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

\begin{enumerate}
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    \item Introduction to the Standard Model (SM), \emph{Conseil Europ\'{e}en pour la Recherche Nucl\'{e}aire}
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        \begin{enumerate}
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            \item Composition 
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                \begin{itemize}
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                    \item Fermions
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                        \begin{enumerate}
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                            \item 3 generations of quarks that come in 6 flavors: up - down, charm -strange, and
                                top - botoom
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                            \item The up and down quarks are the composite quarks of the proton and neutron, with the 
                                electron they make up the dominate parts of matter. 
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                            \item 3 generations or flavors of leptons: Electrons, Muons, Tau Leptons
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                            \item Fermions have $\sfrac{1}{2}$ integer spin and all carry electric charge
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                            \item Quarks carry fractional electric charge: $\sfrac{2}{3}$ for up, charm, and top 
                                and $\sfrac{-1}{3}$ for down, strange, and bottom.
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                            \item Leptons carry integer electric charge: $\pm 1$.
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                            \item Despite the valence quarks contributing to most of the behavior of neutrons and protons,
                                there are additional quarks that interact during particle collisions. The distribution of 
                                the momentum these other quarks carry is defined by the Parton Distribution Function, PDF.
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                            \item The strong force and weak force are the forces of nuclear interactions.
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                                \begin{itemize}
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                                    \item The strong force is  short range force mediate by a boson called the gluon.
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                                    \item Gluons and the strong force are what bind nucleons together and confine the quarks
                                        into a hadron of some sort. They are massless and participate only in particles that 
                                        have color charge as defined by the $SU(3)_{c}$ theory called Quantum Chromodynamics (QCD).
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                                    \item The weak force is a mid-range force mediated by the W and Z bosons.
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                                    \item Weak interactions are responsible for neutrino scattering in the early universe and 
                                        for interactions involving proton and neutron interchange in nucelar fusion and fission.
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                                    \item Weak interactions involve any particle that has weak hypercharge and when combined with 
                                        electromagnetic interactions forms Electroweak theory or Quantum Electrodynamics (QED) 
                                        and has the group structure of $SU(2)_{L} \otimes U(1)_{Y}$.
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                                \end{itemize}
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                        \end{enumerate}
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                    \item Bosons
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                        \begin{enumerate}
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                            \item There are 4 types of bosons that have been observed in accordance with the Standard Model.
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                                \begin{itemize}
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                                    \item The $W^{\pm}$ and $Z$ bosons mediate the Weak interactions through hyperchasrge and facilitate neutron decay.
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                                    \item Gluons $g^{\mu}_{a}$ have color charge and mediate the strong interaction and help bind quarks inside partons.
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                                    \item Photons, $\gamma$, mediate electromagnetic interactions.
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                                    \item The Higgs boson, $h^{0}$, does not mediate any force interactions but does interact with all the fundamental particles that have mass,
                                        save the neutrinos. 
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                                \end{itemize}
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                        \end{enumerate}
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                    \item Neutrinos
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                        \begin{enumerate}
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                            \item Neutrinos are partners to the leptons. They have no magnetic charge and almost zero mass but can interact weakly with $Z$ bosons and through gravity.
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                            \item Neutrinos help conserve lepton number in decays involving a single charged particle and play a part in the current universe as hot dark matter.
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                            \item Because they are not charged they do not interact with the photon, which explains their classification as ``dark'' particles. 
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                            \item They only exist in matter as left-handed particles, no right-handed neutrinos have been observed. It is thought that, likewise, anti-neutrinos only exist in 
                                the left-handed helicity state.
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                        \end{enumerate}
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                        \begin{figure}[H]
                            \begin{center}
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                                \includegraphics[width=0.65\textwidth,keepaspectratio]{./figures/SM_picture_of_particles.png}
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                                \caption[This figure showcases the particles in the SM. The top ring consists of the six quarks]{This figure showcases the particles in the SM. The top ring in green contains the six quarks 
                                with the up-type quarks in the top left and the down-type quarks in the top right.
                                The outer ring underneath shows the leptons on the bottom left and the neutrinos on the bottom right. The blue inner ring shows four gauge bosons, the W and Z mediate the weak force, the 
                                photon, $\gamma$ mediates the electromagnetic force, and the gluon, $g$, mediates the strong force. At the center we complete our picture with the Higgs boson which does not mediate any 
                                force.}
                                \label{SMpicture}
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                            \end{center}
                        \end{figure}
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                \end{itemize}
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            \item Group Structure
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                \begin{enumerate}
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                    \item The group structure of the Standard Model is $SU(3)_{c} \otimes SU(2)_{L} \otimes U(1)_{Y}$.
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                        \begin{itemize}
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                            \item $SU(3)_{c}$ is the group structure for QCD. There are eight unique generators in this group, each generator defines a unique gluon.
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                            \item $SU(2)_{L}$ by itself only defines weak interactions, however, these are not physically obervable as they do not constitute obsevable states.
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                            \item $SU(2)_{L} \otimes U(1)_{Y}$ combine to make the electroweak theory that does contain pysically observable particles, $W^{\pm},\ Z, \ \gamma$.
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                            \item As will be discussed more in the next session, once the $SU(2)_{L} \otimes U(1)_{Y}$ symmetry allows the particles to obtain mass from the Higgs.
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                        \end{itemize}
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                \end{enumerate}
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            \item Higgs Mechanism
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                \begin{enumerate}
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                    \item Since the Higgs boson is a scalar particle, it is sensitive to its potential as well as the interactions with other particles.
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                    \item In this regard we need to minimize the Higgs potential, doing so we find that the symetry that forms this potential is spontaneously broken.
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                        \begin{itemize}
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                            \item The spontaneous breaking of Electroweak Symmetry means it occurs as a natural phenomenon where the field takes on a speciic value when minimized.
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                            \item We call this value the vacuum expectation value or, \vev, 
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                        \end{itemize}
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                        \begin{figure}[H]
                            \begin{center}
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                                \includegraphics[width=0.85 \textwidth, keepaspectratio]{./figures/HiggsPotential.png}
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                                \caption[This is a graphical representation of the complex Higgs portneital, \cref{SMHiggsPotential}]{This is a graphical representation of complex the Higgs potential, 
                                \cref{SMHiggsPotential}, also known as the `Mexican Hat Potential'. 
                                Technically the minimum should be anywhere within the crease seen, however, due to the spontaneous breaking of the electroweak symmetry it takes
                                on a specific value as discussed above.}
                                \label{SMHiggsPotential}
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                            \end{center}
                        \end{figure}
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                \end{enumerate}
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        \end{enumerate}
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    \item Need for New Physics (BSM)
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        \begin{enumerate}
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            \item The Standard Model is the most well tested and precise theory currently known. However, it is incomplete.
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                \begin{itemize}
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                    \item The SM become divergent at large energy scales if we try and include gravity at the quantum level.
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                    \item The SM does not offer any explanation for why there is nearly a large deficit of anti-matter in the observable universe.
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                    \item The SM does not offer any resources for canceling the nearly infinite corrections to the Higgs mass.
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                    \item The SM does not provide a suitable dark matter candidate when dark matter despite the effects of dark matter have been observed.
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                    \item The SM does not offer a good explanation for why neutrinos have mass or why we do not encounter right-handed neutrinos.
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                \end{itemize}
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            \item To address these concerns, theorists have proposed several extensions to and replacements for the SM but so far none have been observed experimentally.
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            \item In SM physics, the most recent observation is of the Higgs boson which leaves the Higgs sector as the newest sector in which to look for deviations from the SM.
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            \item Several extensions to the SM, such as supersymmetry, include extended Higgs sectors as part of their implementation.
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            \item In general, we can then use searches for signs of these extended sectors to help us narrow down which models are more likely to describe the fundamental interactions.
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        \end{enumerate}
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    \item the Two Higgs Doublet Model (2HDM)
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        \begin{enumerate}
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            \item The most natural way in which we can extend the Higgs sector is through the addition of a second Higgs doublet, \cref{HiggsDoublet2}, to the SM Higgs doublet, \cref{HiggsDoublet1}.
                \begin{subequations}
                \begin{align}
                    \setlength{\jot}{8pt}
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                    \Phi_{1} &= \doublet{G^{+}}{\frac{H_{1} + v + i G^{0}}{\sqrt{2}}} \label{HiggsDoublet1} \\
                    \ \notag \\
                    \Phi_{2} &= \doublet{H^{+}}{\frac{H_{2} + i A^{0}}{\sqrt{2}}}   \label{HiggsDoublet2} \ .
                    %
                \end{align}
                \end{subequations}
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            \item The addition of this extra doublet generates new Higgs bosons along with the one predicted in the SM, denoted by $h^0$, and called the \emph{light} higgs.
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            \item The new Higgs bosons are the \emph{heavy Higgs}, $H^0$, the pseudoscalar Higgs, $A^0$, and the two charged Higgs bosons, $H^+ \text{ and } H^-$.
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            \item Most theories that incorporate this extended sector place constraints on their behavior which help explain why we have not seen any sign of these particles yet.
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        \end{enumerate}
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    \item General Formalism
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        \begin{enumerate}
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            \item Though it is more intensive computationally, with a general 2HDM we can also explain the lack of signal while avoiding over constraining the system.
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            \item One potential signature of a general 2HDM system that we can look for at the LHC is in flavor changing neutral currents (FCNC) involving Higgs particles, sometimes
                abbreviated FCNH.
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            \item These currents appear in the part of the Lagrangian where the Higgs field interacts with the fermions, \cref{2HDMLagrangian}, 
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                \begin{align}
                    %
		            {\cal L}_Y =& -\frac{1}{\sqrt{2}} \sum_{F = U,D,L}
		             \bar{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]P_R F \notag \\
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		            & +\frac{1}{\sqrt{2}} \sum_{F = U,D,L}
		             \bar{F} \left [ i \, {\rm sgn}(Q_F)\rho^F A^0 \right ] P_RF \label{2HDMLagrangian} \\
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		            & -\bar{U} \left( V \rho^D P_R - \rho^{U\dagger} V P_L \right) D H^+
		              -\bar{\nu} \left( \rho^L P_R \right) L H^+ + {\rm H.c.} \ , \notag
                    %
                \end{align}
                %
                where $U, \ D, \text{ and } L$ are the up-type quarks, down-type quarks, and leptons respectively; $c_{\beta-\alpha} \equiv \cosba$; and $s_{\beta-\alpha} \equiv \sinba$.
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            \item In the Lagrangian listed in \cref{2HDMLagrangian}, the $\kappa$ matrices are taken to be equivalent to the coupling matrices in the SM while the $\rho$ matrices can have both 
                diagonal and off-diagonal elements.
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            \item With this in mind, as we approach the decoupling limit where we recover the SM we see that $\sinba \sim 1$ and $\cosba \sim 0$. 
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        \end{enumerate}
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    \item Flavor Changing Neutral Currents (FCNC)
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        \begin{enumerate}
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            \item FCNC processes are made possible by the off-diagonal elements of the $\rho$ matrices. The advantage here is that for the Heavy Higgs state such couplings can be enhanced by the
                large \sinba factor while they are suppressed with the SM-like Higgs by the complimentary \cosba factor.
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            \item This allows FCNC processes to occur at the tree-level of this model instead of at the one-loop level or lower. If we impose restrictions on the symmetries present in the 2HDM, 
                however, they can again be suppressed as they are in many SUSY models.
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            \item The advantage is that if w can detect these unique processes then we can rule out some theories entirely and at least further constrain others. 
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            \item The Higgs sector is not the only place in which we could see FCNC processes, In models that incorporate additional neutral bosons, such as a $Z'$, we can achieve similar results.
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            \item The advantage then of the Higgs sector is that it can be well constrained by current experiments and is kinematically similar to processes involving the SM-like Higgs
                that was observed in 2012.
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            \item In the context of the Higgs sector, we expect to see such processes first in the heavier generations since the coupling strength to the Higgs is directly proportional to the 
                mass of the particles it interacts with.
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            \item With this in mind, we will look at interactions involving the Higgs, top and charm in the quark sector; and the Higgs, tau lepton and muon in the leptonic sector.
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            \item The most direct search of such interactions involves producing a Higgs from initial proton proton collisions and having it subsequently decay into the desired channel.
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            \item The largest challenge such an endeavor presents is in the production of the Higgs boson itself. Effective theories allow for the tree level production of Higgs bosons from 
                proton proton collisions but break down as the transverse momentum, $p_{T}$, scale increases. 
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            \item Such a breakdown makes it difficult then to look at large mass scales, $. 2*m_t$ where $m_t$ is the top mass.
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            \item Therefore we first must develop an analytic method for calculating these processes that will not breakdown at high $p_{T}$ scales.
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        \end{enumerate}
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\end{enumerate}
