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

Dear editors,

We thank the referees for their comments. We are resubmitting our
paper with an addition required by the first referee and some
modifications suggested by the second referee.

{\bf Answers to referees' requests}

{\bf First referee (report on 2023-1-31, revised version)}


{\it 1) (cf. question (3) of earlier report) Regarding comment 3, I
understand the explanation for why the seventh-order results are not
included in the draft, but I think it would be useful to provide
more information in the draft. For instance, footnote 10 could be
expanded with estimates in d=3 that compare the six-loop and
seven-loop resummations for the three main critical exponents
considered, giving the central value and uncertainty for each. This
would give the reader a chance to examine the choice of limiting to
six-loop results in the rest of the paper.}

As said in earlier reply, the seventh-order perturbative terms have been
obtained by Schnetz, but were not confirmed by other authors, and there are
some concerns about their validity in the community. Let us look at
the record: sixth-order calculations were independently checked by
Panzer, Kompaniets and Schnetz; fifth-order results were originally
incorrect and had to be modified by subsequent works.

On this basis, we cannot consider perturbative results of such a
complexity which have not been confirmed independently.
Using these data would make us prone to an erratum, while the gain,
assuming correctness, seems minimal.
In footnote 10 we were
rather vague in order to not offend Schnetz; we do not want to say that his
results are incorrect, simply that they need an independent check. We
modified the footnote 10 in a more explanatory way.

\bigskip

{\it 2) On the other hand, the description of resummation methods is
rather heavy on the toy model example, and provides almost no
details on the adaptations of the general methods to the paper at
hand. No intermediate results are given that make it possible to
follow the computation. Neither does the draft have any associated
computational files or software, or clear references to where such
software can be found.}

The resummation methods used in our work are the direct adaptation
to varying $d$ of the analyses in Ref. [40] and [41]. It is not possible
to do justice of the many aspects of these procedures in a short presentation
that would fit in our paper.
Furthermore, Sec. V of Ref. [40] offers a very clear and detailed
description: after the introduction in our paper, the reader can
easily follow all the steps there.

At any rate, in Appendix B.2 we added a survey of Sec. V of Ref. [40],
which can help the reader to find his way in this reference, including
main definitions and main steps, correspondence of notation, etc. We
also added the list of values for the variational parameters entering
the resummation process for any $d$ value considered.
Concerning more technical information, we address the reader to
the supplementary material of Ref. [40] that is available on arXiv:1705.06483.

\newpage

\bigskip

{\bf Second referee (Report on 2023-1-30, revised version)}

{\it I do not understand the choice of plotting Figs. 7, 9 and 13
including only a single point at d=3.5 from reference [21], while
data is available for d=3.0, 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8,
3.9, 3.95. Why one point and why d=3.5? I would add all possible
points since they are available. If the problem is only at the level
of clarity of the plot (because the red triangles are big and there
are many of them), the authors could plot the curve that
interpolates the data of [21] (even without writing the functional
form of such curve), which would in my opinion would be a better
reference "to assess the negligible difference between the two sets
of bootstrap data in the comparison to the epsilon-expansion".}


We put the $d=3.5$ point in Figs. 7, 9 and 13 because it corresponds
to the larger difference between navigator results and our fit, for all
the three critical exponents.
We did not put all the navigator points because they would
have messed up the plots.

Following the referee's suggestion, we put the line linearly interpolating
navigator data in these three figures. Now the differences between the
two sets of bootstrap data are clearer. We also added a remark after
Fig. 13, saying that the resummed perturbative data match
better the navigator results than our ones for $4>d \ge 3.5$.


Best regards,

The Authors (Bonanno, Cappelli, Okuda, Kompaniets, Wiese)

- Sec. 3.2 - 4th paragraph - sentence modified

"A complete account
303 of these methods is too long to be presented here: nonetheless, our introduction, App. B and the
304 paper [38] provide enough background for accessing the original work."

->

"A complete account of these methods is too long
to be presented here; nonetheless, we provide some introductory
material that will allow the reader to assess the
original works. In App.~\ref{appendix:baby_integral}, the resummation
is worked out in a toy model, where one can compare it with the exact result.
In App.~\ref{app:KP17_details}, instead,
a ``reader's guide'' to Ref.~\cite{panzer} is presented, together with
the values of the resummation parameters used here."

- Sec. 3.2 - footnote 10 modified

"Resummations in this section use the 6th-order expansion that received several checks. Contrary to expectation,
the apparent error at 7-loop order seems to be larger than that at 6-loop order, in all resummation schemes we
tried [40, 41]."

->

"Resummations in this section use the
$6$-loop results, that were verified in several independent
works~\cite{panzer,Kompaniets:2019zes,Schnetz:2016fhy}. We do not
use the $7$-loop results of Ref.~\cite{Schnetz:2016fhy}, since they
were not yet checked independently. Past experience, e.g., with the $5$-loop
results, teaches us that involved perturbative calculations require
confirmation."

- Figs. 7, 9, 13 modified + added a sentence in the caption

"We also plot a
solid red line linearly interpolating results of
Ref.~\cite{Henriksson:2022gpa} for $4>d\ge3$."

- Sec. 3.2 - last paragraph - sentence modified

"A drift towards lower
values for the green epsilon-expansion points is seen, as for $\gamma_\epsilon$. "

->

"A systematic difference between
bootstrap and epsilon-expansion points is seen for $d \to 3$,
similar to what was found for $\g_\e$ in Fig.~\ref{fig:e-eps}.
Such a drift is smaller for the navigator
results~\cite{Henriksson:2022gpa} (red line) than for our data,
for $4 >d \ge 3.5$."

- Added new Appendix B.2 (old Appendix B is now Appendix B.1)