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 work we find explicit expressions for these operators to two-loop accuracy going over to QCD in non-integer d=4-2epsilon 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=4-2epsilon 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.
Quantum corrections for the generators of conformal transformations in QCD.
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