Non-perturbative on-shell multiplet structure of SU(N) Yang-Mills fields

TL;DR

The paper addresses non-perturbative on-shell multiplet structure in $SU(N)$ Yang–Mills theory by exploiting the Weyl group $W(SU(N))$ to classify quark and gluon solutions on mass shell. It develops a Weyl-symmetric, gauge-fixed framework that reveals invariant singlet and higher multiplets, notably establishing the existence of a Weyl singlet gluon in $SU(3)$ and constructing a universal Weyl-symmetric Lagrangian for one-particle singlet gluon solutions that generalizes to $SU(4)$ and beyond. It also formulates Weyl-symmetric Abelian projections for theories with quarks, showing how a singlet quark and an $(N-1)$-plet arise from a fundamental quark multiplet via Cartan-Weyl basis, and extends this to general $N$, yielding a consistent decomposition into $ ext{Γ}_1igoplus ext{Γ}_{N-1}$. The work suggests a non-perturbative, color-symmetric basis for fundamental particles that could influence QCD vacuum structure, hadron spectroscopy, and a revised quark model, with implications for non-perturbative QCD and hadron physics.

Abstract

Color multiplets of the gauge fields and fermions on mass shell in SU(N) Yang-Mills theory are classified according to representations of the Weyl group W(SU(N)). The multiplet structure of quark and gluon multiplets has been studied in the framework of a non-perturbative approach by considering complete exact equations of motion of the SU(N) Yang-Mills theory with matter fields. An important case of singlet non-Abelian gluon solutions corresponding to one-dimensional singlet representations of the Weyl group is revised on a rigorous mathematical basis. We demonstrate that Weyl group as a finite color subgroup of SU(N) reveals an inherent color symmetry of quarks and gluons on mass shell which determines a universal color miltiplet structure of quark-gluon solutions in a pure SU(N) Yang-Mills theory and in Abelian projected Yang-Mills theories with quarks. The obtained results allow to introduce strict concepts of fundamental particles, quarks and gluons, which differ drastically from the particle definitions in the conventional perturbative Yang-Mills theory. Possible applications of our results in non-perturbative quantum chromodynamics and hadron physics are discussed.