Determination of the anomalous dimension of gluonic operators in deep inelastic scattering at O(1/N_f)

TL;DR

This work calculates the leading large-$N_f$ correction to the anomalous dimension of the predominantly gluonic twist-2 singlet operator arising in deep inelastic scattering. Using critical-point renormalization group methods, the authors derive a $d$-dimensional, $n$-dependent expression for the gluonic sector's anomalous dimension at $O(1/N_f)$, validating it against known $n$-dependent three-loop MSbar results for $n \le 8$ and extracting the explicit $n$-dependence of higher-order terms. The analysis divides the calculation into a QED-like sector and the $C_2(G)$ gluonic sector, solving a sequence of recurrence relations for intricate loop integrals and employing conformal and IBP techniques to obtain a closed form for the gluonic eigen-operator dimension at the fixed point. The results provide cross-checks for perturbative coefficients, illuminate the structure of operator mixing as a function of $n$, and offer a foundation for extending to higher loops and polarized operators in related theories.

Abstract

Using large N_f methods we compute the anomalous dimension of the predominantly gluonic flavour singlet twist-2 composite operator which arises in the operator product expansion used in deep inelastic scattering. We obtain a d-dimensional expression for it which depends on the operator moment n. Its expansion in powers of epsilon = (4-d)/2 agrees with the explicit exact three loop MSbar results available for n less than or equal to 8 and allows us to determine some new information on the explicit n-dependence of the three and higher order coefficients. In particular the n-dependence of the three loop anomalous dimension gamma_{gg}(a) is determined in the C_2(G) sector at O(1/N_f).