[Paper Review] Where is M-theory in the space of scattering amplitudes?

Andrea L. Guerrieri, Harish Murali+2 more|arXiv (Cornell University)|Nov 30, 2022

Atomic and Subatomic Physics Research4 citations

TL;DR

This paper uses the S-matrix bootstrap to explore the space of unitary, analytic, crossing-symmetric, and supersymmetric graviton scattering amplitudes in 9D, 10D, and 11D, focusing on the Wilson coefficient $\alpha$ that controls the leading correction to maximal supergravity. It finds that the minimal $\alpha$ in extremal amplitudes is very close to, but not equal to, the value in M-theory, suggesting possible inelastic effects beyond the current numerical framework.

ABSTRACT

We use the S-matrix bootstrap to carve out the space of unitary, analytic, crossing symmetric and supersymmetric graviton scattering amplitudes in nine, ten and eleven dimensions. We extend and improve the numerical methods of our previous work in ten dimensions. A key new tool employed here is unitarity in the celestial sphere. In all dimensions, we find that the minimal allowed value of the Wilson coefficient $α$, controlling the leading correction to maximal supergravity, is very close but not equal to the minimal value realized in Superstring theory or M-theory. This small difference may be related to inelastic effects that are not well described by our numerical extremal amplitudes. Although $α$ has a unique value in M-theory, we found no evidence of an upper bound on $α$ in 11D.

Motivation & Objective

• To determine the allowed range of the Wilson coefficient $\alpha$ in maximal supergravity corrections across 9D, 10D, and 11D using the S-matrix bootstrap.
• To investigate whether M-theory corresponds to a unique extremal point in the space of scattering amplitudes.
• To assess the role of unitarity and Regge behavior in constraining UV completions of supergravity.
• To compare numerical extremal amplitudes with predictions from string theory and M-theory.
• To explore the tension between unitarity saturation and high-spin resonance behavior in finite-N approximations.

Proposed method

• Uses a crossing-symmetric, unitary, and analytic ansatz for the scalar amplitude $T(s,t,u)$ with variables $\alpha_{(abc)}$ parameterizing UV corrections.
• Applies unitarity constraints on the celestial sphere via partial-wave projections, enforcing $2\,{\rm Im}\,\mathbb{t} - \mathbb{t}^\dagger\mathbb{t} \succeq 0$.
• Employs a numerical bootstrap approach with finite $N$ to approximate the infinite-dimensional space of amplitudes.
• Maps complex energy variables to the unit disk via $\rho_z$ to control analyticity and large-energy behavior.
• Uses sum rules and spin decomposition to extract contributions from individual partial waves to $\alpha$.
• Compares results with perturbative and non-perturbative string theory predictions, particularly in 11D M-theory and 10D type IIA/B strings.

Experimental results

Research questions

• RQ1 Is the minimal value of the Wilson coefficient $\alpha$ in 11D supergravity equal to the value predicted by M-theory?
• RQ2 Does the space of extremal amplitudes in 9D and 11D exhibit a unique upper or lower bound on $\alpha$?
• RQ3 How do unitarity constraints and high-spin resonances interact in finite-N bootstrap approximations?
• RQ4 To what extent do spin-by-spin contributions to $\alpha$ in extremal amplitudes match those in perturbative string theory?
• RQ5 Can the bootstrap framework distinguish between M-theory and other UV completions of supergravity?

Key findings

• The minimal value of $\alpha$ in 11D extremal amplitudes is $\alpha \approx 0.1028$, matching the M-theory prediction $\alpha = \frac{(2\pi)^2}{3 \cdot 2^7} \approx 0.1028$, but not exactly equal due to small numerical discrepancies.
• In 11D, no upper bound on $\alpha$ is found, indicating the space of allowed amplitudes extends beyond M-theory.
• The minimal $\alpha$ in 10D extremal amplitudes is slightly below the string theory minimum of $\approx 0.1389$ for type IIB, suggesting possible non-perturbative effects.
• Spin-0 partial waves contribute about 84% of the total $\alpha$ in perturbative string theory, and the first four spins capture 99% of the value, indicating dominance of low-spin contributions.
• The bootstrap method reveals a tension between unitarity saturation and the inclusion of high-spin resonances as $N$ increases, indicating limitations in finite-$N$ approximations.
• The extremal amplitudes in 9D and 11D exhibit qualitatively similar resonance structures to those in 10D, with resonances entering the physical sheet as $N$ increases.