\u2019t Hooft anomaly matching condition and chiral symmetry breaking without bilinear condensate

Satoshi Yamaguchi

doi:10.1007/jhep01(2019)014

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 ℤ6 should be spontaneously broken to ℤ2 under the assumption of the confinement, although there cannot be any fermion bilinear condensate in this theory.

Highlights

This analysis we assume that a gauge theory does not reproduce the ’t Hooft anomaly, if the theory is in the confining phase and the global symmetries are not spontaneously broken
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
In the SU(6) gauge theory with a Weyl fermion in the rank 3 antisymmetric representation denoted by the young diagram, the chiral symmetry Z6 is broken to Z2 if it is in the confining phase

Summary

Chiral symmetry

The action is invariant under the U(1) rotation by which ψ and ψare transformed as ψ → eiαψ, ψ → e−iαψ, (α : constant parameter). This symmetry is broken quantum mechanically by the Adler-Bell-Jackiw anomaly [22, 23]. This anomaly is the change of the path-integral measure due to Fujikawa [31]. Ψ → e2πin/ ψ, ψ → e−2πin/ ψ, n ∈ Z. We call this symmetry the “chiral symmetry.”. 1 in this paper is the same one as l(Λ) in [27] and 2T (R) in [28]

Center symmetry

Domain-wall

Example

Confinement

No fermion bilinear

Summary and discussion