\chapter{Conclusion and Outlook}
\label{cha:conclusion}
This thesis presented several contributions to the field of lattice quantum chromodynamics, with a focus on high-precision determinations relevant for tests of the Standard Model.

The \textbf{first part} (Chapters~\ref{cha:QFT}--\ref{cha:data}) laid the theoretical groundwork for lattice field theory and provided a systematic introduction to path integral quantization, fermion discretization and the corresponding symmetry properties and the essential tools for data analysis in lattice simulations.

The \textbf{second part} of this thesis presents research I carried out in cooperation with the Budapest-Marseille-Wuppertal collaboration.




Two methods were employed to estimate the reference scales $w_0$ and $t_0$: one based on the $\Omega^-$ baryon mass, and another using the pion decay constant.  Both approaches led to estimates with unprecedented precision. Both methods of estimating $w_0$ are now used in the most accurate lattice prediction of the hadronic vacuum polarization contribution to the muon anomaly~\cite{Boccaletti:2024guq}.

Furthermore, techniques were explored to investigate the long-distance contributions to the muon $g-2$. Two-pion states with the same quantum numbers as the vector meson were constructed, and a mathematical framework was developed to resolve the mixing between two-pion and vector meson states using a generalized eigenvalue problem. Particular emphasis was placed on the implementation of the contributing diagrams, and a computational strategy was devised to reduce the cost of evaluating the computationally expensive connected diagrams. It was demonstrated that this approach significantly lowers runtime without compromising the quality of the resulting signal.

These techniques were implemented and tested on a large-volume ensemble ($6,\mathrm{fm}^3 \times 9,\mathrm{fm}$, $a \approx 0.13,\mathrm{fm}$). While first results on the reconstruction were obtained, this topic remains rich in open questions and challenges. At the end of this chapter, I provide a brief overview of the current status of the tail reconstruction and discuss several potential strategies to address the remaining issues.

Additionally, a high-precision lattice determination of the pseudo-scalar decay constant ratio was performed. This study included all relevant effects such as finite-volume corrections, strong isospin breaking and a careful treatment of sea and valence quark contributions. A noteworthy feature of this work was the inclusion of a novel fine lattice ensemble with $a = 0.034\,\mathrm{fm}$, which allowed for an unprecedented level of precision. The resulting estimate of $f_K/f_\pi$ directly translates into a highly accurate determination of the relevant CKM matrix elements, and provides a stringent test of CKM unitarity within the Standard Model.
\section*{Outlook}

The results of this thesis contribute to ongoing efforts to reduce the theoretical uncertainties in key observables of the Standard Model. In particular, the reconstruction of the long-distance tail of the vector correlator represents a step forward in lattice determinations of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment. By implementing a method based on a generalized eigenvalue problem and applying it on a large-volume ensemble, this work demonstrated that improved control over the low-energy region is possible. However, in case of staggered fermions there is still a lot of open work as discussed in Section~\ref{sec:conclusion}.

Further progress will require complementary approaches. Data-driven methods remain essential for validating lattice results in the region where the signal is suppressed but still relevant~\cite{Davier:2023cyp,Boccaletti:2024guq}. Moreover, cross-checks using different fermion discretizations are crucial to ensure that systematic effects are under control. These aspects were already discussed in Section~\ref{sec:conclusion}, and future work will need to build upon both the algorithmic and conceptual developments presented here.

An important element of this thesis is the inclusion of a new, exceptionally fine lattice ensemble with a spacing of approximately $0.038\,\mathrm{fm}$. While it played a central role in the precise determination of $f_K/f_\pi$, its usefulness goes far beyond that single application. The ensemble offers an opportunity to perform lattice simulations with significantly reduced discretization errors, which will benefit a broad class of observables ranging from hadron spectroscopy to weak matrix elements.

Finally, the determination of the reference scales $w_0$ and $t_0$ with previously unmatched precision has immediate practical value. These scales serve as key ingredients in setting the overall scale for many lattice calculations and have already been adopted in recent state-of-the-art determinations of $a_\mu^\mathrm{HVP}$. Reliable scale setting is a prerequisite for meaningful comparisons between theory and experiment, and the methods developed here will support future precision studies across a wide range of applications.
