Polarizations of the prompt J/psi's produced in the e(+)e(-) annihilat
ion at CLEO energy (root s = 10.58 GeV) are considered in the framewor
k of the nonrelativistic quantum chromodynamics (NRQCD) factorization
formalism. We find that the J/psi polarization has a strong dependence
on the production mechanism. At CLEO energy, the most dominant J/psi,
production mechanism throughout most of phase space is the color-sing
let mechanism, e(+)e(-) --> J/psi, + gg, for which J/psi's are highly
longitudinally polarized. On the other hand, the dominant J/psi produc
tion mechanism at the upper endpoint of the J/psi energy distribution
is the color-octet process, e(+)e(-) --> (c (c) over bar)((8)) + g, fo
r which J/psi's are almost unpolarized. Thus, the measurement of the p
olarization of the endpoint J/psi's is another test of the color-octet
mechanism and a constraint on the NRQCD matrix elements.