Higher-point positivity

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 = (∂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.

Affine-null metric formulation of general relativity at two intersecting null hypersurfaces

“…This implies β (0) = 0 and thus f A ,w = 0 so that (1),w = 0, therefore we have α ,ww = 0. The latter condition was also found in a coordinate independent calculation by [35]. Manipulation of (3.25) and (3.26) gives us…”

“…It is a particular pleasure for me to thank Jeff Winicour for constant support and teaching me the various facets of the Bondi-Sachs formulation of General Relativity. I am grateful to the authors of [35], in particular K. Prabhu, for communicating their results and comments on the manuscript. Comments from G. Esposito and F. Alessio are also well appreciated.…”

Consistency of the standard model effective field theory

“…where as shown in Refs. [34,55,60] the signs in the speed calculation work out such that when λ 2 < 0, v > 1. In that case, by taking two nonoverlapping bubbles of condensate, giving them a large relative boost (with relative speed v rel satisfying 1 − v rel = O( 2 ) when the signal speed is v = 1 + in the bubble [60]), and sending signals back and forth between the two bubbles [34], one can form a bona fide causal paradox, with the return signal arriving at the sender before the outgoing signal was sent [76,77]; see Fig.…”

“…For simplicity of notation, throughout we will write all bounds on operator coefficients as strict inequalities, though this could be relaxed (i.e., ≥) if we are working at fixed (e.g., tree) order, cf. Refs [34,52,55]3.…”

“…For massive states, the vanishing of the boundary integral at infinity is a consequence of the Froissart bound[71,72]. For massless states, this conclusion can instead be reached from the weak requirement that the amplitude of the UV completion scales more slowly with energy at large momentum than the EFT contribution[55] 5. While the choice of real polarizations is not strictly necessary for placing positivity bounds[45], it will be convenient for our purposes.…”