A theory vade mecum for PSI experiments

REPORT 1
========

Here we list our reply to the various points of Referee 1 and indicate the changes we have made. Equation numbers and line numbers refer to the revised version. For clarity and better connection to the referee's comments, the old numbers are sometimes given as {old NR}.

(i) Yes, Ref[1-23] will be finalized during the proof process, once the detailed information is available.

(ii) We fully agree with this remark but would like to point out that this has been mentioned after (5.35) where we state that the matching in (5.33) or (5.11) and (5.12) is done in the limit \delta r -> 0 (Line 336).

(iii) We have restructured Section 5.2 and at the beginning of Section 5.2.2. introduced a paragraph where we mention low-mass BSM and briefly discuss the axion / ALPs.

(iv) Yes, some partial results are known beyond LL. Our original formulation was indeed somewhat misleading and we have adapted this,
see Lines 180-182. We refrain from adding further references as it is impossible to do justice to the vast literature within our limited
scope.

(v) Indeed, a huge number of original articles were not explicitly mentioned in connection with [41-43] {old 37-39} and in fact many
other places as well. Given the space constraints we have used review articles whenever possible and invite the reader to consult those for further references. Otherwise the number of references would get out of hand.

(vi) We completely agree with the referee and thank him/her for pointing this out. The discussion has been adapted and extended, see
(5.23) - (5.27).

(vii) We are not entirely sure we understand the referee's point here. Refs [59-62] {old 54-57} are all NNLO calculations and we have
omitted all NLO calculations (and calculations in the logarithmic log[mmu/me] approximation) to keep the number of references under
control. We have added a remark (Line 333) to stress the importance of the NNLO QED calculation.

(viii) We have given a second version of (5.35) {old 5.32} where we state the use of the on-shell scheme. We also add a few remarks about the structure of \delta r (Lines 330-332). But a more precise definition of \delta r does in our view not belong into a specific PSI
theory discussion, even though it is of course of utmost importance.

(ix) Apparently our text was not clear enough and we have edited this part (Lines 310 - 343). The statement is that \delta q obtains
hadronic corrections only at NNLO. What the referee has in mind is that \delta r obtains hadronic corrections at NLO. This is true of
course and usually dealt with through the renormalization of \alpha (now mentioned in Line 331). Once more, we would like to point out
that we do not want to discuss the details of electroweak precision fits, as we feel they are not part of theory for PSI experiments. But
we hope with the current formulation the misunderstanding has been cleared up.

(x) This is an interesting question but we feel it is not really part of our article.

(xi) These typos have been corrected.

REPORT 2
========

We gladly take up the two recommendations of Referee 2.

In particular, we have restructured Section 5.2, starting with a brief outline and then structure the section into 3 subsections. In
addition, at the beginning of Section 5.2.2 we have included a brief discussion of low-mass BSM.

Finally, we have extended the introduction a bit and have given illustrative examples of experiments at the intensity frontier and
precision frontier.