Electroweak double-box integrals for Moller scattering

We thank the two referees for the careful reading of the manuscript and their suggestions.
Below we list the modifications we made to the manuscript, following the order in the two referee reports.

Referee 1:

- We see the point of the referee, although section 3 is of course the part where most of the work for this article went into it.
We moved the list of the master integrals to the appendix and added additional explanations to section 3.

- As requested by the referee, we will provide the benchmark values in an electronic file.
In this case there is no longer any need to have them as tables in text.
We therefore shortened section 5 considerably and only give in a short table the numerical values of a few selected integrals.

1. We added a reference, where it is explained how this can be done in practice.

2. The two questions the referee asks are even for mathematicians difficult to answer.
We will not have too much to say about it.
But we would like to stress that the answer to these questions is not relevant in our approach, there
is nothing wrong if the list of differential forms is non-minimal.
For the elliptic case we make sure that a non-minimal set of differential one-forms will not affect the numerical
stability of the Taylor expansion (this is done by first expanding symbolically in the expansion parameter,
then simplifying the coefficients analytically
and only then evaluating the resulting Taylor expansion numerically.)

3. We did use the PSLQ algorithm for a subset of boundary values.
As we have only very few constants to fix, 50 digits was enough.
We added a comment.

4. We choose the boundary point (and the integration path) such that there is no need for analytic continuation.
We modified section 4.2 to make this more clear.

5. Our results are inline with the remark of the referee. We added a comment at the end of section 3.

6. Although a detailed phenomenological analysis is not the main scope of this article, we believe
that the numerical evaluation for the elliptic Feynman integrals outlined in section 4.2 and 5 for the Moller region is close to the optimal one,
both with regard to numerical stability and speed.

7. For topology A we compute the Taylor expansion of the integration kernels "on-the-fly". This allows for arbitrary precision,
as in the Taylor expansion as many terms as needed are included.
But the expansion is only done at run time and contributes to the CPU time.
For topology Btilde we pre-compute the Taylor expansion to a fixed order. This allows only for a fixed precision, but is faster.
For phenomenological applications the latter approach is preferred, if however the master integrals are used as boundary values for
more complicated Feynman integrals, the first approach will be more useful.

8. We adjusted the references.

Referee 2:

General structure and style:

- Section 3 and 5: see above.

- We rephrased the penultimate paragraph of the introduction.

- We corrected "setup".

Requested changes:

- We added the references.

- We added Drell-Yan and quark-pair production in the introduction. We also added a comment in section 3 to emphasize that the results
of section 3 are valid for all crossings.
In section 4.2 we added a comment that from this point onwards we specialise to the kinematic region of Moller scattering.

- The standard approach is to express scattering amplitudes (or gauge-invariant subsets thereof) in terms of master integrals.
The master integrals can be grouped into topologies and each topology can be calculated separately.
This separates different problems.
What we do in this article is to show that some of the more complicated topologies can be computed, removing one bottle neck.
Of course, to get the scattering amplitude more work needs to be done, but this is not the scope of this article.

- We thank the referee for pointing out this inaccuracy. We reworded the abstract, the introduction and the conclusions.

- We are in contact with colleagues, which are very much interested in these logarithms.
Our point in section 6 is that we get these terms at no additional cost from our calculation.

- We removed the word "usual".

- The answer to these questions are given in section 4.2, we added a comment to section 5 where we point the reader to section 4.2.

We fixed the small typos mentioned by the referees. As far as the spelling of M{\o}ller or Moller is concerned,
we follow the convention that whenever we refer to the physicist or the scattering process, we write "M{\o}ller",
while the experiment at Jefferson Lab is written as "Moller" (that's the name of the experiment).

In addition, we corrected small errors and typos:
- Section 6: we corrected "all all" to "and all".
- Definition of $J^B_{13}$: It should read $\mu^4$ (not $\mu^2$).
- Appendix A.1: Colour coding of sector 75, Topo B corrected.