Precise determination of the three-quark potential in SU(3) lattice gauge theory

We investigate the static interquark potential for the three-quark system in SU(3) lattice gauge theory at zero temperature by using Monte Carlo simulations. We extract the potential from the correlation function of the three Polyakov loops, which are computed by employing the multilevel algorithm. We obtain remarkably clean results of the three-quark potential for $O(200)$ sets of the three-quark geometries including not only the cases that three quarks are put at the vertices of acute, right, and obtuse triangles, but also the extreme cases such that three quarks are put in line. We find several new interesting features of the three-quark potential and then discuss its possible functional form.

Figure 1

The three-quark Wilson loop (left) and the three-quark PLCF (right). While the Wilson loop is composed of the three temporal Wilson lines connected by the spatial Wilson lines with a junction, the PLCF is only composed of the three Polyakov loops, where a periodic boundary condition is imposed in the time direction.

Figure 2

A product of two three-link correlators, where repeated Greek indices $\lambda$, $\rho$, and $\sigma$ are to be summed over from one to three.

Figure 3

The three-quark geometries investigated in our numerical simulations. The circles represent the spatial location of quarks. Three Polyakov loops are put at the vertices of (i) acute, (ii) right, (iii) obtuse triangles, and are also put in (iv) line, and are put to be (v) the quark-diquark system.

Figure 4

The definition of distances and angles of a triangle used in our analyses: $x_i$ denotes the location of $i$ th quark, $F$ the Fermat-Torricelli point, so that $\angle x_{1}F{x}_2=\angle x_{2}F{x}_3=\angle x_{3}F{x}_1=120°$. $h_i$ denotes the distance between $F$ and each side. $\mathrm{\Delta }$ and $Y$ are given by $\mathrm{\Delta }=r_1+r_2+r_3$ and $Y=l_1+l_2+l_3$, respectively.

Figure 5

Convergence histories of the PLCF of the equilateral triangle geometries at $\beta =5.85$ (upper), $\beta =6.00$ (middle), and $\beta =6.30$ (lower) as a function of $N_{\mathrm{iupd}}$, where quarks are placed at $(x,0,0)$, $(0,x,0)$, and $(0,0,x)$.

Figure 6

The three-quark potentials of the equilateral triangle geometries at $\beta =6.00$ obtained from one gauge configuration and from the average of 9 gauge configurations as a function of $Y$. The dotted line represents the fit curve to the averaged potential.

Figure 7

The relative error between the two potentials in Fig. 6, $(V_{3q}^{(\mathrm{ave})}-V_{3q})/V_{3q}^{(\mathrm{ave})}$.

Figure 8

The three-quark potential against $\mathrm{\Delta }$ defined in Eq. (16) (upper) and $L_{\mathrm{str}}$ defined in Eq. (21) (lower). Smaller markers denote data from the geometries for $r_{\min }/a\leqq 2$, which may suffer from lattice cutoff effects.

Figure 9

The three-quark potential of acute, right, and obtuse-narrow geometries restricted to $r_{\min }/a>2$ against $L_{\mathrm{str}}$.

Figure 10

The three-quark potential of line geometries against $r_{\min }$.

Figure 11

The quark-diquark and quark-antiquark potentials against $r$. The dashed and dotted lines are the fit curves to Eq. (26), respectively.

Figure 12

The derivatives of the quark-diquark and quark-antiquark potentials with respect to $r$. The horizontal dotted line corresponds to the string tension of the quark-antiquark potential ${\sigma }_{q\overline{q}}{a}^2=0.0449$ (see, Table 12).

Figure 13

The derivatives of the three-quark potential with respect to $L_{\mathrm{str}}$ for acute (upper), right (middle), and line (lower) geometries. The dotted line in each plot corresponds to the string tension of the quark-antiquark potential ${\sigma }_{q\overline{q}}{a}^2=0.0449$ (see Table 12).

Figure 14

The relative error between three-quark potential and the half of the sum of the quark-antiquark potential, $\delta V/V_{3q}$ against $\mathrm{\Delta }$ defined in Eq. (16). As the three-quark potential is from one gauge configuration with no statistical error, the error bars in this plot are purely from that of the quark-antiquark potential from $N_{\mathrm{cnf}}=20$ (see Tables 11 and 13).

Figure 15

The same plot as in Fig. 14, but against ${\theta }_{\max }$ defined in Eq. (14).

Figure 16

The same plot as in Fig. 14, but against $h$ defined in Eq. (23). The fit curve is given by Eq. (31).

Figure 17

The three-quark potential after subtracting the confinement and constant terms $V_{3q}^{(\mathrm{sub}1)}$ in Eq. (32) against the reduced distance $R$ defined in Eq. (22). The open symbols are the three-quark potential from one gauge configuration at $\beta =6.00$, and the filled circles with error bars are from 9 gauge configurations of the equilateral triangle geometries. The dotted line corresponds to $-A_{q\overline{q}}/(2R)$, and the dashed line the fit curve given by Eq. (33).

Figure 18

The same plot as in Fig. 17, but the term $-A_{q\overline{q}}/(2R)$ is further subtracted from the three-quark potential [$V_{3q}^{(\mathrm{sub}2)}$ in Eq. (36)]. The dashed line corresponds to $0.259A_{q\overline{q}}/(2R)+{\tilde{\mu }}_{3q}$.

Figure 19

The three-quark potential as a function of $Y$ in physical unit, where the constant $(3/2){\mu }_{q\overline{q}}$ is subtracted. The dotted line corresponds to the fit curve to Eq. (39), which yields $A_{3q}^{(\mathrm{Y})}\hslash c=0.131(4)[\mathrm{GeV}/\mathrm{fm}]$, ${\sigma }_{3q}=1.02(1)[\mathrm{GeV}/\mathrm{fm}]$, and ${\tilde{\mu }}_{3q}=-0.12(2)[\mathrm{GeV}]$ with ${\chi }^2/N_{\mathrm{df}}=0.02$.

Figure 20

The quark-antiquark potential as a function of $r$ in physical unit, where the constant ${\mu }_{q\overline{q}}$ is subtracted. The dotted line is the fit curve to $V_{q\overline{q}}(r)=-A_{q\overline{q}}\hslash c/r+{\sigma }_{q\overline{q}}r$, which yields $A_{q\overline{q}}\hslash c=0.0671(6)[\mathrm{GeV}/\mathrm{fm}]$, ${\sigma }_{q\overline{q}}=1.025(4)[\mathrm{GeV}/\mathrm{fm}]$ with ${\chi }^2/N_{\mathrm{df}}=0.013$.

Figure 21

The derivative of the potential with respect to $Y$ in physical unit. The dotted line corresponds to the fit curve to the derivative of Eq. (39) with respect to $Y$, which yields $A_{3q}^{(\mathrm{Y})}\hslash c=0.139(4)[\mathrm{GeV}/\mathrm{fm}]$, $\sigma =1.013(7)[\mathrm{GeV}/\mathrm{fm}]$ with ${\chi }^2/N_{\mathrm{df}}=2.0$.

Figure 22

The scale dependence of the constant term of the quark-antiquark and three-quark potentials per one quark. A naive fit to a linear function yields the slope 0.33 for both cases.

Figure 23

The scaling behavior of the remnant of the constant term of the three-quark potential ${\tilde{\mu }}_{3q}$ in Eq. (40). A naive fit to a constant value yields ${\tilde{\mu }}_{3q}=-0.131(7)[\mathrm{GeV}]$.

Figure 24

The three-quark potential as a function of $R$ in physical unit after subtracting the confinement and divergent constant terms expected from the quark-antiquark potential, $V_{3q}^{(\mathrm{sub}1)}$. The dotted line corresponds to the fit curve to $V_{3q}^{(\mathrm{R})}=-A_{3q}^{(\mathrm{R})}\hslash c/R+{\tilde{\mu }}_{3q}$ in Eq. (33), which yields $A_{3q}^{(\mathrm{R})}\hslash c=0.0248(5)[\mathrm{GeV}\mathrm{fm}]$ and ${\tilde{\mu }}_{3q}=-0.132(8)[\mathrm{GeV}]$ with ${\chi }^2/N_{\mathrm{df}}=0.04$.

Figure 25

The quark-antiquark potential at $\beta =6.00$ as a function of the interquark distance $r/a$. The raw data and the fit results are summarized in Tables 11, 12, and 13.