One spatial dimensional finite volume three-body interaction for a short-range potential

In this work, we use McGuire’s model to describe scattering of three spinless identical particles in one spatial dimension; we first present analytic solutions of Faddeev’s equation for scattering of three spinless particles in free space. The three particles interaction in finite volume is derived subsequently, and the quantization conditions by matching wave functions in free space and finite volume are presented in terms of two-body scattering phase shifts. The quantization conditions obtained in this work for the short-range interaction are Lüscher’s formula-like and consistent with Yang’s results [Phys. Rev. Lett. 19, 1312 (1967)].

Figure 1

(a) The disconnected diagram for the third particle as a spectator. (b) and (c) Double scattering contributions from pair (23) and (31) into (12) pair. (d) A triple scattering contribution.

Figure 2

Diagram show six segments from $(I)-(VI)$ in ($r_{12}$, $r_3$) plane, the $\delta$-function potentials are nonzero only at lines, $r_{12}=0$, $r_{23}=0$ and $r_{31}=0$. The arrows show the positive direction of each variable.