Scheme-Independent Series Expansions at an Infrared Zero of the Beta Function in Asymptotically Free Gauge Theories

Abstract

We consider an asymptotically free vectorial gauge theory, with gauge group $G$ and $N_f$ fermions in a representation $R$ of $G$, having an infrared (IR) zero in the beta function at $\alpha_{IR}$. We present general formulas for scheme-independent series expansions of quantities, evaluated at $\alpha_{IR}$, as powers of an $N_f$-dependent expansion parameter, $\Delta_f$. First, we apply these to calculate the derivative $d\beta/d\alpha$ evaluated at $\alpha_{IR}$, denoted $\beta'_{IR}$, which is equivalent to the anomalous dimension of the ${\rm Tr}(F_{\mu\nu}F^{\mu\nu})$ operator, to order $\Delta_f^4$ for general $G$ and $R$, and to order $\Delta_f^5$ for $G={\rm SU}(3)$ and fermions in the fundamental representation. Second, we calculate the scheme-independent expansions of the anomalous dimension of the flavor-nonsinglet and flavor-singlet bilinear fermion antisymmetric Dirac tensor operators up to order $\Delta_f^3$. The results are compared with rigorous upper bounds on anomalous dimensions of operators in conformally invariant theories. Our other results include an analysis of the limit $N_c \to \infty$, $N_f \to \infty$ with $N_f/N_c$ fixed, calculation and analysis of Pad\'e approximants, and comparison with conventional higher-loop calculations of $\beta'_{IR}$ and anomalous dimensions as power series in $\alpha$.