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 nondiffractive 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.

Figure 1

Geometrical description of three-particle scattering in real space. Left: Particles interact on the three planes defined by $x_i=x_j$. Right: Projection along the center-of-mass motion in the $(1,1,1)$ direction. The six sectors correspond to different ordering of the three particles on the line, given by a three-digit code.

Figure 2

Three rays arriving in sector 123 at the same angle, and departing in sector 213, again at the same angle. The paths (a) and (b) contribute to the outgoing wave to the right of the dotted line, while to the left only (c) contributes.

Figure 3

Contour $\gamma ={\gamma }_{+}\cup {\gamma }_{-}$ in the $\alpha$ plane for the Sommerfeld integral given by Eq. (9) [18].

Figure 4

Interpretation of Eq. (14) in terms of scattering amplitudes. Note that angles are measured from the line bisecting each sector.

Figure 5

The three rays considered in Sec. 1c, now labeled with the arguments of the amplitude ${\mathcal{A}}_Q\left(\alpha \right)$ describing each.

Figure 6

All of the rays generated by scattering. Each of the six directions is present in each sector.

Figure 7

Geometry of three-particle scattering in momentum space. Allowed values of the three momenta before and after collision lie on the intersection of the sphere of fixed energy and the plane of fixed momentum. The radius of the resulting circle fixes the center-of-mass momentum $k$, while the angle between the two points defines $\varphi$ and the sector $Q$.