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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.

Theoretical predictions on polarization asymmetry for Drell-Yan process with spin- one deuteron and tensor-polarized structure function $b_1$