Expansion of the Two-NucleonTMatrix Half Off the Energy Shell

Current descriptions of ($p,2p$) reactions, nucleon-nucleon bremsstrahlung, and low-energy pion production use the half-off-shell two-nucleon $T$ matrix in the neighborhood of the on-shell point. We present a method for expanding the amplitude $t_0(p,k,;k^2)$ in powers of $p-k$. After explicitly extracting the contributions arising from the long-range part of the interaction, we obtain a power series with an infinite radius of convergence. The coefficients of this series are expressed in terms of weighted integrals of the difference function, thus permitting us to determine precisely how variations of the interior wave function affect the near off-shell behavior of the half-shell $T$ matrix. This allows treatment of a large class of models for the two-nucleon interaction, including wave-function models as well as local, nonlocal, and energy-dependent potentials. As an example, we apply this method to a wave-function model of the two-nucleon ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ $T$ matrix constructed by the authors in a previous paper. We find that the expansion converges rapidly and that the inclusion of more terms extends the representation farther off the energy shell. The expansion coefficients are smooth, rapidly decreasing functions of energy. The even coefficients and the odd coefficients each form a family of similar functions which fall off rapidly with index. The $T$ matrix arising from this model is very similar to those obtained with realistic potentials. Since the coefficient functions can be tabulated or parametrized quite simply, this expansion should provide a compact, useful method of expressing and comparing the near off-shell behavior of the two-nucleon half-shell $T$ matrices arising from different models while permitting one to maintain both a fixed phase shift and the correct long-range behavior of the potential.