Heavy quark masses from step-scaling

Abstract

We present a determination of the charm- and bottom-quark masses using the heavy-quark step-scaling strategy. Renormalization is performed in small volumes where relativistic bottom quarks can be simulated directly. A sequence of finite-volume simulations connects this calculation to large-volume CLS ensembles, where simulations at physical light and strange quark masses provide reliable control over low-energy hadronic physics. In all but the smallest volume, the B-scale is reached by interpolating between relativistic heavy-quark data and the static limit. The resulting quark masses are obtained with good precision, with subdominant systematic uncertainties that differ from, and thus complement, those of standard large-volume determinations.

Figures (6)

• Figure 1: Left: Strategy for connecting small and large volumes via step-scaling functions, which are obtained from relativistic heavy-quark simulations and in the static limit and hence can be combined by interpolation. Right: Overview of the CLS ensembles used in this work.
• Figure 2: Continuum extrapolations of the ratio $m_H/m_h^{\rm RGI}$ in the volume $L_1$. The left panel shows masses around the bottom-quark region, while the right panel covers the charm region. Different colors correspond to different heavy-quark masses, and the black star in the continuum limit represents the interpolated result from the continuum-extrapolated values.
• Figure 3: Continuum extrapolations for the step-scaling function $\sigma_m$. Left: static limit using two discretizations of the static action. Right: relativistic heavy-quark results. The lower panel shows the continuum extrapolation near the charm mass, while the upper panel displays extrapolations of ratios of step-scaling functions at heavier masses relative to this reference mass.
• Figure 4: Interpolation of heavy-quark observables between relativistic data and the static limit in the continuum limit. The left panel shows the step-scaling function $\sigma_m$ and the right panel $\rho_m^{\rm sym}$. Open circles denote the static-limit results, while black diamonds correspond to relativistic heavy-quark data. The gray band shows a linear interpolation in $1/y$ in the indicated region, whereas the blue curve corresponds to a quadratic interpolation using all data points. Vertical dotted lines indicate the target values for the bottom- and charm-quark masses.
• Figure 5: Determination of $\tau_m(m_\pi,m_K)$. Left: chiral–continuum extrapolation in the static limit. Right: interpolation between the static limit and relativistic heavy-quark data to reach the bottom-quark mass.