Physics of the non-Abelian Coulomb phase: Insights from Padé approximants

We consider a vectorial, asymptotically free $\mathrm{SU}(N_c)$ 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, ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR}}$, to $O({\mathrm{\Delta }}_f^4)$, and the derivative of the beta function, ${\beta }_{\mathrm{IR}}^{\prime }$, to $O({\mathrm{\Delta }}_f^5)$, where ${\mathrm{\Delta }}_f$ 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 ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR}}$ and ${\beta }_{\mathrm{IR}}^{\prime }$ for several $\mathrm{SU}(N_c)$ 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 ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR}}$ in comparison with the exactly known expression in an $\mathcal{N}=1$ supersymmetric gauge theory. It is shown that an expansion of ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR}}$ to $O({\mathrm{\Delta }}_f^4)$ is quite accurate throughout the entire non-Abelian Coulomb phase of this supersymmetric theory.

Figure 1

Plot of ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,[1,2]}$ and ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,[0,3]}$ for $N_c=2$, i.e., SU(2) and $R=F$, together with ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f^s}$ with $1\le s\le 4$, as a function of $N_f$. The vertical axis is labelled generically as ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR}}$. From bottom to top, the curves (with colors online) refer to ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f}$ (red), ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f^2}$ (green), ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f^3}$ (blue), and ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f^4}$ (black). The curves for ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,[1,2]}$ and ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,[0,3]}$ are dashed magenta and dotted magenta.

Figure 2

Plot of ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,[1,2]}$ and ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,[0,3]}$ for $N_c=3$, i.e., SU(3) and $R=F$, together with ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f^s}$ with $1\le s\le 4$, as a function of $N_f$. The vertical axis is labelled generically as ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR}}$. From bottom to top, the curves (with colors online) refer to ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f}$ (red), ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f^2}$ (green), ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f^3}$ (blue), and ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f^4}$ (black). The curves for ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,[1,2]}$ and ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,[0,3]}$ are dashed magenta and dotted magenta.

Figure 3

Plot of ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,[1,2]}$ and ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,[0,3]}$ for $N_c=4$, i.e., SU(4) and $R=F$, together with ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f^s}$ with $1\le s\le 4$, as a function of $N_f$. The vertical axis is labelled generically as ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR}}$. From bottom to top, the curves (with colors online) refer to ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f}$ (red), ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f^2}$ (green), ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f^3}$ (blue), and ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,{\mathrm{\Delta }}_f^4}$ (black). The curves for ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,[1,2]}$ and ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},F,[0,3]}$ are dashed magenta and dotted magenta.