  We study the analyticity properties of amplitudes in theories with nonlocal
vertices of the type occurring in string field theory and a wide class of
nonlocal field theory models. Such vertices are given in momentum space by
entire functions of rapid decay in certain (including Euclidean) directions
ensuring UV finiteness but are necessarily of rapid increase in others. A
parametric representation is obtained by integrating out the loop (Euclidean)
momenta after the introduction of generalized Schwinger parameters. Either in
the original or parametric representation, the well-defined resulting
amplitudes are then continued in the complex space of the external momenta
invariants. We obtain the alternative forms of the Landau equations determining
the singularity surfaces showing that the nonlocal vertices serve as UV
regulators but do not affect the local singularity structure. As a result the
full set of singularities known to occur in local field theory also occurs
here: normal and anomalous thresholds as well as acnodes, crunodes, and cusps
that may under certain circumstances appear even on the physical sheet.
Singularities of the second type also appear as shown from the parametric
representation. We obtain the general Cutkosky discontinuity rule for
encircling a singularity by employing contour deformations only in the finite
plane. The unitarity condition (optical theorem) is then discussed as a special
application of the rule across normal thresholds and the hermitian analyticity
property of amplitudes.
