CERN Accelerating science

Evolution of the chiral condensate (upper-left), pseudoscalar density (upper-right) and plaquette (lower). The different colors and vertical dotted lines indicate the boundaries of the 7 independent Markov chains. Note that measurements of the chiral condensate and pseudo-scalar density were performed only on the subset of configurations corresponding to the first three chains of the lower figure. The last three ensembles in the lower figure were generated using the EOFA technique.

Evolution of the chiral condensate (upper-left), pseudoscalar density (upper-right) and plaquette (lower). The different colors and vertical dotted lines indicate the boundaries of the 10 independent Markov chains. Note that measurements of the chiral condensate and pseudo-scalar density were performed only on the subset of configurations corresponding to the first three chains of the lower figure. The last three ensembles in the lower figure were generated using the EOFA technique.

Evolution of the chiral condensate (upper-left), pseudoscalar density (upper-right) and plaquette (lower). The different colors and vertical dotted lines indicate the boundaries of the 7 independent Markov chains. Note that measurements of the chiral condensate and pseudo-scalar density were performed only on the subset of configurations corresponding to the first three chains of the lower figure. The last three ensembles in the lower figure were generated using the EOFA technique.

Evolution of the chiral condensate (upper-left), pseudoscalar density (upper-right) and plaquette (lower). The different colors and vertical dotted lines indicate the boundaries of the 10 independent Markov chains. Note that measurements of the chiral condensate and pseudo-scalar density were performed only on the subset of configurations corresponding to the first three chains of the lower figure. The last three ensembles in the lower figure were generated using the EOFA technique.

Evolution of the chiral condensate (upper-left), pseudoscalar density (upper-right) and plaquette (lower). The different colors and vertical dotted lines indicate the boundaries of the 7 independent Markov chains. Note that measurements of the chiral condensate and pseudo-scalar density were performed only on the subset of configurations corresponding to the first three chains of the lower figure. The last three ensembles in the lower figure were generated using the EOFA technique.

Evolution of the chiral condensate (upper-left), pseudoscalar density (upper-right) and plaquette (lower). The different colors and vertical dotted lines indicate the boundaries of the 10 independent Markov chains. Note that measurements of the chiral condensate and pseudo-scalar density were performed only on the subset of configurations corresponding to the first three chains of the lower figure. The last three ensembles in the lower figure were generated using the EOFA technique.

The integrated autocorrelation time $\tau_{\rm int}(\Delta_{\rm cut})$ as a function of $\Delta_{\rm cut}$ for the chiral condensate (upper-left), pseudoscalar density (upper-right) and plaquette (lower) superimposed for the different streams. The two very short streams are not included in this analysis. The chiral condensate and pseudoscalar density were measured on only three of the ensembles.

The integrated autocorrelation time $\tau_{\rm int}(\Delta_{\rm cut})$ as a function of $\Delta_{\rm cut}$ for the chiral condensate (upper-left), pseudoscalar density (upper-right) and plaquette (lower) superimposed for the different streams. The two very short streams are not included in this analysis. The chiral condensate and pseudoscalar density were measured on only three of the ensembles.

The integrated autocorrelation time $\tau_{\rm int}(\Delta_{\rm cut})$ as a function of $\Delta_{\rm cut}$ for the chiral condensate (upper-left), pseudoscalar density (upper-right) and plaquette (lower) superimposed for the different streams. The two very short streams are not included in this analysis. The chiral condensate and pseudoscalar density were measured on only three of the ensembles.

The integrated autocorrelation time $\tau_{\rm int}(\Delta_{\rm cut})$ as a function of $\Delta_{\rm cut}$ for the chiral condensate (upper-left), pseudoscalar density (upper-right) and plaquette (lower) superimposed for the different streams. The two very short streams are not included in this analysis. The chiral condensate and pseudoscalar density were measured on only three of the ensembles.

The integrated autocorrelation time $\tau_{\rm int}(\Delta_{\rm cut})$ as a function of $\Delta_{\rm cut}$ for the chiral condensate (upper-left), pseudoscalar density (upper-right) and plaquette (lower) superimposed for the different streams. The two very short streams are not included in this analysis. The chiral condensate and pseudoscalar density were measured on only three of the ensembles.

The integrated autocorrelation time $\tau_{\rm int}(\Delta_{\rm cut})$ as a function of $\Delta_{\rm cut}$ for the chiral condensate (upper-left), pseudoscalar density (upper-right) and plaquette (lower) superimposed for the different streams. The two very short streams are not included in this analysis. The chiral condensate and pseudoscalar density were measured on only three of the ensembles.

Diagrams showing the contractions which contribute to the two-point functions involving the $\pi\pi(\ldots)$ and $\sigma$ operators. The solid dots indicate the positions of the pion two-quark operators and the dotted vertical lines passing through these points indicate the separate 3-dimensional time slices on which these operators are placed, with the nearby pairs of lines separated by four time units as described in Eq.~\eqref{equ:project_angular_mom}. Identical diagrams appear for the $\sigma$ operator only with a single vertical line at the source and/or sink, with the dots now representing the scalar bilinear. The top 4 diagrams are labeled by $C$, $D$, $R$ and $V$ diagrams from left to right, and the lower 4 diagrams are labeled by $C_{\sigma\pi\pi}$, $V_{\sigma\pi\pi}$, $C_{\sigma\sigma}$ and $V_{\sigma\sigma}$ from left to right.

Diagrams showing the contractions which contribute to the two-point functions involving the $\pi\pi(\ldots)$ and $\sigma$ operators. The solid dots indicate the positions of the pion two-quark operators and the dotted vertical lines passing through these points indicate the separate 3-dimensional time slices on which these operators are placed, with the nearby pairs of lines separated by four time units as described in Eq.~\eqref{equ:project_angular_mom}. Identical diagrams appear for the $\sigma$ operator only with a single vertical line at the source and/or sink, with the dots now representing the scalar bilinear. The top 4 diagrams are labeled by $C$, $D$, $R$ and $V$ diagrams from left to right, and the lower 4 diagrams are labeled by $C_{\sigma\pi\pi}$, $V_{\sigma\pi\pi}$, $C_{\sigma\sigma}$ and $V_{\sigma\sigma}$ from left to right.

Diagrams showing the contractions which contribute to the two-point functions involving the $\pi\pi(\ldots)$ and $\sigma$ operators. The solid dots indicate the positions of the pion two-quark operators and the dotted vertical lines passing through these points indicate the separate 3-dimensional time slices on which these operators are placed, with the nearby pairs of lines separated by four time units as described in Eq.~\eqref{equ:project_angular_mom}. Identical diagrams appear for the $\sigma$ operator only with a single vertical line at the source and/or sink, with the dots now representing the scalar bilinear. The top 4 diagrams are labeled by $C$, $D$, $R$ and $V$ diagrams from left to right, and the lower 4 diagrams are labeled by $C_{\sigma\pi\pi}$, $V_{\sigma\pi\pi}$, $C_{\sigma\sigma}$ and $V_{\sigma\sigma}$ from left to right.

Diagrams showing the contractions which contribute to the two-point functions involving the $\pi\pi(\ldots)$ and $\sigma$ operators. The solid dots indicate the positions of the pion two-quark operators and the dotted vertical lines passing through these points indicate the separate 3-dimensional time slices on which these operators are placed, with the nearby pairs of lines separated by four time units as described in Eq.~\eqref{equ:project_angular_mom}. Identical diagrams appear for the $\sigma$ operator only with a single vertical line at the source and/or sink, with the dots now representing the scalar bilinear. The top 4 diagrams are labeled by $C$, $D$, $R$ and $V$ diagrams from left to right, and the lower 4 diagrams are labeled by $C_{\sigma\pi\pi}$, $V_{\sigma\pi\pi}$, $C_{\sigma\sigma}$ and $V_{\sigma\sigma}$ from left to right.

Diagrams showing the contractions which contribute to the two-point functions involving the $\pi\pi(\ldots)$ and $\sigma$ operators. The solid dots indicate the positions of the pion two-quark operators and the dotted vertical lines passing through these points indicate the separate 3-dimensional time slices on which these operators are placed, with the nearby pairs of lines separated by four time units as described in Eq.~\eqref{equ:project_angular_mom}. Identical diagrams appear for the $\sigma$ operator only with a single vertical line at the source and/or sink, with the dots now representing the scalar bilinear. The top 4 diagrams are labeled by $C$, $D$, $R$ and $V$ diagrams from left to right, and the lower 4 diagrams are labeled by $C_{\sigma\pi\pi}$, $V_{\sigma\pi\pi}$, $C_{\sigma\sigma}$ and $V_{\sigma\sigma}$ from left to right.

Diagrams showing the contractions which contribute to the two-point functions involving the $\pi\pi(\ldots)$ and $\sigma$ operators. The solid dots indicate the positions of the pion two-quark operators and the dotted vertical lines passing through these points indicate the separate 3-dimensional time slices on which these operators are placed, with the nearby pairs of lines separated by four time units as described in Eq.~\eqref{equ:project_angular_mom}. Identical diagrams appear for the $\sigma$ operator only with a single vertical line at the source and/or sink, with the dots now representing the scalar bilinear. The top 4 diagrams are labeled by $C$, $D$, $R$ and $V$ diagrams from left to right, and the lower 4 diagrams are labeled by $C_{\sigma\pi\pi}$, $V_{\sigma\pi\pi}$, $C_{\sigma\sigma}$ and $V_{\sigma\sigma}$ from left to right.

Diagrams showing the contractions which contribute to the two-point functions involving the $\pi\pi(\ldots)$ and $\sigma$ operators. The solid dots indicate the positions of the pion two-quark operators and the dotted vertical lines passing through these points indicate the separate 3-dimensional time slices on which these operators are placed, with the nearby pairs of lines separated by four time units as described in Eq.~\eqref{equ:project_angular_mom}. Identical diagrams appear for the $\sigma$ operator only with a single vertical line at the source and/or sink, with the dots now representing the scalar bilinear. The top 4 diagrams are labeled by $C$, $D$, $R$ and $V$ diagrams from left to right, and the lower 4 diagrams are labeled by $C_{\sigma\pi\pi}$, $V_{\sigma\pi\pi}$, $C_{\sigma\sigma}$ and $V_{\sigma\sigma}$ from left to right.

Diagrams showing the contractions which contribute to the two-point functions involving the $\pi\pi(\ldots)$ and $\sigma$ operators. The solid dots indicate the positions of the pion two-quark operators and the dotted vertical lines passing through these points indicate the separate 3-dimensional time slices on which these operators are placed, with the nearby pairs of lines separated by four time units as described in Eq.~\eqref{equ:project_angular_mom}. Identical diagrams appear for the $\sigma$ operator only with a single vertical line at the source and/or sink, with the dots now representing the scalar bilinear. The top 4 diagrams are labeled by $C$, $D$, $R$ and $V$ diagrams from left to right, and the lower 4 diagrams are labeled by $C_{\sigma\pi\pi}$, $V_{\sigma\pi\pi}$, $C_{\sigma\sigma}$ and $V_{\sigma\sigma}$ from left to right.

Diagrams showing the contractions which contribute to the two-point functions involving the $\pi\pi(\ldots)$ and $\sigma$ operators. The solid dots indicate the positions of the pion two-quark operators and the dotted vertical lines passing through these points indicate the separate 3-dimensional time slices on which these operators are placed, with the nearby pairs of lines separated by four time units as described in Eq.~\eqref{equ:project_angular_mom}. Identical diagrams appear for the $\sigma$ operator only with a single vertical line at the source and/or sink, with the dots now representing the scalar bilinear. The top 4 diagrams are labeled by $C$, $D$, $R$ and $V$ diagrams from left to right, and the lower 4 diagrams are labeled by $C_{\sigma\pi\pi}$, $V_{\sigma\pi\pi}$, $C_{\sigma\sigma}$ and $V_{\sigma\sigma}$ from left to right.

Diagrams showing the contractions which contribute to the two-point functions involving the $\pi\pi(\ldots)$ and $\sigma$ operators. The solid dots indicate the positions of the pion two-quark operators and the dotted vertical lines passing through these points indicate the separate 3-dimensional time slices on which these operators are placed, with the nearby pairs of lines separated by four time units as described in Eq.~\eqref{equ:project_angular_mom}. Identical diagrams appear for the $\sigma$ operator only with a single vertical line at the source and/or sink, with the dots now representing the scalar bilinear. The top 4 diagrams are labeled by $C$, $D$, $R$ and $V$ diagrams from left to right, and the lower 4 diagrams are labeled by $C_{\sigma\pi\pi}$, $V_{\sigma\pi\pi}$, $C_{\sigma\sigma}$ and $V_{\sigma\sigma}$ from left to right.

Diagrams showing the contractions which contribute to the two-point functions involving the $\pi\pi(\ldots)$ and $\sigma$ operators. The solid dots indicate the positions of the pion two-quark operators and the dotted vertical lines passing through these points indicate the separate 3-dimensional time slices on which these operators are placed, with the nearby pairs of lines separated by four time units as described in Eq.~\eqref{equ:project_angular_mom}. Identical diagrams appear for the $\sigma$ operator only with a single vertical line at the source and/or sink, with the dots now representing the scalar bilinear. The top 4 diagrams are labeled by $C$, $D$, $R$ and $V$ diagrams from left to right, and the lower 4 diagrams are labeled by $C_{\sigma\pi\pi}$, $V_{\sigma\pi\pi}$, $C_{\sigma\sigma}$ and $V_{\sigma\sigma}$ from left to right.

Diagrams showing the contractions which contribute to the two-point functions involving the $\pi\pi(\ldots)$ and $\sigma$ operators. The solid dots indicate the positions of the pion two-quark operators and the dotted vertical lines passing through these points indicate the separate 3-dimensional time slices on which these operators are placed, with the nearby pairs of lines separated by four time units as described in Eq.~\eqref{equ:project_angular_mom}. Identical diagrams appear for the $\sigma$ operator only with a single vertical line at the source and/or sink, with the dots now representing the scalar bilinear. The top 4 diagrams are labeled by $C$, $D$, $R$ and $V$ diagrams from left to right, and the lower 4 diagrams are labeled by $C_{\sigma\pi\pi}$, $V_{\sigma\pi\pi}$, $C_{\sigma\sigma}$ and $V_{\sigma\sigma}$ from left to right.

Diagrams showing the contractions which contribute to the two-point functions involving the $\pi\pi(\ldots)$ and $\sigma$ operators. The solid dots indicate the positions of the pion two-quark operators and the dotted vertical lines passing through these points indicate the separate 3-dimensional time slices on which these operators are placed, with the nearby pairs of lines separated by four time units as described in Eq.~\eqref{equ:project_angular_mom}. Identical diagrams appear for the $\sigma$ operator only with a single vertical line at the source and/or sink, with the dots now representing the scalar bilinear. The top 4 diagrams are labeled by $C$, $D$, $R$ and $V$ diagrams from left to right, and the lower 4 diagrams are labeled by $C_{\sigma\pi\pi}$, $V_{\sigma\pi\pi}$, $C_{\sigma\sigma}$ and $V_{\sigma\sigma}$ from left to right.

Diagrams showing the contractions which contribute to the two-point functions involving the $\pi\pi(\ldots)$ and $\sigma$ operators. The solid dots indicate the positions of the pion two-quark operators and the dotted vertical lines passing through these points indicate the separate 3-dimensional time slices on which these operators are placed, with the nearby pairs of lines separated by four time units as described in Eq.~\eqref{equ:project_angular_mom}. Identical diagrams appear for the $\sigma$ operator only with a single vertical line at the source and/or sink, with the dots now representing the scalar bilinear. The top 4 diagrams are labeled by $C$, $D$, $R$ and $V$ diagrams from left to right, and the lower 4 diagrams are labeled by $C_{\sigma\pi\pi}$, $V_{\sigma\pi\pi}$, $C_{\sigma\sigma}$ and $V_{\sigma\sigma}$ from left to right.

Diagrams showing the contractions which contribute to the two-point functions involving the $\pi\pi(\ldots)$ and $\sigma$ operators. The solid dots indicate the positions of the pion two-quark operators and the dotted vertical lines passing through these points indicate the separate 3-dimensional time slices on which these operators are placed, with the nearby pairs of lines separated by four time units as described in Eq.~\eqref{equ:project_angular_mom}. Identical diagrams appear for the $\sigma$ operator only with a single vertical line at the source and/or sink, with the dots now representing the scalar bilinear. The top 4 diagrams are labeled by $C$, $D$, $R$ and $V$ diagrams from left to right, and the lower 4 diagrams are labeled by $C_{\sigma\pi\pi}$, $V_{\sigma\pi\pi}$, $C_{\sigma\sigma}$ and $V_{\sigma\sigma}$ from left to right.

Diagrams showing the contractions which contribute to the two-point functions involving the $\pi\pi(\ldots)$ and $\sigma$ operators. The solid dots indicate the positions of the pion two-quark operators and the dotted vertical lines passing through these points indicate the separate 3-dimensional time slices on which these operators are placed, with the nearby pairs of lines separated by four time units as described in Eq.~\eqref{equ:project_angular_mom}. Identical diagrams appear for the $\sigma$ operator only with a single vertical line at the source and/or sink, with the dots now representing the scalar bilinear. The top 4 diagrams are labeled by $C$, $D$, $R$ and $V$ diagrams from left to right, and the lower 4 diagrams are labeled by $C_{\sigma\pi\pi}$, $V_{\sigma\pi\pi}$, $C_{\sigma\sigma}$ and $V_{\sigma\sigma}$ from left to right.

The $t_{\mathrm{min}}$ dependence of the fitted energy $E_\pi$ for the $\pi(111)$(left) and $\pi(311)$(right) cases. Here $E_\pi$ is shown in lattice units with $t_{\mathrm{max}}$ fixed to be 29.

The $t_{\mathrm{min}}$ dependence of the fitted energy $E_\pi$ for the $\pi(111)$(left) and $\pi(311)$(right) cases. Here $E_\pi$ is shown in lattice units with $t_{\mathrm{max}}$ fixed to be 29.

The $t_{\mathrm{min}}$ dependence of the fitted energy $E_\pi$ for the $\pi(111)$(left) and $\pi(311)$(right) cases. Here $E_\pi$ is shown in lattice units with $t_{\mathrm{max}}$ fixed to be 29.

The $t_{\mathrm{min}}$ dependence of the fitted energy $E_\pi$ for the $\pi(111)$(left) and $\pi(311)$(right) cases. Here $E_\pi$ is shown in lattice units with $t_{\mathrm{max}}$ fixed to be 29.

The $t_{\mathrm{min}}$ dependence of fitted ground state energy for the stationary $\pi\pi_{I=2}$ channel (left) with $t_{\mathrm{max}}=25$ and the stationary $\pi\pi_{I=0}$ channel (right) with $t_{\mathrm{max}}=15$. Left: The circles represent the two-operator, two-state fit and downward pointing triangles the one-operator, one-state fit. Right: The pentagons represent the one-operator, one-state fit. The stars and downward pointing triangles show the results from the two two-operator, two-state fits. Finally the circles show the three-operator, three-state fit for $t_{\mathrm{min}}=3$ and 4 while the diamonds show the three-operator, two-state fit for $t_{\mathrm{min}}=5-8$. For the $I=0$ channel, including additional operators (especially the $\sigma$) substantially improves the determination of the ground state energy.

The $t_{\mathrm{min}}$ dependence of fitted ground state energy for the stationary $\pi\pi_{I=2}$ channel (left) with $t_{\mathrm{max}}=25$ and the stationary $\pi\pi_{I=0}$ channel (right) with $t_{\mathrm{max}}=15$. Left: The circles represent the two-operator, two-state fit and downward pointing triangles the one-operator, one-state fit. Right: The pentagons represent the one-operator, one-state fit. The stars and downward pointing triangles show the results from the two two-operator, two-state fits. Finally the circles show the three-operator, three-state fit for $t_{\mathrm{min}}=3$ and 4 while the diamonds show the three-operator, two-state fit for $t_{\mathrm{min}}=5-8$. For the $I=0$ channel, including additional operators (especially the $\sigma$) substantially improves the determination of the ground state energy.

The $t_{\mathrm{min}}$ dependence of fitted ground state energy for the stationary $\pi\pi_{I=2}$ channel (left) with $t_{\mathrm{max}}=25$ and the stationary $\pi\pi_{I=0}$ channel (right) with $t_{\mathrm{max}}=15$. Left: The circles represent the two-operator, two-state fit and downward pointing triangles the one-operator, one-state fit. Right: The pentagons represent the one-operator, one-state fit. The stars and downward pointing triangles show the results from the two two-operator, two-state fits. Finally the circles show the three-operator, three-state fit for $t_{\mathrm{min}}=3$ and 4 while the diamonds show the three-operator, two-state fit for $t_{\mathrm{min}}=5-8$. For the $I=0$ channel, including additional operators (especially the $\sigma$) substantially improves the determination of the ground state energy.

The $t_{\mathrm{min}}$ dependence of fitted ground state energy for the stationary $\pi\pi_{I=2}$ channel (left) with $t_{\mathrm{max}}=25$ and the stationary $\pi\pi_{I=0}$ channel (right) with $t_{\mathrm{max}}=15$. Left: The circles represent the two-operator, two-state fit and downward pointing triangles the one-operator, one-state fit. Right: The pentagons represent the one-operator, one-state fit. The stars and downward pointing triangles show the results from the two two-operator, two-state fits. Finally the circles show the three-operator, three-state fit for $t_{\mathrm{min}}=3$ and 4 while the diamonds show the three-operator, two-state fit for $t_{\mathrm{min}}=5-8$. For the $I=0$ channel, including additional operators (especially the $\sigma$) substantially improves the determination of the ground state energy.

The $t_{\mathrm{min}}$ dependence of the fitted ground state energy for the moving $\pi\pi_{I=2}$ channel (left) and moving $\pi\pi_{I=0}$ channel (right) with $t_{\mathrm{max}}=25$ ($I=2$) and 15 ($I=0$). The upper, middle and lower panels are for total momenta $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$ and $(2,0,0)\frac{\pi}{L}$, respectively. Our final results were obtained from three-operator, three-state fits. Reading from top to bottom the values of $t_{\mathrm{min}}$ for these results were for $I=2$ $t_{\mathrm{min}}=10$, 12, 11 and for $I=0$ $t_{\mathrm{min}}= 6$, 8 and 7.

The $t_{\mathrm{min}}$ dependence of the fitted ground state energy for the moving $\pi\pi_{I=2}$ channel (left) and moving $\pi\pi_{I=0}$ channel (right) with $t_{\mathrm{max}}=25$ ($I=2$) and 15 ($I=0$). The upper, middle and lower panels are for total momenta $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$ and $(2,0,0)\frac{\pi}{L}$, respectively. Our final results were obtained from three-operator, three-state fits. Reading from top to bottom the values of $t_{\mathrm{min}}$ for these results were for $I=2$ $t_{\mathrm{min}}=10$, 12, 11 and for $I=0$ $t_{\mathrm{min}}= 6$, 8 and 7.

The $t_{\mathrm{min}}$ dependence of the fitted ground state energy for the moving $\pi\pi_{I=2}$ channel (left) and moving $\pi\pi_{I=0}$ channel (right) with $t_{\mathrm{max}}=25$ ($I=2$) and 15 ($I=0$). The upper, middle and lower panels are for total momenta $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$ and $(2,0,0)\frac{\pi}{L}$, respectively. Our final results were obtained from three-operator, three-state fits. Reading from top to bottom the values of $t_{\mathrm{min}}$ for these results were for $I=2$ $t_{\mathrm{min}}=10$, 12, 11 and for $I=0$ $t_{\mathrm{min}}= 6$, 8 and 7.

The $t_{\mathrm{min}}$ dependence of the fitted ground state energy for the moving $\pi\pi_{I=2}$ channel (left) and moving $\pi\pi_{I=0}$ channel (right) with $t_{\mathrm{max}}=25$ ($I=2$) and 15 ($I=0$). The upper, middle and lower panels are for total momenta $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$ and $(2,0,0)\frac{\pi}{L}$, respectively. Our final results were obtained from three-operator, three-state fits. Reading from top to bottom the values of $t_{\mathrm{min}}$ for these results were for $I=2$ $t_{\mathrm{min}}=10$, 12, 11 and for $I=0$ $t_{\mathrm{min}}= 6$, 8 and 7.

The $t_{\mathrm{min}}$ dependence of the fitted ground state energy for the moving $\pi\pi_{I=2}$ channel (left) and moving $\pi\pi_{I=0}$ channel (right) with $t_{\mathrm{max}}=25$ ($I=2$) and 15 ($I=0$). The upper, middle and lower panels are for total momenta $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$ and $(2,0,0)\frac{\pi}{L}$, respectively. Our final results were obtained from three-operator, three-state fits. Reading from top to bottom the values of $t_{\mathrm{min}}$ for these results were for $I=2$ $t_{\mathrm{min}}=10$, 12, 11 and for $I=0$ $t_{\mathrm{min}}= 6$, 8 and 7.

The $t_{\mathrm{min}}$ dependence of the fitted ground state energy for the moving $\pi\pi_{I=2}$ channel (left) and moving $\pi\pi_{I=0}$ channel (right) with $t_{\mathrm{max}}=25$ ($I=2$) and 15 ($I=0$). The upper, middle and lower panels are for total momenta $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$ and $(2,0,0)\frac{\pi}{L}$, respectively. Our final results were obtained from three-operator, three-state fits. Reading from top to bottom the values of $t_{\mathrm{min}}$ for these results were for $I=2$ $t_{\mathrm{min}}=10$, 12, 11 and for $I=0$ $t_{\mathrm{min}}= 6$, 8 and 7.

The $t_{\mathrm{min}}$ dependence of the fitted ground state energy for the moving $\pi\pi_{I=2}$ channel (left) and moving $\pi\pi_{I=0}$ channel (right) with $t_{\mathrm{max}}=25$ ($I=2$) and 15 ($I=0$). The upper, middle and lower panels are for total momenta $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$ and $(2,0,0)\frac{\pi}{L}$, respectively. Our final results were obtained from three-operator, three-state fits. Reading from top to bottom the values of $t_{\mathrm{min}}$ for these results were for $I=2$ $t_{\mathrm{min}}=10$, 12, 11 and for $I=0$ $t_{\mathrm{min}}= 6$, 8 and 7.

The $t_{\mathrm{min}}$ dependence of the fitted ground state energy for the moving $\pi\pi_{I=2}$ channel (left) and moving $\pi\pi_{I=0}$ channel (right) with $t_{\mathrm{max}}=25$ ($I=2$) and 15 ($I=0$). The upper, middle and lower panels are for total momenta $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$ and $(2,0,0)\frac{\pi}{L}$, respectively. Our final results were obtained from three-operator, three-state fits. Reading from top to bottom the values of $t_{\mathrm{min}}$ for these results were for $I=2$ $t_{\mathrm{min}}=10$, 12, 11 and for $I=0$ $t_{\mathrm{min}}= 6$, 8 and 7.

The $t_{\mathrm{min}}$ dependence of the fitted ground state energy for the moving $\pi\pi_{I=2}$ channel (left) and moving $\pi\pi_{I=0}$ channel (right) with $t_{\mathrm{max}}=25$ ($I=2$) and 15 ($I=0$). The upper, middle and lower panels are for total momenta $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$ and $(2,0,0)\frac{\pi}{L}$, respectively. Our final results were obtained from three-operator, three-state fits. Reading from top to bottom the values of $t_{\mathrm{min}}$ for these results were for $I=2$ $t_{\mathrm{min}}=10$, 12, 11 and for $I=0$ $t_{\mathrm{min}}= 6$, 8 and 7.

The $t_{\mathrm{min}}$ dependence of the fitted ground state energy for the moving $\pi\pi_{I=2}$ channel (left) and moving $\pi\pi_{I=0}$ channel (right) with $t_{\mathrm{max}}=25$ ($I=2$) and 15 ($I=0$). The upper, middle and lower panels are for total momenta $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$ and $(2,0,0)\frac{\pi}{L}$, respectively. Our final results were obtained from three-operator, three-state fits. Reading from top to bottom the values of $t_{\mathrm{min}}$ for these results were for $I=2$ $t_{\mathrm{min}}=10$, 12, 11 and for $I=0$ $t_{\mathrm{min}}= 6$, 8 and 7.

The $t_{\mathrm{min}}$ dependence of the fitted ground state energy for the moving $\pi\pi_{I=2}$ channel (left) and moving $\pi\pi_{I=0}$ channel (right) with $t_{\mathrm{max}}=25$ ($I=2$) and 15 ($I=0$). The upper, middle and lower panels are for total momenta $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$ and $(2,0,0)\frac{\pi}{L}$, respectively. Our final results were obtained from three-operator, three-state fits. Reading from top to bottom the values of $t_{\mathrm{min}}$ for these results were for $I=2$ $t_{\mathrm{min}}=10$, 12, 11 and for $I=0$ $t_{\mathrm{min}}= 6$, 8 and 7.

The $t_{\mathrm{min}}$ dependence of the fitted ground state energy for the moving $\pi\pi_{I=2}$ channel (left) and moving $\pi\pi_{I=0}$ channel (right) with $t_{\mathrm{max}}=25$ ($I=2$) and 15 ($I=0$). The upper, middle and lower panels are for total momenta $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$ and $(2,0,0)\frac{\pi}{L}$, respectively. Our final results were obtained from three-operator, three-state fits. Reading from top to bottom the values of $t_{\mathrm{min}}$ for these results were for $I=2$ $t_{\mathrm{min}}=10$, 12, 11 and for $I=0$ $t_{\mathrm{min}}= 6$, 8 and 7.

A typical diagram for the decomposition of the ATW effect when the Green's function is constructed from two $O_b$ operators. For the first-order ATW effect, both legs are pions with momentum $(1,1,1)\pi/L$, which means if one of the internal states is the vacuum, e.g., internal state 1, then the other internal state cannot be the vacuum. For the second order ATW effect, we can choose leg A to be $(1,1,1)\pi/L$ and leg B to be $(3,1,1)\pi/L$, while keeping both internal states to be the vacuum state.

A typical diagram for the decomposition of the ATW effect when the Green's function is constructed from two $O_b$ operators. For the first-order ATW effect, both legs are pions with momentum $(1,1,1)\pi/L$, which means if one of the internal states is the vacuum, e.g., internal state 1, then the other internal state cannot be the vacuum. For the second order ATW effect, we can choose leg A to be $(1,1,1)\pi/L$ and leg B to be $(3,1,1)\pi/L$, while keeping both internal states to be the vacuum state.

The t dependence of $\mathcal{N}(t)$, defined in Eq.~\eqref{eq:Norm_Det} for the three dimensional (black, which include all operators) and two dimensional (red, only include $O_a$ and $O_c$ operator) matrix of Green's functions for the stationary $I=0$ case.

The t dependence of $\mathcal{N}(t)$, defined in Eq.~\eqref{eq:Norm_Det} for the three dimensional (black, which include all operators) and two dimensional (red, only include $O_a$ and $O_c$ operator) matrix of Green's functions for the stationary $I=0$ case.

The $t_{\mathrm{min}}$ or $t_0$ dependence of the fitted ground state energy from the GEVP and the usual fit for the $\pi\pi_{I=2}$ (left) and $\pi\pi_{I=0}$ (right) channel with $t_{\mathrm{max}}=20$ ($I=2$) and 15 ($I=0$). The total momenta from the top down are $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$, $(2,0,0)\frac{\pi}{L}$ and 0. Here the $x$-axis represents $t_{\mathrm{min}}$ for the usual fit, and $t_0$ for the GEVP fit. In the legend of the lower right panel, 2D and 3D indicate a $2\times2$ and $3\times3$ GEVP matrix.

The $t_{\mathrm{min}}$ or $t_0$ dependence of the fitted ground state energy from the GEVP and the usual fit for the $\pi\pi_{I=2}$ (left) and $\pi\pi_{I=0}$ (right) channel with $t_{\mathrm{max}}=20$ ($I=2$) and 15 ($I=0$). The total momenta from the top down are $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$, $(2,0,0)\frac{\pi}{L}$ and 0. Here the $x$-axis represents $t_{\mathrm{min}}$ for the usual fit, and $t_0$ for the GEVP fit. In the legend of the lower right panel, 2D and 3D indicate a $2\times2$ and $3\times3$ GEVP matrix.

The $t_{\mathrm{min}}$ or $t_0$ dependence of the fitted ground state energy from the GEVP and the usual fit for the $\pi\pi_{I=2}$ (left) and $\pi\pi_{I=0}$ (right) channel with $t_{\mathrm{max}}=20$ ($I=2$) and 15 ($I=0$). The total momenta from the top down are $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$, $(2,0,0)\frac{\pi}{L}$ and 0. Here the $x$-axis represents $t_{\mathrm{min}}$ for the usual fit, and $t_0$ for the GEVP fit. In the legend of the lower right panel, 2D and 3D indicate a $2\times2$ and $3\times3$ GEVP matrix.

The $t_{\mathrm{min}}$ or $t_0$ dependence of the fitted ground state energy from the GEVP and the usual fit for the $\pi\pi_{I=2}$ (left) and $\pi\pi_{I=0}$ (right) channel with $t_{\mathrm{max}}=20$ ($I=2$) and 15 ($I=0$). The total momenta from the top down are $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$, $(2,0,0)\frac{\pi}{L}$ and 0. Here the $x$-axis represents $t_{\mathrm{min}}$ for the usual fit, and $t_0$ for the GEVP fit. In the legend of the lower right panel, 2D and 3D indicate a $2\times2$ and $3\times3$ GEVP matrix.

The $t_{\mathrm{min}}$ or $t_0$ dependence of the fitted ground state energy from the GEVP and the usual fit for the $\pi\pi_{I=2}$ (left) and $\pi\pi_{I=0}$ (right) channel with $t_{\mathrm{max}}=20$ ($I=2$) and 15 ($I=0$). The total momenta from the top down are $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$, $(2,0,0)\frac{\pi}{L}$ and 0. Here the $x$-axis represents $t_{\mathrm{min}}$ for the usual fit, and $t_0$ for the GEVP fit. In the legend of the lower right panel, 2D and 3D indicate a $2\times2$ and $3\times3$ GEVP matrix.

The $t_{\mathrm{min}}$ or $t_0$ dependence of the fitted ground state energy from the GEVP and the usual fit for the $\pi\pi_{I=2}$ (left) and $\pi\pi_{I=0}$ (right) channel with $t_{\mathrm{max}}=20$ ($I=2$) and 15 ($I=0$). The total momenta from the top down are $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$, $(2,0,0)\frac{\pi}{L}$ and 0. Here the $x$-axis represents $t_{\mathrm{min}}$ for the usual fit, and $t_0$ for the GEVP fit. In the legend of the lower right panel, 2D and 3D indicate a $2\times2$ and $3\times3$ GEVP matrix.

The $t_{\mathrm{min}}$ or $t_0$ dependence of the fitted ground state energy from the GEVP and the usual fit for the $\pi\pi_{I=2}$ (left) and $\pi\pi_{I=0}$ (right) channel with $t_{\mathrm{max}}=20$ ($I=2$) and 15 ($I=0$). The total momenta from the top down are $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$, $(2,0,0)\frac{\pi}{L}$ and 0. Here the $x$-axis represents $t_{\mathrm{min}}$ for the usual fit, and $t_0$ for the GEVP fit. In the legend of the lower right panel, 2D and 3D indicate a $2\times2$ and $3\times3$ GEVP matrix.

The $t_{\mathrm{min}}$ or $t_0$ dependence of the fitted ground state energy from the GEVP and the usual fit for the $\pi\pi_{I=2}$ (left) and $\pi\pi_{I=0}$ (right) channel with $t_{\mathrm{max}}=20$ ($I=2$) and 15 ($I=0$). The total momenta from the top down are $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$, $(2,0,0)\frac{\pi}{L}$ and 0. Here the $x$-axis represents $t_{\mathrm{min}}$ for the usual fit, and $t_0$ for the GEVP fit. In the legend of the lower right panel, 2D and 3D indicate a $2\times2$ and $3\times3$ GEVP matrix.

The $t_{\mathrm{min}}$ or $t_0$ dependence of the fitted ground state energy from the GEVP and the usual fit for the $\pi\pi_{I=2}$ (left) and $\pi\pi_{I=0}$ (right) channel with $t_{\mathrm{max}}=20$ ($I=2$) and 15 ($I=0$). The total momenta from the top down are $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$, $(2,0,0)\frac{\pi}{L}$ and 0. Here the $x$-axis represents $t_{\mathrm{min}}$ for the usual fit, and $t_0$ for the GEVP fit. In the legend of the lower right panel, 2D and 3D indicate a $2\times2$ and $3\times3$ GEVP matrix.

The $t_{\mathrm{min}}$ or $t_0$ dependence of the fitted ground state energy from the GEVP and the usual fit for the $\pi\pi_{I=2}$ (left) and $\pi\pi_{I=0}$ (right) channel with $t_{\mathrm{max}}=20$ ($I=2$) and 15 ($I=0$). The total momenta from the top down are $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$, $(2,0,0)\frac{\pi}{L}$ and 0. Here the $x$-axis represents $t_{\mathrm{min}}$ for the usual fit, and $t_0$ for the GEVP fit. In the legend of the lower right panel, 2D and 3D indicate a $2\times2$ and $3\times3$ GEVP matrix.

The $t_{\mathrm{min}}$ or $t_0$ dependence of the fitted ground state energy from the GEVP and the usual fit for the $\pi\pi_{I=2}$ (left) and $\pi\pi_{I=0}$ (right) channel with $t_{\mathrm{max}}=20$ ($I=2$) and 15 ($I=0$). The total momenta from the top down are $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$, $(2,0,0)\frac{\pi}{L}$ and 0. Here the $x$-axis represents $t_{\mathrm{min}}$ for the usual fit, and $t_0$ for the GEVP fit. In the legend of the lower right panel, 2D and 3D indicate a $2\times2$ and $3\times3$ GEVP matrix.

The $t_{\mathrm{min}}$ or $t_0$ dependence of the fitted ground state energy from the GEVP and the usual fit for the $\pi\pi_{I=2}$ (left) and $\pi\pi_{I=0}$ (right) channel with $t_{\mathrm{max}}=20$ ($I=2$) and 15 ($I=0$). The total momenta from the top down are $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$, $(2,0,0)\frac{\pi}{L}$ and 0. Here the $x$-axis represents $t_{\mathrm{min}}$ for the usual fit, and $t_0$ for the GEVP fit. In the legend of the lower right panel, 2D and 3D indicate a $2\times2$ and $3\times3$ GEVP matrix.

The $t_{\mathrm{min}}$ or $t_0$ dependence of the fitted ground state energy from the GEVP and the usual fit for the $\pi\pi_{I=2}$ (left) and $\pi\pi_{I=0}$ (right) channel with $t_{\mathrm{max}}=20$ ($I=2$) and 15 ($I=0$). The total momenta from the top down are $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$, $(2,0,0)\frac{\pi}{L}$ and 0. Here the $x$-axis represents $t_{\mathrm{min}}$ for the usual fit, and $t_0$ for the GEVP fit. In the legend of the lower right panel, 2D and 3D indicate a $2\times2$ and $3\times3$ GEVP matrix.

The $t_{\mathrm{min}}$ or $t_0$ dependence of the fitted ground state energy from the GEVP and the usual fit for the $\pi\pi_{I=2}$ (left) and $\pi\pi_{I=0}$ (right) channel with $t_{\mathrm{max}}=20$ ($I=2$) and 15 ($I=0$). The total momenta from the top down are $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$, $(2,0,0)\frac{\pi}{L}$ and 0. Here the $x$-axis represents $t_{\mathrm{min}}$ for the usual fit, and $t_0$ for the GEVP fit. In the legend of the lower right panel, 2D and 3D indicate a $2\times2$ and $3\times3$ GEVP matrix.

The $t_{\mathrm{min}}$ or $t_0$ dependence of the fitted ground state energy from the GEVP and the usual fit for the $\pi\pi_{I=2}$ (left) and $\pi\pi_{I=0}$ (right) channel with $t_{\mathrm{max}}=20$ ($I=2$) and 15 ($I=0$). The total momenta from the top down are $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$, $(2,0,0)\frac{\pi}{L}$ and 0. Here the $x$-axis represents $t_{\mathrm{min}}$ for the usual fit, and $t_0$ for the GEVP fit. In the legend of the lower right panel, 2D and 3D indicate a $2\times2$ and $3\times3$ GEVP matrix.

The $t_{\mathrm{min}}$ or $t_0$ dependence of the fitted ground state energy from the GEVP and the usual fit for the $\pi\pi_{I=2}$ (left) and $\pi\pi_{I=0}$ (right) channel with $t_{\mathrm{max}}=20$ ($I=2$) and 15 ($I=0$). The total momenta from the top down are $(2,2,2)\frac{\pi}{L}$, $(2,2,0)\frac{\pi}{L}$, $(2,0,0)\frac{\pi}{L}$ and 0. Here the $x$-axis represents $t_{\mathrm{min}}$ for the usual fit, and $t_0$ for the GEVP fit. In the legend of the lower right panel, 2D and 3D indicate a $2\times2$ and $3\times3$ GEVP matrix.

A graph of our results for the seven phase shifts for $I=0$ and $I=2$ as a function of energy. Shown also on the plot are the corresponding dispersive results~\cite{Colangelo:2001df}. Note: the errors are not shown for the dispersive results.

The overlap amplitudes between the $\pi\pi^{A_1}(111,111)$ and $\pi\pi^{T_2}(111,111)$ operators in the isospin $I=2$ (left) and $I=0$ (right) channels. The overlaps amplitudes are consistent with zero at all time separations which implies negligible cubic symmetry breaking for the $\pi\pi(111,111)$ interpolating operators.

The results for the $I=2$ phase shifts together with the corresponding dispersive results~\cite{Colangelo:2001df} that are shown in Fig.~\ref{fig:final}, but here with an expanded scale.

The overlap amplitudes between the $\pi\pi^{A_1}(111,111)$ and $\pi\pi^{T_2}(111,111)$ operators in the isospin $I=2$ (left) and $I=0$ (right) channels. The overlaps amplitudes are consistent with zero at all time separations which implies negligible cubic symmetry breaking for the $\pi\pi(111,111)$ interpolating operators.

Upper: the overlap amplitudes between the normalized $\pi\pi^{T_2}(111,111)$ operator and the normalized $\pi\pi^{A_1}(311,311)$ operator in the isospin $I=2$ (left) and $I=0$ (right) channels. Lower: the overlap amplitude between the normalized $\sigma$ operator and the normalized $\pi\pi^{T_2}(111,111)$ operator. All overlap amplitudes are consistent with zero with errors more than two orders of magnitude smaller than one which implies a negligible cubic symmetry breaking for these $\pi\pi(311,311)$ and $\sigma$ interpolating operators.

Upper: the overlap amplitudes between the normalized $\pi\pi^{T_2}(111,111)$ operator and the normalized $\pi\pi^{A_1}(311,311)$ operator in the isospin $I=2$ (left) and $I=0$ (right) channels. Lower: the overlap amplitude between the normalized $\sigma$ operator and the normalized $\pi\pi^{T_2}(111,111)$ operator. All overlap amplitudes are consistent with zero with errors more than two orders of magnitude smaller than one which implies a negligible cubic symmetry breaking for these $\pi\pi(311,311)$ and $\sigma$ interpolating operators.

Upper: the overlap amplitudes between the normalized $\pi\pi^{T_2}(111,111)$ operator and the normalized $\pi\pi^{A_1}(311,311)$ operator in the isospin $I=2$ (left) and $I=0$ (right) channels. Lower: the overlap amplitude between the normalized $\sigma$ operator and the normalized $\pi\pi^{T_2}(111,111)$ operator. All overlap amplitudes are consistent with zero with errors more than two orders of magnitude smaller than one which implies a negligible cubic symmetry breaking for these $\pi\pi(311,311)$ and $\sigma$ interpolating operators.

A graph of our results for the seven phase shifts for $I=0$ and $I=2$ as a function of energy. Shown also on the plot are the corresponding dispersive results~\cite{Colangelo:2001df}. Note: the errors are not shown for the dispersive results.

The results for the $I=2$ phase shifts together with the corresponding dispersive results~\cite{Colangelo:2001df} that are shown in Fig.~\ref{fig:final}, but here with an expanded scale.