1,048 research outputs found
Electric Polarizability of Mesons in Semirelativistic Quark Models
The electric polarizability of mesons, in particular, that of the charged
pion, is studied in the framework of a semirelativistic description of hadrons
as bound states of valence quarks in terms of a Hamiltonian composed of the
relativistic kinetic energy as well as a phenomenological potential describing
the strong interactions between the quarks. The quark-core contribution to the
electric polarizability obtainable in quark models of this kind turns out to be
in the semirelativistic approaches even smaller than in the nonrelativistic
limit.Comment: 13 pages, LaTeX, extended version, to appear in Phys. Lett.
Energy bounds for the spinless Salpeter equation
We study the spectrum of the spinless-Salpeter Hamiltonian H = \beta
\sqrt{m^2 + p^2} + V(r), where V(r) is an attractive central potential in three
dimensions. If V(r) is a convex transformation of the Coulomb potential -1/r
and a concave transformation of the harmonic-oscillator potential r^2, then
upper and lower bounds on the discrete eigenvalues of H can be constructed,
which may all be expressed in the form E = min_{r>0} [ \beta \sqrt{m^2 +
P^2/r^2} + V(r) ] for suitable values of P here provided. At the critical point
the relative growth to the Coulomb potential h(r)=-1/r must be bounded by dV/dh
< 2\beta/\pi.Comment: 11 pages, 1 figur
Quality of Variational Trial States
Besides perturbation theory (which clearly requires the knowledge of the
exact unperturbed solution), variational techniques represent the main tool for
any investigation of the eigenvalue problem of some semibounded operator H in
quantum theory. For a reasonable choice of the employed trial subspace of the
domain of H, the lowest eigenvalues of H usually can be located with acceptable
precision whereas the trial-subspace vectors corresponding to these eigenvalues
approximate, in general, the exact eigenstates of H with much less accuracy.
Accordingly, various measures for the accuracy of the approximate eigenstates
derived by variational techniques are scrutinized. In particular, the matrix
elements of the commutator of the operator H and (suitably chosen) different
operators with respect to degenerate approximate eigenstates of H obtained by
variational methods are proposed as new criteria for the accuracy of
variational eigenstates. These considerations are applied to precisely that
Hamiltonian for which the eigenvalue problem defines the well-known spinless
Salpeter equation. This bound-state wave equation may be regarded as (the most
straightforward) relativistic generalization of the usual nonrelativistic
Schroedinger formalism, and is frequently used to describe, e.g., spin-averaged
mass spectra of bound states of quarks.Comment: LaTeX, 7 pages, version to appear in Physical Review
Instantaneous Bethe-Salpeter Equation: Analytic Approach for Nonvanishing Masses of the Bound-State Constituents
The instantaneous Bethe-Salpeter equation, derived from the general
Bethe-Salpeter formalism by assuming that the involved interaction kernel is
instantaneous, represents the most promising framework for the description of
hadrons as bound states of quarks from first quantum-field-theoretic
principles, that is, quantum chromodynamics. Here, by extending a previous
analysis confined to the case of bound-state constituents with vanishing
masses, we demonstrate that the instantaneous Bethe-Salpeter equation for
bound-state constituents with (definitely) nonvanishing masses may be converted
into an eigenvalue problem for an explicitly - more precisely, algebraically -
known matrix, at least, for a rather wide class of interactions between these
bound-state constituents. The advantages of the explicit knowledge of this
matrix representation are self-evident.Comment: 12 Pages, LaTeX, 1 figur
Accuracy of Approximate Eigenstates
Besides perturbation theory, which requires, of course, the knowledge of the
exact unperturbed solution, variational techniques represent the main tool for
any investigation of the eigenvalue problem of some semibounded operator H in
quantum theory. For a reasonable choice of the employed trial subspace of the
domain of H, the lowest eigenvalues of H usually can be located with acceptable
precision whereas the trial-subspace vectors corresponding to these eigenvalues
approximate, in general, the exact eigenstates of H with much less accuracy.
Accordingly, various measures for the accuracy of the approximate eigenstates
derived by variational techniques are scrutinized. In particular, the matrix
elements of the commutator of the operator H and (suitably chosen) different
operators, with respect to degenerate approximate eigenstates of H obtained by
some variational method, are proposed here as new criteria for the accuracy of
variational eigenstates. These considerations are applied to that Hamiltonian
the eigenvalue problem of which defines the "spinless Salpeter equation." This
(bound-state) wave equation may be regarded as the most straightforward
relativistic generalization of the usual nonrelativistic Schroedinger
formalism, and is frequently used to describe, e.g., spin-averaged mass spectra
of bound states of quarks.Comment: LaTeX, 14 pages, Int. J. Mod. Phys. A (in print); 1 typo correcte
Instantaneous Bethe-Salpeter equation: improved analytical solution
Studying the Bethe-Salpeter formalism for interactions instantaneous in the
rest frame of the bound states described, we show that, for bound-state
constituents of arbitrary masses, the mass of the ground state of a given spin
may be calculated almost entirely analytically with high accuracy, without the
(numerical) diagonalization of the matrix representation obtained by expansion
of the solutions over a suitable set of basis states.Comment: 7 page
Stability in the instantaneous Bethe-Salpeter formalism: harmonic-oscillator reduced Salpeter equation
A popular three-dimensional reduction of the Bethe-Salpeter formalism for the
description of bound states in quantum field theory is the Salpeter equation,
derived by assuming both instantaneous interactions and free propagation of all
bound-state constituents. Numerical (variational) studies of the Salpeter
equation with confining interaction, however, observed specific instabilities
of the solutions, likely related to the Klein paradox and rendering (part of
the) bound states unstable. An analytic investigation of this problem by a
comprehensive spectral analysis is feasible for the reduced Salpeter equation
with only harmonic-oscillator confining interactions. There we are able to
prove rigorously that the bound-state solutions correspond to real discrete
energy spectra bounded from below and are thus free of any instabilities.Comment: 23 pages, 3 figures, extended conclusions, version to appear in Phys.
Rev.
Relativistic N-Boson Systems Bound by Oscillator Pair Potentials
We study the lowest energy E of a relativistic system of N identical bosons
bound by harmonic-oscillator pair potentials in three spatial dimensions. In
natural units the system has the semirelativistic ``spinless-Salpeter''
Hamiltonian H = \sum_{i=1}^N \sqrt{m^2 + p_i^2} + \sum_{j>i=1}^N gamma |r_i -
r_j|^2, gamma > 0. We derive the following energy bounds: E(N) = min_{r>0} [N
(m^2 + 2 (N-1) P^2 / (N r^2))^1/2 + N (N-1) gamma r^2 / 2], N \ge 2, where
P=1.376 yields a lower bound and P=3/2 yields an upper bound for all N \ge 2. A
sharper lower bound is given by the function P = P(mu), where mu =
m(N/(gamma(N-1)^2))^(1/3), which makes the formula for E(2) exact: with this
choice of P, the bounds coincide for all N \ge 2 in the Schroedinger limit m
--> infinity.Comment: v2: A scale analysis of P is now included; this leads to revised
energy bounds, which coalesce in the large-m limi
Analytic Quest for Confining Interaction Kernels in Instantaneous Bethe-Salpeter Equations
The Salpeter equation, a standard tool in hadron physics, constitutes a
well-defined three-dimensional approximation to the Bethe-Salpeter formalism
for the description of bound states within quantum field theories. However, if
confinement is implemented in a too careless manner, the Salpeter equation
might predict unstable states where only truly bound states are expected. A
systematic spectral analysis of the Salpeter equation might serve to single out
those Bethe-Salpeter interaction kernels which indeed yield stable bound
states, with associated energy eigenvalues belonging to a real discrete
spectrum bounded from below.Comment: contributed to Quark Confinement and the Hadron Spectrum IX - QCHS IX
(30 August - 3 September 2010, Madrid, Spain), to appear in the proceeding
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