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An Approximation Scheme for Constructing π°π° Amplitudes from ACU Requirements
Author(s) -
Schwarz F.
Publication year - 1980
Publication title -
fortschritte der physik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.469
H-Index - 71
eISSN - 1521-3978
pISSN - 0015-8208
DOI - 10.1002/prop.19800280403
Subject(s) - unitarity , amplitude , symmetry (geometry) , crossing , scattering amplitude , mathematics , set (abstract data type) , scheme (mathematics) , mathematical analysis , physics , quantum mechanics , computer science , geometry , programming language
The requirements of analyticity, crossing symmetry and unitarity (ACU) are used as input to construct amplitudes for the scattering of neutral π mesons. They are obtained explicitly as polynomials in certain conformally transformed variables which ensure the correct analyticity structure. Crossing symmetry is expressed as a set of linear relations between the expansion coefficients. The same is true for the threshold behaviour of the partial waves and the behaviour for large energies of either the partial waves and the amplitudes itself. To enforce the positivity of the partial waves (up to a certain l ) a new set of variables is introduced. It is obtained by solving the system of linear inequalities expressing the positivity condition. In this way one gets explicit expressions for all amplitudes satisfying the ACU requirements to some approximation. The rigorous constraints obtained from these ACU requirements are used to test the accuracy of this construction. The properties of all amplitudes are investigated in detail. It is indicated how this approximation scheme may be improved to any degree of accuracy and how it may be generalized to the full isospin case.