QCD evolution equations

We discuss QCD evolution equations for two and three particle correlation functions of quarks and gluon fields in a hadron which describe development of the momentum distribution of a parton system with a change of the wave length of a probe which resolves it. We show in a general case of two-particle correlators how the four-dimensional conformal algebra and the known pattern of conformal symmetry breaking in QCD can be used to solve the complicated mixing problem of local operators under renormalization and compute economically anomalous dimensions of quark and gluon composite operators. An extension of QCD to N = 1 super Yang-Mills theory and use of superconformal anomalies arising after quantization allows to derive nontrivial relations between the anomalous dimensions. For three-parton systems the conformal symmetry alone is not enough to solve the three-particle problem. We show that in milticolor limit of QCD there arises an extra conserved charge describing the solitonic motion of the system of particles. The problem admits a one-to-one correspondence with certain spin chain models which are exactly solvable. Talk given at the Workshop on The Phenomenology of Large Nc QCD Tempe, Arizona, January 9-11, 2002 QCD EVOLUTION EQUATIONS A.V. BELITSKY Department of Physics University of Maryland at College Park College Park, MD 20742-4111, USA We discuss QCD evolution equations for two and three particle correlation functions of quarks and gluon fields in a hadron which describe development of the momentum distribution of a parton system with a change of the wave length of a probe which resolves it. We demonstrate how broken space-time and hidden QCD symmetries serve to understand the structure and ultimately solve these equations. 1 Unraveling layers of matter: from atoms to partons A wisdom goes back to ancient Greeks who philosophized that the matter consists of tiny particles — atoms. However, the atomic structure remained a puzzle till the beginning of the 20th century when the radioactivity was discovered and used by Rutherford in his seminal experiments on large angle scattering of α particles off atoms which suggested that the atom bears a localized core — the nucleus. On the other hand, electron beams were found to pass through atoms with no or very little deflection forcing Lenard to hypothesize that atoms have wide empty spaces inside. α particles having much larger wave length comparable to the nucleus size scattered more frequently with a low intensity beams than their ‘cousins’ β particles having much smaller wave length and which, for available luminosity, had a very low cross section. Similar experiments but rather with light sources or neutrons are exploited nowadays to study the crystal structure. If one puts a crystal in front of a source of visible light, the object just leaves a shadow on a screen behind it and one does not see elementary building blocks which form it. Obviously, the visible light, having the wave length λlight ∼ 0.4 − 0.7 μm cannot do the job and resolve the internal structure of a crystal. The size of an individual atom, say hydrogen, is of order ratom ∼ (αemme) ∼ (10 KeV)−1. Therefore, to ‘see’ atoms in crystals one has to have photons with a comparable or smaller wave length λγ ≤ ratom, or equivalently, of energy Eγ ≥ r−1 atom. To do this kind of ‘nano-photography’ one need a beam of X-rays. To go beyond and look into the structure of atoms one needs even smaller wave lengths. After the discovery of nucleus’ building blocks — nucleons, i.e., protons and neutrons, the attention has been shifted to the study of these ‘elementary’ particles. However, their elementarity has been questioned starting from 0204047: submitted to World Scientific on April 5, 2002 1