%-----------------------------------------------------------------------------------------------------------------
%       Chapter 1: an Introduction to Dissertation
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\chapter{The Standard Model}

\singlespacing

\epigraph{``\textit{If I have seen further, it is by standing on ye shoulders of giants.}''}{
    --- Sir Isaac Newton, \emph{Letter from Sir Isaac Newton to Robert Hooke}}

\doublespacing % The whole thesis is supposed to be double spaced. The spacing will be switched locally when needed

    The Standard Model (SM) of particle physics is one of the most successful and precise theories in physics to 
    date. It contains the culmination of our knowledge involving the fundamental constituents of matter and how 
    they interact with one another. It also provides a framework on which we can analyze experimental data and 
    make predictions for what happens when these particles collide at ultrarelativistic speeds. Experiments to 
    probe these limits and energy scales exist around the globe as a collaborative effort involving several 
    countries and thousands of scientists and universities. One of the most notable of these collaborations being
    the \emph{Conseil Europ\'{e}en pour la Recherche Nucl\'{e}aire}, (CERN)
    \footnote{About CERN \url{http://home.cern/about}}. Here the largest particle collider ever built, 
    the Large Hadron Collider (LHC), is operated by researchers across the world in several collaborations such 
    as, ATLAS, CMS, Alice, and LHCb to name but a few. It is the LHC and specifically the ATLAS and CMS 
    collaborations that form the focus and impetus of this dissertation. Let us begin our journey with the formation 
    and structure of the lauded SM and why it is we know there is more yet to discover about the fundamentals of nature.

    \section{Composition and group structure}\label{PartsAndStructure}
        
        We begin our dicussion of the SM with a look at its structure and makeup. As mentioned, the SM is not 
        only a collection of the fundamental particles but also describes how those particles interact. To aid 
        with our terminology it is best to start with what are the fundamental particles of nature and later I 
        will address the interactions that are significant to this work in particular.

        \subsection{ Composition }\label{Composition}
            
            Matter, as we observe and interact with on a regular basis, is made up predominately of protons, 
            neutrons, and electrons. Along with these main components, we observe and measure matter, in many 
            cases, through the use of light, electromagnetic waves that interact with matter as packets or 
            ``quanta'' of energy that we call photons. As it turns out, electrons and photons are the first 
            of our fundamental particles. Protons and neutrons, however, have an underlying structure that 
            containing particles classified as fermions and bosons.

            
            \subsubsection{ Fermions }\label{fermions}

                Fermions are particles that have $\sfrac{1}{2}$ integer spin, with the exception of neutrinos, 
                carry electric charge. They are further subdivided into two categories called leptons and quarks. 
                The electron, $e$, that we mentioned before is the first lepton in our picture of nature and has 
                two other compatriots known as the muon, $\mu$, and tau, $\tau$. Each of these leptons 
                has integer charge $\pm e$ and has a corresponding neutral partner called the neutrino: 
                $\nu_{e}, \ \nu_{\mu}, \ \nu_{\tau}$, that have almost no measureable mass. Without much mass and 
                with no electric charge they interact with only a limited number of other particles, which makes 
                them especially difficult to detect. Each pair of lepton and corresponding neutrino form 
                generations and are often organized in doublets,
                %
                \begin{align}
                    %
                    \doublet{\nu_{e}}{e} ,\quad \doublet{\nu_{\mu}}{\mu} ,\quad \doublet{\nu_{\tau}}{\tau} \ .
                    \label{doubletLepton}
                    %
                \end{align}
                %

                Quarks are the other fermionic particles in the SM and are likewise split up into three 
                generations and organized into doublets, \Cref{doubletQuark}. Unlike the leptons, quarks have 
                fractional charge and also have a quantum number known as color charge. The `up-type' quarks:
                up, charm, and top, have a charge of $\sfrac{2}{3} e$ and the `down-type' quarks; down, 
                strange, and bottom; have a charge of $\sfrac{-1}{3} e$. Of these six quarks, it is the up 
                and down quarks that contribute the most to the behavior of the protons and neutrons and are 
                referred to as `valence' quarks. Specifically, the proton has two up quarks and one down quark 
                as its valence quarks and the neutron has one up quark and two down quarks as its valence quarks. 
                The term valence in this respect simply means that of the quarks one might find in a proton or 
                neutron the valence quarks contain the largest fraction of the proton's or neutron's energy. The 
                distribution of the quarks and gluons as functions of the energy fraction they carry from the 
                parent proton or neutron is called, the parton distribution function (PDF) \cite{Hou2016,Martin2009}.
                %
                \begin{align}
                    %
                    \doublet{u}{d} ,\quad \doublet{c}{s} ,\quad \doublet{t}{b} \ .
                    \label{doubletQuark}
                    %
                \end{align}
                %


            \subsubsection{ Bosons }\label{bosons}

                In addition to fermions, which follow Fermi-Dirac statistics and have half-integer spin we also 
                have particles with integer spin that follow Bose-Einstein statistics and are thus called bosons. 
                Their ranks include the gluons, W and Z bosons, the photon, and the Higgs boson. All except for 
                the Higgs are considered gauge bosons and mediate one of the fundamental forces of nature: 
                the strong force, the weak force, and the electromagnetic force. 

                The W and Z bosons; $W^{+}, \ W^{-}, \ Z$; mediate the weak force and interact with any particle 
                that has weak hypercharge and are observed mostly in neutrino scattering and nuclear interactions 
                such as interchange between protons an neutrons. The gluons, $g^{\mu}_{a}$, mediate the strong 
                force and interact with any particle that has color charge. In the SM this means that quarks are 
                the only particles that couple directly to the gluons. The strong force is predominately realized 
                as the force that holds the nucleons together by binding the quarks in the nucleons, protons and 
                neutrons, together. This leads to a phenomenon known as quark confinement where at low energies we 
                do not see any quarks that are not bound inside of some composite object. The last gauge boson that 
                exists in the SM is the photon, $\gamma$, which mediates the well known electromagnetic force and 
                interacts directly with any particles that have electric charge. Gravity remains the only 
                fundamental force that the SM does not incorporate. 

            \subsection{ Group structure }\label{GroupStructure}

                The particles that make up the SM are resonances of fields that are described within 
                a set of mathematics known as group theory. Each group contains information about the particle 
                fields and information on how particle states are created, destroyed, and evolve in space and 
                time. In the SM the group structure is $SU(3)_{c} \otimes SU(2)_{L} \otimes U(1)_{Y}$. The order 
                of each group can tell us how many unique generators, and thus bosons, the group contains. For 
                example, $SU(3)_{c}$ is third order and so contains \( 3^2 -1 = 8 \) unique generators giving us 
                eight gluons, $G^{\mu}_{a}$. Following that example, $SU(2)_{L}$ has three unique generators giving us the three 
                W bosons, $W^{\mu}_1, \ W^{\mu}_2,$ and $W^{\mu}_3$. This leaves $U(1)_{Y}$ which has only one 
                generator that gives rise to the $B^{\mu}$ boson. When we bring these fields together with the 
                other elementary particles we can construct the SM Lagrangian density in its most concise form:
                %
                \begin{align}
                    %
                    \mathscpt{L} =& -\frac{1}{4} \left[ B^{\hspace{.05em}\mu \nu} B_{\mu \nu}   
                     + W^{i \mu \nu} \cdot W_{i \mu \nu} 
                     + G^{\hspace{.05em}\mu \nu}_{a} G_{a \hspace{.05em} \mu \nu} \right] \notag \\
                     %
                     &+ \overline{\psi} \left( i \gamma^{\mu} D_{\mu} - m \right) \psi 
                     + \overline{\psi}_{j} \left( i \gamma_{\mu} D'^{\hspace{.05em}\mu}_{jk} 
                     - M_{j} \delta_{jk} \right) \psi_{k} \notag \\
                     %
                     &+ \left( D^{\mu} \Phi \right)^{\dagger} \left( D_{\mu} \Phi \right) 
                     - \sum_{f} \lambda_{f} \left( \overline{\psi}_{L} \Phi \psi_{R} 
                     + \overline{\psi}_{R} \widetilde{\Phi} \psi_{L} \right) - V(\Phi) \, , \notag \\
                     &i = 1, 2, 3 \hspace{1.5cm} \text{ and } \hspace{1.5cm} a = 1, 2, \ldots, 8 \label{SMLagrangian} .
                     %&- \mu^2 \Phi^{\dagger} \Phi - \lambda \left( \Phi^{\dagger} \Phi \right)^2 \notag
                    %
                \end{align}
                %
                We can expand this further with the following definitions:
                %
                \begin{align}
                    %
                    D_{\mu} &= \partial_{\mu} + i g' \frac{Y}{2} B_{\mu} + i g t^{i} \tensUD[W]{i}{\mu} \, ,
                    \label{EWcovariantDerivative} \\
                    %
                    D'^{\hspace{.05em}\mu}_{jk} &= \delta_{jk} \partial^{\mu} + i g_{3} (T_{a})_{jk} \hspace{.05em} \tensUD[G]{\mu}{a}
                    \label{QCDcovariantDerivative} \, , \\
                    %
                    \overline{\psi} \psi &= \overline{\psi}_{L} \psi_{R} + \overline{\psi}_{R} \psi_{L} \label{psibarpsi} \, , \\
                    %
                    &\text{whe}\text{re} \notag \\
                    %
                    \psi_{L, R} &= \frac{1}{2} ( 1 \mp \Gfive ) \label{psiRelation} \ . 
                    %
                \end{align}
                %
                In the above definitions, we also need to take special care with the scalar field, $\Phi$, which is the Higgs field. 
                It is important to distinguish that,
                %
                \begin{align}
                	\Phi = \doublet{\phi^{+}}{\phi^{0}} \quad &\text{and} \quad 
                    \widetilde{\Phi} = S \Phi = \doublet{ \phi^{0^{*} } }{- \phi^{-} } \label{PhiTilde} \, , \\
                    %
                    &\text{where} \notag \\
                    %
                    S = i \sigma_{2} &= \left(\begin{array}{cc}0 & +1 \\-1 & 0\end{array}\right)
                        \label{IsoSpinOP} \ .
                \end{align}
                %
                All of the rank two tensors above represent the gauge boson field strength, the $\psi$ terms represent the fermion 
                fields, the $\gamma_{\mu}$ term is the Dirac matrices associated with the fermions, the $m \text{ and } M$
                terms represent mass matrices for the fermions, and $\Phi$ is the Higgs field. The coupling between the 
                fermions and Higgs fields is referred to as the Yukawa portion of the Lagrangian. Here the $\lambda_{f}$ 
                terms show the couplings between the Higgs field and the fermions. For leptons $\lambda_{f}$ define the 
                entire coupling, for the quarks, however, they include an additional color tensor, $G^{ab}, F^{ab}$:
                %
                \begin{align}
                    %
                    \text{up-type quarks: } &\lambda_{u} \equiv F^{ab} \label{upTypeLambda} \, , \\
                    %
                    \text{down-type quarks: } &\lambda_{d} \equiv G^{ab} \label{downTypeLambda} \, .
                    %
                \end{align}
                %

                As mentioned before, the total group structure of the SM is $SU(3)_{c} \otimes SU(2)_{L} \otimes U(1)_{Y}$, 
                where $D_{\mu}$, \Cref{EWcovariantDerivative}, is the electroweak covariant derivative and 
                $D'^{\hspace{.05em}\mu}_{jk}$, \Cref{QCDcovariantDerivative}. Together these two terms handle all couplings 
                between the gauge bosons and the other fields of the SM. If you look once more at \Cref{SMLagrangian}, you 
                will notice that the Higgs field is only operated on by the electroweak covariant derivative and is not part 
                of the $SU(3)_{c}$ group structure despite the presence of color matrices in the Yukawa terms. This situation 
                brings about some interesting possibilities, namely in regard to the Higgs Mechanism and the breaking of the 
                electroweak symmetry.
                
                When we combine $SU(2)_{L}$ with $U(1)_{Y}$ we get $SU(2)_{L} \otimes U(1)_{Y}$ which is the group structure 
                of electroweak theory. When the electroweak symmetry is broken, the $W^{\mu}_1$ and $W^{\mu}_2$ to mix and 
                $W^{\mu}_3$ and $B^{\mu}$ mix to generate the physical bosons, $W^{\pm}, \ Z$ and $A$ (photon), 
                \Cref{electroweakMix},
                %
                \begin{subequations}
                    \begin{equation}
                        %
                        \doublet{Z^{\mu}}{A^{\mu}} = \rotmatrix{\theta_{W}} \doublet{W^{\mu}_{3}}{B^{\mu}} \ , 
                        \label{electroweakMix} 
                        %
                    \end{equation}
                    %
                    \begin{equation}
                        \doublet{W^+}{W^-} = \frac{1}{\sqrt{2}} \GENrotMat{1}{-i}{i}{1} \doublet{W^{\mu}_1}{W^{\mu}_2} \ .
                        \label{WpmMix}
                        %
                    \end{equation}
                \end{subequations}
                %
                This formulation can technically exist independent of the electroweak symmetry breaking (EWSB) but is not 
                particularly motivated by anything physical. When EWSB occurs spontaneously, however, the $W$ and $Z$ 
                bosons can acquire a mass while the photon, $A^{\mu}$, remains massless. The breakdown and mixing of the 
                fields then has meaning as the bosons are now distinguishable from one another.

            \section{ The Higgs mechanism }\label{HiggsMechanism}
                
                The Higgs mechanism is a process by which the $SU(2)_{L} \otimes U(1)_{Y}$ symmetry is broken by the vacuum 
                expectation value of the Higgs Field, and by which particles acquire mass without having to break the overall 
                symmetry of the Lagrangian. When the electroweak, $SU(2)_{L} \otimes U(1)_{Y}$, symmetry is broken the Higgs 
                potential acquires a vacuum expectation value (VEV) that is unique and causes the couplings between the fundamental 
                particles and the Higgs boson to be proportional to the particle's mass. This effect is a consequence
                of minimizing the Higgs potential,
                %
                \begin{align}
                    V(\Phi) = \mu^2 \Phi^{\dagger} \Phi + \lambda \left( \Phi^{\dagger} \Phi \right)^2 \ .
                    \label{SMHiggsAlgebraic}
                \end{align}
                %
                When the above potential is minimized according to standard calculus techniques we have:
                %
                \begin{align}
                    %
                    \pd{}{V(\Phi)}{|\Phi|} = \ &2 \mu^2 |\Phi| + 4 \lambda |\Phi|^3 = 0   \label{minPotential} \\
                    %
                    \text{which produces two options} &\text{ for the minimum value of } \Phi \notag \\
                    %
                    |\Phi| &= \pm \left( \frac{-\mu^2}{2\lambda} \right)^{\sfrac{1}{2}} \ . \label{minHiggs}
                    %
                \end{align}
                %
                At this point we seemingly have `two' choices for the minimum of the Higgs potential, call them $\pm V_0$.
                Without losing generality, we can shift the potential with a constant as expected but it does nothing to
                offset the minima as shown in \Cref{realHiggsPotential}. However, if the Higgs field were to make a choice,
                say choose $+V_{0}$ or $-V_{0}$, then the minimum would take on a defined value and now 
                $\bra{0}\Phi\ket{0} \neq 0$ but takes on an expectation value, called $v$ or the VEV. Generally, a break in 
                a symmetry for a particular field would disrupt the overall symmetry of the Lagrangian but in the case of 
                scalar particles that is not true. More specifically, $\mathscpt{L}(\Phi) = \mathscpt{L}(-\Phi)$ but 
                $V(V_{0}) \neq V(-V_{0})$.
                %
                \begin{figure}[H]
                    \begin{center}
                        %
                        \includegraphics[width=0.85 \textwidth, keepaspectratio]{./figures/realHiggsPotential.pdf}
                        %
                        \caption[This is a graphical representation of the real part of the complex Higgs portneital, \Cref{SMHiggsAlgebraic}]{
                            This is a graphical representation of the real part of the complex Higgs potential, \Cref{SMHiggsAlgebraic}, also 
                            known as the `Mexican Hat Potential'. $\pm V_{0}$ indicate the values of the Higgs minimum possible. When 
                            electroweak symmetry is broken, the Higgs field preferentially chooses the $+V_{0}$ minimum.}
                        \label{realHiggsPotential}
                        %
                    \end{center}
                \end{figure}
                %
                
                This is a simplistic picture but instructive in how well it represents the dichotomy of the minimum of the 
                Higgs potential. Realistically, since this is based on a complex field, this is a bit more complicated. What is depicted above, 
                then, is the real part of the Higgs potential. \Cref{complexHiggsPotential} shows a truer picture of how the minimum looks and 
                looks slightly more emblematic of the moniker `` mexican Hat potential," but the results describe are fundamentally no different, 
                if not a bit more striking now. Instead of two choices of minimum value, there are an infinite number of choices creating a circle 
                in the complex plane, however, we often choose a real, positive minimum, $+V_{0}$ from \Cref{realHiggsPotential}, 
                instead of a real, negative or complex value. 
                %
                \begin{figure}[H]
                    \begin{center}
                        %
                        \includegraphics[width=0.85 \textwidth, keepaspectratio]{./figures/complexHiggsPotential.pdf}
                        %
                        \caption[This is a graphical representation of the complex Higgs potential, \Cref{SMHiggsAlgebraic}, potential. 
                        The dark ring in the crease denotes the possible values that minimize the Higgs potential]{This is a graphical 
                        representation of the complex Higgs potential, \Cref{SMHiggsAlgebraic}, potential. The dark ring in the crease 
                        denotes the possible values that minimize the Higgs potential. The $\pm V_{0}$ shown in \Cref{realHiggsPotential},
                        occur as $\imag{\Phi} \to 0$.}
                        \label{complexHiggsPotential}
                        %
                    \end{center}
                \end{figure}
                %
                Once the Higgs field obtains a VEV, its effects on the rest of the particles in the Lagrangian start to become apparent. We can now 
                represent $\Phi$ more completely:
                %
                \begin{subequations}
                    %
                    \begin{equation}
                        %
                        \Phi = \ \frac{1}{\sqrt{2}}\ \doublet{ \sqrt{2} G^{+} }{ \phi^{0} + i G^{0} } \label{smHiggsDoublet} \, ,
                        %
                    \end{equation}
                    %
                    \begin{equation}
                        %
                        \text{where} \notag \\
                        %
                    \end{equation}
                    %
                    \begin{equation}
                        %
                        \phi^{0} = \frac{1}{\sqrt{2}} (H + v) \label{smNeutralHiggs} \, .
                        %
                    \end{equation}
                    %
                    \label{SMHiggs}
                    %
                \end{subequations}
                %
                Here, $H$ or $h^0$ as it is also referred to, is the physical Higgs boson, the discovery of which was announced on July 4, 2012, 
                \cite{Aad2012,Chatrchyan2012}. After the Higgs VEV is obtained, the $W$, $Z$, and fermions all become massive while 
                preserving the symmetry of the Lagrangian. The $G$ terms in the Higgs field above are Goldstone bosons. They are consumed by the 
                electroweak gauge fields and give them mass. In the leptonic sector we must work with a new representation of the Yukawa terms,
                %
                \begin{align}
                	\scpt{L}_{Y} = -\lambda_{e} \ \overline{E}_{L} \Phi \ e_{R} -\lambda_{e} \ \overline{e}_{R} \Phi^{\dagger} \ E_{L} - 
                				F^{ab} \ \overline{Q}^{a}_{L} \Phi \ d^{b}_{R} 
                				- G^{ab} \ \overline{Q}^{a}_{L} \ \widetilde{\Phi} \ u_{R}^{b} \, ,
                	 \label{yukawaReduced}
                \end{align}
                %
                where $\psi_{L,R}$ have been changed to $E_{L}$, $e_{R}$, $Q_{L}$, and $(d/u)_{R}$ to denotes the left-handed lepton doublet, right handed 
                lepton singlet, left-handed quark doublet, and the right-handed quark singlets respectively. The doublet fields are of the
                form,
                %
                \begin{subequations}
                    %
                    \begin{equation}
                        %
                        E_{L} = \doublet{\nu_{\ell}}{e_{\ell}} \label{leptonDoublet} \, ,
                        %
                    \end{equation}
                    %
                    \begin{equation}
                        %
                        Q_{L} = \doublet{u_{L}}{d_{L}} \label{quarkDoublet} \, ,
                        %
                    \end{equation}
                    %
                \end{subequations}
                %
                with $e_{L,R}$ and $(u/d)_{L,R}$ are generic lepton and quark fields that need to be summed over. When we apply the form of the Higgs
                field from \Cref{smNeutralHiggs}, we find that not only do the neutrino fields not interact with the Higgs but we only get pairs of 
                $u_{L}$, $u_{R}$ and $d_{L}$, $d_{R}$ that survive. All cross terms vanish and we are left with 
                %
                \begin{align}
                    %
	                \scpt{L}_{Y} &= -\lambda_{e} \ \overline{e}^{\ -}_{L} \left( \frac{H + v}{ \sqrt{2} } \right) e_{R}
	                			-\lambda_{e} \ \overline{e}^{-}_{R} \left( \frac{H + v}{ \sqrt{2} } \right) e^{-}_{L} \notag \\
	                			&\quad \ - F^{ab} \ \overline{d}^{\ a}_{L} \left( \frac{H +v}{\sqrt{2}} \right) d^{\ b}_{R}
	                			- G^{ab} \ \overline{u}^{\ a}_{L} \left( \frac{H +v}{ \sqrt{2} } \right) u^{b}_{R}
                    \label{yukawaHiggs} \, .
                    %
                \end{align}
                %
                The terms that couple to the VEV then form the mass terms for the fermions,
                %
                \begin{align}
                	-\scpt{L}_{Y} = \overline{e}^{\ -}_{L} \ \widetilde{M}_{\ell} \ e_{R} 
                				+\overline{d}^{\ a}_{L} &\ \widetilde{M}^{ab}_{d} \ d^{b}_{R}
                				+\overline{u}^{\ a}_{L} \widetilde{M}^{ab}_{u} \ u^{b}_{R} \label{SMmassLagrangian} \, , \\
                				&\text{where } \notag \\
                	\widetilde{M}_{\ell} = \frac{ \lambda_{e} v}{\sqrt{2}} \qquad \
                	\widetilde{M}^{ab}_{d} &\ = \frac{v}{\sqrt{2}} F^{ab} \qquad
                    \widetilde{M}^{ab}_{u}   = \frac{v}{\sqrt{2}} G^{ab} \label{SMmassterms} \, .
                \end{align}
                %
                All that remains then is to diagonalize the mass matrices by applying bi-unitary transformations to the lepton and quark fields,
                %
                \begin{align}
                	e'_{L}	  = 		L_{L} \ e_{L} 		
                	\qquad
                	u'^{\ a}_{L} =& \ U^{na}_{L} \ u^{\ a}_{L} 	
                	\qquad
                	d'^{\ b}_{L} = D^{mb}_{L} \ d^{\ b}_{L} 	\label{leftHandedUnitary} \, ,\\
                	%
                	&\text{and}	\notag	 \\
                	%
                	e'_{R}	  = 		L_{R} \ e_{R} 		
                	\qquad
                	u'^{\ a}_{R} =& \ U^{na}_{R} \ u^{\ a}_{R} 	
                	\qquad
                	d'^{\ b}_{R} = D^{mb}_{R} \ d^{\ b}_{R}		\label{rightHandedUnitary} \, .
                \end{align}
                %
                When we diagonalize the mass matrices and sum over $\ell, / a, / b, / m, \text{ and } n$ the eigenvalues fall out naturally and 
                the matrices take on an easily decipherable form,
                %
                \begin{align}
                	L^{\dagger}_{L} \ \widetilde{M}_{\ell} \ L_{R} = M_{\text{L}} 
                	=\left(\begin{array}{ccc}m_{e} & 0 & 0 \\0 & m_{\mu} & 0 \\0 & 0 & m_{\tau}\end{array}\right) \label{leptonMassMat} \, , \\[1.em]
                	%
                	%\ \notag \\
                	%
                	U^{\dagger \ an}_{L} \ \widetilde{M}^{nm}_{u} \ U^{ma}_{R} = M_{U}
                    = \left(\begin{array}{ccc}m_{u} & 0 & 0 \\0 & m_{c} & 0 \\0 & 0 & m_{t}\end{array}\right) \label{UquarkMassMat} \, , \\[1.em]
                	%
                	%\ \notag \\
                	%
                	D^{\dagger \ bn}_{L} \ \widetilde{M}^{nm}_{d} \ D^{mb}_{R} = M_{D}
                	= \left(\begin{array}{ccc}m_{d} & 0 & 0 \\0 & m_{s} & 0 \\0 & 0 & m_{b}\end{array}\right) \label{DquarkMassMat} \, .
                \end{align}
                %
                Feeding these results back into the Yukawa coupling we can finally see the effect of the Higgs VEV on the fermion fields and how 
                the particles `acquire' mass. A more correct statement would be that the strength of the coupling between the particle and the 
                Higgs field is proportional to the particle's mass but it is in some senses a `chicken or the egg' situation due to how closely 
                they are tied together,
                %
                \begin{align}
                	m_{\ell} &= \frac{ v}{\sqrt{2}} \ \lambda_{\ell} \hspace{-1.25in} &\text{ where } 
                	\qquad \ell= e,\ \mu,\ \tau	 \ ;  \label{leptonMasses}\\
                	m_{a} &= \frac{ v}{\sqrt{2}} \ \lambda_{aa} \hspace{-1.25in} &\text{ where } 
                	\qquad a= u, \ c, \ t	 \ ;	  \label{UquarkMasses}\\
                	m_{b} &= \frac{ v}{\sqrt{2}} \ \lambda_{bb} \hspace{-1.25in} &\text{ where } 
                	\qquad b= d, \ s, \ b		  \label{DquarkMasses} \ .
                \end{align}
                %
                We can then define the Yukawa couplings in terms of the particle mass and see directly how strongly the Higgs field couples to 
                each particle, $\displaystyle{ \lambda_{\ell} =  \sqrt{2} \ \frac{ m_{\ell} }{ v } }$, $\displaystyle{ \lambda_{aa} = \sqrt{2} \ 
                \frac{m_a}{v} }$, and  $\displaystyle{ \lambda_{bb} = \sqrt{2} \ \frac{m_a}{v} }$. This rounds out the SM and how the picture of 
                interactions starts to come together but there are still several unanswered questions that leads us to believe there is more than 
                meets the eye, so to speak.


                %
    \section{Need for new physics}\label{BSMmotivation}

    The SM is the most well tested and precise theory to date with the original formulation of Quantum Electrodynamics (QED) in 1948 
    \cite{Schwinger2000,Brown2000,Tomonaga1966,Dyson1996}, to the development of gauge theories in 1954 \cite{Yang1954}, the prediction and later 
    discovery of CP-violation \cite{Branco1999}, the introduction of Quantum Chromodynamics (QCD) in 1973 
    \cite{Seymour2009}, and culminating with the discovery of the Higgs boson in 2012 at the LHC. 
    Despite all of its success, however, the SM does have some limitations. In some ways it is similar to Newtonian mechanics, in the range of its 
    validity and for select topics, the SM expertly explains and predicts the physics of fundamental particle interactions. Just like Newtonian 
    mechanics and relativity, at some point, nature calls for a different approach to particle interactions that can offer a more complete picture 
    of what is going on.

    For instance, If we include gravity in the SM, we generate large divergences at high energy scales \cite{Hooft2015}. While we 
    have observational evidence of dark matter, no candidate for such a particle exists in the SM \cite{VanDenBergh2000}. As mentioned previously, 
    neutrinos do not appear in the Yukawa term of the Lagrangian but are observed to have non-zero mass; but there is no mechanism for this in the SM 
    \cite{Pontecorvo:1957vz}. Cosmologists and astrophysicists have continually verified that there is a large asymmetry between matter and 
    anti-matter in our observable universe but there is no explanation for this in the SM \cite{Bigi2009}. Furthermore, though we 
    have discovered a SM-like Higgs boson matching the predicted mass of $125$ GeV, we do not well understand how the nearly infinite corrections to the 
    Higgs mass term are cancelled \cite{Elias:1980ca}.

    To address these concerns, theorists have proposed several extensions to, and replacements for, the SM but so far none have been observed experimentally.
    Some of the most notable of these theories are supersymmetry (SUSY) \cite{Baer2006}, multi-verse theories \cite{Linde2015}, 
    and string theories \cite{Schwarz2000,Ibrahim2008}. Inside each of these broader areas of study are several select models that contain 
    different features or focus on answering specific questions left open by the SM. In addition to full theoretical frameworks that have been developed 
    there are also highly focused areas where certain sectors are emphasized and the implications of extending those sectors are worked out. In the dark 
    matter realm, for instance, the QCD sector is addressed where Peccei and Quinn introduced a new particle known as the axion and a subsequent symmetry 
    to explain the lack of observed CP-violation in the gluon sector \cite{Peccei1977}. In a similar fashion, much work has been done in 
    extending the Higgs sector in what are called Multi-Higgs Doublet Models (MHDMs) \cite{Branco1999}. One of the most common of which is a 
    Two-Higgs Doublet Model (2HDM), \cite{Gunion1989we}. 
        

    \section{The Two Higgs Doublet Model (2HDM) }\label{2HDMbreakdown}
    
    The extended Higgs sector models are of particular interest as they are also required by all SUSY theories and can have implications for other 
    searches being conducted. As the Higgs boson is the most recently discovered particle, the sector in which the Higgs field operates is one of the 
    least experimentally constrained areas in modern day particle physics. This is partly because of the high mass of the SM Higgs boson but also because 
    of the large QCD background associated with the dominant decay modes of the Higgs \cite{Olive:2016xmw}. In fact, the original discovery channels were in 
    the $ZZ$ and $\gamma\gamma$ channels which collectively only make up about $2.6\%$ of the total decay width of the SM Higgs, meaning $\approx 98\%$ of the 
    time the Higgs will decay into something other than these two states. For these reasons it was a logical place for a pioneering young graduate student to 
    start their search for evidence of new physics. Before we get to that, however, let us take a closer look at how one goes about adding an additional 
    doublet to an area of particle physics.

    In the SM, there is a single Higgs field, $\Phi$, which has the doublet representation shown in \Cref{smHiggsDoublet}. In a 2HDM we append this with an 
    additional doublet of a similar form, so in place of one scalar Higgs field we now have two, $\Phi_{1}$ and $\Phi_{2}$ \cite{Branco2012}, 
    %
    \begin{align}
        %
        \Phi_{a} = \doublet{ \phi^{+}_{a} }{ (v_{a} + \rho_{a} + i \eta_{a}) / \sqrt{2} } , \quad a = 1, \, 2 \ ,
        \label{2HDMdoubletGen}
        %
    \end{align}
    %
    where $\phi$, $\rho$, and $\eta$ are all boson fields. Technically, since the Higgs potential is now dependent on two fields, each field can have its own 
    VEV when the potential is minimized, \cite{Gunion2003}, 
    %
    \begin{align}
        %
        V =& \ m_{11}^2 \PhiPhidagger{1}{1} + m_{22}^2 \PhiPhidagger{2}{2} - \left[ m^{2}_{12} \PhiPhidagger{1}{2} + \hc \right] \notag \\
        &+ \frac{1}{2} \lambda_{1}(\PhiPhidagger{1}{1})^{2} \frac{1}{2} \lambda_{2} (\PhiPhidagger{2}{2})^{2} 
         + \lambda_{3} (\PhiPhidagger{1}{1}) (\PhiPhidagger{2}{2})
         + \lambda_{4} (\PhiPhidagger{1}{2}) (\PhiPhidagger{2}{1}) \notag \\
        &+ \left\{ \frac{1}{2} \lambda_{5} (\PhiPhidagger{1}{2})^{2} 
         + \left[ \lambda_{6} (\PhiPhidagger{1}{1}) + \lambda_{7} (\PhiPhidagger{2}{2}) \right] \PhiPhidagger{1}{2} + \hc \right\} \label{2HDMpotential} \ .
        %
    \end{align}
    %
    However, the two VEV's are not unrelated and we can apply a rotation angle that allows us to talk about one vacuum expectation value in terms of another 
    following the form
    %
    \begin{align}
    	%
        \doublet{ v_{1} }{ v_{2} } = \ &\rotmatrix{\beta} \doublet{ v }{ 0 } \label{VEVmixing} \, , \\
    	%
    	&\text{which results in: }  \notag \\
    	%
        v_{1} = v \cos(\beta)	\hspace{1cm} &\ \hspace{.5cm}	v_{2} = v \sin(\beta)	 \hspace{1.5cm}	\frac{v_{2}}{v_{1}} = \tan(\beta) \, .
    	\label{tanbOrigin}
    	%
    \end{align}
    %
    This transformation is then applied to the remaining fields and allows us to the represent the two Higgs doublets in a more familiar form
    %
    \begin{subequations}
        %
        \begin{equation}
            %
            \Phi_{1} =  \frac{1}{\sqrt{2}} \doublet{ \sqrt{2} \left( G^{+} \cos \beta - H^{+} \sin \beta \right) }{
                v \cos \beta + H_{1} + i \left( G^{0} \cos \beta - A^{0} \sin \beta \right) } \label{HiggsDoublet1} \, , \\
            %
        \end{equation}
        %
        \begin{equation}
            %
            \Phi_{2} =  \frac{1}{\sqrt{2}} \doublet{ \sqrt{2} \left( G^{+} \sin \beta + H^{+} \cos \beta \right) }{
                v \sin \beta + H_{2} + i \left( G^{0} \sin \beta - A^{0} \cos \beta \right) }  \label{HiggsDoublet2} \, , \\
            %
        \end{equation}
        %
    \end{subequations}
    %
    where $G^{+}$ and $G^{0}$ are Goldstone bosons that get absorbed by the $W$ and $Z$ fields as they did in the SM to gain mass. This leaves us 
    now with four unique Higgs fields: $H_{1,2}$, $A^{0}$, $H^{+}$, and one conjugate field, $H^{-}$, \cite{Mahmoudi2010}.

    Separately, the two neutral scalar states, $H_{1}$ and $H_{2}$, can be represented in the physical basis through a similar transformation through 
    another rotation angle, $\alpha$, called the Higgs mixing angle,
    %
    \begin{align}
        %
        \doublet{ H_{1} }{ H_{2} } = \rotmatrix{\alpha} \doublet{H^{0}}{h^{0}}   \label{higgsMixing} \, .
        %
    \end{align}
    %
    To see how this impacts the physics of interactions it is helpful to look at the mass terms in the Yukawa Lagrangian for this new model, 
    %
    \begin{align}
        \scpt{L}_{Y} =& - \rho^{\ell}_{1} \overline{E}_{L} \Phi_{1} \ e_{R} - \rho^{\ell}_{2} \overline{E}_{L} \Phi_{2} \ e_{R} - 
                    \rho^{D}_{1} \ \overline{Q}_{L} \Phi_{1} \ d_{R} - \rho^{U}_{2} \ \overline{Q}_{L} \Phi_{2} \ d_{R} \notag \\
                    &- \rho^{U}_{1} \ \overline{Q}_{L} \ \widetilde{\Phi}_{1} \ u_{R} 
                    - \rho^{U}_{2} \ \overline{Q}_{L} \ \widetilde{\Phi}_{2} \ u_{R}\, ,
    	 \label{2HDMyukawa}
    \end{align}
    %
    where $\rho^{U,D,\ell}$ represent the up-type quark, down-type quark, and lepton coupling matrices respectively. The new fermion masses 
    take on a new form,
    %
    \begin{align}
        %
        M^{F} = \frac{v}{\sqrt{2}} \left( \rho^{F}_{1} \cos \beta + \rho^{F}_{2} \sin \beta \right) \, ,
        \label{2HDMfermionMass}
        %
    \end{align}
    %
    where $F=\left\{ U, D, \ell \right\}$. We can then define a new coupling matrix, $\kappa^{F}$, based on comparing the above mass terms with those in 
    \Cref{leptonMasses,UquarkMasses,DquarkMasses},
    %
    \begin{align}
        %
        \kappa^{F} \equiv \rho^{F}_{1} \cos \beta + \rho^{F}_{2} \sin \beta \, . \label{kappaMatrix} 
        %
    \end{align}
    %
    In addition to this new coupling matrix, $\kappa^{F}$, we also end up creating an orthogonal coupling matrix, $\rho^{F}$, that does not participate in 
    the mass interactions
    %
    \begin{align}
        %
        \rho^{F} \equiv -\rho^{F}_{1} \sin \beta + \rho^{F}_{2} \cos \beta \, . \label{rhoMatrix} 
        %
    \end{align}
    %
    Leading to a new Yukawa Lagrangian,
    %
    \begin{align}
        %
	    {\cal L}_Y =& -\frac{1}{\sqrt{2}} \sum_{F = U,D,L}
	     \overline{F}\left[  \left( \kappa^Fs_{\beta-\alpha}+\rho^Fc_{\beta-\alpha} \right) h^0 +
	     \left( \kappa^Fc_{\beta-\alpha}-\rho^Fs_{\beta-\alpha} \right)H^0 \right] F \notag \\
	     %
	    & +\frac{1}{\sqrt{2}} \sum_{F = U,D,L}
	     \overline{F} \left [ i \, {\rm sgn}(Q_F)\rho^F A^0 \right ] P_RF \notag \\
	     %
	    & -\overline{U} \left( V \rho^D P_R - \rho^{U\dagger} V P_L \right) D H^+
	      -\overline{\nu} \left( \rho^L P_R \right) L H^+ + {\rm H.c.} \ , \label{2HDMLagrangian}
        %
    \end{align}
    %
    where $c_{\beta-\alpha} \equiv \cosba$, $s_{\beta-\alpha} \equiv \sinba$, and $P_{L,R} = (1 \mp \Gfive) / 2$.

    \subsection{ Flavor changing neutral currents }\label{FCNC}
    
    In the Lagrangian above, \Cref{2HDMLagrangian}, we take note that although the $\rho^{F}$ terms do not participate in the mass interactions of the 
    fermions they do still partially govern the coupling of fermions to our Higgs fields. Of particular interest are the couplings involving these 
    $\rho^{F}$ matrices and the neutral Higgs fields: the light Higgs, $h^{0}$, the heavy Higgs, $H^{0}$, and the pseudoscalar Higgs, $A^{0}$. Since the 
    $\rho^{F}$ matrices do not participate in the mass terms they can have off diagonal elements that survive the diagonalization of the mass matrices. 
    It is these off-diagonal terms that produce the phenomenon known as flavor changing currents and since they happen in relation to the neutral Higgs 
    field they are flavor changing neutral current (FCNC) processes. These are of particular interest as they are not possible at the \emph{tree-level}, 
    or leading order level, in the SM and are further suppressed in higher orders by the GIM mechanism involving cancellations between diagrams involving 
    up and charm quarks \cite{Glashow1970}. 

    In many supersymmetric theories FCNCs are suppressed by imposing a symmetry on the Higgs sector where the two Higgs doublets couple preferentially to 
    either up-type quarks or down-type quarks but not both \cite{Mahmoudi2010}. In those theories the suppression due to an additional 
    symmetry is desired so that the lack of such signatures in the SM can be better explained within the SUSY framework. In general, however, this is not 
    necessary as they can be controlled through the mixing angles, $\beta$ and $\alpha$, instead. We call the limit where the 2HDM exactly reproduces 
    the SM, not unlike the low energy limit of relativity, the alignment limit and is benchmarked by $\sinba \to 1$. This effectively destroys any 
    tree-level FCNC process involving the light, or SM-like, Higgs field $h^{0}$, while also removing any effects the heavy Higgs field may have on 
    fermion masses. Current data favors the alignment limit \cite{Craig2013,Carena2014} meaning a heavy Higgs FCNC process is favorable but there 
    has been some evidence to suggest that FCNC processes may be present in the leptonic sector \cite{Collaboration2015}. These processes are the 
    focus of this dissertation and the beginning of our journey starts with how a Higgs boson with a mass of at least $125$ GeV is produced in the 
    energetic collision of two protons deep underground in an area in northern France and southern Switzerland.
        %
        %
