Chapter 3
Efimov’s Universal Three-Body Effect
3.1 Introduction
Efimov [13] discovered a remarkable physical phenomenon where the Hamiltonian of
a three-particle quantum system with the resonantly interacting pairs gives rise to an
attractive long-range ‘effective’ three-body force which can produce a large number
of energy levels in a three-particle spectrum. The original Efimov’s physical argument
is based on the replacement of the two-body potential with a boundary condition,
which is essentially equivalent to considering a two-body zero-range interaction,
and on the introduction of hyper-spherical coordinates. If the resonant condition is
satisfied in these coordinates, the problem becomes separable, and in the equation for
the hyperradius R, the long-range, attractive effective potential of the form, V 0 (R) =
−[|s 0 |
2
+ 1/4]/R
2 , appears.
In the following subsection, we set up the preliminaries by writing the expression
for the scattering amplitude of two particles at low-energy s-wave scattering in terms
of the s-wave phase shifts under the condition when the scattering length ‘a’ is much
larger than the range ‘r 0 ’. The s-wave total cross section is then related by unitarity
through the optical theorem to the s-wave phase shifts which can produce resonance at
large values of scattering length. In the next subsection, a simple derivation to obtain
the expression for attractive long-range effective potential for three identical bosons
using the proper boundary conditions for the resonant pair interaction is presented.
Finally, we highlight the salient features of the Efimov effect.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. S. Bhasin and I. Mazumdar, Few Body Dynamics, Efimov Effect and Halo Nuclei,
SpringerBriefs in Physics,
https://doi.org/10.1007/978-3-030-56171-0_3
31
Efimov’s Universal Three-Body Effect
3.1 Introduction
Efimov [13] discovered a remarkable physical phenomenon where the Hamiltonian of
a three-particle quantum system with the resonantly interacting pairs gives rise to an
attractive long-range ‘effective’ three-body force which can produce a large number
of energy levels in a three-particle spectrum. The original Efimov’s physical argument
is based on the replacement of the two-body potential with a boundary condition,
which is essentially equivalent to considering a two-body zero-range interaction,
and on the introduction of hyper-spherical coordinates. If the resonant condition is
satisfied in these coordinates, the problem becomes separable, and in the equation for
the hyperradius R, the long-range, attractive effective potential of the form, V 0 (R) =
−[|s 0 |
2
+ 1/4]/R
2 , appears.
In the following subsection, we set up the preliminaries by writing the expression
for the scattering amplitude of two particles at low-energy s-wave scattering in terms
of the s-wave phase shifts under the condition when the scattering length ‘a’ is much
larger than the range ‘r 0 ’. The s-wave total cross section is then related by unitarity
through the optical theorem to the s-wave phase shifts which can produce resonance at
large values of scattering length. In the next subsection, a simple derivation to obtain
the expression for attractive long-range effective potential for three identical bosons
using the proper boundary conditions for the resonant pair interaction is presented.
Finally, we highlight the salient features of the Efimov effect.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. S. Bhasin and I. Mazumdar, Few Body Dynamics, Efimov Effect and Halo Nuclei,
SpringerBriefs in Physics,
https://doi.org/10.1007/978-3-030-56171-0_3
31