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Smearing of causality by compositeness divides dispersive approaches into exact ones and precisionlimited ones
by Felipe J. LlanesEstrada, Ra\'ul Rold\'anGonz\'alez
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Submission summary
Authors (as registered SciPost users):  Felipe J. LlanesEstrada 
Submission information  

Preprint Link:  scipost_202010_00009v1 (pdf) 
Date submitted:  20201012 23:13 
Submitted by:  LlanesEstrada, Felipe J. 
Submitted to:  SciPost Physics Core 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational, Phenomenological 
Abstract
Scattering off the edge of a composite particle or finiterange interaction can precede that off its center. In an effective theory treating the particle as pointlike and the interaction as contact, the scattered wave is slightly advanced in violation of causality (the fundamental theory underlying the compositeness is of course causal). In practice, partialwave or other projections of multivariate amplitudes exponentially grow for large imaginary E, so that upper complex plane analyticity is not sufficient to obtain a dispersion relation for them, but only for a slightly modified function. This limits the maximum precision of certain dispersive approaches to compositeness based on Cauchy's theorem leading to partialwave dispersion relations (the modified relations additionally connect different angular momenta). This may be of interest to some dispersive tests of the Standard Model with hadrons, and to unitarization methods used to extend electroweak effective theories. Interestingly, the Inverse Amplitude Method is safe (as the inverse amplitude has the opposite, convergent behavior allowing contour closure). Generically, onedimensional dispersion relations and sum rules such as those for the photon vacuum polarization, onedimensional form factors or the Adler function, to name a few, are not affected by this uncertainty. Likewise, fixedt dispersion relations were cleverly constructed to avoid it and consequences therefrom are solid.
Current status:
Reports on this Submission
Anonymous Report 2 on 20201129 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202010_00009v1, delivered 20201129, doi: 10.21468/SciPost.Report.2241
Strengths
1 Rises an apparent problem
2 Authors are experts in this field
3 Paper is well written
4 Issue being discussed that appears to be of potential relevance
Weaknesses
1 At the end of the day, there is no real problem. Useful or confusing?
2 Authors mix range of interaction with scale of compositeness
3 Maybe misleading to the reader
Report
This manuscript has brought more questions than answers to this referee. I agree with the general discussion in Sect 1, although in my opinion the notation should be improved (see below). However I find the discussion presented in Sect 2 rather confusing and misleading.
In effective theory treatments, pions (or longitudinal W for that matter) are treated as pointlike objects. Period. Then, none of the points discussed in Sect 1 would immediately apply. We of course know that pions are not pointlike or elementary, but Ws could still be to the best of our knowledge. Of course they have a range of interactions (we may take that as 1/M_W in the case of the electroweak bosons, for instance, but this is in principle unrelated to the scale of compositeness. A similar comment could apply to pion scattering.
Being treated as pointlike objects I cannot see the relevance of the discussion displayed in Sect. 1
Sect. 3 is fine, but I do not see much new there.
Requested changes
1. In Sect 1 I would recommend replacing f(x) for a physical amplitude to make the reader understand what we are talking about
2 Authors should propose an example where the connection to the problem presented in Sect. 1 is obvious (not the case now).
3 Some English checks needed: there is a "safe for" on p4 that makes no sense. cm as centerofmass is not defined.
Anonymous Report 1 on 20201118 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202010_00009v1, delivered 20201118, doi: 10.21468/SciPost.Report.2207
Strengths
 Pedagogic introduction
 Elegant tool (dispersion relations)
 Relevant for phenomenology
Weaknesses
 Given existence of valid fixedt dispersion relations, it is not so clear where the problem is
Report
Dear Editor,
the authors discuss dispersion relations that originate from causality and unitarity, and relate the IR and UV parts of the scattering amplitude. They point out a potential problem in fixedangle dispersion relations; in particular they show that the scattering off objects of finite size introduces an apparent violation of causality, despite the microscopic physics being causal.
The topic is in principle interesting, as dispersion relations are an important tool in particole physics. However, as the authors point out, this problem doesn’t appear in fixedt or forward dispersion relations, which are typically employed in particle physics. So, I believe the authors should make more effort to motivate why plain dispersion relations for partial waves are a priori better than fixedt dispersion relations (and show that they contain more information).
Also, it is not clear to me how a computation on the righthandcutonly suffices.
So, rather than pointing out a specific situation in which a specific dispersion relation doesn’t work, I’d see more useful an article that shows what instead works. The authors have some ideas in the 7th paragraph of the conclusions, that should be investigated more.
Requested changes
1) Motivate why plain dispersion relations are important
2) What does the violation of causality imply for the analyticity properties of partial waves?
3) Explain why the authors think the computation on the righthand cut is sufficient. (see p.7)
4) What do the authors mean when they say on p.7 “the important case of QFT”; what other case do they have in mind?
5) The link with LHC physics is a bit poor: can the authors explain better how this would impact the search for SM devviations?
6) extend discussion on polynomials and t and t’ in the conclusions.
Author: Felipe J. LlanesEstrada on 20201124 [id 1063]
(in reply to Report 1 on 20201118)
The referee raises some good points that we quickly address.
We should first like to recall that fixedt dispersion relations were indeed safely constructed in the 50s, partly to circumvent this type of problems. But after six or seven decades, brief mentions in classic articles such as our reference [10] or nonrelativistic discussions in books long out of print such as our reference [7] have largely disappeared from the collective memory, so that practitioners sometimes ignore the distinction between those and other types of dispersion relations. Because partialwave amplitudes are convenient to analyse newphysics or also hadron resonances (keeping in mind that they are sensitive to only one angular momentum channel at a time, isolating new particles), they remain in use. They satisfy good analyticity properties leading the community to expect them to also automatically satisfy good dispersion relations. But as we show, this is not automatic because of divergent behavior deep in the complex plane and care should be exercised (by deploying the dispersion relation for the Inverse Amplitude, for example) .
We do not believe that partialwave nor fixedt dispersion relations are better than the other: they are useful for different problems.
Respect to the comment about the impact of the lefthand cut, a short answer could be that, because physics happens at s>threshold on the right cut, and Cauchy's theorem suppresses contributions far in the complex plane by a 1/(ss') denominator, the integral over the right cut is numerically more important for physical applications. Of course, a more detailed analysis is called for, and a followup preprint by a different author collaboration https://inspirehep.net/literature/1826204 addresses the systematic theory uncertainties in the Inverse Amplitude Method, prominently featuring the leftcut. Interested readers can delve into that work for detail.
That method does work and has been discussed at length in the literature (including that recent preprint and the many references therein). Plain PW dispersion relations can also work in restricted kinematic domains where they can be derived from fixedt dispersion relations, but the community should be continuously aware that restrictions exist and they cannot be blindly applied (and the same for fixedangle dispersion relations or others).
The referee is of course aware that Smatrix theory is more general than local quantum field theory applied to physical kinematics. Non pointlike interactions or objects (e.g. strings) can be cast in the Smatrix formalism and predict scattering amplitudes with the required properties. That was a passing comment and we have no particular interest in pursuing the matter.
The LHC link however is of interest for our phenomenological physics program: if/when separations from the SM are found, they will likely be cast in the form of an Effective Theory for the SM particle content. Dispersion relations can then play a role in extending the reach of those lowenergy measurements to ascertain what may await at higher energies.
We thank all parties for the collegial consideration of our manuscript, and particularly the reviewer for his or her dedication and appraisal of the article.
Anonymous on 20210603 [id 1484]
(in reply to Felipe J. LlanesEstrada on 20201124 [id 1063])
Dear colleague, in preparation for a resubmission of the manuscript, here is the list of changes implemented.
1) Motivate why plain dispersion relations are important
RESPONSE: we have commented about this topic in the introduction, though not extensively, as we believe this article needs to remain of moderate size consistently with the amount of material reported.
2) What does the violation of causality imply for the analyticity properties of partial waves?
RESPONSE: This can be read off equation (4). The lower integration limit needs to be set to 2R/c sin(theta/2) < 0 since the timedependent amplitude only vanishes for earlier times. This change does not affect the convergence at tau> infinity, so that the resulting t_J(E) is still analytic in the upperenergy plane; but it grows (exponentially) for increasingly positive Im(E). That is, a finite target yields an analytic function but a nonconvergent contour integral. Only when the signal can be arbitrarily advanced, and tau> infinity needs to be taken in the time integration, is analyticity completely lost. This comment has now been added to subsection 1.1.
3) Explain why the authors think the computation on the righthand cut is sufficient. (see p.7) RESPONSE: we do not think it is sufficient for a full evaluation of the amplitude, and we discuss it at length in the concurrent extensive paper submitted to SciPost Physics, issued as ePrint: 2010.13709 [hepph]. There is minimum explanation here after Eq.(13). Remember that we are only discussing the uncertainties, and those due to the two cuts should be linearly summed, being conservative.
4) What do the authors mean when they say on p.7 “the important case of QFT”; what other case do they have in mind? RESPONSE: Smatrix theory is more general and can be applied to string theory, to nonrelativistic quantum mechanics and other approaches; this is now explicitly stated at the end of section 2.
5) The link with LHC physics is a bit poor: can the authors explain better how this would impact the search for SM deviations? RESPONSE: We have tried to show it by reorganizing section 2 into subsections and adding minor paragraphs elsewhere as needed.
6) extend discussion on polynomials and t and t’ in the conclusions. RESPONSE: we have added this suggested comment, known from the usage of the Omnès Mushkelishvili method.
Author: Felipe J. LlanesEstrada on 20210603 [id 1485]
(in reply to Report 2 on 20201129)We also thank the second reviewer for his careful reading of the manuscript and suggestions, and we believe that we can address his/her criticism.
REFEREE: In effective theory treatments, pions (or longitudinal W for that matter) are treated as pointlike objects. Period. Then, none of the points discussed in Sect 1 would immediately apply. We of course know that pions are not pointlike or elementary, but Ws could still be to the best of our knowledge.
RESPONSE: The referee is certainly right that for many purposes one can treat Goldstone bosons as pointlike objects. As he states, this is an approximation for pions, and might or might not be correct for W_L. Our point is that, while (RxE) (size x energy) may be small at an accelerator where the energy might be insufficient to probe the object's size, when closing the Cauchy contour in the complex plane, the exponent (Rx Im(E) ) becomes arbitrarily large. We believe that we can restate our confidence in the results of section 1, they are correct in the presence of an underlying structure, even when that structure is not yet probed at a reached energy.
REFEREE: Of course they have a range of interactions (we may take that as 1/M_W in the case of the electroweak bosons, for instance, but this is in principle unrelated to the scale of compositeness. A similar comment could apply to pion scattering. RESPONSE: That statement is correct. The scale of compositeness and the range of interaction can be unrelated, if the mass of the force carrier is very dissimilar to the mass of the source (e.g. the pion and the nucleon). We have improved the logic by clarifying that our considerations apply to a) finite range interactions b) apparently zero range/delta function interactions when the object is composite at any scale.
REFEREE: Being treated as pointlike objects I cannot see the relevance of the discussion in Sect. 1 RESPONSE: A dispersion relation is sensitive to very large energies that probe the underlying scale. One can escape this fact over the real axis because the exponential phase factor is 1, so that the divergence can typically be tamed by subtractions (providing an s^n denominator). But not with an imaginary part of E.
Sect. 3 is fine, but I do not see much new there. RESPONSE: Certainly. Sec. 3 is provided for readers of various backgrounds to clearly apprehend that the type of dispersion relations there discussed are safe; that is, to show with some detail which dispersion relation demonstrations are, and which ones are not (and why) affected by large Im(E) divergences. We think that it is important to keep these comments to avoid confusion by offfield readers that may think that we cast doubt on the entire dispersionrelation approach, but we will agree to drop them if deemed necessary.
Requested changes 1. In Sect 1 I would recommend replacing f(x) for a physical amplitude to make the reader understand what we are talking about. RESPONSE: we have changed it for a partial wave t_J starting in Eq. (2); some rewriting of the text was correspondingly necessary and has been carried out.
2 Authors should propose an example where the connection to the problem presented in Sect. 1 is obvious (not the case now). RESPONSE: Any scattering process in 2 or greater dimension, where one or both objects are of finite size, or where the interaction is finite range, will suffer from that exponential behaviour in the complex plane and care with the formulation of dispersion relations will be needed. Examples include: photonhadron scattering, hadronhadron scattering, photonatom scattering, and so on. This comment has now been incorporated into the manuscript.
3 Some English checks needed: there is a "safe for" on p4 that makes no sense. cm as centerofmass is not defined.
RESPONSE: We have proofread the manuscript and improved as needed. We understand that the expression "safe for" or "save for" may be unfamiliar to some readers (it was imported from french into english, in our understanding, but seems to be correct usage) and have substituted it by the more common "except for". We now have defined "center of mass" explicitly, though this is almost universal in the theoretical physics literature.