Basis transformation properties of anomalous dimensions for hard exclusive processes

TL;DR

This paper establishes explicit similarity transformations between derivative and Gegenbauer operator bases for renormalization in hard exclusive processes, enabling improved computation of anomalous dimensions relevant for non-perturbative parton distributions.

Contribution

It provides explicit formulas for basis transformations for quark and gluon operators, facilitating the calculation of anomalous dimensions in different bases and extending previous results to all orders in certain limits.

Findings

Computed the one-loop gluonic anomalous dimension matrix.

Derived Gegenbauer anomalous dimensions for large quark flavor number $n_f$ to all orders.

Performed a numerical two-loop anomalous dimension calculation beyond the leading-color limit.

Abstract

When considering the renormalization of composite operators for the description of hard exclusive scattering processes, two types of operator basis called the derivative basis and the Gegenbauer basis are often used in the literature. In this work we set up the explicit similarity transformations between these two bases, both for quark and gluon operators. This way, one can use the properties of both bases to their advantage in the computation of the operator anomalous dimensions, which describe the scale dependence of non-perturbative non-forward parton distributions. We provide several applications of our framework. As an application of the gluon transformation formula, we compute the one-loop non-forward purely gluonic anomalous dimension matrix. For the rest of the applications we focus on the transformation formula of the quark operator. We derive the Gegenbauer anomalous dimensions, in the limit of a large number of quark flavors $n_f$, to all orders in the strong coupling $\alpha_s$, extending the computation previously performed in the derivative basis. Next a numeric calculation of the two-loop anomalous dimensions in the derivative basis beyond the leading-color limit is presented. Finally, we discuss a novel way of validating existing computations of the conformal anomaly based on the leading-color anomalous dimensions in the derivative basis.