Instantaneous interquark potential in generalized Landau gauge in SU(3) lattice QCD: A linkage between the Landau and the Coulomb gauges

We investigate in detail “instantaneous interquark potentials,” interesting gauge-dependent quantities defined from the spatial correlators of the temporal link-variable $U_4$, in generalized Landau gauge using SU(3) quenched lattice QCD. The instantaneous $\mathrm{Q}\overline{\mathrm{Q}}$ potential has no linear part in the Landau gauge, and it is expressed by the Coulomb plus linear potential in the Coulomb gauge, where the slope is 2–3 times larger than the physical string tension. Using the generalized Landau gauge, we find that the instantaneous potential can be continuously described between the Landau and the Coulomb gauges, and its linear part rapidly grows in the neighborhood of the Coulomb gauge. We also investigate the instantaneous 3Q potential in the generalized Landau gauge, and obtain similar results to the $\mathrm{Q}\overline{\mathrm{Q}}$ case. $T$-length terminated Polyakov-line correlators and their corresponding “finite-time potentials” are also investigated in generalized Landau gauge.

Figure 1

The lattice QCD result of $⟨({\partial }_i{A}_i^a{)}^2⟩$ in the generalized Landau gauge. As $\lambda \to 0$, this quantity goes to zero monotonically.

Figure 2

The “instantaneous potential” $V_{\lambda }(R)$ in the generalized Landau gauge for typical values of $\lambda$. The symbols denote lattice QCD results and the curves fit results using Coulomb plus linear form, $V_{\lambda }(R)=-A_{\lambda }/R+{\sigma }_{\lambda }R+C_{\lambda }$, in the region of $R\lesssim 0.8\mathrm{fm}$. For $\lambda =1\sim 0.1$, the potential has almost no linear part. For $\lambda \lesssim 0.1$, the linear potential grows rapidly, and ${\sigma }_{\lambda }\simeq 2.6{\sigma }_{\mathrm{phys}}$ at $\lambda =0$.

Figure 3

The instantaneous string tension ${\sigma }_{\lambda }$, the slope of the linear part in the instantaneous potential $V_{\lambda }(R)$ in the generalized Landau gauge. The right upper figure is a close-up near the Coulomb gauge ($\lambda =0$). For $0\le \lambda \lesssim 0.1$, ${\sigma }_{\lambda }$ changes rapidly from $2.6{\sigma }_{\mathrm{phys}}$ to 0, while ${\sigma }_{\lambda }$ is rather small for $0.1\lesssim \lambda \le 1$.

Figure 4

The “finite-time potential” $V_{\lambda }(R,T)$ in the Coulomb gauge ($\lambda =0$) for $T=1$, 2, 3, 4, 5. Here, for the comparison, an irrelevant constant is shifted for each $T$. The curves denote the fit results using the Coulomb plus linear form. The slope of $V_{\lambda }(R,T)$ is clearly changed according to $T$.

Figure 5

The $T$-length dependence of the “finite-time string tension” ${\sigma }_{\lambda }(T)$, the infrared slope of finite-time potential $V_{\lambda }(R,T)$, in the generalized Landau gauge for several typical $\lambda$ values. Near the Coulomb gauge, e.g., for $\lambda \lesssim 0.03$, ${\sigma }_{\lambda }(T)$ goes to the same value for large $T\sim 1\mathrm{fm}$. For $\lambda \gtrsim 0.1$, ${\sigma }_{\lambda }(T)$ is an increasing function of $T$. In fact, even though the instantaneous potential has no linear part, the linear part of $V_{\lambda }(R,T)$ appears and gradually grows, as the Polyakov-line grows.

Figure 6

The finite-time potential $V_{\lambda }(R,T)$ at $\lambda =0.02(\simeq {\lambda }_C)$. For the comparison, an irrelevant constant is shifted for each $T$. The slope of $V_{\lambda }(R,T)$ is almost the same for $T=1,2,\dots 5$.

Figure 7

The instantaneous potential $V_{\lambda }^{3\mathrm{Q}}$ plotted against $L_{\min }$ in the generalized Landau gauge for typical values of $\lambda$. The symbols denote lattice QCD results. For $\lambda =1\sim 0.1$, the potential has almost no linear part. For $\lambda \lesssim 0.1$, the linear potential grows rapidly, and ${\sigma }_{\lambda }^{3\mathrm{Q}}\simeq 2.5{\sigma }_{\mathrm{phys}}$ at $\lambda =0$.

Figure 8

The instantaneous string tension ${\sigma }_{\lambda }^{3\mathrm{Q}}$, the slope of the linear part in the instantaneous potential $V_{\lambda }^{3\mathrm{Q}}$ in the generalized Landau gauge. The right upper figure is a close-up near the Coulomb gauge ($\lambda =0$). For $0\le \lambda \lesssim 0.1$, ${\sigma }_{\lambda }^{3\mathrm{Q}}$ changes rapidly from $2.5{\sigma }_{\mathrm{phys}}$ to 0, while ${\sigma }_{\lambda }^{3\mathrm{Q}}$ is rather small for $0.1\lesssim \lambda \le 1$.

Figure 9

The finite-time 3Q potential $V_{\lambda }^{3\mathrm{Q}}({\overrightarrow{x}}_1,{\overrightarrow{x}}_2,{\overrightarrow{x}}_3,T)$ plotted against $L_{\min }$ in the Coulomb gauge ($\lambda =0$) for $T=1$, 2, 3, 4. Here, for comparison, an irrelevant constant is shifted for each $T$.

Figure 10

The $T$-length dependence of finite-time string tension ${\sigma }_{\lambda }^{3\mathrm{Q}}(T)$, the infrared slope of finite-time 3Q potential $V_{\lambda }^{3\mathrm{Q}}(T)$, in the generalized Landau gauge for several typical $\lambda$ values. Near the Coulomb gauge, e.g., for $\lambda \lesssim 0.03$, ${\sigma }_{\lambda }^{3\mathrm{Q}}(T)$ goes to the same value for large $T\sim 0.6\mathrm{fm}$. For $\lambda \gtrsim 0.1$, ${\sigma }_{\lambda }^{3\mathrm{Q}}(T)$ is an increasing function of $T$. In fact, even though the instantaneous potential has no linear part, the linear part of $V_{\lambda }^{3\mathrm{Q}}(T)$ appears and gradually grows, as the Polyakov line grows.

Figure 11

The finite-time potential $V_{\lambda }^{3Q}({\overrightarrow{x}}_1,{\overrightarrow{x}}_2,{\overrightarrow{x}}_3,T)$ plotted against $L_{\min }$ at $\lambda =0.02(\simeq {\lambda }_C)$. For the comparison, an irrelevant constant is shifted for each $T$. The slope of $V_{\lambda }^{3Q}(L_{\min },T)$ is approximately the same for $T=1$, 2, 3, 4.

Figure 12

The spatial correlator $G_{\lambda }(R)\equiv ⟨\mathrm{Tr}{U}_4^{†}(\overrightarrow{x},a)U_4(\overrightarrow{y},a)⟩$ ($R=|\overrightarrow{x}-\overrightarrow{y}|$) in the Landau gauge (open circles) and the Coulomb gauge (open triangles), together with its asymptotic value of $\frac{1}{3}⟨\mathrm{Tr}{U}_4{⟩}^2$ (solid lines and cross symbols). In the Landau gauge, $⟨\mathrm{Tr}{U}_4⟩\ne 0$, and $G_{\lambda }(R)$ rapidly converges to a constant for $R\gtrsim 0.4\mathrm{fm}$, which leads to a rapid saturation of the instantaneous potential $V_{\lambda }(R)\equiv -\frac{1}{a}\ln G_{\lambda }(R)$. In the Coulomb gauge, $⟨\mathrm{Tr}{U}_4⟩=0$, and $G_{\lambda }(R)$ decreases monotonically to zero as $R$, which leads to $V_{\lambda }(R)=+\infty$ for $R=\infty$.

Figure 13

The expectation value of $\frac{1}{3}⟨\mathrm{Tr}{U}_4⟩$ in the generalized Landau gauge. For $\lambda \ne 0$, $⟨\mathrm{Tr}{U}_4⟩$ takes a nonzero value, and it approaches to zero continuously as $\lambda \to 0$. The value of $⟨\mathrm{Tr}{U}_4⟩$ relates to the infrared behavior of the correlator $G_{\lambda }(R)$ and the instantaneous potential $V_{\lambda }(R)$.

Figure 14

The $T$ dependence of $\frac{1}{3}⟨\mathrm{Tr}L(T)⟩$ in the generalized Landau gauge. In the Coulomb gauge, $⟨\mathrm{Tr}L(T)⟩$ is always zero. For $\lambda \ne 0$, $⟨\mathrm{Tr}L(T)⟩$ is a decreasing function of $T$, and it converges to zero in the large-$T$ limit.

Figure 15

The fit result of the instantaneous potential $V_{\mathrm{inst}}(R)$ in the Landau gauge using the Coulomb plus linear form (solid line) and Yukawa-function form (dashed line). Both forms well reproduce lattice QCD result. The best-fit parameter of the Yukawa mass is $m=0.634(3)\mathrm{GeV}$, which coincides with the infrared effective gluon mass obtained from the Landau-gauge gluon propagator [16].