Transverse Single Spin Asymmetries of charged hadrons at forward and backward rapidity from $p^{\uparrow }+p$, $p^{\uparrow }+\mathrm{A}\mathrm{l}$, and $p^{\uparrow }+\mathrm{A}\mathrm{u}$ collisions in PHENIX

Transverse Single Spin Asymmetries (TSSAs) in transversely polarized proton-proton collisions ($p^{\uparrow }+p$) have been a fruitful source for studying the spin structure of the proton. In the 2015 RHIC data taking periods, collisions of polarized protons with nuclei ($p^{\uparrow }+A$) were made for the first time. The measurements of TSSAs in $p^{\uparrow }+p$ and $p^{\uparrow }+A$ collisions can provide a unique opportunity to investigate the origin of TSSA in a gluon-rich target nucleus and provide a tool to study nuclear effects in $p+A$ collisions. This presentation will report PHENIX results of TSSAs for charged hadrons ($h^{\pm }$) at forward and backward rapidity ($1.4<\mathrm{\mid }\eta \mathrm{\mid }<2.4$) over the transverse momentum ranges $1.25<p_T<7.0\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}c$ and Feynman-$x$ ranges ($-0.2<x_F<0.2$) from $p^{\uparrow }+p$, $p^{\uparrow }+\mathrm{A}\mathrm{l}$, and $p^{\uparrow }+\mathrm{A}\mathrm{u}$ collisions at $\sqrt{s_{\mathrm{N}\mathrm{N}}}=200$ GeV.