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Investigations of the renormalization group approach to the nucleon-nucleon interaction

S. Ramanan

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

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.

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•Dec 13, 2010•0 citations
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.
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.
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.
The Similarity Renormalization Group (SRG) is investigated as a powerful yet practical method to modify nuclear potentials so as to reduce computational requirements for calculations of observables. 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. 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. 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 3H 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 4He 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.
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.
One of the main challenges for ab initio nuclear many-body theory in the coming decade is the growth of computational and storage costs as calculations are extended to increasingly heavy, exotic, and structurally complex nuclei. Here, we investigate the factorization of nuclear interactions as a means to address this issue. We perform singular value decompositions (SVD) of current nucleon-nucleon interactions in partial wave representation and study the dependence of the singular value spectrum on interaction characteristics like regularization schemes and resolution scales. Next, we develop and implement the similarity renormalization group (SRG) evolution of the interaction in terms of the relevant singular vectors, and demonstrate that this SVD-SRG approach accurately preserves two-nucleon observables. We find that low-resolution interactions naturally allow the truncation of the SVD at low rank, and that a small number of relevant components is sufficient to capture the nuclear interaction and perform an accurate SRG evolution, while the Coulomb interaction requires special consideration. The rank is uniform across all partial waves and almost independent of the basis choice in the tested cases. This suggests an interpretation of the relevant singular components as mere representations of a small set of abstract operators that can describe the interaction and its SRG flow. Following the traditional workflow for nuclear interactions, we discuss how the transformation between the center-of-mass and laboratory frames creates redundant copies of the partial wave components when implemented in matrix representation, and we discuss strategies for mitigation. Finally, we test the low-rank approximation to the SRG-evolved interactions in many-body calculations using the in-medium SRG. By including nuclear radii in our analysis, we verify that the implementation of the SRG using the singular vectors of the interaction does not spoil the evolution of other observables.
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.