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 Effective Field Theories and the Many-Body Problem
 (INT program March 23 - June 5, 2009)

  Reported by Calvin Johnson, Dick Furnstahl, Erich Ormand, and Bira van Kolck
  Date posted July 28, 2009

The program on Effective Field Theories and the Many-body problem was held at the INT from March 23 through June 5, 2009. We had close to 90 participants and 54 talks.

The past 15+ years have seen tremendous advances in low-energy nuclear theory, driven by new theoretical approaches as well as more powerful computers. We now have a much more rigorous understanding of the interaction between nucleons, guided largely through effective field theory and the renormalization group; new, more reliable ab initio calculations of low-energy structure through improvements in techniques (no-core shell model, coupled cluster calculations, and improved Monte Carlo methods) as well as better methods for constructing effective interactions that arise through truncation of the model space; and reinvigorated research in density functional theory, as evidenced by the current UNEDF (Universal Nuclear Energy Density Functional) SciDAC project. The goal of this program was, in brief, to foster better communication between these three communities.

In this we succeeded without question. There were intense discussions particularly between the many-body and effective-field theorists, and between many-body and density-functional theorists. Most of the arguments concerned the proper treatment of inter-nucleon interactions within the limited model spaces available for ab-initio calculations: the use and limitations of perturbation theory, the role of cutoffs, and the treatment of center-of-mass motion. In the following, we give a flavor of these discussions. During the program numerous collaborations started, and we will highlight two specific investigations, out of many, that originated during the program.

One particular issue was proper non-perturbative treatment of the interaction. In chiral perturbation theory, one organizes contributions to observables through 'power-counting'; in fitting to scattering data while respecting renormalization group invariance, some leading terms are iterated to all orders while others are used only in finite-order perturbation theory. When done correctly, this allows one to understand the convergence as a function of the cutoff. In most applications to many-body theory (e.g., the no-core shell model or coupled-cluster calculations), however, all terms are iterated to all orders. There were talks by effective-field theorists where the scope of the leading order was discussed. Many-body practitioners gained a better appreciation of the perturbative expansion and agreed to investigate the cutoff dependence from the various orders in the EFT expansion.

The issue of the interpretation of the cutoff, which is in momentum space, was in fact taken more broadly in connection to the effects of truncation in a many-body calculation, which typically uses a harmonic oscillator basis. This issue is central to the implementation of ab initio methods. The error of an ab initio calculation is related to the ability to extrapolate results to the infinite matrix limit with full accounting of the infra-red (IR) and ultra-violet (UV) dependences of the results. J. Vary and S. Coon addressed this issue in the ab initio no core full configuration (NCFC) approach. In particular, they analyzed previously published results for ground state energies and unpublished results for rms radii using a new definitions of the IR and UV regulators that are inherent in the harmonic oscillator basis of the NCFC approach. They found that 4He and 6He results exhibit a significant tendency towards simple scaling in these limits though the scaling behavior is not of the same form as previously published results with simple models. Further work and is in progress to analyze results of other nuclei and to determine how the scaling behavior depends on the Hamiltonian chosen. A paper is being drafted and will be submitted for publication in the Fall.

Figure 1

In no-core shell model calculations, one invokes a energy-truncation scheme that allows one to exactly factor out spurious states from center-of-mass motion. Many other calculations, such as coupled cluster, however, are limited to a orbital truncation scheme, and there were questions about contamination from center-of-mass motion. Inspired by this discussion, G. Hagen, T. Papenbrock, and D. Dean (arxiv:0905.3167) computed the expectation value of the center-of-mass operator for a coupled-cluster calculation in a harmonic oscillator basis. They found a surprising but happy result: the center-of-mass motion is often minimal, that is, zero-point motion resulting in a Gaussian center-of-mass wave function, although the oscillator parameter for that minimal center-of-mass motion was not the same as for the basis states. The main part of Figure 1 shows the expectation value of the generalized center-of-mass Hamiltonian for 16O (computed with a Vlow k interaction from an N3LO EFT potential) as a function of the oscillator frequency of the employed model space; it vanishes and thereby demonstrates that the center-of-mass wave function is a Gaussian. The inset shows the frequency of this wave function. This result indicates that center-of-mass motion may be both less problematic, but also subtler, than previously realized.

As these examples show, there was a enormous amount of activity as well as several specific technical advances that occurred during the program. Exit reports document that participants found this to be a lively and informative program, with some frequent INT visitors stating it was the best program they had ever attended. The reports describe many new collaborative efforts catalyzed by the program. These include: testing similarity renormalization group evolved NN+NNN chiral EFT interactions, which were presented for the first time by E. Jurgenson, to Hartree-Fock and many-body perturbation theory calculations of nuclei; applying Bayesian methods to extract EFT parameters from lattice QCD; testing the stability of energy functionals using RPA theory; providing nuclear structure input for reaction calculations.