The Hopf Skyrmion in QCD with Adjoint Quarks

TL;DR

This paper explores the stability and properties of topological solitons in a modified QCD with adjoint quarks, linking them to Hopf invariants and proposing an extended model to detect exotic hadrons.

Contribution

It identifies the topological origin of stable solitons in adjoint QCD and connects them to Hopf Skyrmions, extending the Skyrme-Faddeev model to include fermions for better understanding.

Findings

Normal hadrons have mass O(1) and specific charge properties.

Exotic hadrons with mass O(N^2) correspond to Hopf Skyrmions.

A shift in fermion number F relates to the Hopf invariant, explaining exotic states.

Abstract

We consider a modification of QCD in which conventional fundamental quarks are replaced by Weyl fermions in the adjoint representation of the color SU(N). In the case of two flavors the low-energy chiral Lagrangian is that of the Skyrme-Faddeev model. The latter supports topologically stable solitons with mass scaling as N^2. Topological stability is due to the existence of a nontrivial Hopf invariant in the Skyrme-Faddeev model. Our task is to identify, at the level of the fundamental theory, adjoint QCD, an underlying reason responsible for the stability of the corresponding hadrons. We argue that all "normal" mesons and baryons, with mass O(N^0), are characterized by (-1)^Q (-1)^F =1, where Q is a conserved charge corresponding to the unbroken U(1) surviving in the process of the chiral symmetry breaking (SU(2) \to U(1) for two adjoint flavors). Moreover, F is the fermion number (defined mod 2 in the case at hand). We argue that there exist exotic hadrons with mass O(N^2) and (-1)^Q (-1)^F = -1. They are in one-to-one correspondence with the Hopf Skyrmions. The transition from nonexotic to exotic hadrons is due to a shift in F, namely F \to F - {\cal H} where {\cal H} is the Hopf invariant. To detect this phenomenon we have to extend the Skyrme-Faddeev model by introducing fermions.