Ghost propagator and ghost-gluon vertex from Schwinger-Dyson equations

We study an approximate version of the Schwinger-Dyson equation that controls the nonperturbative behavior of the ghost-gluon vertex in the Landau gauge. In particular, we focus on the form factor that enters in the dynamical equation for the ghost dressing function, in the same gauge, and derive its integral equation, in the “one-loop dressed” approximation. We consider two special kinematic configurations, which simplify the momentum dependence of the unknown quantity; in particular, we study the soft gluon case and the well-known Taylor limit. When coupled with the Schwinger-Dyson equation of the ghost dressing function, the contribution of this form factor provides considerable support to the relevant integral kernel. As a consequence, the solution of this coupled system of integral equations furnishes a ghost dressing function that reproduces the standard lattice results rather accurately, without the need to artificially increase the value of the gauge coupling.

Figure 1

The fully dressed ghost-gluon vertex.

Figure 2

The SDE for the ghost propagator given by Eq. (2.7). The white blobs represent the fully dressed gluon and ghost propagators, while the black blob denotes the dressed ghost-gluon vertex.

Figure 3

Comparison of the ghost dressing function, $F(p^2)$, obtained as a solution of the ghost SDE when the ghost-gluon vertex is approximate by its bare value, with the lattice data of Refs. 11, 12. The (red) continuous curve represents the case when ${\alpha }_s(4.3\mathrm{GeV})=0.29$, whereas the (blue) dashed curve is obtained when ${\alpha }_s(4.3\mathrm{GeV})=0.22$.

Figure 4

(a) The complete SDE of the ghost-gluon vertex. Notice that we have set it up with respect to the antighost leg. (b) Diagrams included in the skeleton expansion of the ghost-gluon kernel that we will consider in our analysis.

Figure 5

Lattice results for the gluon propagator, $\Delta (q)$, (left panel) and ghost dressing, $F(q)$, (right panel) obtained in Refs. 11, 12 and renormalized at $\mu =4.3\mathrm{GeV}$. The (red) continuous curves represent the corresponding fits for the lattice data.

Figure 6

Left panel: Numerical result for $A(0,-p,p)$, obtained from Eq. (3.12) when ${\alpha }_s(\mu )=0.22$ (violet dashed line). The (blue) dotted curve represents the prediction of Ref. 32 for the same quantity. Right panel: The numerical solution of $F(p)$ compared with the lattice data of Refs. 11, 12. The (red) continuous and the (blue) dotted curves are obtained when the ghost-gluon vertex is approximated by its bare value for ${\alpha }_s(\mu )=0.29$ and ${\alpha }_s(\mu )=0.22$, respectively. The (violet) dashed result is obtained using $A(0,-p,p)$ for the ghost-gluon vertex in the ghost SDE with ${\alpha }_s(\mu )=0.22$.

Figure 7

Left panel: The form factor $A(-k,0,k)$ (circles) and the fit given by Eq. (4.3) (red continuous line). The (black) dotted and (blue) dashed curves represent the OPE models of Ref. 31. Right panel: The numerical solution of $F(p)$ (red continuous line) compared with the lattice data of Refs. 11, 12. Note that the value of ${\alpha }_s$ used when solving the system is ${\alpha }_s(\mu )=0.22$.

Figure 8

The $SU(2)$ lattice data for the gluon propagator, $\Delta (q)$, obtained in Ref. 9 and renormalized at $\mu =2.2\mathrm{GeV}$. The (red) continuous curve represents the corresponding fit for the lattice data.

Figure 9

Left panel: The $SU(2)$ lattice results for $A(-k,0,k)$ of Ref. 38 (square and circles) and the numerical result for $A(-k,0,k)$, obtained from the system of Eqs. (4.1, 4.2) when ${\alpha }_s(\mu )=0.81$ (red continuous line). The (blue) dotted line represents the OPE model of Ref. 32. Right panel: Comparison of $F(p)$ (red continuous line), obtained from the system of Eqs. (4.1, 4.2) with the $SU(2)$ lattice data of Ref. 9.