Kmatrix and unitarity constraints on off-shellTmatrix elements

Abstract

A discussion of off-shell properties of the $T$ matrix is given based on the damping equation and unitarity. The off-shell function ${\varphi }_l(k,q)$ of Baranger et al. is shown to be ${\varphi }_l(k,q)=cos{\delta }_l\mathrm{}\left(k\right)\mathrm{}〈k\left|K_l\right(k^2\left)\right|q〉$, where ${\delta }_l$ and $K_l$ are the phase shift and partial-wave $K$ (reaction) operator. The ratio of the imaginary to the real part of the half-off-shell partial-wave $T$ matrix elements is shown to be $tan{\delta }_l\mathrm{}\left(k\right)\mathrm{}$, thus providing a unitarity constraint to be satisfied by any approximate $T$ matrix elements. In addition, the imaginary part of the fully-off-shell $T$ matrix element is shown to be proportional to the product ${\varphi }_l(p,k){\varphi }_l(k,q)$. These results hold for both one-body and two-channel identical particle scattering.