Abstract
We derive the three-body quantization condition in a finite volume using an effective field theory in the particle-dimer picture. Moreover, we consider the extraction of physical observables from the lattice spectrum using the quantization condition. To illustrate the general framework, we calculate the volume-dependent three-particle spectrum in a simple model both below and above the three-particle threshold. The relation to existing approaches is discussed in detail.
Highlights
On the one hand, there is no such framework for intermediate states with three or more particles, several attempts in this direction have been undertaken
The main features of the three-body problem in the infinite volume can be summarized as follows: (i) All physical observables in the three-particle sector at low energies are parameterized in terms of the two-particle C0, C2, . . . and three-particle D0, D2, . . . couplings
We have thoroughly considered the formulation of the three-particle problem in the infinite volume and have demonstrated that the particle-dimer picture provides an equivalent description of this problem
Summary
In order to simplify the formalism and highlight the central conceptual issues, we consider the interaction of three identical non-relativistic scalars. Because of Galilei invariance, the interaction does not depend on the center-of-mass momentum It is characterized by the relative momenta of the two particles in the final and initial states, p and q, respectively. The validity of the Luscher equation implies that such off-shell terms do not affect the spectrum, which is solely determined by the on-shell S-matrix elements This stems from the existence of two widely separated scales — the box size L and the typical interaction range R, with R L. This means that the two-particle wave function near the boundaries is given by its asymptotic form determined by the phase shift. Where p cot δ(p) is given by the effective-range expansion (2.7)
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