Classifying the CP properties of the ggH coupling in H + 2j production

The authors study an important topic of CP violation which may appear either in
the Higgs boson coupling to a top quark, or in an effective coupling to gluons.
The authors focus on the gluon fusion production of the Higgs boson in association
with two jets. While both experimental and theoretical ideas and techniques for
this analyses have already been established, the authors introduce one new aspect.
They suggest that in addition to the VBF-like topology used in the literature,
a distinct topology of events which do not look like VBF events can be employed.

Unfortunately the authors do not consider non-Higgs background in this topology,
do not perform MC simulation of multiple jet radiation at higher orders in QCD
with careful matching to parton shower, do not consider all sources of systematc
uncertainties or detector effects, and do not properly account for possible CP
violating effects in the other Higgs boson production processes. This makes
numerical prediction of their studies unreliable. Nonetheless, the qualitative
suggestion to investigate a new topology of events in experimental analysis is
worth noting and to be taken for consideration by the ATLAS and CMS collaborations.
There is a possibility that further improvement of experimental results following
this suggestion will result from such a study, but this needs to be confirmed
by a more careful study.

Therefore, I would recommend this paper for publication, provided the wording
claiming significant improvements over the existing experimental results is
changed to a more careful assessment along the lines suggested above, and
comments below are taken into account.

More detailed comments are given below:

Page 3:

Second sentence: quantum numbers of the Higgs boson are mentioned, but ATLAS
and CMS references do not report tests of those.

References in general: Among the many ATLAS and CMS journal references, only one
actually quotes ATLAS Collaboration as the primary author (Ref.[47]). The suggestion
is to always refer to CMS or ATLAS Collaboration first, and then give the first
author and the rest of the journal information. Also suggest to use chronological
order in all groups of references.

Last sentence of the 2nd paragraph: It is not necessary for the Higgs boson to
be a mixed state in order to observe CP violation. In fact, even in the SM, the
purely CP-even Higgs boson could have CP violation in interactions due to three-loop
effects, though those are highly suppressed. The authors may want to refer to
CP violation in interactions instead.

The 4th paragraph assumes two distinct measurements: one using CP-odd observables,
and another using CP-even observables. However, in reality it is not always possible
to clearly separate the two types of observables, and the best measurement would always
come from using complete kinematic information using all observable information, CP-odd
and CP-even. Therefore, the whole concept of this paragraph is not clear. Instead,
the authors may want to introduce the idea of CP-odd and CP-even observables.

In the 5th paragraph: why use reference [8], but not [7]? Why start with Run1 ATLAS
reference [32], but not include the corresponding one from CMS?

page 4:

It is stated that "ggF production in association with one or two jets is more directly
sensitive." While this is correct for the two-jet configuration, the statement about the
one-jet configuration is not correct in general. Perhaps footnote 1 should be added here
already.

While an extensive list of references is given in the first paragraph [55,82-95], not all the
references deal with CP structure of the gluon fusion process. Some references deal primarily
with WVF, and in particular refs. [83-84] calculate CP-even background from gluon fusion.
At the same time, some of the original references for calculation of the CP-odd observables
are missing, in particular:

(1) for calculation of Delta\PhiJJ: https://arxiv.org/abs/hep-ph/0105325
(2) for calculation using matrix elements in gg->HJJ: https://arxiv.org/abs/1309.4819

Overall, references appear in a random order, so the suggestion is to order those chronologically
by submission date.

page 5: In discussion of Fig.1, fractions are quoted, e.g. 72% for gg initial state.
This is probably done in application to SM, but this is not clarified. However, Fig.1
shows non-SM contact interactions of the Higgs boson with gluons. Perhaps it should be
clarified that in this case those interactions are generated by SM loops, most likely
of the top quark.

Eq.(1): while this effective coupling describes the 2nd and 3rd diagrams in Fig.1,
the five-particle effective interaction from the 1st plot cannot be described by this Eq.
Suggest to re-draw the 1st diagram as a three-particle interaction, and leave the
five-particle interaction for separate consideration.

Eq.(4): as footnote-3 points, there are three CP-odd operators in the WBF process. As it has
been shown in Ref. https://arxiv.org/abs/2109.13363, two out of three operators with virtual
photons are less sensitive in this process. Therefore, one can consider one operator as the
primary one with the proper motivation.

Page 7: Any non-Higgs background is neglected in this analysis. While this might be Ok for
conceptual development of CP-odd observables and related studies, this does not provide a
realistic expectation in experimental analysis. An argument that this contribution to uncertainty
is small in the total event rate compared to other experimental uncertainties does not apply
CP asymmetry measurements because many rate uncertainties cancel in the asymmetries.

Appendix-A says that All events are generated at leading order and are scaled to NLO by flat
K-factors. This is not mentioned in the main body on page 7, and the aMC_NLO naming would
make everybody think that this generation is done at NLO. However, this brings a serious concern
about the associated jet modeling used in this paper. Pythia8 parton shower matching to LO
simulation is very unreliable. Even though up to two jets are generated at matrix element level,
additional jets may be modeled incorrectly and result in biased distributions when two jets are
selected for analysis. Perhaps the authors could quote the fraction of events when the two jets
selected are exactly those generated at matrix element level. This fraction could be substantially
smaller than 100% and this may indicate how well one can model such topology.

It is also not clear only two jets are modeled with MadGraph5_aMC@NLO , or if there is an inclusive
modeling of H+0, 1, 2jets with Pythia generating additional jets. Inclusive simulation is more
realistic.

Section 4: two classifies are employed: to separate gg2Fjj from background, and the other
to separate gg2Fjj from VBF. However, earlier it was stated that non-Higgs background was
neglected and only other Higgs production (VBF) was considered as background. The authors
need to clarify what they call backgrounds in this (and other) cases, and what exactly has
been simulated and what not.

The category of events ggF2j-SR has topology different from VBF and we expect the main
contribution to be from non-Higgs events. However, such events are neglected and it is not
clear how one can rely on any prediction of this analysis in regard to events in the ggF2j-SR
category.

Section 4.2: The training of CP classifies (CP even and CP odd) appears similar to that
summarized in Ref. https://arxiv.org/abs/2211.08353, section 2.1.3 in particular. This has
been worked out from first principles for matrix element calculations and then approximated
with machine learning approaches. Connection between the two allows justification of the method.

Fig.6: in the ggF2j-SR category, the main separation power in the shapes of observables
appears to be in tail of the green distribution (left side) of the CP-even observable (left),
which looks almost like a statistical fluctuation. This vision is supported by the large
uncertainty in the edge bins indicated in Appendix C. There is very little difference and
tiny interference contribution in the CP-odd observable (right). Combined with the lack of
background modeling in this category, this leads to a questionable prediction of separation
power in the ggF2j-SR category, which is shown in Section 5.

Section 5.3: comparison to experimental results is absolutely not fair: "In comparison
to the expected limits from existing experimental analyses [37, 38, 45], our combined
limit is significantly stronger." This is because experimental results take all effects
into account, while this work made many approximations and did not consider realistic
MC simulation and all sources of background, statistical, and systematic uncertainties.
The authors of experimental analyses did consider many effects discussed in this paper
and did not find such an improvement in separation power after considering all effects
which degrade performance.

Section 5.4: analysis of separation power of individual observables is somewhat random
since the choice of observables is also random. The key point in choosing input observables
is to represent the full kinematic information and the rest is an arbitrary choice of
the authors. With the present choice, some pattern can be observed, but it is mostly
limited to the delta\Phi_JJ observable which is already known in the literature.

Section 6: the whole section appears to present biased results and wrong picture.
This is especially evident from the statement that the "better" constraints in the
VBF-SR on cg and ~cg are faked from CP violation in the HVV couplings. It is not
possible to obtain better constraints when additional CP violating effects are allowed.
The results can become only worse with additional free parameters if statistical
interpretation is correct. The correct approach would be the following: The model
is extended to allow an additional CP-violalting parameter in the VBF+VH process.
When this parameter is set to zero, old results are recovered. When this parameter
is allowed to float, fits are performed on the same samples as before. The resulting
constraints on cg and ~cg will become weaker when the new CP-odd parameter is floated
in addition to the other parameters. The critical question is how much weaker the
constraints become.

Section 7: comparison to experimental results following Eq.(5) are not fair,
as discussed above. Many effects are neglected in the present analysis, and one
cannot rely on numerical prediction for comparison to the solid experimental
measurements. The same comment applies to Table 2.

1.Interesting analysis that exploit a new final state, H+2jets, to study the CP properties of the ggH coupling.
2.Good introduction with a good description of the interplay of the different couplings, production modes and CP violation origin.
4. Interesting studies on the disentangling of the CP violation in the various couplings.
3.Use of Machine learning to improve the sensitivity and exploit many variables.

1. The paper is not very clear, it is hard to read. Sometime one find in the "Conclusions" the needed explanations.
2. There is a big mistake: in Sec.3, it is written that the background can be subctrated and thus neglected (see later for an explanation).
3. In section 3 there are sentences hard to read, i.e. or are wrong or I do not understand them (see later).
4. There is no explanation on how the "interference events" are generated.
5. The explanation on why the VBF-SR is considered is in the conclusion instead that in sec.4.
6. It is confusing the treatment of Meven and Modd and the corresponding figures 4 and 5
7. Figure 6 and 7 are not well explained and it is unclear how then one can use those distributions
8. There is no definition of "limit" in sec.5.
9. All the comparisons of the result presented in the paper with the experimental results are invalid, since the treatment (and statement) about the background and its subtraction is wrong.

I recommend a revision of the paper before taking a decision. The analysis is interesting, but quite some statements have to be changed or corrected.
It can be an interesting signal-only study.

Section 3.
1. the second line mention that the ggH2j channel offers a unique sensitivity. I suggest to repeat the explanation of why (or at least point to the introduction).
2. line 8: The statement that background from "qq->qq+2" photons can be subctrated is correct only at the level of the "M_{gamma-gamma}" plot, when you want to measure the mass of the Higgs boson or its cross section. I.e. this background subtraction is valid only at statistical level. Event by event you cannot know if the event is signal or background, i.e. you cannot do a CP analysis background free ! I think anyway you can publish this analysis, but as an excercise on signal only (+VH+VBF background). As a consequence you cannot compare with other studies, expecially the one performed on data.
3. As a consequence of comment n.2, line 12 and 13 have to be deleted.
4. line 19: what does it mean "with particle having passed an electron, muon ... before."?
5. line 21: what does it mean : "Photons... are identified by observing electron...". It has no sense to me.
6. 4th paragraph - event generation: please explain how the interference sample is generated/constructed. (Since - to my knowledge- you cannot generate simply the interference term with madgraph, you should explain how you construct it).
7. 4th paragraph: the cut on the invariant mass of the 2 photons does not subctract all the non-Higgs background. Expecially in the gamma-gamma final state, the backgound is by far dominating. See any experimental plot from Atlas or CMS...

Section 4.
8. the procedure and the naming of the samples are quite confusing. The best would be to move the explanation on why you want to use the VBF-SR sample from the "conclusions" to this section, and clarify at the beginning the idea of the two SR. Otherwise it is not clear why the VBF, that is the background to ggH2j, is then considered signal.

Section 4.2
9. This part is very unclear to me: in the first bullet under figure 5 you talk about constructing a CP-even observable, but then while describing figure 6 you instead try to define a CP-odd observable.
Figure 6 and 7 show P(c2) and P+-P- for Meven and Modd (and interference): Meven and Modd have the same shapes, only a different normalisation. How do you use these distribution to discriminate between Meven and Modd and get an estimate of P(c2) ?

Section 5
10. It is not clear the definition of LIMIT. Please define it.
11. why don't you try another CP hypothesys instead of only doing the study for the SM case ?

Section 5.3
12. line 5. add "full line" ---> ".... and the combined SRs (full line)"
13. 3rd paragraph: you cannot compare with experimental analyses since you are not considering the qcd background

Section 5.4
14. line 7. worth --> "worth" (under quotes "")

Section 7.
15 3rd and 4th paragraph: you cannot compare with experimental limit since you are not considering the background.

Conclusions:
Again you cannot compare with other experimental results.