Higher-Point Positivity

TL;DR

This paper extends techniques for bounding higher-dimension operators in quantum effective field theories to higher-point operators, analyzing their positivity constraints and implications for causality and stability.

Contribution

It introduces a systematic approach to bounding higher-point operators in $X=( abla )^2$ theories using causality, analyticity, and unitarity, with new derivations and insights.

Findings

Positivity constraints on coefficients $$ for higher-point operators.

First-principles derivation of propagator numerators for massive higher-spin bosons.

Connections among energy conditions, causality, and stability in effective field theories.

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 $X=(\partial \phi)^2$, 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 $(\partial \phi)^4$. 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.