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

Thomas Barford·Michael C. Birse
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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.

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Thomas Barford, Michael C. Birse. 2004-06-03. Effective theories of scattering with an attractive inverse-square potential and the three-body problem. https://doi.org/10.1088/0305-4470%2F38%2F3%2F009

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