We describe a new set of gauge configurations generated within the CLS effort. These
ensembles have
Article funded by SCOAP3
\tilde \mu_1>\dots>\tilde \mu_{N_\mathrm{sp}}=0$ \begin{align} W_0(\mu_0)=\prod_{i=1}^{N_\mathrm{sp}}\det R(\tilde \mu_{i-1},\tilde \mu_{i})\,; && R(\mu_1,\mu_2)= \frac{(\hat Q^2 + \mu_2^2)^2}{(\hat Q^2+ \mu_1^2)^2} \frac{(\hat Q^2 + 2 \mu_1^2)}{(\hat Q^2 + 2 \mu_2^2)}\,. \end{align} Now each of the factors is evaluated stochastically with $N_\mathrm{r}$ complex-valued Gaussian random fields $\eta$ of unit variance \begin{equation} \tilde R(\mu_1,\mu_2,N_\mathrm{r})= \frac{1}{N_\mathrm{r}} \sum_{i=1}^{N_\mathrm{r}} \mathrm{exp} \{-\left(\eta_i,(R^{-1}(\mu_1,\mu_2)-1) \eta_i\right)\}\,, \end{equation} such that up to an irrelevant constant factor the determinant is retrieved by averaging over the noise fields \begin{equation} \det R(\mu_1,\mu_2) \propto \langle \tilde R(\mu_1,\mu_2,N_\mathrm{r}) \rangle_\eta. \end{equation} Following the initial proposal of ref.~\cite{Luscher-ml-2012av}, it is sufficient to use a single step $N_\mathrm{sp}=1$ with a suitably chosen value of $N_\mathrm{r}$. Its value along with the other parameters of the reweighting can be found in table~\ref{tab:rew}. This is the method implemented in openQCD-1.2. Once the fluctuations in the reweighting factor increase, it is advisable to use intermediate $\tilde \mu$, a possibility given in openQCD-1.4. This is because the distribution of the results for the reweighting factors become long-tailed once exceptionally small eigenvalues of the $\hat Q^2$ are encountered. In this situation it is very difficult to argue about the uncertainty of $W_0$~\cite{Hasenfratz-ml-2002ym}. By splitting the estimate into smaller intervals in $\tilde \mu$, the distribution of each of the factors becomes significantly more regular. For ensemble H105r002 we find precisely such a situation. While with a single step in $\tilde \mu$ the smallest reweighting factors show a distribution which is far from Gaussian, using ten intermediate $\tilde \mu$ the individual factors can be computed reliably to $\mathrm{O}(15\%)$ accuracy using 15 sources each. \begin{table} \centering \renewcommand{\arraystretch}{1.2} \begin{tabular}{|cllc|} \hline id & $N_\mathrm{r}$& $\frac{\mathrm{var}(W_0)}{\langle W_0 \rangle^2} \cdot 10^3$ & $\frac{\mathrm{var}(W_1)}{\langle W_1 \rangle^2} \cdot 10^5$ \\ \hline H101 & 12 & 0.00047(9) & 5.1(2) \\ H102 & 12 & 0.036(4) & 1.88(5) \\ H105 & 36 & 3.2(4) & 7.3(2) \\ H105r005 & 24 & 0.0032(9) & 3.7(2) \\ C101 & 24 & 1.8(1.1) & 1.6(2) \\ C101r014 & 24 & 5.1(2.1) & 1.63(10) \\ \hline H200 & 24 & 0.00018(5) & 4.7(2) \\ N200 & 24 & 0.4(2) & 2.23(7) \\ D200 & 48 & 0.15(5) & 4.9(3) \\ \hline N300 & 24 & 0.00018(2) & 3.0(1) \\ \sl J303 & \sl 24 & \sl 3.7(3.2) & \sl 1.3(2) \\ \hline \end{tabular} ]]>
c_0 e^{-m\frac{T}{2}} \cosh\left\{-m \left(x_{0,\mathrm{min}}-\frac{T}{2}\right) \right\} \,. \end{equation} At the current accuracy of the data, the result of this investigation is that a single $x_{0,{\rm min}}$ is sufficient for each value of $\beta$, as might be expected from figure~\ref{fig:bnd}. The effect of the quark mass is negligible. In particular we have \begin{align} x_{0,\mathrm{min}}(\beta=3.4)/a&=20 \,; & x_{0,\mathrm{min}}(\beta=3.55)/a&=21\,; & x_{0,\mathrm{min}}(\beta=3.7)/a&=24 \,, \end{align} and the final value of $E(t)$ in the vicinity of $t=t_0$ is determined by averaging $E(x_0,t)$ in the corresponding interval. The value of $t_0/a^2$ is then determined by eq.~\eqref{eq:t0}. The results are listed in table~\ref{tab:mpst0}. \begin{figure} \centering \includegraphics[trim = 0 10 0 4, clip, width=0.65\textwidth]{t0_over_t0ref} ]]>