The helicity amplitude approach to Low Energy Theorems: Norman Dombey. University of Sussex, School of Mathematical and Physical Sciences, Brighton BN1 9QH, Sussex, England and N. C. McKenzie, Research Institute for Fundamental Physics, Kyoto University, Kyoto, Japan
- Publisher
- Elsevier Science
- Year
- 1973
- Tongue
- English
- Weight
- 80 KB
- Volume
- 77
- Category
- Article
- ISSN
- 0003-4916
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✦ Synopsis
An eikonal expansion of the potential scattering T matrix is evaluated, without approximation, through third order in the inverse momentum. Based on the results, their correspondence with the WKB approximation and a new statement of the unitarity constraint, we propose a sequence of four approximations to the exact impact parameter (Fourier-Bessel) representation of the scattering matrix. The sequence consists of the Glauber approximation and three systematic corrections to the Glauber approximation. The corrections are analytic functions of the impact parameter for Yukawa and Gaussian potentials; they vanish for a Coulomb potential.
The sequence of eikonal amplitudes is convergent at high energy and is clearly established for small momentum transfer. Validity for all momentum transfer is conjectured based on systematic cancellation, explicitly verified through third order in the expansion, of momentum transfer dependence in the eikonal impact parameter representation. Such cancellation is shown to occur in the explicit construction of the eikonal expansion of the second Born amplitude for a Yukawa potential.
Numerical tests of the sequence of eikonal amplitudes show systematic increase of the angular range of validity by comparison with partial wave results for continuous potentials; the theory is not convergent for discontinuous potentials.
The WKB phase shift formula is shown to produce a systematic connection with eikonal expansion results. From this we deduce a generating function for the eikonal phase corrections of arbitrary order and also conjecture a sum of the eikonal expansion valid in the limit of high energy and arbitrary potential strength.
The Helicity Amplitude Approach to Low Energy Theorems.