Comparison of transverse single-spin asymmetries for forward ${\pi }^0$ production in polarized $pp$, $p\mathrm{A}\mathrm{l}$ and $p\mathrm{A}\mathrm{u}$ collisions at nucleon pair c.m. energy $\sqrt{s_{\mathrm{N}\mathrm{N}}}=200$ GeV

The STAR Collaboration reports a measurement of the transverse single-spin asymmetries, $A_N$, for neutral pions produced in polarized proton collisions with protons ($pp$), with aluminum nuclei ($p\mathrm{A}\mathrm{l}$) and with gold nuclei ($p\mathrm{A}\mathrm{u}$) at a nucleon-nucleon center-of-mass energy of 200 GeV. Neutral pions are observed in the forward direction relative to the transversely polarized proton beam, in the pseudo-rapidity region $2.7<\eta <3.8$. Results are presented for ${\pi }^0$s observed in the STAR FMS electromagnetic calorimeter in narrow Feynman x ($x_F$) and transverse momentum ($p_T$) bins, spanning the range $0.17<x_F<0.81$ and $1.7<p_T<6.0$ GeV/$c$. For fixed $x_F<0.47$, the asymmetries are found to rise with increasing transverse momentum. For larger $x_F$, the asymmetry flattens or falls as $p_T$ increases. Parametrizing the ratio $r(A)\equiv A_N(pA)\mathrm{/}{A}_N(pp)=A^P$ over the kinematic range, the ratio $r(A)$ is found to depend only weakly on $A$, with ${\langle}P{\rangle} = -0.027 \pm 0.005$\mathrm{.}Nosignificantdifferencein$P$ is observed between the low-$p_T$ region, $p_T<2.5$ GeV/$c$, where gluon saturation effects may play a role, and the high-$p_T$ region, $p_T>2.5$ GeV/$c$. It is further observed that the value of $A_N$ is significantly larger for events with a large-$p_T$ isolated ${\pi }^0$ than for events with a non-isolated ${\pi }^0$ accompanied by additional jet-like fragments. The nuclear dependence $r(A)$ is similar for isolated and non-isolated ${\pi }^0$ events.