Probing the dimuon channel of a Z' boson at the HL-LHC using multivariate analysis

Ali Muhammad H. H.; El-sayed A. El-dahshan; S. Elgammal

REVIEW 3 major objections 6 minor 56 references

Probing the dimuon channel of a Z' boson at the HL-LHC using multivariate analysis

T0 review · 3 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A boosted-decision-tree analysis of the dark Higgs Z' dimuon channel finds a statistical significance of 6.77σ at 500 GeV with the full HL-LHC dataset.

desk verdict A clearly documented HL-LHC sensitivity projection for a known dark Higgs model, but the headline z=6.77 is inflated by optimizing the BDT cut on the same events; worth refereeing with a request for a validation split and systematics. read the letter →

arxiv 2505.02739 v2 pith:OM6IIGHR submitted 2025-05-05 hep-ex hep-ph

Ali Muhammad H. H. , El-sayed A. El-dahshan , S. Elgammal This is my paper · ORCID

classification hep-exhep-ph

keywords darkmatterHiggsmodelZ'bosondimuonfinalstateHL-LHCmultivariateanalysisboosteddecisiontreesmissingtransverseenergy

topics Dark Matter

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

The paper asks whether the High-Luminosity LHC, with 3000 $fb^{{-1}}$ at 14 TeV, could discover a dark Higgs model in which a Z' boson decays to a dimuon pair plus missing transverse energy from dark matter. Using a boosted decision tree trained on five kinematic variables, it finds a statistical significance of $z = 6.77$ at a Z' mass of 500 GeV, above the 5σ discovery threshold. It also reports that lighter benchmark points reach 5σ with far less luminosity, while Z' masses from 600 to 1000 GeV stay below 5σ even with the full dataset. A sympathetic reader would take this as evidence that the dimuon-plus-missing-energy channel is a promising early target for the HL-LHC at moderate Z' masses.

What carries the argument

The central mechanism is the trained boosted decision tree (BDT), an ensemble of shallow decision trees that classifies events as signal-like or background-like using five kinematic inputs: the transverse momenta of the two muons, the missing transverse energy, the dimuon invariant mass, and the azimuthal angle between the dimuon system and the missing energy. The analysis scans the BDT response threshold for each benchmark point and takes the cut that maximizes $z = S/\sqrt{S+B}$, where $S$ and $B$ are the expected signal and background counts after the cut. This classifier, trained with five-fold cross-validation, is shown to achieve the largest area under the ROC curve among the three methods compared, and it is the BDT results that produce the quoted significance values.

What would settle it

Apply the same BDT cut to the first 3000 $fb^{{-1}}$ of HL-LHC data and fit the dimuon invariant-mass spectrum around 500 GeV in events with large missing transverse energy; if no excess over the Standard Model prediction appears, the claimed $S=500$ events and $z=6.77$ are refuted.

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Extended reading notes

Core claim

The paper's central claim is that the dark Higgs benchmark point BP4, with $M_{Z'}=500$ GeV and $g_{SM}=0.25$, $g_{DM}=1.0$, would be discovered in the dimuon plus missing transverse energy channel at the HL-LHC: after an optimal cut on the boosted decision tree response, the expected signal is $S = 500$ events against a background of $B = 4951$ events, giving $z = S/\sqrt{S+B} = 6.77$ with 3000 $fb^{{-1}}$ at 14 TeV. For BP1 through BP3, the paper reports 5σ discovery at integrated luminosities of 4.3, 18.8, and 175 $fb^{{-1}}$ respectively. For BP5 through BP9, masses 600 to 1000 GeV, the expected significance falls from 2.81 to 0.48, below the discovery threshold. The BDT classifier outperforms the deep neural network and the likelihood estimator in the comparison, so the quoted sensitivities are specific to the boosted decision tree.

Load-bearing premise

The result assumes the fast detector simulation used for both signal and background reproduces the real HL-LHC's muon reconstruction, isolation, and missing-energy response; if real detectors are less efficient at high pileup, the predicted significance would shrink.

Editorial extensions

If this is right

At $M_{Z'}=500$ GeV, the dark Higgs benchmark BP4 yields a 6.77σ excess in the dimuon plus missing-energy channel with 3000 fb^{-1}, clearing the discovery threshold.
The lighter benchmarks BP1 through BP3 reach 5σ with only 4.3, 18.8, and 175 fb^{-1} respectively, so the channel is sensitive at modest integrated luminosity for low Z' masses.
For Z' masses between 600 and 1000 GeV, the expected significance stays between 2.81 and 0.48, so the full HL-LHC dataset is not enough to discover the model in this channel.
Since the BDT outperforms the DNN and likelihood classifiers on this final state, the reported sensitivities are tied to using the boosted decision tree; the other classifiers would give weaker results.

Reading between the lines

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

A direct extension would replace the counting significance $z = S/\sqrt{S+B}$ with a profile-likelihood fit that includes systematic uncertainties on the background shape and acceptance; that would show how much of the 6.77σ survives real-world effects.
The same BDT recipe, retrained on the dielectron or other leptonic channels, could extend the search and cross-check the dark Higgs interpretation without requiring a new model setup.
Because BP1 through BP3 need less than 500 fb^{-1} for discovery, early HL-LHC data could already test this model, and the analysis could be re-optimized on the first year of collisions.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

What you should know: this is a straightforward, well-documented sensitivity projection for the existing dark Higgs / mono-Z' model, and the central number — BDT significance z=6.77 at M_Z'=500 GeV, 3000 fb^-1 — is likely a bit too good because the cut that maximizes z is chosen on the same sample used to quote S and B. The paper is still worth a serious referee, but the discovery framing should be softened pending a fix.

The genuinely new part is the HL-LHC reach scan: required luminosities for 5-sigma and expected z values for BP4-BP9. The simulation chain is standard and reproducible: MadGraph5_aMC@NLO, Pythia8, Delphes with the CMS pile-up card, TMVA with BDT/DNN/Likelihood, k-fold CV, KS tests, ROC curves. The classifier comparison is competently done, and the background list is sensible. Credit where due: the paper does not oversell the model; it inherits couplings and benchmarks from prior work, which is normal and properly cited.

Now the soft spots, in proportion. The big one is threshold selection bias. The text says (Section VI, Fig. 8, Table VIII) that for each BP they apply the BDT response cut where z is maximum, then report that z. That is maximizing a noisy statistic over many correlated thresholds. The k-fold CV prevents overtraining of the classifier response, but it does not protect against optimizing the cut on the out-of-fold predictions. For BP4, S=500 and B=4951, so Poisson fluctuations are sizeable; the true expected significance at a pre-specified or independently chosen cut will be lower, possibly below 5σ. This is a load-bearing issue for the headline claim.

Second, z = S/sqrt(S+B) is purely statistical. A 10-20% systematic shift in background acceptance or normalization brings z=6.77 down to roughly 5 or lower. The paper acknowledges systematics only as a future step, which is honest as a limitation, but the abstract and conclusion say “strong discovery potential” without that caveat.

The minor issues are just that: the DNN with its chosen hyperparameters underperforms BDT and Likelihood, which is plausible but not deeply investigated; the correlation matrices show some correlated variables used anyway, but the importance ranking justifies keeping them.

Verdict: The paper is a solid, clearly written Monte Carlo sensitivity study with one methodological flaw in the central number. It deserves peer review, not a desk rejection. A referee should ask for a validation split or a pre-specified cut to remove the optimization bias, and a systematic uncertainty treatment or at least a sensitivity scan over background normalization. The discovery claim should be rephrased as a preliminary projection until then.

Referee Report

3 major / 6 minor

Summary. The paper presents a Monte Carlo study of the dark Higgs model in the mono-Z' portal at the HL-LHC, examining the Z'->mu+mu- plus missing transverse energy signature. Events are generated with MadGraph5_aMC@NLO and Pythia8, passed through Delphes with the CMS pile-up card, and analyzed with TMVA classifiers (BDT, DNN, Likelihood). The BDT is selected as the best classifier, and expected significances are computed for dark Higgs signals with M_Z' between 200 and 1000 GeV at sqrt(s)=14 TeV and L=3000 fb^-1. The central numerical result is a BDT-based significance of z=6.77 for M_Z'=500 GeV (BP4), with lighter benchmark points exceeding 5 sigma and heavier points falling below the discovery threshold.

Significance. If the central numerical claim were robust, the paper would be a useful HL-LHC projection for the dark Higgs dimuon channel, providing a concrete comparison of MVA classifiers and per-benchmark significance predictions. Its strengths include the full MadGraph+Pythia+Delphes simulation chain, five-fold cross-validation in training, an explicit hyperparameter table, and a falsifiable set of benchmark results. However, the quoted significances are computed with a simplified statistical formula that ignores systematic uncertainties, and the BDT cut is chosen to maximize the significance on the same sample used to report it. Both effects bias the headline z values upward, so the paper is best viewed as a phenomenological template that needs additional validation before its discovery claim can be taken at face value.

major comments (3)

[Section VI, Eq. (3), Fig. 8, Table VIII] The 'optimal cut' on the BDT response is selected by scanning thresholds and choosing the one that maximizes z on the same out-of-fold predictions used to quote z. As a result, z=6.77 is the maximum of a noisy function over many correlated thresholds, and is systematically larger than the expected significance for a pre-specified or independently chosen cut. With S=500 and B=4951 at the quoted cut, Poisson fluctuations at neighboring thresholds are sizable, so this is a first-order effect. The authors should report significances for a cut fixed on an independent validation set, or use a three-way split (training/validation/test) and quote the test-sample significance at the validation-selected cut.
[Eq. (3) and Section VII] All quoted significances are purely statistical, z=S/sqrt(S+B), with no systematic uncertainty assigned to signal or background rates, acceptances, or shapes. This is load-bearing: a 10-20% background normalization systematic alone moves the BP4 value from 6.8 toward the 5 sigma boundary (roughly 6.5 at 10% and 5.8 at 20%), and the BP5-BP9 values are already below 3 sigma. The paper should either include a nuisance-parameter treatment with correlated systematic uncertainties or clearly label every number as a statistical-only projection, with a matching caveat in the abstract and summary. The final-section sentence deferring systematics to 'future studies' does not adequately qualify the headline discovery claim.
[Sections IV.A, IV.B and Table VIII] The absolute values of S and B, and therefore the significances, are obtained from Delphes simulation using the CMS Pile-Up configuration card. The paper does not cross-check the resulting muon reconstruction, isolation, and missing transverse energy response against public HL-LHC performance studies or against a full simulation sample, even for one benchmark point. Because the acceptance numbers directly determine the central discovery claim, this fast-simulation assumption should either be validated or assigned a quantitative uncertainty; otherwise the absolute z values inherit an unquantified detector-modeling risk.

minor comments (6)

[Section IV.A and Table II caption] There are small language slips: 'tabled I' should be 'Table I', and the Table II caption uses 'Pb' where 'pb' is meant.
[Fig. 4 and surrounding text] The correlation matrix axis labels in Fig. 4 are nearly unreadable due to overlapping text, and the statement that 'most of them are highly uncorrelated' is difficult to reconcile with entries such as the 77-85% correlations visible in the figure. Please clarify the criterion for 'highly uncorrelated' and improve the figure labels.
[Eq. (2)] The equation for the separation power is typeset confusingly, with the integrand appearing as 'Z ( ˆPs(γ)− ˆPb(γ))2'. Please use a standard definition, e.g., 1/2 * integral((Ps-Pb)^2/(Ps+Pb)) dγ, and define all symbols.
[Table VI] The DNN layout string 'TANH|128, TANH|128, TANH|128, LINEAR' is not self-explanatory; please specify the layer sizes and activation functions explicitly in the table or in a footnote.
[Table VIII] The symbol 'NBC' is not defined; please state in the caption that it denotes the number of events after pre-selection but before the BDT cut.
[Section VI, overtraining discussion] The sentence 'This indicates good training performance for all three classifiers, as expected due to the use of CV' is imprecise: cross-validation reduces overfitting but does not by itself guarantee that training and test distributions match. Please report the KS test p-values used for the overtraining check.

Circularity Check

1 steps flagged · score 6.0 of 10

The headline significance z=6.77 is the maximum of a BDT-threshold scan on the same events used to count S and B, so it is a fitted output rather than an independent prediction; the rest of the derivation is self-contained.

fitted input called prediction [Section VI, after Eq. (3), Fig. 8 and Table VIII]
"Consequently, a cut value is applied to the BDT response for BP4 through BP9 to obtain the corresponding numbers of signal events (S) and background events (B) with L = 3000 fb−1 ... Finally, z is calculated for these six BPs using Eq. 3 [56]. z = S/sqrt(S + B) (3) The applied cut value, also known as optimal cut, is where the maximum z is achieved for the BP as shown in Fig. 8 for BP4."

The reported z is the value of Eq. (3) at the cut chosen by maximizing Eq. (3) over the BDT-response threshold. Thus the quoted significance is, by construction, the maximum of the objective over scanned cuts on the same events, not an independent prediction at a pre-specified selection. The k-fold CV and KS tests only guard against classifier overtraining; they do not protect against this threshold-optimization (look-elsewhere) bias, and no validation split is used for the cut choice. Consequently Table VIII's z values are fitted outputs of the scan, with positive bias from Poisson fluctuations in S and B.

full rationale

The model and couplings are inherited from Ref. [26] and from ATLAS recommendations ([28], [47], [48]); these are inputs, not outputs, and the self-citations [27] and [48] are not load-bearing for the numerical claim. The signal and background samples are generated independently with MadGraph, Pythia, and Delphes, and the BDT classifier performance is externally assessed through ROC/AUC curves. The central weakness is the threshold-selection step: the optimal cut is defined as the cut maximizing z, and the same z is then quoted as the achieved significance. This is the fitted-input-called-prediction pattern, and it affects exactly the claimed discovery (z=6.77 for BP4) and the required-luminosity values derived from it. No equation in the paper reduces to a fitted cross-section, and no load-bearing uniqueness theorem is imported from the authors' prior work; however, the headline significance is not an independently predicted number. Since the central claim depends on this optimized quantity, partial circularity is present. The paper's own closing sentence defers systematic uncertainties to future work, which further weakens the headline number but is not itself a circularity step.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

No genuinely new entities are introduced; the Z', dark Higgs, and dark matter particle are inherited from the dark Higgs model of Ref. [26]. The only free choices are the model couplings and the analysis cut.

free parameters (3)

g_SM (dark Higgs model coupling to SM) = 0.25
Chosen by hand, following Ref. [28] recommendation to maximize sigma x Br; it is an input to the simulation, not fitted to this paper's output.
g_DM (dark Higgs model coupling to dark sector) = 1.0
Chosen by hand, following Ref. [28]; input to the simulation.
Optimal BDT cut = varies per BP, e.g., 0.2754 for BP4
Chosen by scanning the BDT response to maximize z on the same sample, which introduces an analysis free parameter and an upward bias in the quoted significance.

assumptions (6)

domain assumption The simplified dark Higgs model with the heavy dark sector mass assumption (M_hD = M_Z' for M_Z' > 125 GeV) is the true signal model.
Adopted in Section II; the entire signal simulation is built on this model.
domain assumption The cross sections computed by MadGraph5_aMC@NLO v3.5.0 at NLO, combined with Pythia8, are accurate for the signal and backgrounds at sqrt(s)=14 TeV.
The cross sections in Tables II and III enter directly into the event yields and hence the significance.
domain assumption Delphes with the CMS pile-up configuration card faithfully approximates the response of the HL-LHC detectors for muons and missing transverse energy.
All event samples are passed through Delphes (Section IV); incorrect detector response would change acceptances and significances.
domain assumption The list of SM background processes in Table III is the complete relevant background.
Backgrounds outside this list, such as Z+jets with mismeasured MET or other diboson processes, are neglected.
ad hoc to paper The significance formula z = S/sqrt(S+B), with no systematic uncertainties, is adequate to assess discovery potential.
Used in Eq. 3; the paper acknowledges that systematics are not included.
domain assumption The MVA classifiers are not overtrained and generalize to unseen data based on the KS test and k-fold cross-validation.
Relying on the KS test and CV described in Section VI.

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Cite this review

Pith. "Pith review of Probing the dimuon channel of a Z' boson at the HL-LHC using multivariate analysis." pith.science (2026). https://pith.science/paper/OM6IIGHR

@misc{pith2026250502739,
  author       = {Pith},
  title        = {Pith review of: Probing the dimuon channel of a Z' boson at the HL-LHC using multivariate analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OM6IIGHR}},
  note         = {Machine review of arXiv:2505.02739}
}

abstract

The upcoming upgrade of the existing LHC facility at CERN is known as the High-Luminosity Large Hadron Collider (HL-LHC). It is designed to extend its physics reach by substantially increasing the integrated luminosity. It will enable more precise measurements of the Standard Model (SM) and improve sensitivity to rare events and possible new physics signatures. This study adopts a multivariate analysis (MVA) approach to effectively discriminate the dark Higgs (DH) signal against the dominant SM background. The analysis targets the leptonic decay mode of the $Z'$ boson, focusing on the dimuon final state at $\sqrt{s} = 14,\text{TeV}$ and $3000,\text{fb}^{-1}$ integrated luminosity corresponding to the HL-LHC. The DH signal is examined using the Toolkit for Multivariate Analysis (TMVA), employing and comparing the performance of three classifiers: Boosted Decision Trees (BDT), Deep Neural Networks (DNN), and Likelihood estimators.

Figures

Figures reproduced from arXiv: 2505.02739 by the authors.

**Figure 2.** FIG. 2: Distribution of dimuon invariant mass [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3: Normalized distributions of kinematics used as features in the MVA classifiers for BP4 of the DH signal, [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4: Linear correlation matrix of the input features used in the MVA for (a) BP4 of the DH signal, taking [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: FIG. 5: The method-unspecific relative importance of [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 6.** Figure 6: FIG. 6: Classifier response distributions for training and testing samples of (a) BDT, (b) Likelihood, and (c) DNN [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

**Figure 8.** Figure 8: FIG. 8: Cut efficiencies verses BDT response thresholds [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]

Discussion (0). Continue with ORCID to comment.

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