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

Abstract

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