Lattice study of the gluon propagator in momentum space

Abstract

We consider pure glue QCD at $\beta =5.7\mathrm{}$, $\beta =6.0\mathrm{}$, and $\beta =6.3\mathrm{}$. We evaluate the gluon propagator both in time at zero three-momentum and in momentum space. From the former quantity we obtain evidence for a dynamically generated effective mass, which at $\beta =6.0\mathrm{}$ and $\beta =6.3\mathrm{}$ increases with the time separation of the sources, in agreement with earlier results. The momentum space propagator $G\left(k\right)\mathrm{}$ provides further evidence for mass generation. In particular, at $\beta =6.0\mathrm{}$, for $300 \mathrm{MeV}\lesssim k\lesssim 1GeV$, the propagator $G\left(k\right)\mathrm{}$ can be fit to a continuum formula proposed by Gribov and others, which contains a mass scale $b$, presumably related to the hadronization mass scale. For higher momenta Gribov's model no longer provides a good fit, as $G\left(k\right)\mathrm{}$ tends rather to follow an inverse $\mathrm{power}\mathrm{law}\approx \frac{1}{k^{2+\gamma }}$. The results at $\beta =6.3\mathrm{}$ are consistent with those at $\beta =6.0\mathrm{}$, but only the high momentum region is accessible on this lattice. We find $b$ in the range of 300 to 400 MeV and $\gamma$ about 0.7. Fits to particle + ghost expressions are also possible, often resulting in low values for ${\chi }_{\mathrm{DF}}^2$, but the parameters are very poorly determined. On the other hand, at $\beta =5.7\mathrm{}$ (where we can only study momenta up to 1 GeV) $G\left(k\right)\mathrm{}$ is best fit to a simple massive boson propagator with mass $m$. We argue that such a discrepancy may be related to a lack of scaling for low momenta at $\beta =5.7\mathrm{}$. From our results, the study of correlation functions in momentum space looks promising, especially because the data points in Fourier space turn out to be much less correlated than in real space.