Physical results from $2+1$ flavor domain wall QCD and SU(2) chiral perturbation theory

We have simulated QCD using $2+1$ flavors of domain wall quarks and the Iwasaki gauge action on a $(2.74\mathrm{fm}{)}^3$ volume with an inverse lattice scale of $a^{-1}=1.729(28)\mathrm{GeV}$. The up and down (light) quarks are degenerate in our calculations and we have used four values for the ratio of light quark masses to the strange (heavy) quark mass in our simulations: 0.217, 0.350, 0.617, and 0.884. We have measured pseudoscalar meson masses and decay constants, the kaon bag parameter $B_K$, and vector meson couplings. We have used SU(2) chiral perturbation theory, which assumes only the up and down quark masses are small, and SU(3) chiral perturbation theory to extrapolate to the physical values for the light quark masses. While next-to-leading order formulas from both approaches fit our data for light quarks, we find the higher-order corrections for SU(3) very large, making such fits unreliable. We also find that SU(3) does not fit our data when the quark masses are near the physical strange quark mass. Thus, we rely on SU(2) chiral perturbation theory for accurate results. We use the masses of the $\Omega$ baryon, and the $\pi$ and $K$ mesons to set the lattice scale and determine the quark masses. We then find $f_{\pi }=124.1(3.6{)}_{\mathrm{stat}}(6.9{)}_{\mathrm{syst}}\mathrm{MeV}$, $f_K=149.6(3.6{)}_{\mathrm{stat}}(6.3{)}_{\mathrm{syst}}\mathrm{MeV}$, and $f_K/f_{\pi }=1.205(0.018{)}_{\mathrm{stat}}(0.062{)}_{\mathrm{syst}}$. Using nonperturbative renormalization to relate lattice regularized quark masses to regularization independent momentum scheme masses, and perturbation theory to relate these to $\overline{\mathrm{MS}}$, we find $m_{ud}^{\overline{\mathrm{MS}}}(2\mathrm{GeV})=3.72(0.16{)}_{\mathrm{stat}}(0.33{)}_{\mathrm{ren}}(0.18{)}_{\mathrm{syst}}\mathrm{MeV}$, $m_s^{\overline{\mathrm{MS}}}(2\mathrm{GeV})=107.3(4.4{)}_{\mathrm{stat}}(9.7{)}_{\mathrm{ren}}(4.9{)}_{\mathrm{syst}}\mathrm{MeV}$, and ${\tilde{m}}_{ud}:{\tilde{m}}_s=1:28.8(0.4{)}_{\mathrm{stat}}(1.6{)}_{\mathrm{syst}}$. For the kaon bag parameter, we find $B_K^{\overline{\mathrm{MS}}}(2\mathrm{GeV})=0.524(0.010{)}_{\mathrm{stat}}(0.013{)}_{\mathrm{ren}}(0.025{)}_{\mathrm{syst}}$. Finally, for the ratios of the couplings of the vector mesons to the vector and tensor currents ($f_V$ and $f_V^T$, respectively) in the $\overline{\mathrm{MS}}$ scheme at 2 GeV we obtain $f_{\rho }^T/f_{\rho }=0.687(27)$; $f_{K^{*}}^T/f_{K^{*}}=0.712(12)$, and $f_{\varphi }^T/f_{\varphi }=0.750(8)$.

Figure 1

(a) One-loop correction to $f_K$, (b) one-loop wave-function-like contribution proportional to $g^2$, (c) diagram contributing to the one-loop wave function (and mass) renormalization. The gray square denotes a flavor-changing axial vector current operator, the black bullets in (b) represent the $P{P}^{*}\pi$ vertex, and the gray circle in (c) denotes a $KK\pi \pi$ vertex from $L_{\pi K}^{(1)}$ in Eq. (22).

Figure 2

One-loop contribution to $B_K$. The gray box denotes the insertion of the $KK\pi \pi$ operator ${\mathcal{O}}_{22}$ as defined in the text.

Figure 3

The left graphs show topological charge, as measured on each ensemble every 5th unit of molecular-dynamics time. The right panels show normalized histograms of toplogical charge. Notice the width of the histograms decreases as the light quark mass is reduced. For both the left and right graphs, the light dynamical quark mass is $m_l=0.005$, 0.01, 0.02, 0.03, from top to bottom.

Figure 4

Integrated autocorrelation time for 12th time slice of the unitary degenerate pseudoscalar correlator with local sources and sinks $⟨P^{LL},P^{LL}⟩$ measured on the $m_l=0.005$ ensemble. Measurements were done every fifth trajectory in the range 900–1530 and a bin factor of 2 was used in the analysis.

Figure 5

Integrated autocorrelation time for 12th time slice of the unitary degenerate pseudoscalar correlator with a local sink and a Gaussian source $⟨P^{LL},P^{HH}⟩$ measured on the $m_l=0.005$ ensemble. Measurements were done every twentieth trajectory in the range 500–4500 and a bin factor of 4 was used in the analysis.

Figure 6

Evolution history of the plaquette for the ensemble $m_l=0.02$.

Figure 7

The residual mass as a function of the valence quark mass. The solid line is a linear fit to the four unitary points with $m_l=0.005$, 0.01, 0.02, and 0.03. The star is the extrapolation to $m_l=0$, which we quote as the residual mass in our simulations.

Figure 8

Comparison of the lattice results for the pseudoscalar decay constants on the ${16}^3\times 32$ and ${24}^3\times 64$ lattices with $m_l=0.01$.

Figure 9

Sea quark mass dependence of the decuplet baryon masses for degenerate valence quarks $m_x=0.03$ (circles) and 0.04 (squares). Linear fits are shown by solid lines. Asterisks denote masses extrapolated to the physical value of the bare, light sea quark mass $m_{ud}$.

Figure 10

Combined $\mathrm{SU}(2)\times \mathrm{SU}(2)$ fits for the meson decay constants (left panels) and masses (right panels) at two different values for the light sea quark mass, and with a cut on the valence masses $m_{\mathrm{avg}}\le 0.01$. Points marked by filled symbols were included in the fit, while those with open symbols were excluded.

Figure 11

$\mathrm{SU}(2)\times \mathrm{SU}(2)$ fits for the kaon sector. Left panel: Kaon decay constant. Right panel: Kaon mass squared for $m_y=0.04$. Points with filled symbols were included in the fit, while those with open symbols were excluded.

Figure 12

Interpolation to the physical strange quark mass (diamond) for $m_K^2$ and $f_K$. Square symbols denote measured values and open symbols are not included in the interpolations. Left panel: Kaon decay constant. Right panel: Squared kaon mass.

Figure 13

Comparison of the size of the LO, NLO, and NNLO contributions in the $\mathrm{SU}(2)\times \mathrm{SU}(2)$ fits for unitary degenerate meson masses squared $m_{ll}^2$ (upper left panel) and decay constants $f_{ll}$ (upper right panel) as a function of the light sea quark mass parameter ${\chi }_l$ (for details see Sec. 6c1). The LO contribution is normalized to one. Vertical dashed lines indicate quark masses of $m_l=m_{ud}$, 0.005, 0.01, and 0.02 (from left to right). The lower panels give a comparison of the size of the LO and NLO contributions in kaon ChPT for the kaon mass squared $m_{lh}^2$ (lower left panel) and kaon decay constant $f_{lh}$ (lower right panel) as a function of the light sea quark mass.

Figure 14

Finite-volume correction factor obtained from the $\mathrm{SU}(2)$ fits for the decay constants (left panels) and squared meson masses (right panels) at sea quark masses $m_l=0.005$ (top panels) and 0.01 (bottom panels). The vertical dashed line denotes the lightest (valence) mass used in this analysis ($m_y=0.001$).

Figure 15

Combined $\mathrm{SU}(3)\times \mathrm{SU}(3)$ fits for the meson decay constants (left panels) and masses (right panels) at two different values for the light sea quark mass, with valence mass cut $m_{\mathrm{avg}}\le 0.01$. Points marked by filled symbols were included in the fit, while those with open symbols were excluded.

Figure 16

Combined $\mathrm{SU}(3)\times \mathrm{SU}(3)$ fits for the meson decay constants (left panels) and masses (right panels) at two different values for the light sea quark mass, with valence mass cut $m_{\mathrm{avg}}\le 0.03$. Points marked by filled symbols were included in the fit, while those with open symbols were excluded.

Figure 17

LO and NLO contributions in the unitary meson decay constant fit in SU(3) PQChPT as a function of the light quark mass parameter (left panels) and the heavy quark mass parameter (right panels). In the plots in the top panels the other quark mass parameter was set to our highest value used in the fits ($m_l=0.01$ or $m_h=0.04$), while in those in the bottom panels the remaining quark mass parameter was set to zero ($m_l$ or $m_h=-m_{\mathrm{res}}$). ${\mathrm{NLO}}_l$ denotes the NLO contribution proportional to $\log {\chi }_l$, and ${\mathrm{NLO}}_{lh}$ the NLO contribution proportional to $\log ({\chi }_l+{\chi }_h)/2$. The LO contribution is always normalized to one. Vertical dashed lines indicate quark masses of $m_l=m_{ud}$, 0.005, 0.01, 0.02, 0.03, and 0.04 (from left to right; left panels only show quark masses up to 0.02).

Figure 18

A plot of our measured values for the decay constant, converted to physical units, versus the degenerate valence pseudoscalar mass squared. The data for degenerate quarks are denoted by filled symbols, and for nondegenerate quarks, open symbols are used. The graph shows that we see small effects for nondegenerate quarks. The results of SU(2) and SU(3) partially quenched ChPT fits to our data are shown, and both fits agree well with the data. The unitary SU(2) chiral extrapolation is also given, along with our value of $f$, and two of our data points lie on this curve, as expected. The value of $f$ differs by $\approx 30\%$ from the decay constant measured at $m_{\pi }=420\mathrm{MeV}$. We also plot the SU(3) chiral limit curve, for which the horizontal axis is the unitary meson mass for three degenerate mass quarks. None of our measured values must lie on this line. The large difference between $f_0$ and our measurements is apparent, showing the poor convergence of NLO SU(3) ChPT, with LECs as determined from our data.

Figure 19

Extrapolation of the pseudoscalar bag parameter $B_{xy}$ in the light quark mass $m_x$ to the physical point $m_{ud}$ for $m_y=0.03$ (left panel) and $m_y=0.04$ (right panel). Points with filled symbols are included in the fit, while those marked by open symbols are excluded. For fit parameters see Table 22.

Figure 20

Linear interpolation in $m_y$ of the pseudoscalar bag parameter $B_{udy}$ to the physical strange quark mass $m_s$ (diamond). The open symbol at $m_y=0.02$ was excluded from the interpolation.

Figure 21

Fitting the pseudoscalar bag parameter $B_{xy}$ to the $\mathrm{SU}(3)\times \mathrm{SU}(3)$ $\chi \mathrm{PT}$ formulas with a cut in the average valence mass value of 0.01 (top panels) and 0.02 (bottom panels). Points marked by open symbols are excluded from the fit. The left panels are for light sea quark masses $m_l=0.005$; the right panels for $m_l=0.01$.

Figure 22

Dependence of the fit parameter $B_{\mathrm{PS}}^{\chi }$ (bars) and the extrapolated physical value of $B_K$ (crosses) on the average valence mass cut, when fitting to the $\mathrm{SU}(3)\times \mathrm{SU}(3)$ fit formula. The horizontal lines indicate the result (with statistical error) for $B_K$ obtained from the $\mathrm{SU}(2)\times \mathrm{SU}(2)$ fit.

Figure 23

Comparison of the pseudoscalar bag parameter $B_{xy}$ for valence quark masses $m_x$, $m_y$ measured on ${16}^3\times 32$ (filled symbols) and ${24}^3\times 64$ (open symbols) lattices at $m_l=0.01$, $m_h=0.04$. (Filled symbols are slightly shifted to the right on this plot.)

Figure 24

The diagram illustrating how $\tau$ decays can be used to deduce $f_{\rho }$ and $f_{K^{*}}$.

Figure 25

Chiral extrapolations for $f_{\rho }^T/f_{\rho }$, $f_{K^{*}}^T/f_{K^{*}}$, and $f_{\varphi }^T/f_{\varphi }$, respectively. The broken lines represent a linear fit to the mass behavior, and the solid lines a quadratic fit.