The Nf=0 heavy quark potential from short to intermediate distances

TL;DR

This work studies the $N_{f}=0$ (quenched) heavy-quark potential from short to intermediate distances using large-scale lattice QCD simulations. It defines and relates the scales $r_{0}$ and $r_{\rm c}$, provides an interpolation for $r_{0}/a$ up to $\beta=6.92$, and demonstrates that continuum extrapolations of the force and potential are dominated by ${O}(a^{2})$ discretization errors when using a tree-level improved force. The nonperturbative results agree with perturbation theory at short distances (no free parameters beyond $\Lambda_{\overline{MS}} r_{0}$) and with a bosonic string description at intermediate distances, with ~1\% accuracy. The study also highlights challenges in extrapolations involving $L_{\rm max}/r_{0}$ where both linear and quadratic $a$-effects are relevant, underscoring the need for nonperturbative ${O}(a)$-improvement or chiral-symmetric formulations. Overall, the paper provides precise nonperturbative insights into the static potential in quenched QCD and refines the scale-setting framework via $r_{0}$ across a wide range of lattice spacings.

Abstract

We study the potential of a static quark anti-quark pair in the range 0.05fm \leq r \leq 0.8fm, employing a sequence of lattices up to 64^4. Lattice artifacts in potential and force are investigated theoretically as well as numerically and continuum quantities are obtained by extrapolation of the results at finite lattice spacing. Consistency of the numerical results with the form of scaling violations predicted by an analysis `a la Symanzik is found. The scale r_0/a is determined for the Wilson action up to beta=6.92.

Figures (7)

• Figure 1: The ratio $r_{\rm c}/r_{0}$ for $5.95\leq\beta\leq 6.57$ in the standard Wilson action (circles) including the continuum extrapolation (solid point). This is an analysis of data in pot:r0_SU3.
• Figure 2: Interpolation of $r_{0}/a$.
• Figure 3: Continuum extrapolation of $r_{\rm c}^2 F(xr_{\rm c})$, for $x=0.4,~0.5,~0.9$ from top to bottom and of $r_{0}^2 F(xr_{0})$, for $x=0.5,~0.6,~1.5$ from top to bottom. The data are from our new computations and from pot:r0_SU3. Filled circles correspond to the naive value $r_{\rm I}=r-\hbox{$a\over2$}$ instead of eq. (\ref{['e_rI']}).
• Figure 4: The force in the continuum limit and for finite resolution, where the discretization errors are estimated to be smaller than the statistical errors. The full line is the perturbative prediction with $\Lambda_{\rm \overline{MS}}\, r_{0} =0.602$. The dashed curve corresponds to the bosonic string model normalized by $r_{0}^2 F(r_{0})=1.65$.
• Figure 5: Continuum extrapolation of $V_{\rm I}(r)$, for $r/r_{0}=1.5,\,0.6,\,0.5$ on the left hand side and for $r/r_{\rm c}=0.9,\,0.4,\,0.3$ on the right hand side.