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Summary

This study introduces a new crossing symmetric dispersion relation for quantum field theories, enabling three-channel symmetry. This method derives new inequalities to locate string states and generalize bounds.

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Area of Science:

  • Theoretical physics
  • Quantum field theory
  • String theory

Background:

  • Standard dispersion relations in quantum field theories lack full crossing symmetry.
  • Existing methods are limited in analyzing multi-channel scattering processes.

Purpose of the Study:

  • To develop a crossing symmetric dispersion relation applicable to quantum field theories.
  • To derive new inequalities for effective field theories and string theory amplitudes.
  • To establish a generalized Froissart bound valid at all energies.

Main Methods:

  • A novel dispersion relation in a transformed variable 'z' is employed.
  • The relation utilizes a parametric cubic transformation of Mandelstam invariants (s, t, u).
  • Geometric rotation in the complex z-plane ensures manifest three-channel crossing symmetry.

Main Results:

  • Simple derivations of known positivity conditions and null constraints for effective field theories.
  • New nonperturbative inequalities are derived, providing two-sided bounds.
  • The first massive string state is located from the four-dilaton amplitude in type II string theory.
  • A generalized numerical Froissart bound is obtained.

Conclusions:

  • The new dispersion relation provides a powerful tool for analyzing scattering processes with full crossing symmetry.
  • The derived inequalities offer new constraints on effective field theories and string theory.
  • This approach unifies the study of scattering amplitudes, positivity conditions, and energy bounds.