Lattice QCD study of the heavy-heavy-light quark potential

We study the heavy-heavy-light quark ($QQq$) potential in SU(3) quenched lattice QCD, and discuss one of the roles of the finite-mass valence quark in the interquark potential. Monte Carlo simulations are performed with the standard gauge action on the ${16}^4$ lattice at $\beta =6.0$ and the $O(a)$-improved Wilson fermion action at four hopping parameters. For statistical improvement, the gauge configuration is fixed with the Coulomb gauge. We calculate the potential energy of $QQq$ systems as a function of the inter-heavy-quark distance $R$ in the range of $R\le 0.8\mathrm{fm}$. The $QQq$ potential is well described with a Coulomb plus linear potential, and the effective string tension between the two heavy quarks is significantly smaller than the string tension $\sigma \simeq 0.89\mathrm{GeV}/\mathrm{fm}$. It would generally hold that the effect of the finite-mass valence quark reduces the inter-two-quark confinement force in baryons.

Figure 1

(a) The $QQq$ Wilson loop. The wavy line represents the light-quark propagator and the straight line the heavy-quark trajectory. (b) The “wall-to-wall $QQq$ Wilson loop.” The gray wavy lines represent the wall-to-wall quark propagator, which is the average of all propagators at a fixed time separation.

Figure 2

The correctors $W_{3Q\mathrm{A}}^{\mathrm{Coul}}$ (left) and $W_{3Q\mathrm{B}}^{\mathrm{Coul}}$ (right) for the $3Q$ potential with the Coulomb gauge. $R_{\mathrm{I}}$ and $R_{\mathrm{II}}$ are the same in Fig. 1a, and $R_{\mathrm{III}}$ is the distance between the two staples and the Wilson line.

Figure 3

The effective mass of the wall-to-wall $QQq$ Wilson loop with the Coulomb gauge and the efficiency of the smearing method. The lightest quark case, $\kappa =0.1380$, is shown. The left graph is the small loop case with $R=2$ and $(R_{\mathrm{I}},R_{\mathrm{II}})=(1,1)$ and the right is the large loop case with $R=8$ and $(R_{\mathrm{I}},R_{\mathrm{II}})=(4,4)$. In each graph, two kinds of data are the result without the smearing method (circle) and with the smearing method of $(\alpha ,N_{\mathrm{smr}})=(2.3,40)$ (square). Thus, the smearing method is found to be effective, especially for large $R$.

Figure 4

The lattice QCD data of $QQq$ potential $V_{QQq}$ with the Coulomb gauge. The results of different four hopping parameters $\kappa$ are shown. The solid curves are the best-fit functions of Eq. (18). All the scales are measured in lattice unit.

Figure 5

The schematic figure of the flux-tube length $L_{\min }$ and the inter-two-quark distance $R$. These are equal, i.e., $L_{\min }=R$, in the $Q\overline{Q}$ system (left), and these are not equal, i.e., $L_{\min }\ne R$, in the $3Q$ or $QQq$ system (right).

Figure 6

Examples of the junction choices of the $QQq$ Wilson loop. The wavy line represents the light-quark propagator and the straight line the heavy-quark trajectory.

Figure 7

The $QQq$ potential $V_{QQq}$ of the different gauge choices: (A) gauge invariant with $\kappa =0.1200$ and $N_{\mathrm{conf}}=4000$, (B) Coulomb gauge with $\kappa =0.1200$ and $N_{\mathrm{conf}}=1000$, (C) Coulomb gauge with $\kappa =0.1380$ and $N_{\mathrm{conf}}=1000$, (D) Landau gauge with $\kappa =0.1380$ and $N_{\mathrm{conf}}=200$. The solid curves are the best-fit functions of Eq. (18).