\chapter{Phenomenology}
\label{cha:Pheno}
The lattice simulations discussed in the previous chapters aim to address tensions between experimental results and theoretical predictions. Beyond this, lattice QCD offers powerful tools to investigate fundamental aspects of the standard model such as the unitarity of the CKM matrix and the anomalous magnetic moment of the muon. Additionally, scale setting - that is, the determination of the lattice spacing - remains a crucial and ongoing challenge in lattice QCD, often leading to discrepancies and debates within the lattice community.
\section{The anomalous magnetic moment of the muon}
\label{sec:gmk}
The anomalous magnetic moment of the muon $a_\mu = \frac{g_\mu -2}{2}$ is currently one of the most important test parameters for the standard model of particle physics. In experiments, it has been measured up to a precision of $0.2\,\mathrm{ppm}$~\cite{Aguillard_2023}, with the current experimental average being
\begin{align}
\label{eq:amuexp}
    a_\mu = 116,592,055(25)\times 10^{-11}.
\end{align}
From the theory point of view its value is dominated by electromagnetic contributions that can be computed using perturbation theory. In addition, there are small contributions from electroweak and hadronic light-by-light effects~\cite{Aoyama_2020,Fodor:2024jyn,Zimmermann:2024uog,Zimmermann:2023tal,Gerardin:2023naa,Verplanke:2022eto,Verplanke:2021gat}.
\begin{figure}[ht]
    \centering
    \includegraphics[width=\linewidth]{Figures/6/2025.png}
    \caption[Comparison of current $g-2$ predictions (experiment and theory)]{Comparison of experimental measurements and theoretical predictions (with or without data-driven inputs). The green data point represents the mean value of experimental measurements of $a_\mu$~\cite{Bennett_2004,Bennett:2006fi,Aguillard_2023,Muong-2:2025xyk} while the other data are theoretical predictions that estimate the HVP contributions in different ways: The purple data points show pure lattice calculations~\cite{Aliberti:2025beg,RBC:2024fic,Djukanovic:2024cmq, Borsanyi_2021} while the blue ones use different data-driven approaches~\cite{Davier:2023fpl,CMD-3:2023alj,BaBar:2009wpw,BaBar:2012bdw,KLOE:2008fmq,KLOE:2010qei,KLOE:2012anl,KLOE-2:2017fda,Davier:2010fmf,Davier:2013sfa}. The best theoretical prediction (red), that is available at the moment uses data-driven approaches for the small tail contributions of the HVP. Figure is taken from~\cite{Boccaletti:2024guq}.}
    \label{fig:gmk_compare}
\end{figure}
Also, hadronic vacuum polarization (HVP) effects contribute to $a_\mu$ by a small amount, which is called $a_\mu^\mathrm{LO-HVP}$. Although the absolute contribution is very small compared to the QED contribution, the error is dominated by the HVP part~\cite{Aoyama_2020}. Currently, there are two methods that are used to obtain a value for $a_\mu^\mathrm{LO-HVP}$: A lattice computation from first principles, and the data-driven approach. With lattice QCD, $a_\mu^\mathrm{LO-HVP}$ can be computed from the vector correlation function via
\begingroup
\allowdisplaybreaks
\begin{align}
    a_\mu^\mathrm{LO-HVP} &= \alpha^2 \int_0^\infty \mathrm{d}t\, K(t)\, C_{1\gamma I}(t),\\
    K(t) &= \int_0^\infty \frac{\mathrm{d}Q^2}{m_\mu^2}\omega\left( \frac{Q^2}{m_\mu^2}\right)\left( t^2- \frac{4}{Q^2}\sin^2\left( \frac{tQ}{2}\right)\right),\\
    \omega(x) &= \frac{\left[x+2-\sqrt{x(x+4)}\right]^2}{\sqrt{x(x+4)}},
\end{align}
\endgroup
where $C_{1\gamma I}(t)$ is the one-photon irreducible part of the vector propagator, $m_\mu$ the mass of the muon and $\alpha$ the electromagnetic fine-structure constant. Using the R-ratio the HVP-contributions to $a_\mu$ can be computed using~\cite{Davier:2023cyp}
\begin{align}
    a_\mu^\mathrm{LO-HVP} &= \left(\frac{\alpha m_\mu}{2\pi}\right)^2 \int_0^\infty \frac{\mathrm{d}s}{s^2}\, \hat K(s) R(s),\\
    \hat K(s) &= \frac{3s^{5/2}}{8m_\mu^4} \int_0^\infty\mathrm{d}t\, e^{-t\sqrt{s}}K(t).
\end{align}
Commonly, the time interval $[0,\infty)$ is split into several windows. This is done because different length scales on the lattice come along with different statistical and systematic subtleties that can be considered separately with this approach. Whenever an integration with respect to the Euclidean time is performed on the lattice, we can pick out a time interval $(t_i,t_f)$ with the (smooth) window function
\begin{align}
    W(t;t_i,t_f,\Delta) &= \begin{cases}
        1,\quad &\textrm{if $t_i = 0$ and $t_f = \infty$}\\
        1-\theta(t-t_f,\Delta),\quad &\textrm{if $0 = t_i< t_f < \infty$}\\
        \theta(t-t_i,\Delta)-\theta(t-t_f,\Delta),\quad &\textrm{if $0 < t_i< t_f < \infty$}\\
        \theta(t-t_i,\Delta),\quad &\textrm{if $0 < t_i< t_f = \infty$}\\
    \end{cases}\\
    \theta(t,\Delta) &= \frac{1+\tanh{\left(\frac{t}{\Delta} \right)}}{2},
\end{align}
where usually $\Delta = 0.15\,\mathrm{fm}$ and the three windows $[0.0\,\mathrm{fm},0.4\,\mathrm{fm}]$ (short distance), $[0.4\,\mathrm{fm},1.0\,\mathrm{fm}]$ (intermediate distance) and $[1.0\,\mathrm{fm},\infty\,\mathrm{fm})$ (long distance) are chosen~\cite{Blum_2018,Davier:2023cyp,Aoyama_2020}.\\
The first theoretical computation of $a_\mu^\mathrm{LO-HVP}$ that reaches the precision of the experiments was done by the BMW collaboration in 2020 (see Figure~\ref{fig:gmk_compare}). At the same time another group of theoretical physicists published a value of $a_\mu^\mathrm{LO-HVP}$ that showed a $5\sigma$-disagreement with the experimental results~\cite{Aoyama_2020}. In this paper the HVP effects were estimated using a data-driven approach that uses the R-ratio of electron-positron scattering for the HVP.
Due to the fact that at the current level of precision there are a lot of contradicting predictions for $a_\mu^\mathrm{LO-HVP}$ this is a very interesting topic for physicists that look for Beyond-Standard-Model (BSM) effects. In order to resolve this problem experimentalists and theorist aim to increase the precision of their respective results.\\
After the first experimental values of $g-2$ from Fermilab were published in 2021~\cite{Abi_2021}, the update came out in 2023 with the precision given in Equation~\eqref{eq:amuexp}~\cite{Aguillard_2023}. The final results with the best precision are expected to be published in 2025. Also, a lot of lattice calculations were published during the last years~\cite{Bazavov:2024eou,FermilabLattice:2024yho,FermilabLatticeHPQCD:2023jof,FermilabLattice:2022izv,Colangelo:2022jxc,RBC:2024fic,Spiegel:2024dec,Djukanovic:2024cmq,Kuberski:2024bcj}.\\
An important aspect of a high-precision computation of $a_\mu^\mathrm{LO-HVP}$ is the setting of the lattice spacing. This is often done using the gradient-flow based observable $w_0$ that has to be determined beforehand up to a very high precision. In the previous chapters the computations of different dimensionless quantities with the $w_0$ were discussed. From this we can extract this quantity in different ways:\\
Using the mass of the $\Omega$-baryon we get a total results of
\begin{align}
    w_0 = 0.17245(22)(46)[51]\,\mathrm{fm}
\end{align}
where the first error is statistical and the second is systematic. Here, strong-isospin breaking contributions of $\sim 0.0001\,$fm are added to the lattice results~\cite{Borsanyi_2021,Boccaletti:2024guq} of $M_\Omega w_0$ after dividing by the physical mass of the $\Omega^-$ baryon~\cite{ParticleDataGroup:2024cfk}.\\
To extract $w_0$ from the $f_\pi w_0$-combination we have to bring the pion decay rate
\begin{align}
     \Gamma(\pi^+ \to \ell \bar \nu_\ell) = \frac{G_F^2}{8\pi}m_\ell^2M_\pi\left(1-\frac{m_\ell^2}{M_\pi^2}\right)^2\times \vert V_{u\bar d}\vert^2 \times f_\pi^2\times (1+\delta_\pi)
\end{align}
to mind. The experimentally estimated value of this quantity is~\cite{ParticleDataGroup:2022pth}
\begin{align}
     \Gamma^\text{exp}(\pi^+ \to \ell \bar \nu_\ell) = 3.8408(7)\times 10^{-7}\,\mathrm{s}^{-1}. 
\end{align}
The pure QCD decay constant is defined by dividing out the kinematical factors as well as the first CKM element $\vert V_{ud}\vert = 0.97373(31)$~\cite{Hardy:2020qwl} and radiative corrections~\cite{DiCarlo:2019thl}:
\begin{align}
    f_\pi^\mathrm{GRS} = 130.65(12)\,\mathrm{MeV}
\end{align}
Thus, we can compute $w_0$ by transforming $f_\pi w_0$ into the GRS-scheme (with the coefficients from Table~\ref{tab:fpiw0}), adding the sea quark effects and dividing out $f_\pi^\mathrm{GRS}$
\begin{align}
    \alpha_\textrm{blind}^{w_0}\,w_0 = \frac{1}{f_\pi^\mathrm{GRS}}\left(A + B(X_l^\mathrm{GRS} - X_l^\mathrm{FLAG})+ C(X_s^\mathrm{GRS} - X_s^\mathrm{FLAG}) + G(B,C)\right),
\end{align}
which results in a total value of 
\begin{align}
  w_0  = 0.17270(21)_{\mathrm{stat.}}(34)_{\mathrm{QCD}}(19)_{\mathrm{valence \;\&\; exp}}(9)_{\mathrm{sea}}[45]_{\mathrm{tot.}}\,\mathrm{fm}\end{align}
In Figure~\ref{fig:w0_fpi}, it can be seen that the overall distribution of $w_0$ exhibits a distinctly Gaussian shape, in contrast to the histogram of $f_\pi w_0$ shown in Table~\ref{tab:fpiw0}. This difference arises from the use of various experimental inputs and radiative corrections, which introduce additional uncertainties. These uncertainties effectively smooth out the statistical and systematic fluctuations, leading to a more Gaussian-like distribution. The estimation beats the precision of the prediction using the $\Omega$ baryon mass and all studies of other lattice groups, including the FLAG average, as can be seen in Figure~\ref{fig:resultw0}. The new $w_0$ value based on the pion decay constant is perfectly compatible with the estimation using the $\Omega^-$ baryon mass and the FLAG average, respectively~\cite{Boccaletti:2024guq,FermilabLattice:2024yho}.\\
To compute the QED contribution to $w_0$ in the FLAG scheme we consider the difference
\begin{align}
w_0^\mathrm{QED, FLAG} =  \,w_0 -  \frac{\left[f_\pi w_0\right]^\mathrm{FLAG}}{f_\pi^\mathrm{FLAG}} =  -0.00026(9)_{\mathrm{stat.}}(1)_{\mathrm{QCD}}(19)_{\mathrm{valence \;\&\; exp}}(9)_{\mathrm{sea}}[22]_{\mathrm{tot.}}\,\mathrm{fm} 
\end{align}
which is nearly compatible with zero at our current precision.
\begin{figure}
    \centering
    \includegraphics[width=0.9\linewidth]{Figures/8/w0_hist_histogram.pdf}
    \caption[$w_0$ histogram based on the pion decay constant]{Total histogram of our $w_0$ estimation based on the pion decay constant.}
    \label{fig:w0_fpi}
\end{figure}
\begin{figure}
    \centering
    \includegraphics[width=0.9\linewidth]{Figures/8/result_comparison_w0.pdf}
\caption[Result comparison of $w_0$]{Comparison of several estimations of $w_0$~\cite{Miller:2020xhy,MILC:2015tqx,Dowdall:2013rya,ExtendedTwistedMass:2020tvp,ExtendedTwistedMass:2021qui,Borsanyi_2021,Boccaletti:2024guq} and the average from the FLAG report 2024~\cite{FlavourLatticeAveragingGroupFLAG:2024oxs}. The data are still blinded, so the result from our study is shifted such that it aligns with the FLAG average. However, the relative errors can be compared due to our multiplicative blinding procedure.}
    \label{fig:resultw0}
\end{figure}
\section{Lattice computation of \texorpdfstring{$\mathcal{F}_K/\mathcal{F}_\pi$}{fkfpi} as a test for the CKM-unitarity}
\label{sec:fkapi}
In the following section we will discuss the unitarity of the Cabibbo-Kobayashi-Maskawa matrix and its relation to the meson decay constants. The topic is of special interest because state-of-the-art calculations show a $2-3\,\sigma$ discrepancy to experimental results. High precision lattice QCD computations~\cite{Bazavov:2017lyh,Dowdall:2013rya,ExtendedTwistedMass:2021qui,Miller:2020xhy,RBC:2014ntl,Follana:2007uv,MILC:2009ltw,Aubin:2008ie,Boyle:2022lsi,Stokes:2022,Ramos:2009hza,Budapest-Marseille-Wuppertal:2010vgx,Ramos:2011wp} can help to further investigate this issue by providing precise values for the mesonic decay constant that --- in combination with experimental measurements~\cite{ParticleDataGroup:2022pth,ParticleDataGroup:2024cfk} --- put the standard model of particle physics to the test.
\subsection{The CKM unitarity and its relation to the meson decay constants}
\label{sec:CKM}
\begin{figure}
 \centering
\begin{tikzpicture}
 \begin{feynman}
  % Pion (pi^+)
  \vertex (a) {\(\pi^+\)};
  \vertex [right=of a] (b);
  
  % W-Boson
  \vertex [right=of b] (c);
  
  % Zerfallsprodukte
  \vertex [above right=of c] (nu) {\(\nu_\ell\)};
  \vertex [below right=of c] (e) {\(\ell^+\)};
  
  % Quarks im Pion mit gekrümmten Linien und manuellen Positionierungen der Labels
  \diagram* {
   (a) -- [fermion, out=45, in=135,edge label=\(u\)] (b) ,
   (a) -- [fermion, out=-45, in=-135,edge label'=\(\bar d\)] (b) ,
   (b) -- [boson, edge label=\(W^+\)] (c),
   (c) -- [fermion] (nu),
   (c) -- [fermion] (e),
  };
 \end{feynman}
\end{tikzpicture}
\begin{tikzpicture}
 \begin{feynman}
  % Pion (pi^+)
  \vertex (a) {\(K^+\)};
  \vertex [right=of a] (b);
  
  % W-Boson
  \vertex [right=of b] (c);
  
  % Zerfallsprodukte
  \vertex [above right=of c] (nu) {\(\nu_\ell\)};
  \vertex [below right=of c] (e) {\(\ell^+\)};
  
  % Quarks im Pion mit gekrümmten Linien und manuellen Positionierungen der Labels
  \diagram* {
   (a) -- [fermion, out=45, in=135,edge label=\(u\)] (b) ,
   (a) -- [fermion, out=-45, in=-135,edge label'=\(\bar s\)] (b) ,
   (b) -- [boson, edge label=\(W^+\)] (c),
   (c) -- [fermion] (nu),
   (c) -- [fermion] (e),
  };
 \end{feynman}
\end{tikzpicture}
\caption[Diagrams of the leptonic pion and kaon decay at tree-level]{The leptonic decay of the positively charged pion $\pi^+$ (left) and kaon (right). To get the diagrams for the negatively charged meson all of the particles have to be replaced by their antiparticles. Nearly $100\%$ of all pions decay into muons, because the tauon is heavier than the pion~\cite{ParticleDataGroup:2024cfk} and the $e/\nu_e$ final state is suppressed because of helicity conservation~\cite{Devanathan2002}.}
 \label{fig:piK_decay}
\end{figure}
With a mass of $M_{\pi^\pm} =139.57039(18)\, \textrm{MeV}$ and $M_{\pi^0} =134.9768(5)\, \textrm{MeV}$~\cite{ParticleDataGroup:2024cfk}, pions are the lightest hadrons.\footnote{Note that the $\pi^\pm$ is not allowed to decay into $\pi^0$ due to charge conservation.} Therefore it is straightforward to conclude that they are stable under strong interaction. However, a pion can decay via weak processes into leptons, as shown in \ref{fig:piK_decay}. On the other hand, the kaon is the lightest hadron with non-zero strangeness ($ M_K =493.677(16)\,\textrm{MeV}$~\cite{ParticleDataGroup:2024cfk}). Since strangeness is a conserved quantity under strong interactions~\cite{halzen1984quarks}, the kaon also only has weak decay channels, but it also has non-leptonic decay products (such as pions). Nevertheless, in this study we only care about the leptonic decay channels, at least for the charged kaon.\\
In weak interactions, the transformations of the quark-flavors (which are necessary for the vertices at which the $W^\pm$ boson is created) happen with probabilities related to the elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix~\cite{10.1143/PTP.49.652,PhysRevLett.10.531}:
\begin{align}
 V^\text{CKM} = \begin{pmatrix}
  V_{u\bar{d}}&V_{u\bar{s}}&V_{u\bar{b}}\\
  V_{c\bar{d}}&V_{c\bar{s}}&V_{c\bar{b}}\\
  V_{t\bar{d}}&V_{t\bar{s}}&V_{t\bar{b}}\\
 \end{pmatrix}.
\end{align}
The unitarity of this matrix is a direct consequence of the fact that each quark of a given up-type flavor can only be transformed into exactly one quark of a given down-type flavor and vice versa.\\
The CKM matrix has four free real parameters,\footnote{A generic $3 \times 3$ complex matrix has 18 real parameters. Here, there are nine constraints from the unitarity. Furthermore, five degrees of freedom are phases that can be absorbed into the quark fields, whereas one overall global phase cannot be absorbed.} and is commonly parametrized with the angles $\theta_{12},\theta_{13},\theta_{23}$ and the phase $\delta$ via
\begin{align}
  V^\text{CKM} = \begin{pmatrix}
  \cos\theta_{12} \cos\theta_{13}& \sin\theta_{12} \cos\theta_{13}& \sin\theta_{13}e^{\mathrm{i}\delta}\\
  -\sin\theta_{12}\cos\theta_{23} - \sin\theta_{23}\cos\theta_{12}\sin \theta_{13}e^{\mathrm{i}\delta} &\cos\theta_{12}\cos\theta_{23} - \sin\theta_{23}\sin\theta_{12}\sin \theta_{13}e^{\mathrm{i}\delta}&\sin\theta_{23}\cos\theta_{12}\\
  \sin\theta_{12}\sin\theta_{23} - \cos\theta_{23}\cos\theta_{12}\sin \theta_{13}e^{\mathrm{i}\delta} &-\cos\theta_{12}\sin\theta_{23} - \cos\theta_{23}\sin\theta_{12}\sin \theta_{13}e^{\mathrm{i}\delta}&\cos\theta_{23}\cos\theta_{12}\\
 \end{pmatrix}.
\end{align}
Using the parameters 
\begin{align}
\label{eq:wolfenstein1}
 \lambda &= \sin \theta_{12}\\
 A &= \frac{\sin \theta_{23}}{\sin^2 \theta_{12}}\\
 \eta &= \frac{\sin \theta_{13}}{\sin \theta_{12}\sin\theta_{23}}\sin\delta\\
 \rho &= \frac{\sin \theta_{13}}{\sin \theta_{12}\sin\theta_{23}}\cos\delta
\end{align}
one can introduce the Wolfenstein parametrization of the CKM matrix~\cite{ParticleDataGroup:2012pjm}:
\begin{align}
\label{eq:wolfenstein2}
  V^\text{CKM} = \begin{pmatrix}
1-\frac{\lambda^2}{2}&\lambda & A \lambda^3(\rho - \mathrm{i}\eta)\\
-\lambda& 1-\frac{\lambda^2}{2} & A \lambda^2\\
 A \lambda^3(1-\rho - \mathrm{i}\eta)&-A\lambda^2 &1
  \end{pmatrix} + \mathcal{O}(\lambda^4).
\end{align}
Within this project a simplified view on the unitarity of the CKM matrix is sufficient. However, we only investigate the first row unitarity, i.e.,
\begin{align}
  \label{eq:1row_unitarity}
 \vert V_{u\bar{d}}\vert^2 + \vert V_{u\bar{s}}\vert^2 + \vert V_{u\bar{b}}\vert^2 = 1.
\end{align}
On the other hand, experimental measurements suggest that the value of $\vert V_{u\bar{b}}\vert^2$ is much smaller than the precision of current lattice computations~\cite{ParticleDataGroup:2024cfk}, therefore the latter equation can be reduced to
\begin{align}
  \label{eq:reduced_1row_unitarity}
  \vert V_{u\bar{d}}\vert^2 + \vert V_{u\bar{s}}\vert^2 \approx 1
\end{align}
and $\lambda$ from the Wolfenstein parametrization in Equation \eqref{eq:wolfenstein1} and \eqref{eq:wolfenstein2} is the remaining quantity of interest.\\
Now the question arises how Equation \eqref{eq:reduced_1row_unitarity} is related to the meson decay constants. In order to clarify this issue we consider the decay width of the processes that are shown in Figure \ref{fig:piK_decay}:
\begin{align}
 \label{eq:pidecaywidth}
 \Gamma(\pi^+ \to \ell \bar \nu_\ell) &= \frac{G_F^2}{8\pi}m_\ell^2M_{\mathrlap{\pi}\phantom{K}}\left(1-\frac{m_\ell^2}{M_{\mathrlap{\pi}\phantom{K}}^2}\right)^2\times \vert V_{u\bar d}\vert^2 \times f_{\mathrlap{\pi}\phantom{K}}^2\times (1+\delta_\pi)\\
  \label{eq:Kdecaywidth}
 \Gamma(K^+ \to \ell \bar \nu_\ell) &= \frac{G_F^2}{8\pi}m_\ell^2M_K\left(1-\frac{m_\ell^2}{M_K^2}\right)^2\times \vert V_{u\bar s}\vert^2\times f_K^2\times (1+\delta_K)
\end{align}
Here $G_F = 1.1663787(6)\,\textrm{GeV}^{-2}$~\cite{fermi_constant} is the Fermi-constant that is related to the coupling constant of the weak interaction $g_w$ via
\begin{align}
 G_F = \frac{\sqrt{2}}{8}\frac{g_w^2}{m_W^2}.
\end{align}
$m_X$ is the mass of a fundamental particle $X$ and $M_Y$ describes the mass of a hadron $Y$. $f_\pi$ and $f_K$ are the decay constants of the respective mesons and the $\delta$'s are the radiative corrections that take care of the inclusion of QCD+QED diagrams, some examples are shown in Figure \ref{fig:radiative_correction}. For the combined QCD+QED decay constant, the notation $\mathcal{F}_H := f_H \sqrt{1 + \delta_H}$ is commonly used. 
\begin{figure}
 \centering
\begin{tikzpicture}
 \begin{feynman}
  % Pion (pi^+)
  \vertex (a) {\(\pi^+\)};
  \vertex [right=of a] (b);
  
  % W-Boson
  \vertex [right=of b] (c);
  
  % Zerfallsprodukte
  \vertex [above right=of c] (nu) {\(\nu_\ell\)};
  \vertex [below right=of c] (e) {\(\ell^+\)};
  \vertex at (0.5,-0.4) (bp);
  \vertex at (3.5, -0.5) (ep);
  % Quarks im Pion mit gekrümmten Linien und manuellen Positionierungen der Labels
  \diagram* {
   (a) -- [fermion, out=45, in=135,edge label=\(u\)] (b) ,
   (a) -- [fermion, out=-45, in=-135,edge label'=\(\bar d\)] (b) ,
   (b) -- [boson, edge label=\(W^+\)] (c),
   (c) -- [fermion] (nu),
   (c) -- [fermion] (e),
   (bp) -- [boson,out=-65,in=270, edge label=\(\gamma\)] (ep),
  };
 \end{feynman}
\end{tikzpicture}
\begin{tikzpicture}
 \begin{feynman}
  % Pion (pi^+)
  \vertex (a) {\(\pi^+\)};
  \vertex [right=of a] (b);
  
  % W-Boson
  \vertex [right=of b] (c);
  
  % Zerfallsprodukte
  \vertex [above right=of c] (nu) {\(\nu_\ell\)};
  \vertex [below right=of c] (e) {\(\ell^+\)};
  \vertex at (0.5,0.4) (ap);
  \vertex at (0.5,-0.4) (bp);

  % Quarks im Pion mit gekrümmten Linien und manuellen Positionierungen der Labels
  \diagram* {
   (a) -- [fermion, out=45, in=135,edge label=\(u\)] (b) ,
   (a) -- [fermion, out=-45, in=-135,edge label'=\(\bar d\)] (b) ,
   (b) -- [boson, edge label=\(W^+\)] (c),
   (c) -- [fermion] (nu),
   (c) -- [fermion] (e),
   (ap) -- [boson,out=-90,in=90, edge label=\(\gamma\)] (bp),

  };
 \end{feynman}
\end{tikzpicture}
\caption[Diagrams of the leptonic pion and kaon decay with radiative corrections]{Examples for diagrams that include the radiative corrections from Equation \eqref{eq:pidecaywidth}. The diagram on the left describes an additional photon exchange between one of the quarks and the lepton, while on the right diagram an additional photon connects the two quarks of the pion. More diagrams that contribute to radiative corrections for meson decays are shown in~\cite{DiCarlo:2019thl}.}
 \label{fig:radiative_correction}
\end{figure}
Combing the Equations \eqref{eq:pidecaywidth} and \eqref{eq:Kdecaywidth} we can immediately observe that the matrix elements of the CKM matrix are related via the formula
\begin{align}
 \vert V _{ud}\vert = \frac{\sqrt{M_K}\left( 1- \frac{m_\ell^2}{M_K^2} \right)}{\sqrt{M_\pi}\left( 1- \frac{m_\ell^2}{M_\pi^2} \right)}\frac{f_K}{f_\pi}\sqrt{\frac{1+\delta_K}{1+\delta_\pi}} \vert V_{us}\vert. 
\end{align}
From the Wolfenstein parametrization in Equation \eqref{eq:wolfenstein2} one can get another connection:
\begin{align}
 \vert V _{ud}\vert = 1- \frac{\vert V _{us}\vert^2 }{2} + \mathcal{O}\left(\vert V _{ud}\vert ^4\right)
\end{align}
Using elementary calculus it can be found out that there is a unique positive solution of both of the latter equations. So the question arises where the $2-3\,\sigma$ tension between theory and experiment comes in. The answer is that there are two more experiments that provide constraint on $\vert V_{ud} \vert $ respectively $\vert V_{us} \vert $ separately. \\
One the one hand, results on the super-allowed $\beta$ decays~\cite{Hardy:2020qwl,PhysRevC.94.011304,GRINYER2010236} give further information on $\vert V_{ud} \vert $, whereas on the other hand, semi-leptonic kaon decays provide a value for $\vert V_{us}\vert $.\\
Nuclear $\beta$ decay is the conversion of a neutron into proton or vice-versa (whatever is energetically favorable) under radiation of an electron(positron) and the corresponding (anti)neutrino. The corresponding Feynman diagrams are shown in Figure \ref{fig:beta_decay}.
\begin{figure}
\centering
\begin{tikzpicture}
 \begin{feynman}
  \vertex (n) {\(n\)};
  \vertex [right=of n] (a);
  
  \vertex [above right=of a] (p) {\(p\)};
  \vertex [below right=of a] (w);

  % Zerfallsprodukte
  \vertex [above right=of w] (nu) {\(\bar{\nu}_e\)};
  \vertex [below right=of w] (e) {\(e^-\)};
  
  \diagram* {
   (n) -- [fermion] (a),
   (a) -- [fermion] (p),
   (a) -- [boson, edge label=\(W^-\)] (w),
   (w) -- [fermion] (nu),
   (w) -- [fermion] (e),
  };
 \end{feynman}
\end{tikzpicture}
\hspace{3cm}
\begin{tikzpicture}
 \begin{feynman}
  \vertex (p) {\(p\)};
  \vertex [right=of p] (a);

  \vertex [above right=of a] (n) {\(n\)};
  \vertex [below right=of a] (w);

  % Zerfallsprodukte
  \vertex [above right=of w] (nu) {\(\nu_e\)};
  \vertex [below right=of w] (e) {\(e^+\)};

  \diagram* {
   (p) -- [fermion] (a),
   (a) -- [fermion] (n),
   (a) -- [boson, edge label=\(W^+\)] (w),
   (w) -- [fermion] (nu),
   (w) -- [fermion] (e),
  };
 \end{feynman}
\end{tikzpicture}

\caption[Diagram of the nuclear $\beta$ decay]{\textit{Left-hand side:} Nuclear $\beta$ decay: A down quark is transformed into an up quark under radiation of a $W^-$ boson. Since the down quark was a parton of a neutron, this neutron is transformed into a proton. The total charge of the nucleus is increased by 1, therefore this nuclear process changed the element. The $W^-$ boson decays into an electron and the corresponding antineutrino. \textit{Right-hand side:} Inverse Nuclear $\beta$ decay: The charge of the nucleus is decreased by 1 and a positron and the corresponding neutrino are emitted. }
 \label{fig:beta_decay}
\end{figure}

A $\beta$ decay is called \textit{superallowed} if the wave-function of the nucleon is unchanged apart from the fact that the flavor of one quark is changed. While the half-lives of the whole set of $\beta$ decays span more than 20 orders of magnitude, the small set of 13 superallowed $\beta$ decays has shown to have a constant intrinsic strength~\cite{superallowed_beta_decay}. From those one can extract a value for $\vert V_{ud}\vert$, Hardy and Towner published in 2020 a value~\cite{Hardy:2020qwl} of
\begin{align}
 \vert V_{ud}\vert = 0.97373 \pm 0.00031.
\end{align}
A value for the up-strange quark-mixing element can be received using date from semi-leptonic kaon decays. The neutral kaons $K^0$ and $\bar K^0$, which consist of a down and a strange quark have decay channels, that consist of a lepton-neutrino pair and a pion, as it can be seen in Figure~\ref{fig:semi_kaon_decay}. From this processes it is possible to give an estimate of~\cite{10.1093/ptep/ptaa104,FlavourLatticeAveragingGroupFLAG:2021npn}
\begin{align}
 \vert V_{us} \vert = 0.2232(6).
\end{align}

\begin{figure}
 \centering
\begin{tikzpicture}
 \begin{feynman}
  \vertex (a) {\(K^0\)};
  \vertex [above right=of a] (b);
  \vertex [right=of b] (pi){\(\pi^-\)}; 
  \vertex [below right=of a] (d);
  % W-Boson
  \vertex [below right=of d] (c);
  
  % Zerfallsprodukte
  \vertex [below right=of c] (e) {\(\ell^+\)};
  \vertex [above right=of c] (nu) {\(\nu_\ell\)};

  % Quarks im Pion mit gekrümmten Linien und manuellen Positionierungen der Labels
  \diagram* {
   (a) -- [fermion, edge label=\(d \)] (b) ,
   (a) -- [fermion, edge label=\(\bar s\)] (d) ,
   (d) -- [boson, edge label=\(W^+\)] (c),
   (c) -- [fermion] (nu),
   (c) -- [fermion] (e),
   (b) -- [plain, edge label = \(u\)] (d) ,
   (b) -- [boson] (pi) ,
  };
 \end{feynman}
\end{tikzpicture}
\hspace{3cm}
\begin{tikzpicture}
 \begin{feynman}
  \vertex (a) {\(\bar K^0\)};
  \vertex [above right=of a] (b);
  \vertex [right=of b] (pi){\(\pi^+\)}; 
  \vertex [below right=of a] (d);
  % W-Boson
  \vertex [below right=of d] (c);
  
  % Zerfallsprodukte
  \vertex [below right=of c] (e) {\(\ell^-\)};
  \vertex [above right=of c] (nu) {\(\bar\nu_\ell\)};

  % Quarks im Pion mit gekrümmten Linien und manuellen Positionierungen der Labels
  \diagram* {
   (a) -- [fermion, edge label=\(\bar d \)] (b) ,
   (a) -- [fermion, edge label=\(s\)] (d) ,
   (d) -- [boson, edge label=\(W^-\)] (c),
   (c) -- [fermion] (nu),
   (c) -- [fermion] (e),
   (b) -- [plain, edge label = \(u\)] (d) ,
   (b) -- [boson] (pi) ,
  };
 \end{feynman}
\end{tikzpicture}
\caption[Diagram of semi-leptonic decay of $K^0$ and $\bar K^0$]{\textit{Left-hand side:} Semi-leptonic decay of the $K^0$. \textit{Right-hand side:} Semi-leptonic decay of the $\bar K^0$. }
 \label{fig:semi_kaon_decay}
\end{figure}
An independent way of measuring $\vert V_{us} \vert$ is given in~\cite{PhysRevC.102.045501,G_miz_2003,G_miz_2005,MALTMAN2009257,Amhis_2021} by considering decays of the $\tau$ lepton as the ones that are shown in Figure~\ref{fig:tau_decay}. In this case the corresponding CKM matrix element is significantly smaller~\cite{ParticleDataGroup:2024cfk}:
\begin{align}
     \vert V_{us} \vert = 0.2207(14).
\end{align}
\begin{figure}
 \centering
\begin{tikzpicture}
 \begin{feynman}
  \vertex (a){\(\tau\)};
  \vertex [right=of a] (b);
  \vertex [above right=of b] (c){\(\nu_\tau\)}; 
  \vertex [below right=of b] (d);
  % W-Boson
  \vertex [right=of d] (e){\(\pi^-\)};
  
 

  % Quarks im Pion mit gekrümmten Linien und manuellen Positionierungen der Labels
  \diagram* {
   (a) -- [fermion] (b) ,
   (b) -- [fermion] (c) ,
   (b) -- [boson, edge label=\(W^-\)] (d),
   (d) -- [fermion, out=45, in=135,edge label=\(u\)] (e) ,
   (d) -- [fermion, out=-45, in=-135,edge label'=\(\bar d\)] (e) ,
  };
 \end{feynman}
\end{tikzpicture}
\hspace{3cm}
\begin{tikzpicture}
 \begin{feynman}
  \vertex (a){\(\tau\)};
  \vertex [ right=of a] (b);
  \vertex [ above right=of b] (c){\(\nu_\tau\)};
  \vertex [below right=of b] (d);
  % W-Boson
  \vertex [right=of d] (e){\( K^-\)};
  
 

  % Quarks im Pion mit gekrümmten Linien und manuellen Positionierungen der Labels
  \diagram* {
   (a) -- [fermion] (b) ,
   (b) -- [fermion] (c) ,
   (b) -- [boson, edge label=\(W^-\)] (d),
   (d) -- [fermion, out=45, in=135,edge label=\(\bar u\)] (e) ,
   (d) -- [fermion, out=-45, in=-135,edge label'=\(s\)] (e) ,
  };
 \end{feynman}
\end{tikzpicture}
\caption[Diagram of hadronic $\tau$ decays]{Diagram of hadronic $\tau$ decays. \textit{Left-hand side:} Strange hadronic $\tau$ decay. \textit{Right-hand side:} Non-strange hadronic $\tau$ decay.}
 \label{fig:tau_decay}
\end{figure}
In all of these cases $2-3\,\sigma$ discrepancies with the CKM unitarity can be seen. Thus, the task in this part is to use a high-precision lattice estimation of the kaon and pion decay constant ratio. This can be used for further investigations of the CKM unitarity. 

\subsection{Phenomenological consequences of the lattice results}
\label{sec:pheno_fkp}

\begin{figure} 
\centering     \includegraphics[width=\linewidth]{Figures/7/result_comparison_fkp.pdf}     \caption[Comparison on the results on $f_K/f_\pi$]{Comparison on the results on $f_K/f_\pi$. The values include the FLAG average~\cite{FlavourLatticeAveragingGroupFLAG:2024oxs}, and existing lattice studies~\cite{Bazavov:2017lyh,Dowdall:2013rya,ExtendedTwistedMass:2021qui,Miller:2020xhy,RBC:2014ntl,Follana:2007uv,MILC:2009ltw,Aubin:2008ie}. For some of the data, the available values do not 	include strong-isospin breaking corrections. Then, the corrections are taken from the FLAG report~\cite{FlavourLatticeAveragingGroupFLAG:2024oxs}, which uses the \(SU(3)\) $\chi$PT formula from~\cite{Cirigliano:2011tm}. The data are still blinded, so the result from our study is shifted such that it aligns with the FLAG average. However, the relative errors can be compared due to our multiplicative blinding procedure. }     \label{fig:fkp_res_comp} \end{figure}

If the isospin symmetric and isospin-breaking contributions to $f_K/f_\pi$ that are given in Section~\ref{sec:res_fkp} are combined properly one gets the full QCD results of $f_K/f_\pi$ in the FLAG scheme which is the common scheme to compare the results with those from other groups. In this study we get the result \begin{align}     \alpha^K_\mathrm{blind}\,\left[\frac{f_K}{f_\pi}\right]^\mathrm{FLAG} = 1.9226(14)(13)[19] \end{align} where the first error is statistical and the second error is systematic. The total error is given in squared brackets. A comparison with the result from other collaborations as well as the averaged results from the 2024 FLAG report is shown in the left column of Figure~\ref{fig:fkp_res_comp}. It can be seen that this study provides the most accurate ($0.10\,\%$) QCD result of the ratio of the kaon and pion decay constants. 
To extract information on the CKM unitarity from the ratio of the pseudo-scalar decay constants we need to re-call the pion and kaon decay widths:
\begin{align}
 \Gamma(\pi^+ \to \ell \bar \nu_\ell) &= \frac{G_F^2}{8\pi}m_\ell^2M_{\mathrlap{\pi}\phantom{K}}\left(1-\frac{m_\ell^2}{M_{\mathrlap{\pi}\phantom{K}}^2}\right)^2\times \vert V_{u\bar d}\vert^2 \times f_{\mathrlap{\pi}\phantom{K}}^2\times (1+\delta_\pi)\\
 \Gamma(K^+ \to \ell \bar \nu_\ell) &= \frac{G_F^2}{8\pi}m_\ell^2M_K\left(1-\frac{m_\ell^2}{M_K^2}\right)^2\times \vert V_{u\bar s}\vert^2\times f_K^2\times (1+\delta_K)
\end{align}
The ratio of these two equations leads to the expression
\begin{align}
    \bigg \vert \frac{V_{ud}}{V_{us}}\bigg \vert^2 = \frac{\Gamma(\pi^+ \to \ell \nu_\ell)}{\Gamma(K^+ \to \ell \nu_\ell)}\frac{M_K}{M_\pi}\frac{\left(1-\frac{m_\ell^2}{M_K^2}\right)^2}{\left(1-\frac{m_\ell^2}{M_\pi^2}\right)^2}&\frac{f_K^2}{f_\pi^2}\frac{1+\delta_K}{1+\delta_\pi} \\
    \eqqcolon C_{\pi,K}&\frac{f_K^2}{f_\pi^2}(1+\delta_{K,\pi}) 
\end{align}
Since we want to use the radiative corrections $\delta_{\pi,K}$ in GRS scheme from RM123~\cite{DiCarlo:2019thl} also the decay constant ratio has to be transformed in the preferred scheme. In order to do so, we can take the values from Table ~\ref{tab:fkp} such that we have a lattice estimate of the decay constant ratio of
\begin{align}
 &\left(\alpha_\mathrm{blind}^K\right)^2\bigg \vert \frac{V_{ud}}{V_{us}}\bigg \vert^2 =  \left(\alpha_\mathrm{blind}^K\right)^2\times \left[\frac{\mathcal{F}_K}{\mathcal{F}_\pi}\right]\times C_{\pi,K}\\
 \label{eq:fkp_scheme_final}
 =&   \left( A + B(X_l^\mathrm{GRS} - X_l^\mathrm{FLAG})+ C(X_s^\mathrm{GRS} - X_s^\mathrm{FLAG}) + D \frac{\delta M^2}{2 f_\pi^\mathrm{GRS}} + G(B,C)\right)^2 \times(1+ \delta_{K,\pi}^\textrm{valence})\times C_{\pi,K} 
\end{align}
The sum in the first brackets of Equation~\eqref{eq:fkp_scheme_final} which is the lattice estimation of $\frac{f_K}{f_\pi}$ in the GRS scheme (including sea quark effects) is computed according to the methods that were introduced in Section~\ref{sec:cov} for estimating the covariance of quantities with systematic and statistical uncertainties. The sea quark effects $G$ are measured for two values of $B$ and $C$, respectively, so they can be interpolated to their values at the corresponding systematic. $\delta_{\pi,K}$ and $C_{\pi,K}$ also have an uncertainty that is respected by taking their upper and lower one-sigma quantile, respectively. \\
Since we include a lattice estimation of the sea quark effects we need to purify the radiative corrections that are given in~\cite{DiCarlo:2019thl} from those contributions:
\begin{align}
    \delta_\pi^\textrm{valence} &= 0.0150 (18) \\
    \delta_K^\textrm{valence} &= 0.0021 (8) \\
    \delta_{K,\pi}^\textrm{valence}  =  \delta R_{K}^\textrm{valence}- \delta R_{\pi}^\textrm{valence} &= 0.0129 (14)
\end{align}
Thus the theoretical prediction of the full QCD+QED decay constants is given by:
\begin{align}
    \alpha_\mathrm{blind}^K \left[\frac{\mathcal{F}_K}{\mathcal{F}_\pi} \right]^\textrm{theory} = 1.9251(15)(27)[31]
\end{align}
which corresponds to a precision of $0.16\%$
% All in all we get a value of $\alpha_\mathrm{blind}^K\frac{V_{ud}}{V_{us}} = 7.000(13)$ that can be used to check the CKM unitarity by using either $V_{ud}$ or $V_{us}$. With the value of the up-down element from Hardy and Towner~\cite{Hardy:2020qwl} one can check if
% \begin{align}
%     \vert V_{ud}^\mathrm{HT} \vert^2 \left(1 +     \bigg \vert \frac{V_{ud}}{V_{us}}\bigg \vert^{-2}\right)
% \end{align}
% is compatible with 1 and using the FLAG value for the up-strange element~\cite{FermilabLattice:2024yho} one can check if
% \begin{align}
%     \vert V_{us}^\mathrm{FLAG} \vert^2 \left(1 +     \bigg \vert \frac{V_{ud}}{V_{us}}\bigg \vert^{2}\right).
% \end{align}
% is compatible with one.
One can check whether the expressions
% \begin{align}
%     \vert V_{ud}^\mathrm{HT} \vert^2 \left(1 +     \bigg \vert \frac{V_{ud}}{V_{us}}\bigg \vert^{-2}\right),\\
%     \vert V_{us}^\mathrm{FLAG} \vert^2 \left(1 +     \bigg \vert \frac{V_{ud}}{V_{us}}\bigg \vert^{2}\right)
% \end{align}
\begin{align}
    \vert V_{ud}^\mathrm{HT} \vert^2 \left(1 +     \bigg \vert \frac{V_{ud}}{V_{us}}\bigg \vert^{-2}\right) \quad \text{and} \quad \vert V_{us}^\mathrm{FLAG} \vert^2 \left(1 +     \bigg \vert \frac{V_{ud}}{V_{us}}\bigg \vert^{2}\right)
\end{align}
are compatible with one by using the value of the up-down element from Hardy and Towner~\cite{Hardy:2020qwl} and the FLAG value for the up-strange element~\cite{FermilabLattice:2024yho}, respectively. A total comparison of our results, the CKM unitarity and the results from the FLAG and Hardy and Towner is shown in Figure~\ref{fig:CKM_constraints}. Since the lattice estimations in this work are not unblinded yet, the results are set to the PDG values using the precision of our result. 
\begin{figure}[H]
    \centering
    \includegraphics[width=0.9\linewidth]{Figures/7/ckm_constraints.pdf}
    \caption[CKM constraints for the pseudo-scalar decay constants]{CKM constraints for the pseudo-scalar decay constants. The results from our work~\cite{FKAPI} are given by the blue band, the super-allowed $\beta$ decay results~\cite{Hardy:2020qwl} are marks by the purple band, the semi-leptonic kaon decays~\cite{FlavourLatticeAveragingGroupFLAG:2024oxs} are given by the yellow band and the dashed line corresponds to the CKM unitarity using the assumption $\vert V_{ub} \vert = 0$ which compatible with our reached precision.}
    \label{fig:CKM_constraints}
\end{figure}
With a relative precision of 2 permille for \( V_{us}/V_{ud} \), my result surpasses the precision of the PDG value~\cite{ParticleDataGroup:2024cfk} for \( V_{us} \), which stands at 3 permille. This is particularly notable given that the \( V_{ud} \) input from Hardy and Towner is already determined with very high precision~\cite{Hardy:2020qwl}.




\section{Discrepancies in scale setting}
\label{sec:scalesetting}
As shown in Figure~\ref{fig:resultt0}, there are significant discrepancies among different lattice determinations of the gradient-flow-based scale $\sqrt{t_0}$, which are not observed for the $w_0$ scale (see Figure~\ref{fig:resultw0}). These various determinations of $\sqrt{t_0}$ employ different discretization schemes and physical input parameters. This raises the question: what is the origin of these discrepancies? Using our data, we are also able to compute this scale and contribute to the discussion.\\
Using the pion decay rate in the FLAG scheme ($f_\pi^\textrm{FLAG} = 130.5\,\textrm{MeV}$~\cite{FlavourLatticeAveragingGroupFLAG:2024oxs}), we can give a value of
\begin{align}
    \alpha_\mathrm{blind}^{t_0}\sqrt{t_0^{\textrm{QCD, FLAG}}} = 0.1791(3)(\substack{+3 \\ -2})[\substack{+4 \\ -4}]{\,\mathrm{fm}},
\end{align}
which has a higher precision than most of the existing lattice studies as can be seen in the right panel of Figure~\ref{fig:resultt0}. However, the QED contributions are missing, but in the previous section we could observe that $w_0^{\textrm{QCD, FLAG}} \sim 0$ at our current level of precision. Therefore, no large differences are expected, even though this assumption still has to be proven.
\begin{figure}[h]
    \centering
    \includegraphics[width=0.9\linewidth]{Figures/8/result_comparison_t0.pdf}
    \caption[Result comparison of $\sqrt{t_0}$]{Comparison of several estimations of $\sqrt{t_0}$~\cite{Miller:2020xhy,MILC:2015tqx,Dowdall:2013rya,RBC:2014ntl,Hudspith:2024kzk,Strassberger:2021tsu,BMW:2012hcm,Bornyakov:2015eaa,Bruno:2016plf} and the average from the FLAG report 2024~\cite{FlavourLatticeAveragingGroupFLAG:2024oxs}. Estimations based on simulations using $N_f = 2+1$ instead of $N_f = 2+1+1$ are marked with $^*$. The data are still blinded, so the result from our study is shifted such that it aligns with the FLAG average. However, the relative errors can be compared due to our multiplicative blinding procedure.}
    \label{fig:resultt0}
\end{figure}
