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Bounding method applied to the local vector current correlation function. The upper bound and lower bound are applied as a function of $t_{\text{max}}/a$ along the horizontal axis. The vertical axis is $a_\mu^{HVP}$ in units of $10^{-10}$ as measured from summing up the timeslices on the lattice. The left plot shows the bounding method with no improvement, and the right plot shows the improved bounding method with a 4-state reconstruction applied. As more states are added to the reconstruction, the upper and lower bounds converge at shorter $t_{\text{max}}$. The red point on the left side of both plots shows the optimal value obtained by picking the timeslice that minimizes the uncertainty (numerical values given in the text). In the right plot, the intermediate values from the $N$-state reconstructions with $N<4$ are shown as well.

Same as Fig.~\ref{fig:gevp2pi}, except for a GEVP with 2 additional $4\pi$ operators. The new states that appear as a result of the addition of the $4\pi$ operators are denoted as black crosses, while the colors and symbols for the other states are kept consistent with Fig.~\ref{fig:gevp2pi}. Even with the additional operators, the spectrum and overlaps are very strongly correlated with those without the additional operators. The large statistical error for $t_0=4$ in the right plot is associated with overlapping error bars, which results in eigenvalues that are not sorted properly.

Left: spectrum obtained from solving the GEVP for a $C(t_0)C^{-1}(t_0+\delta t)$ for fixed $\delta t$. Small values of $t_0$ are subject to contamination from excited states, which is observed as an exponential approach to the asymptotic value at large $t_0$. Right: overlap of the states with the local vector current operator. Both left and right plots should asymptote to the true value in the large $t_0$ limit. In this figure, the GEVP is solved for a 6-operator basis and $\delta t=3$ is used.

Bounding method applied to the local vector current correlation function. The upper bound and lower bound are applied as a function of $t_{\text{max}}/a$ along the horizontal axis. The vertical axis is $a_\mu^{HVP}$ in units of $10^{-10}$ as measured from summing up the timeslices on the lattice. The left plot shows the bounding method with no improvement, and the right plot shows the improved bounding method with a 4-state reconstruction applied. As more states are added to the reconstruction, the upper and lower bounds converge at shorter $t_{\text{max}}$. The red point on the left side of both plots shows the optimal value obtained by picking the timeslice that minimizes the uncertainty (numerical values given in the text). In the right plot, the intermediate values from the $N$-state reconstructions with $N<4$ are shown as well.

Left: Integrand of $a_\mu^{HVP}$ plotted as a function of $t/a$. The local vector current correlation function by itself is plotted as black crosses, and the $N$-state reconstruction obtained from the GEVP are shown in colors. As more states are added to the correlation function reconstruction, the resulting curve shape matches the local vector current down to shorter distance. Right: Ratio of the $N$-state reconstructions normalized by the local vector current correlation function. The uncertainty on the local vector current correlation function is denoted by the gray band. As more states are added, the ratio of reconstruction over local vector current approaches 1, and the 4-state reconstruction gives a reconstruction consistent with the local vector current to within $1\sigma$ after about $t/a=10$.

Left: spectrum obtained from solving the GEVP for a $C(t_0)C^{-1}(t_0+\delta t)$ for fixed $\delta t$. Small values of $t_0$ are subject to contamination from excited states, which is observed as an exponential approach to the asymptotic value at large $t_0$. Right: overlap of the states with the local vector current operator. Both left and right plots should asymptote to the true value in the large $t_0$ limit. In this figure, the GEVP is solved for a 6-operator basis and $\delta t=3$ is used.

Same as Fig.~\ref{fig:gevp2pi}, except for a GEVP with 2 additional $4\pi$ operators. The new states that appear as a result of the addition of the $4\pi$ operators are denoted as black crosses, while the colors and symbols for the other states are kept consistent with Fig.~\ref{fig:gevp2pi}. Even with the additional operators, the spectrum and overlaps are very strongly correlated with those without the additional operators. The large statistical error for $t_0=4$ in the right plot is associated with overlapping error bars, which results in eigenvalues that are not sorted properly.

Left: Integrand of $a_\mu^{HVP}$ plotted as a function of $t/a$. The local vector current correlation function by itself is plotted as black crosses, and the $N$-state reconstruction obtained from the GEVP are shown in colors. As more states are added to the correlation function reconstruction, the resulting curve shape matches the local vector current down to shorter distance. Right: Ratio of the $N$-state reconstructions normalized by the local vector current correlation function. The uncertainty on the local vector current correlation function is denoted by the gray band. As more states are added, the ratio of reconstruction over local vector current approaches 1, and the 4-state reconstruction gives a reconstruction consistent with the local vector current to within $1\sigma$ after about $t/a=10$.