Off-diagonal gluon mass generation and infrared Abelian dominance in maximally Abelian gauge in SU(3) lattice QCD

In SU(3) lattice QCD formalism, we propose a method to extract gauge fields from link variables analytically. With this method, we perform the first study on effective mass generation of off-diagonal gluons and infrared Abelian dominance in the maximally Abelian gauge in the SU(3) case. Using SU(3) lattice QCD, we investigate the propagator and the effective mass of the gluon fields in the maximally Abelian gauge with $\mathrm{U}(1{)}_3\times \mathrm{U}(1{)}_8$ Landau gauge fixing. The Monte Carlo simulation is performed on ${16}^4$ at $\beta =5.7$, 5.8, and 6.0 at the quenched level. The off-diagonal gluons behave as massive vector bosons with the approximate effective mass $M_{\mathrm{off}}\simeq 1.0–1.1\mathrm{GeV}$ in the region of $r=0.3–0.8\mathrm{fm}$, and the propagation is limited within a short range, while the propagation of diagonal gluons remains even in a large range. In this way, infrared Abelian dominance is shown in terms of short-range propagation of off-diagonal gluons. Furthermore, we investigate the functional form of the off-diagonal gluon propagator. The functional form is well described by the four-dimensional Euclidean Yukawa-type function $e^{-m_{\mathrm{off}}r}/r$ with $m_{\mathrm{off}}\simeq 1.2–1.3\mathrm{GeV}$ for $r=0.1–0.8\mathrm{fm}$. This also indicates that the spectral function of off-diagonal gluons has the negative-value region.

Figure 1

The SU(3) lattice QCD results of $G_{\mu \mu }^{\mathrm{Abel}}(r)$ and $G_{\mu \mu }^{\mathrm{off}}(r)$ (top panel), and their logarithmic plots (bottom panel) as the function of $r\equiv \sqrt{(x_{\mu }-y_{\mu }{)}^2}$ in the MA gauge with the $\mathrm{U}(1{)}_3\times \mathrm{U}(1{)}_8$ Landau gauge fixing in the physical unit. The Monte Carlo simulation is performed on the ${16}^4$ lattice with $\beta =5.7$, 5.8 and 6.0. The diagonal-gluon propagator $G_{\mu \mu }^{\mathrm{Abel}}(r)$ takes a large value even at the long distance. On the other hand, the off-diagonal gluon propagator $G_{\mu \mu }^{\mathrm{off}}(r)$ rapidly decreases.

Figure 2

The logarithmic plot of $r^{3/2}{G}_{\mu \mu }^{\mathrm{off}}(r)$ and $r^{3/2}{G}_{\mu \mu }^{\mathrm{Abel}}(r)$ as the function of the four-dimensional Euclidean distance $r$ in the MA gauge with the $\mathrm{U}(1{)}_3\times \mathrm{U}(1{)}_8$ Landau gauge fixing, using the SU(3) lattice QCD with ${16}^4$ at $\beta =5.7$, 5.8 and 6.0. The solid line denotes the logarithmic plot of $r^{3/2}{G}_{\mu \mu }(r)\sim r^{1/2}{K}_1(Mr)$ in the Proca formalism.

Figure 3

The logarithmic plot of $r{G}_{\mu \mu }^{\mathrm{off}}(r)$ and $r{G}_{\mu \mu }^{\mathrm{Abel}}(r)$ as the function of the four-dimensional Euclidean distance $r$ in the MA gauge with the $\mathrm{U}(1{)}_3\times \mathrm{U}(1{)}_8$ Landau gauge fixing, using the SU(3) lattice QCD with ${16}^4$ at $\beta =5.7$, 5.8 and 6.0. For $r{G}_{\mu \mu }^{\mathrm{off}}(r)$, the approximate linear correlation is found.