ANALYSIS OF PARTIAL-WAVE DISPERSION RELATIONS
Questions of existence and uniqueness of solutions of partial-wave dispersion relations are studied, with particular attention to the N/D method. The interaction, assumed to be given, is represented by (i) the strengths and locations of unphysical singularities and (ii) the inelastic partial-wave cross section. A generalization of the N/D method to include part (ii) of the interaction leads to a nonsingular integral equation for ImD. This equation is amenable to the Fredholm theory only if there is a correlation between items (i) and (ii) of the interaction, and only if the increase of inelastic processes at high energies is not too rapid. Certain Cauchy integrals associated with (i) and (ii) must be nonzero at threshold if there is to exist a solution with the normal threshold momentum dependence. Thus, there is no solution for any model in which (i) is constructed from a few partial waves in the two crossed channels. For certain interactions the real part of the phase shift approaches a multiple of pi at large energy, just as in potential scattering. The Castillejo-DalitzDyson (CDD) ambiguity is analyzed in some detail. A uniqueness theorem is proved which asserts that if a solution of a particular type exists, it is the only solution of the problem within the class usually considered. Thus the CDD ambiguity is partially bypassed. In certain cases tue unique solution is found by the ordinary N/D method without subtractions. Some useful results on principal value integrals are obtained. The discussion is carried out for the example of pion- nucleon scattering in the complex plane of w, the certer-of-mass energy. The behavior of the amplitude near w = plus or minus (M-m) is derived from crossing symmetry. (auth)
- Research Organization:
- Univ. of Washington, Seattle
- NSA Number:
- NSA-18-000886
- OSTI ID:
- 4163652
- Journal Information:
- Physical Review (U.S.) Superseded in part by Phys. Rev. A, Phys. Rev. B: Solid State, Phys. Rev. C, and Phys. Rev. D, Vol. Vol: 130; Other Information: Orig. Receipt Date: 31-DEC-64
- Country of Publication:
- Country unknown/Code not available
- Language:
- English
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