The Spectrum and Scale Setting on 2+1-flavor NME Lattices

TL;DR

The paper presents a comprehensive analysis of thirteen 2+1-flavor Wilson-clover lattice ensembles, focusing on meson and baryon spectra, decay constants, and gradient-flow scales to set the lattice scale and extrapolate to the physical point. It employs HB$ olinebreak[4]\chi$PT at NLO and NNLO for octet baryons, with NNLO favored, and uses $f_{iK}$ to determine the physical flow scales $t_0^{Phy}$ and $w_0^{Phy}$, enabling the calculation of $f_K/f_i^{Phy}$. The study also computes topological observables under gradient flow, analyzes autocorrelations, and reports renormalization constants, thereby validating the ensemble set and laying groundwork for future high-precision QCD calculations. While results for some quantities carry larger uncertainties due to CC fits and limited ensemble coverage, the work demonstrates consistency with FLAG averages and external determinations, and outlines clear paths to improved precision with more ensembles and tuned strange-quark masses. Overall, the paper advances scale setting, spectrum analysis, and CP-violation-related observables in 2+1-flavor lattice QCD, contributing valuable data and methodologies for the lattice community.

Abstract

This paper describes the thirteen ensembles, named NME, generated with 2+1-flavor Wilson-clover fermions by the JLab/W\&M/LANL/MIT/Marseille collaborations, and presents an analysis of the meson and baryon spectrum, decay constants $f_π$ and $f_K$, flow scales $t_0$ and $w_0$, and time histories of the $Θ$ and Weinberg operators under gradient flow. Using these quantities, the physical point values of the two flow scales, ${t_0^{\rm Phy}}$ and ${w_0^{\rm Phy}}$, and the ratio $\mathop{f_K / f_π}^{\rm Phy}$ are determined. The masses of the octet and decuplet baryons are analyzed using both the next-to-leading order (NLO) and the next-next-to-leading order (NNLO) ansatz from heavy baryon chiral perturbation theory (HB$χ$PT). The NNLO fit to the octet baryons, $M_N$, $M_Σ$, $M_Λ$ and $M_Ξ$, is preferred while the corresponding fits to the decuplet Omega mass, $M_Ω$, are not distinguished. We also present a study of the autocorrelations in the data and show that there is no evidence, even at large flow time, of the freezing of the topological charge or the Weinberg three-gluon operator.

Figures (23)

• Figure 1: Quark line diagrams for meson (top) and baryon (bottom) two-point functions. The gluon lines have been added only to remind the reader that all possible intermediate states and gluon exchanges are included in lattice calculations, i.e., they are fully non-perturbative.
• Figure 2: Overview of the 11 ensembles in the $\phi_2 - \phi_4$ plane. The blue line corresponds to the choice $\phi_4 = \phi_4^\textrm{Phy}$, while the green line corresponds to $\phi_4 = 4\mathop{t_0^{\rm Phy}}[2(M_K^\textrm{Phy})^2 - (M_\pi^\textrm{Phy})^2] + \phi_2$. Since $m_s\propto 2M_K^2 - M_\pi^2$ in leading order chiral perturbation theory, the strange-quark mass remains close to its physical value on the green line. The two lines intersect at the physical point (black cross), where the pion and kaon have their physical masses.
• Figure 3: The data for the dimensionless quantity $t_0^2 \chi_Q$ is plotted as a function of the flow time $t_{gf} /t_{0}$ for the eleven ensembles. To fit all the data on one plot, the x-axis for the $a117m310$ ensemble has been scaled and should be read as $(t_{gf} /t_{0})/3$. To remove lattice artifacts, these data are fit versus $1/t_{gf}$ using Eq. \ref{['eq:chifit_tf']}. The $t_{gf}/t_0 \to \infty$ values are shown by the horizontal bands that are then extrapolated to the physical point using the chiral-continuum ansatz given in Eq. \ref{['eq:ChiQCC']}. This physical point result is shown by the magenta horizontal band at the bottom.
• Figure 4: Results for the fit parameters $\{A, \ B, \ C \}$ defined in Eq. \ref{['eq:fit_fpiK']} for the 200 bootstrap samples (BS) used for the statistical analysis. The red point on the right in each panel shows the bootstrap average. In these fits, $t_0$ was used to make quantities dimensionless.
• Figure 5: This figure illustrates how well data for $\sqrt{8t_0} f_{\pi K}$ are fit by the ansatz in Eq. \ref{['eq:fit_fpiK']}. The fit shown is for one bootstrap sample with $\mathop{t_0^{\rm Phy}}$ used to set the scale and the $\mathcal{O}(\alpha_s a)$ continuum correction factor. It has $\chi^2/\textrm{dof}=1.20$.