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Positive Geometry for Stringy Scalar Amplitudes.

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We introduce the associahedral grid, a new positive geometry that captures string theory's full dependence on α′ for scalar and pion amplitudes. This geometry reveals infinite resonance structures and connects field theory amplitudes in a novel way.

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Area of Science:

  • Theoretical Physics
  • String Theory
  • High Energy Physics

Background:

  • The Kawai-Lewellen-Tye (KLT) kernel is fundamental in relating gravity and gauge theories in string theory.
  • Understanding the α′ dependence of scattering amplitudes is crucial for quantum field theory and string theory.
  • Positive geometries offer a framework for organizing scattering amplitudes.

Purpose of the Study:

  • Introduce a new positive geometry, the associahedral grid.
  • Provide a geometric realization of the inverse string theory KLT kernel.
  • Capture the full α′ dependence of stringified amplitudes for biadjoint scalar ϕ³ theory and NLSM pions.

Main Methods:

  • Developed the associahedral grid as a novel positive geometry.
  • Demonstrated its ability to represent stringified amplitudes for specific theories.
  • Analyzed the α′→0 limit to connect with field theory amplitudes.
  • Investigated the emergence of the kinematic δ shift within this geometric framework.

Main Results:

  • The associahedral grid geometrically realizes the inverse string theory KLT kernel.
  • It successfully captures the full α′ dependence for ϕ³ and NLSM pion amplitudes.
  • The geometry reveals stringy features like infinite resonance structures.
  • The kinematic δ shift naturally arises as a leading contribution.

Conclusions:

  • Positive geometries can extend beyond rational functions to encode stringy amplitude features.
  • The associahedral grid provides a unified geometric approach to connect string theory and field theory amplitudes.
  • This framework offers new insights into the relationship between scalar Tr(ϕ³) and NLSM pion amplitudes.