Lepton pair production at NNLO in QED with EW effects

Development of a framework for precision predictions of the observables relevant in the precision physics program at Belle.

The description of the perturbative content of the corrections evaluated in this calculation is not precise

The authors of the paper “Lepton pair production at NNLO in QED with EW effects” have nicely answered to my questions. A few details for the precise definition of the subset of corrections studied and their organization is to some extent still missing, in my opinion, but it can be easily fixed.
After such changes, the paper can be accepted for publication.

I take Section 2, Overview of the calculation, as the place where I expect a precise declaration of the content of their work.
I read that NLO-EW corrections are fully included, but it is immediately specified that the authors with NLO-EW refer to the QED corrections, since they specify the different lepton electric charge combinations. Then the weak corrections are introduced, describing their approximation, which contains the first two terms in the expansion in powers of a light scale squared over MZ^2, i.e. Q^2/MZ^2 and (Q^2/MZ^2)^2, meaning that all the box and vertex corrections give a contribution and are included, even if suppressed by two internal massive lines. Is this counting correct?

When the authors discuss the renormalization program, they state that a natural scheme would be with e,MW,MZ in input; but then they trade MW for Gmu, so that their inputs are e,Gmu,MZ; at this stage they predict MW and sin2theta_W, using the muon decay amplitude relation. It is then wrong to say that MW and sin2theta_W are experimental inputs, since these are predictions in this input scheme.

Then the authors say that MCMULE uses as actual inputs MZ and sin2theta_W and specify a numerical value for sin2theta_W. My understanding, but the text is not clear, is that MCMULE uses as an input scheme e,sin2theta_W,MZ. If this is the case, it is not clear to me how the weak mixing angle is defined at NLO-EW and which relation is used to predict MW.
I would recommend to give a shorter and clearer description, indicating which is the choice of three parameters used in input to express (g,g’,v) and to provide the relations and the numerical values of the additional parameters predicted.

At the beginning of Section 2.1 the authors do not mention the top quark. How is it included?

When the authors discuss renormalization and self-energy contributions, they focus on the fermionic subset.
The symbols Sigma are not defined and in the case of the gamma gamma contribution this might lead to confusion.
Since eq.10 gives a representation of the amplitude with self-energy insertions,
and since the fermionic corrections automatically statisfy the transversality condition, such that already at unrenormalized level we have Sigma_{gamma gamma}(0)=Sigma_{gamma Z}(0)=0,
when we go to equation 14 there is potentially a problem: if we drop Sigma(0), then the poles are not subtracted and the renormalized self-energies are still divergent.
The authors, introduce after eq.14 the explicit dependence of the gamma gamma self-energy on q^2, where again Sigma_{gamma gamma, f}(0) and Sigma_{gamma Z, f}(0) are identically vanishing.

The renormalization of the bosonic contributions is not discussed, although it contributes to the definition of the couplings, which are in turn responsible for the observed asymmetries.
Are the Ward identities for the initial- and final-state vertices satisfied, after the expansion of the bosonic contributions?

The diagrammatic description of the bosonic part is misleading: since the electron mass is kept, there are gauge dependent contributions in the massive vector boson propagators and in the corresponding Goldstone propagators, which cancel each other; neglecting the diagrams with the Goldstone, as the authors state, and keeping those with the vector boson is potentially wrong, unless a careful expansion is systematically applied.

At two-loop level , it is not clear why the authors specify that other terms, such as those due to Higgs exchange, are suppressed by 1/MZ^4 and thus discarded: since by definition and their own declaration at two loops they consider only QED corrections with a specific pattern of electric charges, the comment is redundant.

I can not find the definition of the operators of dimension 6 and 8 mentioned in equation 42.

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After fixing these details, I would recommend the paper for publication.

1) The calculation is well motivated in the introduction.
2) The results are implemented in a Monte Carlo code allowing for IR-safe observables.
3) The input and the setup are clearly described.
4) The Monte Carlo code and the numerical results are relevant for the Belle II experiment.

The authors have implemented the changes requested in my previous report and made further improvements. I have an additional proposal for corrections. Nevertheless, I recommend the paper for publication.

I find Equation (42) and the text preceding in confusing. According to this equation, the symmetric term is larger without dimension-six-squared terms, while the sentence before claims the opposite.
Also, I would suggest to remind the reader what dimension-six terms are in this Standard Model calculation.

I think the manuscript can be published in its present form.