Coulomb versus physical string tension on the lattice

From continuum studies it is known that the Coulomb string tension ${\sigma }_C$ gives an upper bound for the physical (Wilson) string tension ${\sigma }_W$ [D. Zwanziger, Phys. Rev. Lett. 90, 102001 (2003)]. How does such a relationship translate to the lattice, however? In this paper we give evidence that on the lattice, while the two string tensions are related at zero temperature, they decouple at finite temperature. More precisely, we show that on the lattice the Coulomb gauge confinement scenario is always tied to the spatial string tension, which is known to survive the deconfinement phase transition and to cause screening effects in the quark-gluon plasma. Our analysis is based on the identification and elimination of center vortices, which allows us to control the physical string tension and study its effect on the Coulomb gauge observables. We also show how alternative definitions of the Coulomb potential may sense the deconfinement transition; however, a true static Coulomb gauge order parameter for the phase transition is still elusive on the lattice.

Figure 1

Gluon propagator for various temperatures at $\beta =2.49$ and anisotropy $\xi =4$. As $T$ increases, the infrared Gribov mass $M\sim D^{-\frac{1}{2}}(0)$ increases as well [30, 33].

Figure 2

Ghost form factor for various temperatures at $\beta =2.49$ and anisotropy $\xi =4$. No flattening is observed as $T$ is increased, except perhaps for the $T=6T_c$ data (the upper curve), which displays a slight decrease in the exponent of the infrared increasing power law as compared to the other data.

Figure 3

Coulomb potential from Eq. (13) for various temperatures at $\beta =2.49$ and anisotropy $\xi =4$; the normalization is such that the intercept at $p\to 0$ equals $8\pi {\sigma }_C$. We find that the extrapolation to $p\to 0$ is compatible with an increase of ${\sigma }_C$ at larger temperatures.

Figure 4

Coulomb potential from Eq. (14) for various temperatures at $\beta =2.49$ and anisotropy $\xi =4$. The slope of the curve at large distances is obviously constant at least up to $1.5T_c$ and increases at $3T_c$.

Figure 5

Creutz ratios for the spatial string tension from the same configurations as in Figs. 3 and 4; the dependence with the temperature is the same. Data at $3T_c$ for $r>7$ have been omitted, since the signal-to-noise ratio was very poor.

Figure 6

Spatial string tension at $\beta =2.6$ for isotropic lattices up to a maximum temperature $1.8T_c$.

Figure 7

Creutz ratios for full center vortex projection (CP) and removal (VR) at $\beta =2.6$. The reference value $\chi =0.01915$ was taken from Ref. [81]. Dotted lines are fits to the formula $\sigma +\frac{2k}{r^2}$; see Ref. [79] for further details.

Figure 8

Creutz ratios for spatial center vortex projection and removal at $\beta =2.6$, with the reference value taken from Ref. [81]. The symbols refer to the temporal (${\sigma }_t$) and spatial (${\sigma }_s$) string tension in the spatial center projected (sCP) and spatial vortex removed (sVR) ensembles, respectively.

Figure 9

Histogram of plaquettes before and after spatial vortex removal.

Figure 10

Renormalized ghost form factor before and after vortex removal at $\beta =2.15$ and $\beta =2.60$. The data were multiplicatively renormalized at a reference scale of $\mu =3\mathrm{GeV}$, whence the curves for the two couplings $\beta$ fall on top of each other.

Figure 11

Ghost form factor after center projection (and restriction to the nonzero subspace) at $\beta =2.15$.

Figure 12

Standard Coulomb potential Eq. (13) in momentum space, normalized such that the intercept at $p\to 0$ yields $8\pi {\sigma }_C$. The relatively sharp drop in the deep infrared for the full $SU(2)$ data made it necessary to take a fairly small coupling $\beta =2.15$ to (roughly) estimate the intercept.

Figure 13

Coulomb potential from Eq. (14) in position space. Linear behavior sets in at moderate distances $r$, so that a fairly large coupling $\beta =2.60$ could be afforded to minimize discretization effects. See the discussion at the end of Sec. 3b for further details.

Figure 14

Coulomb potential from Eq. (14) in position space. Both the spatial center projected and the spatial vortex removed correlators are compatible with a linear rising potential, although the former overestimates the string tension by a large factor.