} | E, \vec{x}, \vec{v}, t_0)
+\end{equation}
+where $E$, $\vec{x}$, $\vec{v}$ represent the initial particle's kinetic
+energy, position, and direction respectively, $t_0$ represents the initial time
+of the event, $\vec{q}$ is the charge seen by each PMT, and $\vec{t}$ is the
+time recorded by each PMT.
+
+In general the right hand side of Equation~\ref{likelihood} is not factorable
+since for particle tracks which scatter there will be correlations between the
+PMT hits. However, to make the problem analytically tractable, we assume that
+the probability of each PMT being hit is approximately independent of the
+others. With this assumption we can factor the right hand side of the
+likelihood as:
+\begin{equation}
+\mathcal{L}(E, \vec{x}, \vec{v}, t_0) = \prod_i P(\text{not hit} | E, \vec{x}, \vec{v}, t_0) \prod_j P(\text{hit}, q_j, t_j | E, \vec{x}, \vec{v}, t_0)
+\end{equation}
+where the first product is over all PMTs which weren't hit and the second
+product is over all of the hit PMTs.
+
+If we introduce the variable $n$ which represents the number of photoelectrons detected we can write the likelihood as:
+\begin{equation}
+\mathcal{L}(E, \vec{x}, \vec{v}, t_0) = \prod_i P(n = 0 | E, \vec{x}, \vec{v}, t_0) \prod_j \sum_{n = 1}^{\infty} P(n, q_j, t_j | E, \vec{x}, \vec{v}, t_0)
+\end{equation}
+
+We can factor the right hand side of the likelihood as:
+\begin{equation}
+\mathcal{L}(E, \vec{x}, \vec{v}, t_0) = \prod_i P(n = 0 | E, \vec{x}, \vec{v}, t_0) \prod_j \sum_{n = 1}^{\infty} P(q_j, t_j | n, E, \vec{x}, \vec{v}, t_0) P(n | E, \vec{x}, \vec{v}, t_0)
+\end{equation}
+
+If we now assume that the charge and time observed at a given PMT are
+independent we can write the likelihood as:
+\begin{equation}
+\mathcal{L}(E, \vec{x}, \vec{v}, t_0) = \prod_i P(n = 0 | E, \vec{x}, \vec{v}, t_0) \prod_j \sum_{n = 1}^{\infty} P(q_j | n, E, \vec{x}, \vec{v}, t_0) P(t_j | n, E, \vec{x}, \vec{v}, t_0) P(n | E, \vec{x}, \vec{v}, t_0)
+\end{equation}
+
+Since there are many photons produced in each event each of which has a small
+probability to hit a given PMT, we will assume that the probability of
+detecting $n$ photons at a given PMT is poisson distributed, i.e.
+\begin{equation}
+P(n | E, \vec{x}, \vec{v}, t_0) = e^{-\mu} \frac{\mu^n}{n!}
+\end{equation}
+
+We can therefore write the likelihood as:
+\begin{equation}
+\mathcal{L}(E, \vec{x}, \vec{v}, t_0) = \prod_i e^{-\mu_i} \prod_j \sum_{n = 1}^{\infty} P(q_j | n, E, \vec{x}, \vec{v}, t_0) P(t_j | n, E, \vec{x}, \vec{v}, t_0) e^{-\mu_j} \frac{\mu_j^n}{n!}
+\end{equation}
+where $\mu_i$ is the expected number of photoelectrons detected at the ith PMT
+(given an initial particle's energy, position, and direction).
+
+First, we'll calculate the expected number of photoelectrons for a single non-showering track which undergoes multiple scattering through small angles. In this case, we can calculate the expected number of photoelectrons as:
+\begin{equation}
+\mu_i = \int_x \diff x \int_\lambda \diff \lambda \frac{\diff^2 N}{\diff x \diff \lambda} P(\text{detected} | E, x, v)
+\end{equation}
+where $x$ is the position along the track and $\lambda$ is the wavelength of
+the light.
+
+If the particle undergoes many small angle Coulomb scatters, the net
+angular displacement of the particle after a distance $x$ will be a Gaussian
+distribution by the central limit theorem\cite{pdg2017}. The distribution of
+the net angular displacement at a distance $x$ along the track is then given
+by\footnote{This distribution will be correlated between different points along the track.}:
+\begin{equation}
+f(\theta,\phi) = \frac{\theta}{2\pi\theta_0^2}e^{-\frac{\theta^2}{2\theta_0^2}}
+\end{equation}
+where
+\begin{equation}
+\theta_0 = \frac{13.6 \text{ MeV}}{\beta c p}z\sqrt{\frac{x}{X_0}}\left[1 + 0.038\ln\left(\frac{x z^2}{X_0 \beta^2}\right)\right]
+\end{equation}
+where $p$, $\beta c$, and $z$ are the momentum, velocity, and charge of the
+particle, and $X_0$ is the radiation length of the particle\cite{pdg2017}.
+
+Now, we integrate over the angular displacement of the track around the original velocity:
+\begin{align}
+\mu_i &= \int_x \diff x \int_\lambda \diff \lambda \frac{\diff^2 N}{\diff x \diff \lambda} \int_\theta \diff \theta \int_\phi \diff \phi P(\text{detected} | \theta, \phi, E, x, v) P(\theta, \phi | E, x, v) \\
+\mu_i &= \int_x \diff x \int_\lambda \diff \lambda \frac{\diff^2 N}{\diff x \diff \lambda} \int_\theta \diff \theta \int_\phi \diff \phi P(\text{detected} | \theta, \phi, E, x, v) f(\theta,\phi)
+\end{align}
+The probability of being detected can be factored into several different compontents:
+\begin{align}
+\mu_i &= \int_x \diff x \int_\lambda \diff \lambda \frac{\diff^2 N}{\diff x \diff \lambda} \int_\theta \diff \theta \int_\phi \diff \phi P(\text{emitted towards PMT i} | \theta, \phi, E, x, v) f(\theta,\phi) P(\text{not scattered or absorbed} | \lambda, E, x, v) \epsilon(\eta) \mathrm{QE}(\lambda) \\
+\label{eq:mui}
+\mu_i &= \int_x \diff x \int_\lambda \diff \lambda \frac{\diff^2 N}{\diff x \diff \lambda} P(\text{not scattered or absorbed} | \lambda, E, x, v) \epsilon(\eta) \mathrm{QE}(\lambda) \int_\theta \diff \theta \int_\phi \diff \phi P(\text{emitted towards PMT i} | \theta, \phi, E, x, v) f(\theta,\phi)
+\end{align}
+where $\eta$ is the angle between the vector connecting the track position $x$
+to the PMT position and the normal vector to the PMT, $\epsilon(\eta)$ is the
+collection efficiency, and $\mathrm{QE}(\lambda)$ is the quantum efficiency of
+the PMT.
+
+The probability that a photon is emitted directly towards a PMT is given by a
+delta function (we make the assumption here that the probability is uniform
+across the face of the PMT):
+\begin{equation}
+P(\text{emitted towards PMT i} | \theta, \phi, E, x, v) = \delta\left(\frac{1}{n(\lambda)\beta} - \cos\theta'(\theta,\phi,x)\right) \frac{\Omega(x)}{4\pi}
+\end{equation}
+where $\theta'$ is the angle between the track and the PMT and $\Omega(x)$ is the solid angle subtended by the PMT.
+
+In a coordinate system with the z axis aligned along the original particle velocity and with the PMT in the x-z plane, the angle $\theta'$ is defined by:
+\begin{equation}
+\cos\theta' = \sin\theta\cos\phi\sin\theta_1 + \cos\theta\cos\theta_1
+\end{equation}
+where $\theta_1$ is the angle between the PMT and the original particle velocity.
+
+We can now solve the integral on the right hand side of Equation~\ref{eq:mui} as:
+\begin{align}
+P(\text{emitted towards PMT i}) &= \int_\theta \diff \theta \int_\phi \diff \phi \delta\left(\frac{1}{n(\lambda)\beta} - \cos\theta'(\theta,\phi,x)\right) \frac{\Omega(x)}{4\pi} \frac{\theta}{2\pi\theta_0^2}e^{-\frac{\theta^2}{2\theta_0^2}} \\
+P(\text{emitted towards PMT i}) &= \frac{\Omega(x)}{4\pi} \frac{1}{2\pi\theta_0^2}\int_\theta \diff \theta \int_\phi \diff \phi \delta\left(\frac{1}{n(\lambda)\beta} - \cos\theta'(\theta,\phi,x)\right) \theta e^{-\frac{\theta^2}{2\theta_0^2}} \\
+P(\text{emitted towards PMT i}) &= \frac{\Omega(x)}{4\pi} \frac{1}{2\pi\theta_0^2}\int_\theta \diff \theta \int_\phi \diff \phi \delta\left(\frac{1}{n(\lambda)\beta} - \sin\theta\cos\phi\sin\theta_1 - \cos\theta\cos\theta_1\right) \theta e^{-\frac{\theta^2}{2\theta_0^2}}
+\end{align}
+
+We now assume $\theta$ is small (which should be valid for small angle scatters), so that we can rewrite the delta function as:
+\begin{align}
+P(\text{emitted towards PMT i}) &= \frac{\Omega(x)}{4\pi} \frac{1}{2\pi\theta_0^2}\int_\theta \diff \theta \int_\phi \diff \phi \delta\left(\frac{1}{n(\lambda)\beta} - \theta\cos\phi\sin\theta_1 - \cos\theta_1\right) \theta e^{-\frac{\theta^2}{2\theta_0^2}}
+\end{align}
+
+We can rewrite the delta function and solve the integral as:
+\begin{align}
+P(\text{emitted towards PMT i}) &= \frac{\Omega(x)}{4\pi} \frac{1}{2\pi\theta_0^2}\int_\theta \diff \theta \int_\phi \diff \phi \frac{1}{\left|\cos\phi\sin\theta_1\right|}\delta\left(\theta - \frac{\frac{1}{n(\lambda)\beta}-\cos\theta_1}{\cos\phi\sin\theta_1}\right) \theta e^{-\frac{\theta^2}{2\theta_0^2}} \\
+&= \frac{\Omega(x)}{4\pi} \frac{1}{2\pi\theta_0^2} \frac{1}{\left|\sin\theta_1\right|} \int_\phi \diff \phi \frac{1}{\left|\cos\phi\right|} \int_\theta \diff \theta \delta\left(\theta - \frac{\frac{1}{n(\lambda)\beta}-\cos\theta_1}{\cos\phi\sin\theta_1}\right) \theta e^{-\frac{\theta^2}{2\theta_0^2}} \\
+&= \frac{\Omega(x)}{4\pi} \frac{1}{2\pi\theta_0^2} \frac{1}{\left|\sin\theta_1\right|} \int_\phi \diff \phi \frac{1}{\left|\cos\phi\right|}\frac{\frac{1}{n(\lambda)\beta}-\cos\theta_1}{\cos\phi\sin\theta_1}H\left(\frac{\frac{1}{n(\lambda)\beta}-\cos\theta_1}{\cos\phi\sin\theta_1}\right)e^{-\frac{1}{2\theta_0^2}\left(\frac{\frac{1}{n(\lambda)\beta}-\cos\theta_1}{\cos\phi\sin\theta_1}\right)^2} \\
+&= \frac{\Omega(x)}{4\pi} \frac{1}{2\pi\theta_0^2} \frac{1}{\left|\sin\theta_1\right|}\sqrt{2\pi}\theta_0 e^{-\frac{1}{2\theta_0^2}\left(\frac{\frac{1}{n(\lambda)\beta}-\cos\theta_1}{\sin\theta_1}\right)^2} \\
+&= \frac{\Omega(x)}{4\pi} \frac{1}{\sqrt{2\pi}\theta_0} \frac{1}{\left|\sin\theta_1\right|} e^{-\frac{1}{2\theta_0^2}\left(\frac{\frac{1}{n(\lambda)\beta}-\cos\theta_1}{\sin\theta_1}\right)^2}.
+\end{align}
+
+To simplify this expression we can write
+\begin{equation}
+P(\text{emitted towards PMT i}) = \frac{\Omega(x)}{4\pi} \frac{1}{\sqrt{2\pi}\theta_0} \frac{1}{\left|\sin\theta_1\right|} e^{-\frac{\Delta^2(\lambda)}{2\theta_0^2}}
+\end{equation}
+where
+\begin{equation}
+\Delta(\lambda) = \frac{\frac{1}{n(\lambda)\beta}-\cos\theta_1}{\sin\theta_1}
+\end{equation}
+
+Plugging this back into Equation~\ref{eq:mui}
+\begin{align}
+\label{eq:mui-exact}
+\mu_i &= \frac{1}{\sqrt{2\pi}\theta_0} \int_x \diff x \frac{\Omega(x)}{4\pi} \frac{1}{\left|\sin\theta_1\right|} \epsilon(\eta) \int_\lambda \diff \lambda \frac{\diff^2 N}{\diff x \diff \lambda} P(\text{not scattered or absorbed} | \lambda, E, x, v) \mathrm{QE}(\lambda) e^{-\frac{\Delta^2(\lambda)}{2\theta_0^2}}
+\end{align}
+
+Ideally we would just evaluate this double integral for each likelihood call,
+however the double integral is too computationally expensive to perform for
+every likelihood call (FIXME: is this true?). We therefore assume that the
+second integral will be dominated by the Bessel function which has a
+singularity when it's argument is zero, and rewrite Equation~\ref{eq:mui-exact}
+as:
+\begin{align}
+\mu_i &= 2 \frac{1}{\sqrt{2\pi}\theta_0} \int_x \diff x \Omega(x) \frac{1}{\left|\sin\theta_1\right|} \epsilon(\eta) P(\text{not scattered or absorbed} | \lambda_0, E, x, v) \mathrm{QE}(\lambda_0) e^{-\frac{\Delta^2(\lambda_0)}{4\theta_0^2}} \int_\lambda \diff \lambda \frac{\diff^2 N}{\diff x \diff \lambda} K_0\left(\frac{\Delta^2(\lambda)}{4\theta_0^2}\right)
+\end{align}
+where $\lambda_0$ is the wavelength at which $\Delta(\lambda) = 0$.
+
+For small values of $\Delta$, the Bessel function may be approximated as:
+\begin{equation}
+K_0(x) \simeq -\log(x) + \log(2) - \gamma
+\end{equation}
+
+We may therefore approximate the expected charge as
+\begin{multline}
+\label{eq:mui-approx}
+\mu_i = 2 \frac{1}{\sqrt{2\pi}\theta_0} \int_x \diff x \Omega(x) \frac{1}{\left|\sin\theta_1\right|} \epsilon(\eta) P(\text{not scattered or absorbed} | \lambda_0, E, x, v) \mathrm{QE}(\lambda_0) e^{-\frac{\Delta^2(\lambda_0)}{4\theta_0^2}} \\
+\int_\lambda \diff \lambda \frac{\diff^2 N}{\diff x \diff \lambda} \left(-\log\left(\frac{\Delta^2(\lambda)}{4\theta_0^2}\right) + \log(2) - \gamma\right)
+\end{multline}
+
+The number of Cerenkov photons produced per unit length and per unit wavelength
+is given by\cite{pdg2017}
+\begin{equation}
+\frac{\diff^2 N}{\diff x \diff \lambda} = \frac{2\pi\alpha z^2}{\lambda^2}\left(1 - \frac{1}{\beta^2 n^2(\lambda)}\right)
+\end{equation}
+where $\alpha$ is the fine-structure constant and $z$ is the charge of the
+particle in units of the electron charge.
+
+We can therefore write the second integral in Equation~\ref{eq:mui-approx} as
+\begin{align}
+\int_\lambda \diff \lambda \frac{\diff^2 N}{\diff x \diff \lambda} \left(-\log\left(\frac{\Delta^2(\lambda)}{4\theta_0^2}\right) + \log(2) - \gamma\right) &=
+2\pi\alpha z^2 \int_\lambda \diff \lambda \frac{1}{\lambda^2}\left(1 - \frac{1}{\beta^2 n^2(\lambda)}\right) \left(-\log\left(\frac{\Delta^2(\lambda)}{4\theta_0^2}\right) + \log(2) - \gamma\right) \\
+\label{eq:lambda}
+&\simeq 2\pi\alpha z^2 \left(1 - \frac{1}{\beta^2 n^2(\lambda_0)}\right) \int_\lambda \diff \lambda \frac{1}{\lambda^2}\left(-\log\left(\frac{\Delta^2(\lambda)}{4\theta_0^2}\right) + \log(2) - \gamma\right)
+\end{align}
+
+Since the $\Delta$ function only depends on the wavelength through the index
+which depends weakly on the wavelength, we can approximate the index of
+refraction as:
+\begin{equation}
+n(\lambda) \simeq a + \frac{b}{\lambda^2}.
+\end{equation}
+
+The integral in Equation~\ref{eq:lambda} may then be solved analytically
+\begin{multline}
+\int_{\lambda_1}^{\lambda_2} \diff \lambda \frac{1}{\lambda^2}\left(-\log\left(\frac{\Delta^2(\lambda)}{4\theta_0^2}\right) + \log(2) - \gamma\right) =
+\left[\log(4\theta_0^2) + \log(\sin^2\theta_1) + \log(2) - \gamma\right]\left(\frac{1}{\lambda_1}-\frac{1}{\lambda_2}\right) + \\
+\left.\left(-4\sqrt{\frac{a}{b}}\arctan\left(\sqrt{\frac{a}{b}}\lambda\right) +
+4\sqrt{\frac{1+a\Delta_0}{b\Delta_0}}\arctan\left(\sqrt{\frac{1+a\Delta_0}{b\Delta_0}}\lambda\right) -
+\frac{1}{\lambda}\log\left[\left(\Delta_0+\frac{\lambda^2}{b+a\lambda^2}\right)^2\right]\right)\right|_{\lambda_1}^{\lambda_2}
+\end{multline}
+where $\lambda_1$ and $\lambda_2$ are chosen to cover the range where the
+quantum efficiency is non-zero, typically between 300 nm and 600 nm.
+
+For simplicity we will write this previous expression as $f(x)$
+\begin{multline}
+f(x) = \left[\log(4\theta_0^2) + \log(\sin^2\theta_1) + \log(2) - \gamma\right]\left(\frac{1}{\lambda_1}-\frac{1}{\lambda_2}\right) + \\
+\left.\left(-4\sqrt{\frac{a}{b}}\arctan\left(\sqrt{\frac{a}{b}}\lambda\right) +
+4\sqrt{\frac{1+a\Delta_0}{b\Delta_0}}\arctan\left(\sqrt{\frac{1+a\Delta_0}{b\Delta_0}}\lambda\right) -
+\frac{1}{\lambda}\log\left[\left(\Delta_0+\frac{\lambda^2}{b+a\lambda^2}\right)^2\right]\right)\right|_{\lambda_1}^{\lambda_2}
+\end{multline}
+
+We can now write Equation~\ref{eq:mui-approx} as
+\begin{equation}
+\mu_i = 2 \frac{1}{\sqrt{2\pi}\theta_0} 2\pi\alpha z^2 \int_x \diff x \Omega(x) \frac{1}{\left|\sin\theta_1\right|} \epsilon(\eta) P(\text{not scattered or absorbed} | \lambda_0, E, x, v) \mathrm{QE}(\lambda_0) e^{-\frac{\Delta^2(\lambda_0)}{4\theta_0^2}} \left(1 - \frac{1}{\beta^2 n^2(\lambda_0)}\right) f(x)
+\end{equation}
+
+The probability that a photon travels to the PMT without being scattered or absorbed can be calculated as follows
+\begin{align}
+P(\text{not scattered or absorbed} | \lambda, x) &=
+P(\text{not scattered} | \lambda, x) P(\text{not absorbed} | \lambda, x) \\
+&= \int_l^\infty\frac{1}{s(\lambda)}e^{-\frac{x}{s(\lambda)}}\int_l^\infty\frac{1}{a(\lambda)}e^{-\frac{x}{a(\lambda)}} \\
+&= e^{-\frac{l}{s(\lambda) + a(\lambda)}}
+\end{align}
+where $l$ is the distance to the PMT from the position $x$, $s(\lambda)$ is the
+scattering length, and $a(\lambda)$ is the absorption length.
+
+We can therefore write the expected charge as
+\begin{equation}
+\mu_i = 2 \frac{1}{\sqrt{2\pi}\theta_0} 2\pi\alpha z^2 \int_x \diff x \Omega(x) \frac{1}{\left|\sin\theta_1\right|} \epsilon(\eta) e^{-\frac{l(x)}{s(\lambda) + a(\lambda)}} \mathrm{QE}(\lambda_0) e^{-\frac{\Delta^2(\lambda_0)}{4\theta_0^2}} \left(1 - \frac{1}{\beta^2 n^2(\lambda_0)}\right) f(x)
+\end{equation}
+
+The last integral is calculated numerically when the likelihood is evaluated.
+
+We now return to the likelihood and calculate the probability of observing a
+given time. In principle, this depends on the number of photons hitting a PMT
+since the PMT hit will only register the \emph{first} photoelectron which
+crosses threshold. However, since this is expected to be a small effect, we
+assume that the probability of observing a given time is independent of the
+number of photons which hit the PMT, i.e.
+\begin{equation}
+P(t_j | n, E, \vec{x}, \vec{v}, t_0) \simeq P(t_j | n \geq 1, E, \vec{x}, \vec{v}, t_0)
+\end{equation}
+
+We first condition on the \emph{true} time at which the photon hits the PMT
+\begin{equation}
+P(t_j | n \geq 1, E, \vec{x}, \vec{v}, t_0) = \int_{t_j'} \diff t P(t_j | t_j') P(t_j' | n \geq 1, E, \vec{x}, \vec{v}, t_0)
+\end{equation}
+where we used the fact that the probability of a measured time only depends on the true PMT hit time.
+
+Now, we integrate over the track
+\begin{equation}
+P(t_j | n \geq 1, E, \vec{x}, \vec{v}, t_0) = \int_{t_j'} \diff t_j' P(t_j | t_j') \int_x \diff x P(t_j', x | n \geq 1, E, \vec{x}, \vec{v}, t_0)
+\end{equation}
+where $x$ here stands for the event that a photon emitted at a distance $x$ along the track makes it to the PMT.
+
+We now use Bayes theorem to rewrite the last probability
+\begin{align}
+P(t_j | n \geq 1, E, \vec{x}, \vec{v}, t_0) &= \int_{t_j'} \diff t_j' P(t_j | t_j') \int_x \diff x P(t_j', x | n \geq 1, E, \vec{x}, \vec{v}, t_0) \\
+&= \int_{t_j'} \diff t_j' P(t_j | t_j') \int_x \diff x P(t_j' | x, n \geq 1, E, \vec{x}, \vec{v}, t_0) P(x | n \geq 1, E, \vec{x}, \vec{v}, t_0) \\
+\end{align}
+The first term in the integral is just a delta function (up to slight differences due to dispersion) since we are assuming direct light
+\begin{align}
+P(t_j | n \geq 1, E, \vec{x}, \vec{v}, t_0)
+&= \int_{t_j'} \diff t_j' P(t_j | t_j') \int_x \diff x \delta\left(\frac{l(x)n(\lambda_0)}{c}-t_j'\right) P(x | n \geq 1, E, \vec{x}, \vec{v}, t_0) \\
+\end{align}
+
+We now use Bayes theorem to rewrite the last term
+\begin{align}
+P(t_j | n \geq 1, E, \vec{x}, \vec{v}, t_0)
+&= \int_{t_j'} \diff t_j' P(t_j | t_j') \int_x \diff x \delta\left(\frac{l(x)n(\lambda_0)}{c}-t_j'\right) \frac{P(n \geq 1 | x, E, \vec{x}, \vec{v}, t_0) P(x | E, \vec{x}, \vec{v}, t_0)}{P(n \geq 1 | E, \vec{x}, \vec{v}, t_0)} \\
+&= \int_{t_j'} \diff t_j' P(t_j | t_j') \int_x \diff x \delta\left(\frac{l(x)n(\lambda_0)}{c}-t_j'\right) \frac{P(x | E, \vec{x}, \vec{v}, t_0)}{P(n \geq 1 | E, \vec{x}, \vec{v}, t_0)} \\
+&= \int_{t_j'} \diff t_j' P(t_j | t_j') \int_x \diff x \delta\left(\frac{l(x)n(\lambda_0)}{c}-t_j'\right) \frac{P(x | E, \vec{x}, \vec{v}, t_0)}{1 - e^{-\mu_j}} \\
+&= \int_{t_j'} \diff t_j' P(t_j | t_j') \int_x \diff x \delta\left(\frac{l(x)n(\lambda_0)}{c}-t_j'\right) \frac{\mu_j(x)}{1 - e^{-\mu_j}} \\
+&= \frac{1}{1 - e^{-\mu_j}} \int_x \diff x \mu_j(x) \int_{t_j'} \diff t_j' P(t_j | t_j') \delta\left(\frac{l(x)n(\lambda_0)}{c}-t_j'\right) \\
+\end{align}
+
+We assume the transit time spread is equal to a gaussian (we ignore the pre and late pulsing)
+\begin{align}
+P(t_j | n \geq 1, E, \vec{x}, \vec{v}, t_0)
+&= \frac{1}{1 - e^{-\mu_j}} \int_x \diff x \mu_j(x) \int_{t_j'} \diff t_j' \frac{1}{\sqrt{2\pi}\sigma_t} e^{-\frac{(t_j-t_j')^2}{2\sigma_t^2}} \delta\left(\frac{l(x)n(\lambda_0)}{c}-t_j'\right) \\
+&= \frac{1}{1 - e^{-\mu_j}} \frac{1}{\sqrt{2\pi}\sigma_t} \int_x \diff x \mu_j(x) e^{-\frac{(t_j-t_0(x))^2}{2\sigma_t^2}}
+\end{align}
+where in the last expression we define
+\begin{equation}
+t_0(x) \equiv \frac{l(x)n(\lambda_0)}{c}
+\end{equation}
+
+\section{Backgrounds}
+\subsection{External Muons}
+Both cosmic ray muons and muons created from atmospheric neutrinos interacting
+in the surrounding rock present a background for this analysis. In both cases,
+it is necessary to cut events which start \emph{outside} the PSUP and enter the
+detector.
+
+During SNO, these events were cut using the MUON cut which tagged events with
+at least 150 hits, 5 or more outward-looking (OWL) PMT hits, and with a time
+RMS of less than 90 nanoseconds. This cut would have a negligible sacrifice for
+any contained atmospheric or dark matter candidate events, but could
+potentially cut events which produce an energetic muon which then exits the
+detector. Therefore, I have slightly modified this cut to \emph{also} require
+that at least 1 OWL tube is both early and has a high charge relative to the
+nearby normal PMTs. We define a early and high charge tube by creating an array
+of the ECA calibrated hit times (we can't use PCA calibrated times since the
+OWL tubes were never calibrated via PCA) and of the best uncalibrated charge
+(FIXME: footnote?) for all normal PMTs within 3 meters of each hit OWL PMT. We
+then compute the median charge and time for these normal PMTs. We then compute
+how many OWL PMT hits are \emph{both} earlier than the median normal PMT time
+and have a higher charge than the surrounding PMTs. If at least 1 OWL PMT hit
+satisfies this criteria and all the other criteria from the SNO MUON cut are
+satisifed (except the time RMS part) then it's tagged as a muon.
+
+\subsection{Noise Events}
+
+There are several sources of noise events which refers to events triggered by
+sources which do not actually create light in the detector. The two most common
+sources are "ringing" after large events and electrical pickup on deck.
+
+These events are tagged by the QvNHIT and ITC cuts which are identical to their
+SNO counterparts aside from minor updates\footnote{The ITC cut uses the pt1
+time which is the time without the charge walk calibration since otherwise the
+cut may fail to tag an event which consists of mostly electronics noise which
+has charge too low to apply PCA. The QvNHIT cut does not require good
+calibrations for the hits for a similar reason.}.
+
+\subsection{Neck Events}
+
+Neck events are caused by light produced in or leaking through the glove box on
+top of the detector\cite{sonley}. The SNO neck event cut is defined
+as\cite{snoman_companion}:
+
+\begin{quotation}
+This cuts events containing neck tubes. It requires that either both tubes in
+the neck fire, or that one of those tubes fires and it has a high charge and is
+early. High charge is defined by a neck tube having a pedestal subtracted
+charge greater than 70 or less than -110. Early if defined by the neck tube
+having an ECA time 70ns or more before the average ECA time of the PSUP PMTS
+with z les than 0. After the cable changes to the neck tubes this time
+difference changes to 15ns. 
+\end{quotation}
+
+Similarly to the MUON cut, I've used these criteria but added an additional
+requirement to avoid tagging high energy upwards going events. The NECK cut I
+use also has a requirement that 50\% of the hit PMTs must have a z coordinate
+of less than 4.25 meters \emph{or} 50\% of the ECA calibrated QHS charge must
+be below z = -4.25 meters.
+
+\subsection{Flashers}
+
+Flashers are probably the most difficult and common source of instrumental
+background for this analysis. A flasher event occurs when there is an
+electrical short in the PMT base or dynode stack which causes light to be
+emitted from the PMT and hit the opposite side of the
+detector\textsuperscript{[citation needed]}. Because this event is caused by
+actual light in the detector it is particularly hard to cut while also
+maintaining a small signal sacrifice.
+
+The cut algorithm is sufficiently complex that it is easier to describe in
+pseudocode. A description of the algorithm is shown in
+Algorithm~\ref{flasher_algorithm}.
+
+\begin{algorithm}
+\caption{Flasher Cut Algorithm}
+\label{flasher_algorithm}
+\begin{algorithmic}
+  \IF{nhit $< 31$} \RETURN 0 \ENDIF
+
+  \COMMENT{This condition is similar to the SNO QvT cut except we require that 70\% of the normal hit PMTs be 12 meters from the high charge channel and that 70\% of the normal hit PMTs be at least 50 ns after the high charge channel.}
+
+  \IF{highest QLX $>$ second highest QLX $+ 80$}
+     \STATE {Collect all hit times from the same slot as the high charge channel and compute the median hit time}
+     \IF{At least 4 hits in the slot \AND 70\% of the normal hit PMTs with good calibration are more than 12 meters from the high charge channel \AND 70\% of the normal hit PMTs with good calibration are more than 50 ns after the median hit time in the slot}
+       \RETURN 1
+     \ENDIF
+  \ENDIF
+  \FOR{All PC with at least 4 hits}
+    \STATE {Collect the QHS, QHL, and QLX charges and the ECA calibrated hit times (EPT) for each PMT in the PC sending charge values below 300 to 4095}
+    \STATE {$t \leftarrow \textrm{median}(\textrm{EPT})$}
+    \STATE {$\textrm{QHS}_1 \leftarrow \textrm{max}(\textrm{QHS})$}
+    \STATE {$\textrm{QHL}_1 \leftarrow \textrm{max}(\textrm{QHL})$}
+    \STATE {$\textrm{QLX}_1 \leftarrow \textrm{max}(\textrm{QLX})$}
+    \STATE {$\textrm{QHS}_2 \leftarrow \textrm{second highest}(\textrm{QHS})$}
+    \STATE {$\textrm{QHL}_2 \leftarrow \textrm{second highest}(\textrm{QHL})$}
+    \STATE {$\textrm{QLX}_2 \leftarrow \textrm{second highest}(\textrm{QLX})$}
+    \IF{$\textrm{QHS}_1 > \textrm{QHS}_2 + 1000$}
+      \IF{70\% of the normal hit PMTs with good calibration are more than 12 meters from the high charge channel \AND 70\% of the normal hit PMTs with good calibration are more than 50 ns after $t$}
+        \RETURN 1
+      \ENDIF
+    \ELSIF{$\textrm{QHL}_1 > \textrm{QHL}_2 + 1000$}
+      \IF{70\% of the normal hit PMTs with good calibration are more than 12 meters from the high charge channel \AND 70\% of the normal hit PMTs with good calibration are more than 50 ns after $t$}
+        \RETURN 1
+      \ENDIF
+    \ELSIF{$\textrm{QLX}_1 > \textrm{QLX}_2 + 80$}
+      \IF{70\% of the normal hit PMTs with good calibration are more than 12 meters from the high charge channel \AND 70\% of the normal hit PMTs with good calibration are more than 50 ns after $t$}
+        \RETURN 1
+      \ENDIF
+    \ELSE
+      \FOR{All normal PMT channels \emph{not} hit in PC}
+        \IF{more hits in slot than surrounding 4 meters or median hit time in slot is 10 ns earlier than PMTs within 4 meters}
+          \IF{70\% of the normal hit PMTs with good calibration are more than 12 meters from the high charge channel \AND 70\% of the normal hit PMTs with good calibration are more than 50 ns after $t$}
+            \RETURN 1
+          \ENDIF
+        \ENDIF
+      \ENDFOR
+    \ENDIF
+  \ENDFOR
+  \RETURN 0
+\end{algorithmic}
+\end{algorithm}
+
+\subsection{Breakdowns}
+
+Breakdowns are very similar to flashers except that they produce \emph{much}
+more light\footnote{In fact, I think there is a continuous spectrum between
+flashers and breakdowns, but the distinction is still helpful since the way to
+tag the two are very different}.
+
+Since breakdowns often cause many of the electronics to saturate, it is
+\emph{very} difficult to find a single common characteristic on which to cut.
+However, the one thing that does seem to be common among almost all breakdowns
+is that the crate with the channel that breaks down all has pickup from the
+breaking down channel and thus comes much earlier in the event than the rest of
+the PMT hits.
+
+Therefore, the breakdown cut tags any event which has at least 1000 PMT hits
+and in which the crate with the highest median TAC has at least 256 hits and is
+500 TAC counts away from the next highest crate (with at least 20 hits).
+
+Occasionally a breakdown is so big that it causes issues with the TAC
+measurement and many of them end up reading outside of the linear TAC region.
+Therefore we also tag any events in which less than 70\% of the PMT hits have a
+TAC value above 400.
+
+\begin{thebibliography}{9}
+\bibitem{grossman2017}
+    Grossman, et al. \textit{Self-Destructing Dark Matter}. \href{https://arxiv.org/abs/1712.00455}{{\tt arXiv:1712.00455}}. Dec 2017.
+\bibitem{pepin2016}
+    M. Pepin. \textit{Low-Mass Dark Matter Search Results and Radiogenic Backgrounds for the Cryogenic Dark Matter Search}. \url{http://hdl.handle.net/11299/185144}. Dec 2016.
+\bibitem{caldwell2015}
+    T. Caldwell. \textit{Searching for Dark Matter with Single Phase Liquid Argon}. \url{https://repository.upenn.edu/dissertations/1632}. 2015.
+\bibitem{pdg2017}
+    C. Patrignani et al. (Particle Data Group), Chin. Phys. C, 40, 100001 (2016) and 2017 update.
+\bibitem{sonley}
+    T. Sonley. \textit{A Measurement of the Atmospheric Neutrino Flux and Oscillation Parameters at the Sudbury Neutrino Observatory}. Feb 2009. \url{https://www.sno.phy.queensu.ca/sno/papers/Sonley_phd_physics_2009.pdf}. Access Date: Oct 5, 2019.
+\bibitem{snoman_companion}
+    \textit{SNOMAN Companion}. Last updated: Nov. 8, 2006. \url{http://hep.uchicago.edu/~tlatorre/snoman_companion/}. Access Date: Oct 5, 2019.
+\end{thebibliography}
+\end{document}