GRIFFIN: A C++ library for electroweak radiative corrections in fermion scattering and decay processes

* GRIFFIN contains a new implementation of all known virtual electroweak corrections to the process e+e- -> Z ->f fbar.
* The framework should facilitate the inclusion of further corrections in the future.

* The embedding into the literature on the current state-of-the-art handling of resonances should be improved.
* Several formal definitions and conventions, such as the resonance expansion and the subtraction of infrared singularities, should be made more explicit.
* The origin of the differences between GRIFFIN and DIZET close to the Z resonance has to be clarified.

The presented program is very valuable in various respects: On the one hand, it is a new implementation of all known virtual electroweak corrections to the process e+e- -> Z ->f fbar, which makes contact to the calculations that were used at LEP for analyzing Z-resonance physics and provides a framework for including further corrections to increase the achieved precision. The program is thus important in the long-term theory preparation for a future e+e- Z-boson factory. Moreover, the program might be used in a future uplift of predictions for neutral-current Drell-Yan processes at the LHC. The manuscript should therefore be published eventually, but every possible effort has to be taken to make the approach to include the higher-order corrections perfectly clear, in order to avoid misunderstanding or confusions. In this respect, some further work has to be invested in the manuscript.

1) In the first paragraph on page 2, the list [12-26] misses some original papers on electroweak corrections to neutral-current four-fermion processes, such as [A,B,C] given below, some of them certainly more important than others included in [12-26]. The authors should carefully check the relevance of all those references.

2) In the second paragraph on page 2 and in the last paragraph of page 3, the list of references [31-34] for the gauge invariance of variables describing resonance poles is not appropriate. At least the inclusion of [33] is questionable, since it only roughly sketches idea of the pole scheme, without further insight in the issue of gauge invariance. On the other hand, refs.[D,E], where the gauge invariance of the location of the complex propagator pole is generally shown, are missing.

3) After Eq.(3) it should be mentioned that the angle \theta is defined in the center-of-mass frame.

4) In the general context of the discussion of corrections to resonances and the corresponding issues with gauge invariance I recommend to include the review [I] as given below, to make better contact to the current state of the art in this subject. Ref.[I] also explains the field-theoretical embedding of Eq.(6) beyond the mere parametrization of the resonance with a fixed or a running width.

5) The Laurent expansion about the complex pole given in Eq.(7) is not uniquely defined, because the kinematics of the process does not only involve the variable s, but also the scattering angle which is not mentioned at all. To uniquely define R_{ij} etc., it has to be specified in terms of which kinematical parameters other than s these variables are defined. For instance, it makes a difference in the definition of the residue R_{ij} (and thus on the full result) whether the scattering angle or the Mandelstam variable t (which is a function of the scattering energy and angle) is used to parametrize R_{ij} etc.
Eq.(7) is also sloppy in the sense that the log(1-s/s_0) terms mentioned after (12) do not fit into the simplified form of Eq.(7).

6) The actual discussion of these log(1-s/s_0) terms is generally not embedded adequately in the literature (see paragrapg before Eq.(20) on page 6). Ref.[F] below is certainly as relevant as the quoted refs.[40,41], but appeared earlier. Moreover, the authors speculate on the cancellation of these log terms between virtual and real corrections beyond NLO. For the order O(\alpha*\alpha_s) this cancellation was explicitly shown in ref.[G], which would be worthwhile to mention.

7) I think Eq.(36), which is key to the proposed way to include corrections beyond NLO, deserves better explanation. It has to be shown that Eq.(36) exactly reproduces all terms for the matrix elements in the unexpanded perturbative series to the desired order (without losing terms and without extra spurious terms). At NLO, this has been shown in ref.[C] (which should be cited in this context as well) for a different implementation of this "pole scheme". Is Eq.(36) equivalent to the procedure described in ref.[C], at least at NLO? Moreover, ref.[C] includes a comparison of results for Drell-Yan production obtained with different resonance schemes, which is certainly worth mentioning.

8) The original reference for the complex-mass scheme is not [48], as quoted in footnote 5 on page 8, but ref.[H] as given below.

9) I agree with previous referees that the details of the subtraction of infrared (soft and collinear) singularities originating from photonic corrections should be made very explicit, either by giving these terms explicitly or by pointing to explicit formulas in the literature. Otherwise the value of GRIFFIN for producing future benchmarking results is significantly reduced.

10) The most serious criticism, which was also raised in earlier reports, however, concerns the surprisingly large numerical differences to the results of DIZET, as shown on the r.h.s. of Fig.1. Corrections of the order of 1-1.5% certainly cannot be explained by missing two-loop effects, as suggested by the authors. In the absence of any particular enhancement factors the typical size of electroweak NLO corrections is 1%, and accordingly the size of missing
NNLO corrections is expected to be at the level of very few 0.1%. Large enhancement factors could only arise from logarithms originating from photons coupling to light fermions, but the photonic effects have been extracted by definition. The origin of the numerical differences has to be clarified.
Maybe a comparison of GRIFFIN and DIZET results without any Dyson resonance might help locating the source for the difference. Splittiung the corrections into well-defined building blocks and comapring them separately would help as well.

[A] U.~Baur, S.~Keller and W.~K.~Sakumoto,
``QED radiative corrections to $Z$ boson production and the forward backward asymmetry at hadron colliders,''
Phys. Rev. D \textbf{57} (1998), 199-215
doi:10.1103/PhysRevD.57.199
[arXiv:hep-ph/9707301 [hep-ph]].

[B] V.~A.~Zykunov,
``Weak radiative corrections to Drell-Yan process for large invariant mass of di-lepton pair,''
Phys. Rev. D \textbf{75} (2007), 073019
doi:10.1103/PhysRevD.75.073019
[arXiv:hep-ph/0509315 [hep-ph]].

[C] S.~Dittmaier and M.~Huber,
``Radiative corrections to the neutral-current Drell-Yan process in the Standard Model and its minimal supersymmetric extension,''
JHEP \textbf{01} (2010), 060
doi:10.1007/JHEP01(2010)060
[arXiv:0911.2329 [hep-ph]].

[D] P.~Gambino and P.~A.~Grassi,
``The Nielsen identities of the SM and the definition of mass,''
Phys. Rev. D \textbf{62} (2000), 076002
doi:10.1103/PhysRevD.62.076002
[arXiv:hep-ph/9907254 [hep-ph]].

[E] P.~A.~Grassi, B.~A.~Kniehl and A.~Sirlin,
``Width and partial widths of unstable particles in the light of the Nielsen identities,''
Phys. Rev. D \textbf{65} (2002), 085001
doi:10.1103/PhysRevD.65.085001
[arXiv:hep-ph/0109228 [hep-ph]].

[F] K.~Melnikov and O.~I.~Yakovlev,
``Final state interaction in the production of heavy unstable particles,''
Nucl. Phys. B \textbf{471} (1996), 90-120
doi:10.1016/0550-3213(96)00151-4
[arXiv:hep-ph/9501358 [hep-ph]].

[G] S.~Dittmaier, A.~Huss and C.~Schwinn,
``Mixed QCD-electroweak $\mathcal{O}(\alpha_s\alpha)$ corrections to Drell-Yan processes in the resonance region: pole approximation and non-factorizable corrections,''
Nucl. Phys. B \textbf{885} (2014), 318-372
doi:10.1016/j.nuclphysb.2014.05.027
[arXiv:1403.3216 [hep-ph]].

[H] A.~Denner, S.~Dittmaier, M.~Roth and L.~H.~Wieders,
``Electroweak corrections to charged-current e+ e- ---\ensuremath{>} 4 fermion processes: Technical details and further results,''
Nucl. Phys. B \textbf{724} (2005), 247-294
[erratum: Nucl. Phys. B \textbf{854} (2012), 504-507]
doi:10.1016/j.nuclphysb.2011.09.001
[arXiv:hep-ph/0505042 [hep-ph]].

[I] A.~Denner and S.~Dittmaier,
``Electroweak Radiative Corrections for Collider Physics,''
Phys. Rept. \textbf{864} (2020), 1-163
doi:10.1016/j.physrep.2020.04.001
[arXiv:1912.06823 [hep-ph]].

The code can be systematically extended to include higher-order corrections and new-physics models.

The code will be relevant for future lepton colliders and supersedes software packages used at LEP 1.

The implementation has been verified against the DIZET library and includes all available higher-order corrections for the leading pole term.

The authors have implemented the changes requested in my previous report and made further improvements. I would have added some references to fermion pair production on the Z resonance from the 1980's, for instance Consoli, M. and Hollik, W. and Jegerlehner, F., Electroweak Radiative Corrections for Z Physics, in CERN-89-08, and references therein.