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

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper argues that symbolic regression can learn analytic, detector-level observables for Higgs CP-violation searches that match or beat black-box networks and classical reconstruction while remaining explicitly checkable for CP parity.

desk verdict A solid, honest methods paper on symbolic regression for Higgs CP observables; referee it, but require a b-jet assignment robustness check and code/data release. read the letter →

arxiv 2507.05858 v1 pith:PEULMNAZ submitted 2025-07-08 hep-ph hep-ex

Henning Bahl , Elina Fuchs , Marco Menen , Tilman Plehn This is my paper
classification hep-phhep-ex
keywords CPviolationsymbolicregressionHiggsbosonoptimalobservableCollins-SoperanglettHproductionvectorfusionLHCphysics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Searching for CP violation in the Higgs sector is a fundamental-symmetry test, and machine-learning classifiers built to find it are effective but hard to control. This paper argues that symbolic regression, which learns closed-form equations instead of opaque networks, can produce detector-level CP-sensitive observables that are both competitive and explicitly checkable. For vector-boson-fusion Higgs production, the learned analytic optimal CP-odd observables match or exceed a boosted-decision-tree classifier, and each formula can be inspected to verify that it is genuinely CP-odd. For top-associated Higgs production, learned formulas reconstruct the parton-level Collins-Soper angle from reconstruction-level data, recovering up to about 80 percent of the parton-level CP information in the most realistic scenario, where a classical top reconstruction keeps roughly 60 percent. A reader should care because an unambiguous probe of a fundamental symmetry requires knowing exactly what the machine learned, and a formula can be checked; a network cannot.

What carries the argument

The central mechanism is symbolic regression itself, in two complementary implementations. PySR is an evolutionary algorithm that mutates and recombines formula trees, growing simple expressions into complicated ones; here it is trained with an added loss term that penalizes non-CP-odd outputs, so the final equation is CP-odd by construction. SymbolNet starts from a densely connected neural network whose activation functions are mathematical operators and prunes it down to a sparse, extractable equation, extended here to a vectorized version whose symbolic layers act on 4-vectors and can apply Lorentz boosts. The training targets are set by the physics: in WBF production the target is the Neyman-Pearson optimal CP-odd observable $\omega_{CP\text{-odd}} = p_o/p_e$, recovered by training a classifier to separate events with positive and negative $c_{H\widetilde W}$; in $t\bar t H$ production the target is the parton-level Collins-Soper angle $\cos\theta^* = \vec p_t\cdot\vec n\,/\,(|\vec p_t||\vec n|)$, the CP-sensitive angle between the $t\bar t$ system and the beam axis, which must be reconstructed from semileptonic decay products without full neutrino information.

What would settle it

Re-run the scenario-6 Collins-Soper reconstruction on events where the two b-jet assignments are deliberately swapped, or enumerated over all b-to-lepton and b-to-quark pairings: if the learned formulas' $\Delta\chi^2$ collapses toward the classical-reconstruction value, the reported advantage is an artifact of the perfect-assignment premise rather than of symbolic regression itself.

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Extended reading notes

Core claim

On the paper's own terms, it establishes that two complementary symbolic-regression algorithms, PySR and an extended vectorized SymbolNet, learn analytic observables for CP searches directly at the detector level. In WBF Higgs production with $H\to\gamma\gamma$, the learned CP-odd observables reach significances around $7\sigma$ for $c_{H\widetilde W}=1$ versus the SM at 300 fb$^{-1}$, slightly above the boosted-decision-tree classifier and the classic parton-level observable $p_{T,j_1}p_{T,j_2}\sin\Delta\phi_{jj}$. In $t\bar t H$ production, the learned expressions for the Collins-Soper angle preserve the CP sensitivity of the parton-level variable: in the most realistic scenario, with an extra jet and detector smearing, SymbolNet and PySR reach $\Delta\chi^2 = 7.628$ and $7.491$ for excluding $\alpha_t = 45^\circ$, against $9.385$ at parton level, while the classical reconstruction captures only about 60 percent of the CP information. The paper further shows the two methods are complementary, with PySR the more data-efficient and stable of the pair and SymbolNet the more accurate when enough data are available, and that the learned formulas retain recognizable parton-level structures.

Load-bearing premise

The load-bearing premise is that in the $t\bar t H$ events the two b-jets have already been correctly assigned to the lepton and the light quarks, and that $b$ and $\bar b$ can be told apart; the learned formulas take the individual $b$ and $\bar b$ momenta as inputs, so the reported $\Delta\chi^2$ values assume this ordering is known.

Editorial extensions

If this is right

  • In WBF Higgs production, the learned analytic observables attain about $7\sigma$ significance for $c_{H\widetilde W}=1$ versus the SM at 300 fb$^{-1}$, matching or slightly exceeding both the BDT and the classic $p_{T,j_1}p_{T,j_2}\sin\Delta\phi_{jj}$ baseline.
  • Because a learned formula is one fast-to-evaluate equation with explicitly checkable CP parity, an observed asymmetry based on it can be certified as genuine CP violation rather than a classifier artifact.
  • PySR's data efficiency means useful CP-odd observables can be learned from as few as 1000 training events, or from training samples with only a small CP-odd component ($c_{H\widetilde W}=\pm 0.1$), in regimes where the BDT and SymbolNet degrade.
  • In $t\bar t H$ production, the learned Collins-Soper angle reconstructions keep roughly 80 percent of the parton-level CP information in the most realistic scenario, versus about 60 percent for classical top reconstruction, giving $\Delta\chi^2 \approx 7.5\text{--}7.6$ against the parton-level value of 9.385.
  • The learned formulas retain recognizable parton-level structures, PySR's around $\sin(\sum_i a_i p_{z,i}/\sum_i b_i E_i)$ and SymbolNet's around a boosted ratio, which the paper reads as evidence that the same analytic skeleton carries the CP information at detector level.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because each learned observable is an entire event-level function, the same formula can be re-evaluated for any future value of the CP-violating coefficient without retraining; a natural extension the paper does not carry out is to apply it to the two companion CP-odd operators listed in Eq. (19) of the paper.
  • The same recipe should transfer to other latent-variable reconstructions at the LHC, wherever a parton-level CP-sensitive quantity needs an analytic, human-checkable proxy built from detector-level inputs, for instance in other $t\bar t H$ decay channels.
  • The gap between the learned formulas and classical reconstruction is the quantity most likely to shrink in a real experimental setting: once combinatorial b-jet assignment is folded in, the 80-percent-versus-60-percent comparison becomes an upper bound that any experimental analysis would have to defend.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. This paper proposes using symbolic regression to obtain analytic, interpretable observables for CP studies. Two SR implementations are used, PySR and an adapted SymbolNet. In WBF Higgs production (H->γγ + 2 jets, generated with MadGraph+Pythia+Delphes, SMEFT operator cHW~), both methods learn detector-level analytic CP-odd observables; their bin-wise asymmetries give significances comparable to or better than a BDT and can be checked analytically for CP parity. In ttH production (semi-leptonic decays, six increasingly realistic scenarios), the SR methods reconstruct the Collins-Soper angle from final-state momenta; in the most realistic scenario 6 (E_T^miss, extra jet, smearing) they recover about 80% of the parton-level Δχ² for α_t=45° versus about 60% for classical reconstruction. The paper emphasizes interpretability, data efficiency, and the complementarity of the two SR algorithms.

Significance. If the results hold, this is a useful contribution: explicit analytic formulas that can be checked for CP parity and used directly, with data-efficiency advantages and a fair comparison against BDT and classical reconstruction. The WBF study is particularly convincing because the CP-odd property of the learned formulas is verified analytically, and the comparison with the known parton-level observable is a good sanity check. The ttH study is well structured into six benchmark scenarios, with explicit formulas in Appendix D and repeated training runs. However, the quantitative detector-level claims are conditioned on idealized simulation and, in ttH, on perfect b-jet assignment and b/bbar discrimination; the absolute numbers should be read as upper bounds rather than as realistic experimental projections.

major comments (3)
  1. [Sec. 4.1, App. A, Fig. 11] The headline ttH result is conditioned on an oracle that correctly assigns the two b-jets to the leptonic and hadronic top decays and distinguishes b from bbar. This assumption is stated in Sec. 4.1 and App. A, but it is not mitigated. The formulas in Appendix D (e.g., scenario 6) use p_z,b, p_z,bbar, p_z,q and p_z,qbar with independent fitted coefficients, so the learned mapping is explicitly not invariant under swapping the two b-jets or under b<->bbar interchange. At the LHC, b-jet charges are not tagged reliably and the assignment of the two b-jets is combinatorial; a detector-level analysis must solve this problem. The quoted Δχ² = 7.628 (SymbolNet) and 7.491 (PySR) in Fig. 11, and the advantage over classical reconstruction in Fig. 12, are therefore upper bounds under perfect assignment, and the paper does not quantify the degradation. Because the "most realistic ttH scenario" result is the central performance claim, this is load-bearing. Please add a misassignment/tagging robustness study or explicitly and consistently label the scenario-6 numbers as idealized upper bounds, in the abstract and conclusions as well as in the figure captions.
  2. [Sec. 3.2, Sec. 4.1, Table 2, Figs. 12-13] The quantitative claims are all made on Monte Carlo events from a single leading-order pipeline: MadGraph LO with a constant K-factor of 1.13 for ttH, Delphes fast simulation for WBF, and only simple smearing (no pileup, no jet clustering, no b-tagging, no lepton isolation) for ttH scenarios 5 and 6. No systematic uncertainties are included in any of the quoted significances, so the numbers in Table 2 and Figs. 7, 12, and 13 are statistical-only projections. The relative ranking of PySR/SymbolNet versus BDT or classical reconstruction may be robust because all methods face the same simplifications, but the abstract's phrase "at the detector level" and the conclusion's "most realistic scenarios" overstate the level of realism. Please either add at least a basic treatment of dominant systematics (jet energy scale, b-tagging/misassignment, PDF/scale uncertainties) or rephrase the claims as idealized, statistical-only benchmarks.
  3. [Sec. 3.3, Table 2] It is not stated which of the two SymbolNet formulas enters the significance comparison in Table 2: the CP-odd formula in Eq. (32) or the non-CP-odd formula in Eq. (33), which the text says discriminates the SM better but would not be a valid CP probe. Since the stated goal is to construct a CP-odd optimal observable and the paper itself warns that a non-CP-odd classifier can inflate significance, each quoted significance should be accompanied by an explicit CP-parity check or at least a statement of which formula was used. Without this, the reader cannot tell whether the slight SymbolNet advantage over PySR in Table 2 reflects genuine CP sensitivity or a CP-even contamination.
minor comments (6)
  1. [Section 4 heading] The heading "Collin-Soper angle" should be "Collins-Soper angle".
  2. [Fig. 5 and Eq. (31)] The axis label in Fig. 5 uses pT,j0 pT,j1 while Eq. (31) defines pT,j1 pT,j2 sin Δφ_jj; the notation should be aligned.
  3. [App. B] In the sentence introducing Eq. (52), "where the is obtained" is missing the words "standard deviation"; please correct.
  4. [Sec. 3.2] The data usage is described as "250k events for training and testing as well as 100k events for validation"; please clarify whether the test set is used only for final evaluation or also for model/formula selection, to rule out selection-on-the-test-set bias.
  5. [Sec. 2.2] The statement that extra checks are needed because operations can render 4-vectors unphysical is never specified; please state how negative Minkowski norms or non-timelike vectors are handled during training and evaluation.
  6. [General] No indication is given that the training pipeline, the modified SymbolNet implementation, or the trained formulas will be released; for a methods paper with many decimal-coefficient formulas, code/data availability would substantially aid reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's learned formulas are supervised fits evaluated on held-out data against parton-level truth and known analytic baselines; the one overlapping-author input (event counts from Ref. [56]) is an external ATLAS-based benchmark, not the paper's target.

full rationale

The central derivations are standard supervised symbolic-regression tasks. In the WBF analysis, PySR/SymbolNet are trained as classifiers to distinguish cHfW=+1 from cHfW=-1 events, and the resulting analytic observables are then compared with the known parton-level observable pT,j1 pT,j2 sin(Delta phi_jj) and with a BDT; the learned constants are fitted to MC data, but the evaluation is a performance comparison, not a derivation of a physics constant from its own prediction. In the ttH analysis, the regression target is the parton-level Collins-Soper angle cos(theta*), and the input features are reco-level momenta; Fig. 10 explicitly states the MSE values are 'evaluated on the test dataset', so the reported performance is not a fit evaluated on its own training target. Scenario 1 recovers the known analytic CS angle only as a sanity check, not as a claimed output. The b-jet assignment and b/bbar-discrimination assumptions are acknowledged experimental idealizations ('we assume that the b-jets have been correctly assigned...', 'we assume that the bottom and anti-bottom can be distinguished. This is experimentally very difficult'), and they limit the detector-level claim, but they do not make any derived quantity equivalent to its input by construction. The only self-citation that enters numerically is Ref. [56] for expected event counts in the CP-sensitivity comparison ('The number of expected events are taken from Ref. [56]. They are based on ATLAS analyses...'); this is an external benchmark input, not a circular justification of the learned formulas, and it does not supply the fitted constants or the reconstructed observable. No equation in the paper is defined in terms of the quantity it is claimed to predict, and no fitted parameter is renamed as a prediction. Accordingly, no circular step meeting the required evidence standard is present.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claims rest on standard collider-physics modeling assumptions and on fitted formula coefficients. The numerical constants in all learned formulas are fit outputs, not predictions. The most fragile assumption is perfect b-jet identification in ttH. No new physical entities are introduced.

free parameters (5)
  • WBF PySR formula coefficients = e.g. 1.8566, 0.3080, 2.5977, 0.6047 in Eq. (32)
    Fitted to MadGraph/Delphes events by symbolic regression; they set the shape of the learned CP-odd observable.
  • WBF SymbolNet formula coefficients = many coefficients in Eq. (32)
    Fitted to simulated events by the SymbolNet training procedure; they define the learned CP-odd observable.
  • ttH learned formula coefficients = e.g. 1.006, 1.001, 1.027 in Eq. (42), and all constants in App. D
    Fitted to simulated ttH events; they define the reconstructed Collins-Soper angle in each scenario.
  • CP-odd loss weight alpha = not specified
    Hand-chosen balance in Eq. (25) between classification loss and CP-odd penalty; it affects whether learned formulas are CP-odd.
  • Inverse Gaussian loss sigma = 8
    Hand-chosen robust-loss scale in Eq. (41) for ttH scenarios; not physically motivated.
assumptions (6)
  • domain assumption SMEFT truncation to the dimension-6 operator cHfW for WBF and the top-Yukawa parameterization for ttH
    All CP sensitivity is modeled through these operators; other BSM contributions are ignored.
  • standard math Neyman-Pearson classifier mapping in Eq. (23) yields the optimal CP-odd observable
    Under a known likelihood ratio, the classifier output is monotonic in the optimal observable; standard result.
  • domain assumption MadGraph + Pythia + Delphes simulation approximates LHC detector response
    Used to generate all training and evaluation data; no full detector simulation, pileup, or systematic uncertainties.
  • ad hoc to paper Perfect b-jet assignment in ttH
    Sec. 4.1 assumes b-jets are correctly assigned to the lepton and light quarks; this is experimentally difficult and load-bearing for the CS-angle reconstruction results.
  • standard math Wilks theorem for the test statistic in Eq. (53)
    Assumes the test statistic q follows a chi-square distribution, used for all quoted significances.
  • domain assumption A classifier trained on +/-cHfW approximates the optimal observable for other cHfW values
    The paper tests this by training on +/-0.1 and +/-1, but it remains a modeling assumption about extrapolation in coupling strength.

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Pith. "Pith review of $\mathcal{CP}$-Analyses with Symbolic Regression." pith.science (2026). https://pith.science/paper/PEULMNAZ

@misc{pith2026250705858,
  author       = {Pith},
  title        = {Pith review of: $\mathcalCP$-Analyses with Symbolic Regression},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PEULMNAZ}},
  note         = {Machine review of arXiv:2507.05858}
}
abstract

Searching for $\mathcal{CP}$ violation in Higgs interactions at the LHC is as challenging as it is important. Although modern machine learning outperforms traditional methods, its results are difficult to control and interpret, which is especially important if an unambiguous probe of a fundamental symmetry is required. We propose solving this problem by learning analytic formulas with symbolic regression. Using the complementary PySR and SymbolNet approaches, we learn $\mathcal{CP}$-sensitive observables at the detector level for WBF Higgs production and top-associated Higgs production. We find that they offer advantages in interpretability and performance.

Figures

Figures reproduced from arXiv: 2507.05858 by the authors.

Figure 1
Figure 1. Exemplary function trees displaying the functions 1.4 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. A crossover operation in which the red-dashed part of the original tree has been [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. SymbolNet architecture with a single symbolic layer. The input parameters are linearly transformed to an input representation for the symbolic layer. The mathmatical oper￾ations are applied in order, and the output of the symbolic layer is linearly transformed to the final prediction. In the second step, the output of the linear operation is passed to a set of pre-defined math￾ematical operations. This replaces the … view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: illustration of a vectorized SymbolNet structure. The input 4-vectors are trans￾formed via one of each layer types to the final prediction. vectors as its input and returns scalars, the pruning implies masked operators return zero, f i (a) → f i (a) θ(1 − tunary,i ) gi…
Figure 5
Figure 5. Figure 5: Normalized distribution of pT,j1 pT,j2 sin∆φj j (left) and its bin-wise asymmetry A defined in Eq. 30 (right) for the SM (corresponding to cHWf = 0) and cHWf = ±1. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: Performance of the learned formulas for the [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: Dependence of the test significance on cHWf. For the diamonds, the observables are trained with cHWf = ±1, while for the crosses the observables are trained with cHWf = ±0.1. For the two lowest cHWf values, the blue crosses are hidden behind the diamond markers. classi…
Figure 8
Figure 8. Figure 8: Test significance as a function of the training statistics. The label [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: Predicted distributions of the CS angle for the six scenarios defined in Tab. [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: MSE values for each scenario. The values are shown for the classical reconstruction [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]
Figure 11
Figure 11. Figure 11: SymbolNet (left) and PySR (right) formulas for scenario 6 with perfect b-ordering. We show the learned CS angle, its calibration normalized to the same maximum count of events in one bin, and the distributions for αt = 45◦ including the expected CP-sensitivity. 22 [P…
Figure 12
Figure 12. Figure 12: ∆χ 2 values for excluding αt = 45◦ based on a measured SM-dataset. We show results for a classical reconstruction (green), PySR (red), and vectorized SymbolNet (blue), compared to the parton-level χ 2 . The central values and error bars correspond to the mean and the …
Figure 13
Figure 13. Figure 13: ∆χ 2 values from the PySR and SymbolNet formulas for scenario 6 by comparing the SM αt = 0 ◦ hypothesis with different amounts of CP violation. metry like CP. We have shown how interpretable and controlled analytical expressions for testing the CP nature of the Higgs …
Figure 14
Figure 14. Figure 14: Distributions of εµνρσ(pt + p¯t ) µ (pt − p¯t ) ν (pl + p¯l ) ρ (pl − p¯l ) σ on parton level (left) and εµνρσ(pb + p¯b ) µ (pb − p¯b ) ν (pl + p¯l ) ρ (pl − p¯l ) σ on detector level (right) for αt = ±45◦ . Next, we train SymbolNet with the detector-level events to c…
Figure 15
Figure 15. Figure 15: Distribution obtained from the equation predicted by [PITH_FULL_IMAGE:figures/full_fig_p027_15.png]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.