Relativistic Constraint Dynamics Research Articles

On the Mass Difference between π and ρ Using a Relativistic Two-Body Model

The big mass difference between the pion (π) and rho meson (ρ) possibly originates from the spin-dependent nature of the interactions in the two states since these two states are similar except for spin. Both π and ρ are quark-antiquark systems which can be treated using the two-body Dirac equations (TBDE) of constraint dynamics. This relativistic approach for two-body system has the advantage over the non-relativistic treatment in the sense that the spin-dependent nature is automatically coming out from the formalism. We employed Dirac’s relativistic constraint dynamics to describe quark-antiquark systems. Within this formalism, the 16-component Dirac equation is reduced to the 4-component 2nd-order differential equation and the radial part of this equation is simply a Schrödinger-type equation with various terms calculated from the basic radial potential. We used a modified Richardson potential for quark-antiquark systems which satisfies the conditions of confinement and asymptotic freedom. We obtained the wave functions for these two mesons which are not singular at short distances. We also found that the cancellation between the Darwin and spin-spin interaction terms occurs in the π mass but not in the ρ mass and this is the main source of the big difference in the two meson masses.

Singularity structures in Coulomb-type potentials in two-body Dirac equations of constraint dynamics

Two-body Dirac equations (TBDE) of Dirac's relativistic constraint dynamics have been successfully applied to obtain a covariant nonperturbative description of QED and QCD bound states. Coulomb-type potentials in these applications lead naively in other approaches to singular relativistic corrections at short distances that require the introduction of either perturbative treatments or smoothing parameters. We examine the corresponding singular structures in the effective potentials of the relativistic Schr\odinger equation obtained from the Pauli reduction of the TBDE. We find that the relativistic Schr\odinger equation leads in fact to well-behaved wave function solutions when the full potential and couplings of the system are taken into account. The most unusual case is the coupled triplet system with $S=1$ and $L={(J\ensuremath{-}1),(J+1)}$. Without the inclusion of the tensor coupling, the effective $S$-state potential would become attractively singular. We show how including the tensor coupling is essential in order that the wave functions be well-behaved at short distances. For example, the $S$-state wave function becomes simply proportional to the $D$-state wave function and dips sharply to zero at the origin, unlike the usual $S$-state wave functions. Furthermore, this behavior is similar in both QED and QCD, independent of the asymptotic freedom behavior of the assumed QCD vector potential. Light- and heavy-quark meson states can be described well by using a simplified linear-plus-Coulomb-type QCD potential apportioned appropriately between world scalar and vector potentials. We use this potential to exhibit explicitly the origin of the large $\ensuremath{\pi}\mathrm{\text{\ensuremath{-}}}\ensuremath{\rho}$ splitting and effective chiral symmetry breaking. The TBDE formalism developed here may be used to study quarkonia in quark-gluon plasma environments.

Relativistic corrections in a three-boson system of equal masses

Three-body systems of scalar bosons are covariantly described in the framework of relativistic constraint dynamics. With help of a change of variables followed by a change of wave function, two redundant degrees of freedom get eliminated and the mass-shell constraints can be reduced to a three-dimensional eigenvalue problem. In general, the reduced equation obtained by this procedure involves the spectral parameter in a nonconventional manner, but for three equal masses a drastic simplification arises at the first post-Galilean order: the reduced wave equation becomes a conventional eigenvalue problem that we treat perturbatively, computing a first-order correction beyond the nonrelativistic limit. The harmonic interaction is displayed as a toy model.

Meson-meson scattering in relativistic constraint dynamics

Dirac's relativistic constraint dynamics have been successfully applied to obtain a covariant nonperturbative description of QED and QCD bound states. We use this formalism to describe a microscopic theory of meson-meson scattering as a relativistic generalization of the nonrelativistic quark-interchange model developed by Barnes and Swanson.

Evolution and covariance in constraint dynamics

Two problems of first-class relativistic constraint dynamics of direct interaction are considered. The first is the relation of the gauge motion to the physical motion. In a manifestly covariant formulation they are shown to be exactly equivalent, the latter duplicating the former. The physical motion arises as a consequence of the explicit dependence on an (invariant) time parameter introduced by the gauge fixations (specification of "equal-time" surfaces). No simple relation exists in general between the evolution generator and the translation generators. The second problem deals with the covariance of the physical world lines. It is satisfied trivially for Lorentz covariance (since the formalism is covariant) and can be satisfied nontrivially for translations in the equal-time surface orthogonal to the total momentum. Sufficient conditions on the fixations are given to ensure such covariant world lines.