The Box Graph In Superstring Theory

TL;DR

This paper introduces an analytic continuation method for interpreting one-loop superstring amplitudes, resolving convergence and singularity issues by reducing them to minimal amplitudes expressed via dispersion relations.

Contribution

It provides a detailed analytic continuation approach that simplifies and generalizes previous methods for understanding superstring one-loop amplitudes.

Findings

Resolved convergence issues of superstring amplitudes

Expressed amplitudes in terms of dispersion relations with spectral densities

Connected superstring amplitudes to superpositions of box graphs in field theories

Abstract

In theories of closed oriented superstrings, the one loop amplitude is given by a single diagram, with the topology of a torus. Its interpretation had remained obscure, because it was formally real, converged only for purely imaginary values of the Mandelstam variables, and had to account for the singularities of both the box graph and the one particle reducible graphs in field theories. We present in detail an analytic continuation method which resolves all these difficulties. It is based on a reduction to certain minimal amplitudes which can themselves be expressed in terms of double and single dispersion relations, with explicit spectral densities. The minimal amplitudes correspond formally to an infinite superposition of box graphs on $\phi ^3$ like field theories, whose divergence is responsible for the poles in the string amplitudes. This paper is a considerable simplification and generalization of our earlier proposal published in Phys. Rev. Lett. 70 (1993) p 3692.