\section{The Standard Model of particle physics}\label{sec:SM}

The structure of matter was a subject of contemplation of philosophers and scientists since ancient history. Particle physics as we know it today, however, exists only since the beginning of the 20th century. It was driven by two breakthrough discoveries: the discovery of electrons by J.\,J.\,Thompson in 1897~\cite{SM-Thompson} and the discovery of the atomic nucleus by E.\,Rutherford in 1911~\cite{SM-Rutherford}. Electrons and protons were considered the main constituents of matter until the 1932, when this picture was expanded by the discovery of the neutron by J.\,Chadwick~\cite{SM-Chadwick} and  by the discovery of the positron by C.\,Anderson~\cite{SM-Anderson}. 

Around this time, the first ideas about interactions between particles emerged: the well-known electro-magnetism was joined by the strong force holding the nuclei together and by the weak-force describing beta radioactivity, discovered in 1896 by H.\,Becquerel~\cite{SM-Becquerel}. Since the weak interaction is very important for the development of the Standard Model (SM) as well as for this work, it will be in the spotlight of this chapter. 

\subsection{The beginning of the Standard Model}\label{sec:SM-hist}

The theory of the beta decay was successfully described by E.\,Fermi in 1933\,\cite{SM-Fermi}, where he predicted the existence of a neutrino\footnote{Fermi's paper suggesting the existence of neutrinos was rejected by Nature, as it was "too remote from physical reality to be of interest to the readers"~\cite{SM-Fermi-reject}.}. In this pioneering work, he suggested a direct interaction of four spin-\oneHalf quantum fields (neutron, proton, electron and antineutrino). This reflects the fact that the weak force has essentially zero-range: unlike electromagnetism, where the photon is the interaction mediator, there is no boson mediator. 

At that time this was a great approximation of the beta decay \emph{at low energies}. Even though Fermi's description is different from quantum electrodynamic (QED), describing the electromagnetic force, Fermi used Lorentz four-vectors to describe the fermion fields appearing in bi-linear combinations (\emph{currents}), similarly to QED. This paved the path to electroweak interaction unification. 

Fermi's theory postulated only beta decays with no change of nucleonic spin ($\Delta S=0$). However, as nuclear spectroscopy became more precise, it was clear that beta decays with a change of nucleonic spin one ($\Delta S=1$) do occur~\cite{SM-Kurie}. A generalization of the Fermi theory explaining this observations was proposed by G.\,Gamow and E.\,Teller in 1936~\cite{SM-Gamow}.  Instead of using only vector currents as Fermi did, one can construct the four-fermion interaction using the whole set of scalars (S), vectors (V), tensors (T), axial vectors (A) and pseudoscalars (P). A- and T-couplings describe a spin-changing interaction, while S- and V- couplings are spin-zero transition operators. 

At that time it was intuitively assumed parity symmetry holds in the quantum world in the same way as in our macroscopic world. However, two particles with very similar masses and the same spin were observed, $\tau$ and $\theta$, decaying into different final states: $\taup\to\pip\pip\pim$ and $\theta^+\to\pip\piz$. As pions have intrinsic parity of $-1$, $\tau$ and $\theta$ had to be different particles or the parity symmetry must be violated\footnote{The angular momentum $J$ is conserved. Therefore the parity of the three-pion system is equal to $(-1)^3(-1)^{J(\tau)}$ and the parity of two-pion system $(-1)^2(-1)^{J(\theta^+)}$.}. In 1954, R.\,H.\,Dalitz looked into the two decays and confirmed $\tau$ and $\phi$ are the same particle~\cite{SM-Dalitz}. Today, we call $\theta$ and $\tau$ neutral kaon $\Kz$.

The theoretical solution to this \emph{$\tau-\theta$ puzzle} was found in 1956, when T.\,D.\,Lee and C.\,N.\,Yang suggested to abandon the idea of parity symmetry conservation~\cite{SM-Lee}. The violation of parity symmetry was quickly observed by three independent measurements: C.\,S.\,Wu measured the rate of electrons originating from the decay of $^{60}$Co atoms aligned by a uniform magnetic field.  If parity is conserved, the same amount of electrons should be emitted in the direction of the nucleus spin and opposite to the spin direction of the nucleus. She observed that the electrons prefer the direction opposite to the spin of the nucleus, confirming the parity violation~\cite{SM-Wu}. The other two experiments studied the pion decay to a muon and a neutrino\footnote{Muons quickly decay to an electron, serving as an analyzer for the muon polarization.}: $\pip\to\mup+\neu$~\cite{SM-Friedman,SM-Lederman}. %$\mup\to\ep+2\neu$, electron serves as an analyzer for the muon polarization.
Since pions have spin zero and spin is conserved, the  muon and the  neutrino have to have opposite spin sign. If parity is conserved, the polarization of the muon would be symmetric along their direction of motion. However, the muon spin direction \emph{favors} the direction of motion.

The general lagrangian proposed by Gamov and Teller expanded by the parity violating term was rather complex. %
This was reduced to only the V-A component in 1958 independently by R.\,Feynman with M.\,Gell-Mann~\cite{SM-Feynman} and R.\,Marschak with E.\,Sudurshan~\cite{SM-Sudarshan}. 
They followed the idea of two-component spinor from 1920's~\cite{SM-Weyl}, applied it to neutrinos, and postulated that any elementary fermion, regardless of its mass, can participate in weak interactions only through the left-handed chiral component of the corresponding spinor field. This lead to the universal current-current form of the weak interaction:%
\begin{equation}
    \mathcal{L}_{int}^{weak} = \frac{G_F}{\sqrt{2}}J^{\rho}J_{\rho}^{\dagger}\,,
\end{equation}
where $\mathcal{L}_{int}^{weak}$ is the weak interaction lagrangian, $G_F$ is the \emph{universal Fermi constant} and $J^{\rho}$ is the weak current (or probability flux). This implies the possibility of interaction of the weak current with itself, opening the path for intermediate vector bosons, similar to QED. 

\clearpage
Given there is no physical reason for this, in the light of today's knowledge, the idea was a rather fortunate but very successful guess. 
The problem of this assumption was that the observed beta decay until then preferred the spin change of either $\Delta S=0$ or  $\Delta S=2$, which is excluded by the two-component theory. On the other hand it described very well all known particles' weak interaction and even predicted some interactions.

Another major problem of this theory is that it is \emph{not renormalizable}. This means it does not hold anymore at high energies of about $\sqrt{\frac{2\pi\sqrt{2}}{G_F}}$~\cite{SM-Lee-2, SM-Okun, SM-Appelquist}. This problem was solved by introducing an electrically charged intermediate vector boson $W$. Nonetheless, this brought a new problem: production of \Wm\Wp pairs in fermion-antifermion annihilation lead to a power-like unitarity violation~\cite{SM-GellMann-1}. 

\subsection{Unification of electromagnetism with weak interaction}

As the \W boson has electromagnetic charge, it has to interact electromagnetically. As the weak interaction when exchanging the $W$ boson violates parity maximally, while QED is parity conserving, one cannot just add QED and weak lagrangians together. Instead of adding terms to the lagrangians, the unification of weak interaction with QED was realized in a very different way. Taking a step back in history, in 1954 C.\,N.\,Yang and R.\,Mills looked into non-abelian (non-commutative) gauge invariance~\cite{SM-Yang-Mills}. They showed that the transformation from global to local symmetry requires a triplet of vector fields, analogous to the photon field. However, contrary to the  photon field, this field also interacts with itself. 

S.\,Glashow showed in 1961 that the minimal representation of the electroweak unification indeed requires four gauge fields: the known photon, \Wp, \Wm and a new neutral vector boson~\cite{SM-Glashow-1}. The new neutral vector boson (called \Z today) is required to bridge the gap between parity-conserving electromagnetism and parity-violating weak interaction. In mathematical terms, the appropriate gauge group is a not simple \SU group\footnote{It is interesting that this \SU electroweak unification was not the only theoretical solution: by introducing new electron-type leptons, one reaches simple $SU(2)$ electroweak unification. This is however not supported by the experimental data.}. This effectively means wo independent coupling constants are needed.

This idea also led to an estimation of the \W boson mass to be 77.7\gev (the currently measured value is $80.38\pm0.012\gev$~\cite{PDG}). While massive \W bosons canceled the main divergences mentioned in the previous subsection, new divergences appeared in the interactions between the vector-boson fields with themselves.


\subsection{Renormalization problem}\label{sec:SM_renormalization}

The renormalization problem was solved by adding a scalar field to the electroweak theory. The initial idea was published by J.\,Goldstone~\cite{SM-Goldstone}, who added the so-called \emph{mexican hat} potential $V(\varphi)$ to the lagrangian density
\begin{equation} \label{eq:mexican_potential}
    V(\varphi) = -\mu^2 \varphi \varphi^* + \lambda (\varphi\varphi*)^2\,,
\end{equation} 
where $\mu$ is a real parameter with dimension of mass,  $\lambda$ is a dimensionless coupling constant and $\varphi$ is a complex scalar field. The potential is sketched in \refFig{mexican_potential}. This potential has a minimum lying on a circle in the complex plane with radius $\sfrac{\mu}{\sqrt{\lambda}}$, therefore the minimal energy is infinitely degenerate. The ratio   $\sfrac{\mu}{\sqrt{\lambda}}$ is commonly denoted as $v$ and referred to as vacuum expectation value. This means the  lagrangian is no longer symmetric at its minimum. The potential effectively describes two real scalar fields with masses $\mu\sqrt{2}$ and $0$.  The appearance of a massles bosonic excitation (Goldstone boson) is referred to as the \emph{Goldstone theorem}~\cite{SM-Goldstone-2}.

\begin{wrapfigure}[20]{r}{0.45\textwidth} \vspace{10pt}
    \centering
    \includegraphics[width=0.5\textwidth]{./StandardModel/higgs_potential.eps}
    \captionof{figure}[Visualization of the Goldstone potential.]{Visualization of the Goldstone potential given by \refEq{mexican_potential}. The full potential is realized by the surface created by rotating the red curve around the y-axis.} \label{fig:mexican_potential}
\end{wrapfigure} 


The Goldstone model was further extended by P.\,Higgs~\cite{SM-Higgs} and others~\cite{SM-Englert,SM-Guralnik}, who described the interaction with an Abelian gauge field in the frame of the Goldstone model. When gauged, the Goldstone boson disappears and the gauge field acquires a mass. This is the famous \emph{Higgs mechanism}. It was shown later by S.\,Weinberg that the Higgs mechanism is actually necessary for tree-level unitarity of the electroweak theory (\ie renormalizability)~\cite{SM-Weinberg}. The application of the  Higgs mechanism on the Glashow model was further expanded by A.\,Salam~\cite{SM-Salam-2} and today we refer to it as the  Glashow-Weinberg-Salam model. They used the Higgs mechanism to generate also lepton and fermion masses, while keeping the electromagnetic interaction parity symmetric and the weak interaction parity violating.

\clearpage

\subsection{Quark model}\label{sec:SM_quark}

The picture of the Standard Model at this point in history is relying on the \SU gauge symmetry and the Higgs mechanism realized via a complex scalar doublet. 

At that time, baryons and mesons were considered to be elementary particles.  That was only until 1961, when M.\,Gell-Mann and independently Y.\,Neeman  proposed the \emph{Eightfold way}. They noticed that the back-then known particles match an $SU(3)$ representation~\cite{SM-Gell-Mann-2,SM-Neeman}. Gell-Mann continued to work on this model, and in 1964 he used the word quark for the first time~\cite{SM-Gell-Mann-3}. Independently of him, G.\,Zweig also proposed that "Both mesons and baryons are constructed from a set of three fundamental particles"~\cite{SM-Zweig-1,SM-Zweig-2}. They postulated that quarks have only a partial charge of $\sfrac{1}{3}$ and $\sfrac{2}{3}$ and are fermions. They called the quarks \emph{up}, \emph{down} and \emph{strange}.  

In the same year, S.\,Glashow and J.\,Bjorken proposed the existence of a fourth - \emph{charm} - quark. This was appealing at that time as the existence of \tauon lepton was yet to be discovered and the existence of two generations of quarks was symmetric to two generations of leptons~\cite{SM-Bjorken}.

The charm quark was later recognized by S.\,Glashow, J.\,Iliopoulos and L.\,Maiani (\emph{GIM}) to be a crucial part of the Standard Model. The problem with the existence of three quarks was the interaction of quarks with the $Z$ boson: the occurrence of strangeness-changing neutral currents was phenomenologically much smaller than expected. They added the fourth quark to the electroweak theory, allowing only for flavor-conserving neutral currents~\cite{SM-GIM}. This gave the basics to the theory of flavor-changing neutral currents, where the divergences are cut-off by a heavy quark exchange in a loop. An example of such diagrams is shown in \refFig{KaonToMuMu}. 

One of the remaining problems of the theory was CP violation. The CP violation was unexpectedly observed in 1964 in the decay of \Kz mesons~\cite{SM-Cronin}. Even though the community at that time was vary of accepting the quark model (the charm quark was still yet to be discovered), M.\,Kobayashi and T.\,Maskawa proposed the existence of two more quarks~\cite{SM-CKM}. The model with two generations of quarks is CP conserving, while the proposed three generation model is not~\cite{SM-CKM}.  The matrix describing the strength of flavor-changing weak interaction is called \emph{CKM} after N.\,Cabibbo\footnote{N.\,Cabibbo postulated a similar matrix with two generation of quarks~\cite{SM-Cabibbo}. The motivation for such matrix was the fact that the $u\leftrightarrow d$, $\electron\leftrightarrow\neue$ and $\muon\leftrightarrow\neum$ transitions had similar measured amplitudes. On top of that, the transitions with strangeness change one ($\Delta s = 1$) have four times larger amplitude than processes with strangeness conserved. This was solved by Cabibbo by postulating weak universality and weak mixing angle $\theta_c$.}, M.\,Kobayashi and T.\,Maskawa.


\begin{figure}[hbt!]  \centering  
	\includegraphics[width=0.33\textwidth]{./Feynman/Kaon_box_cropped.pdf}\hspace{0.25cm}
	\includegraphics[width=0.30\textwidth]{./Feynman/Kaon_penguin1.pdf}\hspace{0.25cm}
	\includegraphics[width=0.30\textwidth]{./Feynman/Kaon_penguin2.pdf}
	\captionof{figure}[Feynman diagrams of kaon decay to a $\mu\mu$ pair including \cquark contribution]
	{
		Feynman diagrams of kaon decay to $\mu\mu$ including \cquark-quark contribution. They were described in\,\cite{SM-Gaillard}. Note that there is also a long distnace contribution from $\KL\to\g\g\to\mup\mun$.  \label{fig:KaonToMuMu}    
	}
\end{figure}

\subsection[\texorpdfstring{${b\rightarrow s l^-l^+}$}{b to sll} transitions]{\texorpdfstring{$\boldsymbol{b\rightarrow s l^-l^+}$}{b to sll} transitions}\label{sec:SM_bsll}

The exchange of heavy quarks in loops in flavor-changing neutral currents (FCNC) is a great tool to probe New Physics at high energies. The loops are sensitive to heavy particles and precision measurement of such processes could lead us to New Physics discovery, similarly as the kaon decay to muons led to the discovery of the charm quark. Higher-order transitions, such as the $\decay{\bquark}{\squark l^-l+}$ transition, are sensitive to New Physics, as they are even more suppressed by the GIM mechanism. The price to pay is that the interactions are rather rarely occurring. The typical decay rate for such a transition is $10^{-6}$. These processes are then referred to as \emph{rare decays}.

Experimentally reachable example of such higher-order FCNC interaction are \bsll transitions. They occur through \emph{box} and \emph{penguin} diagrams, as shown in \refFig{penguin_bsll}. 

\begin{figure}[htb!]  \centering  
    \includegraphics[width=0.42\textwidth]{./Feynman/bsll_box.pdf}\hspace{1cm}
    \includegraphics[width=0.42\textwidth]{./Feynman/bsll_penguin.pdf}
    \captionof{figure}[Feynman diagrams of a \bsll transition.]
    {
        Feynman diagrams of a \bsll transition. The diagram on the left is referred to as \emph{box} diagram, the right diagram is called \emph{penguin} diagram. \label{fig:penguin_bsll}
    }
\end{figure}

The processed are mediated by \g, \Wpm and \Z bosons. One of the experimentally observable variables is the invariant mass squared of the lepton pair, \qsq, as shown in \refFig{q2_theory}. The \bsll transition is dominated by several very different processes depending on the \qsq value. There are two problematic regions of \qsq: around 9\gevgev and  14\gevgev. In these regions, the process is dominated by a tree-level diagram of \bquark\to\squark\jpsi and  \bquark\to\squark\psitwos, where \jpsi or \psitwos decays into two leptons. As this region is dominated by a process with different physics both theory and experiment typically omits these regions in their predictions or measurements.

\begin{figure}[htb!]  \centering  
    \includegraphics[width=0.46\textwidth]{./StandardModel/q2regions.pdf}
    \captionof{figure}[Decay rate of \bsll transition in depence on \qsq.]{
       Decay rate of \bsll transition in depence on \qsq. In the low \qsq region, the decay rate is dominated by the penguin diagram with photon exchange. With increasing \qsq, contribution of other processes increases, until the decay rate is dominated by the \jpsi and \psitwos charm resonances. At very high \qsq, the decay rate is dominated by long distance contributions. For the details about the $C_i\left(\mu\right)$ variables see \refEq{EffHam}. \label{fig:q2_theory}
    }
\end{figure}


\begin{figure}[htb!]  \centering  
	\begin{minipage}{0.4\textwidth}
		\centering
		\includegraphics[width=1.0\textwidth]{./Feynman/bsll_box.pdf}\\
		\includegraphics[width=0.1\textwidth]{./Others/plus_operator.png}\\
		\includegraphics[width=1.0\textwidth]{./Feynman/bsll_penguin.pdf}    
	\end{minipage}
	\begin{minipage}{0.1\textwidth}
		\centering
		\includegraphics[width=0.7\textwidth]{./Others/Rightarrow.png}    
	\end{minipage}
	\begin{minipage}{0.4\textwidth}
		\centering
		\includegraphics[width=1.0\textwidth]{./Feynman/bsll_eff.pdf}	    
	\end{minipage}
	
	\captionof{figure}[Feynman diagrams of an effective \bsll transition.] {
		Feynman diagrams of an effective \bsll transition. Instead of looking at the interaction as a set of diagrams, we can describe the \bsll transition as a point-like four-fermion interaction. \label{fig:bsll_eff}
	}
\end{figure}



Similarly as Fermi described the beta decay as one interaction of four fermions, one can apply this simplification also on these processes. The exchanged energy (smaller than the mass of the \Bu meson) is much lower than the energy scale of the quantum loop (mass of the $W$ boson). Therefore, instead of looking at the interaction from the Standard Model point of view illustrated in \refFig{penguin_bsll}, one can look at it as a point interaction, as shown in \refFig{bsll_eff}. 


This description is commonly referred to as \emph{effective theory}. The effective Hamiltonian of \bsll transition can be expressed as:
\begin{equation}\label{eq:EffHam}
    \mathcal{H}_{eff} = -\frac{4G_F}{\sqrt{2}}V_{tb}V_{ts}^{*}\frac{\alpha_e}{4\pi}\sum_{i}C_i\left(\mu\right)\mathcal{O}_i\left(\mu\right)\,,
\end{equation}
where $G_F$ is the weak decay constant, $V_{tb}V_{ts}^{*}$ are the CKM matrix elements describing the $\bquark\to\tquark$ and $\tquark\to\squark$ transitions (the contributions of the \uquark and \cquark quarks to the loop is negligible), $\alpha_e$ is fine-structure constant, and $\sfrac{1}{4\pi}$ comes from the loop suppression. Wilson coefficients $C_i\left(\mu\right)$ contain all information about short-distance physics in the transition above the renormalization scale $\mu$. 

 % \todo[inline]{Add something about $\mu=M_W$ (matching scale),+ calculation at low-energy scale of $\mu\sim m_b$?} %we need calculation at low-energy scale
The operators $\mathcal{O}_i\left(\mu\right)$ are local four-fermion operators with different Lorentz structures. These currents are all left-handed. Formally, the right-handed $\mathcal{O}'_i$ currents contribute to the Hamiltonian too, however they are very suppressed in the Standard Model due to the parity violating nature of the weak interaction described earlier.

Looking at FCNC transitions,
% $q\to q l^+l^-$ transition, different processes are sensitive to different operators: \Ope1 and \Ope2 are current-current (tree-level) operators, \Ope3  ~- \Ope6 (QCD penguin operators) describe the interactions with gluons and are sensitive to $b \to s qq$ transitions. 
 the operator \Ope7 describes the photon contribution to the decay rate and is constrained by radiative decays of  $q\to q l^+l^-$ transitions at small \qsq. The operators \Ope9 and \Ope10 are V and A currents, respectively. The operator \Ope8 describes gluon contribution to the diagrams. Assuming the SM scale $\mu=M_W$, \Ope8 vanishes in the Standard Model~\cite{SM-Buchalla}.  The operators are given in \refEq{operators}.  

%For \bsll transitions, the \emph{electroweak penguin operators} \Ope7, \Ope8, \Ope9 and \Ope10 are then of importance. The other operators either do not (significantly) contribute to the decay or are strongly constrained by other processes. The operator \Ope7 describes the photon contribution to the decay rate and is constrained by radiative decays of  $q\to q l^+l^-$ transitions at small \qsq. The operators \Ope9 and \Ope10 are V and A currents respectively. The operator \Ope8 describes gluon contribution to the diagrams. Assuming the SM scale $\mu=M_W$, \Ope8 vanishes in the Standard Model~\cite{SM-Buchalla}.  The operators are given in \refEq{operators}.  

In the \refEq{operators},
$e$ is the elementary charge, 
$g$ is the strong coupling constant, and
$m_b$ is the running \bquark quark mass. 
The matrices are denoted as follows: 
$\sigma_{\mu \nu}$ are Pauli matrices,
$\gamma_{\nu,5}$ are Dirac matrices and 
$\lambda^a$ are Gell-Mann matrices.
The quark fields are denoted $\squarkbar$, $\bquark$, 
the muon fields $\mu$, $\bar{\mu}$, 
while $ G^{\mu \nu \, a}$ is the gluon field tensor. 
The electromagnetic tensor is denoted $ F^{\mu \nu}$.

%
%\begin{array}{rl}
%	O^u_1 = & (\bar{s}_L \gamma_{\mu} T^a u_L) (\bar{u}_L \gamma^{\mu} T^a b_L), 
%	\vspace{0.2cm} \\
%	O^u_2 = & (\bar{s}_L \gamma_{\mu}     u_L) (\bar{u}_L \gamma^{\mu}     b_L),
%	\vspace{0.2cm} \\
%	O^c_1 = & (\bar{s}_L \gamma_{\mu} T^a c_L) (\bar{c}_L \gamma^{\mu} T^a b_L),
%	\vspace{0.2cm} \\
%	O^c_2 = & (\bar{s}_L \gamma_{\mu}     c_L) (\bar{c}_L \gamma^{\mu}     b_L),
%	\vspace{0.2cm} \\
%	O_3 = & (\bar{s}_L \gamma_{\mu}     b_L) \sum_q (\bar{q}\gamma^{\mu}     q),     
%	\vspace{0.2cm} \\
%	O_4 = & (\bar{s}_L \gamma_{\mu} T^a b_L) \sum_q (\bar{q}\gamma^{\mu} T^a q),    
%	\vspace{0.2cm} \\
%	O_5 = & (\bar{s}_L \gamma_{\mu_1}
%	\gamma_{\mu_2}
%	\gamma_{\mu_3}    b_L)\sum_q (\bar{q} \gamma^{\mu_1} 
%	\gamma^{\mu_2}
%	\gamma^{\mu_3}     q),     
%	\vspace{0.2cm} \\
%	O_6 = & (\bar{s}_L \gamma_{\mu_1}
%	\gamma_{\mu_2}
%	\gamma_{\mu_3} T^a b_L)\sum_q (\bar{q} \gamma^{\mu_1} 
%	\gamma^{\mu_2}
%	\gamma^{\mu_3} T^a q),
%	\vspace{0.2cm} \\
%	O_7  = &  \f{e}{g^2} m_b (\bar{s}_L \sigma^{\mu \nu}     b_R) F_{\mu \nu},
%	\vspace{0.2cm} \\
%	O_8  = &  \f{1}{g} m_b (\bar{s}_L \sigma^{\mu \nu} T^a b_R) G_{\mu \nu}^a,
%	\vspace{0.2cm} \\
%	O_9  = &  \f{e^2}{g^2} (\bar{s}_L \gamma_{\mu} b_L) \sum_l 
%	(\bar{l}\gamma^{\mu} l),
%	\vspace{0.2cm} \\
%	O_{10} = & \f{e^2}{g^2} (\bar{s}_L \gamma_{\mu} b_L) \sum_l 
%	(\bar{l} \gamma^{\mu} \gamma_5 l),  
%\end{array}
%assuming mu = M_W, C_8 and C_10 vanish
\begin{align}\label{eq:operators} \begin{split}
{\mathcal{O}}_{7} &= \frac{e}{g^2} m_b
(\bar{s} \sigma_{\mu \nu} \frac{1+\gamma_5}{2} b) F^{\mu \nu}\,, \\
{\mathcal{O}}_{8} &= \frac{1}{g} m_b
(\bar{s} \sigma_{\mu \nu} \frac{\lambda^a}{2} \frac{1+\gamma_5}{2} b) G^{\mu \nu \, a}\,, \\
{\mathcal{O}}_{9} &= \frac{e^2}{g^2} 
(\bar{s} \gamma_{\mu} \frac{1-\gamma_5}{2} b)(\bar{\mu} \gamma^\mu \mu)\,,\\
{\mathcal{O}}_{10} &=\frac{e^2}{g^2}
(\bar{s}  \gamma_{\mu} \frac{1-\gamma_5}{2} b)(  \bar{\mu} \gamma^\mu \gamma_5 \mu)\,.\\
\end{split}\end{align}



These operators in combination with corresponding Wilson coefficients dominate in different \qsq regions, as illustrated in \refFig{q2_theory}. As the effective theory allows for any kind of interaction, it can also describe New Physics contributions. If the measured Wilson coefficients values are different to SM expectations, it means the contribution of different SM processes is accompanied by New Physics process.
%The top right diagram, a so called penguin diagram, is related to both C 9 and C 10 .
%The Wilson coefficient C 9 is related to vector like coupling, whereas C 10 is related
%to axial vector like coupling. Therefore the decay described by a virtual photon is
%only possible in the case of C 9 . The bottom diagram, a so called box diagram, is
%associated to both C 9 and C 10 .

Unfortunately, this theory describes free quarks. In experiments, the quarks are bound by the strong force as depicted in \refFig{bsll_meson}, described by non-perturbative quantum chromodynamics (QCD)\footnote{QCD is the theory of the strong interaction between quarks and gluons.}. Despite the fact these effects are very hard to compute, there are several tools that provide these calculations. The most widely used tools are Lattice QCD~\cite{SM-Wilson} and Light-Cone-Sum-Rules (LCSR)~\cite{SM-LCSM}. Moreover, the calculations based on QCD factorisation~\cite{SM-QCDF} are typically performed for low \qsq, for high \qsq (\qsq $\gtrsim$~15\gevgev) the Operator Product Expansion~\cite{SM-OPE} is used.

\begin{figure}[htb!]  \centering  
    \includegraphics[width=0.49\textwidth]{./Feynman/bsll_eff_meson.pdf}
    \includegraphics[width=0.49\textwidth]{./Feynman/bsll_eff_meson_charm.pdf}     \vspace{-50pt}
    \captionof{figure}[ Feynman diagrams of an effective \bsll transition in a meson.] 
    { 
        Feynman diagrams of an effective \BuToKstmm transition. The left diagram shows the \bsll process in the context of the interacting quark being bound in a \Bu meson decaying into \Kstarp\mup\mun. There is also a non-factorizable contribution from charm loops, as shown on the right. Even though its contribution is much smaller, it needs to be correctly treated too.  \label{fig:bsll_meson}
    }
\end{figure}

\clearpage

It is very hard to disentangle QCD processes from the \bsll transition, however by choosing a convenient basis and variables, the form-factor influence can be at least removed at the first order (this will be described later in \refSec{ANA_Theo_BR}). This is a limiting factor in many theory predictions for this process, even though the calculations are constantly improved. %Lattice QCD, Light-Cone-Sum-Rules, operator product expansions, heavy quark expansion, QCD factorisation, Soft Collinear Effective theory and Chiral perturbation theory.

Besides the challenging form-factor contributions, there is another non-factorizable contribution: charm loops~\cite{SM-CharmLoop}. The process is depicted in \refFig{bsll_meson} on the right. A charm loop is coupled to the \bquark  and \squark quarks and to a virtual photon decaying to the muon pair. Their contribution is much smaller than the one of form-factors. However, with increasing precision of both measurements and QCD calculations, their effect becomes significant. Additional gluons can come into play, making the theoretical calculations even harder. This process is included in the \C9 Wilson coefficient and it is therefore important to separate this effect from possible contributions of physics beyond the Standard Model. 

\subsubsection{New Physics}\label{sec:SM_NP}

As the effective theory allows for any kind of interaction, it can also describe New Physics (NP) contributions at large energy scale: they can be integrated out similarly to the electroweak bosons. Other operators that are negligible in the SM can contribute to the effective hamiltonian. An example of such operators $\mathcal{O}_S$, $\mathcal{O}_P$ and $\mathcal{O}_T$ that can bee added \eg from Higgs penguins\footnote{Higgs penguin is a Higgs to fermion-antifermion transition.} is listed in \refEq{operators_BSM}.
%
\begin{align}\label{eq:operators_BSM}\begin{split}
\mathcal{O}_S &= \left( \bar{s}  \frac{1+\gamma_5}{2} b \right) \left( \bar{\mu} \mu\right)\,,\\
\mathcal{O}_P &= \left( \bar{s}  \frac{1+\gamma_5}{2} b \right) \left( \bar{\mu} \gamma_5 \mu\right)\,,\\
\mathcal{O}_T &=  \bar{s} \sigma_{\mu\nu} b \bar{\mu} \sigma^{\mu\nu} \mu
\end{split}
\end{align}
%An example of such a process can be a box diagram as on the left of \refFig{penguin_bsll}) with the exchange of two charged Higgs bosons instead of two $\W$s.

Moreover, the right-handed Wilson coefficients $C^{'}_i$ come into play. Different decays are sensitive to different coefficients, as presented in  \refTab{SM_Wilson_sensitivity}. From the table it becomes clear that the decay of \BuToKstmm is sensitive to most of the coefficients. 

\begin{table}[hbt!] \centering
    \begin{tabular}{l|cccc}
        Decay & \C7 & \C9 & \C10 & $C_{S,P}$  \\ \hline
        $\B\to\left(X_s,\Kstar\right)\g$ & \checkmark & \texttimes & \texttimes & \texttimes \\
        $\B\to\left(X_s,\Kstar\right)\ellm\ellp$ & \checkmark & \checkmark & \checkmark & \texttimes \\
        $\Bs\to\mumu$ & \texttimes & \texttimes & \checkmark & \checkmark \\
    \end{tabular}
    \captionof{table}[Sensitivity of Wilson coefficient for different decays.]{Sensitivity of Wilson coefficient for different decays, where $X_s$ stands for any inclusive decay with an \squark quark.
        \label{tab:SM_Wilson_sensitivity}
    }
\end{table}    


\clearpage

In most of the measurements with the potential to constrain New Physics, there is a good agreement with the SM. However, in several previous measurements, tensions of a few standard deviations appear. These measurements are discussed later in \refSec{ANA_previous}.  All these tensions are in the order of 2-3 standard deviations away from the Standard Model prediction. However, they are all consistent with each other, hinting at possible New Physics contribution to the Wilson coefficients \C9 and \C10. 


%branching ratio of \Bs\to\mumu (Nature 522 (2015) 68), branching ratios of $\B\to X_s\g$ (1301.0836), CP assymetry of \Bd\to\Kstarz\g (https://cds.cern.ch/record/
%1424352) and isospin assymetry of \B\to\Kstarz\g (0906.2177)

%C10 vs C9 State of New Physics in b − s transitions,(2015) arXiv:1411.3161, [1903.09578]

There are numerous NP approaches to explain these tensions, including the supersymmetric theory or the string theory. Among the currently most discussed theories are \emph{portal}, \emph{loop}, and \emph{leptoquark} models. \emph{Portal models} assume a particle responsible for the tensions that can be also involved in the dark matter production in the early universe~\cite{NP-portals,NP-portals2,NP-portals3}. This is typically a \Zprime boson. In these models, the \Zprime boson contributes to the operator \Ope{9} (and sometimes to the operator \Ope{10} ) with flavor violating couplings to quarks and non-universal couplings to leptons. Portal models provide corrections to \C9, however they also mean unwanted contributions to other Wilson coefficients. An example of a Feynman graph with a \Zprime boson is in \refFig{feynman-NP}, left.
%
\emph{Loop} models postulate that the NP contribution comes from loops containing particles. These particles are in some cases potential dark matter candidates~\cite{NP-loops,NP-loops2, NP-loops3}. An example of such a hypothetical loop is in \refFig{feynman-NP}, middle.
%
\emph{Leptoquark} models assume the existence of two (or more) leptoquarks: a particle carrying both lepton and baryon number, allowing leptons and quarks to interact directly, as shown in \refFig{feynman-NP}, right. This model can answer the question why are neutrinos massive, but it can also explain some of the tensions in \bsll decays~\cite{NP-Leptoquarks0, NP-Leptoquarks, NP-Leptoquarks2, NP-Leptoquarks3, NP-Leptoquarks4}. There are also other models combing these approaches~\cite{SM-leptoquarkportal-1,SM-leptoquarkportal-2,SM-leptoquarkportal-3}. 

\begin{figure}[htb!]  \centering  
    \includegraphics[width=0.34\textwidth]{./Feynman/bsll_Zprime.pdf} \hspace{-15pt}
    \includegraphics[width=0.34\textwidth]{./Feynman/bsll_DM.pdf}\hspace{-15pt}
    \includegraphics[width=0.34\textwidth]{./Feynman/bsll_Leptoquark.pdf}
    \captionof{figure}[Potential New Physics Feynman diagrams.]
    {
        Potential New Physics Feynman diagrams. On the left, Feynman diagram with a potential \Zprime gauge boson, in the middle possible loop diagram with a contribution from a dark model particles, on the right diagram of an \bsll interaction through leptoquarks. \label{fig:feynman-NP}
    }
\end{figure}



\clearpage   