  The HJET Polarized Atomic Hydrogen Gas Jet Target polarimeter (HJET)
polarimeter was designed to measure the absolute polarization of the proton
beams at the Relativistic Heavy Ion Collider. In these measurements, the small
scattering angle elastic $pp$ single $A_N(t)$ and double $A_{NN}(t)$ spin
analyzing powers can be precisely determined. The experimental accuracy
achieved at HJET requires corrections to the $A_N(t)$ parametrization,
conventionally used for such studies. In this paper we evaluate the corrections
to the analyzing powers due to (i) the differences between the electromagnetic
and hadronic form factors and (ii) the $m_p^2/s$ terms in the elastic spin-flip
$pp$ electromagnetic amplitude. The corresponding alterations of the evaluated
hadronic spin-flip amplitudes are about the same as the experimental
uncertainties of the HJET measurements. The proposed corrections may have
implications for the elastic $pp$ forward real-to-imaginary amplitude ratios
$\rho$ determined in unpolarized $pp$ experiments.
