Standart Model Of Elementary Particle Interactions. Part Ii

7 Evolution equations p At large energies s an e ective parameter of the perturbation theory apart from em in QED or s in QCD can contain an additional large factor being a certain power of ln s. For example, in QED the e ective parameter is em ln2 s for the electron Sudakov form-factor 1] and for the amplitude of the backward ee-scattering 2]. The corresponding physical quantities in the double-logarithmic approximation (DLA) are obtained by calculating and summing the asymptotic contributions nem ln2n s in all orders of the perturbation theory. Their region of applicability is
1
em ln2 s 1 : (1) In DLA the transverse momenta jkr? j of the virtual and real particles are large and are em implied to be strongly ordered because the rst logarithm ln s in each one-loop diagram is obtained as a result of integration over the energy of a relatively soft particle and another logarithm - over its transverse momentum (or emission angle). Instead of calculating each individual diagram one can initially divide the integration region in several subregions in dependence from the ordering of the particle transverse momenta and to sum subsequently the contributions with the same orderings over all diagrams in the given order of perturbation theory. It gives a possibility to write an evolution equation with respect to the logarithm of the infrared cut-o  3]. Below we consider a simple example of such equation for the case of the radiative corrections to the Z -boson production in the ee collisions. In the Regge kinematics in some cases the integrals over transverse momenta are convergent, which leads to the e ective parameter of the perturbation theory ln s 4] or even 2 ln s 5]. For the inclusive processes at large energies and large momentum transfers Q there is a strong ordering only over transverse momenta of particles. In this case one can use the probabalistic interpretation to express the hadron-lepton processes it terms of the cross-sections for the collision of the point-like objects - partons. We review below the parton model (see also 6]) and discuss its modi cation in QCD, where the parton transverse momenta slowly grow with increasing Q. Later we derive the evolution equations for parton distributions and for fragmentation functions, calculate the splitting kernels and nd the solutions. The relation to the renormalization group in the framework of the Wilson operator product expansion is also discussed. The evolution equations are generalized for the parton correlators describing matrix elements of the so-called quasi-partonic operators. In more details we study evolution equations for the twist-3 quasi-partonic operators describing the large-Q2 behaviour of the structure function g2(x). Next-to-leading corrections to the splitting kernels are reviewed. 7.1 Sudakov resummation of double-logarithmic terms in QED As an example of the necessity to go beyond the usual perturbation th