$\mathcal{CP}$-Analyses with Symbolic Regression

Dear Editor and Referees,

We thank the referees for their careful reviews of our manuscript and the helpful comments. We have addressed the various points raised in the referee's reports and modified the text accordingly to improve the clarity of our manuscript. For convenience, all changes in the text are marked in blue. Below, we answer in detail to the different points raised by the referees.

We hope that with these additions, our article can now be accepted for publication.

Yours sincerely,
H. Bahl, E. Fuchs, M. Menen, T. Plehn

Referee 1

1 please run the paper through an advanced spell checker, I've spotted "mathmatical" there must be more

• Done.

2 table 2 : you can bold face the largest value in each case, this is standard in ML paper to help the reader

• We followed the suggestion.

3 it is great that you compare systematically with BDT. It would have been nice and convincing to compare in addition to a regular dense NN (which I expect to be at the level of the BDT). Actually if you stop the training of Symbolnet after the "default training" (section 2.2), is it not a regular dense NN ? Then you could use this as a NN reference and show it in the results and plots to see "in action" the symbolic layers.

• As suggested, we trained SymbolNet observables without any sparsity loss. We have added the results to Tab. 2 and adapted the description in the text accordingly.

4 I really really like Figure 8, can't you do a similar one for the Collin Soper angle case ?

• We have added the requested Figure (now Fig. 14 in the paper) and a corresponding description. We also slightly changed our conclusions based on the results in this Figure.

5 A large amount of work went into SymbolNet, to have it "understand" 4-momenta. Given that I'm sure you were disappointed by the poorer performance (compared to PySR) for the WBF Higgs production case, and the marginally better one for Collin Soper angle case. Wouldn't it be possible to plug 4-momentum symbolic capabilities to PySR ?

• We considered this possibility. After looking into the PySR code, we, however, realized that adding vector support would require a larger reorganization of the code. In particular, PySR is build upon the Julia package SymbolicRegression which itself is based on the package DynamicExpressions. Adding vector support would require modifying each of these three packages. Since the required amount of work was out of scope for the project, we dismissed this idea. Implementing vector support into SymbolNet on the other hand turned out to be relatively straightforward since SymbolNet is a self-contained package with only a minimal set of application layers.

6 Bibliography : [15] : please add CERN report number CERN-LHCEFTWG-2022-001, CERN-LPCC-2022-05 [27] : very important, this is NOT a NeurIPS paper, this is a paper accepted in ML4PS workshop at NeurIPS which is much easier than proper NeurIPS. Please fix [29] and beyond : 29 has been published in CSBS, please fix, and please check the following references that are only on arXiv [30] something went wrong in formatting the title [33] np-hard => NP-hard [46] XGboost : I believe the standard way (if any) to cite xgboost is to cite the original publication as below, with arXiv:1603.02754 in addition (not twice as now in any case)

@inproceedings{Chen:2016:XST:2939672.2939785, author = {Chen, Tianqi and Guestrin, Carlos}, title = {{XGBoost}: A Scalable Tree Boosting System}, booktitle = {Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining}, series = {KDD '16}, year = {2016}, isbn = {978-1-4503-4232-2}, location = {San Francisco, California, USA}, pages = {785--794}, numpages = {10}, url = {http://doi.acm.org/10.1145/2939672.2939785}, doi = {10.1145/2939672.2939785}, acmid = {2939785}, publisher = {ACM}, address = {New York, NY, USA}, keywords = {large-scale machine learning}, } [49] : this CMS citation is not consistent with the next ones

• We fixed these issues.

Referee 2

title: Could be more informative/detailed, consider adding something about "Higgs", "top" or "collider phenomenology" or similar

• While we agree that the focus of the paper is CP violation in the Higgs sector, the used methods can be straightforwardly applied to the search for CP violation in other sectors (as mentioned in the conclusions). We, therefore, would prefer keeping the more general title.

Introduction

consider add citation to https://arxiv.org/abs/2508.00989

• We followed the suggestion.

Section 2.1

"but poorly initialized." needs further explanation since the term initialization has not been introduced in the paper up to this point

• We have modified the sentence for improved clarity.

Section 2.2

Eq (7): Explain in the text the meaning of Theta (I assume that it's the heaviside function?)

• We have added a comment to Eq (5) where it shows up for the first time.

If the threshold parameter exceeds one and the heaviside function is removed, doesn't this produce null gradients with respect to the threshold parameter during the pruning step? Consider adding a statement about this in the text. (if the Theta function is replaced as in eq(12) and kept in the backpropagation for this case, then explain this)

• For the gradient computation during backpropagation, the theta function is indeed replaced by the function in Eq(12), which is non-zero everywhere and should not produce any null gradients. We have added a comment about the gradient stability.

Fix inconsistent notation (capital vs lowercase) for threshold parameters in equations vs Table 1

• We have fixed this.

Suggest to add comment on the choice and sensitivity to the value kappa=5

• We have cited the original SymbolNet paper as a reference for this value of kappa.

Section 3.1

"the CP-properties of d and omega_CP-odd are the same.": shouldn't this read "of D and omega_CP-odd are the same" ? (ie the comment is about the transformed quantity)

• We have fixed this.

Section 3.2

"at two tagging jets" -> "at least two tagging jets"? (or exactly two tagging jets?)

• At least two jets are required. We clarified this in the text.

Section 3.3

Better explanation needed on the meaning of Eq. 30 (the "#[] notation in particular")

• We have added a clarifying statement.

Eq (33) some explanation about what is different here with respect to (32) would be useful. Different random initialization of the weights?

• Yes, a different initialization is used. We clarified this in the text.

To facilitate comarison between Fig. 7 and Table. 2 suggest explicitly writing in the text that significance = sqrt(chi^2) in this context

• We have added a sentence in the paragraph related to Tab. 2.

Section 4.2

Fig. 10 and surrounding discussion. Most likely the classical reconstruction used doesn't take explicitly into account the expected resolution of the different objects (e.g. from a kinematic fit) which is different for jets, leptons, and missing energy. It's encouraging that the symbolic regression is able to mitigate this (likely by effectively downweighting the less well-measured objects in the combinatin of kinematics). Some more explicit comment on this in the text may be worthwhile.

• We agree that these results indicate a better handling of the different resolutions by the SR algorithms and have added a comment in the description of Fig. 10.

"but these single events can be attributed to statistical fluctuations": this is probably also partly related to the restricted/narrower output range of the PySR-learned function which would tend to "compress" the output space compared to the function learned by SymbolNet

• We agree and have extended our statement about the outliers.

Fig. 11 and surrounding discussion: it would be useful to show the comparison for alpha_t=0 vs alpha_t=45 degrees also for phi*_true

• The true distributions are now included in the respective plots.

Conclusions

Consider adding some comment on possible directions for follow-up work

• We have added a small outlook paragraph.