  We calculate the leading twist contribution to near-forward proton-proton
(and proton-antiproton) elastic scattering with large momentum transfer, in the
multiple scattering (Landshoff) mechanism. The amplitude in the near-forward
region is dominated by singlet exchange for all three valence quark-quark
scatterings. We assume the existence of a hard singlet quark-quark amplitude,
which we estimate to be ${\cal O}(\alpha_s^2/t)$. For a three-quark state whose
transverse size is less than $1/\Lambda_{\rm QCD}$, Sudakov resummation
accounts for both approximate $\d \sigma^{pp}/ \d t \sim t^{-8}$ at moderate
$t$, and $\d \sigma^{pp}/ \d t \sim t^{-10}$ at larger $t$. The transition from
approximate $t^{-8}$ to $t^{-10}$ behavior is strongly correlated with the
transverse size of the valence three-quark state in the proton.
