\chapter{Lattice results} 
\label{cha:results}
In this chapter the total results of the lattice studies are given. Especially, the lattice results to $f_K/f_\pi$ and their IB and QED (sea quark) contributions are given. Furthermore, the lattice estimates of the dimensionless quantities $M_\Omega w_0$, $f_\pi w_0$ and $f_\pi \sqrt{t_0}$ and the sea quark contributions to  $\mathcal{F}_\pi w_0$ are given.
\section{Computation of the pseudo-scalar decay constant ratio}
\label{sec:res_fkp}
Our final results of the fits that were performed during the analysis of $f_K/f_\pi$ are given in Table~\ref{tab:fkp}. In the upper plots one can see different continuum extrapolations of $[f_K/f_\pi]_\textrm{iso}$ as functions of $a^2$ and $\Delta_{KS}$, respectively. The colors denote the different weights of the corresponding fits. In the histogram the overall distribution of the isospin symmetric data is shown. The table next to it shows the error budget of this particular analysis from which it can be concluded that the continuum extrapolation ($\beta$ but, $A$ and $A'$ fit form) is the dominant source of systematic errors. In the table below you can find all of the results of this analysis. On the diagonal there are the results of the isospin-symmetric analysis ($A$) and its derivatives with respect to $X_l$ ($B$) and $X_s$ ($C$). $D$ represents the strong-isospin breaking contributions and $G$ are the sea quark effects. A computation of the valence quark effects can be found in~\cite{DiCarlo:2019thl}.   
\begin{table}     
\centering
\includegraphics[width=\textwidth]{Figures/7/fkp_comp.pdf}
\begin{minipage}{0.45\textwidth}     	  
\includegraphics[width=0.8\textwidth]{Figures/7/fkp_hist_histogram.pdf}     \end{minipage}
\begin{minipage}{0.45\textwidth}
\begin{tabular}{l|l|l} 
\hline\hline
median                         & \multicolumn{2}{c}{  1.9292} \\
statistical error              &   0.0011 & 0.058 \% \\
full systematic error          &   0.0010 & 0.053 \% \\ \hline Plateau region                 & 0.000018 & 0.00095 \% \\
Effective mass d$t$              &  0.00013 & 0.0070 \% \\
Effective mass form            &  0.00016 & 0.0081 \% \\
Hairpin parameters             & 0.0000032 & 0.00017 \% \\
NNLO XPT FV                    & 0.0000022 & 0.00011 \% \\
Beta cut                       &  0.00017 & 0.0088 \% \\
A                              &  0.00062 & 0.032 \% \\
A' fit form                    &  0.00037 & 0.019 \% \\
B fit form                     & 0.000046 & 0.0024 \% \\
C fit form                     & 0.000030 & 0.0016 \% \\ \hline\hline \end{tabular}
\end{minipage}
\begin{tiny}
\begin{tabular}{c||c|c|c|c|c|}
&$A$&$B$&$C$&$D$&$G$\\
\hline\hline 	
$A$&$1.9292(12)(^{+8}_{-12})[^{+14}_{-17}]$&$0.06(0.06)(0.06)$&$0.22(0.1)(0.39)$&$0.08(0.05)(0.11)$ &--\\
\hline 	$B$&&$-0.030(18)(^{+11}_{-8})[^{+21}_{-20}]$&$-0.22(-0.34)(-0.02)$&$0.07(-0.01)(0.17)$&--\\
\hline 	$C$&&&$0.0157(17)(^{+7}_{-13})[^{+18}_{-21}]$&$0.21(0.19)(0.23)$&--\\
\hline 	$D$&&&&$-0.0186(33)(^{+29}_{-16})[^{+44}_{-37}]$&--\\
\hline 	$G$&&&&&$-0.0005(7)(^{+12}_{-4})[^{+14}_{-8}]$\\
\hline 
\end{tabular}
\end{tiny}   
\caption[Blinded lattice results of the $f_K/f_\pi$-study]{Blinded lattice results of the $f_K/f_\pi$-study. \textit{Upper left panel:} Different continuum extrapolations in $a^2$, the points are shifted to physical masses and are averaged if they correspond to the same lattice spacing. The colors denote relative weights of the corresponding fits. \textit{Upper-right  panel:} Different continuum extrapolations in $\Delta_{KS}$. \textit{Center left panel:} Total histogram of the isospin-symmetric $f_K/f_\pi$ continuum extrapolation. \textit{Center right panel: } Error budget of the $f_K/f_\pi$ continuum extrapolation. \textit{Lower panel}: Continuum results (diagonal) and their correlation coefficients with the corresponding statistical and systematic contributions (off-diagonal). The medians from the tables and the histograms differ slightly. This is an artifact of the binning of the histogram. However, for the final results the values from the tables are taken, which use the methods from~\cite{Borsanyi_2021,Boccaletti:2024guq}.}    \label{tab:fkp}
\end{table}


\section{Computation of \texorpdfstring{$M_\Omega w_0$}{wnull} }
\label{sec:res_w0om}
As mentioned before, the physical value of $w_0$ can be obtained by extrapolating the dimensionless product of $w_0$ with either $M_\Omega$ or $f_\pi$, and then dividing by the corresponding physical value.
%\subsection{Computation of  \texorpdfstring{$w_0$}{wnull} using \texorpdfstring{$M_\Omega$}{MOmega}}
The global fit of the combination
\begin{align}
    w_0 = \frac{(w_0M_\Omega)^\text{lat}}{M_\Omega^{\textrm{phys}}},\end{align}
where $M_\Omega^{\textrm{phys}}$ is the experimentally measured mass of the $\Omega$ baryon, can be seen in Figure~\ref{tab:w0_Mo}. The continuum extrapolation is done using polynomials in either $a^2 = \frac{\left[(aM_\Omega)^\text{lat}\right]^2}{\left[ M_\Omega^\text{phys}\right]^2}$ or $\frac{a^2\Delta_{KS}}{\left[(aM_\Omega)^\text{lat}\right]^2}$ up to degrees of order three. Similarly to the $f_K/f_\pi$ global fit in Section~\ref{sec:global_fit} the coarsest three lattice spacings can be excluded, and the physical point is reached via interpolations in $X_l$ and $X_s$. The histogram of all the fits can be seen in Figure~\ref{tab:w0_Mo}. The left-hand side of the histogram has a Gaussian shape, while the right-hand side has a longer tail. This can be explained by the curvature of the data in combination with the third-order polynomial in $a^2$ that shift the continuum limit towards higher values, as evident in the continuum extrapolation above. The error budget is given in Table~\ref{tab:w0_Mo}. The most dominant systematic error source is the choice of the continuum parameter, which makes sense because $\Delta_{KS}$ provides a much smaller extrapolation range than the usual $a^2$ fits. The next largest error source (even larger than the statistical error) is the choice of the fit range for the $\Omega$ propagator. Thus, we can see how important proper analysis of the $\Omega$ baryon mass is in this context. For example, in the global fits for the decay constant the choice of the fit ranges only contributed sub-leading to the total error.  
\begin{table}[h]
    \centering
    \includegraphics[width=0.9\linewidth]{Figures/8/w0_omega.pdf}
    \includegraphics[width=0.9\linewidth]{Figures/8/w0_omega_hist.pdf}
        \begin{tabular}{l|r|r}
    Median & \multicolumn{2}{c}{172.35 am}\\
    \hline
    Total error & 0.51 am & 0.30 \% \\
    Statistical error &  0.22 am & 0.13 \% \\
    Systematic error &  0.46 am & 0.27 \% \\
    \hline
    Pseudo-scalar fit range & 0.01 am & $<$ 0.01 \% \\
    Omega baryon fit range & 0.24 am & 0.14 \% \\
    Physical value of $M_\Omega$ & 0.06 am & 0.03 \% \\
    Lattice spacing cuts & 0.09 am & 0.05 \% \\
    Order of fit polynomials & 0.17 am & 0.10 \% \\
    Continuum parameter ($\Delta_{KS}$ or $a^2$) & 0.30 am & 0.17 \% \\
\end{tabular}
       \caption[Continuum extrapolation, histogram and error budget of $w_0$ computed with the $\Omega$ baron mass]{\textit{Upper left panel:} Continuum extrapolation of the $w_0$ scale, estimated with the $\Omega$ baryon mass, in terms of $a^2$. \textit{Upper right panel:} Continuum extrapolation of the same quantity in terms of $\Delta_{KS}$. \textit{Center panel:} Total histogram of $w_0$ with the one- and two-sigma bands \textit{Lower panel:} Error budget of $w_0$ computed with the mass of the $\Omega$ baryon. Previously published in~\cite{Boccaletti:2024guq}. }
    \label{tab:w0_Mo}
\end{table}
\section{Computation of \texorpdfstring{$f_\pi w_0$}{wnull} }
\label{sec:res_w0fpi}

For the computation of $f_\pi w_0$  we add another systematic ingredient by computing the gradient flow based scale $w_0$ with either the Wilson or the Symanzik improved Zeuthen-flow (see Section~\ref{sec:smoothing}). This systematic is chosen to be a flat-weighted ingredient within the analysis. Furthermore, we skip the finest ensembles ($\beta = 4.3032$) since they were not available when the paper was originally published~\cite{Boccaletti:2024guq}. A comparison of pure $a^2$ and pure $\Delta_{KS}$ fit forms can be seen in Figure~\ref{tab:fpiw0}. Here, the relative weight of the corresponding fit is marked with an associated color. Below that, the total histogram of the distributions of the continuum values of $f_\pi w_0$ and the error budget are given. The error budget is dominated the choice of the gradient flow (Wilson flow vs. Symanzik improved Zeuthen flow). At the bottom, the results of the constituents of the analysis are given. On the diagonal, the FLAG scheme continuum result ($A$) and its derivatives with respect to $X_l$ and $X_s$ ($B$ and $C$) are given. The $G$-coefficient describes the sea quark contributions to the total result. For these quantities, the first error is statistical, the second is systematic and the third (in squared brackets) is the total error. On the off-diagonals of the table, you can find the correlation coefficients of the respective quantities split up into their statistical and systematic contributions.\\

\begin{table}
	\centering
	\includegraphics{Figures/8/fpiw0_comp.pdf}
    \begin{minipage}{0.45\textwidth}
	\includegraphics[width=\textwidth]{Figures/8/fpiw0_wfzf_sy_hist_histogram.pdf}
\end{minipage}
\begin{minipage}{0.45\textwidth}
\begin{tabular}{l|l|l}
\hline\hline
median                         & \multicolumn{2}{c}{ 0.13455} \\
statistical error              &  0.00015 & 0.11 \% \\
full systematic error          &  0.00027 & 0.20 \% \\
\hline
Wilson/Zeuthen(sy)             &  0.00017 & 0.12 \% \\
Plateau region                 & 0.000016 & 0.012 \% \\
Effective mass dt             & 0.00000075 & 0.00056 \% \\
Effective mass form            & 0.000025 & 0.019 \% \\
Hairpin parameters             & 0.0000017 & 0.0013 \% \\
NNLO XPT FV                    & 0.000014 & 0.010 \% \\
Beta cut                       & 0.000063 & 0.047 \% \\
A                              &  0.00011 & 0.085 \% \\
A' fit form                    & 0.000057 & 0.042 \% \\
B fit form                     & 0.0000059 & 0.0044 \% \\
C fit form                     & 0.0000048 & 0.0036 \% \\
\hline\hline
\end{tabular}
\end{minipage}
\begin{tabular}{c||c|c|c|c|}
	&$A$&$B$&$C$&$G$\\
	\hline\hline
	$A$&$0.1346(1)(^{+2}_{-3})[^{+2}_{-4}]$&$	-0.09(0.02)(-0.22)$&$-0.03(0.42)(-0.52)$&--\\ \hline
	$B$&&$0.0069(21)(^{+13}_{-0})[^{+25}_{-21}]$&$-0.19(-0.17)(-0.24)$&--\\ \hline
	$C$&&&$-3(15)(^{+4}_{-9})[^{+16}_{-8}]\times 10^{-5}$&--\\ \hline
	$G$&&&&$9(9)(^{+2}_{-9})[^{+10}_{-13}]\times 10^{-5}$\\ \hline
\end{tabular}
	\caption
	[Blinded lattice results of the $f_\pi w_0$-study]{Blinded lattice results of the $f_\pi w_0$-study. \textit{Upper left panel:} Different continuum extrapolations in $a^2$, the points are shifted to physical masses and are averaged if they correspond to the same lattice spacing. The colors denote relative weights of the corresponding fits. \textit{Upper right panel:} Different continuum extrapolations in $\Delta_{KS}$. \textit{Center left panel:} Total histogram of the isospin-symmetric $f_\pi w_0$ continuum extrapolation. \textit{Center right panel: } Error budget of the $f_\pi w_0$ continuum extrapolation. \textit{Lower panel}: Continuum results (diagonal) and their correlation coefficients (off-diagonal).
	}
    	\label{tab:fpiw0}
\end{table}



\section{Computation of  \texorpdfstring{$f_\pi \sqrt{t_0}$}{tnull} }
\label{sec:res_t0fpi}

The computation of $f_\pi \sqrt{t_0}$ largely follows the same procedure as the computation of $f_\pi w_0$, with only a few differences. Obviously, the target observable is replaced by $f_\pi \sqrt{t_0}$, and we only compute the pure QCD value of $\sqrt{t_0}$ in the FLAG scheme, as we do not have access to data for the sea quark contributions.\\
A comparison of different fit forms depending on only $a^2$ or $\Delta_{KS}$ is shown in Figure~\ref{tab:fpit0}. Continuum extrapolations only depending on $\Delta_{KS}$ do not seem to describe the data properly, while combinations of $a^2$ and $\Delta_{KS}$ contributions are suitable to describe the data. It can also be seen that the overall error is dominated by the continuum extrapolation followed by the choice of the gradient flow. 
\begin{table}
	\centering
	\includegraphics{Figures/8/fpit0_comp.pdf}
    \begin{minipage}{0.45\textwidth}
	\includegraphics[width=\textwidth]{Figures/8/fpit0_wfzf_sy_hist_histogram.pdf}
\end{minipage}
\begin{minipage}{0.45\textwidth}
\begin{tabular}{l|l|l}
\hline\hline
median                         & \multicolumn{2}{c}{ 0.11184} \\
statistical error              &  0.00018 & 0.16 \% \\
full systematic error          &  0.00020 & 0.18 \% \\
\hline
Wilson/Zeuthen(sy)             & 0.000096 & 0.085 \% \\
Plateau region                 & 0.000030 & 0.027 \% \\
Effective mass dt             & 0.0000100 & 0.0089 \% \\
Effective mass form            & 0.000015 & 0.014 \% \\
Hairpin parameters             & 0.0000053 & 0.0048 \% \\
NNLO XPT FV                    & 0.000018 & 0.016 \% \\
Beta cut                       & 0.000067 & 0.060 \% \\
A                              &  0.00012 & 0.11 \% \\
A' fit form                    & 0.000047 & 0.042 \% \\
B fit form                     & 0.0000059 & 0.0053 \% \\
C fit form                     & 0.000025 & 0.022 \% \\
\hline\hline
\end{tabular}
\end{minipage}
\begin{tabular}{c||c|c|c|c|}
	&$A$&$B$&$C$&$G$\\
	\hline\hline
	$A$&$0.1118(2)(^{+2}_{-3})[^{+3}_{-4}]$&$0.03(0.01)(0.07)$&$0.10(-0.05)(0.31)$&--\\ \hline
	$B$&&$0.0069(22)(^{+13}_{-5})[^{+25}_{-24}]$&$-0.20(-0.12)(-0.36)$&--\\ \hline
	$C$&&&$3(2)(^{+0}_{-2})[^{+2}_{-3}]\times 10^{-4}$&--\\ \hline
	$G$&&&&--\\ \hline
\end{tabular}
	\caption
[Blinded lattice results of the $f_\pi \sqrt{t_0}$-study]{Blinded lattice results of the $f_\pi \sqrt{t_0}$-study. \textit{Upper left panel:} Different continuum extrapolations in $a^2$, the points are shifted to physical masses and are averaged if they correspond to the same lattice spacing. The colors denote relative weights of the corresponding fits. \textit{Upper right panel:} Different continuum extrapolations in $\Delta_{KS}$. \textit{Center left panel:} Total histogram of the isospin-symmetric $f_\pi \sqrt{t_0}$ continuum extrapolation. \textit{Center right panel: } Error budget of the $f_\pi \sqrt{t_0}$ continuum extrapolation. \textit{Lower panel}: Continuum results (diagonal) and their correlation coefficients (off-diagonal).}
	\label{tab:fpit0}
\end{table}
