Abstract
Evolution equations for leading-twist operators in high orders of perturbation theory can be restored from the spectrum of anomalous dimensions and the calculation of the special conformal anomaly at one order less using conformal symmetry of QCD at the Wilson-Fisher critical point at noninteger $d=4\ensuremath{-}2ϵ$ space-time dimensions. In this work, we generalize this technique to axial-vector operators. We calculate the corresponding three-loop evolution kernels in Larin's scheme and derive explicit expressions for the finite renormalization kernel that describes the difference to the vector case to restore the conventional modified minimal subtraction scheme. The results are directly applicable to deeply virtual Compton scattering and the transition form factor ${\ensuremath{\gamma}}^{*}\ensuremath{\gamma}\ensuremath{\rightarrow}\ensuremath{\pi}$.
Highlights
The QCD description of hard exclusive reactions in the framework of collinear factorization involves matrix elements of leading-twist operators between hadron states with different momenta—generalized parton distributions (GPDs) or light-cone distribution amplitudes (LCDAs)
Evolution equations for leading-twist operators in high orders of perturbation theory can be restored from the spectrum of anomalous dimensions and the calculation of the special conformal anomaly at one order less using conformal symmetry of QCD at the Wilson-Fisher critical point at noninteger d 1⁄4 4 − 2ε space-time dimensions
The main motivation for this study is provided by the applications to deeply virtual Compton scattering (DVCS), but the results are relevant for reactions of the type γγà → π etc
Summary
The QCD description of hard exclusive reactions in the framework of collinear factorization involves matrix elements of leading-twist operators between hadron states with different momenta—generalized parton distributions (GPDs) or light-cone distribution amplitudes (LCDAs). The complete next-toleading order (NLO) (two-loop) evolution kernels for GPDs were calculated long ago [6,7,8] using an approach developed by Müller [9] These results were later rederived and confirmed [10,11] by a somewhat different technique [10,12] that makes use of (exact) conformal symmetry of QCD at the Wilson-Fisher critical point in noninteger d 1⁄4 4 − 2ε dimensions. The three-loop evolution kernels for GPDs [14] and the corresponding two-loop coefficient functions [15] for DVCS were calculated for flavor-nonsinglet vectorlike distributions The extension of this technique to axial-vector distributions requires special considerations due to known issues with the definition of the γ5-matrix in noninteger dimensions. VII and the rotation matrix in the local operator basis is presented in the Appendix
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