Physical Results from 2+1 Flavor Domain Wall QCD and SU(2) Chiral Perturbation Theory

TL;DR

This study performs 2+1 flavor domain wall QCD simulations on a large volume at a fixed lattice spacing to extract light- and strange-quark observables. By applying SU(2) ChPT to the light sector and SU(2) kaon ChPT for kaons, the authors obtain precise determinations of f_π, f_K, m_ud, m_s, and BK, with nonperturbative renormalization linking lattice to MS-bar masses. They demonstrate that SU(3) ChPT at NLO converges poorly near the physical strange mass, and thus rely on SU(2) ChPT for robust extrapolations, reporting B_K = 0.524(10)(13)_(ren)(25)_(syst) at 2 GeV and f_K/f_π = 1.205(18)(62). The vector-meson couplings f_V^T/f_V are computed to MS-bar at 2 GeV, yielding f_ρ^T/f_ρ = 0.687(27), f_{K^*}^T/f_{K^*} = 0.717(12), and f_φ^T/f_φ = 0.750(8). Overall, the work provides high-precision lattice QCD inputs with controlled chiral and finite-volume systematics, informing CKM phenomenology and hadronic matrix element calculations.

Abstract

We have simulated QCD using 2+1 flavors of domain wall quarks on a $(2.74 {\rm fm})^3$ volume with an inverse lattice scale of $a^{-1} = 1.729(28)$ 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 formulae 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 $Ω$ baryon, and the $π$ and $K$ mesons to set the lattice scale and determine the quark masses. We then find $f_π= 124.1(3.6)_{\rm stat}(6.9)_{\rm syst} {\rm MeV}$, $f_K = 149.6(3.6)_{\rm stat}(6.3)_{\rm syst} {\rm MeV}$ and $f_K/f_π= 1.205(0.018)_{\rm stat}(0.062)_{\rm syst}$. Using non-perturbative renormalization to relate lattice regularized quark masses to RI-MOM masses, and perturbation theory to relate these to $\bar{\rm MS}$ we find $ m_{ud}^{\bar{\rm MS}}(2 {\rm GeV}) = 3.72(0.16)_{\rm stat}(0.33)_{\rm ren}(0.18)_{\rm syst} {\rm MeV}$ and $m_{s}^{\bar{\rm MS}}(2 {\rm GeV}) = 107.3(4.4)_{\rm stat}(9.7)_{\rm ren}(4.9)_{\rm syst} {\rm MeV}$.

Figures (25)

• Figure 1: (a) One loop correction to $f_K$; (b) one-loop wavefunction-like contribution proportional to $g^2$; (c) diagram contributing to the one-loop wavefunction (and mass) renormalization. The grey square denotes a flavor-changing axial-vector current operator, the black bullets in (b) represent the $P P^*\pi$ vertex and the grey circle in (c) denotes a $KK\pi\pi$ vertex from $L_{\pi K}^{(1)}$ in Eq. (\ref{['eq:lkpi1']}) .
• Figure 2: One-loop contribution to $B_K$ . The grey 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 normalised 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 timeslice of the unitary degenerate pseudoscalar correlator with local sources and sinks, $\langle P^{LL},P^{LL} \rangle$, measured on the $m_l=0.005$ ensemble. Measurements were done every fifth trajectory in the range 900-1530 and a bin factor of two was used in the analysis.
• Figure 5: Integrated autocorrelation time for 12th timeslice of the unitary degenerate pseudoscalar correlator with a local sink and a Gaussian source, $\langle P^{LL},P^{HH} \rangle$, measured on the $m_l=0.005$ ensemble. Measurements were done every twentieth trajectory in the range 500-4500 and a bin factor of four was used in the analysis.