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One-loop matrix elements of effective superstring interactions: $α'$-expanding loop integrands

Alex Edison, Max Guillen, Henrik Johansson, Oliver Schlotterer, Fei Teng

TL;DR

This work develops a systematic framework to construct and analyze one-loop matrix elements in type I and type II superstrings with insertions of higher-derivative operators $D^{2k}F^n$ and $D^{2k}R^n$, by uplifting tree-level amplitudes through forward limits and organizing the α'-expansion using ambitwistor-string ideas. The method leverages linearized propagators that can be recombined into quadratic propagators, preserving one-loop monodromy and KLT relations, and yields explicit four- and five-point results for open and closed strings, including higher-order α' corrections with multiple zeta values. The UV structure and the non-analytic discontinuities of these matrix elements are cross-checked against genus-one degenerations and modular graph forms, while conjectures link homology invariants to universal quadratic-propagator representations at all multiplicities. The results offer a new route to understanding the α'-dependent corrections to the string effective action, provide tests for color-kinematics duality at loop level, and connect to the mathematical structure of modular graph forms and elliptic multiple zeta values at cusp degenerations. This framework sets the stage for exploring higher-point, non-planar, and multi-loop extensions, with potential implications for EFT-hedron-type structures and string-number-theoretic relations of loop amplitudes.

Abstract

In the low-energy effective action of string theories, non-abelian gauge interactions and supergravity are augmented by infinite towers of higher-mass-dimension operators. We propose a new method to construct one-loop matrix elements with insertions of operators $D^{2k} F^n$ and $D^{2k} R^n$ in the tree-level effective action of type-I and type-II superstrings. Inspired by ambitwistor string theories, our method is based on forward limits of moduli-space integrals using string tree-level amplitudes with two extra points, expanded in powers of the inverse string tension $α'$. Similar to one-loop ambitwistor computations, intermediate steps feature non-standard linearized Feynman propagators which eventually recombine to conventional quadratic propagators. With linearized propagators the loop integrand of the matrix elements obey one-loop versions of the monodromy and KLT relations. We express a variety of four- and five-point examples in terms of quadratic propagators and formulate a criterion on the underlying genus-one correlation functions that should make this recombination possible at all orders in $α'$. The ultraviolet divergences of the one-loop matrix elements are crosschecked against the non-separating degeneration of genus-one integrals in string amplitudes. Conversely, our results can be used as a constructive method to determine degenerations of elliptic multiple zeta values and modular graph forms at arbitrary weight.

One-loop matrix elements of effective superstring interactions: $α'$-expanding loop integrands

TL;DR

This work develops a systematic framework to construct and analyze one-loop matrix elements in type I and type II superstrings with insertions of higher-derivative operators and , by uplifting tree-level amplitudes through forward limits and organizing the α'-expansion using ambitwistor-string ideas. The method leverages linearized propagators that can be recombined into quadratic propagators, preserving one-loop monodromy and KLT relations, and yields explicit four- and five-point results for open and closed strings, including higher-order α' corrections with multiple zeta values. The UV structure and the non-analytic discontinuities of these matrix elements are cross-checked against genus-one degenerations and modular graph forms, while conjectures link homology invariants to universal quadratic-propagator representations at all multiplicities. The results offer a new route to understanding the α'-dependent corrections to the string effective action, provide tests for color-kinematics duality at loop level, and connect to the mathematical structure of modular graph forms and elliptic multiple zeta values at cusp degenerations. This framework sets the stage for exploring higher-point, non-planar, and multi-loop extensions, with potential implications for EFT-hedron-type structures and string-number-theoretic relations of loop amplitudes.

Abstract

In the low-energy effective action of string theories, non-abelian gauge interactions and supergravity are augmented by infinite towers of higher-mass-dimension operators. We propose a new method to construct one-loop matrix elements with insertions of operators and in the tree-level effective action of type-I and type-II superstrings. Inspired by ambitwistor string theories, our method is based on forward limits of moduli-space integrals using string tree-level amplitudes with two extra points, expanded in powers of the inverse string tension . Similar to one-loop ambitwistor computations, intermediate steps feature non-standard linearized Feynman propagators which eventually recombine to conventional quadratic propagators. With linearized propagators the loop integrand of the matrix elements obey one-loop versions of the monodromy and KLT relations. We express a variety of four- and five-point examples in terms of quadratic propagators and formulate a criterion on the underlying genus-one correlation functions that should make this recombination possible at all orders in . The ultraviolet divergences of the one-loop matrix elements are crosschecked against the non-separating degeneration of genus-one integrals in string amplitudes. Conversely, our results can be used as a constructive method to determine degenerations of elliptic multiple zeta values and modular graph forms at arbitrary weight.

Paper Structure

This paper contains 52 sections, 167 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Inspiration from ambitwistor strings
  3. Comparison with conventional strings
  4. Linearized versus quadratic propagators
  5. Scope of our method and results
  6. Further lines of motivation
  7. Outline
  8. Review
  9. One-loop amplitudes from ambitwistor strings
  10. Linearized versus quadratic propagators
  11. Disk and sphere integrals
  12. Tree-level effective action of type I and II superstrings
  13. One-loop matrix elements for the superstring effective action
  14. a' uplifts for one-loop amplitudes
  15. Relations among one-loop matrix elements
  16. ...and 37 more sections

Figures (8)

  • Figure 1: Contributions to the one-loop matrix elements with single- and double-insertion of the operator $\alpha'^2 {\rm Tr}(F^4)$ in the open-string effective action (\ref{['seffclosed']}).
  • Figure 2: Left panel: $n$-gon diagram associated with the color factor $c_{12\ldots n}$ in (\ref{['rev.5']}). Right panel: the partial-fraction decomposition to be reviewed in (\ref{['ngonPF']}) relates each $n$-gon with a cyclic orbit of the depicted $(n{+}2)$-point tree-level diagram.
  • Figure 3: Diagrammatic representation of the terms $\sim \zeta_2$ in the left panel and the contribution $\sim \frac{s_{13}}{\ell^2 (\ell{+}k_1)^2}$ to the third line in the $\alpha'$-expansion (\ref{['exsec.7']}) in the right panel.
  • Figure 4: Diagrammatic representation of the two-mass bubble in the left panel and the tadpole diagrams in the right panel for the numerators in (\ref{['exsec.10']}).
  • Figure 5: Diagrammatic representation of the contribution to the $\zeta_2$-order of \ref{['t8defs.9']} with numerators in \ref{['t8defs.9a']}. Other $\zeta$ orders will have the same topologies with different operator insertions in place of $F^4$.
  • ...and 3 more figures