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\begin{document}

\title{The Physics Behind No$\nu$a}


\author{Alec Aivazis}
\affil{Affiliation not available}



\date{\today}

\maketitle




\section{Introduction}
In 1934, when the parts of the standard model were first being pieced together, Enrico Fermi introduced a massless particle called a neutrino that is fermionic in nature and does not ineract with baryonic matter, in order to explain how beta decay could convserve fundamental quantities (energy, spin, etc.) \cite{Wilson_1968}. For a while, only electron neutrinos were thought to exist. However, fifty years later in 1988, Lederman, Schwartz, and Steinberger earned the Nobel Prize in physics for work they did in 1962 at the Alternating Gradient Synchotron at the Brookhaven National Laboratory. In their paper, the group from Columbia reported that they had found a second kind of neutrino that did not couple to the electron like the one proposed by Fermi, but instead to muons produced by their beam in upstate New York\cite{Danby_1962}. Another forty years passed before the third generation of neutrino was dicovered in 2000 by the DONUT collaboration at Fermilab near Chicago, Illinois\cite{Kodama_2001}. For a long while, these various "flavors" of neutrinos were thought to not couple with anything apart from their respective fermion. However, people reasoned that it wasn't impossible that these neutrinos could interact with other forms of matter.  


\subsection{Neutrino Oscillations}\label{sect:osc-motiv}
In 1958, Bruno Pontecorvo, an assistant of Fermi's, suggested that if neutrinos did in fact have a mass (unlike what Fermi claimed) then the neutrinos we encounter might be a particle mixture of more fundamental mass states. Consequently, Pontecorvo argued, there is some probability of transitioning between neutrino and its associated anti-neutrino \cite{Pontecorvo_1957}. Around the same time, a variety of experiments reported similar discrepancies between the measured number of neutrinos created in solar rays and what they expected.

Coupled with the discovery of the muon neutrino, Pontecorvo accredited this so-called "solar neutrino problem" to the oscillation of neutrinos between various flavor states due to a non zero mass. In 1967 he wrote that "from the point of view of detection possibilities ... if the oscillation length is much smaller than the radius of the solar region which effectively produces neutrinos ... it will be impossible to detect directly oscillations of the solar neutrino [with current technology]." He continues to say that "the only effect at the surface of the earth would consist in the fact that the flux of observable solar neutrinos would be half as large as the total flux of solar neutrinos" \cite{Pontecorvo_1968}. It was not until 1998 that the oscillation of neutrinos between various states was observed by the Super-Kamiokande Collaboration in Japan, giving substantial evidence to believe that neutrinos do in fact have a non-zero mass like Pontecorvo suggested 40 years earlier \cite{Fukuda_1998}.


\subsection{The NO$\nu$A Experiment}
NO$\nu$A is the latest in the long, rich history of particle detectors specialized to study neutrinos. It is a long baseline experiment managed by the Fermi National Accelerator Laboratory (Fermilab) in Batavia, Illinois and takes advantage of the NuMI (Neutrinos at Main Injector) neutrino beam that was constructed for the MINOS project. With the start of NO$\nu$A, along with other improvements, the NuMI beam was upgraded to nearly twice the power with a new graphite target and magnetic horns to provide a narrow-band neutrino beam with a high intensity whose energy peaks at the maximum probability for neutrino oscillation \cite{Paley_2012}.

\section{Underlying Physics}\label{sect:theory}
For NO$\nu$A, the process being analyzed is 
\begin{equation}
\nu_{\mu} \rightarrow \nu_{e}
\end{equation}
and since a non zero $\theta_{13}$ was recently measured at its predecessor, \footnote{see \cite{Adamson_2011} for more information} the experiment will also see

\begin{equation}
\overline{\nu_{\mu}} \rightarrow \overline{\nu_{e}}
\end{equation}

The beam of neutrinos being used is an "off-axis" \footnote{Fore more information see section \ref{sect:osc-twoflavor}} beam created by colliding protons from Fermilab's Main Injector into a long cylindrical graphite target. While many different particles are generated during these collisions, of most interest are the postive pions that emerge, as seen in figure \ref{fig:pionprod}.
 
 
These pions are selected out of the heap of products using a carefully calibrated magnetic field. After making it past the selector, the relatively short lived particles decay into muon and muon neutrinos, which correspond to the Feynman fiagram seen in figure \ref{fig:piondecay}.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/pionproduction/pionproduction}
\caption{{\label{fig:pionprod}
Pion production via the collision of two protons. Image courtesy of  Lebiedowicz, Piotr et al.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.56\columnwidth]{figures/feynman1/feynman1}
\caption{{\label{fig:piondecay}
The Feynman Diagram showing a pion decaying into a muon and a neutrino through a W boson.%
}}
\end{center}
\end{figure}

\subsection{Kinematics of an Off-Axis Neutrino Beam \footnote {These derivations are adapted from \cite{McDonald_2001}}}

\subsubsection{The Decay}

The relevant decay for the NuMI beam is

\begin{equation}
\pi^{+} \rightarrow \mu^{+} \nu_{\mu}
\end{equation}

Which must conserved energy and momentum according to the 4 vector equation:

\begin{equation} \label{eq:energy-momentum}
\mathbf{\pi^{+}} = \mathbf{\mu^{+}} + \mathbf{\nu_{\mu}}
\end{equation}

Where $\mathbf{\pi^{+}}$, $\mathbf{\mu^{+}}$, and $\mathbf{\nu_{\mu}}$ are the energy-momentum 4-vectors.

Rearranging equation \ref{eq:energy-momentum} as

\begin{equation}
\mathbf{\mu^{+}}  = \mathbf{\pi^{+}} - \mathbf{\nu_{\mu}}
\end{equation}

And squaring both sides, we get

\begin{equation} \label{eq:energy-momentum-expanded}
\mathbf{\mu^{+}}^2  = \mathbf{\pi^{+}}^2 - 2 ( \mathbf{\pi^{+}} \cdot \mathbf{\nu_{\mu}} ) - \mathbf{\nu_{\mu}}^2
\end{equation}

It is important to note that since the magnitude of a 4-vector is a Lorenz invariant, for any energy-momentum 4 vector $\mathbf{p}$, where

\begin{equation}
\mathbf{p} = (E, \vec{p}  )
\end{equation}

one can boost to the rest frame of the particle where $\vec{p} = 0$ and a find the value for $\mid \mathbf{p} \mid ^2$. Since these particles are moving close to the speed of light, in natural units,

\begin{equation}
E^2 = \vec{p}^2 + m^2 
\end{equation}

Then in the rest frame of the particle,

\begin{equation}
E^2 = m^2
\end{equation}

Therefore,

\begin{equation}
\begin{split}
\mid \mathbf{p} \mid^2 &= E^2 + p^2 \\
                       &= m^2
\end{split}
\end{equation}

The neutrinos mass is many orders of magnitude smaller than that of the other particles \cite{Robertson_2008}; to good approximation,

\begin{equation}
\begin{split}
&\mathbf{\nu_{\mu}}^2 = m_{\nu}^2  \\
\implies  &\mathbf{\nu_{\mu}}^2 \approx 0
\end{split}
\end{equation}

Plugging these results into equation \ref{eq:energy-momentum-expanded} we get

\begin{equation} \label{eq:energy-momentum-expanded-more}
m_{\mu}^2 = m_{\pi}^2 - 2(\mathbf{\pi^{+}} \cdot \mathbf{\nu_{\mu}})
\end{equation}

In the rest frame of the pion, with z as the direction of travel for the nutrino

\begin{equation}
\begin{split}
\mathbf{\pi^{+}} &= (m_{\pi} , 0, 0, 0) \\
\mathbf{\nu_{\mu}} &= (E_{\nu}', E_{\nu}' \sin \theta, 0, E_{\nu}' \cos \theta)
\end{split}
\end{equation}

Therefore, the dot-product is given by

\begin{equation}
( \pi \cdot \nu ) = m_{\pi} E_{\nu}'
\end{equation}

Plugging this into equation \ref{eq:energy-momentum-expanded-more},

\begin{equation}
\begin{split}
m_{\nu}^2 = m_{\pi}^2 - 2 m_{\pi}E_{\nu}'
\end{split}
\end{equation}

and solving for the energy of the neutrino in the rest frame of the pion,

\begin{equation}
\begin{split}
E_{\nu}' &= \frac{m_{\pi}^2 - m_{\mu}^2}{2 m_{\pi}} \\
         &= 29.8 MeV
\end{split}
\end{equation}

\subsubsection{Neutrino Decay Angle}

To understand what happens during the decay in the frame of reference of the lab, we need to boost in the beam direction, taken to be in the z-direction. The Lorentz Boost in the z-direction is given by:

\begin{equation}
\begin{bmatrix}
ct' \\
x' \\
y' \\
z' \\
\end{bmatrix} = \begin{bmatrix}
\gamma          & 0 & 0 & - \beta \gamma \\
0               & 1 & 0 & 0              \\
0               & 0 & 1 & 0              \\
-\beta \gamma   & 0 & 0 & \gamma         \\
\end{bmatrix} \begin{bmatrix}
ct \\
x \\
y \\
z \\
\end{bmatrix}
\end{equation} 

With the boost factor $\gamma$ given by

\begin{equation}
\gamma = \frac{E_{\pi}}{m_{\pi}}
\end{equation}

and the velocity $\beta$ given by

Applying this boost to the neutrino 4-vector,

\begin{equation}
\nu_{\rm{lab}} = (\gamma E_{\nu}'(1+\beta \cos \theta), E_{\nu}' \sin \theta, 0, \gamma E_{\nu}' (\beta + \cos \theta))
\end{equation}

Looking at a diagram of the particle decay shows that the neutrino ejects from the collision vertex at an angle $\theta$, where

\begin{equation}
\begin{split}
\tan \theta &= \frac{\nu_{y}}{\nu_{z}} \\
&= \frac{E_{\nu}' \sin \theta '}{\gamma E_{\nu}'(\beta+\cos \theta ')} 
\end{split}
\end{equation}

If we consider the case where the beam has much more energy than the relative masses, then $E_{\pi} \gg m_{\pi}$, $\beta \approx 1$, and $\gamma \gg 1$. In this case, the previous equation becomes

\begin{equation} \label{eq:tan}
\tan \theta \approx \frac{E_{\nu}' \sin \theta'}{E_{\nu}}
\end{equation}

\subsubsection{Neutrino Flux}\label{sect:flux}

Since NO$\nu$A is an off-axis neutrino detector, of most interest to this paper is the total flux of neutrinos at a given energy and angle.

According to \cite{McDonald_2001} since the pion is a spin zero particle, the decay is isotropic in the pion rest frame. Therefore, the volume elements can be seen as concentric cylinders to integrate over. 

Assuming the total neutrino flux is proportional to the $\cos$ of the angle with which they are created and the energy of the beam,

\begin{equation}
\begin{split}
& d^2 N \propto d \cos \theta d E_{\nu} \\
\implies & \frac{d^2 N}{d \cos \theta' d E_{\nu}} \propto 1
\end{split}
\end{equation}

From the point of view of an observer in the lab, the total flux at a given angle and energy can be found using the relation

\begin{equation}\label{eq:money}
\frac{d^2 N}{d \cos \theta d E_{\nu}} = \frac{d^2 N}{d \cos \theta' d E_{\nu}} \frac{d \cos \theta'}{d \cos \theta}
\end{equation}

To understand the behavior of the nutrino flux, notice that

\begin{equation}
\cos \theta = \sqrt{1 - \sin^2 \theta} 
\end{equation}

and from equation \ref{eq:tan} 

\begin{equation}
\sin \theta' \approx \frac{E_{\nu}}{E_{\nu}'} \tan \theta
\end{equation}

Therefore,

\begin{equation}
\begin{split}
\cos \theta' & \approx \sqrt{1-\frac{E_{\nu}^2}{E_{\nu}^2}\tan^2 \theta} \\
             & = \sqrt{1-\frac{E_{\nu}^2}{E_{\nu}^2} \left(\frac{1}{\cos^2  \theta}-1\right)}
\end{split}
\end{equation}

Therefore,
\begin{equation}
\frac{\partial \cos \theta '}{\partial \cos \theta} \approx \frac{E_{\nu}^2}{E_{\nu}^2 \cos \theta'}
\end{equation}

Plugging this into equation \ref{eq:money} its clear that 

\begin{equation}
\frac{d^2 N}{d \cos \theta d E_{\nu}} \propto \frac{E_{\nu}^2}{E_{\nu}^2 \cos \theta'}
\end{equation}

A plot of this quantity can be see in figure \ref{fig:flux-money}. From this plot its clear that while the detector does lose some flux of neutrinos by being slightly off angle, there is a much higher peak in the distribution of neutrinos which allows for much higher sensitivity in measurement.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/fixed-angle/fixed-angle}
\caption{{\label{fig:flux-money}
A plot of the relative flux of neutrinos at various angles off of the beam axis. Photo courtesy of \protect\cite{Levy_2010}%
}}
\end{center}
\end{figure}

\subsection{Neutrino Oscillation}\label{sect:osc-theory}

The motivation for neutrino oscillation was discused in section \ref{sect:osc-motiv}. To see how a neutrino actually "oscillates," lets assume, that we have a neutrino that intially has definite flavor. That is

\begin{equation}
\mid \nu (0) \rangle =  A \mid \nu_{\alpha} \rangle
\end{equation}

Where $\alpha$ ranges over (e, $\mu$, and $\tau$) and stand for the respective flavor state.

Since $\mid \nu_{\alpha} \rangle$ are eigenstates (OF WHAT?), $\mid \nu(t) \rangle$ is given in natural units by

\begin{equation}
\mid \nu (x, t) \rangle = A e^{- i (E_{\alpha} t - \vec{p_{\alpha}} \cdot \vec{x})} \mid \nu_{\alpha} (0) \rangle
\end{equation}

To find the energy of the neutrino, we use the fact that they are traveling close to the speed of light. In this limit, $p_{\alpha} \gg m_{\alpha}$ and we can approximate the energy of the neutrinos using the relativistic equation

\begin{equation}
\begin{split}
E_{\alpha}^2 &= p_{\alpha}^2 + m_{\alpha}^2 \\
             &= p_{\alpha}^2 (1 + \frac{m_{\alpha}^2}{p_{\alpha}^2}) \\
E_{\alpha}   &= p_{\alpha} \sqrt{(1 + \frac{m_{\alpha}^2}{p_{\alpha}^2})} \\
             &\approx p_{\alpha} (1 + \frac{m_{\alpha}^2}{2 p_{\alpha}^2}) \\
             &= p_{\alpha} + \frac{m_{\alpha}^2}{2 p_{\alpha}}
\end{split}
\end{equation}

Since $E^2 = p^2 + m^2$ and $p_{\alpha} \gg m_{\alpha}$, 

\begin{equation}
\begin{split}
p_{\alpha} &\approx E \\
\implies E_{\alpha} &\approx p_{\alpha} + \frac{m_{\alpha}^2}{2 E}
\end{split}
\end{equation}

Plugging this into our expression for $\mid \nu_{\alpha}(x, t) \rangle$

\begin{equation}
\begin{split}
\mid \nu_{\alpha} (x, t) \rangle &= A e^{- i (E_{\alpha} t - \vec{p_{\alpha}} \cdot \vec{x})} \mid \nu_{\alpha} (0) \rangle \\
&= A e^{-i(p_{\alpha} + \frac{m_{\alpha}^2}{2 E})t} e^{- \vec{p_{\alpha}} \cdot \vec{x}} \mid \nu_{\alpha}(0) \rangle \\
&= A e^{- i E_{\alpha} t}e^{i (E_{\alpha} - \frac{m_{\alpha}^2}{2 E}) \cdot \vec{x})} \mid \nu_{\alpha}(0) \rangle \\ 
&\approx A e^{- i E_{\alpha} t}e^{ -i \frac{m_{\alpha}^2}{2 E} \cdot \vec{x}} \mid \nu_{\alpha}(0) \rangle \\ 
\end{split}
\end{equation}

Ignoring the time-dependent phase component, the wavefunction for a neutrino as it travels down the particle accelerator is

\begin{equation}
\mid\nu_{\alpha}(x)\rangle = A e^{-i m_{i}^2 x /2E} \mid \nu_{\alpha}(0) \rangle
\end{equation}

Therefore, the probability to transition from one flavor state to another is given by

\begin{equation}\label{eq:prob-transition}
\begin{split}
P(\nu_{\alpha} \rightarrow \nu_{\beta}) & = \mid \langle \nu_{\alpha} \mid \nu_{\beta} (t) \rangle \mid ^2 \\
& = \mid A_{\alpha}^{*}A_{\beta} e^{-i \cdot m_{\alpha}^2 t /2E} \mid^2 \\
& \equiv \mid U_{\alpha \beta} e^{-i \cdot m_{\alpha}^2 t /2E} \mid^2
\end{split}
\end{equation}

Where $U_{\alpha \beta}$ is a unitary matrix called the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix \cite{Kopp_2006}. As of 2012, the values for this matrix have been experimentally measured to be \cite{Fogli_2012}:

\begin{equation}
U_{\alpha \beta} = \begin{bmatrix}
0.82 & 0.54 & -0.15 \\
-0.35 & 0.70 & 0.61 \\
0.44 & -0.45 & 0.77
\end{bmatrix}
\end{equation}

\subsubsection{Two Flavor Oscillations}\label{sect:osc-twoflavor}

Since $U_{\alpha \beta}$ is a unitary matrix, in the case of two flavor oscillations, it can be seen as a rotation matrix of the form

\begin{equation}
U_{\alpha \beta} = \begin{bmatrix}
\cos \theta & \sin \theta \\
-\sin \theta & \cos \theta \\
\end{bmatrix}
\end{equation}

and equation \ref{eq:prob-transition} turns into:

\begin{equation} \label{eq:twoflavor-result}
\begin{split}
P(\nu_{\mu} \rightarrow \nu_{e}) &= \mid -\sin \theta \cos \theta e^{- im_{i}^2L/2E} + \cos \theta \sin \theta e^{- i m_{i}^2 L / 2 E} \mid ^2 \\
&= \sin^2 \theta \cos^2 \theta \left[ 2 - \cos \frac{(m_{2}^2 - m_{1}^2)L}{2E}\right] \\
&= \sin^2 2\theta \sin^2 \frac{L(m_{1}^2 - m_{2}^2)}{4 E}
\end{split}
\end{equation}

Equation \ref{eq:twoflavor-result} shows that there is a maximum probability for the neutrino to oscillate when it has travelled a distance $L = 4 \pi E / (m_{1}^2 - m_{2}^2)$ \cite{Kopp_2006}.

\subsubsection{CP Violation}\label{sect:cp}

Neutrino oscillations are also useful as probes of the degree to which nature conserves charge and parity. 

While at NO$\nu$A they are looking at $\mu^{+}$ decays into muon neutrinos, if there is no CP violation and nature perfectly conserves charge and parity, then the data gathered at NO$\nu$A could apply equally well to positive or negative muons. 

To encompass this, equation \ref{eq:twoflavor-result} is modified to

\begin{equation}
P(\nu_{\mu} \rightarrow \nu_{e}) = \sin^2 2\theta \sin^2 \frac{L(m_{1}^2 - m_{2}^2)}{4 E} \sin \delta
\end{equation}

where $\delta$ is the "CP violating phase," a measure of the amount by which charge and parity are violated. In the limit that these quantities are conserved, $\delta$ goes to zero.

\section{The NO$\nu$A experiment}

NO$\nu$A is a long baseline experiment based at Fermilab that will emit a concentrated neutrino beam to a 14-kiloton detector in Ash River, Minesota in order to study the phenomenma discussed in section \ref{sect:theory}. 

\subsection{Neutrino source}

The NuMI beam used by NO$\nu$A is fueled by proton collisions that are harnessed by an 8 GeV Booster accelerator and stored in the Main Injector ring. Small portions of the Main Injector beam are diverted using 6 sets of magnets and directed down a 350m long path to the NuMI sector where it is focused to a graphite cylinder with dimensions of $6.4\rm{x}15\rm{x}940 \rm{mm}^3$. At the point of contact, the beam is 1mm in diameter. The graphite target is separated into 47 different fins each individually cooled in a stainless steel vacuum canister \cite{Patterson_2012}. 

The pions that are emitted as a result of the decay are focused by two magnetic horns with maximum of 30 kG of magnetic field, produced by 200 kA of current, direct the positive pions into a 675m long steel pipe, 2m in diameter and evacuated out to ~1 Torr. At the end of the pipe is a beam absorber consisting of water-cooled aluminum cells surrounding a 1m steel block \cite{Patterson_2012}.

For increased adjustability, the target is mounted on a rail system that allows for around 8 feet of movement in the beam direction \cite{Kopp_2004}. Following the decay outlined in section \ref{sect:theory}, the pions that make it through eventually decay into a muon and a muon neutrino. Since there is so much more energy in the beam direction than the rest mass of the particles, the pair can be treated as moving along the beam direction \cite{Patterson_2012}. 



\subsection{The Detectors} \label{sect:detectors}

NO$\nu$A features two detectors in order to reduce uncertainties in neutrino flux, cross section, and event selection efficiencies \cite{Kopp_2004}. The Near Detector is in a cavern close to the existing MINOS Near Detector Hall at Fermilab and is made up of a 206 layers of a lattice made up of highly reflective PVC filled with liquid scintillator. As the neutrino passes through the liquid, it decays and produces a flash of light. This flash is collected and transmitted to the end of the hall by a specially designed fiber optic cable. The far detector uses similar technology but is considerably bigger, with 928 layers of PVC 15.6m wide. See figure \ref{fig:detectors} for size comparison. The cells that make up these layers are designed to make it easier to detect $\pi^0$ decays by using the gap between the position where the neutrino interacted with the liquid and the photon was recorded \cite{Patterson_2012}.

The Far Detector is located 14 mrad off of the NuMI beam axis, exposing the measurements to a relatively narrow band of neutrino flux centered around 2 GeV, the maximum probability for a neutrino to oscillate \footnote{See sections  \ref{sect:flux} and \ref{sect:osc-theory} for more information}. The relation between neutrino flux and energy is summarized in figure \ref{fig:flux-energy}. While the flux is maximized on-axis, peaked at 7 GeV, the sharpness of the peak around 2 GeV gives a much higher certainty when performing measurements as mentioned before and in section \ref{sect:flux} \cite{Patterson_2012}.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.56\columnwidth]{figures/nova-detectors/nova-detectors}
\caption{{\label{fig:detectors}
The different detectors for size comparison. The Near Detector has 206 layers of the PVC scintillator shown in the circle and the Far Detector has 928. Image owned by Fermilab. http://www-nova.fnal.gov/images/nova-detectors.png%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/flux-energy1/flux-energy1}
\caption{{\label{fig:flux-energy}
The flux of neutrinos detected from the NumI beam at various fixed angles away from the beam axis. It is important to note that while the flux is maximized on-axis, the sharpness of the peak centered around 2 GeV gives a good balance between flux and certainty in measurement as well as a maximal probability of muon oscillation. Image courtesy of Fermi National Accelerator Laboratory.%
}}
\end{center}
\end{figure}

\subsection{Upgrades to NuMI}
As part of the NO$\nu$A project, a series of upgrades were installed in the NuMI beam infrastructure to provide a higher intensity neutrino beam than possible before. The first of these is an adaptation of the Recylcer ring from the recently concluded collider run, which used to store anti-protons, to be used as a pre-injector to the Main Injector beam. This reduces the power up delay from 2.2 seconds to 1.3, allowing the Main Injector to ramp up at the same time as the next injection to the loop \cite{Patterson_2012}, resulting in an 80\% increase in the beam power while only increasing the number of protons per bunch by 10\%. In order to support this addition to the Main Injector, the magnets used to deflect the NuMI line must be upgraded to support the additional beam power. Furthermore, the target was upgraded with a new cooling system. 

There are two possible upgrades that are being discussed at Fermilab to NO$\nu$A. The first, "SNuMI", uses the Accumulator ring to stack multiple proton packets from the booster ring in order to increase the beam flux. This would increase the NuMI beam to 1.2 MW. The second, dubbed "Project X" is to replace the Booster ring all togther with a proton linear acceleration to be built which would increase the beam intensity to 2.3 MW \cite{fermilab_nova}


\section{Results}

\subsection{Background Estimation}

The goal of NO$\nu$A is to distinguish electron neutrinos from the muon neutrinos in the beam; therefore, it is important to estimate the amount of background measurements and false positives expected in the data. 

Since the Near Detector (ND) is so close to the NuMI beam, it is able to measure the particle before it has had a chance to oscillate or interact with other matter, providing an idea of what results are reasonable in the Far Detector (FD). The data recorded by the ND is separated into 4 different "channels" that are treated seprately. These channels are: muon neutrinos, charged interactions, neutral interactions, and neutral interactions with small $\nu_{e}$ behavior. 

Muon neutrinos travel much farther in the detector than electron, making them considerably easier to detect. See figure \ref{fig:tracks} for more details. From this, its clear that charged interactions do not make up a large amount of the background measurements. The largest source is from neutral currents whose hadronic showers are occasionally misidentified as an electromagnetic shower similar to that of a muon neutrino \cite{Sachdev_2013}. 

One method, outlined in \cite{Sachdev_2013}, estimates the amount of hadronic showers produced by a neutral interaction by removing from data those events where the Near Detector identified a muon. The muon is identified using a muon particle identifacation (PID) algorithm that is based on the rate at which a particle looses energy as it passes through an individual plane in the detector. It is important to note that sometimes these particles can be misidentified as muons. 

This "Muon-Removed Charged Current" provides a channel that does not contain the main event to be measured by the detectors and provides a data-based way to determine which events would produced charged electromagnetic showers and be identified as an oscillated muon neutrino by the Far Detector even though the Near Detector did not measure it as a muon at the beginning. For an illusration of this technique, see figure \ref{fig:mrcc}.

\subsection{Recent Results}\label{sect:results}

On February 11, 2014 Fermilab announced that NO$\nu$A detected its first signal of a neutrino in the Far Detector, before the detector was finished being built. For the image of the detector track, see figure \ref{fig:results}. While it is too early to draw conclusions from the data, the highly specialized design of the detector arrangement\footnote{see sections \ref{sect:detectors} and \ref{sect:osc-theory} for more information} coupled with the power of ones ability to manipulate the data for higher resolution will give NO$\nu$A a very high level of sensitivity to the neutrino mass hierarchy as well as the CP violating phase \footnote{see section \ref{sect:osc-theory} for more information}

Figure \ref{fig:results-cp} shows the probability of $\nu_{\mu} \rightarrow \nu_{e}$ versus $\overline{\nu_{\mu}} \rightarrow \overline{\nu_{e}}$ at a fixed neutrino energy of 2 GeV. The solid lines represent the possible values as allowed by the standard model and the dotted lines are the neccessary values to fit current solar neutrino data \cite{Friedland_2012}. 

This figure shows a very high level of dependence on the measure probability for $\nu_{\mu} \rightarrow \nu_{e}$ versus $\overline{\nu_{\mu}} \rightarrow \overline{\nu_{e}}$ indicating that the measurements taken at NO$\nu$A are highly dependent on the CP violating phase.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/tracks/tracks}
\caption{{\label{fig:tracks}
The track of various channels in the NO$\nu$A detector.  Each colored point indicates the location of a measurement in a layer of the detector\footnote{see \ref{sect:detectors} for more information}. From this, its clear that electron neutrino currents (center) are not a major background to the muon neutrinos being measured in the Near Detector (top). Since the its goal is to measure the dissappearance of muon neutrinos, this makes the Near Detector a very good way to estimate what would be expected in the Far Detector. Image courtesy of The American Physics Society \protect\cite{Sachdev_2013}.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/mrcc2/mrcc2}
\caption{{\label{fig:mrcc}
An illustration of the Muon Removed Charged Current technique for data driven background estimation. The left diagram is the charged current measured by the Far Detector. By looking at the data without the the muon event (center), one has a good sense of the neutral current present in the data (right). By taking this into account, NO$\nu$A is able to have a higher sensitivity to neutrino oscillations. Image courtesy of the America Physics Society \protect\cite{Sachdev_2013}.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/results/results}
\caption{{\label{fig:results}
The first measurement of a neutrino as it passes through the detectors at NO$\nu$A, announced on February 11, 2014. Image couresy of Fermi National Accelerator Laboratory.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/results-cp/results-cp}
\caption{{\label{fig:results-cp}
This plot shows the probability of $\nu_{\mu} \rightarrow \nu_{e}$ versus $\overline{\nu_{\mu}} \rightarrow \overline{\nu_{e}}$ at a fixed energy of 2GeV as measured by NO$\nu$A. The solid lines represent the values allowed by Standard Model vertices, while the dotted lines have parameters set to fit current solar neutrino data. This diagram shows a large degree in CP violation and indicates that the NO$\nu$A experiment is very dependent on this quantity. Image courtesy of \protect\cite{Friedland_2012}.%
}}
\end{center}
\end{figure}

ectionpagebreak
\section{Conclusions}

With the recent announcement of its first neutrino sightings\footnote{see section \ref{sect:results} for more information}, its clear that NO$\nu$A will bring with it a new level accuracy in experimental neutrino physics. It is the first of its kind to be long enough in order to observe matter effects on neutrino oscillations, in order to uncover more information about the mass hierarchy of the various neutrino flavors. However, as discussed in section \ref{sect:cp} and shown in figure \ref{fig:results-cp}, the probability of oscillating between various flavors is highly dependent on the amount by which charge and parirty conservation hold, therefore NO$\nu$A is actually studying two phenomenon at once. While this is not necessarily bad, it makes it hard to obtain unequivocal data concerning the neutrino mass hierarchy. 

In order to combat, this \cite{Ghosh_2013} suggests using the data from other neutrino experiements like T2K, INO, and other reactor experiments to accent the data coming from NO$\nu$A in order to obtain better information about solar neutrinos since their propagation is not as highly dependent on the CP violating phase. Regardless of the use of the data, it is clear that NO$\nu$A provides a neccessary piece of the experimental puzzle to understanding neutrino interactions.

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