Abstract

We re-examine the constraints imposed by causality and unitarity on the low-energy effective field theory expansion of four-particle scattering amplitudes, exposing a hidden “totally positive” structure strikingly similar to the positive geometries associated with grassmannians and amplituhedra. This forces the infinite tower of higher-dimension operators to lie inside a new geometry we call the “EFT-hedron”. We initiate a systematic investigation of the boundary structure of the EFT-hedron, giving infinitely many linear and non-linear inequalities that must be satisfied by the EFT expansion in any theory. We illustrate the EFT-hedron geometry and constraints in a wide variety of examples, including new consistency conditions on the scattering amplitudes of photons and gravitons in the real world.

Highlights

  • There is a long-appreciated, close connection between vacuum stability/causality/unitarity, and analyticity/positivity properties of scattering amplitudes, going back to the 1960’s Smatrix program

  • We re-examine the constraints imposed by causality and unitarity on the lowenergy effective field theory expansion of four-particle scattering amplitudes, exposing a hidden “totally positive” structure strikingly similar to the positive geometries associated with grassmannians and amplituhedra

  • We take a brief sojourn in the positive geometries relevant to our analysis, giving a pedagogical discussion of convex hulls of moment curves and cyclic polytopes. These geometries will be immediately utilized to define the s-channel EFT-hedron in section 7, where we focus on the theory space for scalar EFTs that allow for preferred ordering and the absence of u-channel thresholds

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Summary

Introduction

There is a long-appreciated, close connection between vacuum stability/causality/unitarity, and analyticity/positivity properties of scattering amplitudes, going back to the 1960’s Smatrix program. There are three fundamental origins of positivity: the positivity of energies (vacuum stability), the sharp localization of signals inside the lightcone (causality) and the positivity of probabilities (unitarity) These basic positivities, together with analyticity properties of scattering amplitudes meant to reflect causality, allow the derivation of more non-trivial positivity constraints on coefficients of higherdimension operators in low-energy effective field theories (as in [1,2,3]). We take a brief sojourn in the positive geometries relevant to our analysis, giving a pedagogical discussion of convex hulls of moment curves and cyclic polytopes These geometries will be immediately utilized to define the s-channel EFT-hedron, where we focus on the theory space for scalar EFTs that allow for preferred ordering and the absence of u-channel thresholds. Other related works can be found in [13,14,15,16]

EFT from the UV
Explicit EFT amplitudes
From local amplitudes to local operators
Dispersive representation for EFT coefficients
Obstructions from the massless poles
Theory space as a convex hull
Hidden total positivity from unitarity and locality
Convex hulls and cyclic polytopes
Hankel matrix total positivity
The Gegenbauer cyclic polytopes
Spinning Gegenbauer cyclic polytope
The s-channel EFT-hedron
Fixed q
The geometry of the gap
Scalar EFT-hedron
Deformed moment curves and the EFT-hedron
Multiple species
The spinning EFT-hedron
Photon EFT
Graviton EFT
10 Explicit EFTs in the EFT-hedron
A2 A2 A3
10.2 Full EFT-hedron
10.3 Living near the boundary of unitary polytopes
11 Running into the EFT-hedron
Λ4μ4 a0 a2β0 4
12 Outlook
A Causality constraints on amplitudes
Full Text
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