Chapter 4
Effective Field Theory
4.1 Introduction
Effective field theory offers a general model independent approach to study the
low energy behavior of physical systems in several different branches of physics;
most importantly in quantum electrodynamics, standard model of elementary particle
physics and condensed matter physics. While most of the applications of effective
field theories are carried out within the framework of quantum field theory, it has
been demonstrated by Lapage [32] that the general principles of effective theory can
equally well be applied to non-relativistic quantum mechanics for two and many
particle systems within the framework of potential approach.
In this chapter, we present a brief description of low energy N-N scattering and
three-body scattering problem with a view to illustrate the basic principles of effective field theory, including the renormalization procedure employed to calculate the
loop diagrams involved in the scattering amplitude. We then derive, starting from
a three-body Schrodinger equation using two-body s-state separable potentials, the
expressions for the scattering amplitude in the framework of effective field theory to
study n–
19 C scattering at low energies (where
19 C is assumed to be a loosely bound
state of n–
18 C system).
4.2 Two-Body Problem: N-N Scattering at Low Energies
in Effective Field Theory
A general principle of effective field theory, used in quantum mechanical problems
employing two body potentials, is that if we are interested only in low energy observables where the large distance behavior of the potential V (r) is dominant, then such an
interaction can be replaced by an effective potential, V eff (r ; λ), where V eff is supposed
© 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_4
41
Effective Field Theory
4.1 Introduction
Effective field theory offers a general model independent approach to study the
low energy behavior of physical systems in several different branches of physics;
most importantly in quantum electrodynamics, standard model of elementary particle
physics and condensed matter physics. While most of the applications of effective
field theories are carried out within the framework of quantum field theory, it has
been demonstrated by Lapage [32] that the general principles of effective theory can
equally well be applied to non-relativistic quantum mechanics for two and many
particle systems within the framework of potential approach.
In this chapter, we present a brief description of low energy N-N scattering and
three-body scattering problem with a view to illustrate the basic principles of effective field theory, including the renormalization procedure employed to calculate the
loop diagrams involved in the scattering amplitude. We then derive, starting from
a three-body Schrodinger equation using two-body s-state separable potentials, the
expressions for the scattering amplitude in the framework of effective field theory to
study n–
19 C scattering at low energies (where
19 C is assumed to be a loosely bound
state of n–
18 C system).
4.2 Two-Body Problem: N-N Scattering at Low Energies
in Effective Field Theory
A general principle of effective field theory, used in quantum mechanical problems
employing two body potentials, is that if we are interested only in low energy observables where the large distance behavior of the potential V (r) is dominant, then such an
interaction can be replaced by an effective potential, V eff (r ; λ), where V eff is supposed
© 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_4
41