Quark-antiquark potential to order $1/m$ and heavy quark masses

An updated heavy quark-antiquark potential is constructed by matching the short-distance perturbative part to long-distance lattice QCD results at an intermediate $r$ scale. The static potential and the order $1/m$ potential are both analyzed in this way. Effects of order $1/m$ in charmonium and bottomonium spectra are discussed in comparison. Charm and bottom quark masses are deduced from the spectra and related to the quark masses of other schemes.

Figure 1

Power series expansion of ${\alpha }_s(q)$ at $\mu =2\mathrm{GeV}$ for $n_f=4$. The thick line shows the full four-loop running according to the renormalization group equation. Note that the sign of ${\alpha }_s(q\to \infty )$ changes order by order.

Figure 2

Static $r$-space potential according to Eq. (7) using $\mu =2\mathrm{GeV}$. The choice ${\mu }_{\mathrm{IR}}^2={\overrightarrow{q}}^2$ has been adopted at next-to-next-to-next-to-leading order (NNNLO).

Figure 3

Static $r$-space potential according to Eq. (7). Progressive orders are shown when $\mu$ is identified with $1/r$. The choice ${\mu }_{\mathrm{IR}}^2={\overrightarrow{q}}^2$ has been adopted at NNNLO.

Figure 4

Static QCD potential (with $n_f=3$) from the restricted numerical Fourier transform (8). Shown is the NNLO potential for different values of ${\mu }_f$. The curves have been shifted by a constant to match at small $r$ values.

Figure 5

Static QCD potential (with $n_f=3$ and ${\mu }_f=1.0\mathrm{GeV}$) from the restricted numerical Fourier transform (8). Different orders of have been matched at 0.01 fm. The choice ${\mu }_{\mathrm{IR}}^2={\overrightarrow{q}}^2$ has been adopted at NNNLO.

Figure 6

Static QCD potential for $n_f=3$, based on Eqs. (8, 1), matched at intermediate distances to a potential from lattice QCD [15]. Dashed-dotted curve: simplest extrapolation using Coulomb-plus-linear $r$ dependence.

Figure 7

The static potential for $n_f=0$ flavors compared to lattice points [16]. The lattice scale $r_0$ has been set to 0.5 fm.

Figure 8

The order $1/m$ potential $V^{(1)}(r,{\mu }_f^{\prime })$ with $n_f=3$ from the restricted numerical Fourier transform (13), for different cutoffs ${\mu }_f^{\prime }$. The curves have been shifted by a constant to match at small $r$ values.

Figure 9

The order $1/m$ potential $V^{(1)}(r,{\mu }_f^{\prime })$ with $n_f=3$, based on Eq. (13), matched at intermediate distances to a potential from lattice QCD [20]. The fit and the matching has been performed using the lattice QCD points with $\beta =5.85$ (circles). Legends are analogous to Fig. 6. Dashed-dotted curve: simple extrapolation using Eq. (14).

Figure 10

Bottomonium spectrum in comparison with experiment. Static plus order $1/m$ results are shown on the left, with additional hyperfine effects (h.f.) added phenomenologically using Eq. (16).

Figure 11

Charmonium spectrum in comparison with experiment. Static plus order $1/m$ results are shown on the left, with additional hyperfine effects (h.f.) added phenomenologically.

Figure 12

Scale dependence of $m_{\mathrm{PS}}({\mu }_f)$ for the bottomonium case. The solid curve shows the numerical ${\mu }_f$ dependence from Eq. (18). The dashed and dashed-dotted curves show the ${\mu }_f$ dependence given by Eq. (19) at different matching points. Point (2) is preferred to point (1) for the determination of ${\overline{m}}_b$.