System stability and truncation schemes to the Dyson-Schwinger Equations

TL;DR

This paper investigates the stability of truncated Dyson-Schwinger Equations for quark and gluon propagators, highlighting the importance of gluon and ghost loops in maintaining system stability and guiding model development.

Contribution

It introduces a stability analysis approach for coupled DSEs, emphasizing the role of specific loop contributions and providing insights for constructing better truncation schemes.

Findings

Gluon and ghost loops are crucial for system stability.

Quark-gluon vertex has a smaller but significant impact.

Stability analysis offers constraints for model building.

Abstract

With decades of years development, although important progresses have been made by the pioneers of this field, providing a sophisticated truncation scheme is still a great challenge up to now if the Dyson-Schwinger Equations(DSEs) of both quark and gluon propagators (or including even more DSEs) remain after truncation. In this work we view the coupled reminiscent DSEs of the gluon and quark propagators after truncation as a system with feedback. Then studying the stability of this equation array gives useful results. Our calculation shows that the sum of the gluon and ghost loops plays the most important role in keeping this system stable and having reasonable solutions. The quark-gluon vertex plays a relative smaller but also important role. Our method also could give constraints and inspirations on fabricating a more sophisticated model of the quark-gluon vertex.