Two-loop conformal generators for leading-twist operators in QCD

TL;DR

This paper derives explicit two-loop conformal generators for leading-twist operators in QCD within a non-integer dimensional regularization scheme, restoring conformal symmetry at the quantum level and enabling advanced evolution equations.

Contribution

It provides the first explicit two-loop expressions for conformal generators in QCD, valid in non-integer dimensions, and shows how to restore conformal symmetry at the quantum level.

Findings

Explicit two-loop conformal generators in QCD derived

Conformal symmetry restored at the quantum level in non-integer dimensions

Constraints on renormalization group equations for composite operators

Abstract

QCD evolution equations in minimal subtraction schemes have a hidden symmetry: One can construct three operators that commute with the evolution kernel and form an $SL(2)$ algebra, i.e. they satisfy (exactly) the $SL(2)$ commutation relations. In this paper we find explicit expressions for these operators to two-loop accuracy going over to QCD in non-integer $d=4-2\epsilon$ space-time dimensions at the intermediate stage. In this way conformal symmetry of QCD is restored on quantum level at the specially chosen (critical) value of the coupling, and at the same time the theory is regularized allowing one to use the standard renormalization procedure for the relevant Feynman diagrams. Quantum corrections to conformal generators in $d=4-2\epsilon$ effectively correspond to the conformal symmetry breaking in the physical theory in four dimensions and the $SL(2)$ commutation relations lead to nontrivial constraints on the renormalization group equations for composite operators. This approach is valid to all orders in perturbation theory and the result includes automatically all terms that can be identified as due to a nonvanishing QCD $\beta$-function (in the physical theory in four dimensions). Our result can be used to derive three-loop evolution equations for flavor-nonsinglet quark-antiquark operators including mixing with the operators containing total derivatives. These equations govern, e.g., the scale dependence of generalized hadron parton distributions and light-cone meson distribution amplitudes.