R. J. Furnstahl

From low-momentum interactions to nuclear structure

We present an overview of low-momentum two-nucleon and many-body interactions and their use in calculations of nuclei and infinite matter. The softening of phenomenological and effective field theory (EFT) potentials by renormalization group (RG) transformations that decouple low and high momenta leads to greatly enhanced convergence in few- and many-body systems while maintaining a decreasing hierarchy of many-body forces. This review surveys the RG-based technology and results, discusses the connections to chiral EFT, and clarifies various misconceptions.

Similarity Renormalization Group for Nucleon-Nucleon Interactions

The similarity renormalization group (SRG) is based on unitary transformations that suppress off-diagonal matrix elements, forcing the Hamiltonian toward a band-diagonal form. A simple SRG transformation applied to nucleon-nucleon interactions leads to greatly improved convergence properties while preserving observables and provides a method to consistently evolve many-body potentials and other operators.

Analysis of chiral mean field models for nuclei

Abstract An analysis of nuclear properties based on a relativistic energy functional containing Dirac nucleons and classical scalar and vector meson fields is discussed. Density functional theory implies that this energy functional can include many-body effects that go beyond the simple Hartree approximation. Using basic ideas from effective field theory, a systematic truncation scheme is developed for the energy functional, which is based on an expansion in powers of the meson fields and their gradients. The utility of this approach relies on the observation that the large scalar and vector fields in nuclei are small enough compared to the nucleon mass to provide useful expansion parameters, yet large enough that exchange and correlation corrections to the fields can be treated as minor perturbations. Field equations for nuclei and nuclear matter are obtained by extremizing the energy functional with respect to the field variables, and inversion of these field equations allows one to express the unknown coefficients in the energy functional directly in terms of nuclear-matter properties near equilibrium. This allows for a systematic and complete study of the parameter space, so that parameter sets that accurately reproduce nuclear observables can be found, and models that fail to reproduce nuclear properties can be excluded. Chiral models are analyzed by considering specific Lagrangians that realize the spontaneously broken chiral symmetry of QCD in different ways and by studying them at theHartree level. The resulting energy functionals are special cases of the general functional considered earlier. Models that include a light scalar meson playing a dual role as the chiral partner of the pion and the mediator of the intermediate-range nucleon-nucleon interaction, and which include a “Mexican-hat” potential, fail to reproduce basic ground-state properties of nuclei at the Hartree level. In contrast, chiral models with a non-linear realization of the symmetry are shown to contain the full flexibility inherent in the general energy functional and can therefore successfully describe nuclei.

QCD sum rules and applications to nuclear physics

Applications of QCD sum-rule methods to the physics of nuclei are reviewed, with an emphasis on calculations of baryon self-energies in infinite nuclear matter. The sum-rule approach relates spectral properties of hadrons propagating in the finite-density medium, such as optical potentials for quasinucleons, to matrix elements of QCD composite operators (condensates). The vacuum formalism for QCD sum rules is generalized to finite density, and the strategy and implementation of the approach is discussed. Predictions for baryon self-energies are compared to those suggested by relativistic nuclear physics phenomenology. Sum rules for vector mesons in dense nuclear matter are also considered.

Improved nuclear matter calculations from chiral low-momentum interactions

We present nuclear matter calculations based on low-momentum interactions derived from chiral effective field theory potentials. The current calculations use an improved treatment of the three-nucleon force (3NF) contribution that includes a corrected combinatorial factor beyond Hartree-Fock that was omitted in previous nuclear matter calculations. We find realistic saturation properties using parameters fit only to few-body data, but with larger uncertainty estimates from cutoff dependence and the 3NF parametrization than in previous calculations.

Vacuum Contributions in a Chiral Effective Lagrangian for Nuclei

A relativistic hadronic model for nuclear matter and finite nuclei, which incorporates nonlinear chiral symmetry and broken scale invariance, is presented and applied at the one-baryon-loop level to finite nuclei. The model contains an effective light scalar field that is responsible for the midrange nucleon-nucleon attraction and which has anomalous scaling behavior. One-loop vacuum contributions in this background scalar field at finite density are constrained by low-energy theorems that reflect the broken scale invariance of quantum chromodynamics. A mean-field energy functional for nuclear matter and nuclei is derived that contains small powers of the fields and their derivatives, and the validity of this truncation is discussed. Good fits to the bulk properties of finite nuclei and single-particle spectra are obtained.

Evolution of Nuclear Many-Body Forces with the Similarity Renormalization Group

The first practical method to evolve many-body nuclear forces to softened form using the similarity renormalization group in a harmonic oscillator basis is demonstrated. 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.

Convergence in the no-core shell model with low-momentum two-nucleon interactions

Abstract The convergence of no-core shell model (NCSM) calculations using renormalization group evolved low-momentum two-nucleon interactions is studied for light nuclei up to 7 Li. Because no additional transformation was used in applying the NCSM framework, the energy calculations satisfy the variational principle for a given Hamiltonian. Dramatic improvements in convergence are found as the cutoffs are lowered. The renormalization group equations are truncated at two-body interactions, so the evolution is only approximately unitary and converged energies for A ⩾ 3 vary with the cutoff. This approximation is systematic, however, and for useful cutoff ranges the energy variation is comparable to natural-size truncation errors inherent from the initial chiral effective field theory potential.

A recipe for EFT uncertainty quantification in nuclear physics

The application of effective field theory (EFT) methods to nuclear systems provides the opportunity to rigorously estimate the uncertainties originating in the nuclear Hamiltonian. Yet this is just one source of uncertainty in the observables predicted by calculations based on nuclear EFTs. We discuss the goals of uncertainty quantification in such calculations and outline a recipe to obtain statistically meaningful error bars for their predictions. We argue that the different sources of theory error can be accounted for within a Bayesian framework, as we illustrate using a toy model.

Low-momentum interactions with smooth cutoffs

Abstract Nucleon–nucleon potentials evolved to low momentum, which show great promise in few- and many-body calculations, have generally been formulated with a sharp cutoff on relative momenta. However, a sharp cutoff has technical disadvantages and can cause convergence problems at the 10–100 keV level in the deuteron and triton. This motivates using smooth momentum-space regulators as an alternative. We generate low-momentum interactions with smooth cutoffs both through energy-independent renormalization group methods and using a multi-step process based on the Bloch–Horowitz approach. We find greatly improved convergence for calculations of the deuteron and triton binding energies in a harmonic oscillator basis compared to results with a sharp cutoff. Even a slight evolution of chiral effective field theory interactions to lower momenta is beneficial. The renormalization group preserves the long-range part of the interaction, and consequently the renormalization of long-range operators, such as the quadrupole moment, the radius and 〈 1 / r 〉 , is small. This demonstrates that low-energy observables in the deuteron are reproduced without short-range correlations in the wave function.