Unquenching the gluon propagator with Schwinger-Dyson equations

In this article we use the Schwinger–Dyson equations to compute the nonperturbative modifications caused to the infrared finite gluon propagator (in the Landau gauge) by the inclusion of a small number of quark families. Our basic operating assumption is that the main bulk of the effect stems from the “one-loop dressed” quark loop contributing to the full gluon self-energy. This quark loop is then calculated, using as basic ingredients the full quark propagator and quark-gluon vertex; for the quark propagator we use the solution obtained from the quark-gap equation, while for the vertex we employ suitable Ansätze, which guarantee the transversality of the answer. The resulting effect is included as a correction to the quenched gluon propagator, obtained in recent lattice simulations. Our main finding is that the unquenched propagator displays a considerable suppression in the intermediate momentum region, which becomes more pronounced as we increase the number of active quark families. The influence of the quarks on the saturation point of the propagator cannot be reliably computed within the present scheme; the general tendency appears to be to decrease it, suggesting a corresponding increase in the effective gluon mass. The renormalization properties of our results, and the uncertainties induced by the unspecified transverse part of the quark-gluon vertex, are discussed. Finally, the gluon propagator is compared with the available unquenched lattice data, showing rather good agreement.

Figure 1

The full PT-BFM gluon self-energy. White (respectively, black) blobs represents connected (respectively, 1-particle irreducible) Green’s functions; the small gray circles on the external legs indicate background gluons.

Figure 2

Definitions and conventions of the auxiliary functions $\Lambda$ and $H$. The color and gauge coupling dependence for the field combination shown, $c^a(p)A_{\mu }^b(r)A_{\nu }^{*c}(q)$, is $g{f}^{acb}$. Gray blobs denote one-particle irreducible (with respect to vertical cuts) Schwinger–Dyson kernels.

Figure 3

The nonlinear propagation of the effect of unquenching the gluon propagator through the addition of dynamical fermions, shown here for the one-loop dressed gluon diagram ($a_1$). Both the internal gluon propagator and the three-gluon vertex get modified (shown here by two representative graphs only); similar modifications occur for all other diagrams.

Figure 4

Comparison between the gluon propagator, $\Delta (q^2)$, obtained from the SDE renormalized at $\mu =4.3\mathrm{GeV}$ (solid red curve) and the lattice data (gray symbols) of Ref. [7].

Figure 5

The full PT-BFM quark-gluon vertex ${\widehat{\Gamma }}_{\mu }^a$. Note that $p_1\leftrightarrow p_2$ with respect to the conventions used in [31, 72].

Figure 6

Diagrammatic representation of the quark-ghost scattering kernel $H(p_1,p_2,p_3)$.

Figure 7

Lattice result for the $SU(3)$ gluon propagator (left panel) and ghost dressing function (right panel) renormalized at three different points: $\mu =4.3\mathrm{GeV}$ (solid red curve), $\mu =3.0\mathrm{GeV}$ (dash-dotted black curve), and $\mu =2.5\mathrm{GeV}$ (dotted blue curve).

Figure 8

Solution of the quark-gap equation: $A^{-1}(k^2)$ (left panel) and $B(k^2)$ (right panel) renormalized at $\mu =4.3\mathrm{GeV}$. Dotted black curves correspond to the improved BC vertex, while dashed blue curves correspond to the improved CP vertex.

Figure 9

The momentum dependence of the dynamical quark mass $\mathcal{M}(k^2)=B(k^2)/A(k^2)$, using the same conventions as in the previous plot.

Figure 10

Individual contributions of the terms proportional to $T_1$ (dashed red curve), $T_2$ (dotted blue line), $T_3$ (dash-dotted green), and $\delta \widehat{X}(q^2)$ (dashed with two dots orange curve), to $\widehat{X}(q^2)$ (left panel) and ${\widehat{X}}^{\mathrm{CP}}(q^2)$ (right panel), respectively. The sum of all contributions produce in both cases the continuous (black with white circles) curve, which represent the full quark-loop contribution to the gluon propagator.

Figure 11

Comparison between the contributions of the quark loop $\widehat{X}(q^2)$ (dotted black curve) and ${\widehat{X}}^{\mathrm{CP}}(q^2)$ (dashed blue curve) to the gluon self-energy with $n_f=2$.

Figure 12

The unquenched gluon propagator ${\Delta }_Q(q^2)$ when no extrapolation is used, i.e. ${\lambda }^2=0$ in Eq. (3.57) (left panel). The dotted black curve represents the unquenched propagator obtained with the BC vertex, whereas the dashed blue curve represents the result for the CP vertex. The result for ${\Delta }_Q(q^2)$ when the extrapolation is performed in the shade area, i.e. from $q^2=0.05{\mathrm{GeV}}^2$ toward the deep IR (right panel).

Figure 13

Comparison between the quenched $\Delta (q^2)$ and the unquenched ${\Delta }_Q(q^2)$ gluon propagators. The yellow striped band shows the possible values that ${\Delta }_Q(0)$ can assume at zero momentum. The two curves delimiting the band represent the extrapolation toward the IR either starting from $0.02{\mathrm{GeV}}^2$ (dash-dotted green line) or $0.07{\mathrm{GeV}}^2$ (dashed orange line). The dashed with two dots magenta curve corresponds to an extrapolation of the numerical result starting from the intermediate value $0.05{\mathrm{GeV}}^2$.

Figure 14

The quenched (solid red curve) and unquenched (dotted black curve) gluon propagators (left panel) and dressing functions (right panel). The unquenched case corresponds to the case where the extrapolation starts at $q^2=0.05{\mathrm{GeV}}^2$.

Figure 15

The unquenched gluon propagator for different number of flavors: $n_f=1$ (dash-dotted green curve), $n_f=2$ (dotted black curve), and $n_f=3$ (dashed blue curve).

Figure 16

Left panel: The unquenched gluon propagator ${\Delta }_Q(q^2)$ for $n_f=2$ (dotted black curve) compared to the one-loop dressed result with a constant quark mass $M=292\mathrm{MeV}$ (dash-dotted blue curve). Right panel: The unquenched gluon propagator ${\Delta }_Q(q^2)$ for $n_f=2$ with different values of constant quark masses: $M=1\mathrm{GeV}$ (dashed green curve) and $M=292\mathrm{MeV}$ (dash-dotted blue curve).

Figure 17

Left panel: The $n_f=2$ unquenched gluon propagator renormalized at different values of $\mu$ and ${\alpha }_s$: $\mu =4.3\mathrm{GeV}$ and ${\alpha }_s=0.295$ (solid red curve), $\mu =3.0\mathrm{GeV}$ and ${\alpha }_s=0.395$ (dash-dotted black curve), $\mu =2.5\mathrm{GeV}$ and ${\alpha }_s=0.461$ (dotted blue curve). Right panel: All curves shown in the left panel renormalized at the same point $\mu =2.5\mathrm{GeV}$ using Eq. (4.4).

Figure 18

The unquenched gluon propagator (left panel) and the gluon dressing function (right panel) obtained in Ref. [5] (dark gray stars), together with the SDE results for two light quarks with $M_{u/d}=15\mathrm{MeV}$ (dashed blue curve) and $M_{u/d}=710\mathrm{MeV}$ (dotted black curve). The quenched lattice results are also displayed for comparison.

Figure 19

The unquenched gluon propagator (left panel) and the gluon dressing function (right panel) obtained in Ref. [6] (dark gray stars), together with the SDE result for two light quarks with $M_{u/d}=292\mathrm{MeV}$ and one heavier with $M_s=500\mathrm{MeV}$ (dash-dotted black curve), and the case where $M_{u/d}=360\mathrm{MeV}$ and $M_s=560\mathrm{MeV}$ (dotted blue curve).

Figure 20

The unquenched gluon propagator for $n_f=2+1$ flavors. The light quarks have masses of $M_{u/d}=292\mathrm{MeV}$, while for the heavier quark: $M_h=292\mathrm{GeV}$ (short dashed blue curve), $M_h=500\mathrm{MeV}$ (dashed orange curve), $M_h=1.0\mathrm{GeV}$ (dotted green curve), and $M_h=1.5\mathrm{GeV}$ (dashed with two dots magenta curve).

Figure 21

The unquenched gluon dressing function for $n_f=2+1$ flavors. The light quarks have masses of $M_{u/d}=292\mathrm{MeV}$, while for the heavier quark: $M_h=292\mathrm{GeV}$ (short dashed blue curve), $M_h=500\mathrm{MeV}$ (dashed orange curve), $M_h=1.0\mathrm{GeV}$ (dotted green curve), and $M_h=1.5\mathrm{GeV}$ (dashed with two dots magenta curve).