Physics of the Non-Abelian Coulomb Phase: Insights from Padé Approximants

Abstract

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 Padé 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, ${\beta }_{IR}^{\prime }$, 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 ${\beta }_{IR}^{\prime }$ 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 $\mathcal{N}=1$ 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.

28 pages, 3 figures

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