Compositeness of gauge boson and asymptotic freedom in non-abelian gauge theory

TL;DR

This paper investigates the conditions under which gauge bosons can be composite in non-abelian gauge theories, revealing a link between compositeness and the loss of asymptotic freedom at certain fermion flavor thresholds.

Contribution

It provides a next-to-leading order calculation of the gauge coupling under the compositeness condition, establishing a criterion for gauge boson compositeness related to fermion flavors and asymptotic freedom.

Findings

Gauge boson compositen occurs when $N_f T(R)/C_2(G) > 11/4$

Gauge coupling is proportional to $1/ oot{4N_f T(R)-11C_2(G)}$

Asymptotic freedom fails at the compositeness threshold

Abstract

In order to investigate the composite gauge field, we consider the compositeness condition (i.e. renormalization constant $Z_3=0$) in the general non-abelian gauge field theory. We calculate $Z_3$ at the next-to-leading order in $1/N_f$ expansion ($N_f$ is the number of fermion flavors), and obtain the expression to the gauge coupling constant through the compositeness condition. Then the gauge coupling constant is proportional to $1/\sqrt{4N_f T(R)-11C_2(G)}$ where T(R) is the index for a representation R of gauge group G, and $C_2(G)$ is the quadratic Casimir. It is found that the gauge boson compositeness take place only when $N_f T(R)/C_2(G) > 11/4$, in which the asymptotic freedom in the non-abelian gauge field theory fails.