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

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.

arXiv comment36 pages

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Cited by in corpus (18)

• Soft-/rapidity- anomalous dimensions correspondence
• Three-loop evolution equation for flavor-nonsinglet operators in off-forward kinematics
• Two-loop evolution equation for the B-meson distribution amplitude
• Two-Loop SL(2) Form Factors and Maximal Transcendentality
• Two-Loop Radiative Jet Function for Exclusive $B$-Meson and Higgs Decays
• Two-loop coefficient function for DVCS: Vector contributions
• The axial-vector contributions in two-photon reactions: pion transition form factor and deeply-virtual Compton scattering at NNLO in QCD
• Evolution of the $B$-Meson Light-Cone Distribution Amplitude in Laplace Space
• Conformal symmetry of QCD in $d$-dimensions
• Two-loop evolution equations for flavor-singlet light-ray operators
• Heavy quark expansion for heavy-light light-cone operators
• Renormalization of the next-to-leading-power $\gamma \gamma \to h$ and $gg\to h$ soft quark functions
• Three-loop off-forward evolution kernel for axial-vector operators in Larin's scheme
• Operator mixing in massless QCD-like theories and Poincare'-Dulac theorem
• Renormalization of non-singlet quark operator matrix elements for off-forward hard scattering
• NNLO anomalous dimension matrix for twist-two flavor-singlet operators
• Basis transformation properties of anomalous dimensions for hard exclusive processes
• Scale-dependence of the $B$-meson LCDA beyond leading order from conformal symmetry