Bethe-Salpeter Equation with Self-Energy Graphs Ⅲ-Numerical Solution and the Abnormal State Problem

Parameter imbedding method for nonlinear parameter integral equation developed in paper I is applied to the BS equation with self-energy graphs discussed in paper 11. Numerical algorithm and computer code are presented. An example problem, which can be treated analytically, is run on the computer VAX-8700 in the USTC, China. The numerical results are excellent. The continuous spectrum and the discrete eigenvalues of the model problem are investigated attentively and calculated on nearly the whole closed complex plane by using the numerical code. Several sets of parameters p (the mass of the exchanged bosons) and μ(the bound energy of the bound state) are given. The numerical results exhibit some strong evidences, which make us come to the suggestion that there are no discrete abnormal states in all the cases we considered when the self-energy is included.     

Bound-state,Bethe—Salpeter solutions with conjugate pairs of complex coupling constants

Solutions are obtained to the Bethe-Salpeter equation describing bound states of two massive scalars interacting via the exchange of a third, massive scalar. Covariance of the equation implies that the interaction is retarded, and in part because the energy appears more than once in the equation, a Hamiltonian for the bound state does not exist. Thus in contrast to the Schrodinger equation, the Bethe-Salpeter equation is solved by specifying the energy and solving for the coupling constant as an eigenvalue. Although the Bethe-Salpeter equation is derived from a Lagrangian with real coupling constants, depending on the value of the energy and the masses of the scalars, some values of the coupling constant that satisfy the Bethe-Salpeter equation are complex and always occur in conjugate pairs. The unexpected existence of solutions with real energy and a complex coupling constant raises the possibility that there are also resonance solutions with real values of the coupling constant and complex energy. Supported by a grant from the Ohio Supercomputer Center. Presented at the 3rd International Workshop “Pseudo-Hermitian Hamiltonians in Quantum Physics”, Istanbul, Turkey, June 20–22, 2005.     

On the Ladder Bethe-Salpeter Equation

The Bethe-Salpeter (BS) equation in the ladder approximation is studied within a scalar theory: two scalar fields (constituents) with mass m interacting via an exchange of a scalar field (tieon) with mass . The BS equation is written in the form of an integral equation in the configuration Euclidean x-space with the kernel which for stable bound states M < 2m is a self-adjoint positive operator. The solution of the BS equation is formulated as a variational problem. The nonrelativistic limit of the BS equation is considered. The role of so-called abnormal states is discussed.The analytical form of test functions for which the accuracy of calculations of bound-state masses is better than 1% (the comparison with available numerical calculations is done) is determined. These test functions make it possible to calculate analytically vertex functions describing the interaction of bound states with constituents.As a by-product a simple solution of the Wick-Cutkosky model for the case of massless bound states is demonstrated.     

The Relativistic Scattering States of the Hulthén Potential with an Improved New Approximate Scheme to the Centrifugal Term

The approximately analytical scattering state solutions of the l-wave Klein-Gordon equation with the unequal scalar and vector Hulthén potentials are carried out by an improved new approximate scheme to the centrifugal term. The normalized analytical radial wave functions of the l-wave Klein-Gordon equation with the mixed Hulthén potentials are presented and the corresponding calculation formula of phase shifts is derived. It is well shown that the energy levels of the continuum states reduce to those of the bound states at the poles of the scattering amplitude. Some useful figures are plotted to show the improved accuracy of our results and two special cases for s-wave (l=0) and for l=0 and equal scalar and vector Hulthén potentials are also studied briefly.     

Determination of the ground state energies of the H 2 + , D 2 + and H 2 + molecular ions taking into account relativistic corrections

On the basis of determination of the asymptotic behavior of correlation functions of the corresponding field currents with the corresponding quantum numbers an analytic method for determination of the energy spectrum of three-body Coulomb system is suggested. Our results show that the constituent masses of particles, which we have defined as masses of particles in a bound state, differ from masses of particles in a free-state. The constituent mass to the free state mass relation for the electron is greater than the same mass relation for the proton, deuteron and triton. It was also found that this constituent electron mass has different values in each systems, i.e. in H ₂ ⁺ , D ₂ ⁺ and T ₂ ⁺ hydrogen molecular ions. The contributions of exchange and self-energy diagrams were taken into account in the determination of the energy spectrum of the three-body Coulomb system. Our results show that the self-energy diagram contribution is inversely proportional to the square of the constituent mass of particles. This contribution is sufficient for the electron and is negligible for the proton, deuteron and triton. When defining the energy and the wave function (WF), it is necessary to take into account the contributions of both the exchange and self-energy diagrams.     

Bound State Effects in Quantum Transport Theory

A quantum kinetical equation for the one-particle density operator for inhomogeneous multi component non-ideal gases is derived taking into account retardation effects and external fields. It is shown that the interactions give rise to additional contributions to the quantities in the transport equation. In some terms (e.g., internal energy, pressure, effective external force) the interaction effects are strong in the presence of bound states and yield contributions of the order exp (—E₁₀/kT) where E₁₀ is the ground state energy of bound pairs of particles. Other terms (e.g. friction forces) are rather insensitive with respect to the presence of bound states. An interpretation of these effects in terms of the mass action law is given using the principle of equivalence between bound states and particles.     

Generalised scale-by-scale energy-budget equations and large-eddy simulations of anisotropic scalar turbulence at various Schmidt numbers

A necessary condition for the accurate prediction of turbulent flows using large-eddy simulation (LES) is the correct representation of energy transfer between the different scales of turbulence in the LES. For scalar turbulence, transfer of energy between turbulent length scales is described by a transport equation for the second moment of the scalar increment. For homogeneous isotropic turbulence, the underlying equation is the well-known Yaglom equation. In the present work, we study the turbulent mixing of a passive scalar with an imposed mean gradient by homogeneous isotropic turbulence. Both direct numerical simulations (DNS) and LES are performed for this configuration at various Schmidt numbers, ranging from 0.11 to 5.56. As the assumptions made in the derivation of the Yaglom equation are violated for the case considered here, a generalised Yaglom equation accounting for anisotropic effects, induced by the mean gradient, is derived in this work. This equation can be interpreted as a scale-by-scale energy-budget equation, as it relates at a certain scale r terms representing the production, turbulent transport, diffusive transport and dissipation of scalar energy. The equation is evaluated for the conducted DNS, followed by a discussion of physical effects present at different scales for various Schmidt numbers. For an analysis of the energy transfer in LES, a generalised Yaglom equation for the second moment of the filtered scalar increment is derived. In this equation, new terms appear due to the interaction between resolved and unresolved scales. In an a-priori test, this filtered energy-budget equation is evaluated by means of explicitly filtered DNS data. In addition, LES calculations of the same configuration are performed, and the energy budget as well as the different terms are thereby analysed in an a-posteriori test. It is shown that LES using an eddy viscosity model is able to fulfil the generalised filtered Yaglom equation for the present configuration. Further, the dependence of the terms appearing in the filtered energy-budget equation on varying Schmidt numbers is discussed.     

New mechanism to cross the phantom divide

Recently, type Ia supernova data appear to support a dark energy whose equation of state w crosses −1, which is a much more amazing problem than the acceleration of the universe. We show that it is possible for the equation of state to cross the phantom divide by a scalar field in gravity with an additional inverse power-law term of the Ricci scalar in the Lagrangian. The necessary and sufficient condition for a universe in which the dark energy can cross the phantom divide is obtained. Some analytical solutions with w<−1 or w>−1 are obtained. A minimally coupled scalar with different potentials, including quadratic, cubic, quantic, exponential and logarithmic potentials are investigated via numerical methods, respectively. All these potentials lead to the crossing behavior. We show that it is a robust result which is hardly dependent on the concrete form of the potential of the scalar.     

Makarov势的Dirac方程的束缚态解

给出了Makarov型标量势与矢量势相等条件下的Dirac方程的束缚态解. Dirac方程的角向方程用因子分解方法求解，在得出角向波函数的过程中，自然地得到了属于同一本征值的不同角向波函数间的递推操作. 径向束缚态波函数用合流超几何函数表示，束缚态的能量方程可由径向波函数满足的边界条件得到. 关键词： Makarov势 Dirac方程 束缚态 因子分解方法     

Exact Solutions of Klein-Gordon Equation with Exponential Scalar and Vector Potentials

We obtain the exact analytical solution of the Klein-Gordon equation for the exponential vector and scalar potentials by using the asymptotic iteration method. For the scalar potential greater than the vector potential case, the exact bound state energy eigenvalues and corresponding eigenfunctions are presented. The bound state eigenfunction solutions are obtained in terms of the confluent hypergeometric functions.     

Problems with vector confinement in 4d QCD

It is shown that vector confinement does not support bound state spectrum in the 4d Dirac equation. The same property is confirmed in the heavy–light and light–light QCD systems. This situation is compared with the confinement in the 2d system, which is generated by the gluon exchange. Considering the existing theories of confinement, it is shown that both the field correlator approach and the dual superconductor model ensure the scalar confinement in contrast to the Gribov–Zwanziger model, where the confining Coulomb potential does not support bound states in the Dirac equation.     

Solving the Bethe-Salpeter equation for two fermions in Minkowski space

The method of solving the Bethe-Salpeter equation in Minkowski space, developed previously for spinless particles (V.A. Karmanov, J. Carbonell, Eur. Phys. J. A 27, 1 (2006)), is extended to a system of two fermions. The method is based on the Nakanishi integral representation of the amplitude and on projecting the equation on the light-front plane. The singularities in the projected two-fermion kernel are regularized without modifying the original BS amplitudes. The numerical solutions for the J = 0 bound state with the scalar, pseudoscalar and massless vector exchange kernels are found. The stability of the scalar and positronium states without vertex form factor is discussed. Binding energies are in close agreement with the Euclidean results. Corresponding amplitudes in Minkowski space are obtained.     

具有一维Coulomb型对称势Dirac方程的精确解

在标量势大于矢量势的情况下,一维Dirac方程的束缚态能级是二重简并的.任意两个不同能量本征值的波函数和同一能量本征值的两个波函数都是相互正交的.对于纯标量场,存在零能量束缚态,存在分数电荷 关键词： Coulomb型对称势 Dirac方程 束缚态 分数电荷     

Model Study of Three-Body Forces in the Three-Body Bound State

The Faddeev equation for the three-body bound state with two- and three-body forces is solved directly as three-dimensional integral equation. The numerical feasibility and stability of the algorithm, which does not employ partial wave decomposition is demonstrated. The three-body binding energy and the full wave function are calculated with Malfliet-Tjon-type two-body potentials and scalar two-meson exchange three-body forces. For two- and three- body forces of ranges and strengths typical of nuclear forces the single-particle momentum distribution and the two-body correlation function are similar to the ones found for realistic nuclear forces.     

Relativistic symmetry of position-dependent mass particle in Coulomb field including tensor interaction

The spatially-dependent mass Dirac equation is solved exactly for attractive scalar and repulsive vector Coulomb potentials including a tensor interaction under the spin and pseudospin symmetric limit. Closed forms of the energy eigenvalue equation and wave functions are obtained for arbitrary spin-orbit quantum number κ. Some numerical results are given too. The effect of the tensor interaction on the bound states is presented. It is shown that the tensor interaction removes the degeneracy between two states in the spin doublets. We also investigate the effects of the spatially-dependent mass on the bound states under the conditions of the spin symmetric limit in the absence of tensor interaction.     

Pseudospin symmetry for modified Rosen-Morse potential including a Pekeris-type approximation to the pseudo-centrifugal term

By a novel algebraic method we study the approximate solution to the Dirac equation with scalar and vector second P?schl-Teller potential carrying spin symmetry. The transcendental energy equation and spinor wave functions with arbitrary spin-orbit coupling quantum number k are presented. It is found that there exist only positive-energy bound states in the case of spin symmetry. Also, the energy eigenvalue approaches a constant when the potential parameter a \alpha goes to zero. The equally scalar and vector case is studied briefly.     

On Internucleon Interaction of Composite Nucleon

We discuss the low energy nucleon-nucleon interaction leading to a bound state, namely the deuteron problem. The currently known method of calculating internucleon interactions is the boson exchange potential model, where the Klein-Gordon equation for a virtual pseudoscalar boson with a single point-like nucleon source is solved using the Green function method. This method is known to be inadequate in particular to the internucleon problem leading to a bound state. As an alternative we propose to solve internucleon potential problems, including the bound state, by solving the Klein-Gordon equation in which the interaction term has been introduced in a more invariant way. In the place of the single source term used in the standard method the interaction term is introduced in the covariant derivative form in the spirit of the minimum coupling scheme. It turns out that this method is not only mathematically satisfactory (gauge and Lorentz invariant formalism aspect), but also gives a more physically satisfactory interpretation of the internucleon interaction mechanism. For a deuteron bound state problem can then be solved approximately using the variational calculus. We obtain the analytic expression for the internucleon potential as a function of internucleon distances. The minimum energy value 2,2 MeV, the binding energy of the deuteron, is found to be at equilibrium distance of r_ab = ?_φ = 2 × 10^?13 cm.     

Statistical-Mechanical Entropy of Garfinkle-Horowitz-Strominger Black Hole

Taking WKB approximation to solve the scalar field equation in the Garfinkle-Horowitz-Strominger (GHS) black hole spacetime, we can get the classical momenta. Substituting the classical momenta into state density equation corrected by the generalized uncertainty principle, we will obtain the number of quantum states with energy less than ω. It is convergent in the neighborhood of the horizon. Then, it is used to calculate the statistical-mechanical entropy of the scalar field in the GHS black hole spacetime. The calculation shows that the entropy is proportional to the horizon area.     

Properties of nuclear and neutron matter in a relativistic Hartree-Fock theory

Relativistic Hartree-Fock (HF) equations are derived for an infinite system of mesons and baryons in the framework of a renormalizable relativistic quantum field theory. The derivation is based on a diagrammatic approach and Dyson's equation for the baryon propagator. The result is a set of coupled, nonlinear integral equations for the baryon self-energy with a self-consistency condition on the single-particle spectrum. The HF equations are solved for nuclear and neutron matter in the Walecka model, which contains neutral scalar and vector mesons. After renormalizing model parameters to reproduce nuclear matter saturation properties, HF results at low to moderate densities are similar to those in the mean-field (Hartree) approximation. Self-consistent exchange corrections to the Hartree equation of state become negligible at high densities. Rho- and pi-meson exchanges are incorporated using a renormalizable gauge-theory model. A chiral transformation of the lagrangian is used to replace the pseudoscalar πN coupling with a pseudovector coupling, for which one-pion exchange is a reasonable first approximation. This transformation maintains the model's renormalizability so that corrections may be evaluated. Pion exchange has a small effect on the HF results of the Walecka model and brings HF results in closer agreement with the mean-field theory. The diagrammatic techniques used here retain the mesonic degrees of freedom and are simple enough to be extended to more refined self-consistent approximations.     

Approximately analytical solutions of the Manning-Rosen potential with the spin-orbit coupling term and spin symmetry

In this Letter the approximately analytical bound state solutions of the Dirac equation with the Manning-Rosen potential for arbitrary spin-orbit coupling quantum number k are carried out by taking a properly approximate expansion for the spin-orbit coupling term. In the case of exact spin symmetry, the associated two-component spinor wave functions of the Dirac equation for arbitrary spin-orbit quantum number k are presented and the corresponding bound state energy equation is derived. We study briefly two special cases; the general s-wave problem and the equal scalar and vector Manning-Rosen potential.