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 Schr\"odinger'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 Schr\"odinger kinetic energy operator. Thus, the Schr\"odinger 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 Schr\"odinger'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.