Comment on "Relativistic description of quark-antiquark bound states"

Abstract

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...

Semirelativistic Hamiltonians of apparently nonrelativistic form

“…A similar observation has already been made in Ref [14]. within a slightly different context[15,16].…”

Electric polarizability of mesons in semirelativistic quark models

“…The derivation of the relativistic corrections to the static interaction potential from the corresponding two-particle scattering problem has been thoroughly analyzed in Refs. [24,25,26] and extensively reviewed in Ref. [13].…”

Nonrelativistic Quark–antiquark Potential: Spectroscopy of Heavy-Quarkonia and Exotic Susy Quarkonia

“…If the coefficients are calculated at tree level; i.e., c 3 (µ, m) = 1, d(µ) = 0, the potential reduces to the Eichten-Feinberg result [66,67]. And if these coefficients are expanded to order α s (µ) then reduced to a one-loop quarkonium spin-spin interaction in the nonrelativistic case [68][69][70] which is responsible for the hyperfine splitting of the mass levels [71][72][73][74][75][76][77][78][79][80][81][82][83][84]…”