Condensates and anomaly cascade in vector-like theories

TL;DR

This paper investigates fermion condensates and anomalies in 4D $SU(N)$ gauge theories, demonstrating how the BC anomaly influences infrared dynamics, especially in theories with higher-order condensates and their behavior under compactification and temperature effects.

Contribution

It introduces the role of the BC anomaly in determining infrared physics and provides a detailed analysis of its effects in vector-like theories, including compactified and finite-temperature scenarios.

Findings

Nonvanishing fermion bilinears are inevitable in certain gapped theories.

The BC anomaly governs the deep infrared dynamics, unlike 0-form anomalies.

The BC anomaly cascades from 4D to 2D under compactification.

Abstract

We study the bilinear and higher-order fermion condensates in $4$-dimensional $SU(N)$ gauge theories with a single Dirac fermion in a general representation. Augmented with a mixed anomaly between the $0$-form discrete chiral, $1$-form center, and $0$-form baryon number symmetries (BC anomaly), we sort out theories that admit higher-order condensates and vanishing fermion bilinears. Then, the BC anomaly is utilized to prove, in the absence of a topological quantum field theory, that nonvanishing fermion bilinears are inevitable in infrared-gapped theories with $2$-index (anti)symmetric fermions. We also contrast the BC anomaly with the $0$-form anomalies and show that it is the former anomaly that determines the infrared physics; we argue that the BC anomaly lurks deep to the infrared while the $0$-form anomalies are just variations of local terms. We provide evidence of this assertion by studying the BC anomaly in vector-like theories compactified on a small spacial circle. These theories are weakly-coupled, under analytical control, and they admit a dual description in terms of abelian photons that determine the deep infrared dynamics. We show that the dual photons talk directly to the $1$-form center symmetry in order to match the BC anomaly, while the $0$-form anomalies are variations of local terms and are matched by fiat. Finally, we study the fate of the BC anomaly in the compactified theories when they are held at a finite temperature. The effective field theory that describes the low-energy physics is $2$-dimensional. We show that the BC anomaly cascades from $4$ to $2$ dimensions.