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

Preprint number: CP3-Origins-2016-39 DNRF90
Authors: Thomas A. Ryttov (CP3-Origins & DIAS) and Robert Shrock (C. N. Yang Institute for Theoretical Physics)
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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 $dbeta/dalpha$ evaluated at
$alpha_{IR}$, denoted $beta’_{IR}$, which is equal to the anomalous dimension
of the ${rm Tr}(F_{munu}F^{munu})$ 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 $alpha$.