Abstract
The presence of a massless spin-2 field in an effective field theory results in a $t$-channel pole in the scattering amplitudes that precludes the application of standard positivity bounds. Despite this, recent arguments based on compactification to three dimensions have suggested that positivity bounds may be applied to the $t$-channel pole subtracted amplitude. If correct this would have deep implications for UV physics and the Weak Gravity Conjecture. Within the context of a simple renormalizable field theory coupled to gravity we find that applying these arguments would constrain the low-energy coupling constants in a way which is incompatible with their actual values. This contradiction persists on deforming the theory. Further enforcing the $t$-channel pole subtracted positivity bounds on such generic renormalizable effective theories coupled to gravity would imply new physics at a scale parametrically smaller than expected, with far reaching implications. This suggests that generically the standard positivity bounds are inapplicable with gravity and we highlight a number of issues that impinge on the formulation of a three-dimensional amplitude which simultaneously satisfies the required properties of analyticity, positivity and crossing symmetry. We conjecture instead a modified bound that ought to be satisfied independently of the precise details of the high energy completion.
Highlights
Over the past few decades effective field theories (EFTs) have proven to be an incredibly powerful tool for studying physical systems at both high and low energies, with applications in all areas of physics ranging from particle physics to cosmology and condensed matter
While it is almost always possible to come up with an EFT valid in a given energy range that correctly describes the physical problem in question, a theoretically more compelling question is whether a given low-energy EFT can be successfully UV completed into another theory valid at higher energies
We find that only new physics at a parametrically low scale Λ ∼ ðMMPlÞ1=2 can ensure the compactified positivity bounds are satisfied and emphasize the far-reaching implications of these arguments, when applied to other fields such as dark matter, which would not have otherwise been expected to couple directly to Standard Model fields at such a low scale
Summary
Over the past few decades effective field theories (EFTs) have proven to be an incredibly powerful tool for studying physical systems at both high and low energies, with applications in all areas of physics ranging from particle physics to cosmology and condensed matter. (i) First, the model we shall propose only has a known partial UV completion and not a full one; its nonrenormalizability arises entirely from graviton exchange/loops This limitation is a weak one as corrections from UV physics will be suppressed by additional powers of MPl. one may argue that our particular example of renormalizable field theory belongs to the ‘swampland,” we will show that these features are common to generic renormalizable field theories including QED itself [32], and applicable to theories which are known to arise from consistent UV completions. We find that only new physics at a parametrically low scale Λ ∼ ðMMPlÞ1=2 can ensure the compactified positivity bounds are satisfied and emphasize the far-reaching implications of these arguments, when applied to other fields such as dark matter, which would not have otherwise been expected to couple directly to Standard Model fields at such a low scale. In Appendix C we discuss the compactification procedure applied to our particular partial UV completion
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