Diffractive scattering of three particles in one dimension: a simple result for weak violations of the Yang--Baxter equation

We study scattering of three equal mass particles in one dimension. Integrable interactions are synonymous with non-diffractive scattering, meaning that the set of incoming momenta for any scattering event coincides with the set of outgoing momenta. A system is integrable if the two particle scattering matrix obeys the Yang--Baxter equation. Nonintegrable interactions correspond to diffractive scattering, where the set of outgoing momenta may take on all values consistent with energy and momentum conservation. Such processes play a vital role in the kinetics of one dimensional gases, where binary collisions are unable to alter the distribution function. When integrability is broken weakly, the result is a small diffractive scattering amplitude. Our main result is a simple formula for the diffractive part of the scattering amplitude, when the violation of the Yang--Baxter equation is small. Although the derivation is given for delta-function interactions, the result depends only on the two-particle scattering matrix, and should therefore also apply to finite-range interactions close to integrable.