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...
COMMENTS Comments are short papers which criticize or correct papers of other authors previously published in the Physical Review. EachComment should state clearly to which paper it refers and must be accompanied by a brief abstract. The same publication schedule as for regular articles is followed, and page proofs are sent to authors.By means of an effective-Hamiltonian method we reconsider the derivation of the effective interaction in a fermion-antifermion system. Furthermore, we point out some errors of Gara and co-workers in their treatment of fermion-antifermion bound states by solving the reduced Salpeter equation in configuration space.PACS number(s): 12.40. Qq, ll.lO.St, In two recent papers [1,2] Gara and co-workers investigated the relativistic description of bound states of quark-antiquark pairs. Their analysis was based on the reduced Salpeter equation. The Salpeter equation [3] is derived from the Bethe-Salpeter equation [4] upon eliminating any dependence on timelike variables in a suitable manner. Some standard and plausible approximations, such as the restriction to positive-energy solutions, then lead to the reduced Salpeter equation.In this Comment we would like to present an alternative way of treating fermion-antifermion bound states relativistically, namely, by construction of an effective Hamiltonian for the two-particle system under consideration. The procedure advocated for consists of two main steps [5,6].(1) Compute the effective interaction potential between the bound-state constituents (at least to the extent you trust in perturbation theory) from the elastic scattering of the involved particles, more precisely, from the Fourier transform o f the corresponding transition amplitude [7,81. -(2) Use this potential in a multiparticle Schrijdinger equation with relativistically correct kinetic Hamiltonian in order to determine the energy eigenvalues and corresponding eigenstate vectors of the bound state under consideration.Obviously, this effective-Hamiltonian method might be regarded as the relativistic generalization of the description of fermion-antifermion bound states in terms of nonrelativistic potential models. As far as the incorporation of relativistic kinematics is concerned, it provides a description of bound states which is of equal quality as the reduced Salpeter equation adopted in Refs. [1,2]. The obvious advantage of our approach is its physical transparency.The basic idea of the proposed effective-Hamiltonian method [5,6] is to approximate by a potential the (perturbatively accessible part of the) interaction between particles which in fact are described by some quantum field theory. T o this end consider the elastic scattering of the involved fermion f and antifermion 7 (with masses m , and m,, respectively). Expressed in terms of Dirac spinors u ( p ,~) and u ( p ,~) , the general form of the corresponding transition amplitude T is where r i , i = 1,2, represent some Dirac matrices. The (unspecified) interaction kernel, which depends on the dynamics of the theory respon...
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