Spatial description of the particle production region in elastic and quasi-elastic processes on the SOμ (2.1) group

• Published in: Physics of particles and nuclei 2009-12, Vol.40 (7), p.1030-1058
• Main Authors: Vall, A. N., Perevalova, I. A., Soldatenko, O. N., Vladimirov, A. A.
• Format: Article
• Language: English
• Subjects: Particle and Nuclear Physics; Physics; Physics and Astronomy
• ISSN: 1063-7796; 1531-8559
• DOI: 10.1134/S106377960907003X

The group-theoretical generalization of the impact parameter for elastic A + B → A + B processes is considered and analyzed in the work. The generalized impact parameter is identified with the vector of the closest approach of two particles. It is shown that after the procedure of standard quantization the components of the impact parameter’s vector, together with the components of the relative orbital’s angular momentum form SO (3.1) algebra from which we identify the SO (2.1) subalgebra. The spectrum of the Casimir operator for this subalgebra defines the allowed values of the impact parameter squared. The expansion coefficients of the elastic amplitude as a function on the SO (2.1) group define the profile function; an analogue of the partial wave on the O (3) group. The expansion itself is the generalization of the eikonal representation of the amplitude and is valid in the entire range of scattering angles. It also correctly describes the region of small impact parameters. The unitarity equation for the amplitude is solved and, as a consequence, the algebraic equation for the profile function, local in the impact parameter, is obtained. The simplest phenomenological models for cross sections (total, elastic, and inelastic) are analyzed in the framework of the solutions obtained. The next step in the development of the group-theoretical approach proposed in the work is associated with construction of a complete orthonormal basis in the one-particle Fock space where the particle is characterized by a particular energy and a particular value of its “impact parameter” relative to a point “O”. This point is related to the target particle’s coordinate (in the laboratory reference frame) or the beam’s collision point (center-of-mass system). This formalism allows for obtaining the relation between the escape parameter’s distribution function of particle C in the A + B → C + D process and the corresponding production amplitude of this particle in an angular interval. The resulting distribution function provides important physical information on the spatial structure of the production region of particle C .