Domain wall QCD with near-physical pions

We present physical results for a variety of light hadronic quantities obtained via a combined analysis of three $2+1$ flavour domain wall fermion ensemble sets. For two of our ensemble sets we used the Iwasaki gauge action with $\beta =2.13$ ($a^{-1}=1.75(4)\mathrm{GeV}$) and $\beta =2.25$ ($a^{-1}=2.31(4)\mathrm{GeV}$) and lattice sizes of ${24}^3\times 64$ and ${32}^3\times 64$ respectively, with unitary pion masses in the range 293(5)–417(10) MeV. The extent $L_s$ for the 5th dimension of the domain wall fermion formulation is $L_s=16$ in these ensembles. In this analysis we include a third ensemble set that makes use of the novel $\mathrm{Iwasaki}+\mathrm{DSDR}$ (dislocation suppressing determinant ratio) gauge action at $\beta =1.75$ ($a^{-1}=1.37(1)\mathrm{GeV}$) with a lattice size of ${32}^3\times 64$ and $L_s=32$ to reach down to partially-quenched pion masses as low as 143(1) MeV and a unitary pion mass of 171(1) MeV, while retaining good chiral symmetry and topological tunneling. We demonstrate a significant improvement in our control over the chiral extrapolation, resulting in much improved continuum predictions for the above quantities. The main results of this analysis include the pion and kaon decay constants, $f_{\pi }=127(3{)}_{\mathrm{stat}}(3{)}_{\mathrm{sys}}\mathrm{MeV}$ and $f_K=152(3{)}_{\mathrm{stat}}(2{)}_{\mathrm{sys}}\mathrm{MeV}$ respectively ($f_K/f_{\pi }=1.199(12{)}_{\mathrm{stat}}(14{)}_{\mathrm{sys}}$); the average up/down quark mass and the strange-quark mass in the $\overline{\mathrm{MS}}$-scheme at 3 GeV, $m_{\mathrm{ud}}(\overline{\mathrm{MS}},3\mathrm{GeV})=3.05(8{)}_{\mathrm{stat}}(6{)}_{\mathrm{sys}}\mathrm{MeV}$ and $m_s(\overline{\mathrm{MS}},3\mathrm{GeV})=83.5(1.7{)}_{\mathrm{stat}}(1.1{)}_{\mathrm{sys}}$; the neutral kaon mixing parameter in the $\overline{\mathrm{MS}}$-scheme at 3 GeV, $B_K(\overline{\mathrm{MS}},3\mathrm{GeV})=0.535(8{)}_{\mathrm{stat}}(13{)}_{\mathrm{sys}}$, and in the RGI scheme, ${\widehat{B}}_K=0.758(11{)}_{\mathrm{stat}}(19{)}_{\mathrm{sys}}$; and the Sommer scales $r_1=0.323(8{)}_{\mathrm{stat}}(4{)}_{\mathrm{sys}}\mathrm{fm}$ and $r_0=0.480(10{)}_{\mathrm{stat}}(4{)}_{\mathrm{sys}}$ ($r_1/r_0=0.673(11{)}_{\mathrm{stat}}(3{)}_{\mathrm{sys}}$). We also obtain values for the SU(2) chiral perturbation theory effective couplings, $\overline{l_3}=2.91(23{)}_{\mathrm{stat}}(7{)}_{\mathrm{sys}}$ and $\overline{l_4}=3.99(16{)}_{\mathrm{stat}}(9{)}_{\mathrm{sys}}$.

Figure 1

Monte Carlo evolution of the average plaquette (top), topological charge (middle), and light-quark pseudoscalar density (bottom) on the $m_l=0.001$ (left) and $m_l=0.0042$ (right) ensembles.

Figure 2

Topological charge distributions for the $m_l=0.001$ (left) and $m_l=0.0042$ (right) ensembles.

Figure 3

The integrated autocorrelation time is shown for the average plaquette, topological charge, the chiral condensate and pseudoscalar density for the light and heavy quark species (labelled “l” and “h” respectively), and the pseudoscalar two-point function at $t=20$, as a function of the upper bound on the integral ${\Delta }_{\mathrm{cut}}$, using data from the $m_l=0.001$ (top) and $m_l=0.0042$ (bottom) ensembles. For those plots on the left we estimated the errors by binning the set of correlations between measurements at fixed MD time separation (the first strategy discussed in the text), and in those on the right we block over the data and measure the correlation between blocks (the second strategy). We chose bin sizes of 25 and 20 on the lighter and heavier ensembles respectively. The pseudoscalar two-point function was only measured every 8 MD time units, hence for both methods we bin these data with a bin size of 24 MD time units. In the right-hand plots the data have been shifted slightly for clarity.

Figure 4

The chiral extrapolation of $m_{\mathrm{res}}^{\prime }$ over the unitary data points at the simulated strange-quark mass, with $m_{\mathrm{res}}$ in the chiral limit denoted by the brown square point (left); and the strange-quark mass dependence of $m_{\mathrm{res}}$ in the chiral limit (right).

Figure 5

Effective unitary pion masses on the $m_l=0.001$ ensemble from the PP LW correlator (top left), PP WW correlator (top right), AP LW correlator (center left), AP WW (center right) and AA LW correlator (bottom). Note the different vertical scale for the WW correlators. The horizontal bands represent the result for the mass from a simultaneous fit.

Figure 6

Effective unitary kaon masses on the $m_l=0.001$ ensemble from the PP LW correlator (top left), PP WW correlator (top right), AP LW correlator (center left), AP WW (center right) and AA LW correlator (bottom). Note the different vertical scale for the WW correlators. The horizontal bands represent the result for the mass from a simultaneous fit.

Figure 7

$Z_V/Z_{\mathcal{V}}$ (top left) and $Z_A/Z_{\mathcal{A}}$ (top right) as a function of time, calculated with unitary quarks on the $m_l=0.001$ ensemble. The bottom figure shows the chiral extrapolation of $Z_V/Z_{\mathcal{V}}$ and $Z_A/Z_{\mathcal{A}}$. In these plots the ratios have been abbreviated to $Z_V$ and $Z_A$.

Figure 8

The left panel displays the fit to the $\Omega$ baryon mass with valence strange mass $m_x=0.045$ on the $m_l=0.001$, $m_h=0.045$ ensemble on the 32ID lattice, showing the quality of the fit with our box source. The right panel shows the $B_{xy}$ matrix element with $m_x=m_y=0.001$ as a function of time on the same ensemble.

Figure 9

The left panel shows the effective potential of the Wilson loops with a spatial extent of $r=2.45$ on the $m_l=0.001$ ensemble at the simulated strange-quark mass, overlaid by the fit to the range $t=3–8$. The right panel shows the static inter-quark potential $V(r)$ on this ensemble, again at the simulated strange-quark mass, as a function of the spatial extent of the Wilson loops, overlaid by the fit to the Cornell form over the range $r=2.00–9.00$.

Figure 10

Ratios of various dimensionless combinations of observables between the 32I and 32ID ensemble sets. The combination of physical quantities is given above or below the corresponding point. A ratio of unity indicates perfect scaling between the two ensemble sets.

Figure 11

Simulated quark masses on each of our three ensemble sets brought into a common normalization with the bare quark masses on our 32I ensemble set using the scaling factors determined in Sec. 5. The top panel shows the light quark mass regime and the bottom panel the heavy quark mass regime. Circular points are used to mark the unitary masses and square points the partially-quenched masses. The physical up/down and strange quark masses are marked with dashed lines.

Figure 12

Global fits obtained using NLO SU(2) chiral perturbation theory with finite-volume corrections for the pion mass (top) and $f_{\pi }$ (bottom) on the 32ID ensembles. Here the left-hand plot of each pair show the data at the simulated strange-quark mass and the corresponding fit curves on the $m_l=0.001$ ensemble, and the right-hand plots those on the $m_l=0.0042$ ensemble. The plots of the pion mass have $m_{\pi }^2/({\tilde{m}}_x+{\tilde{m}}_y)$ on the ordinate axis, a quantity used traditionally to emphasize the chiral curvature of the data.

Figure 13

Global fits using the analytic ansatz with finite-volume corrected data for the pion mass (top) and $f_{\pi }$ (bottom) on the 32ID ensembles. Here the left-hand plot of each pair show the data at the simulated strange-quark mass and the corresponding fit curves on the $m_l=0.001$ ensemble, and the right-hand plots those on the $m_l=0.0042$ ensemble. The plots of the pion mass have $m_{\pi }^2/({\tilde{m}}_x+{\tilde{m}}_y)$ on the ordinate axis, a quantity used traditionally to emphasize the chiral curvature of the data. The circular points are those included in the fit, and the diamond points those excluded by the cut on data with $m_{\pi }\ge 260\mathrm{MeV}$.

Figure 14

Global fits obtained using NLO SU(2) chiral perturbation theory with finite-volume corrections for the square of the kaon mass (left) and $f_K$ (right) on the 32ID ensembles.

Figure 15

The chiral extrapolation of the pion decay constant (left) and Omega baryon mass (right) using the analytic and ChPTFV ansätze. Overlaying these curves we have plotted the unitary data extrapolated to the continuum limit using the $a^2$ dependence of our fit forms. The lighter-shaded points were corrected using the analytic fit form, and the darker points by the ChPTFV form. Here the circular points are those included in the fit, and the diamond points are those excluded by the cuts at 350 MeV (ChPTFV) and 260 MeV (analytic). The upper and lower square points show the continuum predictions obtained using the ChPTFV and analytic ansätze respectively. Note that for $f_{\pi }$, the analytic fit does not include any unitary data points on the 32I and 24I ensembles as they lie above the pion mass cut (cf. Table 11). Note also that the physical limit of the ${\Omega }^{-}$ mass shows no statistical errors and agrees precisely with its physical value because it is this quantity that we use to determine the lattice scale.

Figure 16

Histograms of the deviation of the fit from the data for each quantity on each of the three ensemble sets (32I top, 24I middle and 32ID bottom) with the analytic (left) and ChPTFV (right) ansätze.

Figure 17

$Z_m$ in the two SMOM schemes at a range of scales on the 24I (left) and 32I (right) ensemble sets.

Figure 18

The continuum extrapolations of $Z_{ml}$ (left) and $Z_{mh}$ (right) in the RI/SMOM at 1.4 GeV.

Figure 19

Histograms of the deviation of the fit from the data for $B_K$ on each of the three ensemble sets (32I top, 24I middle and 32ID bottom) with the analytic (left) and ChPTFV (right) ansätze.

Figure 20

The analytic (left) and ChPTFV (right) fit curves overlaying the partially-quenched data on the 32ID ensembles at the simulated strange quark mass. The fits were performed to the data set with corresponding pion masses $m_{\pi }<350\mathrm{MeV}$, with the data renormalized in the $\mathrm{SMOM}(\mathrm{q̸},\mathrm{q̸})$ intermediate scheme.

Figure 21

The left figure shows the chiral extrapolation of $B_K$ in the continuum limit, renormalized in the $\mathrm{SMOM}(\mathrm{q̸},\mathrm{q̸})$ scheme at a scale of 1.4 GeV. The circular and diamond-shaped data points in darker shades show the data corrected to the continuum limit using the ChPTFV fit form, and those in lighter shades via the analytic form. The circular points indicate those data included in the fits, and the diamond points those that were not. The upper and lower curves show the analytic and ChPTFV chiral fit forms and the corresponding square data points the extrapolated values at the physical up/down quark mass. All data and curves are shown at the physical strange quark mass. The right figure shows the data at finite-$a$, adjusted to the infinite volume limit and the physical strange quark mass, overlaid by the ChPTFV fit curves at finite-$a$ and the continuum curve shown in the previous plot (shown without error bands for clarity).

Figure 22

Histograms of the deviation of the fit from the data for $r_0$ and $r_1$ over all three ensemble sets, fitting with the analytic (left) and ChPTFV (right) ansätze to the uncut (top) and cut (bottom) data sets.

Figure 23

The chiral extrapolation of $r_0$ (top) and $r_1$ (bottom) using the analytic and ChPTFV ansätze. The plots on the left show the fits to the full data set and those on the right to the cut data sets. We have overlayed the fit curves with the data points corrected to the continuum limit and physical strange quark mass using each of the aforementioned fit functions; those points shown in bold colors were corrected using the ChPTFV fits and those in pastel colors using the analytic fits. The circular data points are those included in the fits and the diamond points those that were not. The square points show the predicted value at the physical point.

Figure 24

The chiral extrapolation of $r_0$ (left) and $r_1$ (right) using the ChPTFV ansatz applied to the cut data set. Here we have overlayed the fit curves at finite lattice spacing (dashed lines) with the raw data points corrected to the physical strange quark mass. We also show the continuum fit curve (solid line) and the physical point (square). As before the circular data points are those included in the fits and the diamond points those that were not.

Figure 25

The 12 lowest eigenmodes of the 4D Wilson-Dirac operator as a function of the domain wall mass $-M_5$ for positive $M_5$, measured on a single representative configuration of each of three ${16}^3\times 8\times 32$ ensembles; the upper plot on a $\beta =1.95$ ensemble with the Iwasaki gauge action, the lower-left on a $\beta =1.95$ ensemble with the $\mathrm{Iwasaki}+\mathrm{DSDR}$ gauge action, and the lower-right on a $\beta =1.75$ ensemble with the $\mathrm{Iwasaki}+\mathrm{DSDR}$ gauge action.

Figure 26

The fractional difference between the local vector and axial-vector operators, ${\Lambda }_V$ and ${\Lambda }_A$ (top), and also between the scalar and pseudoscalar operators, ${\Lambda }_S$ and ${\Lambda }_P$ (bottom), as a function of the square of the momentum in lattice units. These values were obtained by measuring amputated bilinear vertex functions at a scale defined by the momentum of the incoming quark propagators. The lower-right figure is plotting with logarithmic axes and is overlayed by a line with the expected $1/(ap{)}^6$ dependence.