Higher-point positivity

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