Gluonic-excitation energies and Abelian dominance in SU(3) QCD

We present the first study of the Abelian-projected gluonic-excitation energies for the static quark-antiquark ($\mathrm{Q}\overline{\mathrm{Q}}$) system in SU(3) lattice QCD at the quenched level, using a ${32}^4$ lattice at $\beta =6.0$. We investigate ground-state and three excited-state $\mathrm{Q}\overline{\mathrm{Q}}$ potentials, using smeared link variables on the lattice. We find universal Abelian dominance for the quark confinement force of the excited-state $\mathrm{Q}\overline{\mathrm{Q}}$ potentials as well as the ground-state potential. Remarkably, in spite of the excitation phenomenon in QCD, we find Abelian dominance for the first gluonic-excitation energy of about 1 GeV at long distances in the maximally Abelian gauge. On the other hand, no Abelian dominance is observed for higher gluonic-excitation energies even at long distances. This suggests that there is some threshold between 1 and 2 GeV for the applicable excitation-energy region of Abelian dominance. Also, we find that Abelian projection significantly reduces the short-distance $1/r$-like behavior in gluonic-excitation energies.

Figure 1

Effective mass plots $V_{\mathrm{eff}}(r,t)$ of the SU(3) potential $V(r)$ for (a) the ground state, (b) the first-excited state, and (c) the second-excited state in the static $\mathrm{Q}\overline{\mathrm{Q}}$ system. Here, we display on-axis data of $r=3$, 6, 9, 12, 15 in the lattice unit. Larger $r$ data are a bit shifted horizontally for visibility.

Figure 2

Effective mass plots $V_{\mathrm{eff}}^{\text{Abel}}(r,t)$ of the Abelian potential $V^{\text{Abel}}(r)$ for (a) the ground state, (b) the first-excited state, and (c) the second-excited state in the static $\mathrm{Q}\overline{\mathrm{Q}}$ system. Here, we display on-axis data of $r=3$, 6, 9, 12, 15 in the lattice unit. Larger $r$ data are a bit shifted horizontally for visibility.

Figure 3

(a) The ground-state potential and two excited-state potentials in the static $\mathrm{Q}\overline{\mathrm{Q}}$ system. The circles and the squares denote the SU(3) potentials $V_0$, $V_1$, $V_2$ and the Abelian potentials $V_0^{\text{Abel}},V_1^{\text{Abel}},V_2^{\text{Abel}}$, respectively. The solid curves are the best fits with the Cornell potential $-A/r+\sigma r+C$ for $V_0$ and $V_0^{\text{Abel}}$ for $r>2a$. The dashed lines are the best fits with $\sigma r+C$ for excited-state potentials $V_1$, $V_2$, $V_1^{\text{Abel}}$, $V_2^{\text{Abel}}$ at long distances. For the clear display, an irrelevant overall constant ($+0.2\mathrm{GeV}$) is added to $V_n^{\text{Abel}}$. (b) Gluonic-excitation energies $\mathrm{\Delta }{E}_n(r)\equiv V_n(r)-V_0(r)$. The circles and squares denote gluonic-excitation energies $\mathrm{\Delta }{E}_1$, $\mathrm{\Delta }{E}_2$ and the Abelian parts $\mathrm{\Delta }{E}_1^{\text{Abel}}$, $\mathrm{\Delta }{E}_2^{\text{Abel}}$, respectively. The curves are the best fits with the ansatz $a/r+E^{\mathrm{th}}$ for SU(3) gluonic-excitations.