Chapter1.tex

\chapter{Introduction}\label{chap:1}

As a theory of elementary particles and their interactions, the Standard Model of particle physics (SM) is both extremely impressive and notably incomplete. While capable of extremely precise predictions, which have been extensively verified by experimental observation, it does not account for a number of phenomena such as dark matter, dark energy, particle mass differences, gravity, and the abundance of matter over antimatter in the universe. These omissions in the theory must ultimately be resolved with the discovery of new physics beyond the SM (BSM). 

Searches for new physics employ various strategies: some looking for evidence of exotic processes or particles by use of high energy colliders or low background experiments, and others relying on high precision measurements of SM predicted parameters, looking for evidence of new physics at the point where theory and experiment diverge.

The Fermilab Muon $g-2$ experiment (E989) takes the latter approach, aiming to measure the anomalous magnetic moment of the muon, $a_{\mu}$, to a precision of 140 parts-per-billion (ppb). At the time of writing, the combination of first result for $a_{\mu}$ from E989 \cite{SummaryRun1}, with a precision of 460 ppb, and the final result of previous iteration of the Muon $g-2$ experiment at Brookhaven National Laboratory (BNL E821) \cite{BNLFinalReport} presents $4.2\sigma$ tension with the SM \cite{aMuSM}. If the discrepancy persists beyond $5\sigma$, and if no inaccuracies are discovered in the theoretical calculation of $a_{\mu}$, then new physics will be required to explain the anomaly. Additionally, E989 will conduct a world-leading search for the electric dipole moment (EDM) of the muon, $d_{\mu}$, projecting to improve on the current experimental upper limit of $|d_{\mu}|<1.8\times10^{-19}$ $e\cdot$cm, set by E821 \cite{BNLEDM}, by at least an order of magnitude. While the SM predicts a muon EDM of $\mathcal{O}(10^{-36})$ $e\cdot$cm\footnote{This estimate is based on the upper limit for the electron EDM \cite{ImprovedElectronEDMLimit}, and is discussed further in Chapter \ref{chap:1} Section \ref{sec:TheEDM}.}, which is far beyond the reach of current experimental sensitivity, a large muon EDM is allowed by certain well-motivated extensions to the SM; some of which can simultaneously accommodate the observed discrepancy in $a_{\mu}$. If a muon EDM is discovered at E989, it would signal a BSM source of combined charge conjugation and parity (CP) violation in the lepton sector, which may be critical in explaining the matter-antimatter asymmetry of the universe.

This thesis will focus on efforts towards a muon EDM search at E989. In this introductory chapter, the theory of lepton dipole moments will be outlined and the muon EDM search will be motivated in the context of new physics models. Chapter \ref{chap:2} will introduce the essential physics principles of the experiment, while Chapter \ref{chap:3} will discuss the specific techniques and hardware that comprise E989. Chapter \ref{chap:4} will detail the development and execution of a novel technique for measuring the radial component of the E989 storage ring magnetic field to a precision of $<1$ parts-per-million (ppm). This work ensures that the radial magnetic field, which can mimic an EDM signal, does not present a limiting source of systematic uncertainty in search for $d_{\mu}$ in Run-1, and beyond. Chapter \ref{chap:5} describes the large-scale simulation efforts which are essential to the characterisation of sources of systematic uncertainty in the measurement, and, critically, the momentum dependent dilution of the EDM signal. In Chapter \ref{chap:6}, a blinded search for $d_{\mu}$ in Run-1 using data from straw tracker detectors is presented, including an assessment of sources of systematic uncertainty, which is concluded with an estimation of a preliminary upper limit on $|d_{\mu}|$. Finally, Chapter \ref{chap:7} concludes this thesis, providing a summary of results and discussion as to the outlook of the search for a muon EDM at Fermilab. 

\section{The anomalous magnetic moment of the muon, $a_{\mu}$} \label{sec:TheMDM}
%
The magnetic dipole moment, $\vec{\mu}$, of a subatomic particle with mass $m$, charge $e$, and spin $\vec{s}$ is described by the expression %\footnote{Natural units, where $\hbar=c=1$, will be assumed throughout this thesis, unless stated otherwise.} 
%
\begin{equation} 
  \vec{\mu} = g\left(\frac{Qe}{2m}\right)\vec{s},
  \label{eqn:MDM}
\end{equation}
%
where $Q=\pm1$. $g$ is the $g$-factor: a dimensionless quantity which characterises the strength of the coupling between the magnetic moment and the spin. Classically, where the spin angular momentum may be compared to the total orbital angular momentum of a rotating distribution of charges, the $g$-factor would be equal to one \cite{Jackson}. For particles on the subatomic scale, however, it has been extensively proven that $g\neq1$. %By describing the behaviour of the magnetic moment in an external magnetic field, $\vec{B}$, in terms of the interaction Hamiltonian 

The origin of this deviation arises in part from relativistic effects, which may be reconciled with quantum mechanics via the Dirac equation: giving the prediction that $g=2$ for spin 1/2 particles (fermions). While this prediction holds at leading order, the interaction between the fermion and an external magnetic field includes contributions to the $g$-factor from higher order virtual particle loop diagrams at the fermion-photon vertex, known as radiative corrections, so that $g\neq2$. In the case of the electron and the muon, $g$ is slightly greater than 2. This deviation is the anomalous magnetic moment, $a$, which is defined as 
%
\begin{equation}
  a = \frac{g-2}{2},
  \label{eqn:AnaMagMom}
\end{equation}
% 
allowing Equation \ref{eqn:MDM} to rewritten in the form
%
\begin{equation}
  \vec{\mu} = (1+a)\frac{e\hbar}{2m},
  \label{eqn:MDM2}
\end{equation}
% 
separating the magnetic moment into two parts: the Dirac moment and the \textit{anomaly} (the Pauli moment) \cite{LeptonDipoleMoments}. 

\begin{figure}[b!]
\centering{}
\includegraphics[trim={0 0 0 0},clip,width=0.89\textwidth]{Images/Chapter1/FeynmanDiagrams.pdf}
\caption{SM contributions to $a_{\mu}$. From left to right: leading order QED (Swinger), leading order electroweak, hadronic vacuum polarisation, and hadronic light-by-light scattering. Image reproduced from \cite{SummaryRun1}.}
\label{fig:SMContributions}
\end{figure}

The anomalous magnetic moment of the electron, $a_{e}$, was first measured by Kusch and Foley in 1947 \cite{KuschAndFoley}, with the leading order quantum electrodynamics (QED) radiative correction being famously calculated by Swinger soon after \cite{Swinger}. Since then, $a_{e}$ has become the most precisely predicted \cite{ElectronAnomalyPrediction} and measured \cite{ElectronAnomalyMeasurement} physical quantity ever: a stringent test of QED and a resounding success for the SM.

The anomalous magnetic moment of the muon, $a_{\mu}$, while not as suitable for probing QED as the electron, presents a far more promising avenue in the search for new physics \cite{TheMuonAnomalyAndNewPhysics}. Quantitatively, the reason for this is that a particle with mass $M$, where $M>>m$ (with $m$ being the elementary fermion mass), makes a contribution to the anomaly $a$ on the scale of 
%
\begin{equation}
  \delta a \sim C\cdot\left(\frac{m}{M}\right)^{2}, 
  \label{eqn:NewPhysicsScale}
\end{equation}
%
where $C=\delta m/m$, $\delta m$ being the new physics contribution to the fermion mass, which is highly dependent on the details of the new physics model in question (discussed further in Chapter \ref{chap:1} Section \ref{sec:NewPhysics}). Given that the muon is $\sim200$ times more massive than the electron, this implies that the muon magnetic anomaly is $\sim4\times10^{4}$ times more sensitive to the higher mass-energy scales where BSM physics is most likely to reside.
%
\begin{figure}[t!]
\centering{}
\includegraphics[trim={0 0 0 0},clip,width=0.89\textwidth]{Images/Chapter1/ResultPlotRun1.png}
\caption{Current experimental values of $a_{\mu}$ from Brookhaven E821 and Fermilab E989, as well as the experimental average, compared with the SM prediction. Image reproduced from \cite{SummaryRun1}.}
\label{fig:Run1Result}
\end{figure}
%

Within the SM, $a_{\mu}$ is composed of contributions from all sectors: the electromagnetic (QED), electroweak (EW), and hadronic (had), so that 
%
\begin{equation}
  a_{\mu}^{\text{SM}} = a_{\mu}^{\text{QED}} + a_{\mu}^{\text{EW}} + a_{\mu}^{\text{had}}, % \,( +\, a_{\mu}^{\text{BSM}}), 
  \label{eqn:SMContributions}
\end{equation}
%
where Feynman diagrams summarising the classes of SM contributions to $a_{\mu}$ are shown in Figure \ref{fig:SMContributions}. If there exists a discrepancy between experiment and the SM, $\Delta a_{\mu} = a_{\mu}^{\text{Exp}} - a_{\mu}^{\text{SM}}$, then a fourth (BSM) contribution must included, so that 
%
\begin{equation}
  a_{\mu}^{\text{Exp}} = a_{\mu}^{\text{SM}} + a_{\mu}^{\text{BSM}}. % \,( +\, a_{\mu}^{\text{BSM}}), 
  \label{eqn:AllContributions}
\end{equation}
%
At present, the SM prediction for the muon magnetic anomaly is $a_{\mu}^{\text{SM}}=(116591810\pm43)\times10^{-11}$ \cite{aMuSM}, while the current combined experimental value, which includes input from the first result from Fermilab E989 \cite{SummaryRun1} and the final result from Brookhaven E821 \cite{BNLFinalReport}, is $a_{\mu}^{\text{Exp}}=(116592061\pm41)\times10^{-11}$; so that the discrepancy stands at $\Delta a_{\mu} =(251\pm59)\times10^{-11}$. As illustrated in Figure \ref{fig:Run1Result}, this presents a $4.2\sigma$ tension between theory and experiment which, while not yet conclusive, indicates that new physics extensions to the SM may be required to account for the anomaly. The topic of new physics is expanded upon in Section \ref{sec:NewPhysics}.% Moreover, since E989 has yet to fold the vast majority of its total dataset into the measurement, the discrepancy is set to exceed the 5$\sigma$ threshold for discovery in the near future.
%

\section{The electric dipole moment of the muon, $d_{\mu}$} \label{sec:TheEDM}

In addition to a magnetic dipole moment, Dirac's theory predicts that subatomic particles may also possess an electric dipole moment (EDM), $\vec{d}$ \cite{LeptonDipoleMoments}. Classically, an EDM is a measure of the permanent polarisation of a system of electric charges. For the simplest case of two opposing point charges, $\pm q$, separated by a distance $\vec{r}$, the EDM is given by $\vec{d}=q\vec{r}$. More generally, $\vec{d}$ is equal to the integral of the charge density, $\rho(\vec{r})$, over the charge volume, so that
%
\begin{equation} 
  \vec{d} = \int{\vec{r}\rho(\vec{r})\,d^{3}r},
  \label{eqn:EDMIntegral}
\end{equation}
%
where $\vec{r}$ has it's origin at the centre of charge distribution \cite{Jackson}. In the case of elementary fermions, which are treated as point-particles of no spatial extent within the SM, the classical argument would follow that such particles must have a EDM of zero. In quantum mechanics, however, a non-vanishing EDM is allowed by polarisation of the vacuum field around the particle. Like the magnetic moment, the EDM must be directed along the spin vector, the only vector-like property associated with an elementary particle, and is expressed in form similar to Equation \ref{eqn:MDM}:
%
\begin{equation} 
  \vec{d} = \eta \left(\frac{Qe}{2mc}\right)\vec{s},
  \label{eqn:EDM}
\end{equation}
%
where $c$ is the speed of light in a vacuum. $\eta$ is a dimensionless quantity, analogous to the magnetic $g$-factor, which describes the strength of the coupling between the EDM and the spin. $\eta$ may be expressed in terms of fundamental constants, so that
%
\begin{equation} 
  \eta = \frac{4dmc}{Qe\hbar},
  \label{eqn:eta}
\end{equation}
%
where $d$ assumes the convention
%
\begin{equation} 
  \vec{d} = d\cdot\hat{s}.
  \label{eqn:scalarEDM}
\end{equation}

Unlike its magnetic counterpart, the EDM of the muon and other subatomic particles breaks the symmetries parity, $P$, where $(t,x)\rightarrow(t,-x)$, and time reversal, $T$, where $(t,x)\rightarrow(-t,x)$. This $P$ and $T$ symmetry breaking arises from the differing properties of axial vectors and polar vectors under $P$ and $T$ transformations \cite{LeptonDipoleMoments}. Axial vectors, which describe a rotation about an axis, are even under $P$ (they do not change sign) and polar vectors are odd. Magnetic field vectors, $\vec{B}$, are axial and electric field vectors, $\vec{E}$, are polar. Spin is an axial vector, making the dipole moments $\vec{\mu}$ and $\vec{d}$ axial vectors by Equations \ref{eqn:MDM} and \ref{eqn:EDM}. The third symmetry is charge conjugation, $C$, where $q\rightarrow-q$, under which all quantities mentioned here are odd. The combined symmetry $CPT$ is expected to hold, so it follows that if $\vec{B}$ is even under $P$ then it must be odd under $T$. The same logic applies for $\vec{E}$ and $\vec{s}$. The various transformation properties of $\vec{E}$, $\vec{B}$, $\vec{\mu}$, and $\vec{d}$, are given in Table \ref{tbl:CPT}. 

\begin{table}[h!]
\centering
\begin{tabular}{l|ccc}
\hline
\hline
 & $\vec{E}\quad$ & $\vec{B}\quad$ & $\vec{\mu}$ or $\vec{d}\quad$ \\ 
\hline
 $C\quad$ & $-\quad$ & $-\quad$ & $-\quad$ \\ 
 $P\quad$ & $-\quad$ & $+\quad$ & $+\quad$ \\
 $T\quad$ & $+\quad$ & $-\quad$ & $-\quad$ \\
\hline
\hline
\end{tabular}
\caption{The transformation properties of $\vec{E}$, $\vec{B}$, $\vec{\mu}$, and $\vec{d}$.}
\label{tbl:CPT}
\end{table}

The external magnetic and electric fields, $\vec{B}$ and $\vec{E}$ may be related to their corresponding dipole moments by the interaction Hamiltonian
%
\begin{equation} 
  \mathcal{H} = -\vec{\mu}\cdot\vec{B}-\vec{d}\cdot\vec{E}, 
  \label{eqn:TotalHamiltonian1}
\end{equation}
%
from which it can be deduced that the magnetic part of the interaction is $CP$ even and the electric part, the EDM, is $CP$ odd. This means that subatomic particle EDMs violate $CP$ symmetry: one part of Sakharov's criteria \cite{Sakharov} for a universe dominated by matter rather than antimatter. 

Clearly, particle EDMs provide a unique insight into fundamental physics, and have been considered as such since Purcell and Ramsey first proposed a search for the neutron EDM as a means of testing P symmetry in 1950 \cite{PurcellAndRamsey}. Since then, however, no permanent EDM has been observed for any particle or atom \cite{ChuppEDMReview}. Moreover, the SM predictions for the magnitude of particle EDMs are vanishingly small, well below their current upper limits and out of reach of today's experiments. This is demonstrated Table \ref{tbl:ParticleEDMs}, where the upper limits for the EDM of the proton, neutron, electron, and muon are presented; along with their SM predicted values and corresponding references. 

\begin{table}[t!]
\centering
\begin{tabular}{l|ccc}
\hline
\hline
 Particle & EDM upper limit [$e\cdot$cm] & SM prediction [$e\cdot$cm] & References \\ 
\hline
 Proton & $2.0\times10^{-26}$ (95\% C.L.) & $\mathcal{O}(10^{-32})$ & \cite{199HgEDMLimits}, \cite{ProtonNeutronEDMPred}  \\ 
 Neutron & $1.6\times10^{-26}$ (95\% C.L.) & $\mathcal{O}(10^{-32})$ &  \cite{199HgEDMLimits}, \cite{ProtonNeutronEDMPred}  \\
Electron & $1.1\times10^{-29}$ (90\% C.L.) & $\mathcal{O}(10^{-38})$ & \cite{ImprovedElectronEDMLimit}, \cite{ElectronEDMPred} \\
 Muon & $1.8\times10^{-19}$ (95\% C.L.) & $\mathcal{O}(10^{-36})$ & \cite{BNLEDM}, \cite{ElectronEDMPred} \\
\hline
\hline
\end{tabular}
\caption{The upper limits for the EDM of the proton, neutron, electron, and muon, along with their SM predicted values and corresponding references.}
\label{tbl:ParticleEDMs}
\end{table}

The SM prediction for the magnitude of the muon EDM in Table \ref{tbl:ParticleEDMs} is estimated based on the electron EDM upper limit, under the assumption of minimal flavour violation (MFV) \cite{MFV} which requires that EDMs scale linearly with mass across generations of leptons, as follows
%
\begin{equation}
  d_{\mu} \approx d_{e}\frac{m_{\mu}}{m_{e}}.
  \label{eqn:EDMMassScaling}
\end{equation}
%
If this assumption holds, then the muon EDM is inaccessible by current experiments, and will remain so for the foreseeable future. However, MFV scaling is called into question by the observation of flavour anomalies in $B$-decays, such as the recent result from the LHCb experiment \cite{LHCb2021}, lending support to SM extensions which could allow for a muon EDM within reach of E989. Further discussion on this topic is given in the following section. 

\section{New physics}\label{sec:NewPhysics}

The discovery of a large muon EDM would signal a significant departure from the SM. In particular, it would necessitate a BSM model involving a new complex parameter with a large CP violating phase, far in excess of that allowed within the SM. It would also indicate a breakdown of the linear mass scaling of EDMs between lepton generations, making it incompatible with the assumption of MFV. 

The discrepancy $\Delta a_{\mu}$ and the bounds on a muon EDM in BSM models are related: the effective field Hamiltonian for the $g-2$ interaction involves a complex Wilson coefficient, $c_{R}^{\mu\mu}$, which has a real part relating to the magnetic anomaly and an imaginary part to a non-vanishing EDM. In the various scenarios which resolve $\Delta a_{\mu}$ with the introduction of heavy TeV-scale particles, the parameter $C$ in Equation \ref{eqn:NewPhysicsScale} must be enhanced by allowing the chirality flip of the muon\footnote{In quantum field theory, the operator corresponding to $a_{\mu}$ connects left- and right-handed muons, making $g-2$ a chirality flipping interaction.} to be provided by a new massive fermion. This so-called chiral enhancement automatically introduces an unconstrained CP violating phase in the imaginary part of $c_{R}^{\mu\mu}$, allowing for a large muon EDM. Contours of $|d_{\mu}|$ as a function of the Wilson coefficient phase and $\Delta a_{\mu}$ are shown in Figure \ref{fig:WilsonCoeff} \cite{CombinedExplantionsForaMuAndEDM2018} \cite{CombinedExplantionsForaMuAndEDM2020}, with red bands indicating $\Delta a_{\mu}$ preferred from experiment\footnote{Excluding the most recent result from Fermilab \cite{SummaryRun1}.}, a dark blue band indicating the projected sensitivity of the Fermilab muon EDM search, and a light blue indicating the same for a proposed search experiment at PSI \cite{PSI}. From this, it may be inferred that a muon EDM may be measurable at Fermilab, assuming a contribution to $\Delta a_{\mu}$ from BSM models with chiral enhancement.

\begin{figure}[t!]
\centering{}
\includegraphics[trim={0 0 0 0},clip,width=0.69\textwidth]{Images/Chapter1/WilsonCoeff.png}
\caption{Contours of $|d_{\mu}|$ as a function of the Wilson coefficient phase and $\Delta a_{\mu}$. The red bands indicating $\Delta a_{\mu}$ preferred from experiment (which does not include the most recent Fermilab result \cite{SummaryRun1}), the dark blue band indicating the projected sensitivity of the Fermilab muon EDM search, and the light blue indicating the sensitivity of a proposed experiment at PSI \cite{PSI}. Image reproduced from \cite{CombinedExplantionsForaMuAndEDM2018}.}
\label{fig:WilsonCoeff}
\end{figure}

A well-known example of a model which can both provide chiral enhancement and accommodate $\Delta a_{\mu}$, and violate MFV scaling, is the minimally supersymmetric SM (MSSM). Supersymmetry (SUSY) involves the introduction of so-called superpartners to SM particles, which have values of spin offset by $1/2$ from their SM counterparts, so that SM fermions have a bosonic superpartner and vice versa. In the MSSM, chiral enhancement is provided by the parameter $\tan\beta$, which is the ratio of vacuum expectation values of the two Higgs fields in the model. MSSM parameter space capable of resolving $\Delta a_{\mu}$ has been severely constrained by the persistent lack of evidence from the LHC and dark matter searches, with scenarios involving smuons and heavy charginos requiring unfavourably large values of $\tan\beta$. However, one scenario which could accommodate the discrepancy involves a Bino-like\footnote{The $B$-boson superpartner.} lightest supersymmetric particle (LSP) and slepton loop contribution, allowing for a wide range of mass scales and $\tan \beta$ values \cite{NewPhysicsExplanations2021}. 

A concrete alternative to the MSSM are models which introduce heavy scalar leptoquarks (LQ): BSM bosons which couple simultaneously to SM leptons and quarks, carrying both lepton and baryon number. Scalar LQ models are minimally single particle extensions to the SM, and can resolve $\Delta a_{\mu}$ via virtual loop interactions at the muon-photon vertex with a chiral enhancement factor of $m_{t}/m_{\mu}$, where $m_{t}$ is the top quark mass \cite{NewPhysicsExplanations2021}. While many LQ models have been excluded, they remain well-motivated by both $\Delta a_{\mu}$ and the aforementioned flavour anomalies which also call MFV into question.

All considered, the muon EDM search at Fermilab is well motivated by the current landscape of new physics models, some of which allow for measurable values of $d_{\mu}$ while simultaneously accommodating the discrepancy $\Delta a_{\mu}$. The discussion presented here is far from exhaustive, and further reading on this topic may be found in \cite{CombinedExplantionsForaMuAndEDM2018}, \cite{CombinedExplantionsForaMuAndEDM2020}, and \cite{NewPhysicsExplanations2021}.