Abstract
Correlation functions can be described by the corresponding equations, viz., the gap equation for the quark propagator and the inhomogeneous Bethe-Salpeter equation for the vector dressed-fermion-Abelian-gauge-boson vertex in which specific truncations have to be implemented. The general vector and axial-vector Ward-Green-Takahashi identities require these correlation functions to be interconnected; in consequence of this, truncations made must be controlled consistently. It turns out that, if the rainbow approximation is assumed in the gap equation, the scattering kernel in the Bethe-Salpeter equation can adopt the ladder approximation, which is one of the most basic attempts to truncate the scattering kernel. Additionally, a modified-ladder approximation is also found to be a possible symmetry-preserving truncation scheme. As an illustration of this approximation for application, a treatment of the pion is included. The pion mass and decay constant are found to be degenerate in ladder and modified-ladder approximations, even though the Bethe-Salpeter amplitudes are with apparent distinction. The justification for the modified-ladder approximation is examined with the help of the Gell-Mann-Oakes-Renner relation.
Highlights
The hadron is a composite particle consist of quarks and gluons that are strongly interacting, so it cannot be described by perturbation theory
Correlation functions can be described by the corresponding equations, viz., the gap equation for the quark propagator and the inhomogeneous Bethe-Salpeter equation for the vector dressed-fermion-Abeliangauge-boson vertex in which specific truncations have to be implemented
Our work so far has consisted of setting up a general quark-antiquark scattering kernel in the Bethe-Salpeter equation, the modified-ladder approximation, especially in the case of Λþβ, derived directly from the vector and axial-vector Ward-Green-Takahashi identities
Summary
The hadron is a composite particle consist of quarks and gluons that are strongly interacting, so it cannot be described by perturbation theory. The underlying law governing their application in a simple and elegant way is the preserving of vector and axial-vector Ward-Green-Takahashi identities driven by gauge symmetry They lead to peculiar relationships between truncations made for the quark gluon vertex in the gap equation and the scattering kernel in the BetheSalpeter equation. One of the directions is to go beyond both rainbow and ladder approximations, consistently truncating the quark gluon vertex in DSE and the scattering kernel in BSE [17,18,19,20,21,22,23,24]. We seek for this possibility in this work, assuming the ladder approximation can include a multiplicative factor, which can be recognized as our ansatz for the scattering kernel This introduced factor will be determined by the two nonlinear equations constrained from the preserving of vector and axial-vector Ward-GreenTakahashi identities.
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