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arXiv:nucl-th/0111042

Renormalization Group Equation for Low Momentum Effective Nuclear Interactions

S. K. Bogner, A. Schwenk, T.T.S. Kuo, G. E. Brown

Published Nov 13, 2001• 11 citations
arXivHTMLPDF

Abstract

We consider two nonperturbative methods originally used to derive shell model effective interactions in nuclei. These methods have been applied to the two nucleon sector to obtain an energy independent effective interaction V_{low k}, which preserves the low momentum half-on-shell T matrix and the deuteron pole, with a sharp cutoff imposed on all intermediate state momenta. We show that V_{low k} scales with the cutoff precisely as one expects from renormalization group arguments. This result is a step towards reformulating traditional model space many-body calculations in the language of effective field theories and the renormalization group. The numerical scaling properties of V_{low k} are observed to be in excellent agreement with our exact renormalization group equation.

DOI
10.48550/arxiv.nucl-th/0111042

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We consider two nonperturbative methods originally used to derive shell model effective interactions in nuclei. These methods have been applied to the two nucleon sector to obtain an energy independent effective interaction V_{low k}, which preserves the low momentum half-on-shell T matrix and the deuteron pole, with a sharp cutoff imposed on all intermediate state momenta. We show that V_{low k} scales with the cutoff precisely as one expects from renormalization group arguments. This result is a step towards reformulating traditional model space many-body calculations in the language of effective field theories and the renormalization group. The numerical scaling properties of V_{low k} are observed to be in excellent agreement with our exact renormalization group equation.
This thesis work has investigated the Renormalization Group theory for the nucleonnucleon interaction.Conventional nuclear many-body calculations have the following sources of non-perturbative physics: 1. a strongly repulsive short-range interaction, 2. a tensor force, e.g. from pion exchange, which is highly singular at short-distances, 3. the presence of low-energy bound states or nearly bound states (in the S waves).The RG approach exploits the insensitivity of low-energy processes to the details of the high-energy physics.Using any of the high-precision potentials as input, the highmomentum intermediate states in the Lippmann-Schwinger equation for the T matrix in a particular partial-wave are cut-off at Λ.The details of the physics beyond this limit of resolution are integrated out and included in the potential by requiring that the half-off shell T matrix elements be independent of the cut-off Λ.This requirement leads to a low-momentum potential "V low k ", which is energy independent.The choice of the regulator which cuts off the high momentum intermediate states is investigated.Sharp cut-offs, though straight forward, lead to convergence issues in few-body calculations that are eliminated using smooth regulators.The construction of low-momentum potentials using a smooth regulator is explored in detail.In the course of this study, a three-step process to calculate V low k requiring the cut-off independence of the fully-off shell T matrix elements has been established and this yields better numerical stability than the energy-independent RG.
The shell model is the standard tool for addressing the canonical nuclear many-body problem of nonrelativistic nucleons interacting through a static potential. We discuss several of the uncontrolled approximations that are made in this model to motivate a different approach, one based on an exact solution of the Bloch-Horowitz equation. We argue that the necessary self-consistent solutions of this equation can be obtained efficiently by a Green's function expansion based on the Lanczos algorithm. The resulting effective theory is carried out for the simplest nuclei, d and 3He, using realistic NN interactions such as the Argonne v18 and Reid93 potentials, in order to contrast the results with the shell model. We discuss the wave function normalization, the evolution of the wave function as the "shell model" space is varied, and the magnetic elastic effective operator. The numerical results show a simple renormalization group behavior that differs from effective field theory treatments of the two- and three-body problems. The likely origin of this scaling is discussed.
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.
We summarize recent efforts to develop an angular-momentum-conserving variant of the Density Matrix Renormalization Group method into a practical truncation strategy for large-scale shell model calculations of atomic nuclei. Following a brief description of the key elements of the method, we report the results of test calculations for 48 Cr and 56 Ni . In both cases we consider nucleons limited to the 2p-1f shell and interacting via the KB3 interaction. Both calculations produce a high level of agreement with the exact shell-model results. Furthermore, and most importantly, the fraction of the complete space required to achieve this high level of agreement goes down rapidly as the size of the full space grows.
We show that the renormalization group decimation of modern nucleon potential models to low momenta results in a unique nucleon interaction V_{low k}. This interaction is free of short-ranged singularities and can be used directly in many-body calculations. The RG scaling properties follow directly from the invariance of the scattering phase shifts. We discuss the RG treatment of Fermi liquids. The RG equation for the scattering amplitude in the two particle-hole channels is given at zero temperature. The flow equations are simplified by retaining only the leading term in an expansion in small momentum transfers. The RG flow is illustrated by first studying a system of spin-polarized fermions in a simple model. Finally, results for neutron matter are presented by employing the unique low momentum interaction V_{low k} as initial condition of the flow. The RG approach yields the amplitude for non-forward scattering, which is of great interest for calculations of transport properties and superfluid gaps in neutron star interiors. The methods used can also be applied to condensed matter systems in the absence of long-ranged interactions.
Pionful nuclear effective field theory (NEFT) in the two-nucleon sector is examined from the Wilsonian renormalization group point of view. The pion exchange is cut off at the floating cutoff scale, Λ\LambdaΛ, with the short-distance part being represented as contact interactions in accordance with the general principle of renormalization. We derive the nonperturbative renormalization group equations in the leading order of the nonrelativistic approximation in the operator space including up to O(p2)\mathcal{O}(p^2)O(p2) and find the nontrivial fixed points in the 1S0^1S_01S0​ and 3S1^3S_13S1​--3D1^3D_13D1​ channels which are identified with those in the pionless NEFT. The scaling dimensions, which determine the power counting, of the contact interactions at the nontrivial fixed points are also identified with those in the pionless NEFT. We emphasize the importance of the separation of the pion exchange into the short-distance and the long-distance parts, since a part of the former is nonperturbative while the latter is perturbative.