Gluon-propagator functional form in the Landau gauge in SU(3) lattice QCD: Yukawa-type gluon propagator and anomalous gluon spectral function

We study the gluon propagator $D_{\mu \nu }^{ab}(x)$ in the Landau gauge in SU(3) lattice QCD at $\beta =5.7$, 5.8, and 6.0 at the quenched level. The effective gluon mass is estimated as $400–600\mathrm{MeV}$ for $r\equiv (x_{\alpha }{x}_{\alpha }{)}^{1/2}=0.5–1.0\mathrm{fm}$. Through the functional-form analysis of $D_{\mu \nu }^{ab}(x)$ obtained in lattice QCD, we find that the Landau-gauge gluon propagator $D_{\mu \mu }^{aa}(r)$ is well described by the Yukawa-type function $e^{-mr}/r$ with $m\simeq 600\mathrm{MeV}$ for $r=0.1–1.0\mathrm{fm}$ in the four-dimensional Euclidean space-time. In the momentum space, the gluon propagator ${\tilde{D}}_{\mu \mu }^{aa}(p^2)$ with $(p^2{)}^{1/2}=0.5–3\mathrm{GeV}$ is found to be well approximated with a new-type propagator of $(p^2+m^2{)}^{-3/2}$, which corresponds to the four-dimensional Yukawa-type propagator. Associated with the Yukawa-type gluon propagator, we derive analytical expressions for the zero-spatial-momentum propagator $D_0(t)$, the effective mass $M_{\mathrm{eff}}(t)$, and the spectral function $\rho (\omega )$ of the gluon field. The mass parameter $m$ turns out to be the effective gluon mass in the infrared region of $\sim 1\mathrm{fm}$. As a remarkable fact, the obtained gluon spectral function $\rho (\omega )$ is almost negative definite for $\omega >m$, except for a positive $\delta$-functional peak at $\omega =m$.

Figure 1

Lattice-QCD results of the scalar-type gluon propagator $D(r)\equiv \sum _{a=1}^8\sum _{\mu =1}^4⟨A_{\mu }^a(x)A_{\mu }^a(0)⟩/24$ as the function of the four-dimensional Euclidean distance $r\equiv (x_{\alpha }{x}_{\alpha }{)}^{1/2}$ in the Landau gauge at $\beta =5.7$, 5.8, and 6.0.

Figure 2

Lattice-QCD results of the scalar-type gluon propagator $\tilde{D}(p^2)=\sum _x{e}^{i\widehat{p}·x}D(r)$ plotted against $p\equiv (p_{\mu }{p}_{\mu }{)}^{1/2}$ with the momentum $p_{\mu }=\frac{2}{a}\sin (\frac{\pi n_{\mu }}{L_{\mu }})$, in the Landau gauge at $\beta =5.7$, 5.8, and 6.0. We renormalize the propagator to satisfy the renormalize condition $D(p^2){|}_{p^2={\mu }^2}=1/{\mu }^2$ at the scale $\mu =4\mathrm{GeV}$. The dash-dotted line denotes the tree-level massless propagator, $1/p^2$.

Figure 3

A typical example of the fit analysis of the lattice gluon propagator $D(r)$ with the fit function $D_{\mathrm{mass}}(r)$ of the massive-vector propagator denoted by the dashed line. The fit is done for the lattice data at $\beta =6.0$ in the fit range of $r=0.6–1.0\mathrm{fm}$.

Figure 4

The effective mass $M_{\mathrm{eff}}(t)$ of gluons in the Landau gauge in lattice QCD at $\beta =6.0$, i.e., $a=0.104\mathrm{fm}$.

Figure 5

The fit result of the lattice gluon propagator $D(r)$ with the three functional forms, $D_{\mathrm{Yukawa}}(r)$ (solid line), $D_{\mathrm{dipole}}(r)$ (dotted line), and $D_{\mathrm{mass}}(r)$ (dashed line). The Yukawa-type function $D_{\mathrm{Yukawa}}(r)$ well reproduces the lattice result at $\beta =6.0$ in the whole region of $r=0.1–1.0\mathrm{fm}$.

Figure 6

The ratio of the lattice-QCD data $D_{\mathrm{latt}}(r)$ at $\beta =6.0$ to the fit functions $D_{\mathrm{mass}}(r)$, $D_{\mathrm{Yukawa}}(r)$, and $D_{\mathrm{dipole}}(r)$ on the scalar-type gluon propagator, i.e., $D_{\mathrm{latt}}/D_{\mathrm{mass}}$, $D_{\mathrm{latt}}/D_{\mathrm{Yukawa}}$, and $D_{\mathrm{latt}}/D_{\mathrm{dipole}}$.

Figure 7

The Yukawa-type function $D_{\mathrm{Yukawa}}(r)=Am{e}^{-mr}/r$ (solid line) with $m=0.624\mathrm{GeV}$ and $A=0.162$ obtained by the fit analysis at $\beta =6.0$, and the lattice-QCD data of the scalar-type gluon propagator $D(r)$ in the Landau gauge at $\beta =5.7$, 5.8, and 6.0 in the range of $r=0.1–1.0\mathrm{fm}$. The lower figure is the logarithmic plot of $rD(r)$ and $r{D}_{\mathrm{Yukawa}}(r)$.

Figure 8

The Yukawa-type propagator in the momentum space, i.e., ${\tilde{D}}_{\mathrm{Yukawa}}(p^2)=4{\pi }^{2}Am(p^2+m^2{)}^{-3/2}$ (solid line) with $m=0.624\mathrm{GeV}$ and $A=0.162$, the same values used in Fig. 7. The horizontal axis is $p\equiv (p_{\alpha }{p}_{\alpha }{)}^{1/2}$. The symbols denote the lattice-QCD data of the scalar-type gluon propagator $\tilde{D}(p^2)$ in the Landau gauge at $\beta =6.0$, where the momentum is defined as $p_{\mu }=\frac{2}{a}\sin (\frac{\pi n_{\mu }}{L_{\mu }})$.

Figure 9

The zero-spatial-momentum propagator $D_0(t)$ of gluons in the Landau gauge. The symbols are the lattice-QCD data at $\beta =6.0$, and the solid line denotes the theoretical curve of Eq. (49), derived from the Yukawa-type propagator with $m=0.624\mathrm{GeV}$ and $A=0.162$, the same values in Fig. 7.

Figure 10

The effective mass $M_{\mathrm{eff}}(t)$ of gluons in the Landau gauge. The symbols denote the lattice-QCD data at $\beta =6.0$, and the solid line denotes the theoretical curve of Eq. (55) derived from the Yukawa-type propagator with $m=0.624\mathrm{GeV}$, the same value used in Fig. 7.

Figure 11

The spectral function $\rho (\omega )$ of the gluon field, associated with the Yukawa-type propagator. The unit is normalized by the mass parameter $m\simeq 600\mathrm{MeV}$. As Eq. (67) indicates, $\rho (\omega )$ shows anomalous behaviors: it has a positive $\delta$-functional peak with the residue of $+\infty$ at $\omega =m(+\epsilon )$, and takes negative values for all of the region of $\omega >m$.

Figure 12

A schematic figure for a propagating gluon in the QCD vacuum. The QCD vacuum is filled with color-magnetic fields which are stochastic at an infrared scale, and the gluon propagates in the random color-magnetic fields.

Figure 13

The infrared behavior of the gluon propagator $\tilde{D}(p^2)$. The triangle symbols denote recent huge-volume lattice data taken from Ref. 32. The solid line denotes the Yukawa-type propagator ${\tilde{D}}_{\mathrm{Yukawa}}(p^2)$, and the dashed line the deep-IR-corrected propagator ${\tilde{D}}_{\mathrm{Yukawa}}^{\mathrm{IR}\mathrm{corr}}(p^2)$ with $p_{\mathrm{IR}}=0.45\mathrm{GeV}$.

Figure 14

The Yukawa-type propagator $D_{\mathrm{Yukawa}}(r)$ (solid line), and deep-IR-corrected propagator $D_{\mathrm{Yukawa}}^{\mathrm{IR}\mathrm{corr}}(r)$ (dashed line), together with the lattice data. The difference between them is fairly small in the IR/IM region of $r=0.1–1.0\mathrm{fm}$.