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Multi-loop open string amplitudes and their field theory limit

Lorenzo Magnea, Sam Playle, Rodolfo Russo, Stefano Sciuto

TL;DR

The paper develops a systematic method to extract the field-theory limit of multi-loop string amplitudes by mapping string moduli near degeneration to Schwinger proper-times that label Feynman diagram topologies. Using the Schottky parametrization, it provides an algorithm to decompose open-string amplitudes on D-branes into per-diagram field-theory integrands, clarifying how light string states contribute and how the low-energy gauge theory is recovered (including a non-linear covariant gauge). A key technical advance is showing how to handle supermoduli integration in the NS sector by fixing the bosonic quartic invariant $y$ and matching with the appropriate superperiod matrices, with an explicit two-loop example that reproduces a ghost-plus-gluon diagram in background field $B$. The approach yields a diagram-by-diagram dictionary between string theory and low-energy field theory, applicable to higher genus and broader brane setups, and sets the stage for extending to closed strings and renormalization analyses. Overall, the work provides a principled bridge from covariantly quantized string amplitudes to explicit field-theory Feynman integrands and gauge structures, enabling direct cross-checks and guiding future multi-loop and background-field studies.

Abstract

We study the field theory limit of multi-loop (super)string amplitudes, with the aim of clarifying their relationship to Feynman diagrams describing the dynamics of the massless states. We propose an explicit map between string moduli around degeneration points and Schwinger proper-times characterizing individual Feynman diagram topologies. This makes it possible to identify the contribution of each light string state within the full string amplitude and to extract the field theory Feynman rules selected by (covariantly quantized) string theory. The connection between string and field theory amplitudes also provides a concrete tool to clarify ambiguities related to total derivatives over moduli space: in the superstring case, consistency with the field theory results selects a specific prescription for integrating over supermoduli. In this paper, as an example, we focus on open strings supported by parallel D-branes, and we present two-loop examples drawn from bosonic and RNS string theories, highlighting the common features between the two setups.

Multi-loop open string amplitudes and their field theory limit

TL;DR

The paper develops a systematic method to extract the field-theory limit of multi-loop string amplitudes by mapping string moduli near degeneration to Schwinger proper-times that label Feynman diagram topologies. Using the Schottky parametrization, it provides an algorithm to decompose open-string amplitudes on D-branes into per-diagram field-theory integrands, clarifying how light string states contribute and how the low-energy gauge theory is recovered (including a non-linear covariant gauge). A key technical advance is showing how to handle supermoduli integration in the NS sector by fixing the bosonic quartic invariant and matching with the appropriate superperiod matrices, with an explicit two-loop example that reproduces a ghost-plus-gluon diagram in background field . The approach yields a diagram-by-diagram dictionary between string theory and low-energy field theory, applicable to higher genus and broader brane setups, and sets the stage for extending to closed strings and renormalization analyses. Overall, the work provides a principled bridge from covariantly quantized string amplitudes to explicit field-theory Feynman integrands and gauge structures, enabling direct cross-checks and guiding future multi-loop and background-field studies.

Abstract

We study the field theory limit of multi-loop (super)string amplitudes, with the aim of clarifying their relationship to Feynman diagrams describing the dynamics of the massless states. We propose an explicit map between string moduli around degeneration points and Schwinger proper-times characterizing individual Feynman diagram topologies. This makes it possible to identify the contribution of each light string state within the full string amplitude and to extract the field theory Feynman rules selected by (covariantly quantized) string theory. The connection between string and field theory amplitudes also provides a concrete tool to clarify ambiguities related to total derivatives over moduli space: in the superstring case, consistency with the field theory results selects a specific prescription for integrating over supermoduli. In this paper, as an example, we focus on open strings supported by parallel D-branes, and we present two-loop examples drawn from bosonic and RNS string theories, highlighting the common features between the two setups.

Paper Structure

This paper contains 6 sections, 47 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A parametrization for (super) Riemann surfaces
  3. Multiloop string effective actions
  4. The field theory limit
  5. Extracting Feynman diagrams: an example
  6. Conclusions

Figures (1)

  • Figure 1: The three possible degeneration limits of the open string world-sheet for the two-loop vacuum amplitude. Individual propagators are labelled with their Schwinger proper-time variable $t_i$.