Recent results from STAR for parton distribution functions at low and high $x$ in proton-proton collisions

Abstract: According to perturbative quantum chromodynamic calculations, in $pp$ collisions at $\sqrt{s} = $~200 and 510 GeV studied at RHIC, jet production in mid-pseudorapidity, $|\eta| <$ 1, is dominated by quark-gluon and gluon-gluon scattering processes. Therefore jets at RHIC are direct probes of the gluon parton distribution functions (PDFs) for momentum fractions 0.01 $<x<$ 0.5. Moreover, $W$ boson cross section ratio, $\sigma(W^+)/\sigma(W^-)$, in $pp$ collisions at $\sqrt{s} = 510$~GeV, is an effective tool to explore anti-quark PDFs, $\bar{d}/\bar{u}$. Last but not least, di-$\pi^{0}$ correlation in forward pseudorapidity, $2.6 < \eta <4.0$, is an important indication of the non-linear gluon dynamics at low $x$ where the gluon density is high in protons and nuclei. In this proceeding, we present recent STAR results of mid-pseudorapidity inclusive jet cross-sections at $\sqrt{s} =$~200 and 510 GeV in $pp$ collisions, $W$ boson cross-section ratios at $\sqrt{s} = 510$~GeV in $pp$ collisions, and forward di-$\pi^0$ correlations in $pp$, $p\textrm{Al}$ and $p\textrm{Au}$ collisions at $\sqrt{s_{\textrm{\tiny NN}}} = 200$~GeV.