Aspects of SU(N) gauge theories in the limit of small number of colors

Abstract

We investigate properties of the color space of $SU(N_c)$ gauge theories in the limit of small number of colors $(N_c \to 0)$ and large number of flavors. More generally, we introduce a rescaling of $\alpha_s$ and $n_f$ which assigns a finite limit to colored quantities as $N_c \to 0$, which reproduces their known large-$N_c$ limit, and which expresses them as an analytic function of $N_c^2$ for arbitrary value of $N_c$. The vanishing-$N_c$ limit has an Abelian character and is also the small-$N_c$ limit of $[U(1)]^{N_c-1}$. This limit does not have an obvious quantum field theory interpretation; however, it provides practical consistency checks on QCD perturbative quantities by comparing them to their QED counterparts. Our analysis also describes the two-dimensional topological structure involved in the interpretation of the small $N_c$-limit in color space.