Higher-Point Positivity

TL;DR

This work extends infrared positivity bounds from four-point operators to higher-point operators in $P(X)$ theories with $X=\partial_\mu\phi\partial^\mu\phi$, showing that the first nonzero coefficient $\lambda_n$ must satisfy $\lambda_n>0$ for even $n$ and $\lambda_n<0$ for odd $n$ under weak coupling and mostly-plus signature. The authors derive dispersion-relation bounds via higher-point analyticity, establish causality-based no-superluminality constraints, and connect these results to unitarity through explicit tree-level UV completions with massive higher-spin exchanges, including a general construction of propagator numerators in arbitrary dimension. A detailed treatment of the propagator structure for massive higher-spin bosons and the Källén–Lehmann spectral representation underpins the unitarity bounds, yielding a sign rule that aligns with the analyticity and causality arguments. They also identify subtleties in bounding general $P(X)$ theories beyond the $X^n$ sector and explore the Legendre transform involution as a unifying underpinning, with links to energy conditions. Collectively, the results tighten infrared consistency constraints on EFTs and illuminate the interplay between analyticity, causality, unitarity, and energy conditions in the higher-point regime, with implications for swampland considerations and future generalizations.

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 $X=(\partial φ)^2$, 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 $(\partial φ)^4$. Under weak-coupling assumptions that we clarify, we show using all three methods that in theories in which the coefficient $λ_n$ of the $X^n$ term for some $n$ is larger than the other terms in units of the cutoff, $λ_n$ 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.