  There are four new structure functions for the spin-1 deuteron in comparison
with the ones for the spin-1/2 proton, and they are called $b_1$, $b_2$, $b_3$,
and $b_4$. The twist-2 structure functions $b_1$ and $b_2$ are expressed by
tensor-polarized parton distribution functions in the deuteron. HERMES
measurements of $b_1$ are much different from the prediction of the standard
deuteron model with D-state admixture. It indicates that the structure
functions $b_1$ and $b_2$ probe an interesting new aspect in the deuteron.
There is an approved experiment at JLab to measure $b_1$ and it is expected to
start in 2019. On the other hand, the measurement of tensor-polarized
distributions is under consideration at Fermilab by the Drell-Yan process with
the unpolarized proton beam and tensor-polarized deuteron target. It is
expected to provide crucial information on tensor-polarized antiquark
distributions. Since the distributions are small quantities, it is important to
estimate the tensor-polarized spin asymmetry theoretically to find experimental
feasibility for an actual proposal at Fermilab.
