Abstract Recently, a theoretical framework was set up in Miramontes et al. (Eur Phys J C 85:1055, 2025), 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, in the chiral limit that we consider, this 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.
Ferreira et al. (2026) studied this question.