  In the high-energy $p+p$ collisions, the transverse single spin asymmetry for
very forward neutron production has been interpreted by an interference between
$\pi$ (spin flip) and $a_1$ (spin non-flip) exchange with a non-zero phase
shift. The $\pi$ and $a_1$ exchange model predicted the neutron asymmetry would
increase in magnitude with transverse momentum ($p_{\scriptsize{\textrm{T}}}$)
in $p_{\scriptsize{\textrm{T}}} < 0.4$ GeV/$c$. In June 2017, the RHICf
experiment installed an electromagnetic calorimeter at the zero-degree area of
the STAR experiment at the Relativistic Heavy Ion Collider and measured the
neutron asymmetry in a wide $p_{\scriptsize{\textrm{T}}}$ range of $0 <
p_{\scriptsize{\textrm{T}}} < 1$ GeV/$c$ from polarized $p+p$ collisions at
$\sqrt{s} = 510$ GeV. The RHICf data allows us to investigate the kinematic
dependence of the neutron asymmetry in detail, which not only can test the
$\pi$ and $a_1$ exchange model in the higher $p_{\scriptsize{\textrm{T}}}$
range but also can study the $\sqrt{s}$ dependence by comparing with the
previous measurements. We present the preliminary result and analysis status of
the neutron asymmetry measured by the RHICf experiment. In order to understand
the RHICf result, a theoretical calculation other than Reggeon exchange will
also be discussed.
