As is well known, the s-channel crossed ladder amplitudes calculated in the eikonal approximation (1) present a number of desirable features: i) in the limit of vanishing exchanged (( gluon )) mass /~, these on-shell amplitudes develop poles in the centre-ofmass energy squared s, which, in the nonrelativistic limit of a Coulomb potential, reproduce the Balmer formula; ii) as the invariant momentum transfer t becomes very large, these amplitudes develop a Regge behaviour t ~(*~ which, in the ~3-model, agrees with fhe known behaviour of the ordinary ladder for s>> #2; iii) as s-~ co with t fixed, for vector-gluon exchange these amplitudes show a purely diffractive behaviour; furthermore, iv) the eikonal approximation corresponds to a kind of semi-classical picture (2). Given that many of the presently successful models are based on semiclassical intuit ion (parton models, droplet and optical models), in view of i)-iv) one can reasonably hope that an eikonal formalism might be the appropriate tool for constructing a field theory of hadrons. Unfortunately, available eikonal approximations suffer from some drawbacks, which become particularly serious when radiative corrections and other renormalization effects are included. In order to appreciate these difficulties, consider the simple, superrenorrealizable interaction g~p~p~/2 between a scalar (( quark )) + of mass m and a scalar (~ gluon )) (I) of mass/~. The ~-propagator in the presence of an external field A(x) satisfies the equation