The Similarity Renormalization Group (SRG) is investigated as apowerful yet practical method to modify nuclear potentials so as to reduce
computational requirements for calculations of observables. The SRG
proves to be versatile and robust in its treatment of these
interactions and opens the door to a deeper understanding of the
renormalization process.
Chiral Effective Field Theory (Chiral EFT) provides a consistent and
rigorous parametrization of the inter-nucleon interaction. While
already softer than other available potentials, transformation
via the Similarity Renormalization Group (SRG) brings numerous
computational benefits. The hierarchy of many-body forces inherent
in Chiral EFT's are also treated consistently by the SRG's simple
formalism. The SRG is a natural partner to this modern
program of formulating the nuclear interaction.
The key feature of SRG transformations that leads to computational
benefits is the decoupling of low-energy nuclear physics from
high-energy details of the inter-nucleon interaction. We
examine decoupling quantitatively for two-body observables and
few-body binding energies. The universal nature of this decoupling is
illustrated and errors from suppressing high-momentum modes above the
decoupling scale are shown to be perturbatively small.
As implemented here, the SRG provides freedom to choose the form of
its transformations and can be tailored to a given application. We
explore the impacts of various choices and their decoupling
properties, specifically a block-type transformation inspired by
previous renormalization group techniques. Sharp and smooth block-diagonal
forms of phase-shift equivalent nucleon-nucleon potentials in momentum
space are generated as examples and compared to existing analogous low-momentum
interactions.
To explore the SRG evolution of many-body forces, we use as a
laboratory a one-dimensional system of bosons with short-range
repulsion and mid-range attraction, which emulates realistic nuclear
forces. The free-space SRG is implemented for few-body systems in a
symmetrized harmonic oscillator basis using a recursive construction
analogous to no-core shell model implementations. This approach is
fully unitary up to induced A-body forces when applied with an
A-particle basis (e.g., A-body bound-state energies are exactly
preserved). The oscillator matrix elements for a given A can then be
used in larger systems. Errors from omitted induced many-body forces
show a hierarchy of decreasing contribution to binding energies. An
analysis of individual contributions to the growth of many-body forces
demonstrates such a hierarchy and provides an understanding of its
origins. Several other important sample calculations are explored in
this model for future use in realistic systems.
Building on one-dimensional results we performed the first practical
evolution of three-dimensional many-body forces within the No-Core
Shell Model basis. Results for the triton binding energy are consistent
with previous calculations involving momentum-space evolution of only
two-body forces, and validate expectations from calculations in the
one-dimensional oscillator basis. When applied to Helium-4 calculations,
the two- and three-body oscillator matrix elements yield rapid
convergence of the ground-state energy with a small net contribution
of the induced four-body force. The radius of light nuclei is also
explored in the three-dimensional basis.