On the Isomorphic Description of Chiral Symmetry Breaking by Non-Unitary Lie Groups

TL;DR

This paper explores how chiral symmetry breaking can be described by various non-unitary Lie groups, extending known orthogonal group descriptions and examining implications for Higgs mechanisms and technicolour models.

Contribution

It systematically investigates isomorphic descriptions of chiral symmetry breaking using non-unitary Lie groups, identifying possible orthogonal group patterns and their relevance to Higgs mechanisms.

Findings

Limited alternative descriptions with orthogonal groups.

No evidence of descriptions based on symplectic or exceptional groups.

Connections to technicolour models and fermion-based Higgs particles.

Abstract

It is well-known that chiral symmetry breaking ($\chi$SB) in QCD with $N_{f}=2$ light quark flavours can be described by orthogonal groups as $O(4) \to O(3)$, due to local isomorphisms. Here we discuss the question how specific this property is. We consider generalised forms of $\chi$SB involving an arbitrary number of light flavours of continuum or lattice fermions, in various representations. We search systematically for isomorphic descriptions by non-unitary, compact Lie groups. It turns out that there are a few alternative options in terms of orthogonal groups, while we did not find any description entirely based on symplectic or exceptional Lie groups. If we adapt such an alternative as the symmetry breaking pattern for a generalised Higgs mechanism, we may consider a Higgs particle composed of bound fermions and trace back the mass generation to $\chi$SB. In fact, some of the patterns that we encounter appear in technicolour models. In particular if one observes a Higgs mechanism that can be expressed in terms of orthogonal groups, we specify in which cases it could also represent some kind of $\chi$SB of techniquarks.