Energies of quark-antiquark systems, the Cornell potential, and the spinless Salpeter equation

Abstract

Energy eigenvalues for heavy-quarkonium and heavy-light systems are determined from the spinless Salpeter equation for the Cornell potential. These are calculated by diagonalizing the matrix representation of the Hamiltonian operator in a basis set constructed from the products of centrifugal barrier factors, Laguerre polynomials, and a common exponential. The Salpeter eigenvalues are compared with eigenvalues obtained from Schrodinger's equation and with spin-averaged experimental results. We present analytic expressions for the matrix elements of both the Coulomb and linear parts of the Cornell potential. We also present analytic results for the matrix elements of the Schrodinger kinetic energy operator. Thus, the Schrodinger problem can also be treated as a matrix diagonalization problem. The relativistic kinetic energy operator is evaluated in momentum space. New expressions are derived for the Fourier transforms of the S-and P-state radial functions. We find that the measured energies of the heavy-quark systems are better fit by Salpeter's equation than by Schrodinger's, in agreement with an earlier calculation of Jacobs, Olsson, and Suchyta. We also find this to be true for B-flavor and charmed mesons.

MASS SPECTRA OF HEAVY QUARKONIA AND Bc DECAY CONSTANT FOR STATIC SCALAR-VECTOR INTERACTIONS WITH RELATIVISTIC KINEMATICS

“…The observed mass M Bc = 6.40 ± 0.39 ± 0.13 GeV has inspired new theoretical interest in the problem [23][24][25][26][27][28][29][30][31]. Some preliminary estimates of the cb bound state masses have been made devoted to the description of the charmonium and bottomonium properties.…”

Lorentz-boosted description of a heavy quarkonium

“…(14) we see that this amplitude is a function of only four-dimensional transfered momentum. As we can see from (6,16), a relation between the potentials in different frames can be extracted from the transformation of the gluon momentum k = q−p. In a rest frame, the vector k has a negligible k 0 component (instantaneous approximation), so…”

“…After some trivial algebra, (16) reduces to a simple Schroedinger equation (6). In the front form (16) reduces to the result obtained in [1],…”

“…Quarkonium has been very well understood in the rest frame [9][10][11][12][13][14][15][16][17][18][19][20][21], even for nonzero temperature [22][23][24][25][26][27][28][29]. Experimentally there are data for quarkonia production both on the proton and nuclear targets (see e.g.…”

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

“…Studies with phenomenological potentials, like the Cornell one, but considering a relativistic kinetic energy term, are able to describe the observed spectra of heavy and light hadron systems [14,15]. Non-trivial connections between these relativistic potential models and rigorous numerical results from lattice QCD have been demonstrated [16].…”