SU(2) chiral perturbation theory low-energy constants from $2\mathbf{+}1$ flavor staggered lattice simulations

We extract the next-to-leading-order low-energy constants ${\overline{\ell }}_3$ and ${\overline{\ell }}_4$ of SU(2) chiral perturbation theory, based on precise lattice data for the pion mass and decay constant on ensembles generated by the Wuppertal-Budapest Collaboration for QCD thermodynamics. These ensembles feature $2+1$ flavors of two-fold stout-smeared dynamical staggered fermions combined with Symanzik glue, with pion masses varying from 135 to 435 MeV, lattice scales between 0.7 and 2.0 GeV, while $m_s$ is kept fixed at its physical value. Moderate taste splittings and the scale being set through the pion decay constant allow us to restrict ourselves to the taste pseudoscalar state and to use formulas from continuum chiral perturbation theory. Finally, by dropping the data points near 135 MeV from the fits, we can explore the range of pion masses that is needed in SU(2) chiral perturbation theory to reliably extrapolate to the physical point.

Figure 1

Effective masses (left panel) and amplitudes (right panel) from the pion correlator at $\beta =3.75$, $a{m}_l=0.00172$, $a{m}_s=0.048$, including the plateau fits (blue lines).

Figure 2

Fits to the ratio $(a{M}_{\pi }{)}^2/(a{f}_{\pi }{)}^2$ (left) and the decay constant $a{f}_{\pi }$ (right) at $\beta =3.85$ to obtain the physical light quark mass $a{m}^{\mathrm{phys}}$ and the lattice scale $1/a$. Shown are quadratic (solid blue lines) and rational (dashed red lines) fits which include the black data points; the green data points were excluded from the fits. Error bands are indicated by dotted lines. The dashed gray lines mark the physical value of the ratio and the physical light quark mass extracted therefrom.

Figure 3

Landscape plot of the simulated pion masses $M_{\pi }^2$ and lattice spacings $a$. The horizontal dashed lines indicate the various mass cuts applied in our chiral fits. The full blue line indicates our preferred fit range ($a<0.12\mathrm{fm}$, $M_{\pi }<240\mathrm{MeV}$); see text for details.

Figure 4

Combined fit for all lattice scales, $135\mathrm{MeV}\le M_{\pi }\le 390\mathrm{MeV}$. Left panel: meson decay constant, right panel: squared meson mass divided by quark-mass ratio. Points marked by black symbols are included in the fit; those marked by green symbols are excluded. The physical values are indicated by dashed gray lines.

Figure 5

Results for the fitted parameters (top and middle panels) and ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ (bottom panel) from NLO-ChPT fits without constraints using different mass ranges but including all lattice spacings. The solid, dashed, and dashed-dotted blue lines for the fit parameters denote the central value, statistical, and total (statistical plus systematic) error bands, respectively, from our preferred unconstrained fit (cf. left column of Table ).

Figure 6

Results for the parameters ${\overline{\ell }}_3$, ${\overline{\ell }}_4$ (top panels) and the ratio $f_{\pi }^{\mathrm{phys}}/f$ (bottom panel) from NLO-ChPT fits without constraints using different mass ranges but including all lattice spacings. The solid, dashed, and dashed-dotted blue lines for the fit parameters denote the central value, statistical, and total (statistical plus systematic) error bands, respectively, from our preferred unconstrained fit (cf. left column of Table ).

Figure 7

Fit parameters and ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ from NLO-ChPT fits without constraints to two different mass ranges, where several lattice spacings have been excluded. Vertical dashed lines group points where no, one, two, three, or four (from left to right) lattice spacings have been excluded. The solid, dashed and dashed-dotted horizontal blue lines denote the central value, statistical, and total (statistical plus systematic) error bands, respectively, from our preferred unconstrained fit (cf. left column of Table ).

Figure 8

Results for the fitted parameters (top and middle panels) and ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ (bottom panel) from NLO-ChPT fits without constraints, using different mass ranges and including only lattice spacings $1/a>1.6\mathrm{GeV}$. The solid, dashed, and dashed-dotted blue lines for the fit parameters denote the central value, statistical, and total (statistical plus systematic) error bands, respectively, from our preferred unconstrained fit (cf. left column of Table ).

Figure 9

Results for the parameters ${\overline{\ell }}_3$, ${\overline{\ell }}_4$ (top panels) and the ratio $f_{\pi }^{\mathrm{phys}}/f$ (bottom panel) from NLO-ChPT fits without constraints, using different mass ranges and including only lattice spacings $1/a>1.6\mathrm{GeV}$. The solid, dashed, and dashed-dotted blue lines for the fit parameters denote the central value, statistical, and total (statistical plus systematic) error bands, respectively, from our preferred unconstrained fit (cf. left column of Table ).

Figure 10

Combined fit for lattice scales $1/a>1.6\mathrm{GeV}$ and meson masses $135\mathrm{MeV}\le M_{\pi }\le 240\mathrm{MeV}$. Left panels: meson decay constant, right panels: squared meson mass divided by quark-mass ratio. The top panels show only the points, which have been included in the fit, while the bottom panels show all points. There, points marked by black symbols are included in the fit; those marked by green symbols are excluded. The physical values are indicated by dashed gray lines.

Figure 11

Combined fit for lattice scales $1/a>1.6\mathrm{GeV}$ and meson masses $195\mathrm{MeV}\le M_{\pi }\le 275\mathrm{MeV}$ excluding the nearly physical points. Left panels: meson decay constant, right panels: squared meson mass divided by the quark-mass ratio. The top panels show only the points, which have been included in the fit, while the bottom panels show all points. There, points marked by black symbols are included in the fit; those marked by green symbols are excluded. The physical values are indicated by dashed gray lines.

Figure 12

Extrapolated $f_{\pi }^{\mathrm{phys}}$ (left panel) and $(M_{\pi }^{\mathrm{phys}}{)}^2$ (right panel) from NLO-ChPT fits without constraints, using different mass ranges and including only lattice spacings $1/a>1.6\mathrm{GeV}$. The solid and dashed blue lines denote the central value and total error bands, respectively, as quoted by Refs. 18, 19.

Figure 13

Results for the fitted parameters (top panels) and ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ (bottom right panel) from parameter-reduced NLO-ChPT fits, using different mass ranges and including only lattice spacings $1/a>1.6\mathrm{GeV}$. In this setup ${\overline{\ell }}_3$, ${\overline{\ell }}_4$, and $f_{\pi }^{\mathrm{phys}}/f$ are derived quantities (see the text). The solid, dashed, and dashed-dotted blue lines for the fit parameters denote the central value, statistical, and total (statistical plus systematic) error bands, respectively, from our preferred parameter-reduced fit, cf. right column of Table .

Figure 14

Fit parameters and ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ from parameter-reduced NLO-ChPT fits to two different mass ranges, where several lattice spacings have been excluded. Vertical dashed lines group points where no, one, two, three, or four (from left to right) lattice spacings have been excluded. The solid, dashed, and dashed-dotted horizontal blue lines denote the central value, statistical, and total (statistical plus systematic) error bands, respectively, from our preferred parameter-reduced fit, cf. right column of Table .

Figure 15

Results for the fitted parameters and ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ from NNLO ChPT fits without constraints using different mass ranges and including lattice spacings $1/a>1.6\mathrm{GeV}$. The solid, dashed, and dashed-dotted blue lines for the fit parameters denote the central value, statistical, and total (statistical plus systematic) error bands, respectively, from our NLO fits, cf. Eqs. (16, 17, 18, 19, 20, 21, 22).

Figure 16

Combined NNLO fit for lattice scales $1/a>1.6\mathrm{GeV}$ and meson masses $135\mathrm{MeV}\le M_{\pi }\le 390\mathrm{MeV}$. Left panel: meson decay constant, right panel: squared meson mass divided by the quark-mass ratio. The solid blue lines show the complete (up to NNLO) fit, whereas the dashed red lines show the $\mathrm{LO}+\mathrm{NLO}$ contribution of the full NNLO fit, and the dashed-dotted green lines show the LO contribution. Only data points included in the fit range are depicted in the plots; the physical values are marked by dashed gray lines.

Figure 17

Results for the fitted parameters and ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ from NNLO-ChPT fits with a prior for ${\Lambda }_{12}$ using different mass ranges and including lattice spacings $1/a>1.6\mathrm{GeV}$. The solid, dashed, and dashed-dotted blue lines for the fit parameters denote the central value, statistical, and total (statistical plus systematic) error bands, respectively, from our NLO fits, cf. Eqs. (16, 17, 18, 19, 20, 21, 22). The prior on ${\Lambda }_{12}$ and its width are indicated by the shaded gray area in the respective panel.

Figure 18

Results for the fitted parameters and ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ from NNLO-ChPT fits with priors for ${\Lambda }_{12}$, $k_{M^2}$, and $k_f$ using different mass ranges and including lattice spacings $1/a>1.6\mathrm{GeV}$. The solid, dashed, and dashed-dotted blue lines for the fit parameters denote the central value, statistical, and total (statistical plus systematic) error bands, respectively, from our NLO fits, cf. Eqs. (16, 17, 18, 19, 20, 21, 22). The priors and their widths are indicated by the shaded gray areas in the respective panels.

Figure 19

Combined NNLO fit for lattice scales $1/a>1.6\mathrm{GeV}$ and meson masses $135\mathrm{MeV}\le M_{\pi }\le 340\mathrm{MeV}$ using a prior for ${\Lambda }_{12}$. Left panel: meson decay constant, right panel: squared meson mass divided by the quark-mass ratio. The solid blue lines show the complete (up to NNLO) fit, whereas the dashed red lines show the $\mathrm{LO}+\mathrm{NLO}$ contribution of the full NNLO fit, and the dashed-dotted green lines show the LO contribution. Only data points included in the fit range are depicted in the plots; the physical values are marked by dashed gray lines.

Figure 20

Combined NNLO fit for lattice scales $1/a>1.6\mathrm{GeV}$ and meson masses $135\mathrm{MeV}\le M_{\pi }\le 340\mathrm{MeV}$ using priors for ${\Lambda }_{12}$, $k_{M^2}$, and $k_f$. Left panel: meson decay constant, right panel: squared meson mass divided by the quark-mass ratio. The solid blue lines show the complete (up to NNLO) fit, whereas the dashed red lines show the $\mathrm{LO}+\mathrm{NLO}$ contribution of the full NNLO fit, and the dashed-dotted green lines show the LO contribution. Only data points included in the fit range are depicted in the plots; the physical values are marked by dashed gray lines.

Figure 21

Results for the LECs ${\overline{\ell }}_3$ (left panels) and ${\overline{\ell }}_4$ (right panels) from NNLO fits to ensembles with $1/a>1.6\mathrm{GeV}$ and different mass ranges. The top, middle, and bottom panels show the results using no priors; a prior for ${\Lambda }_{12}$; and priors for ${\Lambda }_{12}$, $k_{M^2}$, and $k_f$, respectively. The solid, dashed, and dashed dotted blue lines indicate the central value, statistical, and total (statistical plus systematic) error bands from our NLO fits, cf. Eqs. (16, 17, 18, 19, 20, 21, 22).

Figure 22

Results for the ratio $f_{\pi }^{\mathrm{phys}}/f$ (left panels) and the LEC ${\overline{\ell }}_{12}$ (right panels) from NNLO fits to ensembles with $1/a>1.6\mathrm{GeV}$ and different mass ranges. The top, middle, and bottom panels show the results using no priors; a prior for ${\Lambda }_{12}$; and priors for ${\Lambda }_{12}$, $k_{M^2}$, and $k_f$, respectively. The solid, dashed, and dashed dotted blue lines indicate the central value, statistical, and total (statistical plus systematic) error bands from our NLO fits, cf. Eqs. (16, 17, 18, 19, 20, 21, 22). The gray shaded areas indicate the used prior and its width (where applicable).