Reconstruction of angular correlations in the associated top quark and the dark matter mediator production

A. Markina; A. Zaborenko; E. Abasov; E. Iudin; G. Vorotnikov; L. Dudko; M. Perfilov; P. Volkov

REVIEW 3 major objections 5 minor 32 references

Reconstruction of angular correlations in the associated top quark and the dark matter mediator production

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper claims that a normalizing-flow network, ν-Flows, reconstructs the invisible momenta in associated top-quark plus dark-mediator production well enough to recover the angular correlation variable that separates signal from…

desk verdict A credible ML comparison showing normalizing flows reconstruct a top-spin-correlation variable much better than an MLP, with code released; the direct-applicability claim outruns the phenomenological evidence. read the letter →

arxiv 2504.14303 v1 pith:KZVWQCMN submitted 2025-04-19 hep-ph

E. Abasov , L. Dudko , E. Iudin , A. Markina , P. Volkov , G. Vorotnikov , M. Perfilov , A. Zaborenko This is my paper · ORCID

classification hep-ph

keywords darkmattermediatorsingletopquarkproductionangularcorrelationsnormalizingflowsneutrinomomentumreconstructionmachinelearningLHCphenomenologyspin

topics Dark Matter

open problems 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 tries to establish that a Normalizing Flows architecture called ν-Flows can reconstruct the momenta of the two invisible particles—the neutrino and the scalar dark-matter mediator—in single-top events, and that the reconstructed momenta reproduce the top-quark angular correlation variable $\cos(\theta_{\bar{l}\bar{d}})$ far more accurately than a multilayer perceptron. This variable is a spin-correlation observable that separates dark-matter production from Standard Model backgrounds in a simplified dark-matter model. At parton level the flow reconstruction gives a histogram mean absolute error of 52.4 and a $\chi^2$ of 335, versus 360.6 and 8985 for the MLP; after detector simulation the improvement persists. The intended payoff is a practical offline tool that experimental searches could apply directly to LHC data.

What carries the argument

The load-bearing object is the ν-Flows architecture, a conditional normalizing flow: an invertible neural transformation that maps a standard normal distribution to the six-dimensional distribution of the neutrino and mediator momenta, conditioned on event observables through an encoding network. The invertible coupling layers split the target variables into two groups in each layer, transform one group with piecewise rational quadratic splines, and use a fully connected network to pass conditioning information between groups. For each event the network is sampled many times and the median is used as the point estimate; the likelihood objective preserves correlations among target variables, which the paper argues is why the reconstructed angular distribution is closer to truth than the MLP's.

What would settle it

Train ν-Flows on events from one generator and test it on events from a second generator with the same truth labels; if the histogram MAE of the reconstructed angular variable rises to the level of the MLP baseline (around 360), the claimed performance is generator-specific. A further test on real LHC data in a Standard Model control region would reveal whether the reconstructed distribution matches the expected background shape within uncertainties.

Watch

Extended reading notes

Core claim

For the process $pp \to t(\to \nu l \bar{b})q$ plus a scalar dark-matter mediator (mass 400 GeV, dark-matter mass 1 GeV), the paper's central claim is that ν-Flows, a normalizing-flow network with coupling layers and piecewise rational quadratic splines, reconstructs the six target momentum components of the neutrino and mediator from observed final-state objects. When these reconstructed momenta are used to build the angular variable $\cos(\theta_{\bar{l}\bar{d}})$ in the top-quark rest frame, the resulting histogram matches the true distribution with histogram MAE 52.4 and $\chi^2$ score 335 at parton level, while the MLP baseline gives 360.6 and 8985. After hadronization and detector smearing, ν-Flows still gives the closest histogram (103.1 and 815.5, compared with 154.6 and 1554.9 for the MLP). The authors therefore conclude that the method 'can be directly applied to collider data' even though the final-state momenta do not uniquely determine the invisible momenta.

Load-bearing premise

The central assumption is that the Monte Carlo sample used for training is faithful to real LHC collisions, because the observed final state does not determine the neutrino and mediator momenta uniquely, so the network can only learn the conditional distribution encoded in that simulation.

Editorial extensions

If this is right

If ν-Flows works on real data, the angular variable $\cos(\theta_{\bar{l}\bar{d}})$ can be used in experimental searches for dark-matter mediators in the single-top final state, where it separates signal from background.
The method's advantage survives detector smearing, so it is suitable for offline analysis on reconstructed objects rather than only at generator level.
The flow's likelihood-based training does not need analytic solutions for the invisible momenta, removing the obstacle that blocked the analytical approach cited in the paper.
Because the network produces a distribution per event, the reconstructed angular variable can be aggregated by taking the median, while the sample spread gives a measure of reconstruction ambiguity.

Reading between the lines

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

The same architecture could be tested on other processes with two invisible particles, such as top-quark pair production with additional new-physics particles, where no unique kinematic solution exists.
A decisive check the paper does not report is cross-generator transportability: training on one Monte Carlo generator and testing on another would reveal how much of the success is tied to simulation-specific features.
The per-event sample spread of the flow could be developed into a systematic uncertainty or an event-level weight for the angular distribution, an extension the paper does not exploit.

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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

Dear colleague,

The one-line take: this is a solid, narrow simulation-based benchmark showing that ν-Flows reconstructs the top-quark spin-correlation variable in single-top + DM events far better than an MLP baseline, and it ships code. The caveat is that the final claim "can be directly applied to collider data" is not supported by anything in the paper.

What's actually new: prior work tried analytic reconstruction [14] and separation studies [23], and the ν-Flows architecture comes from [27] applied to t-tbar. This paper is the first application of normalizing flows to the single-top + DM process with two invisible particles, benchmarked against an MLP. The numbers are striking: histogram MAE drops from 360.6 to 52.4 at parton level, chi-squared from 8985 to 335, and the improvement survives Delphes smearing. The figures show the MLP misses the distribution shape while ν-Flows captures it. Code is on Zenodo, which makes the central comparison reproducible.

The main soft spot is the direct-applicability sentence. Section 2.1 explicitly says real data is not used; the network is trained on a 1:1 SM/DM mixture with fixed mediator mass 400 GeV and DM mass 1 GeV. Because the problem is underdetermined, the flow learns the conditional distribution implied by that training mixture. Real LHC data will have different background composition, signal fraction, and possibly mediator mass, and the paper reports no evaluation on background-only samples, mixed pseudo-data, or varied model parameters. So the final claim is an extrapolation. It may hold, but the authors need to say it is a promise rather than a demonstrated property.

Secondary issue: Table 2 and Figures 5-6 give single-run metrics with no uncertainties or multiple seeds. For a paper whose main evidence is a metric comparison, that's minor but worth fixing. The Basic Flows convergence handling by clipping outputs to [-10,10] is a bit ad hoc but not a problem.

The citation pattern is honest and the build on refs [14,23,27] is explicit. No circular derivation. The central comparative result holds up.

Who this is for: people working on ML-based momentum reconstruction in top/DM searches at the LHC. It is an analysis-tool improvement, not new physics. I would send it to peer review; it deserves referee time even if the final claim needs to be reined in. I would not cite it in my own work unless I were in that exact niche.

Recommendation: engage with the work, but ask the authors to soften the direct-applicability claim, add seed statistics, and ideally show one out-of-distribution test.

Referee Report

3 major / 5 minor

Summary. The paper studies reconstruction of the momenta of the neutrino and the scalar dark matter mediator in associated single-top-plus-DM production at the LHC, with the goal of reconstructing an angular correlation variable cos(θ_{bar l bar d}) that discriminates signal from SM background. Three machine-learning approaches are compared: a multilayer perceptron (MLP), an autoregressive normalizing flow ('Basic Flows'), and a coupling-layer normalizing flow ('ν-Flows'). The authors report that ν-Flows yields the lowest histogram MAE and χ² scores at parton level and retains an advantage after DELPHES detector simulation, and they conclude that the method can be directly applied to collider data.

Significance. If substantiated, the result is a useful demonstration that conditional normalizing flows outperform point-estimate regressors for an underdetermined kinematic inverse problem in top-quark physics, with a concrete downstream benefit for a DM-search variable. The paper has several strengths: the evaluation uses an external MC-truth benchmark from standard generators (CompHEP/MadGraph, Pythia8, DELPHES), so the comparison is not circular; the analysis is presented at both parton and detector level; and the code is publicly released with a DOI. The main scientific claim, however, is stronger than the evidence: the conclusion that the method 'can be directly applied to collider data' is not supported by the phenomenological setup and the fixed training mixture, and the reported performance metrics lack statistical uncertainties.

major comments (3)

[Section 2.1 and Section 4] The manuscript explicitly states in Section 2.1 that 'At this stage, the analysis is phenomenological; real collider data is not used,' yet Section 4 concludes that the method 'can be directly applied to collider data.' This is an extrapolation beyond the tested regime: the flow is trained and evaluated on a 1:1 SM/DM mixture with fixed mΦ = 400 GeV and mχ = 1 GeV, and the inverse problem is underdetermined, so the learned conditional distribution p(pν,pΦ | x) is shaped by the training prior. No out-of-distribution evaluation is reported: no background-only sample, no mixed background-plus-signal pseudo-data, no variation of the mediator mass or signal fraction, and no data/MC closure test. The direct-applicability claim should either be removed or be replaced by a clearly scoped statement that the method is ready for application to simulated signal-region studies, pending validation on more realistic event mixtures.
[Table 2 and Section 3] Table 2 reports MAE, histogram MAE, and χ² score for the three architectures without any statistical uncertainties or multiple-seed statistics. Since neural-network training is stochastic and the χ² values are 335 versus 1557 versus 8985, the reader cannot assess whether the ordering of the methods is stable under retraining or whether the quoted differences are within seed-to-seed variance. The manuscript also does not state the size of the test set or the binning used for the histogram metrics, both of which are needed to interpret the χ² score and to reproduce the comparison. Please add uncertainties over training seeds and specify the test-set size and histogram binning.
[Section 3, Figures 5 and 6] The detector-level comparison is presented only as figures, while the central quantitative table (Table 2) is limited to parton level. Since the conclusion in Section 4 explicitly claims superiority 'after simulation of the detector response,' the detector-level histogram MAE and χ² scores should be reported in the same tabular form as the parton-level results. In addition, the aggregation protocol for flow samples is not specified precisely: Section 3 states that the median is used, but the number of samples per event differs in the captions (5 for ν-Flows, 10 for Basic Flows in Figs. 9–10, and 10,000 in Fig. 7), and this choice can affect both the point-wise MAE and the downstream angular distribution.

minor comments (5)

[Figures 7, 9, 10] The number of flow samples per event is inconsistent across captions: Fig. 7 says 10,000 points are sampled for each Flow-based network, while Figs. 9 and 10 say the median of 10 and 5 points is used. Please reconcile these numbers and state the final sample count used for the reported results.
[Section 2.2.3 and Table 1] The clipping of Basic Flows outputs to the interval [-10, 10] is mentioned in Section 3 but is not included in Table 1 or in the architecture description; please state the clipping range for both flow models and, if the range was tuned, report the tuning range.
[Section 2.1] The list of input features is incomplete: the high-level variables are described only as 'various combinations of low-level variables,' without an explicit enumeration or the construction formulas. This hampers reproducibility independently of the public code; please provide the full feature list.
[Section 2.1 / Figures 5 and 6] The figure captions refer to the 'true_nophi' and 'reconstructed_nophi' distributions, but the term 'nophi' is not defined in the text; clarify which reconstruction uses only the neutrino without the mediator contribution.
[Throughout] There are several typos and formatting issues, including 'T able 1', 'reconstruiction', and the sentence 'the denominator of such a metric may be near zero' in Section 2.2.1; a careful proofread is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the reconstruction is trained on generator truth and benchmarked against the same external truth on a held-out test set; the target variable comes from prior work but is not used to define the result.

full rationale

The paper's central claim is that a ν-Flows normalizing-flow network reconstructs the neutrino and mediator momenta more accurately than an MLP, as measured by MAE on momenta and by histogram MAE and χ² for the angular variable cos(θ_{ℓ̄ d̄}) against Monte Carlo truth. The target variable is taken from the authors' earlier paper [14], and the architecture follows the external ν-Flows recommendation [27], but neither is used as evidence for reconstruction quality. The network outputs are compared with independent generator-level truth from CompHEP/MadGraph and with detector-simulated truth from Pythia8/DELPHES on a held-out test set using a 0.6:0.2:0.2 split. No fitted parameter is renamed as a prediction: the training losses are L1 or log-likelihood, hyperparameters are tuned on a validation χ², and all reported metrics are evaluated on test data. The statement in Section 2.1 that 'the analysis is phenomenological; real collider data is not used' is an explicit limitation, and the conclusion that the method 'can be directly applied to collider data' is an extrapolation beyond the tested MC-only regime; that is a domain-transfer or correctness concern, not a circularity, because the derivation does not reduce to its inputs. The self-citations [14,23] provide the variable and context but are not load-bearing in the sense of forbidding alternatives or importing a uniqueness theorem; the reconstruction benchmark stands on its own external test data.

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

The central claim rests on standard simulation tools and the simplified dark matter model, plus the angular variable from the authors' prior work and a set of tuned hyperparameters. The most fragile inputs are the fidelity of the simulation chain to real LHC data and the learnability of the underdetermined two-invisible-particle system. No new particles or forces are introduced.

free parameters (6)

ν-Flows context size = 31
Chosen by Optuna hyperparameter search using chi-squared score on validation set (Table 1); affects reconstruction accuracy.
Number of ν-Flows blocks = 4
Tuned with Optuna on validation chi-squared (Table 1).
Neurons in ν-Flows inner network layer = 100
Tuned with Optuna on validation chi-squared (Table 1).
Number of ν-Flows inner network layers = 2
Tuned with Optuna on validation chi-squared (Table 1).
Basic Flows output clipping range = [-10, 10]
Hand-chosen to stabilize the inverse transformation; without it test loss tends to infinity (Section 3).
Number of flow samples per event = Basic Flows: 10, ν-Flows: 5 (median)
Hand-chosen aggregation rule used to produce reconstructed momenta from flow distributions (Figs. 8-10).

assumptions (5)

domain assumption The simplified model with scalar mediator (mχ=1 GeV, mΦ=400 GeV, gf=gχ=1) is an adequate benchmark for LHC dark matter searches.
Section 2.1 sets these values following the LHC Dark Matter Working Group; the entire simulation and the angular variable are defined within this model.
domain assumption The angular variable cos(θ_{bar l bar d}) in the top rest frame separates SM and DM processes.
Section 2.2.1 uses this variable as ground truth for evaluation; it is introduced in the authors' previous paper [14], not derived here.
domain assumption The CompHEP/MadGraph + Pythia8 + DELPHES simulation chain faithfully represents LHC events for this process.
Section 2.1 states 'the analysis is phenomenological; real collider data is not used', yet Section 4 concludes the method can be applied to collider data.
domain assumption The six target momentum components are statistically identifiable from the chosen low- and high-level features, despite the underconstrained final state.
Section 2.1 defines the target as pν and pΦ; with two invisible particles the system is not uniquely solvable, so the conditional distribution learned by the flow must be representative.
standard math Normalizing flows preserve a normalized probability density by construction.
Used in Section 2.2.3 to justify using likelihood as the training objective; standard property of invertible transformations.

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Cite this review

Pith. "Pith review of Reconstruction of angular correlations in the associated top quark and the dark matter mediator production." pith.science (2026). https://pith.science/paper/KZVWQCMN

@misc{pith2026250414303,
  author       = {Pith},
  title        = {Pith review of: Reconstruction of angular correlations in the associated top quark and the dark matter mediator production},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KZVWQCMN}},
  note         = {Machine review of arXiv:2504.14303}
}

read the original abstract

For the process of single top quark production within the "simplified model" with a scalar dark matter mediator, a new variable based on angular correlations was presented, for the proper reconstruction of which it is necessary to separate the contributions of two undetectable particles: the neutrino and the mediator. In this work, various machine learning approaches for reconstructing the momenta of these particles are analyzed. A comparison is made between the results obtained using a multilayer perceptron and the Normalizing Flows architectures. The neural networks based on Normalizing Flows, presented in this work, demonstrate a high quality of reconstruction of the target variable and can be used for collider data analysis.

Figures

Figures reproduced from arXiv: 2504.14303 by the authors.

**Figure 3.** For the encoding network in both models, an architecture similar to the baseline model is used but with a reduced number of neurons - 200 instead of 500. 3. RESULTS This section presents comparisons of the results obtained using various machine learning methods: MLP, Normalizing Flows with autoregressive layers, and Normalizing Flows based on coupling layers. Results are provided for both generator-level data and d… view at source ↗

**Figure 1.** Distribution of the cosine of the angle between the lepton and the down-type quark in the top quark rest frame for the processes pp → t(→ νl ¯lb)q (SM) and pp → t(→ νl ¯lb)qχχ¯ (neutrino TRF) with the presence of a scalar mediator in the model with mχ = 1 GeV, mϕ = 400 GeV , gχ = gν = 1. For comparison, the same distribution is shown for the top quark rest frame system that takes the DM mediator into account (neutri… view at source ↗

**Figure 2.** Schematic diagram of Normalizing Flows. Here, Event Observables are variables constructed based on the data from known particles [27] [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

Figures from the paper (8 more)

**Figure 3.** Figure 3: Block of the ν-Flows network [27]. Here, X represents the target momenta of the neutrino and mediator, the Conditioning Tensor is the output of the encoding fully connected network, and RQS denotes the piecewise rational quadratic splines. 0 5 10 15 20 25 30 35 0.28 0.…

**Figure 4.** Figure 4: Loss function curves for different models. L1 loss is used for the MLP, and the logarithm of the likelihood function −2 lnL is used for Normalizing Flows. 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 0 500 1000 1500 2000 2500 3000 3500 Hist MAE: 360.6 2 score: 8984.6 c…

**Figure 5.** Figure 5: Comparison of the reconstructed and original distributions of the cosine of the angle between the lepton and the down-type quark in the top quark rest frame at the parton level. The postfix "nophi" denotes the top quark rest frame reconstruction in which only the neutr…

**Figure 6.** Figure 6: Comparison of the reconstructed and original distributions of the cosine of the angle between the lepton and the down-type quark in the top quark rest frame after detector smearing in DELPHES. The postfix "nophi" denotes the reconstruction of the top quark rest frame u…

**Figure 7.** Figure 7: Comparison of the reconstruction of the z-component of the mediator’s momentum P ϕ z for individual events. For Flow-based networks, 10,000 points are sampled for each event. 1000 750 500 250 0 250 500 750 1000 p x 0 10000 20000 30000 40000 50000 reconstructed true 100…

**Figure 8.** Figure 8: Reconstruction of the momentum components of the neutrino and mediator using the fully connected network [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

**Figure 9.** Figure 9: Reconstruction of the momentum components of the neutrino and mediator using Basic Flows. For each event, the median of 10 points is used as the reconstructed value. 1000 750 500 250 0 250 500 750 1000 p x 0 5000 10000 15000 20000 25000 30000 35000 reconstructed true 1…

**Figure 10.** Figure 10: Reconstruction of the momentum components of the neutrino and mediator using ν-Flows. For each event, the median of 5 points is taken as the reconstructed value [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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Reconstruction of angular correlations in the associated top quark and the dark matter mediator production

INTRODUCTION Convincing astrophysical and cosmological observations, such as the studies of galaxy rotation curves [1] and gravitational lensing during galaxy cluster collisions [2], indicate the presence of a large amount of hidden matter in the Universe, the so-called "dark matter" (DM), which manifests itself through gravitational interactions and inte...

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Data At this stage, the analysis is phenomenological; real collider data is not used

RECONSTRUCTION OF THE SPIN CORRELATIONS OF THE TOP-QUARK 2.1. Data At this stage, the analysis is phenomenological; real collider data is not used. For modeling and numerical calculations at the parton level, the computational packages Com- pHEP4.6rc1 [15, 16] and MadGraph5 [17] were employed. All calculations are presented for proton-proton collisions at...

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histogram

RESULTS This section presents comparisons of the results obtained using various machine learn- ing methods: MLP, Normalizing Flows with autoregressive layers, and Normalizing Flows based on coupling layers. Results are provided for both generator-level data and data after simulating the detector response in DELPHES. The loss function curves for each model...

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Particle Physics and Cosmology

CONCLUSIONS In this work, a new method was proposed for separating the momentum components of the scalar dark matter mediator and the neutrino in single top quark production based on the Normalizing Flows architecture. This approach outperforms multilayered perceptrons, significantly improves the accuracy of reconstructing the top quark’s angular correlat...

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V. C. Rubin and W. K. Ford, Jr., Astrophys. J.159, 379 (1970)

1970