Just how different are the $SU(2)$ and $SU(3)$ Landau-gauge propagators in the infrared regime?

The infrared behavior of gluon and ghost propagators in Yang-Mills theories is of central importance for understanding quark and gluon confinement in QCD. While simulations of pure $SU(3)$ gauge theory correspond to the physical case in the limit of infinite quark mass, the $SU(2)$ case (i.e. pure two-color QCD) is usually employed as a simplification, in the hope that qualitative features be the same as for the $SU(3)$ case. Here we carry out the first comparative study of lattice (Landau) propagators for these two gauge groups. Our data were especially produced with equivalent lattice parameters in order to allow a careful comparison of the two cases. We find very good agreement between $SU(2)$ and $SU(3)$ propagators, showing that in the IR limit the equivalence of the two cases is quantitative, at least down to about 1 GeV. Our results suggest that the infrared behavior of these propagators is independent of the gauge group $SU(N_c)$, as predicted by Schwinger-Dyson equations.

Figure 1

Renormalized gluon propagator as a function of the squared magnitude $k^2$ of the four-momentum $k$, for on-axis momenta $(k,0,0,0)$. We show results for the $SU(3)$ and the $SU(2)$ cases. In the two top plots we report data for the three sets of lattice parameters with approximately the same physical lattice volume $V$. In the two bottom plots we report data for the two sets of lattice parameters with $\beta =6.0$ for $SU(3)$ and $\beta =2.4469$ for $SU(2)$. In all cases we show only data for $k^2\le 5{\mathrm{GeV}}^2$. For larger momenta, the data using different lattice setups agree well.

Figure 2

Renormalized ghost propagator as a function of the squared magnitude $k^2$ of the four-momentum $k$, for on-axis momenta $(k,0,0,0)$. The data are organized as in Fig. 1.

Figure 3

Ratios of $SU(3)$ over $SU(2)$ gluon propagators for the four lattice setups considered.

Figure 4

Ratios of $SU(3)$ over $SU(2)$ ghost propagators for the four lattice setups considered.