Chapter 7
Summary and Future Outlook
This brief report describes a systematic development of theoretical attempts carried
out during the last more than two decades to investigate the structural properties of 2nrich halo nuclei employing three-body approach using two-body separable potentials.
In the first two chapters, we lay down the groundwork to present the essentials of
two-particle scattering theory and highlight the salient features of Faddeev’s threebody scattering formalism. There we also present a comparative study to show how
the basic difficulties of disconnectedness of the kernel and boundary conditions
encountered in the standard Lippmann Schwinger equation for three-body scattering
problem are tackled in a natural way using three-body Schrodinger equation with
separable potentials.
Chapter 3 is devoted to the study of Efimov effect wherein the basic steps to
obtain the expression for attractive long-range effective potential for three identical
bosons using the proper boundary conditions for the resonant pair interactions are
outlined. An important consequence of this attractive long-range three-body potential, 1/R
2 as shown by Efimov, is that as long as the two-body scattering length
|a| is sufficiently large compared to the range r 0 , there is a sequence of three-body
bound states whose binding energies form a geometric series lying in the interval
between
2
/mr
2
0 and
2
/ma
2
. In fact, Efimov effect describes a universal lowenergy behavior of the three-particle system. It is for this reason of the universality
character that the effect can be realized and observed in various physical contexts, e.g.,
in atomic, molecular, nuclear or condensed matter physics. In the context of separable
potential approach, this singular behavior of 1/R
2 type is translated into momentum
space as a singularity of 1/p type in the limiting case as two-body scattering length
|a| → ∞.
Another important area which has attracted in recent years a great deal of attention
in the domain of low-energy few body dynamics is the effective field theory. This
topic has been briefly touched in Chap. 4. The general principle of effective field
theory is illustrated by considering two-nucleon system, where the scattering length
|a| |r eff | ≈ r 0 and the contact or pionless effective field theory can be used. As we
© 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_7
121
Summary and Future Outlook
This brief report describes a systematic development of theoretical attempts carried
out during the last more than two decades to investigate the structural properties of 2nrich halo nuclei employing three-body approach using two-body separable potentials.
In the first two chapters, we lay down the groundwork to present the essentials of
two-particle scattering theory and highlight the salient features of Faddeev’s threebody scattering formalism. There we also present a comparative study to show how
the basic difficulties of disconnectedness of the kernel and boundary conditions
encountered in the standard Lippmann Schwinger equation for three-body scattering
problem are tackled in a natural way using three-body Schrodinger equation with
separable potentials.
Chapter 3 is devoted to the study of Efimov effect wherein the basic steps to
obtain the expression for attractive long-range effective potential for three identical
bosons using the proper boundary conditions for the resonant pair interactions are
outlined. An important consequence of this attractive long-range three-body potential, 1/R
2 as shown by Efimov, is that as long as the two-body scattering length
|a| is sufficiently large compared to the range r 0 , there is a sequence of three-body
bound states whose binding energies form a geometric series lying in the interval
between
2
/mr
2
0 and
2
/ma
2
. In fact, Efimov effect describes a universal lowenergy behavior of the three-particle system. It is for this reason of the universality
character that the effect can be realized and observed in various physical contexts, e.g.,
in atomic, molecular, nuclear or condensed matter physics. In the context of separable
potential approach, this singular behavior of 1/R
2 type is translated into momentum
space as a singularity of 1/p type in the limiting case as two-body scattering length
|a| → ∞.
Another important area which has attracted in recent years a great deal of attention
in the domain of low-energy few body dynamics is the effective field theory. This
topic has been briefly touched in Chap. 4. The general principle of effective field
theory is illustrated by considering two-nucleon system, where the scattering length
|a| |r eff | ≈ r 0 and the contact or pionless effective field theory can be used. As we
© 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_7
121