Crossing Symmetric Spinning S-matrix Bootstrap: EFT bounds

Dear Editor,
We thank the referees for their valuable comments and we are resubmitting after making the changes suggested. We have attached the report with answers to queries by the referees below.

Referee report 3

We thank the referee for his/her valuable comments. We have tried to address the questions raised both in this report and we have also made changes to the draft accordingly which we have highlighted in purple for the ease of the referee. We briefly list the major changes and clarify issues raised below:

1) We thank the referee for pointing this out. The data in table 2 is for $TR_U + PBC$. We have rectified that in the new draft.

2) The referee raises a valid point his/her analysis. The analysis for the range of $a$ has been done in two steps.
a) In section 3.5.2, we try to show that the dispersive part can be put into the Robertson integral representation which requires that the absorptive part $\frac{2}{\p}\int_{-1}^{1}d\xi {\mathcal A}(\xi,s_2(\x,a))$ is positive. Now ${\mathcal A}$ consists of two parts as denoted in the partial wave decomposition eq 3.9- the partial wave coefficient $a_J(s)$ and the Gegenbauer polynomials. In this subsection, we have not imposed any constraints on the partial wave coefficients $a_J(s)$, it is just a kinematic constraint- let all the Gegenbauer polynomials be positive. This is the condition that $\cos \theta \geq 1$. No dynamical information about the theory we are considering is there beyond the fact that the theory is unitary (i.e $a_J(s)>0$). More precisely, this is the most stringent condition one can impose. We are not considering the possibility that bounds on relative ratios of $a_J(s)$s in allowed scalar EFTs might also allow us to have $\cos \theta <1$, in which case Gegenbauers can be negative but overall, the sum can be positive. This leads to the bound eq 3.35

b) In section 3.5.3, we now include the locality constraints (eq 3.14) too in our analysis. This is included by effectively adding ``0" to the amplitude as indicated in eq 3.33. The second equation in the bracket is essentially the locality constraint eq 3.14. Locality constraints are constraints on the partial wave coefficients and hence in a way constrain the allowed space of $a_J(s)$ for scalar EFTs. So imposing them in eq 3.41, we are effectively doing the analysis after imposing dynamical constraints on our partial wave expansion. Now, instead of demanding all the Gegenbauers to be individually positive, we can have $\cos \theta <1$ but still the entire sum remains positive since eq 3.14 put constraints on ratios of different $a_J(s)$. This analysis indeed tells us that we can have such a situation as indicated by eq 3.46.
We have rewritten both sections 3.5.2 and 3.5.3 to address these issues issue and make the LSD analysis more transparent. In particular, the key difference in analysis of 3.5.2 and 3.5.3 is that in the latter section, we have introduced the locality constraints (eq 3.14) into the usual positivity of the absorptive part and the analysis of range of $a$. We have also added a flowchart on page 24 to explain our algorithm better.
5) The locality constraints coming from the crossing symmetric dispersion relation are either same or a linear combination of the null constraints coming out of fixed t dispersion relation in reference [5]. We have an inversion formula for Wilson coefficients and using them we can try to maximize or minimize the Wilson's coefficient using SDPB. This would be similar to Ref [5] and lead to the same bounds.

4) This is because we do not solve the locality constraints completely. We have also not used the non-linear constraints imposed by Toeplitz determinant conditions outlined in section 4 of reference [25]. We make a note of this in our conclusions on page 37. These non-linear constraints will further constrain the Wilson coefficients. Hence solving the locality constraints to higher orders and non-linear constraints will lead to more optimal bounds.

5) We have added a few lines to the draft better explaining the comparison to [22] and [7] for the photon and graviton bounds respectively. We briefly list them here as as well for the perusal of the referee:
Photon case:
The bounds in eq 5.25, 5.26 of the draft can be compared with those in table 1 of [22] and the bounds agree well.
Figure 2 in the draft can be compared with figure 1 in [22] and regions are similar except that the left region in [22] is slightly smaller( with a triangular region in bottom whereas we get a rectangular region) than ours, right region is identical.
The $w_{20}^{(x_1)}$ bound in table 3 is the same as the one in [22].
Graviton case:
The authors of [7] did not impose full three -channel crossing symmetry in their analysis so a direct comparison for Wilson coefficients cannot be made. However we can say the following:
The line plot in figure 6 in the draft is a direct analogue of figure 8 in [7] and we have plotted data points from [7] for comparison.
In figure 7 in the draft we have once again used data points from [7] for comparison.

6) We have thoroughly read and fixed any typos that we could find.

We thank the referee for all his/her valuable comments and suggestions which helped in for improving the accessibility and readability of the manuscript for general readers.

Referee report 4

We thank the referee for his/her valuable comments. We have tried to address the questions raised both in this report and we have also made changes to the draft accordingly which we have highlighted in purple for the ease of the referee. We briefly list the major changes and clarify issues raised below:
a) We have aimed to clarify how our range of $a$ that we get after including the locality constraints in eq 3.41 is indicative of low spin dominance. To this end, the analysis at the end of page 21 is an independent analysis where we truncate the partial wave expansion to some spin $J_c$ and derive the range of $a$ with this truncated amplitude. The table below eq 3.49 shows the possible ranges of $a$ for different possible $J_c$. Note that in this analysis, we don’t use locality constraints and also assume that partial waves for $J>J_c$ cannot ruin the positivity of the truncated expansion. Apriori we cannot say which $J_c$ is relevant for the massless scalar EFTs but we note that the bound 3.46 (which was without the truncation and using locality constraints) coincides with this independent analysis for $J_c=2$. In particular, the first entry in the table below eq 3.49 tells us that eq 3.46 indicates $J_c=2$ for scalar EFTs. Hence this is indicative of spin 2 dominance for massless scalar EFTs- our crossing symmetric analysis naturally leads to such a conclusion once we impose locality constraints in determining the range of $a$. We have rewritten the last part on page 21 to provide more transparency.
b) The referee raises a valid point his/her analysis. Note that all Gegenbauer polynomials are positive for $\cos \theta \geq 1$. So, forcing each term to be non-negative does not imply just $l=0$ contributions. Since $t\geq0$ for $\cos \theta \geq 1$, the higher Gegenbauers are also positive. There are two elements in this analysis- the partial wave coefficient $a_J(s)$ and the Gegenbauer polynomials. In eq 3.35 we have not imposed any constraints on the partial wave coefficients $a_J(s)$, it is just a kinematic constraint- let all the Gegenbauer polynomials be positive, which requires $\cos \theta \geq 1$. Any dynamical information about the theory we are considering is not there beyond the fact that the theory is unitary (i.e $a_ J (s)>0$). Hence, this is the most stringent condition one can impose. We are not considering the possibility that bounds on relative ratios of $a_J(s)$s in allowed scalar EFTs might also allow us to have $\cos \theta <1$, in which case Gegenbauers can be negative but overall, the sum can be positive. Locality constraints are constraints on the partial wave coefficients and hence, in a way, constrain the allowed space of $a_J(s)$ for scalar EFTs further. So imposing them in eq 3.41, we are effectively doing the analysis after imposing dynamical constraints on our partial wave expansion. It does tell us that $\cos \theta <1$ but still the sum is positive because of the relative ratios of $a_J(s)$. To summarize, eq 3.35 is outcome of just kinematical positivity constraint while eq 3.46 uses the dynamical information about UV consistency requirements on relative ratios of low energy partial wave coefficients on top of the usual kinematical positivity.

We have rewritten both sections 3.5.2 and 3.5.3 to address issues raised in the previous two questions better and added a flowchart on page 24.

c) It is true that that we obtain bounds on combinations of Wilson coefficients but that can be systematically disentangled. The discrepancy is, however, due to the following reason. In the crossing symmetric analysis, locality constraints are the dynamical constraints that we impose to get the range of $a$ in our analysis. Note that we only solve a finite number of locality constraints in our analysis (eg: for the scalar case we note this on page 21 footnote 13). However, the low energy expansion contains the relevant Wilson coefficient + an infinite number of such locality constraints as noted in eq (4.2) which we set to 0 by hand. Solving for more accurate range of $a$ with more null constraints will lead to more precise match of data in table 2 with reference [5]. We have also not used the non-linear constraints imposed by Toeplitz determinant conditions outlined in section 4 of reference [25]. We make a note of this in our conclusions on page 37. These non-linear constraints will further constrain the Wilson coefficients.

d) We thank the referee for this interesting question. While the equivalent of the $PB_C$ bounds have been noted in [5], we have not found a way to express $TR_U$ in the language of [5]. The $TR$ properties crucially depend on the CSDR kernel which has no analogue in the fixed $t$ dispersion relation. We also note that the simplest of bounds on $\frac{w_{01}}{w_{10}}$ cannot be derived without using numerics in the approach of [5] while we are able to do it analytically using GFT techniques (see discussion around eq 3.54 and eq 4.1 for the massless scalar case).

e) In general, the GFT techniques, as written will not be applicable for loops since the techniques crucially rely on the expansion in z and a to be meromorphic while loops introduce non-analyticities. It might be possible to apply the same by expanding the crossing symmetric combinations x and y about some non-zero value. This is an interesting direction to pursue.

We thank the referee for all his/her valuable comments and suggestions which helped in improving the accessibility and readability of the manuscript for general readers.