Derivation of a simple coordination number model using a given equation of state and the effective pair potential function

In this work, a simple coordination number (C.N.) model for the (9, 3) Lennard–Jones (LJ) fluid is obtained. It is based on the comparison of the internal pressure derived from a given equation of state (EoS) with the internal pressure derived from the (9, 3) LJ fluid as an effective pair potential (EPP). This model reproduces well the thermodynamic properties of the fluid such as internal energy, and the C.N. which is comparable with the Monte Carlo simulation data for the C.N. in the high-density region. In addition, the obtained C.N. can predict the first shell radial distribution function, g(r), of the fluid as well.     

Derivation of an optical potential for deuterons

A general interpolation scheme is described which allows to determine the different eigenvaluesE _n(κ) for a given value of κ by solving an eigenvalue problem of small rank. The elements of the corresponding matrix, not yet restricted by symmetry requirements, may be determined from calculated energy valuesE _n (κ) along the directions of high symmetry. In addition for different bands connected with one another a new set of Wannier functions is defined.     

Renormalization of the post-Gaussian effective potential

In order to study the effects of renormalization on the variationally calculated φ⁴ effective potential, we employ the Gaussian-effective-potential formalism, nonlinear canonical transformations, and the loop approximation. A quantitative comparison of physically equivalent potentials is carried out in two dimensions. The renormalization procedure in three dimensions, leading after the nonlinear transformation to a manifestly finite energy expectation, is described. Different from finite-dimensional quantum mechanics, the optimization is meaningful only if all divergent sub-graphs generated by the ansatz are identified and renormalized by the bare parameters.     

Derivation of the effective pion-nucleon Lagrangian within heavy baryon chiral perturbation theory

We develop a method for constructing the heavy baryon chiral perturbation theory (HBChPT) Lagrangian, to a given chiral order, within HBChPT. We work within SU(2) theory, with only the pion field interacting with the nucleon. The main difficulties, which are solved, are to develop techniques for implementing charge conjugation invariance, and for taking the nucleon on shell, both within the non-relativistic formalism. We obtain complete lists of independent terms in L_HBChPT through O(q³) for off-shell nucleons. Then, eliminating equation-of-motion (e.o.m.) terms at the relativistic and non-relativistic level (both within HBChPT), we obtain L_HBChPT for on-shell nucleons, through O(q³). The extension of the method (to obtain on-shell L_HBChPT within HBChPT) to higher orders is also discussed.     

Derivation of an effective Lagrangian from a generalized scalar curvature

A spinor Lagrangian invariant under global coordinate, local Lorentz and local chiral SU(n) × SU(n) gauge transformations is presented. The invariance requirement necessitates the introduction of boson fields, and a theory for these fields is then developed by relating them to generalizations of the vector connections in general relativity and utilizing an expanded scalar curvature as a boson Lagrangian. In implementing this plan, the local Lorentz group is found to greatly facilitate the correlation of the boson fields occurring in the spinor Lagrangian with the generalized vector connections.The independent boson fields of the theory are assumed to be the inhomogeneously transforming irreducible parts of the connections. It turns out that no homogeneously transforming parts are necessary to reproduce the chiral Lagrangian usually used as a basis for phenomenological field theories. The Lagrangian in question appears when the gravitational interaction is turned off. It includes pseudoscalar, spinor, vector, and axial vector fields, and the vector fields carry mass in spite of the fact that the theory is locally gauge invariant.     

Dysfunctional methods and the effective potential

The effective potential is a useful and much-studied object. It is known to be both real and convex, but a perturbative calculation often gives a complex and nonconvex result. In this letter we address the apparent conflict between perturbation theory and the convexity of the effective potential.     

Gauge independence of the effective potential revisited

We apply the formalism of extended BRS symmetry to the investigation of the gauge dependence of the effective potential in a spontaneously symmetry broken gauge theory. This formalism, which includes a set of Grassmann parameters defined as the BRS variations of the gauge-fixing parameters, allows us to derive in a quick and unambiguous way the related Nielsen identities, which express the physical gauge independence, in a class of generalized 't Hooft gauges, of the effective potential. We show in particular that the validity of the Nielsen identities does not require any constraint on the gauge-fixing parameters, contrary to some claims found in the literature. We use the method of algebraic renormalization, which leads to results independent of the particular renormalization scheme used.     

Perturbative improvement of the Gaussian effective potential

We improve the Gaussian effective potential by a perturbative expansion up to second order around the mean field solution for the case of the φ⁴-theory in 0+1 and 1+1 dimensions. In 1+1 dimensions, we obtain a second order phase transition in agreement with exact statements.     

Low-temperature effective potential of the Ising model

We study the low-temperature effective potential of the Ising model. We evaluate the three-point and four-point zero-momentum renormalized coupling constants that parametrize the expansion of the effective potential near the coexistence curve. These results are obtained by a constrained analysis of the ε-expansion that uses accurate estimates for the two-dimensional Ising model.     

Variational calculation of the sine-Gordon effective potential

The Gaussian effective potential of the sine-Gordon model is calculated in 1+1 and 2+1 dimensions. Issues like renormalization, vacuum energy and stability of the vacuum are discussed in detail.     

Derivation of collective potential and mass from the generator coordinate method

The Hill-Wheeler equation of the generator coordinate method is approximated by a local collective Schrödinger equation. General expressions for the potential and the mass parameter are obtained by a symmetrized moment expansion. The validity of the approximation is tested for several examples where the exact solution is known. These include the Gaussian overlap with harmonic and anharmonic interaction, the Lipkin model, and monopole resonances of spherical light nuclei. In all cases, surprisingly close agreement with the exact solution is found. Other possible applications of the formalism are indicated.     

On the effective potential minimum versus the tree level potential minimum

We analyze the effective potential minimum (true vacuum of the theory) versus the tree level potential minimum in order to decide under which conditions they coincide. A criterion is found. Applications are made to monomial potentials (Coleman-Weinberg type models) and to theN=1 Supergravity Minimal Model.     

On the effective quark potential in baryons

We analyse the splitting of the non-strange memebers of the first excited level [70,1^?]₁ of baryon resonances. The spin-dependent forces (spin-spin, spin-orbit, tensor) are supposed to arise from the Coulomb term due to one-gluon exchange, from the long-range linearly rising part of the potential, and from additional “hard-core” spin-spin terms which may be generated by higher-order graphs contributing to the qq kernel. For the long range part we either assume that it comes from a superposition of a vector and a scalar kernel of the form ?(γ^μ ? γ_μ ? 1 + (1 ? ?)(1 ? 1 ? 1) (+ permutations), or, alternatively, that it arises from a vector exchange with an anomalous moment κ in the quark-gluon vertex. Values of ? ≈ 0 orκ ≈ ?1 turn out to be favoured. The strong coupling constant and the slope of the linear potential come out in the correct order of magnitude. Very large hard-core spin-spin terms are needed. This fact makes the determination of the effective potential from the underlying theory of quantum chromodynamics as well as the phenomenological analysis of the observed spectra rather problematic.