\chapter{Preface}
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% \emph{Ich betrachte es aber als durchaus möglich, dass die Physik nicht auf dem Feldbegriff begründet werden kann, d.h. auf kontinuierlichen Gebilden.} \
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\epigraph{\texttt{Ich betrachte es aber als durchaus möglich, dass die Physik nicht auf dem Feldbegriff begründet werden kann, d.h. auf kontinuierlichen Gebilden.}}{Albert Einstein~\cite{Foelsing1999}}

\section{Introduction}
Modern particle physics experiments have reached an unprecedented level of precision. For example, the fine-structure constant of quantum electrodynamics (QED) has been measured through the recoil velocity of a Rubidium isotope after an atomic transition to a precision of one part in a billion~\cite{PhysRevLett.96.033001}. Even more remarkably, the magnetic moment of the electron is known to a precision of one part in ten trillion~\cite{PhysRevLett.130.071801}. However, the most widely discussed high-precision experiment today is the $g_\mu - 2$ experiment at Fermilab~\cite{PhysRevAccelBeams.24.044002,PhysRevA.103.042208,PhysRevD.103.072002,PhysRevLett.126.141801,Aguillard_2023,PhysRevD.110.032009}. Such experiments are vital for testing the Standard Model of particle physics.

The Standard Model is a highly developed quantum field theory (QFT) that describes three of the four fundamental forces in nature: electromagnetic, weak, and strong interactions. Gravity remains beyond its scope. Developed during the second half of the 20th century~\cite{Oerter:2006}, the model was solidified by the confirmation of the quark model~\cite{HARARI1975265} and validated through major experimental discoveries, including the top quark (1995)~\cite{Abe_1995}, the $\tau$ neutrino (2000)~\cite{KODAMA2001218}, and the Higgs boson (2012)~\cite{Aad_2012}.

Despite its many successes, the Standard Model is not believed to be a complete theory. Not only does it omit gravity, but it also contains internal inconsistencies, such as the Landau pole in QED~\cite{Landau:1954nau}, where the energy-dependent coupling constant diverges at very high energies. This indicates that the Standard Model is only an effective theory valid at low to intermediate energies.

High-precision experiments allow us to probe the limits of the Standard Model and search for signs of beyond-the-standard-model (BSM) physics. To interpret these experiments, we also need precise theoretical predictions. Calculations within the Standard Model are highly complex. While QED and weak interactions can be computed perturbatively using a series expansion in terms of a small coupling constant~\cite{Peskin}, this approach fails for low-energy strong interactions due to non-convergence.

In 1974, to address this issue, Kenneth Wilson proposed the idea of introducing a hard ultraviolet (UV) cutoff through lattice regularization~\cite{PhysRevD.10.2445}. This technique replaces continuous spacetime with a discrete four dimensional grid with spacing $a$, reducing the number of degrees of freedom and making computations tractable in the strong-coupling limit. One of the most significant advantages of lattice discretization is that it renders the theory finite in volume, making it amenable to numerical evaluation via Monte Carlo methods~\cite{PhysRevLett.43.553}.

Today, lattice calculations are approaching the precision of experimental measurements~\cite{Borsanyi_2021,Boccaletti:2024guq}. To achieve this, sophisticated computational techniques and the use of high-performance computing resources are required. Over the past decade, simulations have begun utilizing GPUs~\cite{Clark:2009wm,Egri:2006zm,Kumar:2025nmb}, and more recently, quantum computing~\cite{Luca_thesis} and machine learning approaches~\cite{Albergo:2019eim}.

With these powerful methods, the lattice physics community provides increasingly accurate predictions that can be directly compared with experimental data. This thesis presents my contributions to this rapidly evolving and exciting field.

\section{Overview of the Thesis}

The \textbf{first part} of this thesis outlines the theoretical foundations underlying lattice field theory. It summarizes the pioneering work of previous generations of physicists who laid the groundwork for modern lattice formulations. The basics of quantum field theory and lattice gauge theory are discussed, along with the numerical and analytical methods used for data analysis.

The \textbf{second part} applies these methods to high-precision QCD studies relevant to tests of the Standard Model. Here, I present my own contributions to the field of lattice gauge theory. These contributions fall into three main categories:

\begin{itemize}
    \item \textbf{Precise determination of the gradient flow scales} $w_0$ and $t_0$, employing multiple methodologies. These determinations achieve a level of precision that, to date, has not been reached in the literature. In particular, the $w_0$ scale plays a key role in our high-precision determination of the anomalous magnetic moment of the muon~\cite{Boccaletti:2024guq}:
    \begin{center}
    A. Boccaletti et al. (Budapest-Marseille-Wuppertal), \emph{High precision calculation of the hadronic vacuum polarisation contribution to the muon anomaly}, \href{https://arxiv.org/abs/2407.10913}{arXiv:2407.10913}.
    \end{center}

    \item \textbf{High-precision determination of the ratio of kaon to pion decay constants}, $f_K / f_\pi$, which provides a sensitive test of CKM matrix unitarity. The precision achieved in this work also surpasses that of previous determinations. The results are to be published as~\cite{FKAPI}:
    \begin{center}
    Fabian J. Frech et al. (Budapest-Marseille-Wuppertal), \emph{Determination of $f_K/f_\pi$ from 2+1+1 flavor 4-stout staggered lattices}.
    \end{center}

    \item \textbf{Studies of tail contributions to the muon $g-2$}, presented at the LATTICE2023 and LATTICE2024 conferences and published in the corresponding proceedings:
    \begin{center}
    Fabian J. Frech et al. (Budapest-Marseille-Wuppertal), \emph{The mixing of two-pion and vector-meson states using staggered fermions}, \href{https://doi.org/10.22323/1.453.0251}{PoS \textbf{LATTICE2023} (2024) 251}, \href{https://arxiv.org/abs/2401.00514}{[arXiv:2401.00514]}.
    \end{center}
    \begin{center}
    Fabian J. Frech et al. (Budapest-Marseille-Wuppertal), \emph{Reconstruction of the vector meson propagator using a generalized eigenvalue problem}, \href{https://doi.org/10.22323/1.466.0133}{PoS \textbf{LATTICE2024} (2025) 251}, \href{https://arxiv.org/abs/2501.19186}{[arXiv:2501.19186]}.
    \end{center}
\end{itemize}



\section{Units and Conventions}

In this thesis, flat spacetime is used, as is customary in particle physics, with the mostly negative metric signature:
\begin{align}
    g_{\mu\nu} = \eta_{\mu\nu} = \begin{pmatrix}
        1 & & & \\
        & -1 & & \\
        & & -1 & \\
        & & & -1
    \end{pmatrix}
\end{align}

Throughout this work, natural units are employed, meaning that the speed of light $c$, the Boltzmann constant $k_B$, the vacuum permittivity $\epsilon_0$ and the reduced Planck constant $\hbar$ are all set to unity. In principle, the gravitational constant $G$ would also be set to one in such a system, however, it does not appear in this thesis.

In some sections, lattice units are used, where the lattice spacing $a$ is set to $1$.
