\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 three lepton flavours are separated over 3 generations (each with two quarks and two leptons in it).
Experimental evidence suggests that exactly three generations of particles exist \cite{akrawy1989measurement}. Each particle of the first generation has higher energy versions of itself with the similar properties, besides their mass, (e.g. $e^- \rightarrow \mu^- \rightarrow \tau^-$) as in other generations. For each following generation, the particles have 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 colour 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 colourless\footnote{Colour confinement}. In conclusion we cannot observe a single quark as they always appear in pairs of two or three to achieve colourlessness.\\
Free particles can achieve being colourless in two ways. Either by having all three colours present in the same amount (one quark of each colour), which creates the characteristic group of baryons($qqq$) and anti-baryons($\bar{q}\bar{q}\bar{q}$) or by having a colour and its anticolour present, which creates the group of mesons($q\bar{q}$).
\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 & Strength & 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 strength of 1 (though it decreases with higher energies as a result of gluon-gluon self interaction loops, which interfere negatively in perturbation theory)\cite{thomson2013modern}. 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 colour 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. Only 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\cite{thomson2013modern}. 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 fundamental bosons to have an inertial mass.\\
First we will discuss the EM neutral Z boson. Even though the Z boson belongs to the weak force, it also has an electromagnetic part additionally to the weak force part\footnote{$Z \rightarrow EM_{part} + W^3$, \cite{thomson2013modern}}. It follows directly, that the Z boson couples weaker to uncharged particles.\\
The other boson of the weak force is the W boson. 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 cannot 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 Z boson, 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 for all other interactions baryon number has to be conserved. So any up-type quark can be changed into a down-type quark and vice versa by emitting or absorbing a W boson. In the end however, there are still 3 quarks which form a baryon\footnote{Pentaquarks($qqqq\bar{q}$) and other exotic states excluded}, 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 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}
\caption{Certain diagrams of decays}
\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 interacts with a W boson, it just changes a lepton to its corresponding lepton neutrino or the other way around (e.g. see Figure \ref{muon-decay_feynman}).\newpage
\section{Physics beyond the SM}
\subsection{Neutrino Oscillation}
Classically the SM considers neutrinos to be massless. While this assumption works well for a lot of cases, we know nowadays that at least two of the three neutrinos have to have mass\footnote{The mass difference between neutrinos is non zero: $m_i - m_j = \Delta m_{i,j} \neq 0$, $\forall j \neq i$}. Neutrinos are known to oscillate between all three states of flavour, as the eigenstates of flavour are not eigenstates of mass. As a consequence $\nu_e$, $\nu_\mu$ and $\nu_\tau$ are not fundamental particle states but a mixture of the mass eigenstates $\nu_1$, $\nu_2$ and $\nu_3$. They are connected through the PMNS matrix:
\begin{equation}
\begin{pmatrix}\nu_e \\ \nu_\mu\\ \nu_\tau\end{pmatrix} =
\begin{pmatrix}
U_{e_1} & U_{e_2} & U_{e_3} \\
U_{\mu_1} & U_{\mu_2} & U_{\mu_1} \\
U_{\tau_1} & U_{\tau_2} & U_{\tau_3}
\end{pmatrix}
\begin{pmatrix}
\nu_1 \\ \nu_2 \\ \nu_3
\end{pmatrix}
\label{PMNS_neutrino}
\end{equation}
As a result, neutrinos propagate as a superposition of all mass eigenstates. Additionally, we can describe the PMNS matrix through three mixing angles $\theta_{12}$, $\theta_{13}$ and $\theta_{23}$ and a complex phase $\delta$ \footnote{Measurements: $\theta_{12} \approx 35^\circ$, $\theta_{13} \approx 10^\circ$, $\theta_{23} \approx 45^\circ$ \cite{abe2008precision}, \cite{adamson2011measurement}}. Using this, the electron superposition looks like this:\\
$\ket{\nu_e} = U_{e_1} \ket{\nu_1} e^{{-i \Phi_1}} + U_{e_2} \ket{\nu_2} e^{{-i \Phi_2}} + U_{e_3} \ket{\nu_3} e^{{-i \Phi_3}}$ with $\Phi_i = E_i \times t$\\
As a result lepton family number is not a conserved quantity anymore as neutrino flavour oscillates over time.\\
We can calculate the probability for a neutrino to transition from flavour $\alpha$ to $\beta$ like this:
\begin{equation}
\begin{split}
P(\nu_\alpha \rightarrow \nu_\beta) = 2 Re \left( U_{\alpha_1} U^\ast_{\beta_1} U^\ast_{\alpha_2} U_{\beta_2} e^{-i(\Phi_1-\Phi_2)}\right)\\
+ 2 Re \left( U_{\alpha_1} U^\ast_{\beta_1} U^\ast_{\alpha_3} U_{\beta_3} e^{-i(\Phi_1-\Phi_3)}\right)\\
+ 2 Re \left( U_{\alpha_2} U^\ast_{\beta_2} U^\ast_{\alpha_3} U_{\beta_3} e^{-i(\Phi_2-\Phi_3)}\right)
\end{split}
\label{neutrino_flavour_change_prob}
\end{equation}
An important thing to note is, that if any elements of the PMNS matrix are complex, this process is not invariant under time reversal ($t \rightarrow -t$)\footnote{\alignLongunderstack{%
\text{The probability does not change, if we add a complex phase to the PMNS}\\ \text{ matrix, just if one of the elements has a phase different from the others}}}\\
$P(\nu_\alpha \rightarrow \nu_\beta) \neq P(\nu_\beta \rightarrow \nu_\alpha)$.
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.8\textwidth]{img/neutrino_oscillation.png}
\caption{Process that violates lepton family number conservation through neutrino oscillation}
\label{neutrino_osc_feyn}
\end{center}
\end{figure}
Nowadays it's a well accepted fact that lepton family number gets violated through neutrino oscillation.\\
However, why should flavour oscillation be exclusive to neutrinos?\\
Maybe there are ways for the EM charged leptons as well to directly transition to another lepton family\footnote{Maybe also possible for quarks?}?
\subsection{New physics}
As a consequence of neutrino oscillation, lepton flavour is a broken symmetry. The SM has to be adapted to include lepton flavour violation (LFV) and massive neutrinos. LFV is also expected for charged neutrinos.\\
Although, it has yet to be determined how LFV exactly works and to which scale it exists.\\
This may raise the question on why charged LFV has never been observed yet. This is especially surprising as the mixing angles of the neutrinos have been measured to be big.\\
There are two reasons why charged LFV is strongly surpressed:
The first is, that charged leptons are much heavier than neutrinos, and the other, that the mass differences between neutrino flavour are tiny compared to the W boson mass.\\
In the classical SM, charged LFV is already forbidden at tree level. Though it can be induced indirectly through higher order loop diagrams using neutrino oscillation. By adding new particles beyond the SM, we generate new ways for LFV in the charged sector to happen. As LFV is naturally generated in many models beyond the SM, finding charged LFV is a strong hint for new physics.
\begin{figure}[H]
\begin{center}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=0.8\textwidth]{img/LFV-neutrino_osc.png}
\caption{LFV through neutrino oscillation}
\label{LFV-neutrino_osc}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=0.8\textwidth]{img/LFV-SUSY.png}
\caption{LFV by using supersymmetric particles}
\label{LFV-SUSY}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=0.8\textwidth]{img/LFV-tree_lvl.png}
\caption{LFV at tree level}
\label{LFV-tree_lvl}
\end{subfigure}
\caption{Charged LFV}
\end{center}
\end{figure}
One way charged LFV can occur is through supersymmetric particles (see Figure \ref{LFV-SUSY}). By observing charged LFV, supersymmetry would gain new importance.\\
Alongside with supersymmetric models, other extensions of the SM, such as left-right symmetric models, grand unified models, models with an extended Higgs sector and models where electroweak symmetry is broken dynamically, are all good candidates to explain charged LFV and most importantly, experimentally accessible in a large region of the parameter space.
