Physics:
- I miss a section explaining bremsstrahlung correction. It is not clear if the numbers you give in Section 3 are before or after correction.
- (385-393) since \epsilon_kin includes bremsstrahlung, it must also be sensitive to the simulation of detector material. How is this accounted for?
- (443-444) Shouldn't the integrals be identical? It should not matter whether efficiencies vary more rapidly. Does it?
- (Tab.2)   I am completely confused: doesn't FSR lead to migration between bins in rapidity? Why are all numbers smaller than 1?
- You use pseudorapidity and rapidity interchangeably, which is correct at large momenta, but give limits 2<eta<4.5 and the say the efficiency is zero above y_Z>4.25. What do you do with Z between 4.25 and 4.5?

General editorial comments:
- You could add many plots in additional material that would be useful for talks. A mass distribution for instance, but also some more you have shown in the approval talk.
- You systematically write "the electron and the positron" but in some occasions you use "electron" to stand for both. For instance L.380.
- Cite the relevant Atlas and CMS papers.
- Use the LHCb symbols everywhere e.g. for TeV (\tev)
- At several places (abstract, L241, L481) you write that you know the luminosity approximately. This is strange as we know the luminosity to an amazing accuracy of 1%. Please remove the word “approximately”. In addition, please quote the lumi with the same number of digits everywhere (preferably 2.0 instead of just 2).

Textual:

- “The acceptance of the measurement … in the range 60-120 GeV”. This sentence does not read well and has to be rewritten. Some suggestions. Write "The acceptance of the detector” or “For this measurement the acceptance is defined …”.  Write “defined to be in the pseudorapidity range 2 < eta < 4.5 and for transverse momenta above 20 GeV for the electrons ..."
- suggest to start a new sentence "Their invariant mass"
- “which is given as the third uncertainty”. The antecedent of which is not clear. As it is now, the antecedent is “luminosity effects”. Suggestion “… , the second is systematic (excluding luminosity effects) and the third arises from the uncertainty on the luminosity.” (in any case, "from" is missing in “arising FROM the luminosity”)
- Remove “also” in “Differential cross-sections are also presented…”. Or move “also” to the start of the sentence.
footnote on title page : the author list should be at the end but is not.

214: "The effective *geometric acceptance* of the LHCb detector". I would not put [1] here.
216: “whose good acceptance”. Good sounds strange here. Not clear what you mean. Maybe use “effective” instead. “extends UP to”
218: Remove “particularly"
220: Z->mumu is not "straightforward". Rephrase.
224: different _sources_ of systematic (else it could mean different magnitudes)
226: “momentum-analysing magnet” sounds a bit strange. The magnet is not analysing. In the second part of the sentence you say that the energy of the electrons is frequently degraded, which sounds weird. Suggest “...before they reach the magnet, and their momentum measurement is therefore degraded by bremsstrahlung. This effect cannot be completely recovered..." or "[...] before they reach the magnet. Therefore, the momentum measurement is degraded by bremsstrahlung, which cannot completely be recovered [...]"
228: "\approx" -> "approximately", or "around"
229: “In consequence the electron direction..”. Suggest “As a consequence the initial electron direction…”
230: “their measured energies” -> “their measured momenta” (in the tracking we measure momenta, not energies). Else it could be understood that we use the calo to determine the electron energies.
230: Add comma after “Therefore”
234+: I would prefer \frac{\tan...}{\cosh...}
237: either remove "respectively" - there is no ambiguity possible - or put commas around it
239: maybe this is obvious, but since this is a production measurement, it may be nice to mention once that Z production is studied in proton-proton collisions
242: "sqrt(s)=7 tev analysis" is jargon. It's not the analysis that is at 7 TeV but the beam CM energy. Rephrase. Same for "measurement...  being" that is also not at 8 TeV (you measure a cross-section, not an energy). This sentence is also too long. Pleas split it.
242-246-247: three instances of "measurement". Avoid repetitions.
244: remove “and”
245: replace first “and” with “, from”. And write “and FROM AN improved modelling…”. Have both the detector MC simulation and the modelling of bremsstrahlung been improved, or is the second a consequence of the first? (if yes, maybe add "consequently")
Footnote 1: shouldn't the second Z be Z^0?
Footnote 2: You need it already in the definition of phi*, as pt/M does has units otherwise. Can you also keep this footnote on the same page?

249-252: This paragraph does not work. Sect 2 and 6 are missing. It now reads as if Sect 3 starts with detector and triggers. Suggest: “Sect. 2 briefly describes …” And end with “by a brief summary in Sect. 6.” Also note that Sect. 2 is called “Detector and simulation”. So simulation should be mentioned in L249 as well.
254-5: Remove “covering the pseudorapidity range 2 < eta < 5”, since you have already discussed the effective range previously. This sentence only confuses the reader (you do no want to explain why it is defined differently here)
260 : “provides a measurement of THE momentum"
268: start a new paragraph for "The hardware trigger stage [...]"
272 : “with a segment of A charged particle track”
273: remove “charged-particle”. This is implied by track.
275: "[...] to prevent *high multiplicity* events from dominating the processing time."
278 : “… to evaluate SOME efficiencies.” -> sloppy! unclear if all relevant ones have been simulated, or all necessary ones, etc. “… to evaluate relevant efficiencies that couldn’t be determined by other means."
278: Remove “also”
286: "refine the sample of candidates for analysis" is jargon. "to reduce the same size used in the analysis" ?
287: "should have induced a *single-electron trigger*" sounds like jargon. Maybe change it into "trigger decision"?
289: both the electron and the positron
292: is $\Delta p/p$ (propose to write $\sigma p/p$) measured before or after bremsstrahlung correction? Maybe adding "curvature significance" can make this clearer
296-8: The clause “where E_ECAL, … respectively” can be removed without any loss of information. The abbreviations were already introduced.
301+303: Add commas or spaces in the yields. E.g. 4,595
302: Typo “aree” -> “are”
Eq.1: Add comma at the end of the equation.
Eq.2: No need to have periods at the end of Kin and Trig
p9: you can remove some of the forward references to Sect. 4 (the one on line 325 is sufficinet)
300: in approximately 0.7% of the cases (EB rule). But better leave out the approximation. 0.7 already means it's somewhere between 0.65 and 0.75.
309: events -> candidates
312: “The choice of binning is informed ….” -> “The choice of binning is DETERMINED by ….”. Does this mean that the phi* and eta resolutions are small wrt the bin widths (and hence migration due to e.g. bremsstrahlung negligible)?
316: propose to remove the first sentence: "\[eff = [...] \cdot eff_trig\], (2) where eff_track is [...] quality requirements, eff_kin gives [...] and \eta, eff_pid is [...] as electrons, eff_GEC is [...].", then it is still very clear what each efficiency exactly means (including the "cumulative" determination)
326-332 : are correlations among bins taken into account? Later on it seems so …
327-8: Not clear what is meant with “electron and positron yield reconstructed tracks”. Not needed either. Suggest: “both tracks satisfy the selection requirements”.
330: Replace “using” with “with” to avoid to write use twice in the sentence.
338: Remove quotation marks around true. You do not have them in L326 either.
339: "[...] (FSR). Therefore the kinematic efficiency [...] and an additional correction is no longer needed." -> or you would double count.
So PHOTOS is not used for FSR in this case?
Why not write Pythia 8.1 as in Section 2?
The discussion of FSR comes a bit confusing at this point. Could it be introduced earlier, when the fiducial region is defined, or in the description of the simulated samples?
345: "which varies significantly with y_Z". “…; this part of the efficiency should be modelled reliably.” -> I’d assume that holds for all parts. Remove entire sentence, or write what is really meant (e.g. needs to be modelled with a better than average precision, or something)
347: “the calorimeter energy requirements” -> the calorimeter acceptance”
Fig.1 misses "LHCb" (all others too). "(left) y_Z and (right) phi*" (EB rule). Please explain what the vertical error means. If you define the symbol epsilon in the caption, you need to put it in the y-axis title as well.
349: Would it be possible to avoid the "GEC" acronym? It is rather specific jargon
354: “apart from the contribution from the leptons”: why not write electrons? Suggest to be more clear: “… after correcting for the additional SPD hits of the electrons”.
357: electrons.
360: maybe you can refer to the Z->mu+mu- paper for the treatment of the SPD multiplicity systematic. Why fit a gamma function?
367: "efficiency --- these"
370: remove "of differential distribution", that's obvious. "and _on_ the integrated"
375: Remove quotes around photon. Instead you could write photon candidate, or better: not call it a photon. Just say it's an ECAL cluster without an associated track.
377: An efficiency is determined by comparing...
379: "spurious candidates" is vague
380: on both the electron and the ECAL cluster.
387-9: "must be accurate" is not resolved by "Reasonable". Are you happy or not? Is there a systematic assigned?
394: "acceptance is purely geometrical". Well, yes, that's the definition -> acceptance is modelled reliably in the simulation (that's a fact, not an assumption).
Fig.2: LHCb missing. A pull plot would help to judge the quality. Why "background-subtracted" in parentheses? That's quite crucial.
397: Remove quotes around probe.
398: what do you mean by the "energy recorded in each of the calorimeters in turn"? I guess it is once without the E_ECAL/p and once without the E_HCAL/p cut, but the formulation should be more explicit
401: “independent of y_Z and phi*”. Assume that you mean the same as on L385, i.e. that it is fully correlated between bins. If so, use the same wording here.
408: “as one of the systematic” -> “as a systematic”
413: “such backgrounds are intended to be …” -> such background are EXPECTED to be …”
415: Remove quotes around probe.
416: It is not the cut that fails, but the candidate fails the cut. Suggest to write: “that it fails the cut on the HCAL energy”
428: "does provide" -> "provides"
431: “statistical error” should be “systematic error”.
436: “The UNCERTAINTY IN THE bins of these ….”
437: remove "assumed to be"
441: remove the commas around f_FSR
449: it seems like most (?) systematic errors are added linearly. From fig. 3 it seems that the results is systematics limited. Is the systematic error evaluation not too cautious?
454: “included some LHC data” sounds as if they purposely neglected available LHC data. Suggest to delete “some”.
460: “too” -> “as well”
469-470: the repetition of "parton shower model (such as/of) Pythia" might cause confusion about the difference between "POWHEG" and "PYTHIA" in your plots. Maybe it could be mentioned that Pythia is then also used for simulating the hard process.
470+6: Write Pythia 8.1 as in Sect.2.
482: Same remark as for abstract.
487: this is discussed as the conclusion of previuos studies ([3], at line 472), but no comparison of phistar with fixed-order calculations is shown in this paper, so either that should be added, or this conclusion removed
Fig.3: what do you mean with "integration errors"?
Table 2 and 3: error -> uncertainty (twice in both tables). Why are there no spaces around \pm?
Fig.4: Add unit to the left-hand y-axis [pb]. In the caption, last-but-one sentence, remove the first occurrence of “are shown”. Add hyphen in “leading-log”.
Fig.5: what is happening to the Pythia prediction at 0.8 in phi*? Are there suppressed large error bars? In the y-axis title, right-hand plot add “ / data": In the caption add hyphen in “leading-log”.
Ref 17: write collaboration with lowercase.
Pg.21: I’m somewhat surprised that the correlation matrix has only positive values. If there is some smearing effect going on, at least some negative values should appear here and there. Is there a single systematic error driving the correlation (e.g. luminosity)?