A remark on the high–energy quark–quark scattering and the eikonal approximation

In this paper we calculate the high–energy quark–quark scattering amplitude, first in the case of scalar QCD, using Fradkin’s approach to derive the scalar quark propagator in an external gluon field and computing it in the eikonal approximation. (This approach was also recently used by Fabbrichesi, Pettorino, Veneziano and Vilkovisky to study the four–dimensional Planckian–energy scattering in gravity.) We then extend the results to the case of “real” (i.e. fermion) QCD, thus deriving again, in a rather direct way, the results previously found by Nachtmann. The abelian case (QED) is also discussed in the Appendix.