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Physical Review C

Stochastic solution of a model meson-nucleon field theory

Brian D. Serot, S. E. Koonin, John W. Negele

Published Oct 1, 1983• 11 citations
Publisher

Abstract

Ground-state properties in a model quantum field theory are calculated by stochastic evaluation of a path integral representation of the many-body propagator. The model consists of nonrelativistic nucleons coupled to vector and scalar mesons in one spatial dimension. Binding energies and density distributions are calculated for bound states of up to twenty nucleons. The binding energy as a function of density, the nucleon-nucleon correlation function, and the meson-meson correlation function are evaluated in nuclear matter. Exact ground-state solutions to the full field theory are shown to differ relatively little from those of the potential theory corresponding to the static limit. The exact solutions differ substantially from those of the mean-field (Hartree) approximation, but are quite similar to those obtained in the Hartree-Fock approximation.NUCLEAR STRUCTURE Meson-nucleon field theory. Monte Carlo solution. Nuclear matter.

DOI
10.1103/physrevc.28.1679

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Ground-state properties in a model quantum field theory are calculated by stochastic evaluation of a path integral representation of the many-body propagator. The model consists of nonrelativistic nucleons coupled to vector and scalar mesons in one spatial dimension. Binding energies and density distributions are calculated for bound states of up to twenty nucleons. The binding energy as a function of density, the nucleon-nucleon correlation function, and the meson-meson correlation function are evaluated in nuclear matter. Exact ground-state solutions to the full field theory are shown to differ relatively little from those of the potential theory corresponding to the static limit. The exact solutions differ substantially from those of the mean-field (Hartree) approximation, but are quite similar to those obtained in the Hartree-Fock approximation.NUCLEAR STRUCTURE Meson-nucleon field theory. Monte Carlo solution. Nuclear matter.
A general method for a microscopic description of nuclear dynamical problems is discussed. The method is based on a functional integral representation for the many-body time evolution operator U(tf,ti)U({t}_{f},{t}_{i})U(tf​,ti​). In the stationary phase limit, a time-dependent mean-field approximation for the matrix elements of UUU is obtained. Using standard procedures this allows for extraction of quantum mechanical information about bound states, tunneling, or scattering phenomena in a many-nucleon system. The approximation represents a natural generalization of the time-dependent Hartree-Fock method.NUCLEAR STRUCTURE, NUCLEAR REACTIONS derived mean-field approximation for matrix elements of $\mathrm{exp}(\ensuremath{-}iHt)$ from exact functional integral; discussed relation to static HF, TDHF, and scattering calculations.
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.
We discuss the description of a many-body nuclear system using Hamiltonians that contain the nucleon relativistic kinetic energy and potentials with relativistic corrections. Through the Foldy-Wouthuysen transformation, the field theoretical problem of interacting nucleons and mesons is mapped to an equivalent one in terms of relativistic potentials, which are then expanded at some order in 1mN\frac{1}{{m}_{N}}mN​1​. The formalism is applied to the Hartree problem in nuclear matter, showing how the results of the relativistic mean field theory can be recovered over a wide range of densities.
In most simulations of nonrelativistic nuclear systems, the wave functions found solving the many-body Schr\odinger equations describe the quantum-mechanical amplitudes of the nucleonic degrees of freedom. In those simulations the pionic contributions are encoded in nuclear potentials and electroweak currents, and they determine the low-momentum behavior. In this work we present a novel quantum Monte Carlo formalism in which both relativistic pions and nonrelativistic nucleons are explicitly included in the quantum mechanical states of the system. We report the renormalization of the nucleon mass as a function of the momentum cutoff, an Euclidean time density correlation function that deals with the short-time nucleon diffusion, and the pion cloud density and momentum distributions. In the two nucleon sector we show that the interaction of two static nucleons at large distances reduces to the one-pion exchange potential, and we fit the low-energy constants of the contact interactions to reproduce the binding energy of the deuteron and two neutrons in finite volumes. We show that the method can be readily applied to light-nuclei.
The method of coherent correlated pairs is applied with a realistic effective Hamiltonian to the four-nucleon system. Three methods of solution are compared for ease of implementation and convergence rates. A Monte Carlo method yields the best upper bound within the coherent correlated pair space. The full variational treatment of the four-nucleon coherent correlated pair problem yields equations which have poor stability properties. An ansatz variational method proves to be rapidly convergent and yields results very close to those of the Monte Carlo method. The results for binding energy and the one body density distribution are compared with Hartree-Fock results using the same effective Hamiltonian in the same model space. As in previous soluble model applications the coherent correlated pair method provides a superior upper bound to the ground state energy.
A method is shown to derive a consistent Hamiltonian of nucleon and meson system in a finite nucleus from a Lorentz-scalar Lagrangian. For this purpose, the Schwinger-Dyson method is applied to a many-body problem of the interacting nucleons and mesons in a finite system. Using this method, closed equations of exact Green's functions for single-particle motions of a nucleon and a meson are derived. It is shown that the interactions of nucleons and mesons generate local dependent potentials in a finite system and that the potentials can be determined self-consistently on the basis of the many-body Schwinger-Dyson formalism. In the method, condensed meson fields are extracted from the meson fields and treated as c-number fields, so that the classical equations of the condensed fields become equivalent to those of the mean-field theory originally proposed by Walecka et al.$---- The present method is regarded as a natural extension of mean-field theory to a full quantum treatment for finite systems. The method provides a self-consistent description of the single-particle motions of nucleons and mesons and their mutual interactions. It allows us to include full quantum effects and to describe not only the ground state but also excited states in a systematic way.