KINEMATICS AND DISPERSION RELATIONS FOR GENERAL PRODUCTION PROCESSES
Technical Report
·
OSTI ID:4766636
On the basis of heuristic arguments it is shown that the amplitude for the reaction at + a/sub 2/ yields b/sub 1/ + b/sub 2/ + b/sub 3/ and the channels associated with it allows a dispersion representation analogous to that given by Mandelstam for processes of type a/sub 1/ +a/sub 2/ yields b/sub 1/ + b/sub 2/, provided that the singularit ies of the amplitude can be assumed to be restricted, in complete analogy to Mandelstam's case, to certain parts of real hyperplanes in the (complex) space of the invariant variables s/sub ik/= (q/sub i/ + q/sub k/)/sup 2/). (Here the q/sub i/(i = 1,2,...,5) are the particle four- momenta.) A Lorentz-invariant description due to Kibble for the boundary of the physical region of the process a/sub 1/ + a/sub 2/ yields b/sub 1/ + b/sub 2/ was generalized for arbitrar y reactions and discussed in terms of scattering angles for some special cases. After suitable generalization of the Breit frame a set of ten one-dimensional dispersion relations analogous to the three one- dimensional relations of Mandelstam was obtained by using a method due to Polkinghorne. Each relation apart from pole terms consists of six dispersion integrals, each of which corresponds to a certain reaction channel. The absorptive parts were obtained from analytic continuation of the unitarity condition in the respective channel. For obtaining such a result it was essential to keep fixed not four variables of type s/sub ik/ but three such variables and a fourth variable which was formerly introduced by Polkinghorne and which is a generalized linear function of those s/sub ik/ which was not kept fixed. Provided that there are no complex singularities each of these one- dimensional dispersion relations can-in a formal way--be derived from a two- dimensional representation, in which certain three variables s/sub ik/ are fixed and which consists of twelve double integrals. It is suggested that if there was analyticity with regard to all variables and only real singularities a possible representation in terms of fivefold dispersion integrals would be of considerable complexity and consist of at least 162 terms. (auth)
- Research Organization:
- California. Univ., Berkeley. Lawrence Radiation Lab.
- DOE Contract Number:
- W-7405-ENG-48
- NSA Number:
- NSA-16-031017
- OSTI ID:
- 4766636
- Report Number(s):
- UCRL-9553
- Country of Publication:
- United States
- Language:
- English
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