  It was found that, using nonrelativistic QCD factorization, the predicted
$\chi_{cJ}$ hadroproduction cross section at large $p_T$ can be negative. The
negative cross sections originate from terms proportional to plus function in
${^{3}\hspace{-0.6mm}P_{J}^{[1]}}$ channels, which are remnants of the infrared
subtraction in matching the ${^{3}\hspace{-0.6mm}P_{J}^{[1]}}$ short-distance
coefficients. In this article, we find that the above terms can be factorized
into the nonperturbative ${^{3}\hspace{-0.6mm}S_{1}^{[8]}}$ soft gluon
distribution function in the soft gluon factorization (SGF) framework.
Therefore, the problem can be naturally resolved in SGF. With an appropriate
choice of nonperturbative parameters, the SGF can indeed give positive
predictions for $\chi_{cJ}$ production rates within the whole $p_T$ region. The
production of $\psi(2S)$ is also discussed, and there is no negative cross
section problem.
