Inclusive Production of $h_c(h_b)$ States via $e^{+}e^{-}$ Annihilation

We calculate the inclusive production of $h_c(h_b)$ at $e^+e^-$ colliders with the center-of-mass energy from the CLEO-c energy to $Z^0$ boson mass at leading order of nonrelativistic QCD. At $Z^0$ boson mass, the cross sections are $39\sim 703$fb for $h_c(1p)$, $37\sim 61$fb for $h_b(1p)$ and $44\sim 73$fb for $h_b(2p)$. At the B-factory, it is $86\sim212$fb for $h_c(1p)$. For $h_c$ at the CLEO-c and $h_b(1p,2p)$ at the B-factory, the perturbative QCD expansion is not good and the results are much smaller than the experimental measurements. It is clearly shown in all the results that the color-octet state $(^1S_0^8)$ contributes dominantly while the color-singlet state $(^1P_1^1)$ contributes small or even negative part, in contrast to the case of inclusive $J/\psi$ production where the color-singlet state is found to contribute dominantly.