Quark-gluon vertex dressing and meson masses beyond ladder-rainbow truncation

We include a generalized infinite class of quark-gluon vertex dressing diagrams in a study of how dynamics beyond the ladder-rainbow truncation influences the Bethe-Salpeter description of light-quark pseudoscalar and vector mesons. The diagrammatic specification of the vertex is mapped into a corresponding specification of the Bethe-Salpeter kernel, which preserves chiral symmetry. This study adopts the algebraic format afforded by the simple interaction kernel used in previous work on this topic. The new feature of the present work is that in every diagram summed for the vertex and the corresponding Bethe-Salpeter kernel, each quark-gluon vertex is required to be the self-consistent vertex solution. We also adopt from previous work the effective accounting for the role of the explicitly non-Abelian three-gluon coupling in a global manner through one parameter determined from recent lattice-QCD data for the vertex. Within the current model, the more consistent dressed vertex limits the ladder-rainbow truncation error for vector mesons to be never more than 10% as the current quark mass is varied from the $u/d$ region to the $b$ region.

Figure 1

Iterative relation for successive terms in the ladder-summed vertex. Large filled circles denote the dressed quark-gluon vertex, letters in parenthesis denote the number of gluon lines contributing to the particular vertices, and small filled circles denote that the propagators are fully dressed. Note that an important non-Abelian term is approximately accounted for by the effective color factor $\mathcal{C}$ as described in the text.

Figure 2

Iterative relation for the enlarged class of dressing diagrams considered in this work. Symbols are the same as in Fig. 1, with $j+k+l+1\equiv i$. The vertex contribution with $i$ internal gluon lines is obtained from vertex contributions with fewer gluon lines.

Figure 3

Vertex skeleton diagrams at $\mathcal{O}(g^5)$. Large filled circle denotes the quark-gluon vertex function dressed to the order of two effective gluon kernel lines. Small filled circles denote that the propagators are fully dressed. Previous work included the ladder structure typified by part (a). The enlarged class of dressing diagrams implemented in this work includes parts (b) and (c) as well. Nonplanar diagrams such as part (d) are not accommodated by the present approach. We use an effective color factor to accommodate a major non-Abelian effect from the three-gluon coupling as described in the text.

Figure 4

Kernel decomposition. Filled triangles represent the meson BSAs; filled circle, the dressed quark-gluon vertex; crossed circle, the Λ function.

Figure 5

Same as Fig. 4, but for Λ function decomposition. Letters in parenthesis denote the number of gluon lines contributing to the particular functions.

Figure 6

Quark propagator amplitude $A(s)$ vs Euclidean $s=p^2$. We use the interaction mass scale $\mathcal{G}=1\mathrm{GeV}$ and the current mass is $m=0.0183\mathcal{G}=18.3$ MeV. $\mathcal{C}$ dependence calculated with converged summation of vertex dressing, for $\mathcal{C}=0.15$ (solid curve), $\mathcal{C}=0.5$ (dashed curve), $\mathcal{C}=-0.125$ (dot-dashed curve), and $\mathcal{C}=-0.25$ (dotted curve).

Figure 7

Same as Fig. 6, but showing the influence of vertex dressing to order $n$ as described in the text. For $\mathcal{C}=0.15,n=0$ yields the solid curve and the result is the ladder-rainbow truncation.

Figure 8

Quark mass function $M(s)$ vs Euclidean $s=p^2$. We use the interaction mass scale $\mathcal{G}=1\mathrm{GeV}$ and the current mass is $m=0.0183\mathcal{G}=18.3$ MeV. $\mathcal{C}$ dependence calculated with converged summation of vertex dressing, for $\mathcal{C}=0.15$ (solid curve), $\mathcal{C}=0.5$ (dashed curve), $\mathcal{C}=-0.125$ (dot-dashed curve), and $\mathcal{C}=-0.25$(dotted curve).

Figure 9

Same as Fig. 8, but showing the influence of vertex dressing to order $n$ as described in the text. For $\mathcal{C}=0.15,n=0$ yields the solid curve and the result is the ladder-rainbow truncation.

Figure 10

Gluon-quark vertex amplitude ${\alpha }_1(s)$ vs Euclidean $s=p^2$, for $\mathcal{C}=0.15$. We use the interaction mass scale $\mathcal{G}=1\mathrm{GeV}$ and the current mass is $m=0.0183\mathcal{G}=18.3$ $\mathrm{MeV}$. $n=0$ (solid curve) results from the bare vertex and is the ladder-rainbow truncation.

Figure 11

Same as Fig. 10, but for ${\alpha }_2(s)$.

Figure 12

Same as Fig. 10, but for ${\alpha }_3(s)$.