Higher-Point Positivity

Understand

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.