One of the embarrassments of covariant string field theory has been the glaring failure to derive the Shapiro-Virasoro amplitude. Modular invariance appears explicitly violated: either the fundamental region is overcounted an infinite number of times, or it is undercounted because of a missing region. We try to approach this problem from a fresh point of view. Conventional wisdom holds that, in string field theory, the Veneziano amplitude can only be derived either in light-cone string field theory or Wittens string field theory. We show that this firmly held belief is actually wrong, that the Veneziano amplitude can actually be derived using vertices of arbitrary lengths. This is a highly nontrivial calculation. Using third elliptic integrals we show that a series of miracles occurs which allow us to cancel scores of unwanted terms in the measure, leaving us with the correct Koba-Nielsen variable and measure. We give three independent proofs of our result. When we generalize our results to closed-string scattering with arbitrary lengths, we find a new surprise, that we can successfully derive the Shapiro-Virasoro amplitude as long as a crucial four-string interaction term is added. We check by explicit computer calculation that we reproduce the correct region of integration for the four-closed-string amplitude. Crucial to the theory is the existence of the missing four-string tetrahedron graph, which precisely fills the missing integration region. We comment on the implications of this for geometric string field theory. © 1990 The American Physical Society.
