The strange quark mass and Lambda parameter of two flavor QCD

TL;DR

This work delivers a nonperturbative determination of the two-flavor QCD Lambda parameter and the strange quark mass, anchoring the scale with the kaon decay constant and updating the hadronic scale $r_0$ as well as the renormalization constant $Z_A$. By employing the Schrödinger functional framework and two complementary chiral extrapolation strategies, the authors demonstrate robust control over systematic effects in the $N_f=2$ theory and extract consistent results for $f_K$, $r_0$, $ ext{and}\, ar m_s$ across lattice spacings. The Lambda parameter is obtained in the SF scheme and converted to $ar{ ext{MS}}$ units, yielding $ ext{Lambda}_{ar{ ext{MS}}}^{(2)}ig|_{ar{ ext{MS}}} ext{(2 GeV)} ext{ and } r_0 ext{~scaled values}$, while the strange quark mass is determined with high precision to be $M_s=138(3)(1)$ MeV and $ar m_s(2 ext{ GeV})=102(3)(1)$ MeV. These results provide a crucial nonperturbative benchmark for the role of $N_f=2$ dynamics in the running of QCD and in connecting lattice scales to physical observables, with clear pathways toward extending to more flavors. The study also highlights the importance of careful scale setting via $f_K$ and shows that cutoff effects are small within the employed framework.

Abstract

We complete the non-perturbative calculations of the strange quark mass and the Lambda parameter in two flavor QCD by the ALPHA collaboration. The missing lattice scale is determined via the kaon decay constant, for whose chiral extrapolation complementary strategies are compared. We also give a value for the scale r_0 in physical units as well as an improved determination of the renormalization constant Z_A.

Figures (11)

• Figure 1: Autocorrelation function of ${F}_\mathrm{K}$ for the O7 lattice. The line gives our estimate for its tail. The standard method of Ref. Wolff:2003sm gives a window $W=W_l$ and $\tau_\mathrm{int}=0.7$, compared to $\tau_\mathrm{int}=4$ including the tail contribution which we add from $W=W_u$, or more than a factor two in the error of the observable.
• Figure 2: The effective pion mass given by $\cosh(M_{\rm eff}(t-T/2))/\cosh(M_{\rm eff}(t+1-T/2))=f_\mathrm{PP}(t)/f_\mathrm{PP}(t+1)$ for the O7 lattice. A two-state fit to data outside the shaded area determines $x_0^\mathrm{min}$. The result of the final one-state fit is given by the error band.
• Figure 3: The left plot shows the ratio $r_0/r_{0{\rm ref}}$ as a function of $x = (r_0 m_{\rm PS})^2$, where $r_{0{\rm ref}}$ is defined at the reference point eq. (\ref{['xref']}), and the cut $x\le1.4$ is applied. The right plot shows our data of $R_0=r_0/a$ with the chirally extrapolated values using a linear function in $x$ (solid lines) applying the cut $x\le1.1$. For comparison a quadratic fit with $c=1.8$ is also shown (dashed lines).
• Figure 4: Left: Trajectories to approach the physical point in the plane of light and strange quark mass $M_{\rm light},\,M_{{\rm s}}$. The dotted line corresponds to strategy 1, i.e., $R_\mathrm{K} = R_\mathrm{K}^\mathrm{phys}$, whereas strategy 2 (full line) holds $M_{\rm s}$ fixed. Right: The two functions $L_\mathrm{K}$ and $L_\pi$, eqs. (\ref{['e:fkstrat1c']}) and (\ref{['e:fpi']}) respectively, in the interval $y_1\in[0,y_{\mathrm{K}}]$.
• Figure 5: Chiral extrapolation of $r_0f_\mathrm{K}$. For the open symbols $r_0$ and $f_\mathrm{K}$ are evaluated at finite quark mass, the dashed lines show eq. (\ref{['e:r0fka']}) for the three values of $\beta$. The filled symbols use the extrapolated value $r_0=r_{0c}$, the dashed dotted lines represent eq. (\ref{['e:r0fkb']}). In both cases, the corresponding solid line gives the continuum result.