  We report recent theoretical progress on a polarization asymmetry in the
proton-deuteron Drell-Yan process with a polarized-deuteron target and the
tensor-polarized structure function $b_1$. Experimental measurements are
possible at JLab for $b_1$ and at Fermilab for the Drell-Yan process. First, we
show a theoretical estimate for the proton-deuteron Drell-Yan asymmetry in the
Fermilab-E1039 experiment. We evolved tensor-polarized parton distribution
functions, which explain existing HERMES $b_1$ data, at $Q^2=2.5$ GeV$^2$ to
the $Q^2$ range of the Fermilab Drell-Yan measurements. Then, we predicted that
the asymmetry is of the order of a few percent. The Drell-Yan experiment has an
advantage to probe the tensor-polarized antiquark distributions, which were
suggested by the HERMES experiment as a finite sum for $b_1$ ($\int dx b_1 (x)
\ne 0$). Second, we predicted $b_1$ for the JLab experiment by the standard
convolution model of the deuteron. Our theoretical $b_1$ structure function
seems to be much different from the HERMES data. Furthermore, a significant
distribution exists at very large $x$ ($>1$) beyond the kinematical limit
$x_{max}=1$ for the proton. Because the standard deuteron-model estimate is
much different from the HERMES data, there could be an interesting development
as a new hadron-physics field if future JLab data will be much different from
our conventional prediction.

High Energy Physics - Phenomenology;High Energy Physics - Experiment;High Energy Physics - Lattice;Nuclear Theory