Complete chapter 2.

Proof read chapter 1 and chapter 2 which are now in first draft.

chap2.tex

@@ -11,16 +11,16 @@
 
 \chapter{Neutrino interactions with atomic nuclei}
 \label{chap:NeutrinoInteractionsAtomicNuclei}
-The neutrino is a strictly weakly interacting particle.  This has difficult implications for any experiment aiming to study neutrinos as particle detectors generally rely on the electromagnetic force.  In fact, the only proven method of neutrino detection is to utilise a high mass target in which the neutrinos can interact with.  Generally speaking, charged particles are produced by this interaction which can be detected by the usual means.  The collected information from these charged final states can then be used to infer information about the incident neutrino.  All neutrino experiments rely on this method and so any attempted measurements (e.g. $\delta$) rely on our understanding on neutrino interactions with atomic nuclei.  Our understanding of such processes is emcompassed in the models we use to simulate the interactions.
+The neutrino is a strictly weakly interacting particle.  This has difficult implications for any experiment aiming to study neutrinos as particle detectors generally rely on the electromagnetic force.  In fact, the only proven method of neutrino detection is to utilise a high mass target in which the neutrinos can interact with.  Generally speaking, charged particles are produced by this interaction which can be detected by the usual means.  The collected information from these charged final states can then be used to infer information about the incident neutrino.  All neutrino experiments rely on this method and so any attempted measurements (e.g. $\delta$) rely on our understanding on neutrino interactions with atomic nuclei.  Our understanding of such processes is encompassed in the models we use to simulate the interactions.
 
 \section{Neutrino interactions at the GeV-scale}
 \label{sec:NeutrinoInteractionsGeVScale}
-As the neutrino is weakly interacting, there are two channels available to a neutrino interacting with a nucleon: the Charge Current (CC) interaction in which are W boson is exchanged and the Neutral Current (NC) interaction in which a Z boson is exchanged.  For neutrino energies below $\sim$1~GeV, the neutrino-hadron interactions are largely Quasi-Elastic (QE)~\cite{RevModPhys.84.1307}.  In such an interaction, the incident neutrino scatters of the nucleon as if it were a single particle, rather than with one of the nucleon's constituent partons.  In the case of a CCQE interaction, the neutrino is converted into its charged lepton equivalent and the target neutron converted to a proton.  In the specific case of an incident $\nu_\mu$, the interaction takes the following form
+As the neutrino is weakly interacting, there are two channels available to a neutrino interacting with a nucleon: the Charge Current (CC) interaction in which a W boson is exchanged and the Neutral Current (NC) interaction in which a Z boson is exchanged.  For neutrino energies below $\sim$1~GeV, the neutrino-hadron interactions are largely Quasi-Elastic (QE)~\cite{RevModPhys.84.1307}.  In such an interaction, the incident neutrino scatters of the nucleon as if it were a single particle, rather than with one of the nucleon's constituent partons.  In the case of a CCQE interaction, the neutrino is converted into its charged lepton equivalent and the target neutron converted to a proton.  In the specific case of an incident $\nu_\mu$, the interaction takes the following form
 \begin{equation}
-\nu_\mu n \rightarrow \mu^- p
+\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 coversion 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 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
 \begin{equation}
 \nu_\mu n \rightarrow \nu_\mu n,
 \label{eq:NCQEInteractionNeutronTarget}
@@ -34,7 +34,7 @@ \section{Neutrino interactions at the GeV-scale}
   \centering
   \subfloat[CCQE.]{\includegraphics[width=8cm]{images/neutrino_interactions/CCQE_FD.eps} \label{fig:CCQEFD}}
   \subfloat[NCQE.]{\includegraphics[width=8cm]{images/neutrino_interactions/NCQE_FD.eps} \label{fig:NCQEFD}}
-  \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 individial parton.}
+  \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}
 \newline
@@ -47,7 +47,7 @@ \section{Neutrino interactions at the GeV-scale}
 \begin{equation}
 N^{*} \rightarrow \pi N',
 \end{equation}
-where $N, N' = n, p$.  An example diagram of $\nu_\mu$-CCRES interaction with a $\pi^+$ in the final state is shown in Fig.~\ref{fig:CCRESFD}.
+where $N, N' = n, p$.  An example diagram of a $\nu_\mu$-CCRES interaction with a $\pi^+$ in the final state is shown in Fig.~\ref{fig:CCRESFD}.
 \begin{figure}%
   \centering
   \includegraphics[width=8cm]{images/neutrino_interactions/CCRES_FD.eps}
@@ -56,7 +56,7 @@ \section{Neutrino interactions at the GeV-scale}
 \end{figure}
 \newline
 \newline
-For neutrinos with energy above the RES-dominant region, the neutrino has enough energy to penetrate the nucleon and scatter off an individual quark.  Because of the nature of the strong force, the scattered quark, and the nucleon remnant, produce a hadronic shower in the final state.  This process is known as Deep Inelastic Scattering (DIS).  For $\nu_\mu$-CCDIS, the interaction takes the following form
+For neutrinos with energy above the RES-dominant region, the neutrino has enough energy to penetrate the nucleon and scatter off an individual quark.  Because of the nature of the strong force, the scattered quark and the nucleon remnant produce a hadronic shower in the final state.  This process is known as Deep Inelastic Scattering (DIS).  For $\nu_\mu$-CCDIS, the interaction takes the following form
 \begin{equation}
 \nu_\mu N \rightarrow \mu^- X,
 \label{eq:CCDIS}
@@ -70,16 +70,16 @@ \section{Neutrino interactions at the GeV-scale}
 \end{figure}
 \newline
 \newline
-While the value of a particular interaction cross-section should depend on the nuclear environment, it is possible to make comparisons of the measured cross-section per nucleon.  Fig.~\ref{fig:CrossSectionMeasurements} shows a comparison of $\nu_\mu$ CC cross-section measurements per nucleon from different experiments, all of which sample a different neutrino energy range.  There are large uncertainties for many of the cross-section meausrements, particularly for the ones sampling the lower energy ranges.  The T2K beam energy is $\sim$700~MeV, which sits in the region of higher uncertainty.
-\begin{figure}%
+While the value of a particular interaction cross-section should depend on the nuclear environment, it is possible to make comparisons of the measured cross-section per nucleon.  Fig.~\ref{fig:CrossSectionMeasurements} shows a comparison of $\nu_\mu$ CC cross-section measurements per nucleon from different experiments, all of which sample a different neutrino energy range.  There are large uncertainties for many of the cross-section measurements, particularly for the ones sampling the lower energy ranges.  The T2K beam energy is $\sim$700~MeV, which sits in the region of higher uncertainty.
+\begin{figure}[b]%
   \centering
   \includegraphics[width=8cm]{images/neutrino_interactions/CrossSectionMeasurements.pdf}
-  \caption{$\nu_mu$ CC cross-section measurements per nucleon for a range of energies, showing the QE, RES and DIS contributions~\cite{RevModPhys.84.1307}.}
+  \caption{$\nu_\mu$ CC cross-section measurements per nucleon for a range of energies, showing the QE, RES and DIS contributions~\cite{RevModPhys.84.1307}.}
   \label{fig:CrossSectionMeasurements}
 \end{figure}
 \newline
 \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
+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}
 \label{eq:FAFormFactor},
@@ -91,7 +91,36 @@ \section{Neutrino interactions at the GeV-scale}
   \caption{The CCQE cross-sections measured by the MiniBooNE and NOMAD experiments.  The solid and dashed lines represent models with different values of $M_A$~\cite{PhysRevD.81.092005}.}
   \label{fig:CCQECrossSectionMiniBooNENOMAD}
 \end{figure}
-A popular explanation for this discrepancy is a lack of understanding of the nuclear environment.  Because the neutrino is not scattering of a free nucleon, but rather a nucleon in a strongly contained system, experiments actually measure an effective $M_A$.  It is possible that the nuclear effects cause a modification to the effective $M_A$ that the experiments measure.  This possible explanation has placed a heavier emphasis on nuclear modelling in neutrino interaction experiments. 
+A popular explanation for this discrepancy is a lack of understanding of the nuclear environment.  Because the neutrino is not scattering of a free nucleon, but rather a nucleon in a strongly contained system, experiments actually measure an effective $M_A$.  It is possible that the nuclear effects cause a modification to the effective $M_A$ that the experiments measure.  This possible explanation for the discrepancy has placed a heavier emphasis on nuclear modelling in neutrino interaction experiments. 
+\section{Neutrino interactions with heavy nuclei}
+\label{sec:NeutrinoInteractionsHeavyNuclei}
+As introduced above, consideration of nuclear effects in cross-section measurements is important.  This is especially true for neutrino interactions on heavy target nuclei.  As one can imagine, the presence of a nucleus can dramatically effect the interactions that are observed in a detector.  A popular model for the nucleus is the Relativistic Fermi-Gas (RFG) model~\cite{Smith:1972xh}.  The RFG model treats the nucleus as a collection of non-interacting nucleons sitting in a potential well.  The nucleons are stacked in the potential well according to the Pauli exclusion principle.  This leads to a uniform momentum distribution of the nucleons up to the Fermi momentum $p_F$.  Importantly, the Pauli exclusion principle has a further effect.  Because the final state nucleon is forbidden from occupying a state taken by another nucleon in the potential well, the energy transfer of the neutrino to the nucleon must result in a final state nucleon with a momentum above $p_F$, resulting in a reduction of the cross-section.
+\newline
+\newline
+The RFG can only model the effect of the nucleus on the initial neutrino interaction which creates the final states.  However, these final states are created within the nucleus and so additional interactions of the final states with the nucleus can occur.  The Final-State Interactions (FSI) can significantly alter the momentum and direction of the final-state particles.  As the final-state particles are used to infer neutrino properties, the FSI effects can alter the interpretation of the reconstructed events.  In simulation, variations of the cascade model are typically used.  This involves pushing the final-state particles through the nucleus in discreet steps and, at each step, probabilistically updating the particle properties.  If at any point a final-state particles knocks out another nucleon, the additional nucleon is also pushed through the nucleus in parallel.  The discreet stepping occurs until all relevant particles have escaped the nucleus.
+\newline
+\newline
+To test such cross-section models, including nuclear effects, it is necessary to compare prediction with collected data.  However, collected cross-section data for heavy nuclei is relatively sparse.  In the case of lead, only two experiments have performed cross-section measurements.  The first measurement was performed by the CHORUS~\cite{CHORUS_XSEC} experiment.  The CHORUS detector, exposed to the CERN SPS beam with a wide-band $\nu_\mu$ beam of 27~GeV average energy, measured a cross-section for lead, iron, marble and polyethylene.  However, because the absolute flux was not measured in the experiment, all of the cross-section measurements were normalised to a common constant.  Their results are summarised in Fig.~\ref{fig:CHORUSXSec}.
+\begin{figure}%
+  \centering
+  \includegraphics[width=8cm]{images/neutrino_interactions/CHORUS_XSec.pdf}
+  \caption{Measured values of the $\nu_\mu$ CC relative cross-section for several elements.  The black and white points are the collected data and prediction respectively.  The solid and dashed lines are the linear best fit lines to the data and prediction respectively.  Going from left to right, the points represent data/prediction for lead, iron marble and polyethylene.}
+  \label{fig:CHORUSXSec}
+\end{figure}
+The second measurement was made by the MINER$\nu$A experiment~\cite{PhysRevLett.112.231801}, which used the Fermilab NuMI beam with a 8~GeV average energy, to measure the relative $\nu_\mu$ CC cross-section on lead to that of plastic scintillator as a function of neutrino energy.  Their results, shown in Fig.~\ref{fig:MINERvAXSec}, largely agreed with the prediction.
+\begin{figure}%
+  \centering
+  \includegraphics[width=8cm]{images/neutrino_interactions/MINERvA_XSec.pdf}
+  \caption{Ratio of the measured $\nu_\mu$ CC inclusive cross-section on lead to plastic scintillator as a function of neutrino energy.  The error bars on the simulation (data) are statistical (systematic) uncertainties~\cite{PhysRevLett.112.231801}.}
+  \label{fig:MINERvAXSec}
+\end{figure}
+\newline
+\newline
+As neutrino oscillation physics enters the precision era, it is becoming increasingly important that our understanding of neutrino cross-sections is improved.  To achieve this goal, more cross-section measurements across a range of nuclear targets are needed.  This thesis presents a measurement of the $\nu_\mu$ CC inclusive cross-section on lead using the electromagnetic calorimeters contained in the near detector of the T2K experiment.  
+
+
+
+
 
 
 

mythesis.bib

@@ -850,3 +850,44 @@ @article{NOMAD-CCQE
 pages={355-381},
 language={English}
 }
+
+@article{Smith:1972xh,
+      author         = "Smith, R.A. and Moniz, E.J.",
+      title          = "{NEUTRINO REACTIONS ON NUCLEAR TARGETS}",
+      journal        = "Nucl.Phys.",
+      volume         = "B43",
+      pages          = "605",
+      doi            = "10.1016/0550-3213(72)90040-5",
+      year           = "1972",
+}
+
+@article{CHORUS_XSEC,
+year={2003},
+issn={1434-6044},
+journal={The European Physical Journal C - Particles and Fields},
+volume={30},
+number={2},
+doi={10.1140/epjc/s2003-01292-3},
+title={Measurement of the Z/A dependence of neutrino charged-current total cross-sections},
+url={http://dx.doi.org/10.1140/epjc/s2003-01292-3},
+publisher={Springer-Verlag},
+pages={159-167},
+language={English}
+}
+
+@article{PhysRevLett.112.231801,
+  title = {Measurement of Ratios of ${\ensuremath{\nu}}_{\ensuremath{\mu}}$ Charged-Current Cross Sections on C, Fe, and Pb to CH at Neutrino Energies 2\char21{}20 GeV},
+  author = {Tice, B. G. and Datta, M. and Mousseau, J. and Aliaga, L. and Altinok, O. and Barrios Sazo, M. G. and Betancourt, M. and Bodek, A. and Bravar, A. and Brooks, W. K. and Budd, H. and Bustamante, M. J. and Butkevich, A. and Martinez Caicedo, D. A. and Castromonte, C. M. and Christy, M. E. and Chvojka, J. and da Motta, H. and Devan, J. and Dytman, S. A. and D\'{i}az, G. A. and Eberly, B. and Felix, J. and Fields, L. and Fiorentini, G. A. and Gago, A. M. and Gallagher, H. and Gran, R. and Harris, D. A. and Higuera, A. and Hurtado, K. and Jerkins, M. and Kafka, T. and Kordosky, M. and Kulagin, S. A. and Le, T. and Maggi, G. and Maher, E. and Manly, S. and Mann, W. A. and Marshall, C. M. and Martin Mari, C. and McFarland, K. S. and McGivern, C. L. and McGowan, A. M. and Miller, J. and Mislivec, A. and Morf\'{i}n, J. G. and Muhlbeier, T. and Naples, D. and Nelson, J. K. and Norrick, A. and Osta, J. and Palomino, J. L. and Paolone, V. and Park, J. and Patrick, C. E. and Perdue, G. N. and Rakotondravohitra, L. and Ransome, R. D. and Ray, H. and Ren, L. and Rodrigues, P. A. and Savage, D. G. and Schellman, H. and Schmitz, D. W. and Simon, C. and Snider, F. D. and Solano Salinas, C. J. and Tagg, N. and Valencia, E. and Vel\'asquez, J. P. and Walton, T. and Wolcott, J. and Zavala, G. and Zhang, D. and Ziemer, B. P.},
+  collaboration = {(MINERvA Collaboration)},
+  journal = {Phys. Rev. Lett.},
+  volume = {112},
+  issue = {23},
+  pages = {231801},
+  numpages = {6},
+  year = {2014},
+  month = {Jun},
+  publisher = {American Physical Society},
+  doi = {10.1103/PhysRevLett.112.231801},
+  url = {http://link.aps.org/doi/10.1103/PhysRevLett.112.231801}
+}
+

thesis.bbl

@@ -150,4 +150,15 @@ A.~A. Aguilar-Arevalo {\em et~al.},
 V.~Lyubushkin {\em et~al.},
 \newblock The European Physical Journal C {\bf 63}, 355 (2009).
 
+\bibitem{Smith:1972xh}
+R.~Smith and E.~Moniz,
+\newblock Nucl.Phys. {\bf B43}, 605 (1972).
+
+\bibitem{CHORUS_XSEC}
+The European Physical Journal C - Particles and Fields {\bf 30}, 159 (2003).
+
+\bibitem{PhysRevLett.112.231801}
+(MINERvA Collaboration), B.~G. Tice {\em et~al.},
+\newblock Phys. Rev. Lett. {\bf 112}, 231801 (2014).
+
 \end{thebibliography}

thesis.blg

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