Precision computation of the strange quark's mass in quenched QCD

TL;DR

The paper presents a non-perturbative lattice QCD determination of the renormalization-group invariant combination $M_s+\hat{M}$ in the quenched approximation, using kaon observables and a non-perturbatively renormalized mass $Z_M$. By employing chiral perturbation theory to relate current quark masses to pseudoscalar observables via the ratio $R(m_i,m_j)$ and performing a continuum extrapolation with ${\cal O}(a)$-improved actions, it yields $(M_s+\hat{M})/F_K = 0.874(29)$ and, when running to the ${\overline{\rm MS}}$ scheme at $\mu=2\,$GeV, $\bar{m}_s(2\,\mathrm{GeV})=97(4)$ MeV. The study also quantifies quenched-scale ambiguities—approximately 10%—when setting the scale with different physical inputs (e.g., $F_K$ vs. the nucleon mass), highlighting inherent limitations of quenched QCD. Overall, the work demonstrates that high-precision (3%) quark-mass determinations are achievable in quenched QCD while emphasizing the systematic ambiguities that would persist in the real world without dynamical fermions. It also sets the stage for extending the methodology to full QCD and to extracting coefficients of the chiral Lagrangian from lattice data.

Abstract

We determine the renormalization group invariant quark mass corresponding to the sum of the strange and the average light quark mass in the quenched approximation of QCD, using as essential input the mass of the K-mesons. In the continuum limit we find $(M_s + M_{light})/F_K=0.874(29)$, which includes systematic errors. Translating this non-perturbative result into the running quark masses in the $\msbar$-scheme at $μ=2 GeV$ and using the quark mass ratios from chiral perturbation theory, we obtain $\mbar_s(2 GeV)=97(4) MeV$. With the help of recent results by the CP-PACS Collaboration, we estimate that a 10% higher value would be obtained if one replaced $F_K$ by the nucleon mass to set the scale. This is a typical ambiguity in the quenched approximation.

Figures (5)

• Figure 1: The functions $Q(m,y)$. $Q_{\rm M}$ denotes the case of pseudoscalar masses and $Q_{\rm F}$ their decay constants. The average of $m_i$ and $m_j$ is between $\approx 1.4\, m_{\rm ref}$ and $\approx 2.4\, m_{\rm ref}$.
• Figure 2: Mass dependence and extrapolations for the two smallest values of the lattice spacing. The ratio $R$ is defined in eq. (\ref{['e_defR']}).
• Figure 3: Continuum limit extrapolations of several observables. Full symbols show the extrapolated values. Dashed lines represent the extrapolation function, which are continued outside the fit range towards larger lattice spacings.
• Figure 4: Mass dependence of pseudoscalar decay constant and vector meson mass.
• Figure 5: Quark mass dependence of flavour non-singlet vector meson masses. Experimental masses are shown as asterisks and their width is indicated by a dotted line.