\section{Standard Model}
\subsection{Elementary particles and forces}
\label{intro_elem_part}
The Standard Model(SM) describes all known elementary particles as well as three of the four known forces\footnote{Strong, weak and electromagnetic forces}.\\
The elementary particles that make up matter can be split into two categories, namely quarks and leptons. There are 6 types of quarks and six types of leptons. The type of a particle is conventionally called flavour. The six quark flavours and the six lepton flavours are separated over 3 generations (each which two quarks and two leptons in it).
Experimental evidence suggests that there exist exactly three generations of particles. Each particle of the first generation has higher energy versions of itself with the same characteristics (e.g. $e^- \rightarrow \mu^- \rightarrow \tau^-$)as in other generations. Contrary, each following generation has a higher mass than the generation before.\\
\begin{table}[H]
\begin{center}
\caption{Quarks in the Standard Model} \label{Quark_SM_table}
\begin{tabular}{l l l l l}
&&Quarks&&\\\hline
& Particle& & Q$[e]$ & $\frac{mass}{GeV}$\\ \hline
1. Gen. & up & u & $-\frac{1}{3}$ & $0.003$\\
& down & d & $\frac{2}{3}$ & $0.005$\\
2. Gen. & strange & s & $-\frac{1}{3}$ & $0.1$\\
& charm & c & $\frac{2}{3}$ & $1.3$\\
3. Gen. & bottom & b & $-\frac{1}{3}$ & $4.5$\\
& top & t & $\frac{2}{3}$ & $174$\\
\end{tabular}
\end{center}
\end{table}
One category consists of quarks($q$)(see Table \ref{Quark_SM_table}). In this, we differentiate between up-type quarks, with charge $-\frac{1}{3}e$, and down-type, quarks with charge $\frac{2}{3}e$. Quarks interact with all fundamental forces.\\
Each quark carries a property called colour-charge. The possible color charges are red(r), green(gr), blue(bl) in which anti-quarks carry anti-colour. Quarks can only carry one colour, whilst every free particle has to be colorless\footnote{Colour confinement}. In conclusion we cannot observe a single quark.\\
Free particles can achieve being colourless in two ways. Either by having all three colors present in the same amount (one quark of each color), which creates the characteristic group of baryons($qqq$) and anti-baryons($\bar{q}\bar{q}\bar{q}$) or by having a color and its anticolor present, which creates the group of mesons($q\bar{q}$).\\\newpage
\begin{table}[H]
\begin{center}
\caption{Leptons in the standard model} \label{Lepton_SM_table}
\begin{tabular}{l l l l l}
& & Leptons & &\\ \hline
& Particle& & Q$[e]$ & $\frac{mass}{GeV}$\\ \hline
1. Gen. & electron & $e^-$ & $-1$ & $0.005$\\
& neutrino & $\nu_e$ & $0$ & $<10^{-9}$\\
2. Gen. & muon & $\mu^-$ & $-1$ & $0.106$\\
& neutrino & $\nu_{\mu}$ & $0$ & $<10^{-9}$\\
3. Gen. & tau & $\tau^-$ & $-1$ & $1.78$\\
& neutrino & $\nu_{\tau}$ & $0$ & $<10^{-9}$
\label{Lepton_table}
\end{tabular}
\end{center}
\end{table}
The other group consists of leptons(l)(see Table \ref{Lepton_SM_table}). They only interact through the weak and the electromagnetic force. Each generation consists of a lepton of charge -1 and a corresponding EM neutrally charged neutrino. The electron has the lowest energy of all charged leptons. This makes the electron stable while the higher generation particles decay to lower energy particles. \\
\\
The leptons of one generation, namely the charged lepton and its corresponding neutrino are called a lepton family. A lepton of a family counts as 1 to its corresponding lepton family number whilst a anti-lepton counts as -1.
\begin{table}[H]
\begin{center}
\caption{Fundamental forces} \label{fund_forces_table}
\begin{tabular}{l l l l l l l}
Force & Strengh & Boson & & Spin & Charge & $\frac{mass}{GeV}$ \\\hline
Strong & 1 & gluon & $g$ & 1& 0& 0 \\
Electromagnetism & $10^{-3}$ & photon & $\gamma$ & 1& 0& 0 \\
Weak & $10^{-8}$ & Z boson & $Z$ & 1& 0& 80.4\\
& $10^{-8}$ & W boson & $W^{\pm}$& 1& $\pm 1$& 91.2\\
\end{tabular}
\end{center}
\end{table}
The particles of the SM interact through the 3 fundamental forces of the SM. In these interactions, particles called bosons are being exchanged which are the carriers of their respective force (see Table \ref{fund_forces_table}).\\
As mentioned above, only quarks can interact through the strong force, in which they exchange gluons. Gluons are massless and EM neutrally charged. The strong force has the biggest coupling strengh of 1 (though it decreases with higher energies as a result of gluon-gluon self interaction loops, which interfere negatively in perturbation theory)\footnote{Mark Thomson - Modern Particle physics - 10.5.2}. A gluon carries colour charge and hence can change the colour of a quark but it conserves its flavour. The strong interaction has an underlying gauge symmetry of SU(3). Therefore, it can be derived that color charge is conserved through the strong interaction\footnote{E.g. through Gell-Mann matrices}.\\
The electromagnetic(EM) force is propagated through the photon. It carries zero charge and no invariant mass. Exclusively charged particles can interact through the electromagnetic force. The coupling strength is $\alpha \approx \frac{1}{137}$, contrary to the strong force the coupling constant increases with higher energies\footnote{Mark Thomson - Modern Particle physics - 10.5.1}. This difference stems from the fact that photon-photon interaction loops are not allowed whereas gluon-gluon interaction loops are. In perturbation theory this results in only positive terms being added to the coupling strength. The underlying gauge symmetry is of SU(1). The electromagnetic force also conserves flavour.\\
The weak force has two types of bosons. The bosons of the weak force are the only bosons to have an inertial mass.\\
First we will discuss the EM neutrally charged Z boson\footnote{Discovered at Super Proton Synchrotron accelerator - Cern - 1983}. Even though the Z boson belongs to the weak force it, it also has an electromagnetic part additionally to the weak force part\footnote{$Z \rightarrow EM_{part} + W^3$, Modern Particle Physics - Blababla}. It follows directly, that the Z boson couples weaker to uncharged particles.\\
The other boson of the weak force is the W boson\footnote{Discovered at Super Proton Synchrotron accelerator - Cern - 1983}. In the classical SM, the only way particles can change flavour is through the weak force by emitting or absorbing W boson. It is important to notice that, besides of having an invariant mass, the W boson is the only boson with a non zero charge ($Q_{W^\pm} = \pm 1e$). In the gauge symmetry of the weak force the $W^\pm$ are actually the creation and annihilation operators of said symmetry\footnote{$W^\pm = W_1 \pm i W_2$}.\\
An important characteristic of the weak force is that it exclusively couples to lefthanded(LH) particles and righthanded(RH) antiparticles (describing chirality states)\footnote{In the ultrarelativistic limit helicity and chirality eigenstates are the same}.\\
The chirality operators for left- and righthandedness are: \\\\
LH: $\frac{1}{2}(1-\gamma^5)$, RH: $\frac{1}{2}(1+\gamma^5)$\\\\
As a consequence RH particles and LH anti-particles cant couple to the W boson at all. This also results in charged RH particles and LH anti-particles to couple to the Z boson only through the electromagnetic part of the itself, while uncharged RH particles and LH anti particles (e.g. RH $\nu$, LH $\bar{\nu}$) don't couple with the EM force nor the weak force.
\subsection{Interaction rules}
Now we will establish the general rules for interactions in the SM.\\\\
\textbf{Baryon number is conserved}\\
As we already established before, the only interaction that can change flavour is the weak force through the W boson. We directly see that all other interactions baryon number has to be conserved. So any up-type quark can be changed to a down-type quark and backwards by emitting or absorbing a W boson. In the end however, there are still 3 quarks which form a baryon\footnote{I exclude $q\bar{q}$ pair-antipair production to form a pentaquark($qqqq\bar{q}$) and other exotic states}, even though it changed its type and charge. A well known example is the beta decay, where a down quark in a neutron decays into a an up quark to form now a proton(e.g. see Figure \ref{beta-decay_feynman}). We easily see that the baryon number is conserved.\\\\
\begin{figure}[H]
\begin{center}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=0.8\textwidth]{img/beta_decay_feynman.png}
\caption{Feynman diagram of the $\beta$-decay}
\label{beta-decay_feynman}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=0.8\textwidth]{img/muon-decay-feynman.png}
\caption{Feynman diagram of a $\mu$-decay}
\label{muon-decay_feynman}
\end{subfigure}
\end{center}
\end{figure}
\textbf{Lepton family number is conserved}\\
According to the SM lepton family number is conserved. As all interactions beside the W conserve particle flavour, it is easy to see that lepton family number is conserved.\\
Whenever a lepton interaction with a W boson, it just changes a lepton to its corresponding lepton neutrino and or the other way around (e.g. see Figure \ref{muon-decay_feynman}).
