  We perform the analysis of quark-antiquark Reggeon exchange in meson-meson
scattering, in the framework of the gauge/gravity correspondence in a confining
background. On the gauge theory side, Reggeon exchange is described as
quark-antiquark exchange in the t channel between fast projectiles. The
corresponding amplitude is represented in terms of Wilson loops running along
the trajectories of the constituent quarks and antiquarks. The paths of the
exchanged fermions are integrated over, while the "spectator" fermions are
dealt with in an eikonal approximation. On the gravity side, we follow a
previously proposed approach, and we evaluate the Wilson-loop expectation value
by making use of gauge/gravity duality for a generic confining gauge theory.
The amplitude is obtained in a saddle-point approximation through the
determination near the confining horizon of a Euclidean "minimal surface with
floating boundaries", i.e., by fixing the trajectories of the exchanged quark
and antiquark by means of a minimisation procedure, which involves both area
and length terms. After discussing, as a warm-up exercise, a simpler problem on
a plane involving a soap film with floating boundaries, we solve the
variational problem relevant to Reggeon exchange, in which the basic geometry
is that of a helicoid. A compact expression for the Reggeon-exchange amplitude,
including the effects of a small fermion mass, is then obtained through
analytic continuation from Euclidean to Minkowski space-time. We find in
particular a linear Regge trajectory, corresponding to a Regge-pole singularity
supplemented by a logarithmic cut induced by the non-zero quark mass. The
analytic continuation leads also to companion contributions, corresponding to
the convolution of the same Reggeon-exchange amplitude with multiple elastic
rescattering interactions between the colliding mesons.
