SciPost Submission Page
Adding subtractions: comparing the impact of different Regge behaviors
by Brian McPeak, Marco Venuti, Alessandro Vichi
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Brian McPeak |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2310.06888v1 (pdf) |
| Date submitted: | April 17, 2025, 6:52 p.m. |
| Submitted by: | Brian McPeak |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Dispersion relations let us leverage the analytic structure of scattering amplitudes to derive constraints such as bounds on EFT coefficients. An important input is the large-energy behavior of the amplitude. In this paper, we systematically study how different large-energy behavior affects EFT bounds for the $2 \to 2$ amplitude of complex scalars coupled to photons, gravity, both, or neither. In many cases we find that singly-subtracted dispersion relations (1SDRs) yield exactly the same bounds as doubly subtracted relations (2SDRs). However, we identify another assumption, which we call "$t$-channel dominance," that significantly strengthens the EFT bounds. This assumption, which amounts to the requirement that the $++ \to ++$ amplitude has no $s$-channel exchange, is justified in certain cases and is analogous to the condition that the isospin-2 channel does not contribute to the pion amplitude. Using this assumption in the absence of massless exchanges, we find that the allowed region for the complex scalar EFT is identical to one recently discussed for pion scattering at large-$N$. In the case of gravity and a gauge field, we are able to derive a number of interesting bounds. These include an upper bound for $G$ in terms of the gauge coupling $e^2$ and the leading dispersive EFT coefficient, which is reminiscent of the weak gravity conjecture. In the $e \to 0$ limit, we find that assuming smeared 1SDRs plus $t$-channel dominance restores positivity on the leading EFT coefficient whose positivity was spoiled by the inclusion of gravity. We interpret this to mean that the negativity of that coefficient in the presence of gravity would imply that the global $U(1)$ symmetry must be gauged.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
1) Understanding unitarity/causality implications for EFTs is an important and timely topic 2) The manuscript explores systematically the IR EFT implications of different UV assumptions. In this way it also clarifies the origin of several results in the literature. 3) It studies many different EFTs, covering more or less all possible situations of interest (for scalar scattering), including also gravity and electromagnetism. The methods used to study these scenarios are widely different (e.g smeared vs forward dispersion relations). 4)The article is in general well written, it makes links with broad fields in high energy physics, including the weak gravity conjecture and swampland program. It also provides a UV perspective of the theories that saturate the bounds.
Weaknesses
1) The approach is slightly enciclopedic: the result of the article is a series of bounds on couplings. It lacks a sharp question to motivate it -- it is rather an exploration. (Illustrated by the first sentence of sec 4: "We can apply these dispersion relations with differing numbers of subtractions, and we can turn off G, e, or both, so there are a number of scenarios to consider.)
2) It is difficult to appreciate the t-channel dominance assumption in a generic context -- in this sense having a concreter scenario in mind (like the large-n in pion scattering) would have helped.
Report
Requested changes
1) Below 2.12: did the authors try to explicitely construct the scalar amplitude that satisfies 0SDR Regge behaviour ? What about M/((m - s) (M - u)) + all crossings, where M is sent to infinity and m not? It wuld be nice to have these amplitudes explicitely.
2) Fig1. Is it always possible to find a plot for which spin-J is extremal?
3) I find the paragraph below 2.13 confusing: "If we require that the amplitude is a polynomial times Astu, then this is the unique such amplitude" - say more on why? Also specify better what is meant with "single spectral density". Explain better why it is unsurprising that such amplitudes are extremal.
4) End of p13 it is referred to section4, but it is already sec 4. Which null constraints do they mean? That paragraph is hard to read (even more so without ref 53 at hand).
5) Fig 9 - how many SDR? Specify in caption
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Report #1 by Ahmadullah Zahed (Referee 1) on 2025-5-27 (Invited Report)
Strengths
- The manuscript is well written.
- Given the current developments of EFT positivity bounds, knowing how different high-energy behaviours affect the EFT bounds is interesting.
- The connection between the weak gravity conjecture and t-channel dominance is intriguing.
Weaknesses
Report
I believe that it meets the criteria to be published in SciPost Physics upon answering the following:
Requested changes
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2SDR and 1SDR give similar bounds: In the introduction on page 3, there is a statement that 1SDR gives "useless sum rules". In section 4, this statement is not clearly established when the sum rules are introduced. Maybe for positivity bounds, these are not useful, but these should give non-trivial constraints in case of full non-linear unitarity. Some clarification will be useful.
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Changing the normalization Lambda(v_e)=1 (on page 29) does not give bounds: Doesn't it mean fixing a normalization of the test function phi? Why is this affecting the numerics? Do extremal solutions using different normalizations violate any of the input assumptions? If not, then why different normalizations are not valid choices? How are the bounds from Lambda(v_e)=1 trustworthy?
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1SDR+"t-channel dominance" leads to weak gravity conjecture: The assumption is that rho2=0. Does this violate crossing relations? Namely, writing the crossing equations and showing how rho2=0 affects them. What changes are expected in the bounds for rho2=non-zero but small in the hope that full crossing symmetry can be recovered?
Eq 5.24 says, "if g01<0, then e can't be zero if we want G>0"-- is clear. But the following statement in the same paragraph, "g01<0, implies G=0", is unclear how it will appear from the eq 5.24. Clarification will be helpful.
I also have an open-ended question: 1. Is it possible to say something about the converse of the theorem, or any intuition that the weak gravity conjecture will lead to t-channel dominance?
Minor suggestions: 1. End of line on page 13, maybe the authors mean the next sections of section 4. 2. The caption of Fig. 11 could be clearer.
Recommendation
Ask for minor revision
Author: Brian McPeak on 2025-12-04 [id 6103]
(in reply to Report 1 by Ahmadullah Zahed on 2025-05-27)
We thank the reviewer for their careful report and interesting questions. We have made all the requested changes, summarized below:
(1) We see the reviewer’s point here. The fact that g_01 can’t be bounded is discussed below equation 4.5. This is primarily what the line in the intro referred to. But the fact that these null constraints are useless in general is more of an empirical fact – including 1SDR constraints (without t-channel dominance) led to identical plots as when we didn’t include them. We have changed the wording and added a footnote on page 3 about how they might be useful in the non-linear context.
(2) That footnote was misleading – it is meant to say that normalizing any other Lambda[v_i] other than Lambda[v_e] does not lead to a bound, not that the actual value of the normalization matters. We have reworded this to clarify.
(3) No, this doesn’t violate crossing, at least not the crossing symmetry that we’ve assumed for the complex scalar (which is t-u crossing symmetry). To see this, consider the Lovelace-Shapiro amplitude, discussed in more detail in 2203.11950. This amplitude satisfies 1SDR, t-channel dominance, and crossing. T-channel dominance can be thought of as what happens at large N in 2203.11950, so small but non-zero rho_2 might be relevant for finite but large N. I don’t think anyone has looked at that.
That last statement near 5.24 is just wrong, I’m not sure what it was supposed to say but we’ve removed it.
Open ended question: we have thought about this but we don’t see how to make this argument. T-channel dominance really seems to be related to the large N limit for pions. In AdS / CFT large N is vaguely related to gravity being weak in the bulk… so maybe there is a connection in some way. But I suppose you mean to try to show that having e^2 < G requires rho_2 = 0? I don’t see a way to prove this analytically this work but it would be interesting to try numerically. For instance, one could perhaps fix G and e^2 and then maximize rho_2 at different values. We added a few sentences about this as a possible future direction in the discussion on page 38.
End of page 13 – fixed (should have said 5.2).
Caption of figure 11: we’ve added a brief explanation in the caption
Author: Brian McPeak on 2025-12-04 [id 6102]
(in reply to Report 2 on 2025-06-05)We thank the reviewer for their thoughtful report and feedback. We have made all of the requested changes, which are listed below:
(1) I think this is a misunderstanding – we did construct all of the amplitudes that appear in the plot, and indeed the amplitude that the reviewer suggests is proportional to the amplitude in 2.11. Possibly this is unclear since we originally called that amplitude A_stu instead of A_stu-pole. We have changed this for clarity.
(2) This is a good question that we wondered about, but we don’t know. We guess that the answer is yes and we made a few other plots internally that we didn’t include in the paper that had e.g. spin 4 and spin 6 extremal. We added a brief speculation about this at the very end of section 2
(3) Single spectral density means that all spectral densities are zero except one. In general, there is only a single amplitude that is 1) single spectral density and 2) (up to addition of contact terms and rescaling by a constant). This probably isn’t particularly deep but we wanted to point it out anyways. We also elaborated that these are extremal in the sense that all of the spectral densities except one have been dialed to their minimum value, which is zero. The value of the remaining spectral density doesn’t matter since these bounds only care about ratios. We have rephrased the paragraph to be more clear.
(4) This was a typo – we meant section 5.2, which is where we elaborate. This paragraph is meant to summarize the issue with the null constraints from 0SDR in the context of pion scattering. The forward limit 0SDRs imply g_00 > 0 but pions are goldstone bosons so they must have g_00 = 0 (they’re derivatively coupled). We’ve fixed the typo and rephrased the paragraph.
(5) We added that these figures were made assuming 1SDR and t-channel dominance.