Unraveling the proton spin composition, comprising intrinsic spins and orbital angular momenta of quarks and gluons, stands as one of the most fundamental and challenging questions in QCD. Tremendous progress has been made since the first surprising result by the EMC experiment in the late 1980s, which significantly contribute to our understanding of this question. However, the helicity distributions of strange quarks and anti-quarks are still not well constrained by the experimental data.
Measurement of the longitudinal double spin asymmetry, $A_{LL}$, of $\Lambda$, $\overline{\Lambda}$ and $K_S^0$ in the longitudinally polarized proton-proton collisions may shed light on the strange quark and anti-quark helicity distributions. In addition, measuring the longitudinal spin transfer $D_{LL}$ of $\Lambda$ and $\overline{\Lambda}$ is not only sensitive to the helicity distribution of strange quark but can also shed light on the polarized fragmentation functions. Meanwhile, $A_{LL}$ of the inclusive jets tagged with a $\pi^+/\pi^-$ carrying high jet momentum fraction, $z$, in proton-proton collisions can provide further constraints on the gluon helicity distribution in the proton.
In this talk, we will present the results of the $A_{LL}$ for $\Lambda$, $\overline{\Lambda}$ and $K_S^0$, and the inclusive jets tagged with a high-$z$ $\pi^{\pm}$ and the measurements of $D_{LL}$ for $\Lambda$ and $\overline{\Lambda}$. These results are based on the longitudinally polarized proton-proton collisions at $\sqrt{s}=200$ $\rm{GeV}$ collected by the STAR experiment with an integrated luminosity of about 52 $\rm{pb^{-1}}$.