John Dirk Walecka

The Relativistic Nuclear Many Body Problem

These lectures are based on a book “THE RELATIVISTIC NUCLEAR MANY-BODY PROBLEM” written with Brian Serot which will appear as volume 16 of the series Advances in Nuclear Physics edited by J. W. Negele and E. Vogt [R1]. I am distributing copies of the table of contents.

A theory of highly condensed matter

Abstract To discuss properties of cold, condensed stellar objects such as neutron stars, it is necessary to know the stress tensor Tμν, the source in Einsteins field equations, from nuclear matter densities upwards. To overcome some of the difficulties with the conventional many-body approach to this problem, a model relativistic, many-body, quantum field theory composed of a baryon field, a neutral scalar meson field coupled to the scalar density ψ ψ , and a neutral vector meson field coupled to the conserved baryon current i Ψ γλψ is developed. For a uniform system of given baryon density ϱB, the linearized theory obtained by replacing the scalar and vector fields by their expectation pectation values, φ → φ0, Vλ → iδλ4V0 can be solved exactly. The resulting equation of state for nuclear matter exhibits nuclear saturation, and if the two dimensionless coupling constants in this theory are matched to the binding energy and density of nuclear matter, predictions are obtained for all other systems at all densities. In particular, neutron matter is unbound and the equation of state for neutron matter at all densities is presented; it extrapolates smoothly into the relativistic form P = ϵ. Comparison is made with some conventional many-body calculations. The full field theory is developed by expanding the fields about the condensed values φ0, V0, and the unperturbed hamiltonian is shown to correspond to the linearized theory. The energy shift due to these quantum fluctuations in the fields is related to the baryon Greens function. V0 is related directly to ϱB; φ0, however, must be determined through a self-consistency relation involving the baryon Greens function. The Feynman rules for this theory are developed. Expressions for the lowest-order contributions of the quantum fluctuations to the energy shift and φ0 are derived. It is shown that the terms qμqν in the vector-meson propagator do not contribute to these expressions, and a prescription involving assumptions on the limiting form of the theory as ϱB → 0 is presented which ensures that these lowest-order quantum fluctuations will yield finite results.

Recent progress in quantum hadrodynamics

Quantum hadrodynamics (QHD) is a framework for describing the nuclear many-body problem as a relativistic system of baryons and mesons. Motivation is given for the utility of such an approach and for the importance of basing it on a local, Lorentz-invariant lagrangian density. Calculations of nuclear matter and finite nuclei in both renormalizable and nonrenormalizable, effective QHD models are discussed. Connections are made between the effective and renormalizable models, as well as between relativistic mean-field theory and more sophisticated treatments. Recent work in QHD involving nuclear structure, electroweak interactions in nuclei, relativistic transport theory, nuclear matter under extreme conditions, and the evaluation of loop diagrams is reviewed.

PROPERTIES OF NUCLEAR MATTER

Abstract An account is given of recent theories by Brueckner, Eden, Swiatecki, Bethe, and co-workers about the properties of nuclear matter, which is identical in content, but different in its method of presentation. This presentation tries to clarify the reasons for the applicability of the independent-particle model and to determine what particular features of the nuclear forces are responsible for the validity of this model.

ELECTRODYNAMIC PROCESSES WITH NUCLEAR TARGETS

Abstract It is known that two general form factors depending on energy loss and momentum transfer characterize inelastic electron scattering from nuclei in the first Born approximation in α = 1 137 . The same two form factors appear in all electrodynamic processes connected by one photon exchange with nuclei. This observation is used to compute cross sections and to discuss experiments which are aimed at probing electrodynamics by scattering or pair producing electrons or muons from nuclear targets.

On the theory of the optical potential

Abstract We consider elastic scattering of a projectile by a complex many-body system. The incident energy is assumed high enough so that closure may be used on the target, and the problem is reduced to computing the target ground-state expectation value of the scattering amplitude from A fixed scattering centers. We solve for this A -body scattering amplitude exactly, using a model of a sum of separable potentials (the approach is extended to a more general class of potentials in an appendix). The exact amplitude exhibits many interesting properties. We then make a multiple scattering expansion of this amplitude, and simultaneously, an expansion of the ground-state density in terms of single-particle densities, two-particle densities, and so on. By comparing the resulting multiple scattering amplitude term by term with the scattering series from a one-particle Schrodinger equation for the projectile, we can identify an equivalent, or optical potential. We identify a lowest-order optical potential, stating precisely under what conditions it will generate the correct scattering amplitude. By summing additional classes of multiple scattering graphs, we include modifications due to two-particle correlations and multiply-struck target particles (“local field” corrections). The limiting cases of S -wave scattering and very high energy projectiles are discussed in detail. Some calculations of the high-energy optical potential based on more realistic elementary-particle scattering amplitudes are presented. The effects of spin and isospin are discussed, and an appendix is included on center-of-mass motion.

A calculation of the level spectrum of O18 from the free two-nucleon potential

Abstract The two-neutron binding energy and excitation spectrum of O 18 are computed using the free nucleon-nucleon potential of Brueckner-Gammel-Thaler (which contains a hard core). The problem is discussed within the framework of the Bethe-Goldstone theory using harmonic oscillator wave functions as the unperturbed solutions. The interaction is diagonalized among the degenerate states and matrix elements are computed by a transformation to relative and center-of-mass coordinates. The energies in relative s -states are obtained by a numerical integration of the relative s -wave Schrodinger equation and then corrected for the presence of the filled levels. The ordering of the first five states of O 18 is given correctly and the binding energy and level spacings are quite close to the experimental values.

ELECTROPRODUCTION OF NUCLEON RESONANCES

Abstract We compute the differential cross section for the process e + p → e + pr where pR is a nucleon resonance characterized by parity πR, spin J, and mass MR. The two inelastic form factors describing this cross section are expressed in terms of three amplitudes characterizing the (p, pR) electromagnetic vertex. The kinematic and analytic structure of these three amplitudes as a function of q2 are discussed. The case of the 33 resonance is discussed in some detail.

Properties of Finite Nuclei in a Relativistic Quantum Field Theory

Abstract A renormalizable relativistic quantum field theory is used to study finite nuclei in the mean-field Thomas-Fermi approximation. Mass formula parameters are calculated and proton densities illustrated for 40 Ca and 208 Pb. The Dirac equation is solved to determine single-particle spectra.

LEVEL SPECTRA OF A = 6 NUCLEI FROM THE FREE NUCLEON-NUCLEON INTERACTION

Abstract The calculational procedure recently discussed by Dawson, Talmi, and Walecka for calculating level spectra of double-magic-plus-two-nucleon nuclei from the free nucleon-nucleon interaction is applied to the A = 6 system. A Bethe-Goldstone equation is written for the two external nucleons and by working within an harmonic oscillator framework, the equation can be transformed to relative coordinates and solved. The free nonsingular nucleon-nucleon interaction (independent of the form used) is found to give a good fit to all the known levels, and the more sophisticated Brueckner-Gammel-Thaler and Hamada potentials are found to give essentially the same results, after careful examination of the effect of the strong tensor coupling in the T = 0 states. This, in a sense, justifies previous calculations of spectra of light nuclei with nonsingular interactions. Somewhat too much absolute binding energy is found, presumably because of our use of harmonic oscillator single-particle potentials. Quite good agreement is found for the magnetic moment of Li6 and the fτ value for the ground state β decay He6 → Li6 (both within