Cancellation in the off-shell eikonal watson series

Abstract

An explicit evaluation of the off-shell 3-body scattering amplitude in the eikonal approximation shows that the first two terms of the Watson multiple-scattering series,i.e. terms corresponding to the single and double scattering, if averaged with a bound-state wave function, give the exact answer. We find that $$ \ll p'q'|\tilde U_{\beta \beta } |pq \gg _d = \ll p'q'|\tilde T_\alpha + \tilde T_\gamma - \frac{1}{{\left( {2\pi } \right)^2 \mu }}\left( {\tilde T_\alpha \tilde G_0 \tilde T_\gamma + \tilde T_\gamma \tilde G_0 \tilde T_\alpha } \right)|pq \gg _d ,$$ , where $$ \ll p'q'|\tilde U_{\beta \beta } |pq \gg _d \int {d^3 q'} \int {d^3 q\psi _d^* \left( {q'} \right)\left\langle {p'q'|\tilde U_{\beta \beta } |} \right\rangle pq\psi _d \left( q \right)} ,$$ , Ũββ is the 3-body Lovelace scattering amplitude, and\(\tilde T\)α\(\tilde T\)γ are 2-body scattering amplitudes, written in the 3-body space. All these amplitudes are calculated in the appropriate eikonal approximation. It is essential to have on the right-hand side of our formula the whole eikonal free Green’s function, containing both the principal part and theδ-part. When the on-shell 3-body amplitude is calculated, we obtain the ordinary Glauber formula, and we recover the cancellation pointed out by Harrington.