Off-Energy-Shell Partial-Wave Amplitudes

Abstract

We present dispersion relations which give the full off-energy-shell $T$-matrix elements ${T}_{l{l}^{\ensuremath{'}}}(p,{p}^{\ensuremath{'}};s)$ for all values of the parametric energy $s$ in terms of bound-state form factors, a subtraction constant, and the half-off-shell $T$-matrix elements ${T}_{l{l}^{\ensuremath{'}}}(p,{s}^{\frac{1}{2}};s)$ in the scattering ($s>0$) region. Study of the half-off-shell $T$ matrix for $s>0$ shows that it can be written as the product of a real matrix ${H}_{l{l}^{\ensuremath{'}}}(p,s)$ and the on-shell $T$ matrix. We combine these results to obtain a representation of the full off-energy-shell $T$-matrix elements in terms of experimental on-shell $T$-matrix elements, the real half-off-shell factors ${H}_{l{l}^{\ensuremath{'}}}(p,s)$, a subtraction constant and bound-state form factors. Our results are based only on assumptions of time-reversal invariance, off-energy-shell unitarity, analyticity, and asymptotic behavior. The results are independent of any specific dynamical assumptions. We conclude with a discussion of the special case of uncoupled partial waves and the advantages of a separable representation of the half-off-shell factors.