Significance of relativistic wave equations for bound states

Abstract

We scrutinize the relevance of relativistic wave equations for the description of quark-antiquark bound states. By comparing the predictions of the nonrelativistic Schrodinger equation (with only the lowest-order relativistic corrections of the famous Breit-Fermi Hamiltonian), the spinless Salpeter equation, and a new semirelativistic wave equation (which incorporates relativistic kinematics and the complete relativistic corrections to the static interaction potential) for light and heavy quarkonia within three different potential models, we discuss the extent to which the use of relativistic wave equations is reasonable or necessary in order to reproduce the experimentally observed meson mass spectra. We are forced to conclude that-contrary to one's physical intuition-a relativistic treatment of bound states in a potential model provides no improvement at all compared to the corresponding nonrelativistic description. PACS number(s): 1 l.lO.St, 03.65.Ge, 1 l.lO.Qr, 12.40.Qq I. INTRODUCTIOKOne of the most popular approaches to hadrons is to describe them by means of a nonrelativistic Schrodinger equation as bound states of (constituent) quarks which interact via some effective potential. The overwhelming success of these nonrelativistic potential models not only for heavy quarkonia but also for light mesons remains up to now a miracle in hadron spectroscopy [I]. In principle, at least bound states consisting of light constituents should be dealt with in a relativistic framework. In this paper we would like to address the question of to what extent the employment of a relativistic equation of motion for the description of fermion-antifermion bound states is meaningful or even unavoidable.The strategy of our investigation (and simultaneously the outline of this paper) is the following. We solve the nonrelativistic Schrodinger equation, the spinless Salpeter equation, and a new semirelativistic wave equation, all of them introduced in Sec. 11, with the help of the numerical method briefly sketched in Sec. I11 for some typical interquark potentials presented in Sec. IV. Comparing in Sec. V the output of our fits to the experimentally observed meson mass spectra, we are, in Sec. VI, unambiguously led to the conclusion that, as far as the confrontation with experiment is concerned, an increase of the relativistic consistency of the bound-state wave equations makes things worse. WAVE EQUATIONSWe are interested in Schrodinger-type eigenvalue equations of the form where H is the Hamiltonian governing the dynamics of the bound state under consideration and $(x) the configuration-space representation of the corresponding state vector. In the center-of-momentum system of the bound state, the energy eigenvalue arising from this equation is, of course, nothing else but the mass M of the composite particle.We shall consider bound states consisting of fermions with equal masses m , = m 2 = m and spins S,,S2, respectively, which interact via a spherically symmetric potential V ( r ) , r 1x1, where x denotes the relative coordinate...

Bc AND HEAVY MESON SPECTROSCOPY IN THE LOCAL APPROXIMATION OF THE SCHRÖDINGER EQUATION WITH RELATIVISTIC KINEMATICS

“…for heavy quarkonia (hydrogen-like trial functions) [23]. All these predictions for the masssquared difference are independent of the mass of the particles which constitute the bound state.…”

“…The spin dependent correction (25), which is responsible for the hyperfine splitting of the mass levels, in the short-range is generally used in the form for S-wave (L = 0) (cf. e.g., [23,30]):…”

“…We also present solution for the Schrödinger equation to determine the bound state masses of the cc, bb, and cb mesons taking into account the spinspin, spin-orbit and tensor interactions [22][23][24][25][26][27][28][29]. The spin effects are treated as perturbation to the static potential.…”

Untitled

“…This has been illustrated by considering the commutator of the Hamiltonian of the spinless Salpeter equation-which represents the first step from the nonrelativistic Schrödinger formalism towards incorporation of relativistic effects and which is used for the semirelativistic description of the spin-averaged mass spectra of bound states of "constituent" quarks within the framework of potential models (for analyses of these attempts see, e.g., Refs. [23,24,25])-and the generator of dilations.…”

Hadron spectra from a non-relativistic model with confining harmonic potential

“…Non-trivial connections between these relativistic potential models and rigorous numerical results from lattice QCD have been demonstrated [16]. Some recent works have tried to understand why the non-relativistic treatment works and allows useful predictions even for relativistic systems [17,18].…”