Effective theories of scattering with an attractive inverse-square potential and the three-body problem

Abstract

A distorted-wave version of the renormalisation group is applied to scattering by an inverse-square potential and to three-body systems. In attractive three-body systems, the short-distance wave function satisfies a Schroedinger equation with an attractive inverse-square potential, as shown by Efimov. The resulting oscillatory behaviour controls the renormalisation of the three-body interactions, with the renormalisation-group flow tending to a limit cycle as the cut-off is lowered. The approach used here leads to single-valued potentials with discontinuities as the bound states are cut off. The perturbations around the cycle start with a marginal term whose effect is simply to change the phase of the short-distance oscillations, or the self-adjoint extension of the singular Hamiltonian. The full power counting in terms of the energy and two-body scattering length is constructed for short-range three-body forces.

19 pages (RevTeX), 2 figures

References in corpus (3)

• The three-body problem with short-range forces: renormalized equations and regulator-independent results
• An effective field theory for coupled-channel scattering
• Log-periodic behavior of finite size effects in field theories with RG limit cycles

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