Impact-Parameter Representations in Potential Scattering

Abstract

Starting from the Lippmann-Schwinger equation the off-energy-shell generalizations of the Glauber amplitude and the Blankenbecler-Goldberger amplitude a r e derived for potential scattering. The Lippmann-Schwinger equation for partial waves treated a s a Fredholm equation i s solved and the high-energy limit studied.

Coupled-channel integral-equation approach to antiproton–hydrogen collisions

“…This set takes into account the off-the-energy-shell effects through the integration over the intermediate particle's momentum q α . A similar equation but for potential scattering was given by Kamal and Chavda [33]. At sufficiently high energies of the projectile, the interaction between the antiproton and the target proton cannot markedly change the incident velocity of the projectile.…”

“…where the P l m are Pollaczek polynomials in the variable x simply related to the momentum κ and the ϕ ml (r) are Laguerre functions similar to ξ ml (r) of equation (33) except that the Lagerre polynomials in the definition are replaced by L 2l+1 m (r). The details of deriving (A.1) are given in [42].…”

“…We emphasise that the impact-parameter representation as used in the present approach is merely a transformation from the momentum-transfer space into the mathematically equivalent impact-parameter space. The momentum-transfer (with or without partial-wave expansion) and impact-parameter representations are complementary and, in principle, transform into each other without any limitation or approximation (see [30][31][32][33] and references therein). We do, however, assume in the present work that the main contribution comes when the heavy projectile is scattered into small angles.…”