Physics of the Non-Abelian Coulomb Phase: Insights from Pad\'e Approximants

We consider a vectorial, asymptotically free SU($N_c$) gauge theory with $N_f$ fermions in a representation $R$ having an infrared (IR) fixed point. We calculate and analyze Pad\'e approximants to scheme-independent series expansions for physical quantities at this IR fixed point, including the anomalous dimension, $\gamma_{\bar\psi\psi,IR}$, to $O(\Delta_f^4)$, and the derivative of the beta function, $\beta'_{IR}$, to $O(\Delta_f^5)$, where $\Delta_f$ is an $N_f$-dependent expansion variable. We consider the fundamental, adjoint, and rank-2 symmetric tensor representations. The results are applied to obtain further estimates of $\gamma_{\bar\psi\psi,IR}$ and $\beta'_{IR}$ for several SU($N_c$) groups and representations $R$, and comparisons are made with lattice measurements. We apply our results to obtain new estimates of the extent of the respective non-Abelian Coulomb phases in several theories. For $R=F$, the limit $N_c \to \infty$ and $N_f \to \infty$ with $N_f/N_c$ fixed is considered. We assess the accuracy of the scheme-independent series expansion of $\gamma_{\bar\psi\psi,IR}$ in comparison with the exactly known expression in an ${\cal N}=1$ supersymmetric gauge theory. It is shown that an expansion of $\gamma_{\bar\psi\psi,IR}$ to $O(\Delta_f^4)$ is quite accurate throughout the entire non-Abelian Coulomb phase of this supersymmetric theory.