Are Leptons, Quarks or Both Highly Relativistic Bound States of a Minimally Interacting Fermion and Scalar?

The possibility that leptons, quarks or both might be highly relativistic bound states of a spin-0 and spin-1/2 constituent bound by minimal electrodynamics is discussed in the context of the two-body, bound-state Bethe?CSalpeter equation. For most interactions, strongly bound solutions exist only when the coupling constant is on the order of or greater than unity. However, for the bound-state system discussed here, in the strong-binding limit there exist two classes of boundary conditions that could yield solutions with coupling constants on the order of the fine structure constant. In both classes only bound states with spin 1/2 can exist, thus providing a possible explanation for the absence of higher spin leptons and quarks. Also, a mechanism for the suppression of the decay??? ?? e?+??? exists.     

Relativistic bound-state equations for fermions with instantaneous interactions

In thispaper three types of relativistic bound-state equations for a fermion pair with instantaneous interaction are studied, viz., the instantaneous Bethe-Salpeter equation, the quasi-potential equation, and the two-particle Dirac equation. General forms for the equations describing bound states with arbitrary spin, parity, and charge parity are derived. For the special case of spinless states bound by interactions with a Coulomb-type potential the properties of the ground-state solutions of the three equations are investigated both analytically and numerically. The coupling-constant spectrum turns out to depend strongly on the spinor structure of the fermion interaction. If the latter is chosen such that the nonrelativistic limits of the equations coincide, an analogous spectrum is found for the instantaneous Bethe-Salpeter and the quasi-potential equations, whereas the two-particle Dirac equation yields qualitatively different results.     

Bound-State Solutions of the Klein-Gordon Equation for the Generalized <Emphasis Type="Italic">PT</Emphasis>-Symmetric Hulthén Potential

The one-dimensional Klein-Gordon equation is solved for the PT-symmetric generalized Hulthén potential in the scalar coupling scheme. The relativistic bound-state energy spectrum and the corresponding wave functions are obtained by using the Nikiforov-Uvarov method which is based on solving the second-order linear differential equations by reduction to a generalized equation of hypergeometric type. PACS numbers: 03.65.Fd, 03.65.Ge     

Spinor Bethe-Salpeter equation with generalized ladder kernels and the Goldstein problem

The spinor Bethe-Salpeter equation is investigated for tightly bound fermion pairs. The covariant interaction kernel contains contributions of vector and axial-vector gluons within the framework of the Stückelberg formalism. The free gluon propagators of the strict ladder approximation are replaced by a convenient spectral form. This generalized ladder model can be extended to a large class of gauge field theories by specifying the spectral functions. The model-independent O(4) analysis of the Wick-rotated wave functions is carried out by using a complete set of four-dimensional scalar, vector, and tensor spherical harmonics. At vanishing center-of-mass energy, the radial Bethe-Salpeter equations can be classified in six disconnected sectors. All these equations are recorded in a general form which provides a study of the gauge dependence of the wave functions at short distances. Illustrative calculations are based on a simple Abelian field theory. In two Goldstein equations the leading singular term of the kernel may be absent by cancellation. In addition, one obtains a generalized Goldstein equation in which the kernel includes a gauge-independent marginally singular term. It is discussed how corrections of the large-distance behavior of the singular Goldstein kernel can lead to normalizable bound-state solutions without introducing a short-distance cutoff. Exact and numerical solutions are presented by using a simple parametrization of the kernel. In other sectors, the noncanonical angular behavior of the solutions may be avoided by prescribing a complex mass for the Stückelberg ghosts.     

Energy levels of a scalar particle in a static gravitational field close to the black hole limit

The bound-state energy levels of a scalar particle in the gravitational field of finite-sized objects with interiors described by the Florides and Schwarzschild metrics are found. For these metrics, bound states with zero energy (where the binding energy is equal to the rest mass of the scalar particle) only exist when a singularity occurs in the metric. Therefore, in contrast to the Coulomb case, no pairs are produced in the non-singular static metric. For the Florides metric the singularity occurs in the black hole limit, while for the Schwarzschild interior metric it corresponds to infinite pressure at the center. Moreover, the energy spectrum is shown to become quasi-continuous as the metric becomes singular.     

Bound states in gauge theories as the Poincaré group representations

The bound-state generating functional is constructed in gauge theories. This construction is based on the Dirac Hamiltonian approach to gauge theories, the Poincaré group classification of fields and their nonlocal bound states, and the Markov-Yukawa constraint of irreducibility. The generating functional contains additional anomalous creations of pseudoscalar bound states: para-positronium in QED and mesons inQCDin the two-gamma processes of the type of γ + γ → π ₀ +para-positronium. The functional allows us to establish physically clear and transparent relations between the perturbativeQCD to its nonperturbative low-energy model by means of normal ordering and the quark and gluon condensates. In the limit of small current quark masses, the Gell-Mann-Oakes-Renner relation is derived from the Schwinger-Dyson and Bethe-Salpeter equations. The constituent quark masses can be calculated from a self-consistent nonlinear equation.     

Numerical Method for Solving Two-Body Bound-State Bethe-Salpeter Equations for Unequal Constituent Masses

 A theoretical technique is developed for obtaining finite-energy numerical solutions to a class of two-body, bound-state Bethe-Salpeter equations in the ladder approximation when the constituent masses are unequal. The class of equations is restricted to those for which the Bethe-Salpeter equation can be written as a differential equation and to situations where the coupling constant is real. Such equations can result when the binding force is created by the exchange of a massless quanta. The theoretical technique is tested numerically by obtaining finite-energy solutions of the partially-separated Bethe-Salpeter equation describing the unequal-mass Wick-Cutkosky model in the ladder approximation. Received February 19, 1997; Revised April 2, 1998; accepted for publication October 30, 1998     

Singularity-Free Two-Body Equation with Confining Interactions in Momentum Space

We are developing a covariant model for all mesons that can be described as quark-antiquark bound states in the framework of the Covariant Spectator Theory (CST) in Minkowski space. The kernel of the bound-state equation contains a relativistic generalization of a linear confining potential which is singular in momentum space and makes its numerical solution more difficult. The same type of singularity is present in the momentum-space Schrödinger equation, which is obtained in the nonrelativistic limit. We present an alternative, singularity-free form of the momentum-space Schrödinger equation which is much easier to solve numerically and which yields accurate and stable results. The same method will be applied to the numerical solution of the CST bound-state equations.     

Exact solution to the one-dimensional Dirac equation of linear potential

In this paper the one-dimensional Dirac equation with linear potential has been solved by the method of canonical transformation. The bound-state wavefunctions and the corresponding energy spectrum have been obtained for all bound states.     

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 Importance of Boundary Conditions in Solving the Bound-State Bethe-Salpeter Equation

The characteristic of bound-state, Bethe-Salpeter equations that makes them so difficult to solve numerically can be overcome, in some if not many cases, by expanding solutions in terms of basis functions that obey the boundary conditions that are satisfied by the solutions. The utility of such basis functions is demonstrated by calculating the zero-energy, bound-state solutions of a spin-0 boson and a spin-&frac; fermion with unequal masses. The constituents interact via scalar electrodynamics and are described by the Bethe-Salpeter equation in the ladder approximation. Although the Bethe-Salpeter equation that is solved is separable in the zero-energy limit, the feature that typically prevents solutions from being obtained numerically is still present. A technique for calculating boundary conditions, which is readily generalized to other Bethe-Salpeter equations, is discussed in detail.Supported by a grant from the Ohio Supercomputer CenterReceived January 31, 2003; accepted April 4, 2003 Published online August 25, 2003     

Meson-hyperon couplings in the bound-state approach to the Skyrme model

Kaon and pion coupling constants to hyperons are calculated in the bound-state approach to strangeness in the Skyrme-soliton model. The pion and kaon coupling constants are properly defined as matrix elements of source terms of the mesons sandwiched between two single-baryon states. Numerical calculation of the coupling constants shows that the bound-state approach well reproduces the empirical values.     

Fermion masses in a strong Yukawa coupling model

For strong enough Yukawa coupling the electroweak standard model fermion finds it energetically advantageous to transform itself into a bound state in the hedgehog background of the Higgs field in the semiclassical approximation. By considering that the bound states give the masses for the lepton and quark, it is found that all fermion masses can be described by the strongly Yukawa coupling constants which tend to a unitary constant.     

Quantum binding of the BCS interaction model

The Schrödinger (as opposed to the Cooper or BCS-gap) equation is solved without approximation in momentum space for the BCS interaction model to obtain the quantum bound-state spectrum of an isolated pair of fermions in one, two, and three dimensions. Regardless of dimensionality, there is never more than a single bound state (in analogy with the nucleon-nucleon interaction), but a threshold value of the potential strength is needed to support this state in any dimension. For very low densities one recovers previously known formulas for two and three dimensions which are consistent in this limit with the more familiar properties of quantum binding for simple, purely attractive wells. Results are illustrated for typical conventional, cuprate, and superconducting semiconductors having controversially low carrier densities.     

Physical particles of the massive Schwinger model

The massive Schwinger model is considered in the infinite momentum frame. By assuming its physical particles consist of two fermion bound states, we compute a spectrum. For fermions with large bare masses, the method is reliable. For low-mass fermions, we find we must include states of higher fermion number to adequately describe excited states of the fundamental boson of the theory. We do this for the scalar state in the limit of small bare fermion mass. This representation of the theory provides a unified treatment of both the weak and strong coupling limits, remaining in the fermion representation throughout. We have checked our numerical results with exact calculations wherever possible, and find good agreement.     

Nonlocal anomaly of the axial-vector current for bound states

We argue that the amplitude does not vanish in the limit of zero quark masses. This represents a new kind of violation of the classical equation of motion for the axial current and should be interpreted as the axial anomaly for bound states. The anomaly emerges in spite of the fact that the one loop integrals are ultraviolet finite as guaranteed by the finite size of bound-state wave functions. As a result, the amplitude behaves like approximately 1/p(2) in the limit of a large momentum p of the current. The observed effect requires the modification of the classical equation of motion of the axial-vector current by nonlocal operators.     

Mass-Imbalanced Three-Body Systems in 2D: Bound States and the Analytical Approach to the Adiabatic Potential

Three-body systems in two dimensions with zero-range interactions are considered for general masses and interaction strengths. The problem is formulated in momentum space and the numerical solution of the Schrödinger equation is used to study universal properties of such systems with respect to the bound-state energies. The number of universal bound states is represented in a form of boundaries in a mass-mass diagram. The number of bound states is strongly mass dependent and increases as one particle becomes much lighter than the other ones. This behavior is understood through an accurate analytical approximation to the adiabatic potential for one light particle and two heavy ones.     

On the convergence of reflectionless approximations to confining potentials

The construction of reflectionless potentials supporting a prescribed spectrum of Schrödinger bound states is discussed and related to the inverse problem for confining potentials. A simple formula is derived for the Jost solution in a one-dimensional reflection-less potential with N bound states. This leads to compact expressions for the potential and the bound-state wavefunctions in terms of the bound-state energies. For symmetric potentials, N-fold product formulas are obtained for bound-state wavefunctions and their slopes at the origin. Corresponding quantities in a confining potential are given by infinite products. Comparison of the finite-product and infinite-product expressions allows a demonstration of the convergence of the reflectionless results to the confining potential results as N → ∞. Several sum rules satisfied by the reflectionless potential at the origin are applied to numerical studies of convergence.