move sddm.tex to doc/

1 files changed, 0 insertions, 652 deletions

diff --git a/sddm.tex b/sddm.tex
deleted file mode 100644
index 1f3fa96..0000000
--- a/sddm.tex
+++ /dev/null
@@ -1,652 +0,0 @@
-\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}