On currents in the $O(n)$ loop model

Changes according to the report 1:

1. We have added a section for the review on the spectrum of the $O(n)$ CFT as Section 2.

2. We have sketched the derivation for the identities of $q_{r,s}$ in (4.17) [3.17 in the old version]. See the details between the equation (4.17) and (4.19). 



The main idea here is that the ratios of structure constants from four-point functions of $J$ and $\bar J$ in (4.19) are completely determined by the degenerate-shift equation, which also coincides with the ratios of the corresponding reference structure constants.


3. We have summarized the main ideas of the numerical bootstrap techniques required for our results as bullet points on page 18.


4. We have added different colors to different $O(n)$ labels on diagrams in (A.28).


5. We have updated reference [25].


Clarifications according to the report 1:

1. What we call reference structure constants are universal factors of structure constants that are independent of model’s global symmetry, that is to say reference structure constants serve as references for structure constants of primary fields with the same dimensions, but transform in different $O(n)$ representations. 



We have added the above clarification above the equation (4.14a).



From our results (4.8), we expect that the three-point functions $C_{JJV}$ is a product between polynomials in $n$ and the reference structure constants, as written in (4.24). This expectation seems to make sense because the model’s symmetry is, a priori, a product between $O(n)$ and conformal symmetry.



2. The equation (4.16) [3.16 in the old version] is a conjecture for the bounds of the polynomials’ degree for any value of $r,s$: this has been now stressed above (4.16), however we have only explicitly checked the inequalities in (4.16) for $r<=5$.



In general, there are certainly polynomials that do not saturate the bounds (4.16). Unfortunately, our results on the polynomials and their degrees were obtained based only on the numerical observations, and we do not yet know how to determine their degrees by analyzing the crossing-symmetry equation.



Above the title of Section 4.2, we have added a paragraph to stress that our results on the polynomials were obtained based on purely numerical observations.


3. We have considered $<\bar J\bar J\bar J\bar J>$ instead of $<JJJJ>$ due to purely technical reasons.



Our numerical bootstrap program, initially proposed in the paper (*), was designed for four-point functions of primary fields with positive Kac indices. Recall that $\bar J$ carries the Kac indices $(1,1)$ whereas we have $(1,-1)$ for $J$.



In practice, interchiral blocks of those two four-point functions take slightly different forms due to the singularities in the degenerate-shift equations and logarithmic blocks, and the details on how to regularize those singularities is given in Section (3.1) of the paper (*). And we chose to write the program that fits with the case $<\bar J\bar J\bar J\bar J>$



However, since structure constants in $<\bar J\bar J\bar J\bar J>$ and $<JJJJ>$ are related by the degenerate-shift equation, it is enough to consider only 1 of them, and we chose to consider $<\bar J\bar J\bar J\bar J>$



*=https://arxiv.org/pdf/2111.01106.pdf

Clarifications according to report 2:

1. In each interchiral block, we apply the truncation to any fields, both primaries and descendants, such that the remaining fields obey our desired bound. We have stressed this point in the first bullet point on page 18.



That is to say, we truncate each interchiral block to be a finite sum of truncated Virasoro blocks wherein we only include the contribution from the descendant fields up to some certain levels.


2. In practice, it is more convenient to discuss the analyticity of (4.8) in terms of the parameter $\beta^2$ because four-point functions of the $O(n)$ CFT depends explicitly on $\beta^2$. Recall that $\beta^2$ is related to $n$ through equation (2.1).



The analytic structure constants (4.8) [3.8 in the old version] are only valid for non-rational value of $\beta^2$ because (4.8) could have zeros and poles at rational value of $\beta^2$. The latter case also includes some integers $n$, for instance $\beta^2 = 1$ corresponds to $n=2$. 



We expect that crossing symmetric solutions for rational $\beta^2$ can be obtained as rational limits of (4.8) wherein we expect that those poles in rational $\beta^2$ always cancel each others. 



While we do not know yet the complete mechanism of pole cancellation, we believe that this should be true because the four-point functions in the $O(n)$ loop model on the finite-size lattice exist for generic $n$, including $\beta^2$.

We have added the above clarification in the first paragraph of page 14.