Are the dressed gluon and ghost propagators in the Landau gauge presently determined in the confinement regime of QCD?

The gluon and ghost propagators in Landau gauge QCD are investigated using the Schwinger-Dyson equation approach. Working in Euclidean spacetime, we solve for these propagators using a selection of vertex inputs, initially for the ghost equation alone and then for both propagators simultaneously. The results are shown to be highly sensitive to the choices of vertices. We favor the infrared finite ghost solution from studying the ghost equation alone where we argue for a specific unique solution. In order to solve this simultaneously with the gluon using a dressed-one-loop truncation, we find that a nontrivial full ghost-gluon vertex is required in the vanishing gluon momentum limit. The self-consistent solutions we obtain correspond to having a masslike term in the gluon propagator dressing, in agreement with similar studies supporting the long-held proposal of Cornwall.

Figure 1

The Schwinger-Dyson equation for the gluon in the absence of quarks.

Figure 2

The Schwinger-Dyson equation for the ghost.

Figure 3

The ghost-gluon vertex indicating the momentum definition we adopt; the outgoing ghost momentum is $q$ and the gluon momentum is $k$.

Figure 4

A sketch of typical dressing functions. The momentum scale is arbitrary, but can be thought of as $\sim 1\mathrm{GeV}$.

Figure 5

Examples of ghost solutions on a log-log plot subtracting at zero momentum. Only the specified subtraction value $\mathcal{G}h(0)$ is varied between the solutions which can be read off. The $\mathcal{G}h(p^2=0)$ value is fixed and the ghost equation solved with the depicted gluon until the ghost inputs and outputs are self-consistent. The gluon dressing $\mathcal{G}\ell (p^2)$ is the dashed curve and the solid curves are the different ghost dressings $\mathcal{G}h(p^2)$. The units of $p^2$ are arbitrary since we have not fixed the coupling to the physical value, but may be considered to be $\mathcal{O}(1{\mathrm{GeV}}^2)$.

Figure 6

The dotted curves correspond to the zero-momentum subtracted solutions and the colors match Fig. 5. The dashed curve is the gluon input, $\mathcal{G}\ell (p^2)$, and the solid curve is the physically relevant solution of the ghost equation, $\mathcal{G}h(p^2)$. The units of $p^2$ are arbitrary.

Figure 7

A comparison of two solutions normalized using the perturbative condition. The (red) dotted curve is the ghost solution, $Gh(p^2)$, with a bare vertex or transverse vertex. The (blue) dashed curve is the ghost dressing obtained using Eq. (14). The (black) solid curve is the fixed gluon dressing input, $\mathcal{G}\ell (p^2)$ from Eq. (13). The units of $p^2$ are arbitrary internal units.

Figure 8

The singular ghost solutions and violation of transversality in the infrared. The dashed gluon and dotted ghost curves are obtained first using the $\mathcal{A}$ term and then we switch to the $\mathcal{B}$ term and we find that transversality is broken since the curves differ. The units of $p^2$ are arbitrary internal units.

Figure 9

The range of solutions obtained using the possible vertex combinations. The label $(i,j)$ refers to the vertices used in obtaining the solutions corresponding to ${\Gamma }_{\mu }^{(i)}$ for the ghost-gluon vertex and ${\Gamma }_{\mu \nu \rho }^{(j)}$ for the triple-gluon vertex. The missing curve corresponds to the ${\Gamma }_{\mu }^{(1)}$ and ${\Gamma }_{\mu \nu \rho }^{(B)}$; self-consistent solutions were not obtained there. The parameters are not varied between the solutions, only the vertices. The units of $p^2$ are arbitrary since we have not fixed the coupling to a physical value.

Figure 10

Gluon polarization functions, $p^2{\Pi }_{2c}(p^2)$ (positive values) and $p^2{\Pi }_{2g}(p^2)$ (negative values). Left: IR region, Right: UV region. In the UV, the curves are indistinguishable on this scale. In the IR, the ghost-loop curves for multiple solutions lie on top of each other. The bare ghost-gluon vertex, not shown here, always gives a vanishing IR contribution given an IR vanishing gluon dressing and an IR finite ghost dressing. The functions $p^2{\Pi }_{2c}(p^2,{\mu }^2)$ are labeled according to the input vertices with the contributions from ${\Gamma }_{\mu }^{(i)}$ and ${\Gamma }_{\mu \nu \rho }^{(j)}$ labeled as $(i,j)$ in the plot. The units of $p^2$ are arbitrary.

Figure 11

Self-consistent solutions tuned to lattice solutions, showing the dressing functions. The solid curves depict a smooth fit to the lattice data. The heavier region is where the functions are represented by lattice data and the feint region represents the natural extrapolation. The broken curves are the tuned solutions. Left: The blue peaked curves correspond to the gluon dressing function $\mathcal{G}\ell (p^2)$, which vanishes as $p^2\to 0$, the red monotonic curves correspond to the ghost dressing function $\mathcal{G}h(p^2)$. Right: The upper (blue) curves correspond to $\mathcal{G}\ell (p^2)/p^2$, which is $\sim 10$ as $p^2\to 0$, the lower (red) curves correspond to the ghost dressing function again.

Figure 12

The running coupling ${\alpha }_T(p^2)$ from Eq. (22) is shown as the black solid curve. The running coupling ${\alpha }_{EC}(p^2)$ from Eq. (23) is shown as the green solid curve. The gluon dressing $\mathcal{G}\ell (p^2)$ is the dashed blue curve and the dotted red curve is the ghost dressing function $\mathcal{G}h(p^2)$. The units of $p^2$ are now very close to ${\mathrm{GeV}}^2$.