  Conventional superstring perturbation theory based on the world-sheet
approach gives divergent results for the S-matrix whenever the total center of
mass energy of the incoming particles exceeds the threshold of production of
any final state consistent with conservation laws. Two systematic approaches
have been suggested for dealing with this difficulty. The first one involves
deforming the integration cycles over the moduli space of punctured Riemann
surfaces into complexified moduli space. The second one treats the amplitude as
a sum of superstring field theory Feynman diagrams and deforms the integration
contours over loop energies of the Feynman diagram into the complex plane. In
this paper we establish the equivalence of the two prescriptions to all orders
in perturbation theory. Since the second approach is known to lead to unitary
amplitudes, this establishes the consistency of the first prescription with
unitarity.
