Selection rules for the S-Matrix bootstrap

Dear Editor,
Please find below a detailed list of responses to the comments from the referees. We thank the referees for their valuable comments which have no doubt improved the readability of our paper.

Sincerely,
Anjishnu Bose,
Aninda Sinha,
Shaswat S Tiwari

Anonymous Report 1 on 2021-4-9 Invited Report

1-Figures have been rearranged. Fig 10 is now Fig 3. R^2 and slope for odd spins have been added to fig 6. It is difficult to demonstrate non-linearity because S-matrices away from A, B, C and D have one or more than one missing peaks.
2- We think this deviation is related to the imaginary part of the resonance masses. It does not match with experimentally observed values for the \rho-river. However, this may just be because we are operating at low N_max and L_max. We need to investigate more to decode the significance of this new river.
3- Fixed. The footnotes point correctly now.
4- Some changes to the intro have been made. We hope that the reviewer is satisfied.
5- LS is just for comparison, like \chi PT. It was exciting to see the higher loop \chi PT drift towards the kink. In the same light, it is fascinating (and completely unexpected) for the LS line to have an intersection near the kink. Even though it is non-unitary, it does have good regge behaviour.
6- This has been mentioned separately in footnote 6.
7-Changes to introduction have been made.
8- We have included the reference to equation A.7 in Appendix A where scattering lengths and effective ranges have been defined.
9-Further description and an example reference has been included.
10- We have added this to the caption.
11- An explanation has been added to footnote 7.
13- Fixed
14- We have added \sigma (s_0,s_2). However, in section 4 we deal with the boundaries (where s_2(s_0)), hence ultimately, we do have \sigma(s_0).
15- The red curve gives E, the blue curve gives \sigma_{\pi^0 +\pi^0 to \pi^0+\pi^0} and black curve is for \sigma_{ \pi^0 +\pi^0 to \pi^0+\pi^0} (as mentioned in the captions). The local minimum in the red curve is correlated with the global minimums of the blue curve and the black curve, while the global red minimum occurs elsewhere for both upper and lower boundary. This signals a complex relationship between E and \sigma.
16- Figure 14 (a) and (c) for s_cut=375 are there in Fig 4 (given by the blue lines). We think Fig 4 will become cluttered if we include the variation with s_cut.
17- A justification has been attempted in the line below equation 4.3. We wish to include the contributions of all experimentally known resonances (the peaks for \ell=7 occur in the range 300-360) while averaging the cross section.
18- We have added more details.
19- Yes, we find the results good (as mentioned in 1st paragraph of section 3). The points are close to the experimental range.
20- The dots represent all detected peaks for the S-matrix at a given s_0 (U stands for upper boundary and L stands for lower boundary). The odd peaks are now coloured red for better clarity.
21- They are joined just to assist the eye in tracing the behaviour across the upper boundary. You may notice that a few s_0 values either have only black dots or only red dots. Such S-matrices either have missing even peaks or missing odd peaks.
22- New Convergence graphs have been included (check Fig 12)
23- You are correct. Since we are only imposing unitarity up to ell=19, we cannot go very high. \ell=8 suffices in distinguishing between different regions (A, B, C and D).

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Report 2 by Yifei He on 2021-4-11 Invited Report

1-The data used was taken from PDG. Appropriate citation has been added to fig 2 and fig 9 (previously fig 8).
2- 1909.06495 has been cited.
3- A different convergence graph has been added. Hope it is clearer to understand. About the 2nd point, indeed elastic unitarity does get satisfied for low energy and low spins. In fact upto ell=3, we barely notice any change upon division by |S_ell(s)|^2. However beyond that, resonances do occur at a higher energy and for higher spins. Since we are imposing unitarity upto ell=19 for the majority of S-matrices, elastic unitarity cannot be guaranteed at higher ell and higher s. This is where division by |S_ell(s)|^2 becomes relevant.

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Anonymous Report 3 on 2021-4-16 Invited Report

1-D does not show peaks for \ell=8. hence it is not of the same quality as A, B and C. Furthermore, only the tip (approximately near s_0=0.63) shows any linearity, which is smaller than the linear regions of A, B and C. However, we do not expect the LS line to uniquely determine the regge behaviour along the boundary. LS is a non-unitary, zero width resonance theory which should only be considered as a reference. The fact that there is even some correlation between the LS line and the observed regge is non-trivial.
2- Yes, the low energy constants have uncertainties. We have calculated the uncertainty corresponding to 1-loop (orange cross) and 2-loop (red cross) in fig 1. The s_0 uncertainty is around 0.12 and s_2 uncertainty is around 0.08 (mentioned in footnote 3).
3- Yes, ideally one should have elastic unitarity to observe a connection between the zeros and peaks. However, it is not yet possible to impose elastic unitarity in the framework of bootstrap. Since, we do not have elastic unitarity, the breit wigner formulation holds only approximately. This is the reason we choose to check the peaks of |f_{\ell}|^2/|S_{\ell}|^2 to improve our results.
4-The peaks here are approximate. Their real part is close to the expt, but the imaginary part is off. We expect convergence in the decay width for larger values of N_max and L_max. It is not within our numerical abilities to find the zero corresponding to the pole. This is something we intend to do in the future. However, using the partial wave peaks, one does get good values of \sigma resonance and its deviation from experimentally measured value is directly correlated with decrease in R^2 as we move across the river boundary.
5- If you want SU(2) isospin to be exact, then yes we require m_up=m_down. However, the mass difference in Pions can be shown to be a QED effect in leading order and receives only small corrections from QCD effects of m_up!=m_down. Since we are not considering QED at all, to the leading order you can still ignore the quark mass difference arXiv:hep-ph/9611331v1 .
6-This has been fixed.
7- Black corresponds to even spins and red corresponds to odd spins (now mentioned in the caption).