In the high-energy polarized $p+p$ collisions, a spin-involved diffractive particle production mechanism can be understood by measuring transverse single-spin asymmetries $A_{\scriptsize{\textrm{N}}}$ of forward particles that are produced at pseudorapidity larger than 6.
Since $A_{\scriptsize{\textrm{N}}}$ of forward neutron had only been studied in a narrow transverse momentum range of $p_{\scriptsize{\textrm{T}}}<0.4$~GeV/$c$, the RHICf Collaboration has extended the previous measurements up to 1.0~GeV/$c$ at $\sqrt{s}=510$~GeV to study the kinematic dependence of the neutron $A_{\scriptsize{\textrm{N}}}$ in more detail.
The resulting $A_{\scriptsize{\textrm{N}}}$s reach a plateau in the low longitudinal momentum fraction $x_{\scriptsize{\textrm{F}}}$ range, but explicitly increase in magnitude with $p_{\scriptsize{\textrm{T}}}$ in the high $x_{\scriptsize{\textrm{F}}}$ range.
The $A_{\scriptsize{\textrm{N}}}$s show little $x_{\scriptsize{\textrm{F}}}$ dependence in the low $p_{\scriptsize{\textrm{T}}}$ range.
A clear $x_{\scriptsize{\textrm{F}}}$ dependence is observed for higher $p_{\scriptsize{\textrm{T}}}$ range in the intermediate $x_{\scriptsize{\textrm{F}}}$ region.
The results are consistent with the previous measurements at $\sqrt{s}=200$~GeV, which suggests no $\sqrt{s}$ dependence of the neutron $A_{\scriptsize{\textrm{N}}}$.
A theoretical model based on $\pi$ and $a_1$ exchange between two protons could reproduce
the current results only in a limited kinematic region.
An additional mechanism is necessary to understand the measured neutron $A_{\scriptsize{\textrm{N}}}$s over the whole kinematic region.