Anomalous dimensions of operators in polarized deep inelastic scattering at O(1/N_f)

TL;DR

This work addresses the calculation of twist-2 operator anomalous dimensions in polarized and unpolarized deep inelastic scattering using the leading term of the large-$N_f$ expansion at a nontrivial fixed point. The authors develop and apply a critical-point framework, with massless propagators and self-consistent Dyson equations, to obtain $n$-dependent exponents and to diagonalize singlet operator mixing directly, including a careful treatment of $\gamma^5$. The results reproduce known two- and three-loop perturbative calculations for both non-singlet and singlet cases and provide new all-orders-in-$1/N_f$ insights, particularly for polarized operators and the singlet axial current anomaly via finite renormalization. The study also clarifies how to connect these critical-exponent results to DGLAP splitting functions and highlights future extensions to higher-order $1/N_f$ corrections and fixed-point physics in QCD-like theories.

Abstract

Critical exponents are computed for a variety of twist-2 composite operators, which occur in polarized and unpolarized deep inelastic scattering, at leading order in the 1/N_f expansion. The resulting d-dimensional expressions, which depend on the moment of the operator, are in agreement with recent explicit two and three loop perturbative calculations. An interesting aspect of the critical point approach which is used, is that the anomalous dimensions of the flavour singlet eigenoperators, which diagonalize the perturbative mixing matrix, are computed directly. We also elucidate the treatment of gamma^5 at the fixed point which is important in simplifying the calculation for polarized operators. Finally, the anomalous dimension of the singlet axial current is determined at O(1/N_f) by considering the renormalization of the anomaly in operator form.