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Physical Review C

Auxiliary potential dependence of the Brueckner-Bethe-Goldstone theory

R. Sartor

Published Dec 1, 1984• 3 citations
Publisher

Abstract

It is observed that unlike the versions which rely on auxiliary potentials designed to cancel diagrams exactly, the continuous version of the Brueckner-Bethe-Goldstone theory can be used to calculate the properties and in particular the ground state energy of repulsive many body systems. The only proviso amounts to a trivial requirement on the effective mass at the Fermi momentum. These results are related to those concerning the predictions of the corresponding versions of Green's function theory for the analytical properties of the mass operator. The case of attractive many body systems is also considered.

DOI
10.1103/physrevc.30.2036

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It is observed that unlike the versions which rely on auxiliary potentials designed to cancel diagrams exactly, the continuous version of the Brueckner-Bethe-Goldstone theory can be used to calculate the properties and in particular the ground state energy of repulsive many body systems. The only proviso amounts to a trivial requirement on the effective mass at the Fermi momentum. These results are related to those concerning the predictions of the corresponding versions of Green's function theory for the analytical properties of the mass operator. The case of attractive many body systems is also considered.
For a nonlocal, separable interaction which describes low-energy nucleon-nucleon scattering, the scattering-theory solution to the Brueckner-Bethe-Goldstone (BBG) equation is obtained analytically. The associated scattering amplitudes are evaluated for all momenta of the interacting pair. The off-energy-shell amplitudes are defined by analytic continuation to complex values of the relative momentum of the pair. The positions of singularities in the amplitudes are investigated for various values of both the two-body kinematical parameters and the density of the passive BBG Fermi gas. A comparison of the amplitudes calculated with and without incorporating the effects of the exclusion principle indicates that even for repulsive interactions the latter may differ from the former by as much as 50% for Fermi-gas densities as low as 1500\frac{1}{500}5001​ of the observed nuclear density.
It is shown that starting with a correlated wave function for nuclear matter of a general form one may make direct correspondence to the conventional diagrammatic theory of Brueckner, Bethe, and Goldstone. In particular, in this work we show that the expression for the three-body-cluster energy is the same as that obtained by Bethe. Also, it is shown that the current approximation which neglects the potential for particle states in the diagrammatic approach corresponds to a neglect of a coupling between the two-body (Bethe-Goldstone) equation and the equation for the three-body-cluster wave function (Bethe-Faddeev). The theory presented here provides a clarification of the relation of the nnn-body-cluster wave functions and the structure of the wave function of the entire system.
We present calculations of the energy per particle of pure neutron and symmetric nuclear matter with simplified Argonne nucleon-nucleon potentials for different many-body theories. We compare critically the Brueckner-Hartree-Fock results to other formalisms, such as the Brueckner-Bethe-Goldstone expansion up to third order, self-consistent Green's functions, auxiliary field diffusion Monte Carlo, and Fermi hypernetted chain. We evaluate the importance of spin-orbit and tensor correlations in the equation of state and find these to be important in a wide range of densities.
This paper contains a critique of the Brueckner-Goldstone perturbation series for the ground-state energy of an interacting gas of fermions. We have calculated this energy by first constructing the grand partition function at finite temperature, and then carefully taking the limit as $T\ensuremath{\rightarrow}0$. In general this leads to a series which differs from that of Brueckner and Goldstone. An exception is the case where both the unperturbed single-particle energy as well as the interaction potential have spherical symmetry. Reasons for the breakdown of the Brueckner-Goldstone formalism are briefly discussed.
In order to investigate the validity of the Brueckner and the simple sum of the ladder diagrams approximations to the energy of an infinite system of fermions, we have calculated as a function of density all the terms in the perturbation expansion through the fourth order in the strength of the potential. We have done this for a repulsive, square-well, two-body potential spin-\textonehalf{} fermions. We are able to construct rigorous upper and lower bounds from the coefficients for the value of the ladder approximation and find that the standard solution procedures give reasonably accurate results (within a few percent). We find that the error in the solution as obtained in practice to the equations of the Brueckner approximation is large compared to the size of its departure from the ladder approximation. We further find for low-to-moderate densities and for low-to-moderate potential strengths that the Brueckner approximation both as a sum of a certain class of diagrams and as computed in practice lies above the ladder approximation while the complete perturbation theory lies below it. This result arises from the neglect of the ring diagrams by the Brueckner approximation.
A nonperturbative derivation of the Brueckner approximation to the ground-state energy of nuclear matter (with standard choice of hole and particle energies) is given within the framework of the method of correlated basis functions. The structure of the factor-cluster (FIY) expansion of the expectation value of the Hamiltonian with respect to a trial function incorporating short-range correlations in a general fashion, is analyzed for a uniform extended medium of fermions. With a special choice of correlated wave function, the Brueckner approximation is extracted from the FIY expansion by selective summation of two-body combination terms to all orders in the smallness parameter $\ensuremath{\kappa}$ associated, in this case, with both Brueckner-Bethe-Goldstone and FIY energy expansions. A reexamination of the numerical comparison of conventional Brueckner and Jastrow methods carried out earlier for two simple models of nuclear matter leads to the conclusion that it is inadvisable, at least in a Jastrow calculation satisfying the average Pauli condition, to incorporate the $O(\ensuremath{\kappa})$ contribution of the analog of the dispersion correction of reaction-matrix theory without at the same time including the $O(\ensuremath{\kappa})$ contribution of the analog of the Bethe-Faddeev term. For the uniform extended medium for the Jastrow choice of wave function, there is a formal cancellation of these two three-body contributions, the scale of this cancellation being governed by the size of $\ensuremath{\kappa}$. In the numerical example considered, $\ensuremath{\kappa}$ is about 0.22 and the cancellation goes something like +14 MeV-15 MeV=-1 MeV. These findings, when juxtaposed with the standard Brueckner results for the simple models in question, suggest that a departure from the standard prescription of hole and particle energies in reaction-matrix theory may be necessary when $\ensuremath{\kappa}$ is sizeable-as in liquid 3He^{3}\mathrm{He}3He and the high-density neutron gas, as well as equilibrium nuclear matter described with some potentials.