Benchmarking the Ising Universality Class in $3 \le d < 4$ dimensions

The paper by Bonanno, Cappelli, Kompaniets, Okuda, Wiese [BCKOW] studies the Wilson Fisher fixed point in d=4-epsilon dimensions for various values of epsilon using the conformal bootstrap and compares the results to other methods.

The paper adds 4 grid points in d in the range 3<d<4 (d=3.875, 3.75, 3.5, 3.25) to the 10 grid points for 2<d<3 already computed in a previous reference [19]. For each value of d, various conformal dimensions and OPE coefficients are estimated. For each data point a non-rigorous estimate of the error is also provided. The computation is done with a single correlator bootstrap setup (with c-minimization) which the authors themselves admit “has been surpassed by more recent implementations”.

The main result is a fit in d of the conformal bootstrap observables. The paper also provides many checks of the consistency of the fits with other techniques as resummation of the perturbative series, Monte Carlo simulation and other bootstrap approaches.

The paper is well written, clear and the results are interesting.
However unfortunately, reference [20] appeared a few months before with the same type (but better quality) of bootstrap results. In particular [20] studies the following values of d: d=2.6, 2.7, 2.8, 2.9, 3.0, 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.9, 3.95. It thus gives 10 new data points in the range 3<d<4 (instead of only 4). Moreover the results of [20] are obtained with a much more sophisticated technique (one of the “more recent implementations” above), which are expected to be far more precise: [20] is in fact using a multiple-correlator setup combined with the recently introduced navigator method.
While [20] is believed to be far more accurate than any single-correlator bootstrap, admittedly the authors of [20] do not estimate the errors in their data points so a honest comparison is not easy to perform (this is also hard since there is no way to know if the estimated error in [BCKOW] is correct).
However some plots of [BCKOW] clearly show that the bootstrap results of [BCKOW] are far less precise than [20]. For example in figure 20 it is shown that [20] finds two operators which lie (with good agreement) on the epsilon expansion results, while [BCKOW] only resolves a single operator which interpolates between the two curves.
One should also stress that when there are discrepancies between the results of [BCKOW] and [20], there is a good reason to prefer the ones of [20]. Indeed the latter are by construction inside an allowed island provided by the mixed correlator setup, while the estimates of [BCKOW] together with their full uncertainty can possibly live in the disallowed region and so they could be mathematically ruled out.

Because of the existence of [20], the bootstrap results of [BCKOW] are outdated. However they may be still instructive to show the power/precision of the single correlator bootstrap, which is much simpler to implement. [BCKOW] also give estimates for some OPE coefficients which are not presented in other works. The paper also provides fits of the bootstrap data which may be of interest. The thorough comparison with the resummation techniques is also a valuable addition to the literature.
I would thus recommend [BCKOW] for publication under major revision.

My main problem with the paper is that it is using outdated bootstrap results treating them on the same footing as the new state of the art results of [20]. A reader who is not expert in the bootstrap may likely think that the bootstrap results of [BCKOW] are comparable or even more accurate than [20]. This is misleading. In particular I do not agree with the choice of phrasing the paper as a comparison of many techniques the bootstrap results of [BCKOW]. The paper would be more useful if the comparison was made with the new data of [20]. This means that the authors could provide a polynomial fit of the data of [20] and use that to define the origin of their plots. They could then plot their 1-correlator results and show that these are close enough to the origin given by the fit of [20].
The broad logic of the paper (which should be stated from the abstract to the conclusion) should be:
- provide a fit of the bootstrap results of [20] and compare it to other methods like resummation, Monte Carlo, etc.
- provide a fit of the much simpler 1-correlator bootstrap and show that this is not too far from [20], thus showing that this technique could be used in other cases where the results of [20] are not yet available

In order to achieve this logic I propose the following changes:

1) Write a fit of the data of [20] for all observables for which this is available. For example formula (10) should be also compared with a new formula for a polynomial interpolating the data of [20]. Similarly for all other fits.

2) Change the origin of all plots to the one provided by the fit of [20] (when available).

3) Throughout the paper it should be made clear that the 1-correlator results are not the best ones available in the literature, but that they are simpler to achieve. For example in lines 91-95 only d=3 implementations are mentioned, while it should be stated that reference [1602.02810] already used a mixed correlator setup in fractional dimensions and that [20] uses mixed correlator with the navigator, thus surpassing the bootstrap results of [BCKOW]. The fact that the bootstrap results of [BCKOW] are surpassed should also be clear from the introduction and the conclusions where the works of [1602.02810] and [20] are not even mentioned in the current version.

4) In figure 4 (and similar) we can see that the data of [20] lies consistently below the shaded area of the fit of [BCKOW]. This suggests that the bootstrap data points of [BCKOW] produced by c-minimization have a systematic error, which is not taken into account in the paper.
If it is possible it would be useful to estimate such error.
An option is to simply increase the error in order to contain the points predicted by [20] (this is also a safe measure to be sure that future rigorous bootstrap bound will not completely rule out the 1-correlator fits of [BCKOW]). If these changes are not implemented, at least it should be stated that the method likely suffers from a systematic error which is hard to estimate.

Other comments/questions follow:

5) Since one of the most valuable parts of the paper is the comparison with resummation techniques, it would be useful to expand on the latter. Can you review the details of the various resummation methods used in the paper and how the errors are estimated?

6) Did you try to use other non-polynomial fits of the bootstrap data? For example, inspired by the Padé approximations of the perturbative series, one could try rational functions. It would be instructive to know if other fits have advantages.

See points 1)-6) above.

1- Well-written manuscript
2- Improved estimates for many operator dimensions and structure constants for $3<d<4$
3- Clear appendix with a worked example of Borel transform

1- Bootstrap study of a well-known system with no significant novelty
2- Discussion on Borel transform and resummation in the main text lacks details
3- No discussion of systematic errors

The paper concerns the $d=4-\epsilon$ expansion for the Ising CFT, and the aim of the paper is to compare predictions from this expansion with results from the numerical conformal bootstrap at finite $\epsilon$.

The method is a bootstrap scheme consisting of bounds from a single correlator together with central charge minimization, followed by an extraction of the extremal spectrum. This gives access to the leading critical exponents and a set of CFT-data (dimensions and structure constants with sigma) for low-lying operators. Errors are estimated by comparing the position of the kink in the bound and the position of the minimum value of the central charge.

The main conclusions are that the unresummed $\epsilon$ expansions give good agreement for $3.8<d<4$, and that resummed expansions give good agreement in the whole studied range $3\leq d<4$. The paper also gives a collection of estimates for CFT-data that is a nice contribution to the literature.

The manuscript is well-written and conveys its main results in a clear way. However, I have doubts about the novelty of the work, for the following reason:

The bootstrap method using a single correlator is a direct application of the method used by some of the authors in 2018 (ref. 19). But more refined techniques have been available for a while and applied to the Ising CFT. Specifically, in the range $3<d<4$, bootstrap islands have been known since 2016 (see 1602.02810; the authors may choose to include this paper in their reference list). The only novelty with respect to the bootstrap method is a denser sampling of points near the kink, and an additional point $d=3.875$. Also the estimates from resummation appear to be a direct application of previously developed techniques, however there is a lack of details that makes it hard to judge the potential difficulties behind the produced estimates.

Based on this, I make the judgement that the current manuscript does not meet the acceptance criteria of the journal, section "Expectations". Since the results of the study may still be a useful addition to the literature, I would reconsider my recommendation after major revisions. One way to meet the criteria would be to make a more elaborate discussion on the resummation methods, which are not given enough room in the current draft. By expanding on that aspect, the paper could become a useful resource for future projects comparing perturbative series and non-perturbative results.

1) Could you give some more details on how the resummation was computed? How were the errors estimated? What value does the parameter $a$ take? Which variational parameters were used? Note the “General acceptance criterion” number 5 of this journal.

2) You mention another resummation technique (hypergeometric resummation of ref. 55). Could you compare against this technique and give recommendations based on the comparison? Also, it appears like the self-consistent resummation for $3<d<4$ was only performed for $\Delta_\epsilon$ and not the other quantities.

3) Footnote 8 comments on the inclusion of seventh-order results, saying that they give larger errors than sixth-order results. Could you provide more details?

I also have the following remarks/questions:

4) Comparing with the "Navigator" of ref. 20 (red triangles in the draft) there appear to be systematic errors of the same size, or larger, than the errors given. This is the case at least in figures 4 and 17. Could you discuss the origin of these systematic errors and, possibly, give a method to estimate their size?

5) What does 190 components correspond to in terms of the value $\Lambda$ that is commonly given in the bootstrap literature?

6) In table 1, comparing the size of the errors in $d=3.875$ and $d=3.75$, precision for spin 4 is much higher in $d=3.875$, and for spin 0 and spin 2 much higher in $d=3.75$. Is there any explanation for this?

7) In appendix B, you mention that convergence for $|z|<1$ is difficult to prove. Does it not follow from the assumptions made on the analytic structure in the $t$ plane?

See points 1) to 7) in the report