't Hooft anomaly matching condition and chiral symmetry breaking without bilinear condensate

TL;DR

This work uses 't Hooft anomaly matching between a discrete chiral symmetry $\mathbb{Z}_{\ell}$ and center symmetry $\mathbb{Z}_{q}$ to constrain chiral-symmetry breaking in 4d SU($N$) gauge theories with a Weyl fermion in a self-conjugate representation. By introducing background 2-form $B$ and 1-form $C$ fields with $qB=dC$ and performing an anomaly inflow analysis, the authors show that in the confining phase the chiral symmetry is typically broken further to $\mathbb{Z}_{\ell/q'}$, with $q'>1$ in many cases. The SU($6$) example with a rank-3 antisymmetric Weyl fermion yields $\ell=6$, $c=3$, giving $q=3$ and $q'=3$, hence $\mathbb{Z}_{6} \to \mathbb{Z}_{2}$; crucially, this breaking occurs without a fermion bilinear condensate since the Lorentz-invariant bilinear vanishes identically. The domain walls between vacua realize anomaly inflow via a 3d theory, illustrating how conformal or topological sectors on the wall reproduce the center anomaly, and highlighting a nonperturbative mechanism for chiral-symmetry breaking without bilinear condensation.

Abstract

We explore 4-dimensional SU(N) gauge theory with a Weyl fermion in an irreducible self-conjugate representation. This theory, in general, has a discrete chiral symmetry. We use 't Hooft anomaly matching condition of the center symmetry and the chiral symmetry, and find constraints on the spontaneous chiral symmetry breaking in the confining phase. The domain-walls connecting different vacua are discussed from the point of view of the 't Hooft anomaly. We consider the SU(6) gauge theory with a Weyl fermion in the rank 3 anti-symmetric representation as an example. It is argued that this theory is likely to be in the confining phase. The chiral symmetry $\mathbb{Z}_6$ should be spontaneously broken to $\mathbb{Z}_2$ under the assumption of the confinement, although there cannot be any fermion bilinear condensate in this theory.