Relativistic Bound-state Equations Research Articles

The Spectroscopy and Form Factors of Nucleon Resonances from Superconformal Quantum Mechanics and Holographic QCD

The superconformal algebraic approach to hadronic physics is used to construct a semiclassical effective theory for nucleons which incorporates essential nonperturbative dynamical features, such as the emergence of a confining scale and the Regge resonance spectrum. Relativistic bound-state equations for nucleons follow from the extension of superconformal quantum mechanics to the light front and its holographic embedding in a higher dimensional gravity theory. Superconformal algebra has been used elsewhere to describe the connections between the light mesons and baryons, but in the present context it relates the fermion positive and negative chirality states and uniquely determines the confinement potential of nucleons. The holographic mapping of multi-quark bound states also leads to a light-front cluster decomposition of form factors for an arbitrary number of constituents. The remarkable analytical structure which follows incorporates the correct scaling behavior at high photon virtualities and also vector dominance at low energies.

Light-front two-dimensional QED: Self-field approach

The self-field approach to quantum electrodynamics (QED) is used to study the bound state problem in light-front two-dimensional QED with massive matter fields. A composite matter field describing bound states is introduced and the relativistic bound state equation for the composite field including a self-potential is obtained. The Hamiltonian form of the bound state equation in terms of the invariant mass squared operator is given. The eigenvalue problem of this operator is solved for a fixed value of the self-potential, the corresponding eigenfunctions and the mass spectrum are found. In the case of massless matter fields, there are no self-field terms in the bound state equation, and the invariant mass spectrum can be evaluated explicitly. Possible ways of deriving more complete information about the bound state spectrum are briefly discussed.

Relativistic bound-state equations in three dimensions.

First, a systematic procedure is derived for obtaining three-dimensional bound-state equations from four-dimensional ones. Unlike ``quasipotential approaches'' this procedure does not involve the use of delta-function constraints on the relative four-momentum. In the absence of negative-energy states, the kernels of the three-dimensional equations derived by this technique may be represented as sums of time-ordered perturbation theory diagrams. Consequently, such equations have two major advantages over quasipotential equations: They may easily be written down in any Lorentz frame, and they include the meson-retardation effects present in the original four-dimensional equation. Second, a simple four-dimensional equation with the correct one-body limit is obtained by a reorganization of the generalized ladder Bethe-Salpeter kernel. Third, our approach to deriving three-dimensional equations is applied to this four-dimensional equation, thus yielding a retarded interaction for use in the three-dimensional bound-state equation of Wallace and Mandelzweig. The resulting three-dimensional equation has the correct one-body limit and may be systematically improved upon. The quality of the three-dimensional equation, and our general technique for deriving such equations, is then tested by calculating bound-state properties in a scalar field theory using six different bound-state equations. It is found that equations obtained using the method espoused here approximate the wave functions obtained from their parent four-dimensional equations significantly better than the corresponding quasipotential equations do. \textcopyright{} 1996 The American Physical Society.

Pion and rho meson masses from quark and gluon condensates

Nonzero quark and gluon condensates generate nonzero value of pion mass, even in the zero limit of current quark mass. In turn, the nozero ϱ meson width is due to the nonzero pion mass. Nonzero quark and gluon condensates are also necessary conditions for permanent confinement of quarks and gluons. The notion of permanent confinement means: i) the nonexistence of any asymptotic quark or gluon states, ii) the nonexistence of any asymptotic continuum partonic states, and iii) the nonexistence of any colourfull bound systems. Only colourless hadrons composed of permanently confined partons are present. These hadrons must be calculated as solutions of truly relativistic bound state equations. Masses of constituent quarks are defined in hadrons, and crucially depend on the magnitude of a space-like Wightman-Garding relative momentum. The dominating binding potential of constituents in hadrons is the QCD analog of Coulomb interaction. There is no place for any interaction between constituents which could increase indefinitely with a space-like separation of constituents. For example, a linear interaction, with singularity (q-q′) −4 in momentum space, is ruled out on two grounds: i) as contradicting Dyson-Schwinger equations, and ii) as being in conflict with the cluster property of local QCD, if there is a nonzero mass gap. For QCD with quark condensate for up and down quarks the Goldstone theorem fails. If it would hold it would require massless pion for massless quarks, and the possibility of approximating the pion field by a local field. Nonperturbative QCD with permanently confined quarks says “no” to both of these claims.

Relativistic bound state equation for unstable fermions and the toponium width

The bound state problem for a fermion-antifermion system is considered taking into account a finite decay width of the constituents. We propose an exactly solvable relativistic zero order equation similar to that of Barbieri and Remiddi, but including a constant width. We focus especially on the t t system for which we reconsider our recent calculation of the bound state corrections to the toponium width, which was performed in the narrow width approximation and needed the use of second order Bethe-Salpeter perturbation theory. We show that one obtains the same result already in first order BS perturbation theory if one uses our present approach. Furthermore the large cancellations of gauge dependent terms is demonstrated to be a consequence of a Ward identity. This cancellation mechanism is shown to be valid for general fermion-antifermion systems.

Relativistic equations for the $$N\bar N$$ bound statesbound states

We have studied the nucleon-antinucleon interaction at low energy, in view of the possible existence of bound states or resonances in\(N\bar N\) reactions. We have employed a relativistic treatment of the two-body bound-state problem. Such a treatment is most easily carried out in momentum space, and leads to a relativistic bound-state equation in momentum space. We have solved this equation initially considering an interaction potential with a long and medium range, Yukawa-like, meson exchange part (OBEP) and then considering a phenomenological, data-driven annihilation potential. The features of the\(N\bar N\) spectrum are discussed and compared with the non-relativistic approaches and with the available experimental data.

Predictability of charmonium levels from a range of good fits.

A QCD-motivated potential with five parameters and a cutoff on high virtual momentum is used in a relativistic bound-state equation to examine the predictive power of valence quark potential models by computing fits to the seven well-established charmonium levels. The predictions of potential models are really ranges for new levels given by the ranges of parameters that give good fits to established levels, rather than the single best fit. The range of parameters that give good fits is presented in three-dimensional plots by isosurfaces of fits with fixed theoretical standard deviations in energy from the known levels. Next, for each value of a predicted splitting we show the minimum deviation found over all the fits that lead to that value of splitting. We also show correlations between predictions by finding for each pair of values for two predictions the minimum deviation among the fits that give those predicted values, and making contour plots. For the $^{1}P_{1}\ensuremath{-}^{3}P_{\mathrm{c}.\mathrm{g}.}$ splitting we find the predictions to cover a range of several MeV both positive and negative. A correlation of the spin-spin $^{1}P_{1}\ensuremath{-}^{3}P_{\mathrm{c}.\mathrm{g}.}$ splitting with the similar ${\ensuremath{\psi}}^{\ensuremath{'}}\ensuremath{-}{\ensuremath{\eta}}_{c}^{\ensuremath{'}}$ splitting prediction is presented, and values for ${\ensuremath{\psi}}^{\ensuremath{'}}\ensuremath{-}{\ensuremath{\eta}}_{c}^{\ensuremath{'}}$ have a range of order 20-50 MeV centered about 35 MeV. To range as high as the single experimental observation of 92 MeV for ${\ensuremath{\psi}}^{\ensuremath{'}}\ensuremath{-}{\ensuremath{\eta}}_{c}^{\ensuremath{'}}$, such a poor fit to the seven levels is encountered that the model lacks any significant predictive power. Predictions for the $^{3}D_{2}$ and $^{1}D_{2}$ levels are also given.

Solution of two-body relativistic bound-state equations with confining plus Coulomb interactions.

Studies of meson spectroscopy have often employed a nonrelativistic Coulomb plus linear confining potential in position space. However, because the quarks in mesons move at an appreciable fraction of the speed of light, it is necessary to use a relativistic treatment of the bound-state problem. Such a treatment is most easily carried out in momentum space. However, the position-space linear and Coulomb potentials lead to singular kernels in momentum space. Using a subtraction procedure we show how to remove these singularities exactly and thereby solve the Schr\"odinger equation in momentum space for all partial waves. Furthermore, we generalize the linear and Coulomb potentials to relativistic kernels in four-dimensional momentum space. Again we use a subtraction procedure to remove the relativistic singularities exactly for all partial waves. This enables us to solve three-dimensional reductions of the Bethe-Salpeter equation. We solve six such equations for Coulomb plus confining interactions for all partial waves.

Relativistic treatment of light quarks in D and B mesons and W-exchange weak decays.

A relativistic bound-state equation is applied to formulate the nonspectator (or W-exchange) decay rates of D, D/sub s/, and B mesons, and their leptonic decay constants. The nonspectator decay rates are also calculated here in the Dirac-equation limit. In this limit we show that up to a logarithm the nonspectator rate grows with heavy-quark mass like m/sup 6/, faster than the spectator rate of m/sup 5/. Including an analysis of experimental D/sup +- /, D/sup 0/, and D/sub s/ decay rates shows that the enhancement of D/sup 0/ decays from the nonspectator diagram is about a quarter of the contribution from the spectator diagram. The relativistically calculated weak decay constant f/sub B/ is in the 260--300-MeV range. Predictions are also made for nonspectator B, B/sub s/, and B/sub c/ decay rates including the ratio tau/sub B//sub <//sub +- //sub =//tau/sub =/approx. =2. .AE

Numerical solution of Salpeter's equation

A method for solving Salpeter's relativistic bound-state equation is presented. The interaction can be given in either momentum space or configuration space and may have various Lorentz-Dirac properties. The operators of the equation are represented as matrices in a basis of nonrelativistic harmonic-oscillator states. The resulting non-Hermitian matrix is diagonalized for various values of the oscillator frequency, a variational parameter. To reduce the size of the matrices a two-particle Foldy-Wouthuysen transformation is applied. As an example, the charmonium and $b$-quarkonium mass spectra are calculated using a linear confining potential plus one-gluon exchange. The effects of the coupling between the positive- and negative-energy components are examined and found to be important for the light mesons.

Relativistic treatment of quark-antiquark spectra in quantum chromodynamics

Author(s): Bander, M; Silverman, D; Klima, B; Maor, U | Abstract: A fully relativistic bound-state equation is used to study the spectra (and leptonic widths) of charmonium, b-quarkonium, D mesons, F mesons, B mesons, and mesons composed of light quarks. The short-distance behavior of the interaction is that of asymptotically free QCD, while the largedistance interaction is a linear confining one whose Lorentz nature is either pure scalar or an equal mixture of vector and scalar. Multiquark states are accounted for in an approximate manner. An interesting dependence of the quark masses on the Lorentz nature of the long-distance interaction and on the overall mass scale of the system is obtained. © 1984 The American Physical Society.

1/ c expansion of bound-state energies in QED

A new method for calculating bound-state energies in quantum electrodynamics is proposed. It starts from a non-relativistic Schrödinger equation as a zeroth approximation and treats the relativistic corrections by a systematic expansion. The result is expressed as a sum over Feynman-like graphs. No relativistic bound-state equations are used. Only the case of positronium is considered explicitly, but the method generalizes easily to other cases.

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.

Formal Equivalence of the Hydrogen Atom and Harmonic Oscillator and Factorization of the Bethe-Salpeter Equations

We investigate the possible factorizations of the confluent hypergeometric equation and show that this leads to a formal equivalence of the usual hydrogen atom problem with that of a set of multidimensional harmonic oscillators of appropriate classical frequency in a space of varying (even) dimensionality ranging from 4l + 4 to 2. We next use this method to investigate the solutions of some relativistic bound-state equations. As specific examples we consider Goldstein's eigenvalue problem and the Wick-Cutkosky scalar meson equation. A formal similarity also exists between the Goldstein problem and the scalar meson equation in the zero binding limit. The factorization of either of these equations leads to an infinite ladder of nonsquare integrable solutions. On the other hand, for arbitrary nonvanishing binding energies a complete solution of the scalar meson problem has been obtained by investigating the zero-energy factorization of the equation.