Pion physics with dressed quark-gluon vertices

TL;DR

This paper develops a simplified, symmetry-preserving framework for studying pion properties using dressed quark-gluon vertices, leading to accurate numerical results and insights into meson dynamics within quantum chromodynamics.

Contribution

It introduces a tractable truncation of the quark-gluon interaction that preserves chiral symmetry and simplifies calculations of meson properties in quantum field theory.

Findings

The truncation satisfies chiral Ward-Takahashi identities.

Numerical analysis shows the two-loop diagram is crucial for pion masslessness.

The quark mass function and pion wave function are accurately related.

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.