Scalar field theory limits of bosonic string amplitudes

TL;DR

This work develops a detailed prescription to extract scalar field theories from the zero-slope limit of bosonic open string theory, using the Schottky parametrization to analyze multiloop amplitudes up to two loops. By matching string couplings to field-theory couplings and translating string moduli into Schwinger parameters, the authors reproduce cubic and quartic scalar interactions in their renormalizable dimensions, with correct planar and non-planar color structures. They show that different corners of moduli space correspond to distinct Feynman diagrams, yielding correct combinatorial and normalization factors, thereby validating the string master formula as a unified tool for field-theory amplitudes. The results lay groundwork for extending the technique to gauge theories, while noting technical and conceptual challenges in transitioning to Yang–Mills and spin-1 states.

Abstract

We describe in detail the techniques needed to compute scattering amplitudes for colored scalars from the infinite tension limit of bosonic string theory, up to two loops. These techniques apply both to cubic and quartic interactions, and to planar as well as non-planar diagrams. The resulting field theories are naturally defined in the space-time dimension in which they are renormalizable. With a careful analysis of string moduli space in the Schottky representation we determine the region of integration for the moduli, which plays a crucial role in the derivation of the correct combinatorial and color factors for all diagrams.

Figures (11)

• Figure 1: Two tree--level six--point diagrams with cubic vertices.
• Figure 2: A sample tree--level six--point diagram with quartic vertices.
• Figure 3: The annulus in the Schottky representation.
• Figure 4: Multi--leg one--loop one--particle--irreducible diagram in a $\Phi^3$ theory.
• Figure 5: Multi--leg one--loop one--particle--irreducible diagram in a $\Phi^4$ theory.