  We calculate the single transverse spin asymmetry $A_N(t)$, for inclusive
neutron production in $pp$ collisions at forward rapidities relative to the
polarized proton in the energy range of RHIC. Absorptive corrections to the
pion pole generate a relative phase between the spin-flip and non-flip
amplitudes, leading to a transverse spin asymmetry which is found to be far too
small to explain the magnitude of $A_N$ observed in the PHENIX experiment. A
larger contribution, which does not vanish at high energies, comes from the
interference of pion and $a_1$-Reggeon exchanges. The unnatural parity of $a_1$
guarantees a substantial phase shift, although the magnitude is strongly
suppressed by the smallness of diffractive $\pi p\to a_1 p$ cross section. We
replace the Regge $a_1$ pole by the Regge cut corresponding to the $\pi\rho$
exchange in the $1^+S$ state. The production of such a state, which we treat as
an effective pole $a$, forms a narrow peak in the $3\pi$ invariant mass
distribution in diffractive $\pi p$ interactions . The cross section is large,
so one can assume that this state saturates the spectral function of the axial
current and we can determine its coupling to nucleons via the PCAC
Goldberger-Treiman relation and the second Weinberg sum rule. The numerical
results of the parameter-free calculation of $A_N$ are in excellent agreement
with the PHENIX data.
