Three-Body Unitarity and Khuri-Treiman Amplitudes

The $s$-wave Khuri-Treiman (K-T) amplitude for three-body production or decay processes with final-state interactions satisfies a linear single-variable integral equation. The inhomogeneous term is the Watson final-state enhancement factor (or isobar term), while the integral part represents rescattering corrections. It is proved that the presence of this term guarantees that the amplitude satisfies a form of three-body unitarity, in which the three bodies interact in pairs only. This means that two-body unitarity, analyticity, and crossing have generated three-body unitarity (and a three-body amplitude). If the result can be generalized to higher partial waves, it suggests that one can unitarize (in this sense) any inhomogeneous term, provided it has no three-body branch point itself, by simply adding on the integral term, a procedure which could be applied, for instance, to the isobar, peripheral, or stripping models. This is by way of alternative to the usual $\frac{N}{D}$ method in the three-body channel, and seems to imply equations a good deal easier to handle numerically. A comparison between the unitarity relation obtained from the K-T equation and the one normally assumed discloses a difference between them, which would be important if one wished to consider continuations of the relation in the three-body mass.