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Nuclear Physics A

The renormalization of single-particle states in nuclear field theory

Daniel R. Bes, G.G. Dussel, R.P.J. Perazzo, Hugo M. Sofia

Published Dec 1, 1977• 12 citations
Publisher
DOI
10.1016/0375-9474(77)90103-8

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We have examined the ground state of the many-fermion system, using perturbation theory, with particular attention to Brueckner's method. A study of the single-particle propagators in the ground state of the system sheds some light on the relation between the single-particle energies used, for example, in the optical model of nuclear reactions, and Brucekner's self-consistent energies. This leads naturally to the use of a renormalization factor to allow for propagation off the energy shell. It is possible to formulate the perturbation theory entirely in terms of graphs with no fermion self-energy parts, and this method is used to examine the approximations used for this problem, and to relate more closely the calculations of the ground-state energy and of other properties of the ground state.

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The renormalization of the effective field theories (EFTs) in many-body systems is the most pressing and challenging problem in modern nuclear ab initio calculation. For general non-relativistic EFTs, we prove that the renormalization group (RG) invariance can be achieved if and only if all single-particle momenta are regulated with a universal cutoff Λ. For a numerical demonstration, we construct a series of N^{2}LO chiral forces with Λvarying from 250 MeV to 400 MeV. With all low energy constants fixed in two- and three-nucleon systems, we reproduce the experimental binding energies of ^{4}He and ^{16}O nearly independently of Λ. In contrast, all recent nuclear EFT constructions regulate the relative momenta for Galilean invariance, thus inherently break the RG invariance. This explains the unpleasantly strong cutoff-dependences observed in recent ab initio calculations. Our method can also be used to build RG-invariant EFTs with non-perturbative interactions.
In the low-energy limit, non-relativistic particles with short-range interactions exhibit universal behavior that is largely independent of microscopic details. This universality is typically described by effective field theory, in which the two-body interaction is renormalized to a single parameter-the scattering length. For systems of identical bosons, the three-body problem reveals the Efimov effect, a novel phenomenon proposed that necessitates the introduction of an additional three-body parameter. However, the exact relation between this three-body parameter, the coupling constants in the effective field theory, and the binding energies of Efimov states remains unresolved. In this Letter, we address this question through a comprehensive analysis of the Skorniakov-Ter-Martirosian equation with a finite cutoff. We establish an exact renormalization relation for the three-body parameter and determine its connection to the energies of Efimov bound states. These results are validated through high-precision numerical simulations. We expect our findings to be of fundamental interest across various fields, including atomic, nuclear, condensed matter, and particle physics, and to have broad applications in both few-body and many-body physics.
We have examined the ground state of the many-fermion system, using perturbation theory, with particular attention to Brueckner's method. A study of the single-particle propagators in the ground state of the system sheds some light on the relation between the single-particle energies used, for example, in the optical model of nuclear reactions, and Brucekner's self-consistent energies. This leads naturally to the use of a renormalization factor to allow for propagation off the energy shell. It is possible to formulate the perturbation theory entirely in terms of graphs with no fermion self-energy parts, and this method is used to examine the approximations used for this problem, and to relate more closely the calculations of the ground-state energy and of other properties of the ground state.