On The Contribution of the P and D Partial-Wave States to the Binding Energy of the Triton in the Bethe-Salpeter-Faddeev Approach

The influence of the partial-wave states with nonzero orbital moment of the nucleon pair on the binding energy of the triton T(nnp) in the relativistic case is considered. The relativistic generalization of the Faddeev equation in the Bethe-Salpeter formalism is applied. Two-nucleon t matrix is obtained from the Bethe-Salpeter equation with separable kernel of nucleon-nucleon interaction of the rank one. The kernel form factors are the relativistic type of the Yamaguchi functions. The following two-nucleon partial-wave states are considered: ¹S₀, ³S₁, ³D₁, ³P₀, ¹P₁, ³P₁. The system of the integral equations are solved by using the iteration method. The binding energy of the triton and three-nucleon amplitudes are found. The contribution of the P and D states to the binding energy of triton is given.

     

Final state interaction effects in electrodisintegration of the deuteron in the Bethe-Salpeter approach

The electrodisintegration of the deuteron is considered in a relativistic model of nucleon-nucleon interaction based on the Bethe-Salpeter approach with a separable interaction kernel. The exclusive cross section is calculated in the impulse approximation under various kinematic conditions. Final state interactions between the outgoing nucleons are taken into account. The comparison of nonrelativistic and relativistic calculations is presented. Partial-wave states of the neutron-proton pair with total angular momentum J = 0, 1 are considered.     

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.     

Ionization of ytterbium atoms from an excited state

The cross sections of the photoionization and the electron impact-induced ionization of Yb atoms from the excited 6s6p(³ P ₁) state are numerically calculated. Matrix elements are computed in multielectron relativistic and nonrelativistic approximations with allowance for the superposition of configurations and a relaxation effect. The radial part of the electron wavefunction in a continuous spectrum is calculated using the solutions to one-configuration Hartree-Fock and Dirac-Fock equations. The cross sections calculated by a relativistic method are compared to those for a nonrelativistic approximation. The ratios of the radiation reduced matrix elements and the phase shifts of the wavefunctions of a continuous spectrum calculated for the 6p ɛs and 6p → ɛd transitions are compared to the values obtained by approximating the experimental dependences of the angular distribution of photoelectrons for the photoionization by ultraviolet radiation from an oriented excited state.     

Dynamical relativistic corrections to the leptonic decay width of heavy quarkonia

We calculate the dynamical relativistic corrections, originating from radiative one-gluon exchange, to the leptonic decay width of heavy quarkonia in the framework of a covariant formulation of light-front dynamics. Comparison with the non-relativistic calculations of the leptonic decay width of J = 1 charmonium and bottomonium S-ground states shows that relativistic corrections are large. Most importantly, the calculation of these dynamical relativistic corrections legitimate a perturbative expansion in , even in the charmonium sector. This is in contrast with the ongoing belief based on calculations in the non-relativistic limit. Consequences for the ability of several phenomenological potentials to describe these decays are described. Received: 6 December 2001 / Published online: 5 April 2002     

Structure of quantum-mechanical relativistic two-body interactions for spinning particles

Relativistic constraint mechanics yields consistent systems of coupled Dirac equations for pairs of spinning particles. We explicitly connect these equations to the Bethe-Salpeter equation of quantum field theory and to the interactions of classical Fokker-Tetrode dynamics (and hence to classical relativistic field theory) to obtain versions of these equations governed by systems of (possibly noncoulombic) relativistic potentials whose detailed structures contain important relativistic effects like correct Darwin interactions. We recast the defining pair of Dirac equations in a number of equivalent but important forms—“external potential,” Sazdjian, hyperbolic, and Breit— and examine their interconnection. Since the potentials in these equations are no more singular than — 1/4r² we are able to solve appropriate versions of them nonperturbatively for the qˉq system to obtain a very good fit to the entire meson spectrum and for the e ⁺ e ⁻ system to calculate the positronium spectrum of QED correct through order α ⁴ .     

ψ Phenomenology and the nature of quark confinement

The nonrelativistic Schrödinger equation is used with confinement potentials that are either fractional power laws or logarithmic functions of the radial coordinate to investigate the spectrum of states, leptonic decay widths, and radiative decays of the ψ family of resonances. The spectrum of states and the leptonic decay widths are in good agreement with the data for the entire class of potentials considered here. The radiative decays are still somewhat large for the standard model but could be brought into agreement with the data when threshold and relativistic effects are taken into account.     

Global existence proof for relativistic Boltzmann equation

The existence and causality of solutions to the relativistic Boltzmann equation inL ¹ and inL _loc ¹ are proved. The solutions are shown to satisfy physically naturala priori bounds, time-independent inL ¹. The results rely upon new techniques developed for the nonrelativistic Boltzmann equation by DiPerna and Lions.     

Measurement of Δ1J(199Hg, 31P) in [HgPCy3(OAc)2]2 and relativistic ZORA DFT investigations of mercury–phosphorus coupling tensors

Using ³¹P solid-state NMR spectroscopy, anisotropy in the indirect ¹⁹⁹Hg-³¹P spin–spin coupling tensor (ΔJ) for powdered [HgPCy₃(OAc)₂]₂ (1) has been measured as 4700±300 Hz. Zeroth-order regular approximation (ZORA) density functional theory (DFT) calculations, including scalar and spin-orbit relativistic effects, performed on 1 and a series of other related compounds show that ΔJ(¹⁹⁹Hg, ³¹P) arises entirely from the ZORA Fermi-contact–spin-dipolar cross term. The calculations validate assumptions made in the spectral analysis of 1 and in previous determinations of ΔJ in powder samples, namely that J is axially symmetric and shares its principal axis system with the direct dipolar coupling tensor (D). Agreement between experiment and theory for various ¹⁹⁹Hg, ³¹P spin–spin coupling anisotropies is reasonable; however, experimental values of ¹J(¹⁹⁹Hg, ³¹P)_iso are significantly underestimated by the calculations. The most important improvements in the agreement were obtained as a result of including more of the crystal lattice in the model used for the calculations, e.g., a change of 43% was noted for ¹J(¹⁹⁹Hg, ³¹P)_iso in [HgPPh₃(NO₃)₂]₂ depending on whether the two or three nearest nitrate ions are included in the model. Finally, we have written a computer program to simulate the effects of non-axial symmetry in J and of non-coincidence of the J and D on powder NMR spectra. Simulations clearly show that both of these effects have a pronounced impact on the ³¹P NMR spectrum of ¹⁹⁹Hg–³¹P spin pairs, suggesting that the effects should be observable experimentally if a suitable compound can be identified.     

Relativistic and nonrelativistic oscillator strengths in the Na isoelectronic sequence

We report the general trend of the nonrelativistic Hartree-Fock relativistic Hartree-Fock, nonrelativistic configuration-interaction, and relativistic configuration-interaction length form of the oscillator strengths for the resonance excitation 1s ²2s ²2p⁶3s ² S _J ^e 1s ²2s ²2p ⁵3s ^{2 2} P _J ^o . transition, which leads to autoionization (Auger transition), in Na, Mg¹⁺, Al²⁺, Si³⁺, P⁴⁺, S⁵⁺, Cl⁶⁺, Ar⁷⁺, K⁸⁺, Ca⁹⁺, Sc¹⁰⁺, Ti¹¹⁺, Fe¹⁵⁺, Cu¹⁸⁺, Zn¹⁹⁺, Br²⁴⁺, and Kr²⁵⁺ ions of the sodium isoelectronic sequence.     

Deuteron photodisintegration in the Bethe-Salpeter formalism

A relativistically invariant analysis of the deuteron-photodisintegration reaction is performed. The reaction γd → pn is considered in the plane-wave approximation on the basis of the Bethe-Salpeter equation with a separable kernel. The results obtained in this way are compared with the results based on nonrelativistic models and on the quasipotential approximation. It is shown that the most important relativistic contributions to the relevant angular distributions are associated with the dependence of the Bethe-Salpeter amplitude on the relative energy of nucleons and on the boost transformation of this amplitude.     

Electron-positron scaling in bound state models

We investigate scaling assuming a generalized vector meson dominance picture. The vector mesons are described as relativistic quark-antiquark bound states by a Bethe-Salpeter equation which yields the mass spectrum and the coupling to e⁺e^? pairs. We discuss the spin structure and find that scaling can occur only for a γ_μ type amplitude. We solve the BS equation using a generalized WKB approximation and find scaling, independent of the detailed shape of the interaction. This means that scaling in e⁺e^? annihilation does not select a particular “confinement potential”. The scaling constant depends on the current renormalization constant and on the details of the relativistic spin structure.     

The unified BS wavefunctions of mesons with natural JPC

From the Bethe-Salpeter equation with a spin-independent kernel, a unified wavefunction is derived for mesons with natural J^PC. Masses of vector mesons calculated from this wavefunction yield a spectrum which agrees with the observed one.     

Nonrelativistic particles and wave equations

This paper is devoted to a detailed study of nonrelativistic particles and their properties, as described by Galilei invariant wave equations, in order to obtain a precise distinction between the specifically relativistic properties of elementary quantum mechanical systems and those which are also shared by nonrelativistic systems. After having emphasized that spin, for instance, is not such a specifically relativistic effect, we construct wave equations for nonrelativistic particles with any spin. Our derivation is based upon the theory of representations of the Galilei group, which define nonrelativistic particles. We particularly study the spin 1/2 case where we introduce a four-component wave equation, the nonrelativistic analogue of the Dirac equation. It leads to the conclusion that the spin magnetic moment, with its Landé factorg=2, is not a relativistic property. More generally, nonrelativistic particles seem to possess intrinsic moments with the same values as their relativistic counterparts, but are found to possess no higher electromagnetic multipole moments. Studying galilean electromagnetism (i.e. the theory of spin 1 massless particles), we show that only the displacement current is responsible for the breakdown of galilean invariance in Maxwell equations, and we make some comments about such a nonrelativistic electromagnetism. Comparing the connection between wave equations and the invariance group in both the relativistic and the nonrelativistic case, we are finally led to some vexing questions about the very concept of wave equations.     

A Generally Covariant Wave Equation for Grand Unified Field Theory

A generally covariant wave equation is derived geometrically for grand unified field theory. The equation states most generally that the covariant d'Alembertian acting on the vielbein vanishes for the four fields which are thought to exist in nature: gravitation, electromagnetism, weak field and strong field. The various known field equations are derived from the wave equation when the vielbein is the eigenfunction. When the wave equation is applied to gravitation the wave equation is the eigenequation of wave mechanics corresponding to Einstein's field equation in classical mechanics, the vielbein eigenfunction playing the role of the quantized gravitational field. The three Newton laws, Newton's law of universal gravitation, and the Poisson equation are recovered in the classical and nonrelativistic, weak-field limits of the quantized gravitational field. The single particle wave-equation and Klein-Gordon equations are recovered in the relativistic, weak-field limit of the wave equation when scalar components are considered of the vielbein eigenfunction of the quantized gravitational field. The Schrödinger equation is recovered in the non-relativistec, weak-field limit of the Klein-Gordon equation). The Dirac equation is recovered in this weak-field limit of the quantized gravitational field (the nonrelativistic limit of the relativistic, quantezed gravitational field when the vielbein plays the role of the spinor. The wave and field equations of O(3) electrodynamics are recovered when the vielbein becomes the relativistic dreibein (triad) eigenfunction whose three orthonormal space indices become identified with the three complex circular indices (1), (2), (3), and whose four spacetime indices are the indices of non-Euclidean spacetime (the base manifold). This dreibein is the potential dreibein of the O(3) electromagnetic field (an electromagnetic potential four-vector for each index (1), (2), (3)). The wave equation of the parity violating weak field is recovered when the orthonormal space indices of the relativistic dreibein eigenfunction are identified with the indices of the three massive weak field bosons. The wave equation of the strong field is recovered when the orthonormal space indices of the relativistic vielbein eigenfunction become the eight indices defined by the group generators of the SU (3) group.     

A relativistic quark model for mesons based on numerical solutions of the Bethe-Salpeter equation

A study is made to determine if the results of the nonrelativistic quark model can be reproduced by a fully relativistic model of deeply bound $\text{spin}\text{-}\text{1}\text{2}$ quarks. It is found that the relativistic model does not reproduce the nonrelativistic results, even when the quarks have nonrelativistic momenta. However, the model is rather successful in accounting for the known properties of mesons.Numerical solutions to the Bethe-Salpeter equation are obtained for pseudoscalar and vector bound states of equal mass quark-antiquark pairs, with either a scalar, pseudoscalar, or neutral vector exchange interaction. The interaction function corresponds to single particle exchange, with the addition of either one or two regulating terms. It is found that the second regulator allows the internal quark momentum to be nonrelativistic, but that the spinor structure of the wave function remains highly relativistic.Only the scalar interaction can account for the observed spectrum of states. The pseudoscalar interaction yields a vector state of lower mass than the pseudoscalar state, and the vector interaction leads to a vector state which lies approximately one quark mass above the pseudoscalar state. The λ quark is taken as slightly heavier than the p and n, and the perturbation treatment of the mass difference leads to a quadratic mass formula.The decay amplitudes for π, K → μν are calculated, and it is found, independent of parameters, that $\text{?}{}_{\mathrm{\pi }}\text{≈ ?}{}_{\mathrm{K}}$ for either a scalar or vector interaction, in agreement with experiment and in contrast with the nonrelativistic model. The amplitudes for ?^o, ω, φ → e⁺e^?, μ⁺μ^? are also calculated, but in this case the ratios (again parameter independent) are in minor discrepancy with experiment.The question of the additivity of quark amplitudes is examined by calculating (with significant restrictions) the magnetic moments of the vector mesons and the amplitudes for magnetic transitions such as ω → π^oγ. The magnetic moments of the vector mesons have the same (trivial) ratios to each other as in the nonrelativistic model, but they are strongly enhanced over the sum of the quark magnetic moments. The amplitude for magnetic transitions, however, is related to the quark magnetic moments in approximately the same ratio as in the nonrelativistic model.The model is also used to obtain parameter dependent predictions for the masses and decay amplitudes. These predictions are not experimentally correct, but are generally well within an order of magnitude for a wide range of the parameters.The most significant defect discovered of the model is the presence of ghost states (the daughters of the vector mesons, with J^PC = 0^+?) with masses of about 2 BeV.     

On quantum electrodynamics of two-particle bound states containing spinless particles

We develop here the general treatment arising from the Bethe-Salpeter equation for a two-particle bound system in which at least one of the particles is spinless. It is shown that a natural two-component formalism can be formulated for describing the propagators of scalar particles. This leads to a formulation of the Bethe-Salpeter equation in a form very reminiscent of the fermion-fermion case. It is also shown, that using this two-component formulation for spinless particles, the perturbation theory can be systematically developed in a manner similar to that of fermions. Quantum electrodynamics for scalar particles is then developed in the two component formalism, and the problem of bound states, in which one of the constituent particles is spinless, is examined by means of the means of the Bethe-Salpeter equation. For this case, the Bethe-Salpeter equation is cast into a form which is convenient to perform a Foldy-Woutyhuysen transformation which we carry out, keeping the lowest-order relativistic corrections to the nonrelativistic equation. The results are compared with the corresponding fermion-fermion case. It is shown, as might have been expected, that the only spin-independent terms that occur for the fermion-fermion system which do not occur for bound scalar particle cases, is the zitterbewegung contribution. The relevance of the above considerations for systems that are essentially bound by electromagnetic interactions, such as kaonic hydrogen, is discussed.     

Pseudoscalars in the charmonium model

We take the recently found charmonium state at 2.976 GeV to be the η _c and show that it can be included in a charmonium model with relativistic corrections which reproduces the s-wave spectrum, the leptonic widths Γ(V→e ⁺ e^?) and the p-wave splittings. The upsilon spectrum is discussed as are the effects of radial and pseudoscalar mixing.     

Quantum Mechanical Virial Theorem in Systems with Translational and Rotational Symmetry

Generalized virial theorem for quantum mechanical nonrelativistic and relativistic systems with translational and rotational symmetry is derived in the form of the commutator between the generator of dilations G and the Hamiltonian H. If the conditions of translational and rotational symmetry together with the additional conditions of the theorem are satisfied, the matrix elements of the commutator [G,H] are equal to zero on the subspace of the Hilbert space. Normalized simultaneous eigenvectors of the particular set of commuting operators which contains H, J ², J _z and additional operators form an orthonormal basis in this subspace. It is expected that the theorem is relevant for a large number of quantum mechanical N-particle systems with translational and rotational symmetry.