Lattice QCD analysis for Faddeev-Popov eigenmodes in terms of gluonic momentum components in the Coulomb gauge

We analyze the relation between Faddeev-Popov eigenmodes and gluon-momentum components in the Coulomb gauge using SU(3) lattice QCD. In the Coulomb gauge, the color-Coulomb energy is largely enhanced by near-zero Faddeev-Popov eigenmodes, which would lead to the confining potential. By the ultraviolet-momentum gluon cut, the color-Coulomb energy and the Faddeev-Popov spectrum are almost unchanged. In contrast to the ultraviolet insensitivity, the color-Coulomb energy and the Faddeev-Popov eigenmodes drastically change by infrared-momentum gluon cut. Without infrared gluons, the color-Coulomb energy tends to become nonconfining, and near-zero Faddeev-Popov eigenmodes vanish. We also investigate the full Faddeev-Popov eigenmodes, and find that infrared gluons widely influence both high and low Faddeev-Popov eigenmodes.

Figure 1

The instantaneous potential $V_{\mathrm{inst}}^{\Lambda }(R)$ with the IR/UV-momentum gluon cut for ${16}^4$ lattice with $a\simeq 0.15\mathrm{fm}$, i.e., $a_p\equiv 2\pi /La\simeq 0.50\mathrm{GeV}$. The original (no momentum cut) instantaneous potential is added with the fitting curve of Coulomb plus linear form. The statistical error is small and the error bars are hidden in the symbols. (a) IR-momentum cut with ${\Lambda }_{\mathrm{IR}}/a_p=1$ and $2$. (b) UV-momentum cut with ${\Lambda }_{\mathrm{UV}}/a_p=12$ and 8. An irrelevant constant is shifted.

Figure 2

Low-lying 250 FP spectrum $\rho (\lambda )$ with the IR/UV momentum cut and original spectrum for ${16}^4$ lattice with $a\simeq 0.15\mathrm{fm}$, i.e., $a_p\equiv 2\pi /La\simeq 0.50\mathrm{GeV}$. The binwidth is taken as $\Delta \lambda =0.005a^{-2}$. (a) IR-momentum cut with ${\Lambda }_{\mathrm{IR}}/a_p=1$, 2, and 3. The vertical bars denote the nonzero free-field spectrum. (b) UV-momentum cut with ${\Lambda }_{\mathrm{UV}}/a_p=12$ and 8.

Figure 3

The scatter plot of the diagonal matrix element $F(\lambda )\equiv ⟨\lambda |-{\nabla }^2|\lambda ⟩$ with the IR/UV momentum cut, and original matrix element, for ${16}^4$ lattice with $a\simeq 0.15\mathrm{fm}$, i.e., $a_p\equiv 2\pi /La\simeq 0.50\mathrm{GeV}$. (a) IR-momentum cut with ${\Lambda }_{\mathrm{IR}}/a_p=1$, 2, and 3. The solid-box symbols denote for non-zero free-field values, and the solid line for free-field value in continuum limit, i.e., $F(\lambda )=\lambda$. (b) UV-momentum cut with ${\Lambda }_{\mathrm{UV}}/a_p=12$ and 8.

Figure 4

The full FP spectrum $\rho (\lambda )$ with the IR-momentum cut for $8^4$ lattice at $\beta =5.6$, i.e., $a\simeq 0.25\mathrm{fm}$ and $a_p\equiv 2\pi /La\simeq 0.62\mathrm{GeV}$. The binwidth is taken as $\Delta \lambda =0.1a^{-2}$. The solid curve denotes original spectrum. (a) IR-momentum cut with ${\Lambda }_{\mathrm{IR}}/a_p=1$, (b) ${\Lambda }_{\mathrm{IR}}/a_p=2$, (c) ${\Lambda }_{\mathrm{IR}}/a_p=3$, (d) free field spectrum with rescaled by $1/10$.

Figure 5

The full FP spectrum $\rho (\lambda )$ with the UV-momentum cut at ${\Lambda }_{\mathrm{UV}}/a_p=4$ for $8^4$ lattice at $\beta =5.6$, i.e., $a\simeq 0.25\mathrm{fm}$ and $a_p\equiv 2\pi /La\simeq 0.62\mathrm{GeV}$. The binwidth is taken as $\Delta \lambda =0.1a^{-2}$. The solid curve denotes original spectrum.

Figure 6

The color-Coulomb energy $V_{\mathrm{Coul}}(R)$ obtained with Eq. (21) using low-lying 50 FP eigenmodes apart from trivial zero-modes. For ${16}^4$ lattice at $\beta =5.8$, i.e., $a\simeq 0.15\mathrm{fm}$ and $a_p\equiv 2\pi /La\simeq 0.50\mathrm{GeV}$, we show the IR-momentum cut with ${\Lambda }_{\mathrm{IR}}/a_p=1$, the UV-momentum cut with ${\Lambda }_{\mathrm{UV}}/a_p=12$, and original color-Coulomb energy. For comparison, the instantaneous potential is also added with the fitting curve of Coulomb plus linear form. An irrelevant constant is shifted.