Sivers Mechanism for Gluons

We study T-odd gauge invariantly defined T-odd unintegrated gluon densities in QCD. The average $\perp$ momentum of the gluons in a $\perp$ polarized nucleon can be nonzero due to the Sivers mechanism. However, we find that the sum of the transverse momenta due to the Sivers mechanism from gluons and all quark flavors combined vanishes, i.e. $<{\bf k_{\perp,g}}> + \sum_q < {\bf k_{\perp,q}}> =0$.