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

TL;DR

This paper develops scheme-independent series expansions for quantities at an infrared zero of the beta function in asymptotically free gauge theories, providing new calculations of anomalous dimensions and comparisons with other methods.

Contribution

It introduces general formulas for scheme-independent expansions at the IR zero and applies them to compute derivatives and anomalous dimensions up to high orders for various gauge groups and representations.

Findings

Calculated $eta'_{IR}$ to order $ riangle_f^4$ for general theories.

Computed anomalous dimensions of fermion operators up to order $ riangle_f^3$.

Compared results with conformal bounds and higher-loop calculations.

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 $\alpha$.