Add small corrections

chap1.tex

@@ -15,7 +15,7 @@ \chapter{Introduction}
 \begin{pmatrix}
 1 & 0 & 0 \\
 0 & c_{23} & s_{23} \\
-3 & -s_{23} & c_{23}
+0 & -s_{23} & c_{23}
 \end{pmatrix}
 \begin{pmatrix}
 c_{13} & 0 & s_{13}e^{-i\delta} \\
@@ -48,7 +48,7 @@ \chapter{Introduction}
 
 \section{The state of the field}
 \label{sec:StateOfTheField}
-Data provided from a wide range of experiments show excellent agreement with the theory of neutrino oscillation and with a \Yoshi{ADDRESSED - three-}{was ``3''}flavour neutrino picture.  Global fits applied to the data provided by these experiments gives best fit values for the oscillation parameters, which are summarised in table~\ref{table:NeutrinoOscillationParameterValues}~\cite{Agashe:2014kda}.  The experiments which provided the data inputs to the global fit generally fall into one of four categories, with each category sensitive to a different subset of the neutrino oscillation parameters.
+Data provided from a wide range of experiments show excellent agreement with the theory of neutrino oscillation and with a \Yoshi{three-}{ADDRESSED - was ``3''}flavour neutrino picture.  Global fits applied to the data provided by these experiments gives best fit values for the oscillation parameters, which are summarised in table~\ref{table:NeutrinoOscillationParameterValues}~\cite{Agashe:2014kda}.  The experiments which provided the data inputs to the global fit generally fall into one of four categories, with each category sensitive to a different subset of the neutrino oscillation parameters.
 \begin{table}
   \begin{tabular}{l c }
     Parameter & best-fit $(\pm1\sigma)$ \\ \hline \hline

chap2.tex

@@ -9,21 +9,21 @@ \section{Neutrino interactions at the GeV-scale}
 \nu_\mu n \rightarrow \mu^- p.
 \label{eq:CCQEInteraction}
 \end{equation}
-For NCQE interactions, the incident neutrino remains after the interaction has occurred and no nucleon conversion takes place.  Because of this fact, the target nucleon in a NCQE interaction need not be a neutron.  So, for $\nu_\mu$ NCQE interactions, there are two channels available
+For Neutral Current Elastic (NCE) interactions, the incident neutrino remains after the interaction has occurred and no nucleon conversion takes place.  Because of this fact, the target nucleon in a NCE interaction need not be a neutron.  So, for $\nu_\mu$ NCE interactions, there are two channels available
 \begin{equation}
 \nu_\mu n \rightarrow \nu_\mu n,
-\label{eq:NCQEInteractionNeutronTarget}
+\label{eq:NCEInteractionNeutronTarget}
 \end{equation}
 \begin{equation}
 \nu_\mu p \rightarrow \nu_\mu p.
-\label{eq:NCQEInteractionProtonTarget}
+\label{eq:NCEInteractionProtonTarget}
 \end{equation}
 The two kinds of QE interaction are shown in Fig.~\ref{fig:QEFD}\Yoshi{}{ADDRESSED - In the figure, the W isn't really a W$^+$; it could be going either way in time. You can only call it a W}.
 \begin{figure}%
   \centering
   \subfloat[CCQE.]{\includegraphics[width=7.5cm]{images/neutrino_interactions/CCQE_FD.eps} \label{fig:CCQEFD}}
   \hspace{1em}
-  \subfloat[NCQE.]{\includegraphics[width=7.5cm]{images/neutrino_interactions/NCQE_FD.eps} \label{fig:NCQEFD}}
+  \subfloat[NCE.]{\includegraphics[width=7.5cm]{images/neutrino_interactions/NCQE_FD.eps} \label{fig:NCEFD}}
   \caption{Quasi-Elastic (QE) interactions of a $\nu_\mu$ with a nucleon.  The small ellipse represents the neutrino interacting with the nucleon as a whole, rather than with an individual parton.}
   \label{fig:QEFD}
 \end{figure}
@@ -71,7 +71,7 @@ \section{Neutrino interactions at the GeV-scale}
 \newline
 CCQE interactions are experimentally the most interesting and this is the interaction region where most recent measurements have been focused.  Because of the simplicity of the CCQE topology, the interaction can be treated as a two-body scatter.  So, by applying simple conservation rules, the neutrino energy can be kinematically reconstructed.  In such interactions, the nucleon structure is parameterised using a set of form factors, the most interesting of which is the axial-vector form factor, $F_A(Q^2)$. $F_A(Q^2)$ has been, and still is, assumed to take a dipole form
 \begin{equation}
-F_A(Q^2) = \frac{F_A(0)}{(1-Q^2/M_A^2)^2}
+F_A(Q^2) = \frac{F_A(0)}{(1+Q^2/M_A^2)^2}
 \label{eq:FAFormFactor},
 \end{equation}
 where $Q^2$ is the negative of the squared four-momentum transfer of the lepton to the hadron, $F_A(0) = 1.2694\pm0.0028$~\cite{0954-3899-37-7A-075021}, and $M_A$ is known as the axial mass.  Recent measurements of the CCQE cross-section by the MiniBooNE~\cite{PhysRevD.81.092005} and NOMAD~\cite{NOMAD-CCQE} experiments have sparked interest by reporting measured cross-sections which are in tension with one another, the results of which shown in Fig.~\ref{fig:CCQECrossSectionMiniBooNENOMAD}.  

chap7.tex

@@ -1,13 +1,19 @@
 \chapter{Selection of neutrino interactions in the ECal}
 \label{chap:NeutrinoInteractionSelection}
-This analysis presents a measurement of the CC inclusive interaction cross-section of $\nu_\mu$ with lead nuclei using the ND280 \Yoshi{Tracker}{ADDRESSED - I would say that ``Tracker'' is a proper noun, the name of the detector, so it should be capitalised throughout. The detector name is then ``Tracker ECal''} ECals.  To make such a measurement, a sample of neutrino interaction vertices within the ECal must be found.  The selection of events is based on the enhanced reconstruction outlined in chapter~\ref{chap:EnhancedECalReconstruction}, which was specifically designed to \Yoshi{be sensitive to}{ADDRESSED - introduce} track multiplicity.  As a result of this method, vertices in the ECal are naturally separated into topologies defined by the number of reconstructed tracks.  Any selection development should take advantage of this situation and tailor cuts to be specific to each topology, which should result in a higher overall sample purity.  The 3D track matching aspect of the reconstruction was only tuned to handle up to three tracks simultaneously.  However, this analysis aims to measure a CC-inclusive cross-section so it should not be biased against any neutrino energy range.  \Yoshi{There is a deep connection between the number of reconstructed tracks and the energy of the neutrino that created them.  The number of reconstructed tracks should correlate with the number of final state particles involved with the neutrino interaction and the number of final state particles correlates with the energy of the interacting neutrinos.  Therefore it is important to not reject events based solely on the number of reconstructed tracks}{ADDRESSED - state the implied connection between number of prongs and neutrino energy}.  Bearing this information in mind, the selection should separate out the events into the following topologies:
+This analysis presents a measurement of the CC inclusive interaction cross-section of $\nu_\mu$ with lead nuclei using the ND280 \Yoshi{Tracker}{ADDRESSED - I would say that ``Tracker'' is a proper noun, the name of the detector, so it should be capitalised throughout. The detector name is then ``Tracker ECal''} ECals.  To make such a measurement, a sample of neutrino interaction vertices within the ECal must be found.  The selection of events is based on the enhanced reconstruction outlined in chapter~\ref{chap:EnhancedECalReconstruction}, which was specifically designed to \Yoshi{be sensitive to}{ADDRESSED - introduce} track multiplicity.  As a result of this method, vertices in the ECal are naturally separated into topologies defined by the number of reconstructed tracks.  Any selection development should take advantage of this situation and tailor cuts to be specific to each topology, which should result in a higher overall sample purity.  The 3D track matching aspect of the reconstruction was only tuned to handle up to three tracks simultaneously.  However, this analysis aims to measure a CC-inclusive cross-section so it should not be biased against any neutrino energy range.  \Yoshi{There is a deep connection between the number of reconstructed tracks and the energy of the neutrino that created them.  The number of reconstructed ECal tracks  correlates with the number of final state particles involved with the neutrino interaction as shown in Fig.~\ref{fig:ENuVsProngsAllEventsAllECals} and the number of final state particles correlates with the energy of the interacting neutrinos.  Therefore it is important to not reject events based solely on the number of reconstructed tracks}{ADDRESSED - state the implied connection between number of prongs and neutrino energy}.  Bearing this information in mind, the selection should separate out the events into the following topologies:
 \begin{itemize}
   \item 1 prong topology
   \item 2 prong topology
   \item 3 prong topology
   \item 4+ prong topology
 \end{itemize}
-The definition of a prong is a reconstructed track associated with a reconstructed vertex.
+The definition of a prong is a reconstructed track associated with a reconstructed vertex.  It should also be noted that there exists a 0 prong topology in which a neutrino interacts but does not have sufficient energy to produce any recognisable signal in the ECal.  This topology is automatically rejected by the reconstruction.
+\begin{figure}
+  \centering
+  \includegraphics[width=12cm]{images/selection/ENuVsNProngs_AllEvents_AllECals.pdf}
+  \caption{The average energy of neutrino interactions seen in the ECal against the number of reconstructed tracks in the ECal that the neutrino final states created.  The uncertainty on the neutrino energy is calculated as the standard error on mean of the neutrino energy.  There is a clear trend between the two variables.}
+  \label{fig:ENuVsProngsAllEventsAllECals}
+\end{figure}
 \newline
 \newline
 However, as described in section~\ref{sec:ReconOutput}, the output of the enhanced reconstruction has been kept generic and is not specifically tailored to this task.  The reconstruction outputs a set of clusters which contain a set of 3D tracks and every pairwise crossing that said tracks make.  While it is true that in some situations the reconstruction will accurately represent a vertex ``out of the box''---e.g., when only two tracks are reconstructed in the cluster---there will be many situations where extra reconstruction steps are needed before any further selection can take place.  \Yoshi{Hence}{ADDRESSED - I wouldn't start a sentence with ``So,''---it seems like spoken English to me. I'd say ``Hence'' with no comma} the structure of this chapter is as follows: the definition of signal is described first along with the sample used to develop the selection, followed by a discussion of the vertex reconstruction.  After the final reconstruction steps have been discussed, a full discussion of the neutrino selection follows.
@@ -271,7 +277,7 @@ \section{Track merging}
   \phi^{\textrm{merge}} = \epsilon \eta^2
   \label{eqn:TrackMergingTuningMetric}
 \end{equation}
-where $\epsilon$ is the efficiency of the track merging to keep signal events reconstructed as two track vertices and $\eta$ is the purity of the events that remain as two track vertices after merging has taken place.
+where $\epsilon$ is the efficiency of the track merging to keep signal events reconstructed as two track vertices and $\eta$ is the purity of the events that remain as two track vertices after merging has taken place.  \DomTC{The reason for the metric defined in equation~\ref{eqn:TrackMergingTuningMetric}, rather than the more commonly used $\epsilon \eta$, is that the track merging should aim to only merge tracks when there is great evidence that the merge is correct.  Specifically, if the merging algorithm used relaxed parameters and passed more two track events for merging, a large amount of signal events which fell into the 2 prong topology would move to the 1 prong topology, where there is less reconstruction information to separate signal from background}.
 \newline
 \newline
 The first merging condition identified is the cosine of the opening angle, $\cos\theta$, subtended by the two constituent tracks bounded between 0 and 1 which is shown in Fig.~\ref{fig:TrackMergingConditionCosTheta}.  The opening angle is clearly a powerful discriminator.  The distribution for incorrect matches is very flat across the full angular range whereas there is a clear build up of correct matches as $\cos\theta \rightarrow 1$.
@@ -809,7 +815,7 @@ \subsection{Performance of the selection}
   \caption{The number of selected events in the Monte Carlo sample, separated out into the prong topologies.  Each event is categorised by the associated truth information from the simulation.  The effect of the developed selection can be clearly seen in all prong topologies; each topology is now signal dominated.}
   \label{fig:ProngStackSelected}
 \end{figure}
-The number of selected events for each prong topology is shown in Fig.~\ref{fig:ProngStackSelected}, with each topology broken down by truth categories.  For both the barrel and DS ECal, signal events dominate each prong topology.  The most impure topology for both detectors is the 1 prong topology.  This is primarily due to having an insufficient number of prongs to really benefit from what the reconstruction can provide.  Despite this, it is clear that the selection results in a pure sample of events.  
+The number of selected events for each prong topology is shown in Fig.~\ref{fig:ProngStackSelected}, with each topology broken down by truth categories.  For both the barrel and DS ECal, signal events dominate each prong topology.  The most impure topology for both detectors is the 1 prong topology.  This is primarily due to having an insufficient number of prongs to really benefit from what the reconstruction can provide.  Despite this, it is clear that the selection results in a pure sample of events.  \DomTC{The variation in sample purities for each prong topology also highlights the benefit of prong topology separation in the MC selection.  The 2 prong topology in both detectors sees a very high sample purity.  This is because multi-track reconstruction information is available but the number of reconstructed tracks is sufficiently small such that the coarse detector granularity does not significantly mask the track information.  As you move to the higher prong topologies, it becomes harder to discern all of the tracks clearly, resulting in less information available to preciesly reject background}.
 \newline
 \newline
 The selection efficiencies and purities for each prong topology are shown in table~\ref{table:SelEfficiency} and table~\ref{table:SelPurity} respectively.  The final purities and efficiencies are generally good.