  Intensive theoretical and experimental efforts over the past decade have
aimed at explaining the discrepancy between data for the proton electric to
magnetic form factor ratio, $G_{E}/G_{M}$, obtained separately from cross
section and polarization transfer measurements. One possible explanation for
this difference is a two-photon-exchange (TPEX) contribution. In an effort to
search for effects beyond the one-photon-exchange or Born approximation, we
report measurements of polarization transfer observables in the elastic
$H(\vec{e},e'\vec{p})$ reaction for three different beam energies at a fixed
squared momentum transfer $Q^2 = 2.5$ GeV$^2$, spanning a wide range of the
virtual photon polarization parameter, $\epsilon$. From these measured
polarization observables, we have obtained separately the ratio $R$, which
equals $\mu_p G_{E}/G_{M}$ in the Born approximation, and the longitudinal
polarization transfer component $P_\ell$, with statistical and systematic
uncertainties of $\Delta R \approx \pm 0.01 \mbox{(stat)} \pm 0.013
\mbox{(syst)}$ and $\Delta P_\ell/P^{Born}_{\ell} \approx \pm 0.006
\mbox{(stat)}\pm 0.01 \mbox{(syst)}$. The ratio $R$ is found to be independent
of $\epsilon$ at the 1.5% level, while the $\epsilon$ dependence of $P_\ell$
shows an enhancement of $(2.3 \pm 0.6) %$ relative to the Born approximation at
large $\epsilon$.
