A A couple of long questions on positivity bounds for UV-complete EFTs

[Suekdccia]

TL;DR Summary

1. Can the EFT-hedron be applied to non-weakly-coupled EFTs UV-completions?

2. Could a generalized EFT-hedron (perhaps a negative-geometry EFT-hedron) be used to study EFTs with a non-standard UV-completion violating positivity bounds?

I had two rather long questions about the recent programe on searching for UV-completions of EFTs through positivity bounds (that is, UV completions that obey fundamental constraints given by QFTs: unitarity, locality, causality, analyticity and Lorentz invariance)

Question #1:

The EFT-hedron developed by Arkani-Hamed and collaborators encode EFTs that can be UV-complete (https://arxiv.org/abs/2012.15849) constrained by positivity bounds already mentioned that are encoded by the positive geometry of the EFT-hedron itself

But does it only encode EFTs that can be completed by weakly coupled UV-completions (like weakly coupled string theories)? Or can it also be applied for stronlgy coupled UV-completions (like M-theory)?

This question came up after asking one of the authors of this paper (https://doi.org/10.1103/PhysRevLett.127.081601) about the relation between their approach and the EFT-hedron. He said that the EFT-hedron would be a much more limited way of studying UV-completions implying that it would only be applied to weakly coupled UV-completions of EFTs:

>About the EFT-hedron. That formalism describes a very special class of theories: those that admit weakly coupled UV completions.
Instead, what we did was to explore the space of all possible consistent scattering theories in a model-independent and nonperturbative setup.
It is rather the opposite: the theories described in the EFT-hedron are a small corner of the space of all possible theories which was the goal of our research.
Example1: the EFT-hedron cannot contain M-theory, but can describe weakly coupled string theories.
Example2: there are physical theories like QCD that have a nice EFT low energy expansion and can violate the EFT-hedron.
Our bounds are much harder to derive, and therefore a few people work on this, but they are more general.

But is this right? Would the EFT-hedron only be applied to weakly coupled UV-completions? Or is it agnostic to the coupling strength of the UV-completion?


Question #2:

These positive geometries like the EFT-hedron or the Amplituhedron have been used for example to carve out the space of possible UV-completions of EFTs that obey positivity constraints (the ones previously mentioned) for example in the case of the EFT-hedron

However, these authors working on these problems have also studied "negative geometries" which would obey "negativity constraints" (mentioned in this talk: Integrated negative geometries in ABJM: https://www.ias.edu/sites/default/files/Amplitudes_2024_Gong_Show_combined-compressed.pdf).

Also, in this presentation (https://pcft.ustc.edu.cn/_upload/ar...489b/d9cc4fe1-4c19-404b-9f0d-ed2349230b18.pdf), the author mentions that positivity constraints (mutual positivity) are the condition where no constraints are assumed and "substracting" the negativity constraints (mutual negativity), implying that all constraints given by the positivity bounds would not be obeyed by negative geometries.

Therefore, would it be possible to construct other geometries for EFTs or generalize the EFT-hedron so that theories that would not obey positivity bounds (like these ones*) would be also encoded in some geometry (like a negative geometry, for instance)?

Could it be generalized to consider non-standard UV-completions of EFTs, as it is studied in this paper: https://scoap3-prod-backend.s3.cern.ch/media/files/67371/10.1088/1674-1137/abcd8c.pdf ?





*https://arxiv.org/abs/2412.08634
https://arxiv.org/abs/2305.16422
https://arxiv.org/abs/2007.15009

 

[mitchell porter]

I have no time to think about this, but I will just mention that coincidentally, a very similar issue came up in a physics discussion the other day. It was about Eric Weinstein's theory of everything, and the other person insisted that "any gravitational theory requires infinite higher spin towers for unitarisation". It sounded odd because if that was strictly true, if it was actually a theorem, you could rule out asymptotic safety a priori. But as your correspondent suggests, I think this claim assumes the possibility of weak coupling (or, equivalently, the possibility of a perturbative calculation) all the way into the UV.

 

[SergejMaterov]

Regarding question 2, I believe that it is mathematically possible to “invert” the signs of all the restrictions of the EFT-hedron and obtain a different geometry, but then there will be no convexity and canonical forms, with all the ensuing consequences.

 

[Suekdccia]

Regarding question 2, I believe that it is mathematically possible to “invert” the signs of all the restrictions of the EFT-hedron and obtain a different geometry, but then there will be no convexity and canonical forms, with all the ensuing consequences.

Has this been done? Could then all theories that would violate positivity bounds (causality, unitarity, analyticity, locality, Lorentz invariance), like non-standard (non-wilsonian) UV-complete theories be entailed by such structure then?

 

[SergejMaterov]

Without convexity and canonical form you will not get a powerful geometric tool for calculation or classification. It is possible to make a "negative EFT-hedron", but its usefulness is unlikely to be comparable to the positive one: the main properties are lost. I have not yet seen a single fully functional negative analogue of EFT-hedron in the spirit of Arkani-Hamed & Co. Practical implementations to date have privileged weakly coupled examples, because those are the only ones we can compute. Maybe someone will do this in the coming years - but for now this is a pure prospect, not a working tool.And anyway: Why reinvent the wheel without wheels?