Three-dimensional Relativistic Equation Research Articles

Green's function and pointwise space-time behaviors of the three-dimensional relativistic Boltzmann equation

The pointwise space-time behavior of the Green's function of the three-dimensional relativistic Boltzmann equation is studied in this paper. It is shown that the Green's function has a decomposition of the macroscopic diffusive waves and Huygens waves with the speed a2+b2 at low-frequency, the singular kinetic wave and the remainder term decaying exponentially in space and time. In addition, we establish the pointwise space-time estimate of the global solution to the nonlinear relativistic Boltzmann equation with non-smooth initial data based on the Green's function.

Special relativistic effects revealed in the Riemann problem for three-dimensional relativistic Euler equations

We consider the Riemann problem of three-dimensional relativistic Euler equations with two discontinuous initial states separated by a planar hypersurface. Based on the detailed analysis on the Riemann solutions, special relativistic effects are revealed, which are the variations of limiting relative normal velocities and intermediate states and thus the smooth transition of wave patterns when the tangential velocities in the initial states are suitably varied. While in the corresponding non-relativistic fluid, these special relativistic effects will not occur.

Rigorous relativistic equation for quark–antiquark bound states at finite temperature derived from thermal QCD formulated in the coherent-state representation

A rigorous three-dimensional relativistic equation for quark–antiquark bound states at finite temperature is derived from the thermal QCD generating functional which is formulated in the coherent-state representation. The generating functional is derived newly and given a correct path-integral expression. The perturbative expansion of the generating functional is specifically given by means of the stationary-phase method. Especially, the interaction kernel in the three-dimensional equation is derived by virtue of the equations of motion satisfied by some quark–antiquark Green functions and given a closed form which is expressed in terms of only a few types of Green functions. This kernel is very suitable to use for exploring the deconfinement of quarks. To demonstrate the applicability of the equation derived, the one-gluon exchange kernel is derived and described in detail.

Glueball spectrum based on a rigorous three-dimensional relativistic equation for two-gluon bound states. II. Calculation of the glueball spectrum

In the preceding paper, a rigorous three-dimensional relativistic equation for two-gluon bound states was derived from QCD with massive gluons and represented in the angular momentum representation. In order to apply this equation to calculate the glueball spectrum, in this paper, the equation is recast in an equivalent three-dimensional relativistic equation satisfied by the two-gluon positive energy state amplitude. The interaction Hamiltonian in the equation is exactly derived and expressed as a perturbative series. The first term in the series describes the one-gluon exchange interaction which includes fully the retardation effect in it. This term plus the linear confining potential are chosen to be the interaction Hamiltonian and employed in the practical calculation. With the integrals containing three and four spherical Bessel functions in the QCD vertices being analytically calculated, the interaction Hamiltonian is given an explicit expression in the angular momentum representation. Numerically solving the relativistic equation with taking the contributions arising from the retardation effect and the longitudinal mode of gluon fields into account, a set of masses for the ${0}^{++}{,0}^{\ensuremath{-}+}{,1}^{++}{,1}^{\ensuremath{-}+}{,2}^{++},$ and ${2}^{\ensuremath{-}+}$ glueball states are obtained and are in fairly good agreement with the predictions given by the lattice simulation. In addition, some new glueball states are predicted.

Glueball spectrum based on a rigorous three-dimensional relativistic equation for two-gluon bound states. I. Derivation of the relativistic equation

A rigorous three-dimensional relativistic equation satisfied by two-gluon bound states is derived from the QCD with massive gluons. With the gluon fields and the quark fields being expanded in terms of the gluon multipole fields and the spherical Dirac spinors respectively, the equation is well established in the angular momentum representation and hence is much convenient for solving the problem of two-gluon glueball spectra. In particular, the interaction kernel in the equation is exactly derived and given a closed expression which includes all the interactions taking place in the two-gluon glueballs. The kernel contains only a few types of Green's functions and commutators. Therefore, it is not only easily calculated by the perturbation method, but also provides a suitable basis for nonperturbative investigations.

Relativistic field-theoretical representation of the Schrödinger equation and field-theoretical generalization of the inverse scattering method

The relativistic three-dimensional equation ( 2 E p ′ − 2 E p ) 〈 p ′ / Ψ p 〉 = − ∫ V ( t ) d 3 p ″ 〈 p ″ / Ψ p 〉 with E p = m 2 + p 2 and t = ( E p′ − E p″ ) 2−( p′− p″) 2 is transformed into Schrödinger equation (Δ r + k 2)〈 r|ϕ k 〉=− v( r)〈 r|ϕ k 〉 with E p = k 2/2 m+ m using the Fourier transformation. The potential v( r) and the wave function 〈 r|ϕ k 〉 differ from the corresponding Fourier images of the relativistic potential V( t) and the relativistic wave function 〈 p′| Ψ p 〉 with the kinematical factors only. In the framework of the standard field-theoretical approach the three-dimensional relativistic equations for the NN scattering amplitude with the total NN potential V tot = V( t)+ V nonl are derived. The nonlocal part of this NN potential V nonl is constructed from the πNN vertex functions which are determined from the πN phase shifts. The other part of the NN potential V( t) consists of the π-, σ-, ρ-, ω-, … meson exchange diagrams and for the NN contact (overlapping) terms. In the case of the renormalizable Lagrangians V( t) reproduces exactly the One Boson Exchange (OBE) Bonn model of the NN potential. The inverse scattering problem for this V( t) potential is reduced to the construction of the potential v( r) of the ordinary Schrödinger equation from the NN phase shifts.

RETARDATION TERMS IN THE ONE-GLUON EXCHANGE POTENTIAL

It is pointed out that the retardation terms given in the original Fermi–Breit potential vanish in the center-of-mass frame. The retarded one-gluon exchange potential is rederived in this paper from the three-dimensional one-gluon exchange kernel which appears in the exact three-dimensional relativistic equation for quark–antiquark bound states. The retardation part of the potential given in the approximation of order p2/m2 is shown to be different from those derived in the previous literature. This part is off-shell and no longer vanishes in the center-of-mass frame.

Bc mesons in a relativistic three-dimensional two-particle quasipotential model

Abstract We apply three-dimensional relativistic equation for the system of two-particles with unequal masses to the B c mesons family. The method of ℏ-expansion is applied for deriving the masses of B c mesons on the basis of three-dimensional relativistic two-particle equation. Numerical calculations for the Cornell potential are presented.

Numerical solution of two-body relativistic equations for the bound-state problem with confining and Coulomb potentials

Recent investigations of the meson spectroscopy often have been based on models considering mesons as quark–antiquark bound states. These models are described by three-dimensional relativistic equations with various generalizations of Coulomb and confining potentials in momentum space. Algorithms and codes are developed for numerical investigations of these equations. The modified Generalized Continuous analogue of the Newton method is used. The numerical results demonstrate the efficiency of the created software. The code allows obtaining a numerical solution with a required accuracy. The numerical results are compared with the same results obtained by other authors.

Boundqq¯systems in the framework of the different versions of the three-dimensional reductions of the Bethe-Salpeter equation

Bound $q\overline{q}$ systems are studied in the framework of different three-dimensional relativistic equations derived from the Bethe-Salpeter equation with the instantaneous kernel in the momentum space. Except the Salpeter equation, all these equations have a correct one-body limit when one of the constituent quark masses tends to infinity. The spin structure of the confining $\mathrm{qq}$ interaction potential is taken in the form $x{\ensuremath{\gamma}}_{1}^{0}{\ensuremath{\gamma}}_{2}^{0}+(1\ensuremath{-}{x)I}_{1}{I}_{2},$ with $0<~x<~1.$ At first stage, the one-gluon-exchange potential is neglected and the confining potential is taken in the oscillator form. For the systems $(u\overline{s}),(c\ifmmode \bar{u}\else \={u}\fi{}),(c\overline{s})$ and $(u\ifmmode \bar{u}\else \={u}\fi{}),(s\overline{s})$ a comparative qualitative analysis of these equations is carried out for different values of the mixing parameter x and the confining potential strength parameter. We investigate (1) the existence/nonexistence of stable solutions of these equations, (2) the parameter dependence of the general structure of the meson mass spectum and leptonic decay constants of pseudoscalar and vector mesons. It is demonstrated that none of the three-dimensional equations considered in the present paper simultaneously describes even general qualitative features of the whole mass spectrum of $q\overline{q}$ systems. At the same time, these versions give an acceptable description of the meson leptonic decay characteristics.

Role of retardation in three-dimensional relativistic equations

Equal-time Green's function is used to derive a three-dimensional integral equation from the Bethe-Salpeter equation. The resultant equation, in the absence of antiparticles, is identical to the use of time-ordered diagrams, and has been used within the framework of ${\ensuremath{\varphi}}^{2}\ensuremath{\sigma}$ coupling to study the role of energy dependence and nonlocality when the two-body potential is the sum of $\ensuremath{\sigma}$ exchange and crossed $\ensuremath{\sigma}$ exchange. The results show that nonlocality and energy dependence make a substantial contribution to both the on-shell and off-shell amplitudes.

Relativistic effect on low-energy nucleon-deuteron scattering.

The relativistic effect on differential cross sections, nucleon-to-nucleon and nucleon-to-deuteron polarization transfer coefficients, and the spin correlation function, of nucleon-deuteron elastic scattering is investigated employing several three-dimensional relativistic three-body equations and several nucleon-nucleon potentials. The polarization transfer coefficients are found to be sensitive to the details of the nucleon-nucleon potentials and the relativistic dynamics employed, and prefer trinucleon models with the correct triton binding energy. (To appear in Phys. Rev. C)

Relativistic Three-Particle Dynamical Equations II. Application to the Trinucleon System

We calculate the contribution of relativistic dynamics on the neutron-deutron scattering length and triton binding energy employing five sets trinucleon potential models and four types of three-dimensional relativistic three-body equations suggested in the preceding paper. The relativistic correction to binding energy may vary a lot and even change sign depending on the relativistic formulation employed. The deviations of these observables from those obtained in nonrelativistic models follow the general universal trend of deviations introduced by off- and on-shell variations of two- and three-nucleon potentials in a nonrelativistic model calculation. Consequently, it will be difficult to separate unambiguously the effect of off- and on-shell variations of two- and three-nucleon potentials on low-energy three-nucleon observables from the effect of relativistic dynamics.

Calculation of baryon spectra based on an improved quark potential

An improved QCD-inspired quark potential is introduced and employed to calculate baryon spectra. This potential has some novel features: (i) the one-gluon exchange potential in it is derived from the exact three-dimensional relativistic equation and contains a retardation term which is logarithmic; (ii) the confining part of the potential is generated from a linear potential Bethe-Salpeter (BS) kernel and a general Lorentz structure of the confinement and includes therefore various spin-dependent terms added to the linear potential; (iii) the off-shell correction part of the potential which is related to the negative energy component of the BS amplitude automatically gives rise to a harmonic oscillator term. We analyse the contributions of all parts of the potential to the baryon spectrum and find that the potential we use makes some improvements to theoretical baryon masses, especially for the Roper resonance.

Three-dimensional relativistic structure for quarkonium states.

A three-dimensional relativistic equation is used to calculate the quarkonium spectra. The kernel is the one obtained by projecting the Bethe-Salpeter (BS) kernel with both fermions being on the mass shell. The potential in the momentum representation is assumed to have a Coulomb part with highmomentum cutoff and a confinement part regularized by a small mass parameter. A complete Dirac bilinear covariant set is used to describe the spin structure of the potential. The quarkonium wave function is expanded in terms of the free Dirac spinors which are classified by the irreducible representation of the Lorentz group. Most of the quarkonium states are embedded in the continuum which exists due to the finite barrier of the confinement potential. Several sets of parameters for qualitatively fitting the experimental data are considered. The results show that the axial-vector and tensor bilinear covariants have to be included in the spin structure of the original BS kernel to explain the spectra of the quarkonium system. This property is different from that in the Schr\"odinger formalism, where the spin-dependent forces originate from relativistic corrections of vector covariants. The mixing of $S$ and $D$ components in the wave function of the $\frac{J}{\ensuremath{\psi}}$ system is shown to be sensitive to the presence of the axial-vector and tensor components of the potential.

One-time relativistic equation and its application to nucleon-nucleon scattering.

A three-dimensional relativistic equation with real time variables is derived from the Bethe-Salpeter equation. The kernel is uniquely determined by the Bethe-Salpeter kernel through an infinite series. The one-time kernel in the momentum representation is represented by a superposition of Feynman amplitudes with the particle 1 or the antiparticle 2 being on the mass shell. The numerical analysis shows that the one-time equation is equivalent to the Gross equation in the N-N scattering case.

Examination of retardation effect on meson spectra

Based on the exact three-dimensional relativistic equation for qq bound states, an off-shell retarded one-gluon exchange potential is derived and employed to calculate meson spectra. The calculated results show that the retardation effect is important for the improvement of the theoretical values of lighter meson masses.

Exact Three-Dimensional Relativistic Equation for Bound states11The project supported by National Natural Science Foundation of China.

Starting with the QCD generating functional, a new derivation and a new form of the Dirac-Schrödinger (D-S) equation, satisfied by the equal-time Bethe-Salpeter amplitude of quark-antiquark bound states have been given. In this equation, the effective interaction kernel has a compact expression and can be directly calculated by means of the conventional QCD Feynman rules. Furthermore, an equivalent reduction of the above equation to a Pauli-Schrödinger equation has also been achieved and a closed form of the effective interaction Hamiltonian appearing in the latter equation has been explicitly written out.

Equal-time Equation of the Complex Scalar Field

By using the time displacement operators and rearranging the irreducible kernel, the equal-time equation of the two-component complex scalar field is established. From the equations of the matrix elements the equation of the usual two composite bosons is separated and a three-dimensional relativistic equation is obtained. The most important feature is that the ghost states disappear in our equation.

Self-consistent retardation in a three-dimensional relativistic equation.

A new technique for approximating solutions of the two-body Bethe-Salpeter equation is presented. Coupled equations for the relative energy dependence and the relative three-momentum dependence of the relativistic $T$ matrix are derived. These equations are solved self-consistently for the Wick-rotated $T$ matrix in a simple model problem and the numerical results are compared with exact as well as usual three-dimensional reduction results.