Higher-Point Positivity

Abstract

We consider the extension of techniques for bounding higher-dimension operators in quantum effective field theories to higher-point operators. Working in the context of theories polynomial in $X=(\partial \varphi {)}^2$, we examine how the techniques of bounding such operators based on causality, analyticity of scattering amplitudes, and unitarity of the spectral representation are all modified for operators beyond $(\partial \varphi {)}^4$. Under weak-coupling assumptions that we clarify, we show using all three methods that in theories in which the coefficient ${\lambda }_n$ of the $X^n$ term for some $n$ is larger than the other terms in units of the cutoff, ${\lambda }_n$ must be positive (respectively, negative) for $n$ even (odd), in mostly-plus metric signature. Along the way, we present a first-principles derivation of the propagator numerator for all massive higher-spin bosons in arbitrary dimension. We remark on subtleties and challenges of bounding $P(X)$ theories in greater generality. Finally, we examine the connections among energy conditions, causality, stability, and the involution condition on the Legendre transform relating the Lagrangian and Hamiltonian.

25 pages

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• On the Geometry of the String Landscape and the Swampland
• Causality, Analyticity and an IR Obstruction to UV Completion
• Proof of the Weak Gravity Conjecture from Black Hole Entropy
• Beyond Amplitudes' Positivity and the Fate of Massive Gravity
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Cited by in corpus (27)

• Consistency of the Standard Model Effective Field Theory
• Causality, Unitarity, and the Weak Gravity Conjecture
• Finite Quantum Gravity Amplitudes -- no strings attached
• Natural Selection Rules: New Positivity Bounds for Massive Spinning Particles
• Causal Effective Field Theories
• Causality Constraints on Black Holes beyond GR
• Positivity Bounds without Boosts
• Graviton-Mediated Scattering Amplitudes from the Quantum Effective Action
• Dispersion Formulas in QFTs, CFTs, and Holography
• Unambiguously Testing Positivity at Lepton Colliders
• Sum Rules in the Standard Model Effective Field Theory from Helicity Amplitudes
• Positivity bounds from multiple vacua and their cosmological consequences
• Swampland Conditions for Higher Derivative Couplings from CFT
• Scalar Fields Near Compact Objects: Resummation versus UV Completion
• Positivity in the effective field theory of cosmological perturbations
• The shape of Scalar Gauss-Bonnet Gravity
• Spinning Sum Rules for the Dimension-Six SMEFT
• Causality and quasi-normal modes in the GREFT
• Effective Field Theories as Lagrange Spaces
• Multipositivity Bounds for Scattering Amplitudes
• Cosmological Decoherence from Thermal Gravitons
• Two-body problem in theories with kinetic screening
• Positively Identifying HEFT or SMEFT
• Spinning amplitudes from scalar amplitudes
• Five-point Superluminality Bounds
• Strings from Almost Nothing
• Positivity vs. Lorentz-violation: an explicit example