Nuclear field theory

By using path integral techniques nuclear field theory (NFT) is developed for Fermi systems interacting via a general two-body force. The NFT Lagrangian is strictly derived. As a by-product, the corresponding graphical rules are obtained. The relation between the NFT and the conventional Feynman diagrammatic many-body perturbation theory is established for processes connecting initial and final states, too.     

Non local effects induced by the particle-vibration coupling

Dynamical corrections to the single particle potential are (perturbatively) evaluated. The corrections tom ^*/m resulting from the coupling of independent (effective) fermions to collective vibrations (low energy modes) are analyzed within the NFT formalism. The results for the²⁰⁸Pb are discussed.     

Two-pion interferometry for a partially coherent evolution source

We give the formulas of two-pion Hanbury-Brown-Twiss (HBT) correlation function for a partially coherent evolution pion-emitting source,using quantum probability amplitudes in a path-integral formalism.The multiple scattering of the particles in the source is taken into consideration based on Glauber scattering theory.Two-pion interferometry with effects of the multiple scattering and source collective expansion is examined for a partially coherent source of hadronic gas with a finite baryon density and evolving hydrodynamically.We do not find observable effect of either the multiple scattering or the source collective expansion on HBT chaotic parameter.     

Two-pion interferometry for a partially coherent evolution source

We give the formulas of two-pion Hanbury-Brown-Twiss (HBT) correlation function for a partially coherent evolution pion-emitting source, using quantum probability amplitudes in a path-integral formalism. The multiple scattering of the particles in the source is taken into consideration based on Glauber scattering theory. Two-pion interferometry with effects of the multiple scattering and source collective expansion is examined for a partially coherent source of hadronic gas with a finite baryon density and evolving hydrodynamically. We do not find observable effect of either the multiple scattering or the source collective expansion on HBT chaotic parameter.     

Nuclear effective interactions and spin excitations

In view of the one-boson-exchange model for the nucleon-nucleon interaction and the Hartree-Fock (HF) interaction, we formulate the effective interactions for particle-hole states in terms of the exchange of the fields which are confined in the nucleus. This theory, as an extension to the nuclear field theory (NFT), takes into account the propagation of the fields which is neglected in NFT. The effective interactions thus obtained reproduce the energies of a sequence of electric giant resonances and the core polarizabilities associated with the resonances. It is found that the coupling constants of the σ- and ω-fields are suppressed for the particle-hole interaction by 60% with respect to the HF interaction. As for the effective interactions involving nucleon spins, we consider the fields coupled to nucleon spins. The effective interactions obtained, essentially different from those in NFT, have a tensor component. We analyse the energies and cross sections for excitation of stretched spin particle-hole states which are the most sensitive to the tensor force. The effective interaction responsible for the stretched spin states is shown to be consistent with that for the magnetic resonances observed in the (p, n) reactions.     

The boson-fermion hybrid representation formulation,I

A boson-fermion hybrid representation is presented. In this framework, a fermion system is described concurrently by the bosonic and the fermionic degrees of freedom. A fermion pair in this representation can be treated as a boson without violating the Pauli principle. Furthermore the “bosonic interactions” are shown to originate from the exchange processes of the fermions and can be calculated from the original fermion interactions. Both the formulation of the BFH representations for the even and odd nuclear systems are given. We find that the basic equation of the nuclear field theory (NFT) is just the usual Schrödinger equation in such a representation with the empirical NFT diagrammatic rules emerging naturally. This theory was numerically checked in the case of four nucleons moving in a single-j shell and the exactness of the theory was established.     

The renormalization of collective states and the improper initial or final states in NFT

The collective lines in a given diagram are renormalized by including higher order processes. The problem is cast into the form of a conventional linear algebraic matrix equation that allows a simple treatment of the normalization conditions. It is shown that the states entering in the renormalization of the phonons become improper initial or final states, if dressed phonons are used in the intermediate states. A simple extension of this argument allows one to justify one of the rules given in the formulation of the NFT.     

Collective Multipole Expansions and the Perturbation Theory in the Quantum Three-Body Problem

The perturbation theory with respect to the potential energy of three particles is considered. The first-order correction to the continuum wave function of three free particles is derived. It is shown that the use of the collective multipole expansion of the free three-body Green function over the set of Wigner D-functions can reduce the dimensionality of perturbative matrix elements from twelve to six. The explicit expressions for the coefficients of the collective multipole expansion of the free Green function are derived. It is found that the S-wave multipole coefficient depends only upon three variables instead of six as higher multipoles do. The possible applications of the developed theory to the three-body molecular break-up processes are discussed.     

Microscopic analysis of nuclear collective motions in terms of the boson expansion theory: (I). Formulation

A normal-ordered linked-cluster boson expansion theory, previously worked out by one of the authors (T.K.) and Tamura, has been developed further by reformulating it in a “physical” quasiparticle subspace which contains no spurious particle-number excitation modes. The expansion coefficients of the collective hamiltonian for low-lying quadrupole motions are determined starting from a microscopic fermion hamiltonian including self-consistent higher-order (many-body) interactions derived in our previous work. The contributions from the non-collective states with all possible non-collective one-boson excitations having I^π = 0⁺− 4⁺, which can directly couple to the collective states with one or two phonons, are taken into account in a systematic and compact way.     

Statistical mechanics of a classical one-component coulomb gas using collective coordinates

We develop a new expansion for the logarithm of the canonical partition function ln Q for the classical one-component Coulomb gas, using collective coordinates. Our initial use of collective coordinates is similar to that of Iuknovskii 1958, and our expansion resembles that of Abe (1959). Our result for the lowest-order correction to the Debye-Huckel theory is the same as these earlier results, while our next order correction is different. From our expansion for ln Q we obtain an expansion for the grand function $\text{Ω = F ? μN = ?pV}$. The ultimate purpose of this work is to develop a new mathematical technique for obtaining thermodynamic properties of an ionized gas from quantum statistical mechanics.     

Diagrammatic analysis of correlations in polymer fluids: Cluster diagrams via Edwards’ field theory

Edwards’ functional integral approach to the statistical mechanics of polymer liquids is amenable to a diagrammatic analysis in which free energies and correlation functions are expanded as infinite sums of Feynman diagrams. This analysis is shown to lead naturally to a perturbative cluster expansion that is closely related to the Mayer cluster expansion developed for molecular liquids by Chandler and co-workers. Expansion of the functional integral representation of the grand-canonical partition function yields a perturbation theory in which all quantities of interest are expressed as functionals of a monomer-monomer pair potential, as functionals of intramolecular correlation functions of non-interacting molecules, and as functions of molecular activities. In different variants of the theory, the pair potential may be either a bare or a screened potential. A series of topological reductions yields a renormalized diagrammatic expansion in which collective correlation functions are instead expressed diagrammatically as functionals of the true single-molecule correlation functions in the interacting fluid, and as functions of molecular number density. Similar renormalized expansions are also obtained for a collective Ornstein-Zernicke direct correlation function, and for intramolecular correlation functions. A concise discussion is given of the corresponding Mayer cluster expansion, and of the relationship between the Mayer and perturbative cluster expansions for liquids of flexible molecules. The application of the perturbative cluster expansion to coarse-grained models of dense multi-component polymer liquids is discussed, and a justification is given for the use of a loop expansion. As an example, the formalism is used to derive a new expression for the wave-number dependent direct correlation function and recover known expressions for the intramolecular two-point correlation function to first-order in a renormalized loop expansion for coarse-grained models of binary homopolymer blends and diblock copolymer melts.     

Perturbative construction and rigorous proof of locality of effective lattice actions for continuum theories

We construct an effective lattice action for a continuum theory, by fixing a set of collective coordinates which play the role of lattice variables. As opposed to Symanzik's improvement program our method involves no expansion in powers of the lattice spacing; in other words it simultaneously yields all “irrelevant” operators generated by the renormalization group to a given order in the continuum coupling constant. We are thus able to rigorously establish that the effective lattice action, for both smooth and singular collective coordinates, is local in the sense that long-range couplings decay exponentially over a distance independent of the mass gap of the theory; for asymtotically free theories this is interpreted as an existence proof of Wilson's infrared stable trajectories. Our methods are for convenience described in the context of dimensional φ⁴, but can be easily extended to any theory with a set of collective coordinates which (i) are renormalizable and (ii) provide an infrared cutoff. Application to the 2-dimensional O(N) σ-model is, in particular, discussed; the technical problems of renormalization posed by gauge invariance are, on the other hand, not dealt with in this paper, although our treatment of singular coordinates is meant as a prelude to them. A by-product of our proofs is the derivation of an interesting factorization property of Zimmermann-subtracted diagrams.     

Quasiclassical Approach to Collective Nuclear Phenomena

We point out that large-amplitude collective excitations, tunneling phenomena, condensation of higher particle clusters, and excitation spectra beyond Landau's theory of Fermi liquid are most conveniently described in terms of successive effective actions. These are functionals depending explicitly on two-particle, four-particle, etc. correlations which have a simple quasi-classical expansion. Their extrema account for the above described phenomena. Contrary to the path integral approach to collective phenomena, the lowest approximation contains exchange and pairing effects and is therefore suited for systems in which the time-dependent Hartree-Fock-Bogoljubov equations are required for a proper understanding of the phenomena.     

Singlet collective variables for the Yang-Mills field

The Yang-Mills theory is reformulated in terms of singlet collective variables. The corresponding effective action may serve as a starting point for the N^?1 expansion.     

Semiclassical inertia of nuclear collective dynamics

For the low-lying collective excitations in nuclei, the transport coefficients, such as the stiffness, the inertia, and the friction, are derived within the periodic-orbit theory in the lowest orders of semiclassical expansion corresponding to the extended Thomas—Fermi approach. The multipole vibrations near the spherical shape are described in the mean-field approximation through the infinitely deep square-well potential and Strutinsky averaging of the transport coefficients. Owing to the consistency condition, the collective inertia for sufficiently increased particle numbers and temperatures is substantially larger than that of irrotational flow. The average energies of collective vibrations, reduced friction, and effective damping coefficients are in better agreement with experimental data than those found from the hydrodynamic model. The text was submitted by the authors in English.     

General semi-classical method for collective motion and its connection with the Rowe-Basserman and marumori theories,and adiabatic limits

In recent work, we have shown that in the adiabatic limit (large amplitude, small momentum), time-dependent Hartree-Fock theory (TDHF) yields a well-defined theory of large-amplitude collective motion which provides an essentially unique construction for a collective hamiltonian. An alternative theory, put forward by Rowe and Basserman and by Marumori is, apparently, not restricted to small momenta. We describe a general framework for the study of collective motion in the semi-classical limit without limitation on the size of coordinates or momenta, which includes all previous methods as limiting cases. We find it convenient, as in the past, to consider two general systems: first, a system with n degrees of freedom and no special permutation symmetry, and, second, a system of fermions described in TDHF. For both systems the problem can be formulated as a search for a hamiltonian flow confined to a finite-dimensional hypersurface in a phase space, which itself may be finite- or infinite-dimensional. Though, in general, there are no exact solutions to this problem, we can formulate consistent approximation schemes corresponding to both the adiabatic and Rowe-Basserman, and Marumori limits. We also show how to extend the momentum expansion, which underlies the adiabatic approximation, to higher orders in the momentum. We thereby confirm the structure of the theory found in our previous work.     

Spatial anisotropy: a method to study anisotropic flow in heavy ion collisions

We introduce a method to study anisotropic flow parameter v<,n> as a collective probe to Quark Gluon Plasma in relativistic heavy ion collisions. The emphasis is put on the use of the Fourier expansion of initial spatial azimuthal distributions of participant nucleons in the overlapped region. The coefficients ε<,n> of Fourier expansion are called the spatial anisotropy parameter for the n-th harmonic. We propose that collective dynamics can be studied by v<,n>/ε<,n>. In this paper, we will discuss in particular the second (n=2) and the fourth (n=4) harmonics.     

Green's function approach to the quantum many-body theory of the solid state

TheGreen's function approach is used to develop a quantum many-body theory of the solid state which should work at low temperatures as well as in the neighbourhood of phase transition points. The theory is applicable also in those cases where the traditional expansion of the potential in powers of the atomic displacements is entirely inadequate (crystalline helium). The starting point of our approach is the concept of broken symmetry since the invariance of the equilibrium ensemble under the continuous group of infinitesimal translations is reduced in a crystalline solid to the invariance under finite translations through a lattice vector. A homogeneous integral equation is derived which has nontrivial solutions in the crystalline state. By this equation it is shown that the umklapp phonons are the symmetry restoring collective modes expected due to a general theorem ofGoldstone. The single particle excitations and the structure of the Dyson mass operator in the crystalline state are discussed. It is further shown that the homogeneous Bethe-Salpeter equation for the linear response to an external disturbance possesses symmetry breaking solutions which are connected to the lattice dynamics of the solid state. These collective excitations (phonons) are exhibited in RPA and tight-binding approximation for monoatomic cubic crystals with a Bravais lattice in order to demonstrate how the present theory reproduces well-known results.