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Positivity of the Veneziano Amplitude in Ten Dimensions

Gareth Mansfield

Abstract

The Veneziano amplitude describing the tree-level scattering of four open superstrings is expected to be consistent with unitarity in ten spacetime dimensions. While this follows indirectly from the no-ghost theorem, a direct proof at the level of the amplitude has only been found for $D\leq 6$. In this article, we close this gap by providing a complete proof for the partial-wave positivity of the Veneziano amplitude in $D\leq 10$, derived directly from its definition in terms of the Euler beta function. We also demonstrate that this proof can be modified to show positivity of a wider family of amplitudes relevant to the $S$-matrix bootstrap.

Positivity of the Veneziano Amplitude in Ten Dimensions

Abstract

The Veneziano amplitude describing the tree-level scattering of four open superstrings is expected to be consistent with unitarity in ten spacetime dimensions. While this follows indirectly from the no-ghost theorem, a direct proof at the level of the amplitude has only been found for . In this article, we close this gap by providing a complete proof for the partial-wave positivity of the Veneziano amplitude in , derived directly from its definition in terms of the Euler beta function. We also demonstrate that this proof can be modified to show positivity of a wider family of amplitudes relevant to the -matrix bootstrap.

Paper Structure

This paper contains 10 sections, 3 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Positivity of the Veneziano Amplitude
  3. Review of Positivity in $D\leq 6$
  4. Positivity in $D\leq 10$
  5. Positivity of $Q^4_{\mu\nu}$
  6. Other Amplitudes
  7. Virasoro-Shapiro
  8. The Massless Hypergeometric Amplitude in $D=5$
  9. Discussion
  10. Proof of the Low-Order $Q^j_{\mu\nu}$ Bound

Key Result

Lemma 1

The $G_k$ coefficients satisfy the following bounds:

Theorems & Definitions (3)

  • Lemma 1
  • Lemma 2
  • Lemma 3