Higher-point positivity

A bstract 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 = (∂ 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 (∂ 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.