Conformal constraints for anomalous dimensions of leading twist operators

Leading-twist operators have a remarkable property that their divergence vanishes in a free theory. Recently it was suggested that this property can be used for an alternative technique to calculate anomalous dimensions of leading-twist operators and allows one to gain one order in perturbation theory so that, i.e., two-loop anomalous dimensions can be calculated from one-loop Feynman diagrams, etc. In this work we study feasibility of this program on a toy-model example of the $\varphi^3$ theory in six dimensions. Our conclusion is that this approach is valid, although it does not seem to present considerable technical simplifications as compared to the standard technique. It does provide one, however, with a very nontrivial check of the calculation as the structure of the contributions is very different.Comment: 14 pages, 6 figure

Two-loop evolution equations for light-ray operators

QCD in non-integer d=4-2 epsilon space-time dimensions possesses a nontrivial critical point and enjoys exact scale and conformal invariance. This symmetry imposes nontrivial restrictions on the form of the renormalization group equations for composite operators in physical (integer) dimensions and allows to reconstruct full kernels from their eigenvalues (anomalous dimensions). We use this technique to derive two-loop evolution equations for flavor-nonsinglet quark-antiquark light-ray operators that encode the scale dependence of generalized hadron parton distributions and light-cone distribution amplitudes in the most compact form.Comment: 13 pages, 1 figur

Baryon octet distribution amplitudes in Wandzura-Wilczek approximation

We study higher twist distribution amplitudes for the SU_F(3) baryon octet. We identify independent functions for all baryons in the isospin symmetry limit and calculate the Wandzura-Wilczek contributions to the twist-4 and 5 distributions amplitudes.Comment: 7 page

Evolution equations beyond one loop from conformal symmetry

We study implications of exact conformal invariance of scalar quantum field theories at the critical point in non-integer dimensions for the evolution kernels of the light-ray operators in physical (integer) dimensions. We demonstrate that all constraints due the conformal symmetry are encoded in the form of the generators of the collinear sl(2) subgroup. Two of them, S_- and S_0, can be fixed at all loops in terms of the evolution kernel, while the generator of special conformal transformations, S_+, receives nontrivial corrections which can be calculated order by order in perturbation theory. Provided that the generator S_+ is known at the k-1 loop order, one can fix the evolution kernel in physical dimension to the k-loop accuracy up to terms that are invariant with respect to the tree-level generators. The invariant parts can easily be restored from the anomalous dimensions. The method is illustrated on two examples: The O(n)-symmetric phi^4 theory in d=4 to the three-loop accuracy, and the su(n) matrix phi^3 theory in d=6 to the two-loop accuracy. We expect that the same technique can be used in gauge theories e.g. in QCD.Comment: 19 pages, 3 figure

Conformal symmetry of the Lange-Neubert evolution equation

The Lange-Neubert evolution equation describes the scale dependence of the wave function of a meson built of an infinitely heavy quark and light antiquark at light-like separations, which is the hydrogen atom problem of QCD. It has numerous applications to the studies of B-meson decays. We show that the kernel of this equation can be written in a remarkably compact form, as a logarithm of the generator of special conformal transformation in the light-ray direction. This representation allows one to study solutions of this equation in a very simple and mathematically consistent manner. Generalizing this result, we show that all heavy-light evolution kernels that appear in the renormalization of higher-twist B-meson distribution amplitudes can be written in the same form.Comment: 8 page

Evolution equation for the higher-twist B-meson distribution amplitude

We find that the evolution equation for the three-particle quark-gluon B-meson light-cone distribution amplitude (DA) of subleading twist is completely integrable in the large $N_c$ limit and can be solved exactly. The lowest anomalous dimension is separated from the remaining, continuous, spectrum by a finite gap. The corresponding eigenfunction coincides with the contribution of quark-gluon states to the two-particle DA $\phi_-(\omega)$ so that the evolution equation for the latter is the same as for the leading-twist DA $\phi_+(\omega)$ up to a constant shift in the anomalous dimension. Thus, ``genuine'' three-particle states that belong to the continuous spectrum effectively decouple from $\phi_-(\omega)$ to the leading-order accuracy. In turn, the scale dependence of the full three-particle DA turns out to be nontrivial so that the contribution with the lowest anomalous dimension does not become leading at any scale. The results are illustrated on a simple model that can be used in studies of $1/m_b$ corrections to heavy-meson decays in the framework of QCD factorization or light-cone sum rules.Comment: Extended version, includes new results on the large momentum limit and a detailed study of the evolution effects in a simple mode