Lattice gluon propagator in renormalizable $\xi$ gauges

We study the SU(3) gluon propagator in renormalizable $R_{\xi }$ gauges implemented on a symmetric lattice with a total volume of $(3.25\mathrm{fm}{)}^4$ for values of the gauge fixing parameter up to $\xi =0.5$. As expected, the longitudinal gluon dressing function stays constant at its tree-level value $\xi$. Similar to the Landau gauge, the transverse $R_{\xi }$ gauge gluon propagator saturates at a nonvanishing value in the deep infrared for all values of $\xi$ studied.

Figure 1

Number of iterations required to meet our convergence goal $\theta <10^{-15}$ with a FFT steepest descent minimization procedure (similar results hold for the OVR and STR minimization procedures). We illustrate the case of 200 uncorrelated configurations for the value $\xi =0.3$. Notice that at 50000 iterations the minimization is considered to have failed.

Figure 2

(Top panel) The ${32}^4$ values of $\nabla ·A^4$ evaluated for a configuration gauge fixed at $\xi =0.5$, grouped in 5000 bins, compared with a Gaussian with standard deviation $\sqrt{\xi }\simeq 0.316$. (Bottom panel) Plot of $d=\nabla ·A^4-{\mathrm{\Lambda }}^4$; the two distributions coincide within $\sqrt{\theta }$ precision.

Figure 3

(Left panel) The $R_{\xi }$ longitudinal dressing function $q^2{\mathrm{\Delta }}_{\mathrm{L}}\equiv \xi$; a fit of the data to a constant yield $\xi =0.103(2)$, 0.203(2), 0.302(3), 0.402(3), and 0.502(3) respectively. (Right panel) The $R_{\xi }$ gluon transverse dressing function $q^2{\mathrm{\Delta }}_{\mathrm{T}}$. Landau gauge results obtained for a symmetric lattice of $L=80$ and $\beta =6.0$ (gray crosses) are also plotted [9].

Figure 4

(Left panel) The $R_{\xi }$ transverse propagator ${\mathrm{\Delta }}_{\mathrm{T}}$ renormalized at $\mu =4.317[\mathrm{GeV}]$. The gray crosses (the same as Fig. 3 right [9]) provide an estimate for the volume effects expected at $q^2=0$. (Right panel) The ratio ${\mathrm{\Delta }}_{\mathrm{T}}(q^2)/{\mathrm{\Delta }}_{\mathrm{T}}^{\xi =0}(q^2)$.