chapter1.tex

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\chapter{NEUTRINO PHYSICS AND ITS DEPENDENCE ON NUCLEAR PHYSICS}
\label{chap:0vbb}
\begin{comment}
Neutrino physics is fascinating because ??????.
Maybe I will talk about particles in general - but no, that's probably a bit too much.
I will mention the standard model table and talk about why neutrinos are important in physics today?

The neutrino was first proposed as a undetectable particle that carried away energy in nuclear decay processes \citep{Pauli}.  Twenty-six years later (fact check!), Reines and Cowan detected inverse beta decay to detect anti-neutrinos streaming out of a nearby nuclear reactor \citep{poltergeist}.  Detecting neutrinos is difficult: with an interaction volume of 10$^{26}$ potential targets, Reines and Cowan saw 1 event per week (FACT CHECK!!) \citep{poltergeist}.  Simply confirming the existence of the particle hypothesized to participate in beta decay was significant enough to merit the Nobel Prize \citep{CowanNobel}.

Figure: beta decay spectrum, with and without neutrino.  See F.A. Scott, Phys Rev 48, 391 (1935)

While detecting neutrinos is a difficult endeavor, it's also a lucrative one: particles that interact extremely rarely carry information about where they're created, no matter what they have to travel through to get to us.  With a detector that counts neutrinos, one can begin to imagine interrogating the cosmos: ``How many neutrinos are you making?  And you?  And you?''  Bahcall was interested in finding out how many neutrinos the Sun made.  And so he and Ray Davis set out to make a good neutrino counter. 

(AMY!  You're neglecting to talk about the experiment that demonstrated the left-handedness of neutrinos by Goldhaber in 1958.  This is kind of relevant to \zvbb!)
\end{comment}

The neutrino was first proposed by Pauli to be a chargeless, nearly massless fermion \citep{Pauli}.  This particle was a means to preserve energy, momentum, and angular momentum conservation in nuclear beta decay.  The hypothesized process taking place within the nucleus was
\begin{equation}
n \rightarrow p + e^- + \overline{v}_e,
\end{equation}
where a neutron $n$ decays into a proton $p$, an electron $e^-$, and also an electron anti-neutrino $\overline{v}_e$, allowing the continuous electron energy spectrum that was observed. The existence of this difficult-to-detect particle was not confirmed until 26 years later, when Reines and Cowan used a nuclear reactor as a source of anti-neutrinos and observed inverse beta decay of proton targets \citep{poltergeist}.  Since the first detection of electron anti-neutrinos, an much has been learned about these elusive particles.  This chapter will begin by discussing what is currently known about neutrinos, in particular that there are three, unique flavors and that while they are very light, they do have mass.  The remainder of the chapter discusses how neutrinos get their mass in the Standard Model (SM) framework and the experiments currently underway to help determine why the neutrino mass scale is so small.

\subsection{Neutrino Oscillation}
Neutrinos interact very weakly with matter, and while this makes their detection difficult, it also makes them a potentially valuable source of information.  Studying the interior of systems that produce neutrinos becomes possible with a neutrino detector, while other forms of radiation would be absorbed by the surrounding matter.  Raymond Davis and John Bahcall recognized the neutrino could be used to test the theory that nuclear fusion was the Sun's energy source.  The neutrino detector built by Davis consisted of 100,000 gallons of liquid dry cleaning fluid, which contained $^{37}$Cl \citep{DavisInitial}.  Solar neutrinos interacting with $^{37}$Cl initiate inverse beta decay, leaving the radioactive $^{37}$Ar as a detectable signal.  Years of careful data taking yielded a count of $\sim$7 neutrinos per two weeks \citep{DavisInitial}, only $\sim\frac{1}{3}$ the rate predicted by Bahcall \citep{BahcallSun}.  Further refinements to the experiment and to the calculations confirmed the discrepancy \citep{Davis}.

While the Davis experiment continued to collect data, other experiments were begun to explore other properties of the neutrino.  Originally imagined as a single particle, it was found that there are three distinct flavors of neutrinos, each associated with a lepton partner.  It is important to note that the beta decay which transforms nucleus $A$ to $A'$,
\begin{equation}
A(Z,N) \rightarrow A'(Z-1,N+1) + \overline{l} + v_l,
\end{equation}
where $l$ is an electron, muon, or tau, requires the mass difference between the initial and final nuclei to be larger than the mass of the lepton $l$.  While electrons have a mass of only 0.51~MeV, muons are considerably heavier, having a mass of 105.7~MeV.  Tau leptons have a mass of 1776.8~MeV, comparable to a light nucleus. At most, nuclear reactions in the Sun provide $\sim$11~MeV, so that the Sun can produce only electron neutrinos.  Energetic pion beams at Brookhaven National Laboratory (BNL) were used to make the first direct measurement of muon neutrinos \citep{muonNeutrino}.  Later, more than 25 years after the discovery of the tau lepton \citep{tauDiscovery} in 1975, tau neutrinos were successfully detected at Fermilab \citep{tauNeutrino}.  The inclusion of the neutrino into the SM as a participant in weak interactions mediated by the $W^{\pm}$ and $Z^0$ bosons suggested that experiments determining the lifetime of the $Z^0$ boson could determine the number of interacting neutrinos.  The $Z^0$ has known lepton and hadron decay modes \citep{PDG} and can also decay to a neutrino-antineutrino pair of any flavor provided the neutrinos have a mass less than $M_Z/2$.  More flavors of light neutrinos should therefore reduce the $Z^0$ lifetime while fewer should increase it.   An electron-positron collider experiment at CERN measured the lifetime of the $Z^0$ and determined the number of neutrino flavors to be $2.92\pm0.05$ \citep{PDG}.

That there are three flavors of neutrinos, each associated with a different-mass lepton, is significant because the Davis experiment was sensitive only to electron neutrinos.  Other radiochemical neutrino experiments, also only sensitive to electron neutrinos, confirmed Davis' results \citep{SNO_Sun,SuperK_Sun,PDG}.  An idea suggested by Pontecorvo, that neutrinos have mass \citep{Pontecorvo}, showed a way forward.  Neutrinos had been incorporated into the SM as massless, making it impossible for their flavor to vary with time.  If neutrinos were massive, neutrinos could change flavor.  The hypothesis was that the radiochemical experiments, sensitive only to electron neutrinos, were measuring a deficit because $\sim\frac{2}{3}$ of the electron neutrinos from the Sun had changed flavor and could not be detected.  SNO, an experiment designed to be sensitive to all three neutrino flavors, measured the predicted number of solar neutrinos \citep{SNO_Sun}, confirming that neutrino flavors change with time and therefore that neutrinos must be massive.

That neutrino flavor oscillation implies a massive neutrino is a general property of combining states with different energy eigenvalues.  This can be seen by imagining some two states, $\psi_{\alpha}$ and $\psi_{\beta}$, neither of which are energy eigenstates.  These states can be written in terms of the energy eigenstates: 
\begin{align}
|\psi_{\alpha}\rangle &= U_{\alpha 1}|\psi_1\rangle + U_{\alpha 2}|\psi_2\rangle \\
|\psi_{\beta}\rangle &= U_{{\beta}1}|\psi_1\rangle + U_{{\beta}2}|\psi_2\rangle, 
\end{align}
where $\hat{H}|\psi_1\rangle = E_1|\psi_1\rangle$ and $\hat{H}|\psi_2\rangle = E_2|\psi_2\rangle$.  Then for an initial state $|\psi_{\alpha}\rangle$, the probability of measuring $|\psi_{\beta}\rangle$ some time $t$ later is
\begin{align}
\begin{split}
P(\alpha\rightarrow\beta) &=  |\langle\psi_{\beta}|\hat{T}|\psi_{\alpha}\rangle|^2 \\
                             &=  |\langle\psi_{\beta}|e^{i\hat{H}t / \hbar}|\psi_{\alpha}\rangle|^2 \\
                             &=  U_{{\alpha}1}U_{{\alpha}2}U_{{\beta}1}U_{{\beta}2} \times \frac{\cos((E_1 - E_2)t/\hbar)}{2} 
\end{split}
\end{align}
This calculation is not exactly analogous to neutrino mixing because there are three mass eigenstates, not two.  However, the modulation of the probability of detecting a different flavor state is, as in this test case, dependent on the energy difference.  In the case of the neutrino, the energy eigenstates are also its mass eigenstates, and it can be shown that in vacuum, the neutrino oscillation phase between components $\nu_i$ and $\nu_j$ is \citep{PDG,neutrinoOscillations}
\begin{equation}
\phi = (m_i^2 - m_j^2)\frac{L}{2E},
\end{equation}
where $m_i$ and $m_j$ are the masses of $\nu_i$ and $\nu_j$, respectively, $L$ is the distance between the neutrino source and the detector, and $E$ is the neutrino energy.  Neutrino oscillation experiments are therefore sensitive to the differences between neutrino masses but not to the absolute mass scale.  It is important to note that the oscillation phase for neutrinos traveling through matter is still dependent only on the mass differences \citep{MSW}.  Cosmological measurements and beta-decay experiments, while not sensitive to neutrino oscillation, can place limits on the absolute neutrino mass scale.  Cosmological limits are sensitive to $m_1+m_2+m_3$ and constrain this quantity to be less than $0.3-1.3$~eV at the 95\% confidence level \citep{cosmoNuMassLimit}.  Experiments designed to measure the endpoint of beta decay are sensitive to the quantity $\sqrt{|U_{e1}|^2m_1^2 + |U_{e2}|^2m_2^2 + |U_{e3}|^2m_3^2}$ currently limit this value to less than 2.05~eV \citep{tritiumEndpoint}. 

The neutrino mixing matrix $U$ can be written with three angles, one Dirac CP-violating phase, and two Majorana CP-violating phases:
\begin{multline}
\begin{bmatrix}
c_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i\delta} \\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta} & c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta} & s_{23}c_{13} \\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta} & -c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta} & c_{23}c_{13} 
\end{bmatrix} 
\\
\times 
\begin{bmatrix}
1 & 0 & 0 \\
0 & e^{\frac{i}{2}\alpha_{21}} & 0 \\
0 & 0 & e^{\frac{i}{2}\alpha_{31}}
\end{bmatrix}
,
\end{multline}
where $c_{ij} = \cos{\theta_{ij}}$ and $s_{ij} = \sin{\theta_{ij}}$, $\delta$ is the Dirac CP-violating phase, and the Majorana CP-violating phases $\alpha_{ij}$ are only relevant if the neutrino is a Majorana fermion as discussed in {\sect}~\ref{sec:mass}.  Several generations of long-baseline neutrino experiments using solar, atmospheric, and reactor neutrinos have constrained the mixing parameters and mass differences.  A summary of the parameters is given in {\tab}~\ref{tab:neutrinoParameters}.  It is significant that the absolute values of the mass differences are measured; because the oscillation depends on the cosine of the phase, current measurements are insensitive to the sign of the mass difference, and the ordering of the mass eigenstates is unknown.  Three different ``mass hierarchies'' are possible: the normal hierarchy (NH) where $m_1 < m_2 < m_3$, the inverted hierarchy (IH) where $m_3 < m_1 < m_2$, and the quasi-degenerate hierarchy (QD) where the mass scale is close to the current limit so that $m_1 \approx m_2 \approx m_3$.  A diagram of the three mass hierarchies is shown in {\fig}~\ref{fig:massScale}.

Long-baseline neutrino experiments have provided a comprehensive picture of neutrino mixing, but they cannot provide access to important information about the neutrino such as the absolute mass scale, CP-violating phases, or the origin of its small mass.  These will be discussed in the next section.

\begin{table}[hp]
\ra{1.1}
%\centering
\begin{center}
\caption[\uppercase{Neutrino oscillation parameters}]{\\\uppercase{Neutrino oscillation parameters} \label{tab:neutrinoParameters}}
\begin{tabular}{lll}\toprule
Parameter & Best Fit ($\pm$ 1$\sigma$) & 3$\sigma$ \\
\midrule
${\Delta}m^2_{12}$ [$10^{-5}$ eV$^2$] & $7.58^{+0.22}_{-0.26}$ & 6.99 - 8.18 \\
$|{\Delta}m^2_{31}|$ [$10^{-3}$ eV$^2$] & $2.35^{+0.12}_{-0.09}$ & 2.06 - 2.67 \\
$\sin^2{\theta_{12}}$ & $0.312^{+0.018}_{-0.015}$ & 0.265 - 0.364 \\  
$\sin^2{\theta_{23}}$ & $0.42^{+0.08}_{-0.03}$ & 0.34 - 0.64 \\  
$\sin^2{\theta_{13}}$ & $0.025^{+0.007}_{-0.008}$ & 0.005 - 0.050 \\   
\bottomrule  
\end{tabular}
\begin{flushleft}
{\footnotesize NOTE: 
Three-neutrino oscillation parameters, determined by a global fit to relevant neutrino data.   The mixing angles $\sin^2{\theta_{12}}$ and $\sin^2{\theta_{13}}$ were determined using reactor $\overline{\nu}_e$ spectra calculated in {\refref}~\citep{reactorNeutrinoSpectrum}.  Note that while it is known that $m_1 < m_2$, the sign of ${\Delta}m^2_{31}$ is not known.  The table is from {\refref}~\citep{PDG}.}
\end{flushleft}
\end{center}
\end{table}

\begin{figure}[hp]
\centering
%\includegraphics[height=0.8\textwidth,angle=-90]{figures/mass_scale.eps}
\includegraphics[width=1.0\textwidth]{figures/mass_scale.eps}
\caption[Neutrino mass hierarchies.]{The three possible mass hierarchies: normal hierarchy (NH), inverted hierarchy (IH), and quasi-degenerate (QD).  The dashed line indicates one-third the current limit for the absolute mass scale, $\sim\frac{2}{3}$~eV.}
\label{fig:massScale}
\end{figure}  

\FloatBarrier

\section{Massive Neutrinos in the SM}
\label{sec:mass}
\begin{comment}
Discuss mechanisms by which neutrinos could get their mass.  
\end{comment}
In the standard model, fermions are four-component spinors that can be written in a chiral basis so that there is a ``left-handed'' component of the fermion $\psi_L$ and a ``right-handed'' component $\psi_R$, where $\psi_L$ and $\psi_R$ are two-component spinors.  The chiral basis is particularly useful because the weak bosons $W^{\pm}$ and $Z^0$ have been experimentally observed to only interact with the left-handed component of the fermion field.  The chiral basis is also helpful in understanding two possible ways to give neutrinos mass in the standard model.  Leptons acquire their mass by interacting with the Higgs field; the electron is the lightest because its coupling to the Higgs field is weaker than that of the muon.  The tau is strongly coupled to the Higgs field, making it the most massive of the leptons.  The diagram in {\fig}~\ref{fig:leptonMass} gives a heuristic picture of the lepton fields' interaction with the Higgs background.  
\begin{figure}[hp]
\centering
\includegraphics[width=0.8\textwidth]{figures/leptonMass.eps}
\caption[Lepton mass via the Higgs interaction.]{Leptons acquire mass by interacting with the Higgs field.  Vertices marked with $\times$ indicate an interaction with the background Higgs field.  The electron has a smaller coupling to the Higgs field than the tau and therefore interacts less.  In a Feynman diagram, lepton lines indicate already-massive leptons.  The lines here represent the  massless lepton fields and the entire diagram is analogous to a solid lepton line in a typical Feynman diagram.  Figure from {\refref}~\citep{neutrinoMass}.}
\label{fig:leptonMass}
\end{figure}

When neutrinos were thought to be massless, they were introduced into the SM as two-component spinor fields with no right-handed component.  The Higgs, which changes the chirality of the particle it interacts with, could not interact with the neutrino because it had no right-handed state to convert to.  The ansatz of a solely left-handed neutrino, then, created a massless neutrino in the SM.  As experimental evidence has overwhelmingly favored a massive neutrino, it became necessary to modify the theoretical treatment of the neutrino.  One approach is to assume that the neutrino, like the SM leptons, is a Dirac fermion and has a left-handed component as well as a right-handed component, allowing the Higgs field to interact with the neutrino as it does for the leptons.  There is another approach to generate massive neutrinos that is not solely dependent on the Higgs field.  If neutrinos are Majorana fermions, that is, if unlike Dirac fermions they are their own antiparticles, then the right-handed component of the neutrino field can introduce a mass term independent of the Higgs interaction.  When left-handed neutrinos interact with the background Higgs field, the right-handed neutrino with mass $M$ can only exist for a short time without violating the Pauli principle if $M$ is very large.  The right-handed neutrino quickly interacts with the Higgs background, transforming back into a left-handed neutrino.  The mass of the neutrino should then scaled by $m/M$, where $m$ is the mass due to interaction with the Higgs field.  See {\fig}~\ref{fig:neutrinoMass} for a picture of these different neutrino theories.  The advantage of the Majorana neutrino is that the scale of its interaction with the Higgs field can be comparable to that of other leptons; its small mass can be achieved by assuming a large $M$.  Majorana neutrinos are also attractive because a very massive neutrino could provide an explanation for the observed baryon asymmetry \citep{baryogenesis_Fukugita}.  
\begin{figure}[hp]
\centering
\includegraphics[width=0.8\textwidth]{figures/neutrinoMass.eps}
\caption[A depiction of massless, Dirac, and Majorana models of the neutrino.]{A representation of a massless neutrino, a Dirac neutrino, and a Majorana neutrino.  Vertices marked with $\times$ indicate an interaction with the background Higgs field.  Figure from {\refref}~\citep{neutrinoMass}.}
\label{fig:neutrinoMass}
\end{figure}  

Long-baseline experiments have determined that neutrinos are massive and have measured their mass differences and mixing angles, but much of their fundamental nature is still not understood.  That they could potentially play a significant role in many areas of physics provides significant motivation to build experiments that are sensitive to these neutrino ``parameters.'' One type of experiment that is sensitive to the Dirac or Majorana nature of the neutrino is a search for a process called neutrino-less double-beta decay (\zvbb).  Two-neutrino double-beta decay (\tvbb) is a process that has been observed for a number of nuclei and is the simultaneous beta-decay of two neutrons into two protons.  If the neutrino is a Majorana fermion, it would be possible for the neutrino to become an internal line in the Feynman diagram as shown in {\fig}~\ref{fig:zvbb}.  This process would be impossible if the neutrino were not its own antiparticle; observation of \zvbb would confirm the Majorana nature of the neutrino. 
\begin{figure}[hp]
\centering
\includegraphics[width=0.8\textwidth]{figures/feynman2.eps}
\caption[Feynman diagrams describing \tvbb and \zvbb.]{Both the observed \tvbb (left) and the hypothesized \zvbb (right) modes are shown.  Nuclear matter is an ideal environment for spatially-close neutrons.  Figure from {\refref}~\citep{zvbbReview_Elliott}.}
\label{fig:zvbb}
\end{figure}
It should be noted that measured lifetimes for \tvbb are extremely long, on the order of $10^{20}$~yr.  The expected lifetime for \zvbb, if it exists, will be even larger due to the suppression of the right-handed component of the neutrino; the IGEX experiment places the current limit at $>1.57\times 10^{25}$~yr \citep{IGEX}.  This long lifetime makes \zvbb searches experimentally challenging.  However, they are currently the only way to explore crucial properties of the neutrino.  Aspects of these experiments are discussed in the next section.
\FloatBarrier

\section{\zvbb searches}
\begin{comment}
Discuss \zvbb process and sensitivity to nature of neutrino.
Discuss concurrent sensitivity to hadron part
I feel like I should discuss ongoing searches but not in much detail?  Relevant information is: expected lifetime, mass, expected counts/year, expected limits?
Okay, yes.  Here is how this section could go: discuss the process and the resulting equation for the lifetime, and then talk about each of the components of the equation.  START with the discussion of the lifetime - can include details of ongoing experiments there.
\end{comment}

Searches for \zvbb are of interest not only because an observation would conclusively demonstrate that neutrinos are Majorana, but also because an observed rate gives information on the absolute mass scale of the neutrino.  The lifetime of \zvbb, assuming it results from the exchange of light Majorana neutrinos, is 
\begin{equation}
(T^{0\nu}_{1/2})^{-1} = G_{0\nu}(Q_{\beta\beta},Z)|M^{0\nu}|^2 {\langle}m_{\beta\beta}{\rangle}^2,
\end{equation}
where $G_{0\nu}(Q_{\beta\beta},Z)$ is a phase space factor, $|M^{0\nu}| = |{\langle}f|O|i{\rangle}|$, where $O$ is the \zvbb operator, is the nuclear matrix element, and $\displaystyle {\langle}m_{\beta\beta}{\rangle} \equiv |\sum_{i}m_i U_{ei}^2|$ is the effective Majorana mass.  The phase space factor can be readily calculated \citep{hadron_zvbb_Suhonen}, although it should be noted that in most current calculations, a scaling factor is introduced to this term so that \NME is a dimensionless quantity \citep{scalingFactorNME}.  The mass term is the effective mass of the electron neutrino and, unlike long-baseline experiments, is sensitive to the mass scale of the lightest neutrino particle.  This dependence is shown in {\fig}~\ref{fig:effectiveMajoranaMass} for the three possible hierarchy schemes.  
\begin{figure}[hp]
\centering
\includegraphics[width=1.0\textwidth]{figures/effectiveMajoranaMass.eps}
\caption[The effective Majorana mass as a function of the smallest neutrino mass.]{The effective Majorana mass as a function of the smallest neutrino mass $m_{min}$ for the inverse hierarchy (IH), normal hierarchy (NH), and quasi-degenerate (QD) scheme.  The values are shown in bands indicating the $2\sigma$ uncertainty.  For all values, it is assumed that $\sin^2{\theta}_{13} = 0.0236$ and $\delta = 0$.  Colors distinguish between different CP-violating scenarios of the Majorana phases.  Red bands require that $\alpha_{31}-\alpha_{21}$ as well as either $\alpha_{21}$ or $\alpha_{31}$ have a CP-violating phases.  Blue and green bands require that both $\alpha_{21}$ and $\alpha_{31}$ have CP-conserving phases.  Overlapping regions are shown hatched.  Figure from {\refref}~\citep{PDG}.}
\label{fig:effectiveMajoranaMass}
\end{figure}

The hadronic dependence of the lifetime, \NME, is sensitive to the initial and final nuclear wavefunctions.  Details of these calculations are discussed in {\chap}~\ref{chap:nucl}, but it should be noted that calculated \NME values for most candidate nuclei vary by as much as a factor of 5.  This uncertainty in \NME directly affects the limits that can be placed on the neutrino mass scale if \zvbb is observed.  Transfer reactions, discussed in {\chap}~\ref{chap:nucl}, offer valuable experimental data that can be used to understand which models are most appropriate for these nuclei and improve the accuracy of the calculations.  The topic of this thesis is the two-proton transfer reaction, which is also discussed in {\chap}~\ref{chap:nucl}.
\begin{figure}[hp]
\centering
\includegraphics[width=0.8\textwidth]{figures/differentNME.eps}
\caption[Uncertainty in current calculated values of \NME.]{Calculated \NME for candidate \zvbb nuclei.  The models used for these calculations are the Interacting Shell Model (ISM) \citep{ISM}, self-consistent renormalized quasi-particle random phase approximation (SRQRPA) \citep{FaesslerReview}, proton-neutron quasi-particle random phase approximation (PNQRPA) \citep{pnQRPA_Suhonen}, generating coordinate method (GCM) \citep{GCM}, the interacting boson model (IBM) \citep{IBM_Iachello}, and the projected Hartree-Fock-Bogoliubov (pHFB) \citep{pHFB}.  Figure taken from \citep{zvbbReviewSchwingenheuer}.  Note that \NME is a dimensionless quantity; this is accomplished by multiplying by the nuclear radius, typically chosen to be $1.2A^{1/3}$~fm.  For details on the use of this factor in \NME calculations see {\refref}~\citep{scalingFactorNME}.}
\label{fig:differentNME}
\end{figure}

Nuclei that are suitable for \zvbb experiments are those that are stable against single beta decay but energetically allowed to double-beta decay.  This is the same group of nuclei in which \tvbb has been studied.  The \zvbb peak is at the endpoint of the \tvbb spectrum, and in fact the \tvbb process is a significant background to \zvbb due to finite detector resolution.  The nuclei $^{48}$Ca, \Ge{76}, \Se{82}, $^{100}$Mo, $^{130}$Te, $^{136}$Xe, and $^{150}$Nd have been used to search for the process in past and present experiments.  See {\tab}~\ref{tab:experiments} for a list of past and present experiments.  Because of the uncertainties on \NME for these nuclei (see {\fig}~\ref{fig:differentNME}, it is not clear which, if any, would enjoy a shorter $T^{0\nu}_{1/2}$.  So many different experiments have arisen because each candidate nucleus offers different advantages in experimental design.  \Ge{76} is an appealing candidate because the active volume also serves as the detector, and Ge crystals are well understood.  Experiments using \Ge{76} are also important because the Heidelberg-Moscow experiment, which claimed an observed \zvbb signal \citep{KlapdorKleingrothaus}, used \Ge{76} crystals.  Other nuclei are appealing because they have high abundance or are easy to obtain.  This is the case for $^{136}$Xe, which is currently being used by the EXO-200 collaboration \citep{EXO200}.  Experiments using $^{136}$Xe typically use time projection chambers (TPC's), which are able to strongly reduce background by reconstructing the momenta of particles in a decay.  Using $^{136}$Xe is particularly appealing because large quantities are readily available, reducing the cost of increasing the mass scale of the experiment.  Another candidate nucleus, $^{130}$Te, is frequently used in experiments using bolometry to detect \zvbb.  A summary of \zvbb searches is shown in {\tab}~\ref{tab:experiments}.

Searches for \zvbb offer access to unique areas of neutrino physics.  Confirmation of the process would demonstrate that neutrinos are Majorana in nature and would also provide a measurement of the absolute mass scale of the electron neutrino.  The dependence of the lifetime on \NME poses a difficulty because the current uncertainty in calculations limits the sensitivity to the neutrino mass scale and also increases the difficulty of planning experiments that search for the process.  Nuclear transfer experiments provide information that can help reduce the uncertainty of \NME calculations, and this thesis focuses on two-proton transfers in the Ge nuclei.  The impact of such a transfer experiment is discussed in {\chap}~\ref{chap:nucl}.

\begin{sidewaystable}
\ra{1.1}\small
\centering
\caption[\zvbb \uppercase{experiments}]{\\\zvbb \uppercase{experiments}}
\label{tab:experiments}

\begin{tabular}{@{}rlllllll@{}}\toprule
\multicolumn{1}{l}{Experiment} & Isotope & Mass [kg] & Method & $T^{2\nu}_{1/2}$ [yr] & $T^{0\nu}_{1/2}$ [yr] & Start - End & {\refref} \\
\midrule
\multicolumn{1}{l}{\textbf{Past Experiments}} \\
Heidelberg-Moscow & \Ge{76} & 11 & ionization & $(1.74\pm0.18)\times 10^{21}$ & $1.19^{+2.99}_{-0.5}\times 10^{25}$ &1990 - 2003 & \citep{KlapdorKleingrothaus} \\
Cuorcino & $^{130}$Te & 11 & bolometer & & $> 2.8\times 10^{24}$ & 2003 - 2008 & \citep{Cuorcino} \\
NEMO-3 & $^{100}$Mo & 7 & track + calorim. & $(0.716\pm0.055)\times 10^{19}$ & $> 1.0\times 10^{24}$ & 2003 - 2009 & \citep{NEMO3} \\
NEMO-3 & $^{82}$Se & 1 & track + calorim. & $(9.6\pm1.1)\times 10^{19}$ & $> 3.2\times 10^{23}$ & 2003 - 2009 & \citep{NEMO3} \\
\noalign{\vskip 0.3cm}

\multicolumn{1}{l}{\textbf{Current Experiments}} \\
EXO-200 & $^{136}$Xe & 175 & liquid TPC & $(2.1\pm0.2)\times 10^{21}$ & $>1.6\times 10^{25}$ & 2011 - & \citep{EXO200} \\
Kamland-Zen & $^{136}$Xe & 330 & liquid scint. & $(2.38\pm0.14)\times 10^{21}$ & $>5.7\times 10^{24}$ & 2011 - & \citep{KamLAND_Zen} \\
GERDA-I/GERDA-II & \Ge{76} & 15/35 & ionization & $(1.88\pm0.1)\times 10^{21}$ & & 2011/2013 - & \citep{Gerda} \\
CANDLES & $^{48}$Ca & 0.35 & scint. crystal & & & 2011 - & \citep{CANDLES} \\
\noalign{\vskip 0.3cm}

\multicolumn{1}{l}{\textbf{Funded Experiments}} \\
NEXT & $^{136}$Xe & 100 & gas TPC & & & 2015 - & \citep{NEXT} \\
Cuore0/Cuore & $^{130}$Te & 10/200 & bolometer & & & 2012/2015 - & \citep{Cuore} \\
Majorana Demo & \Ge{76} & 30 & ionization & & & 2013 - & \citep{Majorana} \\
SuperNEMO Demo & \Se{82} & 7 & track + calorim. & & & 2014 - & \citep{SuperNEMO} \\
SNO+ & $^{150}$Nd & 44 & liquid scint. & & & 2013 - & \citep{SNO} \\
\bottomrule
\end{tabular}
\begin{flushleft}
{\footnotesize NOTE:
Past, present, and future \zvbb experiments.  Detecting a signal in several different isotopes would greatly improve the likelihood of a Majorana process.  From \citep{zvbbReviewSchwingenheuer}.}
\end{flushleft}
\end{sidewaystable}



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