Three-body forces produced by a similarity renormalization group transformation in a simple model

Abstract

A simple class of unitary renormalization group transformations that force hamiltonians towards a band-diagonal form produce few-body interactions in which low-and high-energy states are decoupled, which can greatly simplify many-body calculations. One such transformation has been applied to phenomenological and effective field theory nucleon-nucleon interactions with success, but further progress requires consistent treatment of at least the three-nucleon interaction. In this paper we demonstrate in an extremely simple model how these renormalization group transformations consistently evolve two-and three-body interactions towards band-diagonal form, and introduce a diagrammatic approach that generalizes to the realistic nuclear problem.

Similarity renormalization group evolution of many-body forces in a one-dimensional model

“…[8] is where each bar denotes a commutator with T rel . We remind the reader that dT rel /ds = 0 by construction.…”

“…In Ref. [8], a diagrammatic approach to the SRG equation was introduced, which organized the independent evolution of two-and three-body (and higher-body) potentials. This formalism is necessary in a momentum basis to avoid "dangerous" delta functions from spectator particles.…”

Density matrix expansion for low-momentum interactions

“…These are not yet available (although SRG methods show promise of providing them in the near future [10,11,12]), and are instead approximated by adjusted chiral N 2 LO three-body potentials [7]. The validity of this approximation relies on the RG methods modifying only the short-distance part of the potential and is supported by the observation that the EFT hierarchy of many-body forces appears to be preserved by the RG running [7].…”

“…This can be accomplished by lowering a momentum cutoff Λ [4,5,6,7] or performing a series of unitary transformations that drive the hamiltonian toward the diagonal [10,11,12]. The UCOM transformations of Ref.…”

“…Recent progress in evolving chiral effective field theory (EFT) interactions to lower momentum using renormalization group (RG) methods [4,5,6,7,8,9,10,11,12] (see also [13,14]) makes feasible a microscopic calculation of a universal nuclear energy density functional (UNEDF) [15]. The evolution weakens or largely eliminates sources of non-perturbative behavior in the two-nucleon sector such as strong short-range repulsion and the tensor force from iterated pion exchange [9], and the consistent three-nucleon interaction is perturbative at lower cutoffs [7].…”