Systematizing and addressing theory uncertainties of unitarization with the Inverse Amplitude Method

1) The paper addresses a longstanding open question about the estimation of systematic uncertainties for unitarization methods, in particular the popular Inverse Amplitude Method, applied to effective field theories
2) The estimated "a priori" uncertainties are in good agreement with the "a posteriori" ones obtained in Hadron Physics.
3) It reinforces the interest and relevance of these methods for near future results at LHC.
4) It opens the way for similar attempts for other unitarization methods.

1) The authors emphasize the interest for new Physics in case only non-resonant deviations from are found in the longitudinal gauge boson and Higgs sector of the SM. However, if that is the case, it is very likely that the uncertainties in those deviations will largely swamp the systematic effects studied here, although it would be comforting to know that the method does not contribute with significant uncertainties. It would have been nice to estimate the expected size of the uncertainties in effective parameters in such scenario and compare with those of the method itself.
2) The uncertainties are illustrated for a particular chice of pole mass motivated by Hadron Physics. It would be desirable to have estimates for other masses and an estimate of how high the mass of new resonances can be for the IAM to still provide reliable predictions.

As for the SciPost expectatins criteria, to my view the meanuscript presents a breakthrough on a previously-identified and long-standing research stumbling block, which is the estimation of uncertainties in unitarization methods and it opens a new pathway in similar studies for other unitarization methods applied to different effective theories.

Concerning the general acceptance criteriafor SciPost, I think they will be easily met after the authors address properly the comments that I detail next:

a) The authors make a valuable effort to ascertain the systematic uncertainties in the IAM and one of their main motivations is the prediction of New Physics states or resonances that may lie out of reach of LHC but whose non-resonant effect may be observed at low energies. Here I see several issues that should be commented or addressed by the authors.

a.1) Their generic estimates are made in comparison with a rho-like scenario. This imposes a size for the low-energy constants (LECS), given by Resonance Saturation and the ratio f/Mrho. However, if the LECS were much smaller because the new states lied much higher compared to v (246 Gev), the equivalent ratio v/Mnew would be much smaller and I suspect that all the relative uncertainties calculated here would be much larger. It should be easy for the authors to estimate how high the resonances should be so that the IAM becomes useless due to its systematic uncertainties. Although the authors explicitly say that their estimates are obtained in a rho-like scenario, it is relevant to provide at least another example with higher masses to illustrate the scaling of the uncertainties. (There is also the issue that the predicted resonance could be as wide as massive so that it may be hard to identify as a resonance)

a.2) The work deals with longitudinal gauge boson and/or Higgs scattering. It is however possible that some of these states may be better seen in other processes (in the case of a rho-like state, the resonance may be much more easily seen through vector form factors starting from gluon fusion without the need for "initial" longitudinal gauge bosons). In such case, non-resonant behavior may not have been clearly identified in scattering but a resonance may have been seen in another process related through final state interactions. The specific scenarios to which the authors refer ç, where the IAM would suggest new resonant physics should be commented with more detail. In particular, what if the resonance is not seen in scattering but in other processes? What to do with the IAM then?

a.3) There is the issue of the uncertainties in the LECs as well. If the resonance lies out of reach of the LHC and only its "low-energy tail" is seen as a non-resonant effect, the corresponding LECS will have, I am afraid, a rather large uncertainty. This actually happened in the first applications of the IAM to the rho, when the uncertainties in the LECS were so big that their propagation to the rho pole using the IAM covered very well the correct description of data, around the resonance peak, dwarfing the systematic IAM uncertainties. The question is, then, whether the uncertainty in the measured (non-resonant) deviations from the SM will dominate over the IAM uncertainty. This should be commented and, of course, that will change with the mass of the resonance (see a.1 above)

b) CDD poles are not so much, or not only, a source of uncertainty as an impediment to apply the naive IAM, questioning the validity of the assumptions needed to derive it. The examples about CDD poles in Appendix A.2 need some clarification and some more physical insight of why this is so. I understand the need for a simple model, but the fact that f is both the resonance pole mass (Eq.80) , the scale for the chiral expansion (Eq.81 in s/f^2) and of the same order as the new dynamical scale $M_0$ makes the discussion somewhat confusing. In particular:

b.1) The authors state that Eq.81 is a well-behaved chiral expansion. I agree that it is a low-energy expanion, but why they claim is chiral? How would one arrive to that expression from a chiral Lagrangian? Surprisingly the LO seems to depend on more parameters than just f. How would $M_0$ appear there?. Why f is to be identified with a chiral scale? From what it seems, it is just the resonance mass, and if it is also the chiral scale then the resonance is very light compared to the hadronic cases, (almost by an order of magnitude compared to the rho). Actually, the fact that $t_0$ and Re$t_1$ cancel before $s=f^2$ is not just evidence for the breakup of the series? Certainly we can construct an amplitude like that, but zeros appear for some dynamical reason, so the authors are setting the new dynamical scale at $M_0$ (below f), similar to $f$ but much below the "chiral scale" $4\pi f$ which is the order of magnitude where the relevant degrees of freedom of the theory are integrated out to buid the EFT. But nowthere is another dynamical scale $M_0$. I am not so sure this is a chiral expasion in the usual sense, the authors should definitely comment about this.

b.2) The authors are assuming f and $M_0$ to be close, with $M_0$ somewhat smaller... how justified is then the expansion in Eq.80?

b.3) How well the expansion in 81 converges compared to the usual ChPT case?. In ChPT the LECS are of order 10-2, or, in other words, we expect a suppression with powers of $4\pi f$ (as the authors remark in several places in their work). Where is that suppression in Eq.81?, it seems that the suppressing $4\pi$ factors are gone. Actually, the presence of $M_0/f$ factors is diminishing the LO wrt the NLO in the Hadron Physics case.

b.4) Is really the IAM to be blamed for not reproducing well the resonance? Or is it just that the NLO low-energy expansion is the one not able to reproduce a zero at $s=M_0^2$ (the real parts do cancel, but the imaginary part does not and its coefficient is order one, if I understand correctly).
I think the the blame can also be put on the effective theory, which is a very bad approximation to the zero at $M_0$. Of course, if one removes the CDD pole, the rest of the amplitude has a convergent expansion on which unitarization can provide meaningful results. But it seems more a problem of the expansion itself than the unitarization. This should also be discussed an commented.

b.5) Another way of seeing the previous comments is that, naively, the IAM reconstructs a resonance from the information on its low-energy tail. This information can be encoded in the LECS. Indeed, for Hadron Physics, it is known that the value of the combination of $l_1$ and $l_2$ parameters in the rho channel are dominated by the rho contribution itself (This is known as Resonance Saturation). The smallness of the combination that appears in the scalar channel and the enhanced factor of the loops there make the sigma a very different kind of resonance, but if the LECS had a different value it would look relatively similar to the rho again. The authors should discuss whether by placing the CDD pole "right before" the resonance (Since $M_0$ is slightly less than f), the information about the resonance in the LECS becomes subdominant, which hinders its reconstruction with the IAM unless the additional and dominant dynamical information about the presence of the $M_0$ scale is provided and removed. Unfortunately, with the model used by the authors, where the resonance mass seems to be given by f, it is hard to disentangle in the EFT series what is the information about the resonance and what is the scale of the chiral breaking (assuming they explain why that is a chiral series, see b.1 above).

b.6) The main example in the appendix, although simple, at first sight does not seem to fit the motivation of the authors for a measurement of a non-resonant behavior without seeing the resonance, since the resonance would be located roughly at $f\simeq 250$GeV, and I guess it would be seen even better than the non-resonant effects. In the same appendix, the authors provide then another example with HEFT. But here the CDD pole appears {\it past} the resonance. So the mechanism looks rather different to the previous example where the zero seem to be "hiding" the resonance from threshold. So, the question is wheter, and how, is it also cancelling the resonance saturation of the LECS. Certainly here the LO and NLO do not cancel at low energies, which does not seem to compromise the good convergence of the series.

b.7) How well is the criterion of "narrow resonance", which the authors decided to apply in their systematic estimates, apply to the resonance in Fig.8?
It does not look as narrow as the rho-like in Figure 6. Although the CDD zero distorts its shape, it looks like its width is comparable to the mass. Has the use of narrow-resonance criteria an additional effect hindering the reconstruction of that resonance or the estimation of uncertainties in this case?

c) When discussing the presence of the CDD pole Eq.24, obtained at NLO, is often used as implying the presence of a zero in the full (all orders) partial wave. But, at first sight,it only implies the vanishing of the real part of the LO+NLO. I understand that if there is a CDD pole one would expect that relation to hold approximately somewhere near the real CDD pole position. But I do not see the reciprocal implication. The authors also talk about its "correct position". That is known if you know the model a priori,and is certainly correct for the example. But... is it "correct" in general or the position is only approximated up to a given order in the series ?(As it happens with the modified IAM and the Adler zero). The authors should elaborate more on these questions.

d) In section 4.2. I guess $\sigma_i(s)$ is the same as $\phi_2$ or proportional to it. Please, for clarity, write the relation between the two quantities. I understand then the need for a dimensional factor to make a meaningful comparison with $\phi_4$. But, why the authors decide to multiply $\phi_2$ by powers of $f_\pi$?. They are making an estimate based on phase space, i.e. kinematics, whereas $f_\pi$ is a dynamical feature. I understand that using $m_\pi$, which is a kinematic quantity would give problems in the chiral limit, and $f_\pi$ is the only dimensionful quantity around. However,
the issue here is how much the four-pion state yields an inelasticity in two-pion scattering, i.e, their contribution to Im t(s). Why then$f_\pi$ and not $4\pi f_\pi$?. After all, the 4 pions appear in a two-loop diagram when they contribute to the pion-pion scattering amplitude. The authors even consider later $4\pi f_\pi$ as the suppresion scale for other factors. Please explain why this difference.

e) There is an important caveat about the comment section 4.6 on crossing symmetry violation. The Roskies relations are certainly appropriate to quantify such violation. However, in the works of Cavalcante and Sa Borges, the IAM is used {\it once it has been fit to data}. As a consequence, what those authors have actually tested with their approach is how well {\it those data in the fit} satisfy crossing symmetry, not the IAM violation of crossing. In other words, any other curve/method/model/freehand-drawing fitting those data would yield the same values for the Roskies relations. Thus, that approach, {\it as applied by Cavalcante and Sa Borges}, does not test the IAM, nor any model, just the data. The authors should reconsider the writing of this subsection in view of this comment.

f) I also have some very minor additional comments/suggestions related to footnotes and references, which I guess are very simple to address:

f.1) In the second note on page 2, there seem to be stronger reasons than that not to adopt Roy.Eqs. As a matter of fact, in the derivation of the IAM, the dispersive integrals also extend to infinity as in Roy equations. Thus, the IAM would be affected by this same caveat. Of course, in the IAM the elastic approximation is taken for the inverse as a simplification. One could think about making the same approximations for Roy eqs. formulated for inverse amplitudes. Thus, whatever happens in the right cut is not the most pressing problem. The real interest in Roy eqs. is to deal with the left cut as precisely as possible and this entails an infinite series over the partial waves in the crossed channels, although in practice reduced to the few first. It si also not clear how to formulate that for the inverses.

f.2) The authors claim that the IAM predicts the rho at 710 MeV with the "pure" ChPT LECS. They should provide a reference for this result and what do they mean by "pure", and why it is more reliable than the other several references about the IAM that the authors quote, where the rho seems to come out fairly well with different values of the LECS in apparently fair agreement with determinations from standard ChPT.

f.3) Page 1: in the references [5,6] the $f_0(980)$ pole is certainly found, but that is not the complete IAM as derived in the manuscript . Maybe the authors may consider adding J.R.Pelaez. Mod.Phys.Lett.A 19 (2004) 2879-2894. Moreover, when discussing the coupled-channel IAM in page 7, and its "less crisp theoretical basis" it would be appropriate to refer the reader to the full IAM works where the theoretical complications were highlighted . I suggest at least: F. Guerrero and J.A. Oller, Nucl.Phys.B 537 (1999) 459-476 as well as A.Gómez Nicola and J.R. Pelaez , Phys.Rev.D 65 (2002) 054009.

f.4) Page one. When mentioning variations of Bethe-Salpeter equations, the authors may consider citing J. Nieves, A. Ruiz Arriola. Nucl.Phys.A 679 (2000) 57-117

f.5) Page 2: when discussing the reconstruction of a heavy scalar particle with the IAM, a relevant reference is : A. Dobado, Phys.Lett.B 237 (1990) 457-462

f.6) On page 6: First paragrpah of Section 3. Ref. 15 is not about scattering and ref 25 has no disperion relations. Thus they do not fit well with what is being discussed in the sentence. For the IAM derivation, that function was introduced in refs 13 and 61, in both cases for scattering using dispersion relations. Actually, in ref.25 the IAM is introduced as a Padé approximant, an interpretation that the authors may consider mentioning in passing.

1) Provide an estimate of the highest mass resonances that the IAM could reliably predict, given the systematic uncertinties found here.
2) Provide at least another example of relative uncertainties evaluated for a resonance with different mass than the rho, possibly higher, to illustrate how the relative systematic uncertainty scales.
3) Comment on how the IAM systematic uncertainty compares with the expected uncertainty propagated from the error bars in the low-energy constants if the are obtained from non-resonant physics.
4) Clarify the appearance of $M_0$ from a chiral Lagrangian and why Eq.81 is a "well behaved {\it chiral} expansion", as well as the convergece of Eq.80, given that f and $M_0$ can be rather similar in size.
5) Clarify if it is the IAM that fails or is it the low-energy expansion that fails to reproduce the zero at $M_0$.
6) Clarify or comment the connection between Resonance saturation of the LECS and the presence of a CDD pole. Identify what is the resonance contribution to the LECS in the example.
7) Check if/how the presence of the CDD pole is invalidating the resonance saturation of the LECS in the HEFT example given in appendix A.2.
8) Clarify why/if the cancellation of the real part of the LO+NLO necessarily implies the existence of a CDD pole.
9) Explain why sometimes $f_\pi$ or sometimes $4\pi f_\pi$ are used as the suppresion parameters.
10) Please consider the suggestions for footnotes/references.