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The Boostless Bootstrap: Amplitudes without Lorentz boosts

Enrico Pajer, David Stefanyszyn, Jakub Supeł

TL;DR

This work develops a boostless bootstrap for massless, luminal particles by analyzing three-particle on-shell amplitudes and enforcing consistent four-particle factorization in Minkowski space. It shows that the presence of a massless spin-2 particle enforces Lorentz-invariant three-point graviton couplings and constrains higher-spin interactions, yielding gravity as universal at low energies and prohibiting S>2 self-interactions or minimal coupling beyond GR. Photon and scalar couplings exhibit boost-breaking structures only under higher-derivative terms, with the four-particle test sharply restricting possible vertices; when gravity is included, these boost-breaking couplings are further suppressed or eliminated, reinforcing Lorentz invariance for physically relevant interactions. The authors also discuss how IR effects in curved spacetimes (cosmology) relax some flat-space constraints, indicating that the boostless relations are IR-sensitive and that cosmological correlators require careful treatment beyond Minkowski bootstrap. Overall, the results advance a Lorentz-violation-agnostic, on-shell program that connects flat-space amplitudes to cosmological observables and clarifies the role of gravity in enforcing Lorentz symmetry.

Abstract

Poincaré invariance is a well-tested symmetry of nature and sits at the core of our description of relativistic particles and gravity. At the same time, in most systems Poincaré invariance is not a symmetry of the ground state and is hence broken spontaneously. This phenomenon is ubiquitous in cosmology where Lorentz boosts are spontaneously broken by the existence of a preferred reference frame in which the universe is homogeneous and isotropic. This motivates us to study scattering amplitudes without requiring invariance of the interactions under Lorentz boosts. In particular, using on-shell methods and assuming massless, relativistic and luminal particles of any spin, we show that the allowed interactions around Minkowski spacetime are severely constrained by unitarity and locality in the form of consistent factorization. The existence of an interacting massless spin-2 particle enforces (analytically continued) three-particle amplitudes to be Lorentz invariant, even those that do not involve a graviton, such as cubic scalar couplings. We conjecture this to be true for all n-particle amplitudes. Also, particles of spin S > 2 cannot self-interact nor can be minimally coupled to gravity, while particles of spin S > 1 cannot have electric charge. Given the growing evidence that free gravitons are well described by massless, luminal relativistic particles, our results imply that cubic graviton interactions in Minkowski must be those of general relativity up to a unique Lorentz-invariant higher-derivative correction of mass dimension 9. Finally, we point out that consistent factorization for massless particles is highly IR sensitive and therefore our powerful at-space results do not straightforwardly apply to curved spacetime.

The Boostless Bootstrap: Amplitudes without Lorentz boosts

TL;DR

This work develops a boostless bootstrap for massless, luminal particles by analyzing three-particle on-shell amplitudes and enforcing consistent four-particle factorization in Minkowski space. It shows that the presence of a massless spin-2 particle enforces Lorentz-invariant three-point graviton couplings and constrains higher-spin interactions, yielding gravity as universal at low energies and prohibiting S>2 self-interactions or minimal coupling beyond GR. Photon and scalar couplings exhibit boost-breaking structures only under higher-derivative terms, with the four-particle test sharply restricting possible vertices; when gravity is included, these boost-breaking couplings are further suppressed or eliminated, reinforcing Lorentz invariance for physically relevant interactions. The authors also discuss how IR effects in curved spacetimes (cosmology) relax some flat-space constraints, indicating that the boostless relations are IR-sensitive and that cosmological correlators require careful treatment beyond Minkowski bootstrap. Overall, the results advance a Lorentz-violation-agnostic, on-shell program that connects flat-space amplitudes to cosmological observables and clarifies the role of gravity in enforcing Lorentz symmetry.

Abstract

Poincaré invariance is a well-tested symmetry of nature and sits at the core of our description of relativistic particles and gravity. At the same time, in most systems Poincaré invariance is not a symmetry of the ground state and is hence broken spontaneously. This phenomenon is ubiquitous in cosmology where Lorentz boosts are spontaneously broken by the existence of a preferred reference frame in which the universe is homogeneous and isotropic. This motivates us to study scattering amplitudes without requiring invariance of the interactions under Lorentz boosts. In particular, using on-shell methods and assuming massless, relativistic and luminal particles of any spin, we show that the allowed interactions around Minkowski spacetime are severely constrained by unitarity and locality in the form of consistent factorization. The existence of an interacting massless spin-2 particle enforces (analytically continued) three-particle amplitudes to be Lorentz invariant, even those that do not involve a graviton, such as cubic scalar couplings. We conjecture this to be true for all n-particle amplitudes. Also, particles of spin S > 2 cannot self-interact nor can be minimally coupled to gravity, while particles of spin S > 1 cannot have electric charge. Given the growing evidence that free gravitons are well described by massless, luminal relativistic particles, our results imply that cubic graviton interactions in Minkowski must be those of general relativity up to a unique Lorentz-invariant higher-derivative correction of mass dimension 9. Finally, we point out that consistent factorization for massless particles is highly IR sensitive and therefore our powerful at-space results do not straightforwardly apply to curved spacetime.

Paper Structure

This paper contains 30 sections, 1 theorem, 209 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction and summary
  2. Motivations
  3. Cosmology
  4. Cosmological correlators
  5. Gravitational waves
  6. Summary of the main results
  7. Notation and conventions
  8. Note added
  9. On-shell methods: symmetries and bootstrap techniques
  10. Symmetries and on-shell conditions for free particles
  11. Little group scaling and the spinor helicity formalism
  12. Unitarity, analyticity and the four-particle test
  13. Three-particle amplitudes
  14. Non-perturbative structure for all spins
  15. Identical particles: symmetric and alternating polynomials
  16. ...and 15 more sections

Key Result

Theorem 2.1

(Factorization Theorem) Singularities of codimension $1$ in $4$-particle amplitudes may appear at vanishing energies ($E_i = 0$) or else are at most simple poles in the momenta. Each singularity of the latter type is in one-to-one correspondence with an exchange diagram (Fig. fig:FD), in the limit w where $s$ is the propagator of the intermediate particle, and $s \to 0$ corresponds to the intermed

Figures (4)

  • Figure 1: The figure shows CMB dipole at the level of $3$ mK align with the $\pm \beta_{\parallel}$ direction. The two perpendicular directions $\pm \beta_{\times}$ and $\pm \beta_{\perp}$ are also shown for reference. This observation highlights the existence of a preferred frame in our universe and hence implies the spontaneous breaking of boost invariance.
  • Figure 2: Exchange diagram. Circles represent non-perturbative, exact 3-particle amplitudes.
  • Figure 3: $s$, $t$ and $u$-channel exchange diagrams, respectively.
  • Figure 4: Two choices for the helicity configuration of the exchanged particle.

Theorems & Definitions (1)

  • Theorem 2.1