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

TL;DR

The work addresses the existence of Nambu solutions to the QCD rainbow-ladder gap equation at infinite renormalization scale. It develops a fixed-point framework using Krasnoselskii-Guo cone compression on the trace form $u(p)=r(p)B(p)$, extended to the coupled $M(p)$ and $Z(p)$ system via Schauder theory, to prove a second-order dynamical chiral symmetry breaking transition when the kernel strength crosses a critical threshold. The key result is that for positive, asymptotically perturbative $L^1$ kernels with $\lambda_{\max}=1$ the mass function vanishes at the critical point, while for $\lambda_{\max}>1$ there exists a positive, continuous decreasing $u(p)$ for all current-quark masses $m\ge0$, with a parallel decreasing $M(p)$ in the coupled system. The analysis extends to representative QCD RL models (e.g., Qin_2011) and demonstrates a rigorous, kernel-structure-driven existence landscape for Nambu solutions, informing nonperturbative modeling of the QCD gap equation.

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.