  Four-particle tree-level scattering amplitudes in string theory are magically
consistent with unitarity, reflected in the non-trivial fact that beneath the
critical dimension, the residues of the amplitudes on massive poles can be
expanded in partial waves with all positive coefficients. While this follows
(rather indirectly) from the no-ghost theorem, the simplicity of the statement
and its fundamental importance for the physical consistency of string theory
begs for a more direct and elementary understanding. In this note we take a
step in this direction by presenting a new expression for the partial wave
coefficients of string amplitudes, given by surprisingly simple double/triple
contour integrals for open/closed strings. This representation allows us to
directly prove unitarity of all superstring theories in $D \leq 6$ spacetime
dimensions, and can also be used to determine various asymptotics of the
partial waves at large mass levels.
