The Bethe-Salpeter equation with fermions

The Bethe-Salpeter (BS) equation in the ladder approximation is studied within a fermion theory: two fermion fields (constituents) with mass m interacting via an exchange of a scalar field with mass μ. The BS equation can be written in the form of an integral equation in the configuration Euclidean x-space with the symmetric kernel K for which Tr K ² = ∞ due to the singular character of the fermion propagator. This kernel is represented in the form K = K ₀ + K _I . The operator K ₀ with Tr K ₀ ² = ∞ is of the “fall at the center” potential type and describes a continuous spectrum only. Besides the presence of this operator leads to a restriction on the value of the coupling constant. The kernel K _I with Tr K _I ² < ∞ is responsible for bound fermion-fermion states. Our approach is that the eigenvalue problem of the equation $\Lambda\Psi = g^2(K_0 + K_I)\Psi \qquad {\rm with}\qquad \Lambda = 1The Bethe-Salpeter (BS) equation in the ladder approximation is studied within a fermion theory: two fermion fields (constituents) with mass m interacting via an exchange of a scalar field with mass μ. The BS equation can be written in the form of an integral equation in the configuration Euclidean x-space with the symmetric kernel K for which Tr K ² = ∞ due to the singular character of the fermion propagator. This kernel is represented in the form K = K ₀ + K _I . The operator K ₀ with Tr K ₀ ² = ∞ is of the “fall at the center” potential type and describes a continuous spectrum only. Besides the presence of this operator leads to a restriction on the value of the coupling constant. The kernel K _I with Tr K _I ² < ∞ is responsible for bound fermion-fermion states. Our approach is that the eigenvalue problem of the equation can be rewritten in the form The kernel of the last equation is finite for g ² < g _c ² and the variational procedure of calculations of eigenvalues and eigenfunctions can be applied. The quantum pseudoscalar and scalar mesodynamics is considered. The binding energy of the state 1⁺ (deuteron) as a function of the coupling constant is calculated in the framework of the procedure formulated above. It is shown that this bound state is absent in the pseudoscalar mesodynamics and does exist in the scalar mesodynamics. A comparison with the non-relativistic Schr?dinger picture is made. Correspondence: G. V. Efimov, Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia     

Simplified Bethe-Salpeter equation for positronium

The Bethe-Salpeter equation for a fermion-antifermion system, coupled by photons, is considered in the Feynman gauge. The kernel is that resulting from exchange of a single photon. The usual reduction of the sixteen B-S spinor amplitudes in terms of tensors leads to 16 coupled integro-differential equations. By straightforward application of charge conjugation-, parity-, and Lorentz-invariance, the system of coupled equations is reduced to ones involving no more than eight and as few as three scalar structure functions for the various parity, charge conjugation, and total angular momentum states. The results hold for arbitrary coupling strength. As a check of the equations obtained, a perturbation theory is carried out for the Coulomb interaction. It leads to effective potentials in agreement with those obtained previously to order mα⁴ for positronium.     

Solving the Bethe-Salpeter equation for positronium

We propose a Coulomb-like kernel for the relativistic two-fermion Bethe-Salpeter equation, to be used as the lowest-order approximation in systematic perturbative calculation of bound-state energy levels in QED. The kernel is symmetric in the two fermions and for the exchange of in and out momenta. The resulting equation is exactly soluble, unlike previously considered unperturbed kernels. We give explicitly the Green function and eigen-functions. We also discuss the problem of the behaviour of the wave functions at zero relative coordinate in connection with the contribution to energy levels from the one-photon annihilation channel in QED.     

Solving Bethe-Salpeter equation in Minkowski space

We develop a new method of solving the Bethe-Salpeter (BS) equation in Minkowski space. It is based on projecting the BS equation on the light-front (LF) plane and on the Nakanishi integral representation of the BS amplitude. This method is valid for any kernel given by the irreducible Feynman graphs. For massless ladder exchange, our approach reproduces analytically the Wick-Cutkosky equation. For massive ladder exchange, the numerical results coincide with the ones obtained by Wick rotation.     

Bethe-Salpeter equation with the sine-Gordon interaction

An attempt is made to study the interaction Hamiltonian,H _int =Gψ ²(x)U(φ(x)) in the Bethe-Salpeter framework for the confined states of theψ particles interactingvia the exchange of theU field, whereU(φ) = cos (gφ). An approximate solution of the eigenvalue problem is obtained in the instantaneous approximation by projecting the Wick-rotated Bethe-Salpeter equation onto the surface of a four-dimensional sphere and employing Hecke’s theorem in the weak-binding limit. We find that the spectrum of energies for the confined states,E =2m+B (B is the binding energy), is characterized byE ∼n ⁶, wheren is the principal quantum number.     

The Bethe-Salpeter equation for spin-1 particles

We develop here the general treatment of the Bethe—Salpeter equation for the bound state of two spin-l particles interacting through an electromagnetic interaction. The treatment here, which can be generalized to strong interactions, combines the two-component approach utilized previously by the author in conjunction with spontaneous symmetry breaking. This is done by using a Lagrangian having SU(2)×U(1) symmetry (without fermions) and then choosing the ′t Hooft gauge. In this way, a renormalizable theory for the interaction of two spin-l particles via an electromagnetic interaction is ensured.     

Hölder-Banach space analysis of the Bethe-Salpeter equation

We find a Hölder Banach space in which the Bethe-Salpeter equation is a compact integral equation as it stands. We study the properties of the solution in preparation for an analysis of linear field theory models of 3-body amplitudes. In particular we study the properties of the Regge poles of the solution and prove the existence and uniqueness of on mass shell scattering amplitudes.     

Application of the Riemann method to the Bethe-Salpeter equation

The Bethe-Salpeter equation describing the interaction of two scalar particles via the exchange of a third scalar particle with mass 0 is in configuration space a hyperbolic partial differential equation of fourth order which will be studied with the help of the Riemann method. This method yields two Volterra equations the solutions of which are special solutions of the Bethe-Salpeter equation. The wave function is a superposition of the special solutions. For the coefficients one gets a system of two integral equations. The Fredholm determinant of the system is the generalization of the nonrelativistic Jost function.     

Heavy-quark bound states in potentials with the Bethe-Salpeter equation

The spinless Bethe—Salpeter equation is solved for three attractive static quark-antiquark potentials of the form V(r)=–ar^–+br+c, 01, and the effective non-Coulombic power-law potential of the formV(r)=ar ^0.1+c to obtain the spin-averaged energy levels in bottomonium (b ) and charmonium (c ) families. The shifted 1/N expansion technique is used. Calculations of the energy eigenvalues are carried out up to third order and parameters of each potential are adjusted to obtain the best agreement with the experimental spin-averaged data (SAD). Flavor-dependent and flavor-independent cases are considered in this work.     

Bethe-Salpeter type relativistic fermion-antifermion equation in the WKB approximation

We investigate the relativistic fermion-antifermion Bethe-Salpeter type equation whose potential is the sum of Coulomb and linear terms in the WKB approximation. It is shown that in the particular case of an attractive Coulomb potential, the discrete energy spectrum lies in the interval (0,2), and in the case of a repulsive linear potential, in the interval (2, ).Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 21–27, September, 1989.     

Numerical solution of the spinor Bethe-Salpeter equation and the goldstein problem

The spinor Bethe-Salpeter equation describing bound states of a fermion-antifermion pair with massless-boson exchange reduces to a single (uncoupled) partial differential equation for special combinations of the fermion-boson couplings. For spinless bound states with positive or negative parity this equation is a generalization to nonvanishing bound-state masses of the equations studied by Kummer and Goldstein, respectively. In the tight-binding limit the Kummer equation has a discrete spectrum, in contrast to the Goldstein equation, while for loose binding only the generalized Goldstein equation has a nonrelativistic limit. For intermediate binding energies the equations are solved numerically. The generalized Kummer equation is shown to possess a discrete spectrum of coupling constants for all bound-state masses. For the generalized Goldstein equation a discrete spectrum of coupling constants is found only if the binding energy is smaller than a critical value.     

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.     

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.     

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.     

Bethe-Salpeter equation for non-self conjugate mesons in a power-law potential

We develop an approach to the solution of the spinless Bethe-Salpeter equation for the different-mass case. Although the calculations are developed for spinzero particles in any arbitratry spherically symmetric potential, the non-Coulombic effective power-law potential is used as a kernel to produce the spin-averaged bound states of the non-self-conjugate mesons. The analytical formulas are also applicable to the self-conjugate mesons in the equal-mass case. The flavor-independent case is investigated in this work. The calculations are carried out to the third-order correction of the energy series. Results are consistent with those obtained before.