Dispersion Theory Model of Three-Body Production and Decay Processes

We study three-body production and decay processes, $A+B\to a+b+c$ and $A\to a+b+c$. We assume that the amplitudes are determined in both cases solely by the final-state interactions among pairs of $a$, $b$, and $c$, and that the dynamics is given by single-variable dispersion terms in each two-body channel (that is, a representation of the Khuri-Treiman type). Applying a method introduced by Anisovich, we obtain a linear integral equation which gives the three-body amplitude in terms of the two-body one $g$. In an effective-range approximation for $g$, the kernel can be easily evaluated; it is the sum of the discontinuities of the triangle graph in perturbation theory, when considered as a function of an internal mass. In the $S$-wave case, the resulting integral equation has a unique solution depending on one arbitrary parameter (a subtraction constant). In the nonrelativistic limit, the equation coincides with Anisovich's; this in turn is analogous to the equation derived by Skornyakov and Ter-Martyrosyan for a three-body problem in potential theory, with delta-function forces. It is suggested that the present theory is therefore a relativistic analog of a potential model with short-range forces. As applications, we mention the determination of the final-state pion spectra in $K\to 3\pi$ decay, and the possibility of studying the generation of resonances in the total center-of-mass energy, induced by interactions of real particle exchange processes (the Peierls mechanism).