$A_k \bar F$ Chiral Gauge Theories

TL;DR

This paper analyzes the conditions for asymptotic freedom and possible infrared behaviors of chiral SU(N) gauge theories with fermions in antisymmetric and conjugate fundamental representations, identifying ranges of N for different k.

Contribution

It classifies the asymptotic freedom and infrared phases of $A_k ar F$ chiral gauge theories for various N and k, and shows the non-existence of certain symmetric representation theories.

Findings

$A_2 ar F$ theories are asymptotically free for all N ≥ 5.

$A_3 ar F$ and $A_4 ar F$ theories are asymptotically free only within specific N ranges.

No asymptotically free theories with $k \, extgreater \, 4$ in the antisymmetric case.

Abstract

We study asymptotically free chiral gauge theories with an SU($N$) gauge group and chiral fermions transforming according to the antisymmetric rank-$k$ tensor representation, $A_k \equiv [k]_N$, and the requisite number, $n_{\bar F}$, of copies of fermions in the conjugate fundamental representation, $\bar F \equiv \overline{[1]}_N$, to render the theories anomaly-free. We denote these as $A_k \, \bar F$ theories. We take $N \ge 2k+1$ so that $n_{\bar F} \ge 1$. The $A_2 \, \bar F$ theories form an infinite family with $N \ge 5$, but we show that the $A_3 \, \bar F$ and $A_4 \,\bar F$ theories are only asymptotically free for $N$ in the respective ranges $7 \le N \le 17$ and $9 \le N \le 11$, and that there are no asymptotically free $A_k \, \bar F$ theories with $k \ge 5$. We investigate the types of ultraviolet to infrared evolution for these $A_k \, \bar F$ theories and find that, depending on $k$ and $N$, they may lead to a non-Abelian Coulomb phase, or may involve confinement with massless gauge-singlet composite fermions, bilinear fermion condensation with dynamical gauge and global symmetry breaking, or formation of multifermion condensates that preserve the gauge symmetry. We also show that there are no asymptotically free, anomaly-free SU($N$) $S_k \, \bar F$ chiral gauge theories with $k \ge 3$, where $S_k$ denotes the rank-$k$ symmetric representation.