Gluon propagator in Feynman gauge by the method of stationary variance

The low-energy limit of pure Yang-Mills $SU(3)$ gauge theory is studied in Feynman gauge by the method of stationary variance, a genuine second-order variational method that is suited to deal with the minimal coupling of fermions in gauge theories. In terms of standard irreducible graphs, the stationary equations are written as a set of coupled nonlinear integral equations for the gluon and ghost propagators. A physically sensible solution is found for any strength of the coupling. The gluon propagator is finite in the infrared, with a dynamical mass that decreases as a power at high energies. At variance with some recent findings in Feynman gauge, the ghost dressing function does not vanish in the infrared limit and a decoupling scenario emerges as recently reported for the Landau gauge.

Figure 1

The two-point vertices in the interaction $S_2$ of Eq. (15) are shown in the first line. The ghost vertex and the three- and four-gluon vertices of Eqs. (17) are shown in the second line. In the last line the ghost (straight line) and gluon (wavy line) trial propagators are displayed.

Figure 2

First- and second-order two-point graphs contributing to the ghost self energy and the gluon polarization. Second- order terms include nonirreducible graphs.

Figure 3

The gluon propagator $D(p)/D(0)$ as a function of the Euclidean momentum for several values of the bare coupling $g=0.25$, 0.35, 0.45, 0.55, 0.65, 0.75, 0.85, 0.95, 1 (from the top to the bottom). For each bare coupling the energy scale is fixed by taking $M=0.5\mathrm{GeV}$. The Landau gauge lattice data of Ref. [37] ($g=1.02$, $\mathrm{L}=96$) are reported as filled circles.

Figure 4

Log-log plot of the renormalized propagator $D_R(p)$ as obtained by appropriate scaling of the bare propagator for the bare coupling $g=0.35$, 0.40, 0.65, 0.75, 0.90, 1. The scale is arbitrary because of scaling: all curves have been scaled in order to fall on top of the $g=1$ bare propagator of Fig. 3. Energy is in units of ${\mathrm{\Lambda }}_{g=1}$ so that $g(1)=1$ (for $g=1$ the curve is not rescaled). The dotted straight line is a fit of the asymptotic behavior by $D(p)\approx z/p^2$ and $z=0.085$.

Figure 5

The renormalized propagator $D_R(p)$ in physical units for the bare coupling $g=0.35$, 0.40, 0.65, 0.75, 0.90, 1. Scale factors are the same as in Fig. 4 but the energy scale has been fixed in order to fit the lattice data of Ref. [37] ($g=1.02$, $\mathrm{L}=96$) that are displayed as filled circles.

Figure 6

The physical dynamical mass $m^2(p)$ is evaluated by Eq. (58) and displayed in a log-log plot for $g=0.9$ and 1. The leading behavior $m^2(p)\approx (p^2{)}^{-\eta }$ is shown by the dotted straight line for $\eta =0.045$.

Figure 7

The bare second-order propagator $D_{(2)}(p)/D_{(2)}(\mathrm{\Lambda })$ is shown in a log-log plot for $g=0.4$, 0.65, 0.9, 1.15, 1.4, 1.65, 1.9, 2.15, in units of the cutoff $\mathrm{\Lambda }$.

Figure 8

The renormalized second-order propagator is shown in physical units for the same couplings of Fig. 7. By scaling, all the curves fall one on top of the other. The energy scale is fixed by a rough fit of the Landau-gauge lattice data of Ref. [37] ($g=1.02$, $\mathrm{L}=96$) that are dispayed as filled circles.

Figure 9

The second-order dynamical mass of Eq. (67) is reported in a log-log plot with the same scaling factors and energy scale of Fig. 8, for $g=0.9$, 1.15 and 1.4. The straight dotted line shows the power behavior $\sim (p^2{)}^{-\eta }$ with the exponent $\eta =1.5$.

Figure 10

The polarization functions ${\mathrm{\Pi }}^{\prime }$, $-{\mathrm{\Pi }}^{\prime \prime }$ for the ghost-loop graph $(2a)$ and the gluon-loop graph (2b) are displayed for a bare coupling $g=1.2$, in units of the cutoff. The functions ${\mathrm{\Pi }}^{\prime }$ have been shifted by a constant in order to have ${\mathrm{\Pi }}^{\prime }(0)=0$. The transversality condition of Eq. (71), requiring that ${\mathrm{\Pi }}^{\prime }\approx -{\mathrm{\Pi }}^{\prime \prime }$, is not satisfied by the single terms.

Figure 11

Functions ${\mathrm{\Pi }}^{\prime }$, ${\mathrm{\Pi }}^{\prime \prime }$ for the total polarization function, including all the 1PI second-order graphs of Fig. 2. The function ${\mathrm{\Pi }}^{\prime }$ is shifted by the constant term $\delta m^2$ in order to have ${\mathrm{\Pi }}^{\prime }(0)=0$. The opposite of ${\mathrm{\Pi }}^{\prime \prime }$, namely $-{\mathrm{\Pi }}^{\prime \prime }$, is also shown as a dotted line for a direct comparison. We observe that ${\mathrm{\Pi }}^{\prime }\approx -{\mathrm{\Pi }}^{\prime \prime }$, and the transversality condition Eq. (71) is approximately satisfied when all graphs are added together.

Figure 12

The bare second-order ghost dressing function ${\chi }_{(2)}$ for several values of the bare coupling $g$, in units of the cutoff.