Nonlocal vertices and analyticity: Landau equations and general Cutkosky rule

Abstract

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.