QCD static energy at next-to-next-to-next-to leading-logarithmic accuracy

We compute the static energy of QCD at short distances at next-to-next-to-next-to-leading logarithmic accuracy in terms of the three-loop singlet potential. By comparing our results with lattice data we extract the value of the unknown piece of the three-loop singlet potential.

Figure 1

Ultrasoft contributions to the self-energy of the octet field in a covariant gauge, at the order of interest. Symmetric graphs are understood for (c)–(i).

Figure 2

Infrared divergent diagram in the matching calculation of $V_A$ and $V_B$.

Figure 3

Static potential $r_0{V}_s(r)$, in the RS scheme, as a function of $r/r_o$. The dotted blue curve is at tree level, the dotted-dashed magenta curve is at one loop, the dashed brown curve is at two loop plus leading ultrasoft logarithmic resummation, and the long-dashed green curve is at three loop (Padé estimate) plus next-to-leading ultrasoft logarithmic resummation. The solid black curve is also at three loop plus next-to-leading ultrasoft logarithmic resummation but using $c_0=250.7$, see Sec. 4d.

Figure 4

Illustration of the hierarchy of scales relevant for the static energy. The scales are: $1/r$ (dotted red), $(V_o-V_s)$ (dotted-dashed green), ${\alpha }_s(V_o-V_s)$ (dashed black) and ${\Lambda }_{\overline{\mathrm{MS}}}=0.602/r_0$ (solid blue). $(V_o-V_s)$ is taken at tree level, $(V_o-V_s)=N_c{\alpha }_s/(2r)$, and the one-loop running of ${\alpha }_s={\alpha }_s(1/r)$ is used.

Figure 5

Impact of the variation of $c_0$, $K_2$, ${\Lambda }_{\overline{\mathrm{MS}}}$ and effect of higher-order terms in the ${\mathrm{N}}^3\mathrm{LL}$ expression for the static energy, respectively, represented as the green bands. $c_0$ has been varied by 35% around the Padé value $c_0=313$. $K_2$ has been varied from $-5$ to 5. ${\Lambda }_{\overline{\mathrm{MS}}}$ has been varied according to ${\Lambda }_{\overline{\mathrm{MS}}}{r}_0=0.602(48)$ [38], and the effect of higher-order terms has been estimated by adding the term $\pm 10C_F{\alpha }_s^5/r$ to the ${\mathrm{N}}^3\mathrm{LL}$ curve. $c_0=313$ and $K_2=0$ have been used as central values. The red points are the lattice data of 35.

Figure 6

$({\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.{)}_{{\mathrm{N}}^3\mathrm{LL}}/({\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.{)}_{{\mathrm{N}}^2\mathrm{LL}}$ for different values of the $K_2$ and $c_0$ parameters. The color of each point in the $c_0-K_2$ plane represents the value of the ratio, lighter colors correspond to higher values and darker colors to lower values. We show values of the ratio up to 1.

Figure 7

$({\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.{)}_{{\mathrm{N}}^3\mathrm{LL}}/({\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.{)}_{{\mathrm{N}}^2\mathrm{LL}}$ for different values of $c_0$ (left) or $K_2$ (right), the other parameter is fitted.

Figure 8

Comparison of the singlet static energy with lattice data. We plot $r_o(E_0(r)-E_0(r_{\min })+E_0^{\mathrm{latt}}(r_{\min }))$ as a function of $r/r_0$ and the lattice data of 35 (red points). The dotted blue curve is at tree level, the dotted-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, using the best fit values of Eq. (39).

Figure 9

Impact of the variation of $c_0$, $K_2$, ${\Lambda }_{\overline{\mathrm{MS}}}$ and effect of higher-order terms in the static energy, respectively, represented as the gray bands. $c_0$ has been varied in the interval (215 350), $K_2$ in the interval $(-2,2)$, ${\Lambda }_{\overline{\mathrm{MS}}}$ has been varied according to ${\Lambda }_{\overline{\mathrm{MS}}}{r}_0=0.602(48)$ [38] and the effect of higher-order terms has been estimated by adding the term $\pm 10C_F{\alpha }_s^5/r$ to the ${\mathrm{N}}^3\mathrm{LL}$ curve. All the bands are obtained by fixing the other parameters at their central or best fit values. The curves are the same as in Fig. 8. The red points are the lattice data of 35.