diff --git a/chap1.tex b/chap1.tex index 484ae8f..6f1040a 100644 --- a/chap1.tex +++ b/chap1.tex @@ -4,7 +4,7 @@ \chapter{Introduction} %% Restart the numbering to make sure that this is definitely page #1! \pagenumbering{arabic} -The field of neutrino physics is currently undergoing a revolution. With its tenuous postulation~\cite{PauliOpenLetter} acting as a future omen, the neutrino's mark on history would not become apparent from its discovery~\cite{Cowan20071956, PhysRevLett.9.36, Kodama2001218}, but rather from a spate of surprising discoveries at the end of the 20th century~\cite{PhysRevLett.81.1562, PhysRevLett.87.071301, PhysRevLett.90.021802} which conclusively proved that the Standard Model, while very successful, was incomplete. This revelation was experimental proof of Maki, Nagakawa and Sakata's extension~\cite{Maki01111962} to Pontecorvo's theory of neutrino oscillation ~\cite{Pontecorvo} with the inclusion of the Mikheyev-Smirnov-Wolfenstein (MSW) effect~\cite{PhysRevD.17.2369,Mikheev:1986gs}. The findings were groundbreaking as the underlying theory requires massive neutrinos, which is in direct contradiction to the Standard Model. The, now, standard theory of neutrino oscillation defines three neutrino flavours and three neutrino masses. However, the map between flavour and mass is not one-to-one, but rather a rotation of mass space onto flavour space. The main consequence of this rotation is that the flavour eigenstates are a superposition of mass eigenstates, namely +The field of neutrino physics is currently evolving very rapidly. With its tenuous postulation~\cite{PauliOpenLetter} acting as a future omen, the neutrino's mark on history would not become apparent from its discovery~\cite{Cowan20071956, PhysRevLett.9.36, Kodama2001218}, but rather from a spate of surprising discoveries at the end of the 20th century~\cite{PhysRevLett.81.1562, PhysRevLett.87.071301, PhysRevLett.90.021802} which conclusively proved that the Standard Model, while very successful, was incomplete. This revelation was experimental proof of Maki, Nagakawa and Sakata's extension~\cite{Maki01111962} to Pontecorvo's theory of neutrino oscillation ~\cite{Pontecorvo} with the inclusion of the Mikheyev-Smirnov-Wolfenstein (MSW) effect~\cite{PhysRevD.17.2369,Mikheev:1986gs}. The findings were groundbreaking as the underlying theory requires massive neutrinos, which is in direct contradiction to the Standard Model. The, now, standard theory of neutrino oscillation defines three neutrino flavours and three neutrino masses. However, the map between flavour and mass is not one-to-one, but rather a rotation of mass space onto flavour space. The main consequence of this rotation is that the flavour eigenstates are a superposition of mass eigenstates, namely \begin{equation} \ket{\nu_\alpha} = \sum^{3}_{i=k}U^{\ast}_{\alpha k}\ket{\nu_k}, \label{eq:NeutrinoEigenstates} @@ -35,7 +35,7 @@ \chapter{Introduction} P(\nu_\alpha \rightarrow \nu_\beta) = |\braket{\nu_\beta|\nu\left(t\right)}|^2 = |U_{\beta k} e^{-iE_{k}t} U^{\ast}_{\alpha k}|^2, \label{eq:NeutrinoOscillationProbability} \end{equation} -where $\nu\left(t\right)$ is the time-dependent neutrino mass eigenstate and $E_k$ is the energy of the $\nu_k$. For an accelerator-based neutrino oscillation experiment, the beam will be $\nu_\mu$ dominated. So, the $\nu_\mu$ survival probability, $P(\nu_\mu \rightarrow \nu_\mu)$, and $\nu_e$ appearance probability, $P(\nu_\mu \rightarrow \nu_e)$, have are typically of interest and can be approximated in the following forms +where $\nu\left(t\right)$ is the time-dependent neutrino mass eigenstate and $E_k$ is the energy of the $\nu_k$. For an accelerator-based neutrino oscillation experiment, the beam will be $\nu_\mu$ dominated. So, the $\nu_\mu$ survival probability, $P(\nu_\mu \rightarrow \nu_\mu)$, and $\nu_e$ appearance probability, $P(\nu_\mu \rightarrow \nu_e)$, which are typically of interest, can be approximated in the following forms \begin{equation} P(\nu_\mu \rightarrow \nu_\mu) \approx 1 - \cos^{4}\theta_{13}\sin^{2}2\theta_{23}\sin^2\left(1.27\frac{\Delta m^{2}_{23}}{\left(\textrm{eV}^2\right)}\frac{L}{\left(\textrm{km}\right)}\frac{\left(\textrm{GeV}\right)}{E}\right) \label{eq:MuonNeutrinoSurvivalProbability} @@ -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 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}. +Data provided from a wide range of experiments show excellent agreement with the theory of neutrino oscillation and with a 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 @@ -65,7 +65,6 @@ \section{The state of the field} \caption{The best-fit values of the 3-neutrino oscillation parameters. $\Delta m^2 \equiv m^2_3 - \left(m^2_2 - m^2_1\right)/2$. The values (values in brackets) correspond to $m_1 < m_2 < m_3$ ($m_3 < m_1 < m_2$)~\cite{Agashe:2014kda}.} \label{table:NeutrinoOscillationParameterValues} \end{table} -The experiments which provided the data inputs to the global fit generally fall into one of four catagories, with each catagory sensitive to a different subset of the neutrino oscillation parameters. \newline \newline Solar neutrino experiments detect neutrinos generated in the core of the Sun as a result of nuclear reaction chains. Such experiments are primarily sensitive to $\theta_{12}$ and $\Delta m^{2}_{12}$ which are often referred to as the solar mixing parameters. The final state neutrinos created in the Sun's core are MeV-scale $\nu_e$ but, because of propagation through the core's surrounding matter, the MSW effect results in a highly pure state of $\nu_2$ at the Sun's surface. As $\nu_2$ is a mass eigenstate, no oscillation occurs between the surface of the Sun and the Earth. Homestake~\cite{0004-637X-496-1-505}, SAGE~\cite{PhysRevC.80.015807} and SNO~\cite{PhysRevLett.87.071301} are examples of such experiments. @@ -74,26 +73,26 @@ \section{The state of the field} Reactor neutrino experiments measure $\bar{\nu}_e$ disappearance provided by inverse $\beta$ decay in nuclear reactors with an average neutrino energy of 3~MeV. The baseline for oscillations varies between experiments, but a baseline of around 1~km provides excellent sensitivity to $\theta_{13}$. Examples of reactor experiments are CHOOZ~\cite{CHOOZ}, Double CHOOZ~\cite{Abe201366}, Daya Bay~\cite{PhysRevLett.108.171803} and RENO~\cite{PhysRevLett.108.191802}. \newline \newline -Atmospheric neutrino experiments detect neutrinos which are produced from $\pi$ and $K$ mesons, created bycosmic rays interactions with the upper atmosphere of the Earth, decay. The neutrinos produced are a mixture of $\nu_\mu$, $\bar{\nu}_\mu$, $\nu_e$ and $\bar{\nu}_e$. Because the cosmic ray flux is fairly uniform, atmospheric neutrino experiments are exposed to neutrinos from all directions, which results in a very wide range of oscillation baselines. The oscillation parameters that such experiments are sensitive to are $\theta_23$ and $\Delta m^2_{13}$. Super-Kamiokande~\cite{PhysRevLett.81.1562} is an example of an atmospheric neutrino experiment. +Atmospheric neutrino experiments detect neutrinos which are produced when $\pi$ and $K$ mesons, created by cosmic rays interactions with the upper atmosphere of the Earth, decay. The neutrinos produced are a mixture of $\nu_\mu$, $\bar{\nu}_\mu$, $\nu_e$ and $\bar{\nu}_e$. Because the cosmic ray flux is fairly uniform, atmospheric neutrino experiments are exposed to neutrinos from all directions, which results in a very wide range of oscillation baselines. The oscillation parameters that such experiments are sensitive to are $\theta_{23}$ and $\Delta m^2_{13}$. Super-Kamiokande~\cite{PhysRevLett.81.1562} is an example of an atmospheric neutrino experiment. \newline \newline -Accelerator neutrino experiments produce beams of high purity $\nu_\mu$ (or $\bar{\nu_\mu}$) at GeV-scale energy with wide ranging baselines which are generally $\mathcal{O}\left(100~\textrm{km}\right)$. The highly man-made nature of such experiments allows almost complete control over $L/E$ allowing careful tuning of parameter sensitivity. Accelerator neutrino experiments are generally sensitive to $\theta_{13}$, $\theta_{23}$, $\Delta m^{2}_{13}$ and $\delta$. K2K~\cite{PhysRevD.74.072003}, MINOS~\cite{PhysRevLett.97.191801}, T2K~\cite{PhysRevLett.112.061802} and NO$\nu$A~\cite{Ayres:2004js}are examples of such experiments. +Accelerator neutrino experiments produce beams of high purity $\nu_\mu$ (or $\bar{\nu}_\mu$) at GeV-scale energy with wide ranging baselines which are generally $\mathcal{O}\left(100~\textrm{km}\right)$. The highly man-made nature of such experiments allows almost complete control over $L/E$ allowing careful tuning of parameter sensitivity. Accelerator neutrino experiments are generally sensitive to $\theta_{13}$, $\theta_{23}$, $\Delta m^{2}_{13}$ and $\delta$. K2K~\cite{PhysRevD.74.072003}, MINOS~\cite{PhysRevLett.97.191801}, T2K~\cite{PhysRevLett.112.061802} and NO$\nu$A~\cite{Ayres:2004js}are examples of such experiments. \section{The future} \label{sec:NeutrinoFieldFuture} It should be clear that an immense amount of progress has been made in the field, with remarkable contributions to the picture coming only in the last 20 years. However, there are several key questions which remain unanswered. \newline \newline -By far the most sought after answer is whether CP violation occurs in the leptor sector. The magnitude of CP violation is encapsulated in the CP violating phase $\delta$ and so it is this parameter which current and future long-baseline experiments are aiming towards. Currently, T2K and NO$\nu$A can provide the strongest constraints on $\delta$. The future long-baseline experiments, Hypker-Kamiokande~\cite{Abe:2014oxa} and DUNE (formerly LBNE)~\cite{Adams:2013qkq} are being designed with a possible measurement of $\delta$ as a primary goal. +By far the most sought after answer is whether CP violation occurs in the lepton sector. The magnitude of CP violation is encapsulated in the CP violating phase $\delta$ and so it is this parameter which current and future long-baseline experiments are aiming towards. Currently, T2K and NO$\nu$A can provide the strongest constraints on $\delta$. The future long-baseline experiments, Hyper-Kamiokande~\cite{Abe:2014oxa} and DUNE (formerly LBNE)~\cite{Adams:2013qkq} are being designed with a possible measurement of $\delta$ as a primary goal. \newline \newline -The second question still to be answered is the ordering of the mass eigenstates. Specifically is $m_3 \gg m_2 > m_1$ (the normal mass hierarchy) or $m_2 > m_1 \gg m_3$ (the inverted mass hierarchy)? The matter effects introduced by the MSW effect are mass hierarchy dependent. So, for very long-baseline experiments, there is mass hierarchy sensitivity. Currently NO$\nu$A is set to resolve the mass hierarchy problem. However, both Hyper-Kamiokande (via atmospheric measurements) and DUNE have measurement of the hierarchy as a primary goal. +The second question still to be answered is the ordering of the mass eigenstates. Specifically is $m_3 \gg m_2 > m_1$ (the normal mass hierarchy) or $m_2 > m_1 \gg m_3$ (the inverted mass hierarchy)? The matter effects introduced by the MSW effect are mass hierarchy dependent. So, for very long-baseline experiments, there is mass hierarchy sensitivity. Currently NO$\nu$A is set to resolve the mass hierarchy problem. However, both Hyper-Kamiokande (via atmospheric measurements) and DUNE have measurement of the mass hierarchy as a primary goal. \newline \newline -Oscillation experiments only have the capability to measure the square of the mass splitting differences. This means that all oscillation experiments have no sensitivty to the absolute neutrino mass scale. This means an entirely different type of neutrino experiment is required. Neutrinos are one of the final states associated with $\beta$ decay and the mass of the neutrino should appear as a cut off in the $\beta$ spectrum. The visibility of the cut-off entirely depends on the mass scale. So, the KATRIN experiment~\cite{Weinheimer2002141} will attempt to utilise the $\beta$ decay feature, with a neutrino mass sensitivity of $0.2$~eV. +Oscillation experiments only have the capability to measure the square of the mass splitting differences. This means that all oscillation experiments have no sensitivity to the absolute neutrino mass scale and an entirely different type of neutrino experiment is required. Neutrinos are one of the final states associated with $\beta$ decay and the mass of the neutrino should appear as a cut off in the $\beta$ spectrum. The visibility of the cut-off entirely depends on the mass scale. So, the KATRIN experiment~\cite{Weinheimer2002141} will attempt to utilise the $\beta$ decay feature, with a neutrino mass sensitivity of $0.2$~eV. \newline \newline It is not currently known whether neutrinos are their own anti-particle, otherwise known as Majorana neutrinos. A number of experiments are currently investigating this, all by searching for neutrinoless double $\beta$ decay. A large neutrinoless double $\beta$ decay experiment effort is ongoing, including EXO~\cite{PhysRevLett.109.032505}, SuperNEMO~\cite{1742-6596-375-4-042012} and SNO+~\cite{Chen:2008un}. \newline \newline -While the long-baseline neutrino oscillation programme has been very successful, the short-baseline programme has seen several anomalies~\cite{ShortBaseLineAnomaly}. The LSND experiment found evidence of $\bar{\nu}_e$ in a $\bar{\nu_\mu}$ beam, which was consitent with neutrino oscillations~\cite{Aguilar:2001ty}. However the data suggested a mass-squared splitting of 0.2-10~eV$^2$. This large splitting is consitent with a fourth species of neutrino. Because the data suggesting 3 flavours of weakly-interacting neutrino is strong, this postulated fourth species must be sterile. More recently, the Mini-BooNE experiment observed a similar short baseline excess of $\bar{\nu}_e$ in a $\bar{\nu}_\mu$ beam with a mass-squared splitting of 0.01-1.0~eV$^2$~\cite{Aguilar-Arevalo:2013pmq}, further suggesting the sterile hypthosesis. New experiments are now under development which aim to thest this hypothesis, which include MicroBooNE~\cite{2011arXiv1110.1604I} and SBND (formally LAr1-ND)~\cite{Adams:2013uaa}. +While the long-baseline neutrino oscillation programme has been very successful, the short-baseline programme has seen several anomalies~\cite{ShortBaseLineAnomaly}. The LSND experiment found evidence of $\bar{\nu}_e$ in a $\bar{\nu}_\mu$ beam, which was consistent with neutrino oscillations~\cite{Aguilar:2001ty}. However the data suggested a mass-squared splitting of 0.2-10~eV$^2$. This large splitting is consistent with a fourth species of neutrino. Because the data suggesting 3 flavours of weakly-interacting neutrino is strong, this postulated fourth species must be sterile. More recently, the MiniBooNE experiment observed a similar short baseline excess of $\bar{\nu}_e$ in a $\bar{\nu}_\mu$ beam with a mass-squared splitting of 0.01-1.0~eV$^2$~\cite{Aguilar-Arevalo:2013pmq}, further suggesting the sterile hypothesis. New experiments are now under development which aim to test this hypothesis, which include MicroBooNE~\cite{2011arXiv1110.1604I} and SBND (formally LAr1-ND)~\cite{Adams:2013uaa}. diff --git a/chap2.tex b/chap2.tex index 46824f2..1442f89 100644 --- a/chap2.tex +++ b/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. + + + + diff --git a/images/neutrino_interactions/CHORUS_XSec.pdf b/images/neutrino_interactions/CHORUS_XSec.pdf new file mode 100644 index 0000000..e0cb475 Binary files /dev/null and b/images/neutrino_interactions/CHORUS_XSec.pdf differ diff --git a/images/neutrino_interactions/MINERvA_XSec.pdf b/images/neutrino_interactions/MINERvA_XSec.pdf new file mode 100644 index 0000000..03cb972 Binary files /dev/null and b/images/neutrino_interactions/MINERvA_XSec.pdf differ diff --git a/mythesis.bib b/mythesis.bib index 09cb177..8099b62 100644 --- a/mythesis.bib +++ b/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} +} + diff --git a/thesis.bbl b/thesis.bbl index 1ef0b05..73c2a83 100644 --- a/thesis.bbl +++ b/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} diff --git a/thesis.blg b/thesis.blg index 89292e0..73c8905 100644 --- a/thesis.blg +++ b/thesis.blg @@ -3,44 +3,46 @@ Capacity: max_strings=35307, hash_size=35307, hash_prime=30011 The top-level auxiliary file: thesis.aux The style file: h-physrev.bst Database file #1: mythesis.bib -You've used 37 entries, +Warning--empty author in CHORUS_XSEC +You've used 40 entries, 1946 wiz_defined-function locations, - 867 strings with 45232 characters, -and the built_in function-call counts, 7417 in all, are: -= -- 510 -> -- 358 + 887 strings with 47010 characters, +and the built_in function-call counts, 7992 in all, are: += -- 549 +> -- 379 < -- 0 -+ -- 133 -- -- 72 -* -- 565 -:= -- 965 -add.period$ -- 37 -call.type$ -- 37 ++ -- 141 +- -- 76 +* -- 616 +:= -- 1030 +add.period$ -- 40 +call.type$ -- 40 change.case$ -- 1 chr.to.int$ -- 0 -cite$ -- 37 -duplicate$ -- 397 -empty$ -- 748 -format.name$ -- 109 -if$ -- 1739 +cite$ -- 41 +duplicate$ -- 430 +empty$ -- 811 +format.name$ -- 115 +if$ -- 1876 int.to.chr$ -- 0 -int.to.str$ -- 37 -missing$ -- 36 -newline$ -- 151 -num.names$ -- 37 -pop$ -- 168 +int.to.str$ -- 40 +missing$ -- 39 +newline$ -- 162 +num.names$ -- 39 +pop$ -- 179 preamble$ -- 1 purify$ -- 0 quote$ -- 0 -skip$ -- 355 +skip$ -- 386 stack$ -- 0 -substring$ -- 437 -swap$ -- 103 +substring$ -- 476 +swap$ -- 112 text.length$ -- 0 text.prefix$ -- 0 top$ -- 0 type$ -- 0 -warning$ -- 0 -while$ -- 66 -width$ -- 39 -write$ -- 279 +warning$ -- 1 +while$ -- 71 +width$ -- 42 +write$ -- 299 +(There was 1 warning) diff --git a/thesis.log b/thesis.log index 85c87a9..eb9e855 100644 --- a/thesis.log +++ b/thesis.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014) (preloaded format=pdflatex 2015.5.13) 3 JUL 2015 18:26 +This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014) (preloaded format=pdflatex 2015.5.13) 6 JUL 2015 19:46 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -1187,22 +1187,22 @@ LaTeX Font Info: Font shape `OT1/ppl/bx/n' in size <17.28> not available LaTeX Font Info: Font shape `OT1/ppl/bx/n' in size <10.95> not available (Font) Font shape `OT1/ppl/b/n' tried instead on input line 65. -Underfull \hbox (badness 10000) in paragraph at lines 51--81 +Underfull \hbox (badness 10000) in paragraph at lines 51--80 [] -Underfull \hbox (badness 10000) in paragraph at lines 51--81 +Underfull \hbox (badness 10000) in paragraph at lines 51--80 [] -Underfull \hbox (badness 10000) in paragraph at lines 51--81 +Underfull \hbox (badness 10000) in paragraph at lines 51--80 [] -Underfull \hbox (badness 10000) in paragraph at lines 51--81 +Underfull \hbox (badness 10000) in paragraph at lines 51--80 [] @@ -1210,29 +1210,29 @@ pdfTeX warning (ext4): destination with the same identifier (name{page.2}) has been already used, duplicate ignored \relax -l.81 +l.80 [2] [3]) (./chap2.tex -Underfull \hbox (badness 10000) in paragraph at lines 84--9 +Underfull \hbox (badness 10000) in paragraph at lines 83--9 [] -Underfull \hbox (badness 10000) in paragraph at lines 84--9 +Underfull \hbox (badness 10000) in paragraph at lines 83--9 [] -Underfull \hbox (badness 10000) in paragraph at lines 84--9 +Underfull \hbox (badness 10000) in paragraph at lines 83--9 [] -Underfull \hbox (badness 10000) in paragraph at lines 84--9 +Underfull \hbox (badness 10000) in paragraph at lines 83--9 [] -Underfull \hbox (badness 10000) in paragraph at lines 84--9 +Underfull \hbox (badness 10000) in paragraph at lines 83--9 [] @@ -1253,7 +1253,7 @@ ns/CCQE_FD-eps-converted-to.pdf images/neutrino_interactions/CCQE_FD.eps> (epstopdf) \includegraphics on input line 35. 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