Existence of Decreasing Nambu Solutions to the Rainbow Ladder Gap Equation of QCD by Cone Compression

TL;DR

This paper proves the existence of decreasing Nambu solutions to the rainbow-ladder gap equation in QCD, demonstrating how the mass function emerges continuously from zero as the interaction strength crosses a critical threshold, using advanced fixed point theorems.

Contribution

It introduces a novel mathematical proof for the existence of decreasing Nambu solutions in QCD's gap equation using cone compression and fixed point theorems, applicable to physically relevant models.

Findings

Mass function emerges continuously from zero at critical interaction strength.

Existence of positive, decreasing Nambu solutions for all current quark masses.

Applicable to a class of models including popular QCD models.

Abstract

Studying Nambu solutions of the rainbow-ladder gap equation in QCD at zero temperature and chemical potential, we prove that the mass function emerges continuously from zero as the interaction strength is increased past the critical point for all positive, asymptotically perturbative kernels almost everywhere continuous in $L^1$ using the Krasnosel'skii-Guo Cone Compression Theorem. We prove that the coupled system of equations must have a positive, continuous Nambu solution with decreasing mass function for all current quark masses for a class of models which includes the physical point of a popular model of QCD by using a hybrid Krasnosel'skii-Schauder Fixed Point Theorem.