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

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 $1\mathrm{/}{N}_f$ 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 $1/\sqrt{4N_f T®-11C_2(G)}$ where T® 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 $N_{f}T(R)\mathrm{/}{C}_2(G)>11\mathrm{/}4$, in which the asymptotic freedom in the non-abelian gauge field theory fails.