Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Where is String Theory?

Andrea Guerrieri, Joao Penedones, Pedro Vieira

Abstract

We use the S-matrix bootstrap to carve out the space of unitary, crossing symmetric and supersymmetric graviton scattering amplitudes in ten dimensions. We focus on the leading Wilson coefficient $α$ controlling the leading correction to maximal supergravity. The negative region $α<0$ is excluded by a simple dual argument based on linearized unitarity (the desert). A whole semi-infinite region $α\gtrsim 0.14$ is allowed by the primal bootstrap (the garden). A finite intermediate region is excluded by non-perturbative unitarity (the swamp). Remarkably, string theory seems to cover all (or at least almost all) the garden from very large positive $α$ -- at weak coupling -- to the swamp boundary -- at strong coupling.

Where is String Theory?

Abstract

We use the S-matrix bootstrap to carve out the space of unitary, crossing symmetric and supersymmetric graviton scattering amplitudes in ten dimensions. We focus on the leading Wilson coefficient controlling the leading correction to maximal supergravity. The negative region is excluded by a simple dual argument based on linearized unitarity (the desert). A whole semi-infinite region is allowed by the primal bootstrap (the garden). A finite intermediate region is excluded by non-perturbative unitarity (the swamp). Remarkably, string theory seems to cover all (or at least almost all) the garden from very large positive -- at weak coupling -- to the swamp boundary -- at strong coupling.

Paper Structure

This paper contains 11 sections, 74 equations, 9 figures, 1 table.

Table of Contents

  1. Super Unitarity
  2. One Loop Unitarization
  3. Dispersive formula for $\alpha$
  4. Superstring theory
  5. Ansatz Details
  6. Large Spin Conditions
  7. Large Energy Conditions
  8. A simple example of matching
  9. The expansion of $J^{bc}_n(s)$ integrals
  10. Fits in $L$ and in $N$
  11. Inverse Amplitude Method

Figures (9)

  • Figure 1: Minimum $\alpha_\text{min}(N,L)$. We see that the curves nicely converge towards a plateau whenever $L$ is large enough. The larger $N$ is, the further we need to go in $L$ to reach this plateau. For each value of $N$ we extrapolate these plateaus to estimate $\alpha_\text{min}(N,\infty)$ which we plot in the next figure.
  • Figure 2: Minimum $\alpha_\text{min}(N,\infty)$ obtained by extrapolating the various plateaus in figure \ref{['AllNs']}. We estimate the error bars here by scanning over a large number of such fits as explained in appendix \ref{['numDet']}. We then extrapolate these points to estimate $\alpha^\text{Boot}_\text{min}=\alpha_\text{min}(\infty,\infty) \simeq 0.13$ with an uncertainty represented by the green strip. It nicely embraces the strong coupling string prediction depicted by the solid blue line.
  • Figure 3: String Theory covers all or almost all the allowed quantum gravity theory space.
  • Figure 4: Spin zero phase shift as we increase $N$ from $18,\dots,23$ (in gray) until $24$ (in red). Between $N=20$ and $N=21$ a zero enters the physical sheet -- it is the lightest resonance we encounter. Our best numerics with $N=24$ seem to be close to converging in this energy range and hint at a mass of around $m^2 \ell_P^2 \simeq 3.2 + 0.3 i$.
  • Figure 5: Integration contour leading to equation \ref{['idCauchy']}. The large arc does not contribute because $g(z)\to 0$ when $|z|\to \infty$.
  • ...and 4 more figures