Precision Determination of $r_0{\Lambda }_{\overline{\mathrm{MS}}}$ from the QCD Static Energy

We use the recently obtained theoretical expression for the complete QCD static energy at next-to-next-to-next-to leading-logarithmic accuracy to determine $r_0{\Lambda }_{\overline{\mathrm{MS}}}$ by comparison with available lattice data, where $r_0$ is the lattice scale and ${\Lambda }_{\overline{\mathrm{MS}}}$ is the QCD scale. We obtain $r_0{\Lambda }_{\overline{\mathrm{MS}}}={0.622}_{-0.015}^{+0.019}$ for the zero-flavor case. The procedure we describe can be directly used to obtain $r_0{\Lambda }_{\overline{\mathrm{MS}}}$ in the unquenched case, when unquenched lattice data for the static energy at short distances becomes available. Using the value of the strong coupling ${\alpha }_s$ as an input, the unquenched result would provide a determination of the lattice scale $r_0$.

Figure 1

(a) Comparison of the singlet static energy with lattice data. We plot $r_o\mathbf{(}{E}_0(r)-E_0(r_{\min })+E_0^{\mathrm{latt}.}(r_{\min })\mathbf{)}$ as a function of $r/r_0$ and the lattice data of Ref. [23] (red points). The dotted blue curve is at tree level, the dot-dashed magenta curve is at one loop, the dashed brown curve is at two loop plus leading ultrasoft logarithmic resummation and the solid black curve is at three loop plus next-to-leading ultrasoft logarithmic resummation. The yellow (lighter) band is obtained by varying $r_0{\Lambda }_{\overline{\mathrm{MS}}}$ according to $r_0{\Lambda }_{\overline{\mathrm{MS}}}=0.602\pm 0.048$ [24] and the green (darker) band is obtained by adding the term $\pm C_F{\alpha }_s^5/r$, for the solid black curve. (b) Same but using $r_0{\Lambda }_{\overline{\mathrm{MS}}}={0.622}_{-0.015}^{+0.019}$ (see text).

Figure 2

Fit values of $r_0{\Lambda }_{\overline{\mathrm{MS}}}$ for different values of $\rho$. The range of $\rho$ displayed in the plot corresponds to that obtained after step (3) (${\mathrm{N}}^3\mathrm{LL}$ points) or step (4) (${\mathrm{N}}^3\mathrm{LL}$ points) of the procedure described in the text. The size of each point is proportional to its assigned weight.