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
Summary
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]
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