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arXiv:1208.1734 · Physical Review C

High-momentum tails from low-momentum effective theories

S. K. Bogner, Dietrich Roscher

Published Dec 5, 2012• 40 citations
arXivHTMLPDF

Abstract

In a recent work, Anderson et al. [Phys. Rev. C 82, 054001 (2010)] used the renormalization group (RG) evolution of the momentum distribution to show that, under appropriate conditions, operator expectation values exhibit factorization in the two-nucleon system. Factorization is useful because it provides a clean separation of long- and short-distance physics, and suggests a possible interpretation of the universal high-momentum dependence and scaling behavior found in nuclear momentum distributions. In the present work, we use simple decoupling and scale-separation arguments to extend the results of Anderson et al. to arbitrary low-energy $A$-body states. Using methods that are reminiscent of the operator product expansion (OPE) in quantum field theory, we find that the high-momentum tails of momentum distributions and static structure factors factorize into the product of a universal function of momentum that is fixed by two-body physics, and a state-dependent matrix element that is the same for both and is sensitive only to low-momentum structure of the many-body state. As a check, we apply our factorization relations to two well studied systems, the unitary Fermi gas and the electron gas, and reproduce known expressions for the high-momentum tails of each.

DOI
10.1103/physrevc.86.064304

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In a recent work, Anderson et al. [Phys. Rev. C 82, 054001 (2010)] used the renormalization group (RG) evolution of the momentum distribution to show that, under appropriate conditions, operator expectation values exhibit factorization in the two-nucleon system. Factorization is useful because it provides a clean separation of long- and short-distance physics, and suggests a possible interpretation of the universal high-momentum dependence and scaling behavior found in nuclear momentum distributions. In the present work, we use simple decoupling and scale-separation arguments to extend the results of Anderson et al. to arbitrary low-energy AAA-body states. Using methods that are reminiscent of the operator product expansion (OPE) in quantum field theory, we find that the high-momentum tails of momentum distributions and static structure factors factorize into the product of a universal function of momentum that is fixed by two-body physics, and a state-dependent matrix element that is the same for both and is sensitive only to low-momentum structure of the many-body state. As a check, we apply our factorization relations to two well studied systems, the unitary Fermi gas and the electron gas, and reproduce known expressions for the high-momentum tails of each.
The momentum distribution in a Fermi gas with two spin states and a large scattering length has a tail that falls off like 1/k4 at large momentum k, as pointed out by Tan. He used novel methods to derive exact relations between the coefficient of the tail in the momentum distribution and various other properties of the system. We present simple derivations of these relations using the operator product expansion for quantum fields. We identify the coefficient as the integral over space of the expectation value of a local operator that measures the density of pairs.
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 nucleon momentum distribution nA(k){n}_{A}(k)nA​(k) for A=2A=2A=2, 3, 4, 16, and 40 nuclei is systematically analyzed in terms of wave functions resulting from advanced solutions of the nonrelativistic Schr\"odinger equation, obtained within different many-body approaches based upon different realistic bare nucleon-nucleon (NNNNNN) interactions featuring similar short-range repulsion and tensor interactions. Particular attention is paid to the separation of the momentum distributions into the mean-field and short-range correlation (SRC) contributions. It is shown that although at high values of the momentum kkk different approaches lead to some quantitative differences, these do not hinder the general conclusion that the high-momentum behavior ($k\ensuremath{\gtrsim}1.5--2$ fm${}^{\ensuremath{-}1}$) of all nuclei considered are very similar, exhibiting the well-known scaling behavior with the mass number AAA, independently of the used many-body approach and the details of the bare NNNNNN interaction. To analyze and understand the frequently addressed question concerning the relationships between the nucleus, nA(k){n}_{A}(k)nA​(k), and the deuteron, nD(k){n}_{D}(k)nD​(k), momentum distributions, the spin (SSS)-isospin (TTT) structure of few-nucleon systems and complex nuclei is analyzed in terms of realistic NNNNNN interactions and many-body approaches. To this end, the number of NNNNNN pairs in a given (STSTST) state, viz., (ST)=(10)(ST)=(10)(ST)=(10), (00), (01), and (11), and the contribution of these states to the nucleon momentum distributions are calculated. It is shown that, apart from the (00) state, which has very small effects, all other spin-isospin states contribute to the momentum distribution in a wide range of momenta. It is shown that for all nuclei considered the momentum distributions in the states T=0T=0T=0 and T=1T=1T=1 exhibit at $k\ensuremath{\gtrsim}1.5--2$ fm${}^{\ensuremath{-}1}$ very similar behaviors, which represents strong evidence of the AAA-independent character of SRCs. The ratio nA(k)/nD(k){n}_{A}(k)/{n}_{D}(k)nA​(k)/nD​(k) is analyzed in detail, stressing that in the SRC region it always increases with the momentum and the origin of such an increase is discussed and elucidated. The relationships between the one- and two-body momentum distributions, considered in a previous paper, are discussed and clarified, pointing out the relevant role played by the center-of-mass motion of a correlated pair in the (10) state. Eventually, the values of the the probability of high-momentum components in nuclei and the per nucleon probability a2{a}_{2}a2​ of deuteronlike configurations in nuclei are calculated, and the relationship of the present approach with the many-body methods based upon low-momentum effective interactions is briefly discussed.
In this thesis, cold gases as well as the electron gas are analyzed with two expansion techniques that have, in essence, the same aim: To describe parameter regimes where the few-body aspect dominates over the collective many-body behavior.The Operator Product Expansion (OPE) of quantum field theory is used to extract high-frequency properties of the correlation function that determines the radio-frequency (RF) response of a two-dimensional Fermi gas with zero-range interactions.Exact sum rules that relate the zeroth and first moment of the radio-frequency spectrum to the particle number and the contact of the gas, are derived from these properties.A quantum cluster (virial-) expansion is employed to study the properties of the spectral function of the two-dimensional Fermi gas at small to moderate degeneracy.From the spectral function, observables like the density of states and the momentum distribution are calculated and are found to be in excellent agreement with Tan's universal relations.In the second part of the thesis, the cluster expansion is applied to both bosonic and fermionic three-dimensional quantum gases.Virial coefficients, contacts and momentum distribution are calculated up to third order in the fugacity.This includes three-particle correlations exactly.The results are are in excellent agreement with known universal relations.In the third part of this thesis, an OPE is applied to the problem of the interacting electron gas in both two and three dimensions.A power-law decay of the momentum distribution and the static structure factor at high momenta is found, compatible with earlier findings by J. Kimball.
Short-range interactions between the constituent nucleons of atomic nuclei usually generate strong coupling between high-and low-momentum degrees of freedom in a nuclear Hamiltonian.This coupling leads to significant difficulty in computing the properties of nuclei using traditional quantum many-body techniques.The Similarity Renormalization Group (SRG) flow equations can unitarily soften these nuclear Hamiltonians by decoupling high-energy intermediate state contributions to low-energy observables.Analogous flow equations can be used to consistently evolve operators so that observables other than the binding energy can be calculated.The question in practice is whether the advantages of a softer Hamiltonian and less correlated wave functions are offset by complications in approximating and applying other operators.We begin by examining the properties of SRG-evolved operators in the deuteron, and lead into methods for few-body systems.The nuclear Hamiltonian contains a hierarchy of many-body forces.The most significant I would like to express my sincerest thanks to Richard Furnstahl, my thesis advisor.He has provided me a thorough grounding in the study of nuclear physics.Thanks to him, I have been involved not only in research, but also the larger nuclear physics community since my first year of study at The Ohio State University (OSU).However, more than that, he has set an exceptional personal and professional example.He has cultivated an overwhelmingly positive atmosphere in our research group.He has an infectious energy for learning and research.He clearly loves what he does and sharing his knowledge with others.The level of care and personal attention he exhibits has been exceptional, and I am especially thankful for this.I would also like to thank the many other outstanding teachers I have had throughout my education here at OSU, the University of Kentucky, and the school system of Campbell County, Kentucky.With their help and dedication I have gained the tools and motivation needed to continue on this path.There are too many, unfortunately, to list here all by name.However, I must thankfully acknowledge the exceptional time, support, and positive encouragement provided by my undergraduate research