Into the EFThedron and UV constraints from IR consistency

Abstract

Recently it was proposed that the theory space of effective field theories with consistent UV completions can be described as a positive geometry, termed the EFThedron. In this paper we demonstrate that at the core, the geometry is given by the convex hull of the product of two moment curves. This makes contact with the well studied bi-variate moment problem, which in various instances has known solutions, generalizing the Hankel matrices of couplings into moment matrices. We extend these solutions to hold for more general bi-variate problem, and are thus able to obtain analytic expressions for bounds, which closely match (and in some cases exactly match) numerical results from semi-definite programing methods. Furthermore, we demonstrate that crossing symmetry in the IR imposes non-trivial constraints on the UV spectrum. In particular, permutation invariance for identical scalar scattering requires that any UV completion beyond the scalar sector must contain arbitrarily high spins, including at least all even spins ℓ ≤ 28, with the ratio of spinning spectral functions bounded from above, exhibiting large spin suppression. The spinning spectrum must also include at least one state satisfying a bound $$ {m}_J^2<{M}_h^2\frac{\left({J}^2-12\right)\left({J}^4-32{J}^2+204\right)}{8\left(150-43{J}^2+2{J}^4\right)} $$ m J 2 < M h 2 J 2 − 12 J 4 − 32 J 2 + 204 8 150 − 43 J 2 + 2 J 4 , where J2 = ℓ(ℓ+1), and Mh is the mass of the heaviest spin 2 state in the spectrum.

Positivity bounds from multiple vacua and their cosmological consequences

“…When comparing with observational constraints in section 3, we will find that (2. 19) and (2.20) taken together are significantly more constraining than (2.19) alone.…”

“…Using (2.21) and imposing the above stability conditions, the two positivity bounds (2. 19) and (2.20), respectively, then imply…”

“…For the residual parameter space we then ask, whether the positivity bounds (2. 19) and (2.20) are satisfied-regions failing this test (hatched gray regions) can be as large as 2/3 of the residual The three panels from left to right correspond to the cubic, quartic and quintic Galileon theories, respectively, where only the respective c i and c 2 are non-zero and the kinetic term has been normalised by setting c 2 = −1 (in anticipation of the next section and hence ruling out a stable vacuum solution φ = 0). For any given choice of (α, β) we solve the tadpole condition (2.4) to obtain the free c i .…”

“…where our convention is that a p a = 0 (and recall from section 2 that gµν = diag −1, c 2 s , c 2 s , c 2 s ). These obey the usual relation, s+t+u = 0, since on-shell fluctuations satisfy the free equation of motion 19 p µ gµν p ν = 0. In terms of these variables, the 2 → 2 scattering amplitude from the cubic and quartic interactions (2.5) and (2.6) is,…”

“…Note that the Galileon 1 In this work we focus on linear positivity bounds for a single scalar field. There has been much progress recently developing similar (and in some cases stronger) bounds for spinning particles [13][14][15][16], non-linear bounds from moment theorems [17][18][19] and exploiting full crossing symmetry [20][21][22][23][24], generalised bounds for EFTs with multiple fields [25,26], and bounds including the effects of gravity [27][28][29][30][31][32].…”