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

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, ${\mathrm{\Delta }}_f$. First, we apply these to calculate the derivative $d\beta \mathrm{/}d\alpha$ evaluated at ${\alpha }_{IR}$, denoted ${\beta }_{IR}^{\mathrm{\prime }}$, which is equivalent to the anomalous dimension of the $\mathrm{T}\mathrm{r}(F_{\mu \nu }{F}^{\mu \nu })$ operator, to order ${\mathrm{\Delta }}_f^4$ for general $G$ and $R$, and to order ${\mathrm{\Delta }}_f^5$ 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 ${\mathrm{\Delta }}_f^3$. 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 ${\beta }_{IR}^{\mathrm{\prime }}$ and anomalous dimensions as power series in $\alpha$.