Bethe-Salpeter Equation For Bound States Research Articles

Solving the three-body bound-state Bethe-Salpeter equation in Minkowski space

The scalar three-body Bethe-Salpeter equation, with zero-range interaction, is solved in Minkowski space by direct integration of the four-dimensional integral equation. The singularities appearing in the propagators are treated properly by standard analytical and numerical methods, without relying on any ansatz or assumption. The results for the binding energies and transverse amplitudes are compared with the results computed in Euclidean space. A fair agreement between the calculations is found.

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–Salpeter 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 μ → e + γ exists.

Zero-Energy Solutions of the Bethe-Salpeter Equation for a Spinor-Scalar System Exchanging Photons

Zero-energy, bound-state solutions are obtained in the ladder approximation to the Bethe-Salpeter equation describing a spinor and a scalar with arbitrary masses that interact via quantum electrodynamics. The coupling constant is found to be almost independent of the masses of the two constituents whereas the solutions themselves depend significantly on the constituent masses. Even though the Bethe-Salpeter equation is separable in the zeroenergy limit, the feature that makes obtaining numerical solutions difficult is still present: it does not appear to be possible to discretize the Bethe-Salpeter equation in such a way that the coupling constant, which is real in the Lagrangian, is also always real when calculated as an eigenvalue of the discretized Bethe-Salpeter equation. Nevertheless, solutions with real coupling constants are obtained using a systematic, numerical method that yields both zero- and finite-energy solutions to many, if not all, two-body, bound-state Bethe-Salpeter equations. As part of the process of obtaining solutions, boundary conditions satisfied by the solutions are determined analytically. At large momenta such boundary conditions are often determined only within ranges. Because eigenvalues are sensitive to the behavior of solutions at large momenta, an iterative, numerical procedure is used to ensure that the input boundary conditions and the boundary conditions actually obeyed by each numerical solution agree. Among the solutions that are found, those that are the zero-energy limit of normalizable, finite-energy solutions are identified.

General Relativistic Bound States of a Fermion and a Scalar Interacting via a Massive Scalar

A complete set of numerical solutions is obtained, in the ladder approximation, to the Bethe-Salpeter equation describing bound states of a spin-\(\frac{1}{2}\) fermion and spin-0 boson with arbitrary masses that interact via the exchange of a massive, spin-0 boson. The equation has been used previously, without solutions actually being calculated, to derive some properties of nucleons by treating the physical nucleon as a bound state of a “bare” nucleon and a “bare” meson. It is likely that most, if not all, two-body, bound-state Bethe-Salpeter equations can be solved in the ladder approximation using the method discussed here.

General, Relativistic, Bound-State Solutions of a Spin-0 and Spin-1/2 Constituent Interacting via Scalar Electrodynamics

General, bound-state solutions are obtained to the Bethe-Salpeter equation describing a spin-0 boson and a spin-1/2 fermion with arbitrary masses that interact via scalar electrodynamics. When the energy is non-vanishing, the equation is solved in the ladder approximation using a systematic, numerical method that yields solutions to many, if not all such, two-body, bound-state Bethe-Salpeter equations. Solutions to the two-body, bound-state Bethe-Salpeter equation 1) are usually obtained (in the ladder approximation) either when the energy is zero, implying separability of the equation, or when the energy is nonzero and the masses of the two constituent particles are equal. When the energy is non-vanishing and the masses of the constituents are arbitrary, to the author’s knowledge solutions have previously been obtained only to two equations: the Wick-Cutkosky Model 2),3) and the ScalarScalar Model. 4) Here solutions are obtained to the Bethe-Salpeter equation describing a spin-0 boson and a spin-1/2 fermion with arbitrary masses that are bound by scalar electrodynamics. This complete set of numerical solutions is apparently only the third such set calculated and the first when one of the constituents has nonzero spin. Although this Bethe-Salpeter equation does not appear to be completely separable, solutions are nevertheless determined using a systematic, numerical method previously developed by the author that has been employed to calculate solutions to two-body, bound-state Bethe-Salpeter equations in the ladder approximation both when the energy is zero and nonzero. 5)–8) For all of the solutions discussed or referenced here, the Bethe-Salpeter equation is first Wick-rotated 2) to eliminate a singularity in the kernel. Also, since no external fields are present, the Bethe-Salpeter equation is invariant under rotations in threedimensional space, implying that two angular variables can be separated. Thus the two-body, bound-state Bethe-Salpeter equation can be expressed in terms of two variables in Euclidean space. The Wick-Cutkosky Model, 2),3) which is completely separable, 2),3),9)–12) consists of two scalars with arbitrary masses that interact via a massless scalar and are

Bethe-Salpeter Solutions with Complex Coupling Constants

Using a general, numerical method for solving two-body, bound-state Bethe-Salpeter equations recently developed by the author, a complete set of solutions is obtained, in the ladder approximation, to the Bethe-Salpeter equation describing bound states of two massive scalars bound by the exchange of a third, massive scalar. Solutions with either a complex or real coupling constant and either zero or nonzero angular momentum are calculated and explicitly shown to satisfy the Bethe-Salpeter equation. A knowledge of boundary conditions is a prerequisite for calculating solutions, necessitating the development of a general technique for determining boundary conditions when the binding quanta are massive.

The role of boundary conditions in solving finite-energy, two-body, bound-state Bethe–Salpeter equations

The difficulties that typically prevent numerical solutions from being obtained to finite-energy, two-body, bound-state Bethe–Salpeter equations can often be overcome by expanding solutions in terms of basis functions that obey the boundary conditions. The method discussed here for solving the Bethe–Salpeter equation requires only that the equation can be Wick rotated and that the two angular variables associated with rotations in three-dimensional space can be separated, properties that are possessed by many Bethe–Salpeter equations including all two-body, bound-state Bethe–Salpeter equations in the ladder approximation. The efficacy of the method is demonstrated by calculating finite-energy solutions to the partially-separated Bethe–Salpeter equation describing the Wick–Cutkosky model when the constituents do not have equal masses.

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-½ 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.

Solving the two-body, bound-state Bethe–Salpeter equation

By expanding the solution of the the two-body, bound-state Bethe–Salpeter equation in terms of basis functions that obey the boundary conditions, solutions can be obtained to some, if not many, equations that have heretofore proved intractable. The utility of choosing such basis functions is demonstrated by calculating the zero-energy, bound-state solutions of a spin-0 boson and a spin-1/2 fermion with unequal masses that interact via scalar electrodynamics and are described by the Bethe–Salpeter equation in the ladder approximation. The equation is solved by first making a Wick rotation and then projecting four-dimensional Euclidean space onto the surface of a unit, five-dimensional sphere. Solutions are expanded in terms of basis functions, each of which obeys the boundary conditions and can be expressed in terms of hyperspherical harmonics in five-dimensional space. The Bethe–Salpeter equation is discretized by requiring that the coefficient of each hyperspherical harmonic vanish. All integrations are performed analytically, yielding a generalized matrix eigenvalue equation that is solved numerically. Although the Bethe–Salpeter equation is separable in the zero-energy limit, the feature of Bethe–Salpeter equations that often prevents solutions from being obtained numerically is still present in the equation that is solved.

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.

Relativistic oscillator in a Bethe-Salpeter model: Nonexistence of discrete bound states

We investigate the solutions of the covariant Bethe-Salpeter equation for bound states of a scalar quark-antiquark system with an effective confining kernel which in coordinate four-space behaves as ${R}^{2}$. Using a suitable set of new variables we separate the equation into one-dimensional equations and study the eigenvalue problem. We find that the covariant bound-state Bethe-Salpeter equation with the above kernel possesses only a continuous spectrum.

Relativistic nucleon exchange in backwardp−dscattering

The nucleon-exchange contribution to backward-angle $p\ensuremath{-}d$ scattering is evaluated as a Feynman diagram, in which the $d\ensuremath{-}n\ensuremath{-}p$ vertex function is obtained from recent numerical solutions to a bound-state Bethe-Salpeter equation. Comparisons are made to experimental cross section and polarization data, as well as to previous relativistic and nonrelativistic calculations.NUCLEAR REACTIONS Backward-angle $p\ensuremath{-}d$ scattering and polarization. Relativistic and negative-energy effects. Bethe-Salpeter equation.

Dynamical Symmetry Breaking and the Sigma-Meson Mass in Quantum Chromodynamics

The spontaneous breakdown of chiral symmetry is analyzed dynamically via bound-state Bethe-Salpeter equations. While in general spontaneous mass generation is linked to a massless ${0}^{\ensuremath{-}}$ pion and no specific constraint on a massive ${0}^{+}$ meson, for the particular theory of asymptotically free QCD it is shown that a ${0}^{+}$ $\ensuremath{\sigma}$ meson should exist with mass ${m}_{\ensuremath{\sigma}}\ensuremath{\approx}600 \mathrm{to} 700$ MeV.

Approximation methods in the context of the bound-state Bethe-Salpeter equation

Numerical approximation schemes of the Wick-rotated scalar Bethe-Salpeter equation are discussed for general local potentials with special emphasis on mesh-point methods. Convergence properties are obtained by considering the analytic properties of the kernel. To this end, the four-dimensional partial wave equations are formulated in a new representation-independent way. The close relationship of variational and mesh-point methods is demonstrated and the difficulties which arise if singular potentials are introduced are discussed. For marginal singular potentials those difficulties are overcome in a new way by redefining the corresponding two-particle Green's function. Numerical examples for this case are given.

Simple connection between covariant Feynman formalism and time-ordered perturbation theory in the infinite-momentum frame

We introduce a parametrization of loop momenta which allows us to perform one of the Feynman integrations in a very transparent way. This leads to expressions which can easily be related to terms resulting from time-ordered perturbation theory in the infinite-momentum frame. To exemplify our method we consider some simple Feynman integrals. As another example we discuss the covariant expressions of Landshoff, Polkinghorne, and Short for the scaling graph and the electromagnetic form factor. We indicate how to substitute the Sudakov parametrization in their work in order to simplify their discussion and to make comparisons with the work of Gunion, Brodsky, and Blankenbecler more convenient. Finally we derive an elegant form of the bound-state Bethe-Salpeter equation in which one of the Feynman integrations is performed.

Microcausality, Kramers' theorem and the bound-state Bethe-Salpeter equation

The explicit role of microcausality in exact bound-state Bethe-Salpeter schemes is examined. It is found to lead to covariant analyticity properties of Bethe-Salpeter momentum-space amplitudes which, it is suggested, can be used to judge the structural consistency of approximate schemes. It is argued ingeneral that the structure of the exact scheme is retained by those approximations obtained by finite order truncation of graphical series. The additional implications of time-reversal invariance are also considered and a relativistic extension of the configuration-space Kramers theorem is established. Finally, explicit examination of the Wick-Cutkosky model solutions is made: all such solutions, both normal and abnormal, are shown to display the requisite analyticity, microcausality, and time-reversal properties.