Collapse to the Center and Ambiguity in the Asymptotic Behavior of the Off-Shell Scattering Amplitude in Singular Three-Body Problems

TL;DR

This paper investigates the asymptotic behavior of off-shell scattering amplitudes in singular three-body problems, revealing a collapse phenomenon similar to the two-body falling to the center, with infinite bound states and unbounded energies.

Contribution

It demonstrates the collapse to the center phenomenon in various three-body equations, highlighting the emergence of infinite bound states with unbounded energies.

Findings

Infinite number of bound states appear at large momentum.

Energy spectrum is not bounded from below.

Collapse phenomenon is analogous to the two-body falling to the center.

Abstract

We discuss some examples of equations of the three-body problem with the oscillating asymptotics at large momentum: (i) the fixed-center approximation, (ii) the unitarized equation in the fixed-center approximation, (iii) Skornyakov--Ter-Martirosyan equation, and (iv) equations with operators used in the effective field theory, i.e., which can be expanded in power-series with positive powers of momentum. We show that in the aforementioned three-body problems the situation analogous to the falling down to the center in the two-body problem takes place -- there appears an infinite number of bound states. The energy of these states is not bounded from below. In that sense the situation is close to the falling down to the center in the two-body problem.