Pion physics with dressed quark-gluon vertices

TL;DR

This paper develops a symmetry-preserving truncation for Schwinger-Dyson and Bethe-Salpeter equations that includes the full, dressed quark-gluon vertex in a tractable way. By dropping gluon-axial-vector dependent diagrams and replacing the vertex inside one-loop diagrams with a simple symmetric-input form factor V, the authors obtain a pion BSE composed of three diagrams (two one-loop dressed and one two-loop crossed) that preserves the axial Ward-Takahashi identities. They solve the coupled SDE-BSE system in the chiral limit and verify that the pion is massless via an exact WTIs-consistent relation, with the three diagrams contributing roughly 66%, 33%, and 1% to the kernel. The results demonstrate robustness of the truncation, showing the pion properties are tied to the dynamical quark mass function and remain stable under variations of V. This framework advances the nonperturbative description of pion structure beyond rainbow-ladder and sets the stage for including finite quark masses and complex momenta.

Abstract

Recently, a theoretical framework was set up in [1], which allows for the symmetry-preserving inclusion of full quark-gluon vertices in the description of the meson dynamics. In the present work, we develop a special truncation within this approach, which leads to a tractable set of functional equations that satisfy the fundamental chiral Ward-Takahashi identities. Specifically, the truncation allows us to simplify considerably the quark-gluon Schwinger-Dyson equation, without significant loss of quantitative accuracy. Importantly, this implies a substantial reduction of complexity of the renormalized Bethe-Salpeter equation: it is composed by a pair of one-loop diagrams that contain the full quark-gluon vertex, and a single two-loop diagram that is instrumental for the masslessness of the pion in the chiral limit. A detailed numerical analysis reveals that the incorporation of the aforementioned two-loop diagram is instrumental for the corresponding eigenvalue to reach unity. The key relation between the quark mass function and the pion wave function is shown to be satisfied to within the numerical precision of the loop integrals, which is at the level of about one percent or better. The field-theoretic ingredients required for the extension of this analysis beyond the chiral limit are briefly discussed.

Figures (18)

• Figure 1: Panel A: Diagrammatic representation of the SDE (gap equation) that determines the quark propagator $S(p)$. White circles denote full propagators, red circles stand for full quark-gluon vertices, while small black circles indicate tree-level quark-gluon vertices. Panel B: The SDE of the quark-gluon vertex (red circle) obtained within the 3PI formalism. The gray circle denotes the fully-dressed three-gluon vertex.
• Figure 2: Panel A: The SDE of the axial-vector vertex, $\Gamma_{\!5}^\mu$ (blue circles). The graph ($b_5$) contains the "gluon-axial-vector" vertex, $G_5^{ab \mu\nu}$ (yellow circle), which is crucial for the preservation of the axial WTI. Panel B: The one-loop dressed representation of the gluon-axial-vector vertex; we refer to the graphs $d_1,~d_2,~d_3$ as $G$- independent, while to the $g_1,~g_2,~g_3,~h_1,~h_2$ as $G$- dependent.
• Figure 3: Panel A: The truncated version of the SDE for the vertex $G_5^{\mu\nu}$, composed only by $G$-independent graphs. The green circles denote the component $V_{\mu}$, defined in Eq.(\ref{['eq:qgsym']}). Panel B: The SDE of the quark-gluon vertex (cyan circle), which is compatible with the truncation of the $G_5^{\mu\nu}$ shown in the panel above.
• Figure 4: Panel A: The SDE for the axial-vector vertex, emerging once the truncation for the $G_5^{\mu\nu}$ shown in Panel A of \ref{['fig:1ld_green']} has been implemented. Panel B: The pion BSE, obtained after contracting the SDE in the panel above with $P_\mu$ and then taking the limit $P \to 0$.
• Figure 5: Panel A: The diagrammatic formation of $\Gamma_{\!{ Q}}$, once the effective renormalization has been carried out. Panel B: The final form of the BSE after renormalization. Diagram ($a$) is denominated "dressed RL" because it corresponds to the standard RL diagram, but now dressed with a full quark-gluon vertex, diagram ($b$) is called "quantum" because it contains the quantum part of the same vertex, while diagram ($c$), which is named "crossed" due to its geometry, contains only the $V$.