\chapter{Theoretical Framework}\label{ch:intro}
\section{Overview}\label{sec:overview}
The aim of particle physics is to understand the structure and behavior of the universe in terms of fundamental building blocks known as elementary particles. Understanding the natural universe requires three basic ingredients: particles that constitute matter, the forces those particles feel, and finally the influence of those forces on those particles. Many elementary particles are commonly not observed in nature. Rather, they are created during collisions between particles at sufficiently high energies either at particle accelerators or in cosmic-ray interactions. All particles and their interactions are currently described by a collection of quantum field theories known as the Standard Model of Particle Physics. Details of the Standard Model will be discussed in Section~\ref{sec:SM}.\\
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.8\textwidth]{figure/StandardModel.pdf}
\caption{\small{The Standard Model of elementary particles (more schematic depiction), with the three generations of matter, gauge bosons are in the fourth column, and the Higgs boson in the fifth column~\cite{wiki:sm}.}}
\label{fig:standardmodel}
\end{center}
\end{figure}
All matter in the universe is composed of spin-$\frac{1}{2}$ particles called fermions. There are two types of fermions, leptons and quarks. In analogy with the electric charge, quarks have an additional charge-like property known as color charge. Unlike electric charge, the color charge comes in three different varieties, Red(R), Green(G), and Blue(B). These color charges are responsible for the strong interaction, which is mediated through the exchange of spin-1 gauge bosons called gluons. Gluons bind quarks into hadrons and nucleons. Similarly, a gauge boson known as the photon is the particle mediating electromagnetic interactions. In addition to gluons and photons there are two other types of gauge bosons that transmit the weak interaction, responsible for some nuclear decays, the $W$ and $Z$ bosons. Unlike gluons and photons, these bosons have mass ($M_{W}\approx $ 80 GeV and $M_{Z} \approx$ 91 GeV) and are the force carriers of the weak interactions. The gauge bosons, along with the fermions and the Higgs boson, are shown in Table~\ref{table:partons}.\\
%%%table showing the leptons and quarks of the standard model
\begin{table}[!ht]
\begin{center}
\caption{\small{Forces and Gauge Bosons~\cite{GordonKane:1993fk}.}}
\begin{tabular}{|c|c|c|}
\hline
Force & Acts On & Transmitted by \\
\hline
Gravity & All particles & graviton\\
& & (massless, spin-2)\\
& & \\
\hline
Electromagnetism & All electrically & Photons($\gamma$ \\
& charged particles & (massless, spin-1)\\
& &\\
\hline
Weak interaction & quarks leptons, & $W^{\pm}$, $Z^{0}$ \\
& electroweak gauge bosons & (heavy spin-1)\\
& & \\
\hline
Strong interaction & All colored particles & Eight gluons(g)\\
(QCD) & (quarks and gluons) & (massless spin-1)\\
& & \\
\hline
\end{tabular}
\label{table:partons}
\end{center}
\end{table}
With the recently discovered Higgs boson~\cite{HiggsPaper:2012} at the Large Hadron Collider (LHC), the standard model of particle physics is complete. Detailed discussion about the Higgs boson and its role is presented in the Section~\ref{ssec:ewk}.\\
\section{Standard Model}
\label{sec:SM}
The Standard Model (SM) is one of the most successful theories in physics. There is extraordinary matching between theoretical predictions and experimental observations. With carefully defined symmetry arguments in the context of special relativity and quantum mechanics, we can derive the entire theory known as the Standard Model. In the following section we will derive the Lagrangian for spin $\frac{1}{2}$ fermions from the Dirac approach and again from symmetry arguments described by the gauge group $\emph{U(1)}$. Later, the entire SM will be derived from gauge invariance under the $SU(3)\times SU(2)\times U(1)$ local gauge symmetry group.\\
\subsection{Quantum Electrodynamics(QED)}
\label{ssec:qed}
The operator version of the standard relativistic relation $ E^2 = m^2c^4 + \vec{p}^2c^4$ can be written in natural units ($\hslash =1, c =1$) as the \textbf{Klein Gordon} equation~\cite{GordonKane:1993fk}
\begin{equation}
(\partial^{2} - m^{2})\phi = 0.
\label{eq:kgeq}
\end{equation}
A major problem with the Klein Gordon equation is that energy eigenvalues have both negative and positive solutions $ E = \pm\sqrt{m^2 + \vec{p}^2}$. The negative energy eigenvalues are physically meaningless because negative energy means we don't have a true vacuum. As a result particles can cascade down forever, yielding an infinite amount of radiation. \\
Dirac invented the "anti-particle" to explain negative energy eigenvalues. The Lagrangian for spin-$\frac{1}{2}$ particles in terms of Dirac spinor can be written as
\begin{equation}
\mathcal{L}_{D} = \bar{\psi}(i\gamma^{\mu}\partial_{\mu} - m) \psi.
\label{eq:diraceq}
\end{equation}
The classical electromagnetic Lagrangian with quadratic kinetic terms is given by the expression,
\begin{equation}
\mathcal{L}_{EM} = -\frac{1}{4} F_{\mu\nu}F^{\mu\nu} - J^{\mu}A_{\mu},
\label{eq:lagrangianem}
\end{equation}
where $J^{\mu}$ and $A_{\mu}$ are the 4-vector current and potential, respectively. $F^{\mu\nu}$ is an anti-symmetric electromagnetic field strength tensor given by the expression $\partial^{\mu}A^{\nu} - \partial^{\nu}A^{\mu}$.\\
Looking at the Dirac Lagrangian (Eq~\ref{eq:diraceq}) with complex spinor fields $\psi$ and $\psi^{\dagger }$, we can make the transformation $\psi \rightarrow e^{i\alpha}\psi $ and $\psi^{\dagger} \rightarrow e^{-i\alpha}\psi^{\dagger}$, where $\alpha$ is an arbitrary real number. We call this transformation the $U(1)$ transformation. We notice that the Lagrangian is invariant under $U(1)$ symmetry, with current
\begin{equation}
j^{\mu} = \bar{\psi}\gamma^{\mu}\psi.
\label{eq:current}
\end{equation}
In the example above we observed that the symmetry was a \textbf{Global Symmetry}. That is a symmetry that changes the field at all points in space.\\
It is also seen that the Lagrangian in Eq.~\ref{eq:diraceq} along with the electromagnetic part in Eq.~\ref{eq:lagrangianem} have no terms in common, meaning there is no interaction in the theory. In the real world, particles interact with with each other so we need to add an interaction term into the electromagnetic Lagragian as
\begin{equation}
\mathcal{L} = \mathcal{L}_{D} + \mathcal{L}_{EM} + \mathcal{L}_{int},
\end{equation}
so that
\begin{eqnarray}
\mathcal{L} = \bar{\psi}(i\gamma^{\mu}\partial_{\mu} - m) \psi -\frac{1}{4} F_{\mu\nu}F^{\mu\nu} - J^{\mu}A_{\mu} - qj^{\mu}A_{\mu}, \nonumber \\
= \bar{\psi}(i\gamma^{\mu}\partial_{\mu} - m) \psi -\frac{1}{4} F_{\mu\nu}F^{\mu\nu} - (J^{\mu} +qj^{\mu})A_{\mu}.
\label{eq:lagrangianEM}
\end{eqnarray}
$\mathcal{L}$ is still invariant under $U(1)$ and the current is unchanged, i.e. $\bar{\psi}\gamma^{\mu}\psi$. Setting $q = e$, the magnitude of electron's charge, then Eq.~\ref{eq:lagrangianEM} is the Quantum Electrodynamics Lagrangian (QED). In the next couple of paragraphs, we derive the same QED Lagrangian in a more fundamental way.\\
%
% Alternative derivation of QED lagrangian
%
%
As we discussed before, the Dirac Lagrangian in Eq.~\ref{eq:diraceq} is invariant under $U(1)$ transformation and the underlying symmetry was essentially \emph{global}. But we want our symmetry to be \emph{local}, so that $\alpha$ is space-time dependent. Thus, the differential operator now acts not only on $\psi$, but also on $\alpha(x)$. That essentially leaves the Lagrangian with one extra term. We still want invariance of $\mathcal{L}$ under a local $U(1)$ transformation, so we replace the derivative $\partial_{\mu}$ with a covariant derivative,
\begin{equation}
D_{\mu} = \partial_{\mu} - ig_{1} A_{\mu},
\label{eq:covDer}
\end{equation}
where the $g_{1}$ is the coupling strength and $A_{\mu}$ is the invariant vector field under $U(1)$ if it transforms as
\begin{equation}
A_{\mu} \rightarrow A_{\mu} + \frac{1}{g_{1}}\partial_{\mu}\alpha.
\label{eq:vecField}
\end{equation}
Now our new Lagrangian is
\begin{equation}
\mathcal{L}= \bar{\psi}(i\gamma^{\mu}\partial_{\mu} - m -\frac{1}{g_{1}}\gamma^{\mu}A_{\mu}) \psi.
\label{eq:newLag}
\end{equation}
Using Eq.~\ref{eq:vecField} in Eq.~\ref{eq:newLag} followed by some algebra, it is clear that $\mathcal{L}$ is now invariant under $U(1)$ and the conserved quantity is charge $j^{\mu} = \bar{\psi}\gamma^{\mu}\psi$. A field $A_{\mu}$ has no kinetic term, and therefore there is no kinetic energy in the system. But in reality, we can not imagine any physical field without the kinetic energy. So we introduce a gauge-invariant kinetic term for an arbitrary field $A_{\mu}$ as
\begin{eqnarray}
\mathcal{L}_{Kin, A} = -\frac{1}{4} F_{\mu\nu}F^{\mu\nu}, \nonumber
\end{eqnarray}
where $F^{\mu\nu}$ is expressed in terms of a covariant derivative
\begin{equation}
F^{\mu\nu} = \frac{i}{g_{1}}[D^{\mu},D^{\nu}].
\label{eq:fmunuCo}
\end{equation}
For any physical field $A_{\mu}$, it is natural to assume that there is some source causing the field, which we simply call $J^{\mu}$. This makes our final Lagrangian
\begin{equation}
\mathcal{L} = \bar{\psi}(i\gamma^{\mu}D_{\mu} - m) \psi -\frac{1}{4} F_{\mu\nu}F^{\mu\nu} - J^{\mu}A_{\mu},
\label{eq:finalLag}
\end{equation}
which is the same as we derived in Eq.~\ref{eq:lagrangianEM}.\\
We started with the Lagrangian for spin $\frac{1}{2}$ particles with a global $U(1)$ symmetry, added kinetic terms, and finally required that the gauge symmetry be local. The field $A_{\mu}$, upon quantization describes a spin-1 gauge boson known as the photon~\cite{Niessen:2008hz}.\\
\subsection{Quantum Chromodynamics(QCD)}
\label{ssec:qcd}
Inspired by success of local gauge theory for QED, now we try to generalize this approach to QCD~\cite{GordonKane:1993fk}, the theory that describes the strong interactions and is responsible for the binding of quarks into mesons and baryons. Mesons are quark anti-quark bound states while baryons are three-quarks bound states. The $\Delta^{++}$ particle is a particle with a "++" electric charge and a spin of 3/2. A spin 3/2 particle composed of half-integer-spin quarks must include a hidden degree of freedom to satisfy Fermi-Dirac statistics. The hidden degree of freedom must come in three distinct varieties, one for each of the three quarks, which we call color. Baryons can thus be thought of as half-integer-spin composite states composed of three quarks, each with a different color, while mesons are composite states composed of quark anti-quark pairs, each representing color combinations that are colorless or color-singlets. The underlying theory of QCD is a non-abelian gauge theory with an $SU(3)$ symmetry group.
To cope with a non-Abliean theory, consider a Lagrangian $\mathcal{L}$ that is invariant under $SU(3)$ symmetry
\begin{equation}
\psi^{j} \rightarrow U^{jk}\psi^{k},
\label{eq:qcdtransform}
\end{equation}
where $U^{jk}$ is a $3\times3$ unitary matrix of $SU(3)$. $U^{jk}$ can be written as an exponential function of Hermition operators that are linear combination of the generators of a Lie-algebra. Eq.~\ref{eq:qcdtransform} can be written as,
\begin{eqnarray}
\psi \rightarrow U\psi = e^{i\theta^{a}(x)T^{a}}\psi \equiv e^{i\vec{\theta}.\vec{T}}\psi,
\label{eq:qcdtransform1}
\end{eqnarray}
where $\theta^{a}$ are the $3^2-1 =8$ parameters of the $SU(3)$ group, and $T^{a}$ are generator matrices of the group. These commutators obey the commutation relation
\begin{eqnarray}
[T^a, T^b] = if_{abc}T^c, \nonumber
\end{eqnarray}
where $f_{abc}$ is called the structure constant of the group. \\
To construct the QCD Lagrangian we use the same arguments used to motivate QED but now the result is more complicated due to a larger symmetry group. The covariant derivative takes the form
\begin{eqnarray}
D_{\mu} = \partial_{\mu} - ig_{2}T^{a}A_{\mu}^{a},
\label{eq:covQCD}
\end{eqnarray}
where the term $g_{2}$ is the strong coupling constant and the field $A^{a}_{\mu}$ is a field that transforms as
\begin{eqnarray}
A^{a}_{\mu} \rightarrow A^{a}_{\mu} - \frac{1}{g_{2}}\partial_{\mu}\theta_{a(x)} - f_{abc}\theta_{b}A_{\mu}^{a}.
\label{eq:fieldQCD}
\end{eqnarray}
The extra term in the transformed field will cancel the additional terms introduced by gauge transformation of the covariant derivative. Now adding the kinetic energy term completes the Lagrangian, which is still invariant under the gauge transformation. The field strength tensor takes the form
\begin{eqnarray}
F_{\mu\nu}^{a} = \partial_{\mu}A_{\nu}^{a}- \partial_{\nu}A_{\mu}^{a} -g_{2}f_{abc}A_{\mu}^{b}A_{\nu}^{c}.
\label{eq:fieldtensor}
\end{eqnarray}
Finally, with Eqs.~\ref{eq:covQCD}, ~\ref{eq:fieldQCD} and~\ref{eq:fieldtensor}, we can write the complete Lagrangian for QCD as
\begin{equation}
\mathcal{L} = \bar{\psi}(i\gamma^{\mu}D_{\mu} - m) \psi -\frac{1}{4} F_{\mu\nu}F^{\mu\nu}.
\label{eq:QCDlag}
\end{equation}
This has the exact same form as the QED Lagrangian but with extra complexity due to the $SU(3)$ symmetry hidden in the covariant derivative and the stress energy tensor. These extra terms give rise to additional interaction terms between the gauge bosons as these have the color charge and can thus interact with each other as well as with the colored quarks.\\
\subsection{Electroweak Theory and Higgs Mechanism}
\label{ssec:ewk}
QED describes the physics of electrically charged fermions interacting with each other and the electromagnetic field, while QCD describes the physics of particles with color charge that interact with each other and a gluonic field. What QED and QCD fail to describe are some types of decays of heavier particles into lighter ones. For example, in beta decay a nucleus can transmute into another nucleus by the emission of a beta particle (either a positron or electron) and a neutrino. This type of process is not described either by QED nor QCD and is of primary importance in the nuclear reactions that powers our sun. The list of reactions that are not explained by QCD or QED is large. A necessity for a unified theory of electromagnetic and weak interactions was realized and those problems were elegantly solved by yet another gauge theory that combines the electromagnetic and weak interaction into a theory known as the electroweak theory~\cite{Weinberg:1967uq}. \\
In the theory of electroweak interactions, left handed fermions are represented as doublets and right handed fermions as singlets. The $SU(2)$ symmetry corresponds to an unbroken weak theory with two "weak" charges, in analogy with the $SU(3)$ of the strong interaction. $U(1)$ has a single hypercharge that is analogous to the elerctric charge of QED. Initially the theory has four massless vector bosons, three that correspond to the $SU(2)$ part and one to the $U(1)$ part. These vector bosons will acquire mass through the Higgs Mechanism~\cite{HiggsMechanicsm:1964fq}. We will discuss the Higgs mechanicsm later in this section. The massive vector bosons have all been observed. The $W$ and $Z$ bosons were discovered at CERN by the UA1 and UA2 collaborations at the super proton sychrotron collider~\cite{WZBosons}. \\
Getting the Standard Model gauge bosons from group theory is a monumentally significant result, but the problem is they are all massless. This is acceptable for photons and gluons but it has been experimentally verified that the weak bosons have a mass. As mentioned previously, left-handed fermions are $SU(2)$ doublets while right-handed fermions are $SU(2)$ singlets, so the mass term takes the form
\begin{equation}
m\bar{\psi}\psi = m(\bar{\psi_{L}}\psi_{R} + \bar{\psi_{R}}\psi_{L}).
\label{eq:fermionmass}
\end{equation}
This mass term in the Lagrangian would break chiral symmetry~\cite{Niessen:2008hz}. It implies that Dirac fermions are massless but it is clear from observations that they posses a non-zero rest mass. So we now introduce another field $\phi$, which is an $SU(2)$ doublet and a complex scalar
\begin{equation}
\phi = \begin{pmatrix}
\phi^{+}\\
\phi^{0}
\end{pmatrix}.
\label{eq:scalarPhi}
\end{equation}
The covariant derivative takes the form
\begin{equation}
D_{\mu} = \partial - ig^{'}\frac{1}{2}YB_{\mu} - igTW_{\mu},
\label{eq:covEWK}
\end{equation}
where $g$ and $g^{'}$ are coupling constants of different strengths, $B_{\mu}$ is the gauge field of the unbroken $U(1)$ symmetry while $W_{\mu}$ is the gauge field of the unbroken $SU(2)$ symmetry, and $T$ is a vector of Pauli matrices that satisfies the commutation relation,
\begin{equation}
[\sigma_{i}, \sigma_{j}] = 2\epsilon_{ijk}\sigma_{k}.
\label{eq:paulimatrices}
\end{equation}
These Pauli matrices generate the $SU(2)$ group.
The covariant derivative of Eq.~\ref{eq:covEWK} acts on left-handed doublets as
\begin{eqnarray}
D_{\mu}\begin{pmatrix}
\psi_{u}\\
\psi_{d}
\end{pmatrix}_{L} = (\partial - igTW_{\mu} + ig^{'}\frac{1}{2}B_{\mu}) \begin{pmatrix}
\psi_{u}\\
\psi_{d}
\end{pmatrix}_{L},
\end{eqnarray}
and the right-handed singlets as
\begin{eqnarray}
D_{\mu}\psi_{R} = (\partial + ig^{'}B_{\mu})\psi_{R}.
\end{eqnarray}
Right-handed and left-handed fermions couple with each other via the Yukawa interaction. The Lagrangian for the unbroken electroweak theory can be written in the same fashion as in QCD:
\begin{eqnarray}
\mathcal{L} & = &\bar{\psi}(i\gamma^{\mu}D_{\mu} - m)\psi - \frac{1}{4}W_{\mu\nu}W^{\mu\nu} - \frac{1}{4}B_{\mu\nu}B^{\mu\nu},
\label{eq:LagrangianEWK}
\end{eqnarray}
where
\begin{eqnarray}
W_{\mu\nu}^{i} = \partial_{\mu}W_{\nu}^{i} - \partial_{\nu}W_{\mu}^{i} + g\epsilon^{ijk}(W_{\mu}^{j} \times W_{\nu}^{k}), \nonumber
\end{eqnarray}
and
\begin{eqnarray}
B_{\mu\nu} = \partial_{\mu}B_{\nu} - \partial_{\nu}B_{\mu}. \nonumber
\end{eqnarray}
Now with the field we introduced in Eq.~\ref{eq:scalarPhi}, we write the Lagrangian as
\begin{equation}
\mathcal{L} = (D_{\mu}\phi)^{\dagger}(D_{\mu}\phi) - V(\phi).
\label{eq:newLagrangianEWK}
\end{equation}
In order to explain the massive $W$ and $Z$ bosons observed in the lab, electroweak symmetry must be spontaneously broken. Thus, we can construct the potential energy term from a complex scalar field as mentioned in Eq.~\ref{eq:scalarPhi} as
\begin{eqnarray}
V(\phi)& = &- \mu^{2}\phi^{\dagger}\phi + \lambda(\phi^{\dagger}\phi)^{2},
\label{eq:higgsPotential}
\end{eqnarray}
where $\mu^{2}$ , $\lambda$ $\in$ $\mathbb{R}$. With such a representation of the potential, it can take basically two forms as shown in Fig.~\ref{fig:twoPotential}.\\
\begin{figure}[!tbp]
\begin{subfigure}[b]{0.4\textwidth}
\includegraphics[width=\textwidth]{figure/potentail_zero_min.pdf}
\caption{$\mu^{2} > 0$}
\label{fig:minZero}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.4\textwidth}
\includegraphics[width=\textwidth]{figure/potential_mexHat.pdf}
\caption{$\mu^{2} < 0$}
\label{fig:minNonZero}
\end{subfigure}
\caption{Two possible shapes of the potential in Eq.~\ref{eq:higgsPotential}. A minimum potential (a) indicates no interesting physics but (b) a indicates nonzero vacuum expectation value (VEV)~\cite{samthesis}.}
\label{fig:twoPotential}
\end{figure}
The bounded nature of the potential from below means $\lambda$ is positive and $\mu^{2}$ is negative. The potential energy then looks like that shown in Fig.~\ref{fig:minNonZero}. This is referred to as the Mexican hat potential where the minimum is not at V($\phi$) = 0 but at some non-zero $\phi$ as shown in Fig.~\ref{fig:minNonZero}. This scenario illustrates a non-zero vacuum expectation value (VEV) with a two-fold degeneracy. If we now allow the potential to have an additional degree of freedom, in an orthogonal dimension $\alpha$, the two-fold degeneracy become continuous. Writing the potential $\phi$ in terms of the expectation value $\langle\phi\rangle$, $\alpha$, and massless scalar $\beta$, the Lagrangian will have five terms with no $U(1)$ symmetry. Essentially we broke the global symmetry by writing the field, $\phi$, in terms of the quantum fluctuation $( \alpha + i\beta)$ around $\langle\phi\rangle$ and we get massless bosons, known as Goldstone Bosons.\\
Now let us force a symmetry to be a local with the same potential as in Eq.~\ref{eq:higgsPotential} such that the vacuum expectation value is $\phi = \langle\phi\rangle$ and expanding $\phi$ as
\begin{equation}
\phi = \langle \phi \rangle + h.
\label{eq:localfix}
\end{equation}
With the new covariant derivative in Eq.~\ref{eq:covEWK} we can expand the Lagrangian in Eq.~\ref{eq:newLagrangianEWK} around the minimum $\phi$ and we end up with a Lagrangian of the form
\begin{equation}
\mathcal{L}_{SM} = \mathcal{L}_{QCD} + \mathcal{L}_{EW} + \mathcal{L}_{Higgs} + \mathcal{L}_{Yukawa},
\end{equation}
\begin{eqnarray}
\mathcal{L}_{SM} &&= \frac{1}{2}(\partial_{\mu}h)^{2} + \frac{1}{2}\mu^{2}h^{2} - \lambda\langle\phi\rangle h^{3} \nonumber \\ && - \frac{\lambda}{4}h^{4} + \underline{\frac{1}{4}g^{2}\langle\phi\rangle ^{2}B_{\mu}^{2}}
+ \frac{1}{4}g^{2}\langle\phi\rangle B_{\mu}^{2}h \nonumber \\ &&+ \frac{1}{4}g^{2}B_{\mu}^{2}h^{2} + \underline{\frac{1}{4}g^{'2}\langle\phi\rangle ^{2}(\vec{T}.\vec{W_{\mu}})^{2}} + \frac{1}{2}g^{'2}\langle\phi\rangle ^{2}(\vec{T}.\vec{W_{\mu}})^{2}h^{2} \nonumber \\ &&+ \frac{1}{2}g^{'2}\langle\phi\rangle ^{2}(\vec{T}.\vec{W_{\mu}})^{2}h^{4} + \sum_{i, j =1}^{9}\psi_{i}y_{ij}\psi_{j}h + \underline{\langle\phi\rangle\sum_{i, j =1}^{9}\psi_{i}y_{ij}\psi_{j}} \nonumber \\ &&+ \text{Gluonic field} + \mathcal{L}_{kinetic}.
\label{eq:smLagrangian}
\end{eqnarray}
The underlined terms above are quadratic in the field, which are interpreted as mass terms for the corresponding SM field. The last underlined term contains the mass terms for the SM fermions, which is quadratic in the fermion fields and is multiplied by the VEV and corresponding Yukawa coupling. The second underlined term corresponds to the origin of mass for gauge bosons. Here the VEV has been added to the Lagrangian and upon diagonalizing the 2x2 matrix, one of the eignevalues is zero and corresponds to the photon while the other is non-zero and corresponds to the massive gauge bosons in the broken theory:
\begin{eqnarray}
W_{\mu}^{\pm} = \frac{1}{\sqrt{2}}(W_{\mu}^{1} - iW_{\mu}^{2}),\nonumber
\end{eqnarray}
\begin{eqnarray}
Z_{\mu} = \cos \theta_{W}W_{\mu}^{3} - \sin\theta_{W}B_{\mu},\nonumber
\end{eqnarray}
and
\begin{eqnarray}
A_{\mu} = \cos\theta_{W}B_{\mu} + \sin\theta_{W}W_{\mu}^{3}, \nonumber
\end{eqnarray}
where $\theta_{W} = \tan^{-1}(\frac{g}{g^{'}})$ is called the Weinberg angle~\cite{pdg}. We get the mass of vector bosons from
\begin{eqnarray}
M_{Z} = \frac{1}{2}\langle\phi\rangle\sqrt{g^{2} + g^{'2}},
\end{eqnarray}
\begin{eqnarray}
M_{Z} = \frac{1}{2}\langle\phi\rangle g,
\end{eqnarray}
and
\begin{eqnarray}
M_{A} = 0.
\end{eqnarray}
We generalize the mass terms for all vector bosons $V_{\mu}$ with
\begin{equation}
\mathcal{L}_{m} = \frac{1}{2}V_{\mu}\textbf{M}^{2}V^{\mu},
\end{equation}
Where the mass matrix $\textbf{M}^{2}$ is
\begin{eqnarray}
\textbf{M}^{2} = \frac{v^{2}}{4}
\begin{bmatrix}
g^{'2}&0&0&0\\
0&g^{'2}&0&0\\
0&0&g^{'2}&-gg^{'}\\
0&0&-gg^{'}&g^{2}
\end{bmatrix}.
\label{eq:massMatrix}
\end{eqnarray}
All mass eigenstates mentioned in the Lagrangian of Eq.~\ref{eq:smLagrangian} are obtained by diagonalizing $\textbf{M}^{2}$. The mass of the resulting bosons and the zero mass of the photon are exactly observed in nature.\\
This mechanism of introducing mass in a gauge invariant way is called the Higgs Mechanism~\cite{HiggsMechanicsm:1964fq}, named after Peter Higgs, who along with Englert, and Bourt first described the mechanism in a relativistic context. The resulting field $h$ is called the Higgs field and the excitation state is called the Higgs boson whose mass is equal to $\mu\sqrt{2}$. The Higgs boson was discovered with the Large Hadron Collider(LHC) in 2012~\cite{HiggsPaper:2012}. This discovery completed the Standard Model. The overall process of generating masses in a gauge invariant way is what the Higgs mechanism brings to the standard model. It incorporates the spontaneous breaking of the $SU(2)_{L}\times U(1)_{Y}$ symmetry to a $U(1)_{EM}$ symmetry and generates masses for all SM particles in a gauge invariant way, which is a requirement for a quantum field theory to be renormalizable or self-consistent and allows for solutions that are sensible and free of infinities. \\
\section{Shortcoming of the Standard Model}
Despite an excellent agreement between the SM and almost all experimental measurements concerning elementary particles, the SM is not the final theory of everything. For one thing the SM does not include gravity in any way. Additionally some observations are beyond the scope of the SM to explain. For example there is no dark matter candidate within the SM. The standard model can't tell why there are exactly three families of leptons and quarks. It does not fully explain as to why there is more matter than antimatter. There is also the fine-tuning problem~\cite{finetuning} in the standard model. The fine-tuning problem is where quantum loop corrections to the Higgs mass term cancel each other in miraculous way to keep the Higgs mass low and stable. \\
Take gravity for example; it was the first force understood and it is known to have influence over very large distances. It is however, very poorly understood at short distances, the domain of the particle physics. At the microscopic level the gravitational force is very much weaker than any of the other forces so is typically ignored when trying to describe the physics of fundamental particles, which are close to massless when compared to the Plank mass ($\sim10^{19}$ GeV). No modern day colliders (the LHC collision energy is about $10^{3}$ GeV) or for that matter possible future collider can reach anywhere near the Plank scale, so ignoring the effects of gravity in particle collisions is justified. \\
From a wide spectrum of astrophysical data, it is observed that baryonic matter accounts for $5\%$ of the universe while dark matter accounts for about 26$\%$, The rest of the energy density is in the form of a mysterious force called dark energy, which acts like anti-gravity and is responsible for the observation of the accelerated expansion of the universe~\cite{DMresults:2008}. If dark matter is composed of particles they must be stable, massive, and weakly interacting. The neutrino is the one SM particle that could be considered a candidate for dark matter, however, hot dark matter (hot here meaning that they would be moving close to the speed of light) have been ruled out as dark matter candidates from astrophysical observations~\cite{Hannestad:2010yi}.\\
Almost all SM propagators and vertices are proportional to the original parameters of the theory. This means if tree-level parameters are small, original parameters will stay small. The underlying reason is symmetry. Symmetries protect parameters from being too big even if the symmetry is broken. Essentially, gauge boson masses are protected by gauge symmetry and fermion masses are protected by chiral symmetry. The only particle's mass that is not protected against higher order correction is the Higgs boson. There are basically two contributions to the Higgs mass; the bare mass parameter $m_{0}$ and quantum corrections. The latter when summed together with the bare mass, results in the observed mass. In real experiments we do not measure the bare mass. We split self energy (effective mass due to interactions between the particles and its system of particles) into a finite and a divergent part. The finite part is the bare mass and the divergent part is a correction to the bare mass. So, the measured mass, expressed in terms of the bare mass and the correction is
\begin{equation}
m_{H}^{2} = m_{0}^{2} + \delta m_{C}^{2}.
\label{eq:masssplitting}
\end{equation}
Out of many possible loop corrections to the Higgs boson mass, the largest correction comes from the heaviest fermion, the top quark. This contribution is proportional to the Yukawa coupling $y$, as
\begin{equation}
\delta m_{C}^{2} \propto - |y|^{2}\Lambda^{2}.
\end{equation}
In the Standard model, the Yakawa coupling is not related to any other interaction. There are no other diagrams to cancel this divergence. If a very large renormalalization parameter $\mu$ is chosen, the Higgs mass doesn't survive re-normalization, which means that its value is not definite and can take on any value whatsoever. However, with the recent observation of the Higgs boson with a mass of 125 GeV, we know that the Higgs boson exists and has a finite mass. But our best theory, the SM, does not predict the mass of the Higgs. The Higgs mass is not protected by a symmetry, it can in fact, take on any value up to the Planck mass at $10^{19}$ GeV. This riddle is still unsolved in the standard model and is known as the Hierarchy Problem. \\
Moreover, there is a huge difference between fermion masses. For example, the electron is about 200 times lighter than the muon and 3500 times lighter than the tau while in the SM, neutrinos are massless. It is now an experimentally established fact that neutrinos have a mass. It is not clear if the neutrino masses would arise in the same way that the masses of other fundamental particles do in the Standard Model. %Amrit : Add Matter antimatter symmetry
\\
These issues alone motivate us to look for new physics beyond the Standard Model. There are many other models and theories that provide some solutions to problems of the Standard Model. In the next section we will discuss one possible class of theories that builds on lessons learned from the SM and provides a natural fix to the many problems mentioned above. \\
\section{Super Symmetry (SUSY)}
In previous sections we have encountered symmetry arguments that can have a tremendous role in the successful construction of a theory. It is a firmly established fact that almost all of the possible fundamental symmetries are preserved by the SM. The only remaining symmetry not exploited by the SM is the symmetry between the fermions and the bosons. The fundamental difference between fermions and bosons is spin. Supersymmetry is essentially a symmetry between the bosons and fermions. We can think of an operator $\hat{Q}$ whose action is
\begin{eqnarray}
\hat{Q}\ket{Fermion} = \ket{Boson}, \\ \\
\hat{Q}\ket{Boson} = \ket{Fermion}.
\label{eq:SUSYOperator}
\end{eqnarray}
Those spinors are intrinsically complex objects. $Q$ and $Q^{\dagger}$ are symmetry generators and fermionic operators. As $\ket{Boson}$ and $\ket{Fermion}$ states differ by spin $\frac{1}{2}$, the fermionic operator $Q$ itself carries spin $\frac{1}{2}$ and we can find $Q$ following an algebra of anti-commutation relations:
\begin{equation}
\left\{Q, Q^{\dagger}\right\} = P^{\mu},
\end{equation}
\begin{equation}
\left\{Q, Q\right\} = 0 , \left\{Q^{\dagger},Q^{\dagger}\right\} = 0,
\end{equation}
and
\begin{equation}
[P^{\mu}, Q] = 0 , [P^{\mu}, Q^{\dagger}] = 0,
\end{equation}
where $P^{\mu}$ is the four-momentum generator of space-time.\\
In a supersymmetric theory, all single particle states like $\ket{F}$ and $\ket{B}$ are combined into single objects called super multliplets. Each supermultliplet contains both bosons and fermions, which have the same gauge charges. In the SM there are no particles that share all of their quantum numbers and are of the same mass but differ by 1/2 spin. So in the SUSY framework there are predicted to be twice as many particles as we currently observe. If supersymmetry was an unbroken theory then the masses of the SUSY partner would be identical to the mass of the SM counterpart. Because we do not observe SUSY partners the symmetry must be broken by some mechanism at some high-energy scale such as in the early stages of the evolution of the universe. \\
\textbf{SUSY Nomenclature:} All bosonic SUSY partner names are preceeded by an "s" in their name. For ferminoic SUSY partners an "ino" is attached to the end of their name. We end up with amusing names for particles. The name of the SUSY partner of any quark will be "squark" and is represented by a tilda above the symbol. For example the SUSY partner of the top quark ($t$) will be the stop ($\tilde{t}$). Similarly the $W$ boson will get it's partner the Wino ($\tilde{W}$). Another category of SUSY particles are the neutralinos, which consist of the photino, Higgsino, and the Zino. The charginos are linear combinations of the charged Wino and charged Higgsinos.\\
\subsection{SUSY Lagrangian}
\label{ssec:susyLagrangian}
To consider SUSY as a serious replacement for the SM, we have to start by making supermultliplets that preserve SM symmetries and rules. So we create "chiral supermultliplets" combining Weyl fermions~\cite{weylfermions} with complex scalar particles and "gauge supermultiplets" by combining gauge bosons and Weyl fermions. To deal with unequal dimensions of fields in the chiral and gauge supermultiplets, we introduce auxiliary fields "$F^{a}$" and "$D^{a}$" such that,
\begin{equation}
\mathcal{L}_{F} = F^{a}*F^{a}
\label{eq:auxF}
\end{equation}
and
\begin{equation}
\mathcal{L}_{D} = \frac{1}{2}D^{a}D^{a}.
\label{eq:auxD}
\end{equation}
The "$F$" and "$D$" have no kinetic term. The next step in creating SUSY multiplets is to introduce a superpotential to specify interactions of the supermultiplets. As in the SM, we start with fermions that are now chiral supermultiplets labeled with indices $i$ and $j$. The Lagrangian for the scalar ($\phi_{i}$), fermion ($\psi_{i}$), and auxiliary ($F_{i}$) fields and their interaction terms, provided that the Lagrangian can be renormalized, takes the form,
\begin{eqnarray}
\mathcal{L}_{chiral} = (-\frac{1}{2}W^{ij}\psi_{i}\psi_{j} + W^{i}F_{i}) + \text{c.c},
\label{eq:firstsusyLagrangian}
\end{eqnarray}
with
\begin{equation}
W^{ij} = \frac{\partial^{2}W}{\partial\phi_{i}\partial\phi_{j}}, W^{i} = \frac{\partial{W}}{\partial\phi_{i}}. \nonumber
\end{equation}
$W$ is a superpotential of the form
\begin{equation}
W = \mathcal{L}^{i}\phi + \frac{1}{2}M^{ij}\phi_{i}\phi_{j} + \frac{1}{6}y^{ijk}\phi_{i}\phi_{j}\phi_{k}, \nonumber
\end{equation}
where $M^{ij}$ is the fermion mass term, $y^{ijk}$, known as the Yukawa interaction. Finally $L^{i}$ are the parameters with dimension of $[mass]^{2}$, which affects only the scalar potential part of the Lagrangian.\\
Similarly, the Lagrangian for the gauge supermultiplet is written as:
\begin{eqnarray}
\mathcal{L}_{gauge} = -\frac{1}{4}F_{\mu\nu}^{a}F^{\mu\nu a} + i\lambda^{\dagger a}\bar{\sigma}^{\mu}\Delta_{\mu}\lambda^{a} + \frac{1}{2}D^{a}D^{a},
\label{eq:gaugesupermultiplet}
\end{eqnarray}
where,
\begin{equation}
F_{\mu\nu}^{a} = \partial_{\mu}A_{\nu}^{a} - \partial_{\nu}A_{\mu}^{a} + gf^{abc}A_{\mu}^{b}A_{\nu}^{c} \nonumber
\end{equation}
is the usual Yang-Mills field strength, and
\begin{equation}
\Delta_{\mu}\lambda^{a} = \partial_{\mu}\lambda^{a} + gf^{abc} A_{\mu}^{b}\lambda^{c}. \nonumber
\end{equation}
$A_{\mu}^{a}$ is the covariant derivative for the gaugino field and transforms as
\begin{equation}
A_{\mu}^{a} \rightarrow A_{\mu}^{a} + \partial_{\mu}\Lambda^{a} + gf^{abc}A_{\mu}^{b}\Lambda^{c},
\end{equation}
where $\lambda^{a}$ is the two-component gaugino with an index $a$ that runs over all Weyl fermion adjoint representations of the gauge group. \\
\textbf{U(1) Transformation:} As in the SM, we want the theory to be gauge invariant. This means that we will get an interacting theory through gauge bosons just like the SM. But because of SUSY, we also get the superpartners of the gauge bosons called gauginos along with some $D$ terms and their interactions. We have to take into consideration the interaction between gauge and chiral supermultliplets. The details are worked out in~\cite{susyPrimer:2016} and give
\begin{equation}
F_{i} = -W_{i}^{*} \text{, } F^{*i} = -W^{i} \text{and } D^{a} = -g\phi^{*}T^{a}\phi,
\end{equation}
with $T^{a}$ as the generator of the group, $g$ being the gauge coupling, and $W$ are the gauginos. With these terms in Eqs.~\ref{eq:firstsusyLagrangian},~\ref{eq:auxD}, and~\ref{eq:auxF}, we can see the complete Lagrangian with trilinear and quadratic interactions introduced by auxiliary fields $F$ and $D$. \\
%Amrit Find lagrangian in susy primer
\subsection{SUSY Breaking}
\label{ssec:susybreaking}
One of the motivations that led to supersymmetry was the hierarchy problem. This can be turned around and used to explore how SUSY is broken. SUSY required us to introduce scalar fields for each SM Dirac fermion that cancel out the quadratic divergent terms in the Higgs mass. A simple way to generalize this statement is to show that loop corrections for fermions and bosons are of opposite sign. The introduction of the SUSY field helps to cancel the divergent terms in the Higgs mass as their contribution to the correction are opposite in sign. If SUSY was an unbroken symmetry then there will be no mass difference between the SM particles and SUSY particles. This lead to the situation where theories in SUSY can't explain mass at all. \\
%Failure to maintain would lead to quadratic divergence of correction of Higgs scalar mass of form,
%\begin{equation}
%\Delta m_{H}^{2} \sim \frac{1}{8\pi^{2}} (\lambda_{s} - \left|\lambda_{f}\right|^{2}) \Lambda_{+}^{2}
%\end{equation}
So the effective Lagrangian can be broken into two parts; one $\left(\mathcal{L}_{susy}\right)$ part that contains gauge and Yukawa interaction and preserves SUSY invariance and another softly broken term that contains the mass and interaction terms:
\begin{equation}
\mathcal{L} = \mathcal{L}_{susy} + \mathcal{L}_{soft}.
\end{equation}
Sometimes, this type of breaking of SUSY is called "soft" supersymmetry breaking~\cite{susyPrimer:2016}. Soft breaking consists of several possible terms:
\begin{equation}
\mathcal{L}_{soft} = - \frac{1}{2} M_{a} \tilde{\lambda}_{a} \tilde{\lambda}_{a}- \frac{1}{6} b_{ijk} \tilde{f}_{i} \tilde{f}_{j} \phi_{k} - \frac{1}{2} b_{ij} \tilde{f}_{i} \tilde{f}_{j} - m_{ij}^{2} \tilde{f}_{i}^{*} \tilde{f}_{j},
\label{eq:susysoft}
\end{equation}
where, \\
$\tilde{\lambda} \rightarrow $ super partner field of gauge bosons \\
$\tilde{f} \rightarrow $ super partner field of fermions.\\
$ \phi \rightarrow $ scalar field \\
The first term in the equation represents the gaugino mass, the last two terms represent the sfermions mass, and the second term indicates the triple-scalar interaction.\\
These terms break SUSY preserving R-parity~\footnote{ R-parity is a symmetry acting on the Minimal Supersymmetric Standard Model (MSSM) and defined as:
\begin{equation}
P_{R}= (-1)^{3B + L +2s} \nonumber
\end{equation}
where $s$ is spin, $B$ is a baryon number, and $L$ is a lepton number. All Standard Model particles have R-parity of \{+1\} while supersymmetric particles have R-parity of \{-1\}.}.
In the upcoming section, we will expand generic soft breaking terms within the context of the Standard Model.\\
\subsection{Minimal Supersymmetric Standard Model}
\label{ssec:mssm}
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
&&&& \\
Names& & spin 0 & spin $\frac{1}{2}$ & $SU(3)_{C}, SU(2)_{L}, U(1)_{Y}$ \\
&&&&\\
\hline
&&&&\\
squarks, quarks & Q & ($\tilde{u}_{L}\text{ }\tilde{d}_{L}$) & ($u_{L} \text{ } d_{L}$) & ($3, 2, \frac{1}{6}$)\\
&&&&\\
(×3 families) & $\bar{u}$ & $\tilde{u}_{R}^{*}$ & $u^{\dagger}_{R}$ & ($\bar{3}, 1, -\frac{2}{3}$)\\
&&&&\\
& $\bar{d}$ & $\tilde{d}_{R}^{*}$ & $d^{\dagger}_{R}$ &($\bar{3}, 1, \frac{1}{3}$)\\
&&&&\\
\hline
&&&&\\
sleptons, leptons & L & ($\tilde{\nu}, \tilde{e}_{R}$) & ($\nu, e_{R}$) & ($1, 2, -\frac{1}{2}$)\\
&&&&\\
(×3 families) & $\bar{e}$ & $\tilde{e}_{R}^{*}$ & $e^{\dagger}_{R}$ & ($1, 1, 1$)\\
&&&&\\
\hline
&&&&\\
Higgs, higgsinos & $H_{u}$ & ($H_{u}^{+}, H_{u}^{0}$)& ($\tilde{H}_{u}^{+}, \tilde{H}_{u}^{0}$) &($1, 2, +\frac{1}{2}$)\\
&&&&\\
& $H_{d}$ & ($H_{d}^{-}, H_{d}^{0}$)& ($\tilde{H}_{d}^{-}, \tilde{H}_{d}^{0}$) &($1, 2, -\frac{1}{2}$)\\
&&&&\\
\hline
\end{tabular}
\end{center}
\caption{\small{Chiral supermultiplets in the Minimal Supersymmetric Standard Model. The spin-0
fields are complex scalars, and the spin-$\frac{1}{2}$ fields are left-handed (with subscript $L$) and right-handed (with subscript $R$) two-component Weyl fermions.}}
\label{table:chiralsupermultiplets}
\end{table}
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
&&& \\
Names& spin $\frac{1}{2}$& spin 1 & $SU(3)_{C}, SU(2)_{L}, U(1)_{Y}$ \\
&&&\\
\hline
&&&\\
gluino, gluon & $\tilde{g}$ & $g$& (8, 1, 0) \\
&&& \\
\hline
&&&\\
winos, W bosons & $\tilde{W}^{\pm}, \tilde{W}^{0}$& $W^{\pm}, W^{0}$ & (1, 3, 0) \\
&&&\\
\hline
&&&\\
bino, B boson & $\tilde{B}^{0} $ & $B^{0} $ &(1, 1, 0)\\
&&&\\
\hline
\end{tabular}
\end{center}
\caption{\small{Gauge supermultiplets in the Minimal Supersymmetric Standard Model.}}
\label{table:gaugesupermultiplets}
\end{table}
The MSSM is an extension to the Standard Model. The word minimal refers to the model with the smallest number of new particle states and interactions consistent with existing theoretical model and experiments. Table~\ref{table:chiralsupermultiplets} and~\ref{table:gaugesupermultiplets} form the particle list for MSSM. Note that there are two Higgs supermultiplets. The second Higgs supermultiplet is required to give "up" type fermions their mass after spontaneous symmetry breaking. It also prevents gauge any anamoly~\cite{susyPrimer:2016}~\cite{HEHaber:1993b}. The superpotential for MSSM is written as
\begin{equation}
W_{mssm} = \bar{u}Y_{u}Q H_{u} - \bar{d}Y_{d}Q H_{d} - \bar{e}Y_{r}LH_{d} + \mu H_{u}H_{D}.
\label{eq:mssmsuperpotential}
\end{equation}
To describe MSSM completely, we take the generic soft breaking terms in Eq.~\ref{eq:susysoft} and express them in terms of the super fields listed in Tables~\ref{table:chiralsupermultiplets} and~\ref{table:gaugesupermultiplets}. All soft breaking terms are~\cite{susyPrimer:2016}:
% Some equations
\begin{itemize}
\item \textbf{Gaugino mass } -$\frac{1}{2}(M_{3} \tilde{g}\tilde{g} + M_{2} \tilde{W}\tilde{W} +M_{1} \tilde{B}\tilde{B}$)
\item \textbf{SFermion masses } $-\tilde{Q}^{\dagger}\hat{M}_{\tilde{Q}}^{2}\tilde{Q} - \tilde{L}^{\dagger}\hat{M}_{\tilde{L}}^{2}\tilde{L} - \tilde{\bar{u}}m_{\tilde{\bar{u}}}^{2} \tilde{\bar{u}}^{\dagger} - \tilde{\bar{d}}m_{\tilde{\bar{d}}}^{2} \tilde{\bar{d}}^{\dagger} - \tilde{\bar{e}}m_{\tilde{\bar{e}}}^{2} \tilde{\bar{e}}^{\dagger}$, where mass matrices are $3\times3$ Hermition matrices.
\item \textbf{Triple scalar coupling } -($\tilde{\bar{u}}\hat{a}_{u}\tilde{Q}H_{2} + \tilde{\bar{d}}\hat{a}_{d}\tilde{Q}H_{1} + \tilde{\bar{e}}\hat{a}_{e}\tilde{L}H_{1} + c.c $ where scalar couplings are $3\times3 $ complex matrices
\item \textbf{Higgs masses and mixing }$ -m_{H_2}^{2}H_{2}^{\dagger}H_{2} - -m_{H_1}^{2}H_{1}^{\dagger}H_{1} - (bH_{2}H_{1} + \text{c.c.})$
\end{itemize}
These four sets contain 107 unknown parameters along with four from the Higgs doublet making for a total of 111 parameters in the MSSM. These 111 parameters are unspecified and can only be extracted from measurements. This vast parameter space is impossible to explore, at least for the time being.\\
\subsection{Minimal Super Gravity( mSUGRA)}
\label{ssec:msugra}
By imposing some assumptions on MSSM, we can reduce the number of the parameters to some workable number. These assumptions reduce the SUSY breaking parameters from above one hundred to five. Essentially, the idea is that at very high energies (the GUT and Strong unification scale of $\text{10}^{19}$ GeV), all super partners become mass degenerate. This assumption allows us to set the mass of all sfermions to a single value at very high energy and we do the same for the gaugino mass and the higgsino mass. Imposing additional constraints such as CP-violation~\cite{CPViolationSUSY} and inserting small off-diagonal elements in the mass matrix in Eq.~\ref{eq:susysoft} leads to the universality of SUSY breaking. Under this universality, all mass matrices are proportional to the unit matrix, triple scalar couplings are proportional to the Yukawa matrix, and breaking parameters have no complex phase. The five remaining parameters after the mSuGRA assumptions are:
\begin{equation}
M_{1} = M_{2} =M_{3} = M_{\frac{1}{2}},
\end{equation}
\begin{equation}
\hat{m}_{\tilde{Q}}^{2} = \hat{m}_{\tilde{L}}^{2} = \hat{m}_{\tilde{\bar{u}}}^{2} = \hat{m}_{\tilde{\bar{d}}}^{2} = \hat{m}_{\tilde{\bar{e}}}^{2} = M_{0}^{2}\hat{1},
\end{equation}
\begin{equation}
M_{0}^{2} = m^{2}_{H_1} = m^{2}_{H_2},
\end{equation}
\begin{equation}
a_{u} = A_{0}Y_{u}, a_{d} = A_{0}Y_{d}, a_{e} = A_{0}Y_{e},
\end{equation}
and
\begin{equation}
b= B_{0}\text{sign}\mu,
\end{equation}
where $\text{sign}\mu$ is a sign of $\mu$ SUSY conserving Higgsino mass parameter and takes the value $\pm$1. Given the small parameter space, this model can be extrapolated from the experimental data. Also, since off-diagonal elements in the mass matrix are non-zero in the SM and in CP-violation we allow a complex phase in the quark-mixing matrix. Moreover, the theoretical motivation for grand unification is mainly aesthetical. These two assumptions are weakly motivated from theoretical point of view, the reason to go along with these assumptions is practicality. \\
%Generally speaking, the most motivating aspect of mSUGRA is it's predictive power as theory with more than hundred parameters would have lost predictive power.: We need to find a more elegant way of saying this. The small number of free parameters in msugra makes it... Look at papers to see how to better describe this. %
\subsection{Simplified Model Spectra (SMS)}
\label{ssec:sms}
A simplified model is defined by a set of hypothetical particles and a sequence of their production and decay. In the simplified models under consideration, only the production process for two primary particles is considered. Each primary particle can undergo a direct decay or a cascade decay through an intermediate new particle. Each particle decay chain ends with a neutral, undetected particle, denoted LSP (lightest supersymmetric particle) and one or more SM particle. The masses of the primary particle and the LSP are free parameters. The simplified models with a T1-, T3-, and T5-prefix are all models of gluino pair production and those with a T2- and T6-prefix are models of squark-antisquark production. In this document, only the T1tttt and T2tt models are considered for interpretation. Simplified models will be described in Chapter~\ref{ch:analysis}. Detailed description of the SMS is presented in~\cite{sms}.\\
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.8\textwidth]{figure/susyxslhc}
\caption{\small{NLO+NLL production cross sections for the case of equal degenerate squark and gluino masses as a function of mass at $\sqrt{s}$ = 13 TeV}~\cite{susyxslhc}}
\label{fig:susyxslhc}
\end{center}
\end{figure}
This analysis focuses on a search for supersymmetric particles produced in two specific decay chains associated with a simplyfied SUSY model: gluino and stop pairs produced at a center of mass energy of 13 TeV in proton-proton collisions. Furthermore, our search assumes the stop and gluino masses are around 2 TeV, with an inclusive cross section of approximately 10$^{-2}$ pb. Theoretical expectations for SUSY cross sections as a function of mass at 13 TeV are shown in Fig.~\ref{fig:susyxslhc}. This is the mass scale the LHC is capable of exploring.