These are Long-Lived Particles (LLPs), and they are a central prediction of numerous theories that extend beyond the Standard Model. In a new result recently submitted to Physics Letters B – an effort I had the privilege of coordinating – the ATLAS Collaboration presents a comprehensive hunt for these elusive travelers, utilizing a dataset collected during the first three years of LHC Run 3 (2022–2024), amounting to 164 inverse femtobarns of data.

The Challenge of Displaced Events

Searching for LLPs is analogous to looking for a ghost that only becomes visible after it has left the room. Most standard reconstruction algorithms are designed for particles originating from the primary proton–proton collision point. Particles that decay further out – leaving displaced vertices (DVs) or displaced tracks – are frequently filtered out by standard software as noise or misreconstructions.

To overcome this, we have implemented several technical innovations. In Run 3, ATLAS introduced a dedicated displaced-muon trigger. This specialized hardware and software combination allows us to identify muons that do not point back to the collision point with significantly higher efficiency. By requiring the coincidence of these displaced muons with a high-mass (massive) displaced vertex, we can suppress the background from traditional prompt physics and focus on the signals that truly stand out from the Standard Model expected rates.

Why Long-Lived?

The stability (or lifetime) of a particle is typically determined by the strength of its interactions and the mass of the mediators – the messenger particles that carry forces between others. In the Standard Model (our current best theory of the subatomic world), the neutron is relatively long-lived because its decay is suppressed by the high mass of the W boson. In theories such as Supersymmetry – which proposes a superpartner for every known particle in nature – we encounter a similar phenomenon.

If a symmetry known as R-parity is slightly broken – a scenario referred to as R-parity violation – the Lightest Supersymmetric Particle is no longer stable. While this particle is often considered a dark matter candidate in stable theories, in models with broken symmetry it can decay into quarks and leptons. If this instability is rather subtle (that is, the coupling is weak), the particle will travel a measurable distance through the detector before finally revealing itself.

In our analysis, we probed some of these superpartners with R-parity violation couplings. The search was optimized for two primary supersymmetric scenarios:

1. Higgsinos: The superpartners of the Higgs bosons. In our benchmark models, these can decay into muons and quarks via processes that break the fundamental rules that usually keep the number of leptons (like electrons) or baryons (like protons) balanced in the universe.
2. Top Squarks (Stops): The superpartners of the top quark. When these decay via R-parity-violating couplings, they produce a distinctive signature: a b-quark jet and a muon, both originating from a common vertex located far from the initial collision point.

Figure 1: Benchmark signal models for long-lived supersymmetric particles, showing the characteristic displaced vertex and displaced muon signature. (Image: ATLAS Collaboration/CERN)

The Art of Reconstruction

Reconstructing these “displaced” events is an immense feat of engineering. The ATLAS detector is constructed like a giant onion, with concentric layers of sensors. When a particle decays inside these layers – rather than at the geometric center – it creates a complex hit pattern that requires specialized logic to interpret.

We employ two distinct tracking passes. The first pass captures the standard, “prompt” particles. Any detector hits not used in the initial pass are then analyzed by a second, more computationally intensive pass. This large-impact-parameter pass is designed specifically for tracks with a large transverse impact parameter – essentially a miss distance that quantifies how far a track deviates from the center of the collision. These tracks are then algorithmically grouped to form a Displaced Vertex (DV).

To ensure the signal is robust, a vertex must meet stringent criteria: it must have at least four associated tracks and a high invariant mass. We categorize these vertices into two regions based on their distance from the collision point: near DVs, which occur between 1–4 mm from the center and require a mass of at least 40 GeV, and far DVs, which occur more than 4 mm out and require a mass of at least 20 GeV. These mass thresholds are powerful discriminators, as random track crossings or hadronic interactions with detector material typically result in much lower masses.

Cleanliness through Data-Driven Methods

One of the most significant hurdles in any exotic search is the “background” – the billions of Standard Model events that can mimic the signal. For LLPs, the backgrounds are particularly diverse:

• Heavy-flavor decays: Quarks such as “bottom” and “charm” have finite lifetimes and can naturally produce displaced muons.
• Cosmic rays: High-energy muons from space that happen to traverse the detector during a collision.
• Algorithmic fakes: Random combinations of hits that the reconstruction software mistakenly identifies as a coherent track or vertex.

To address this, we developed a fully data-driven background estimate. Rather than relying on potentially imprecise simulations of these rare events, we measure the background rates directly in the data. By defining validation regions where we relax certain selection requirements, we can accurately measure the rates of individual background sources. We then extrapolate these into our signal region using the transfer factor method.

This method relies on the statistical independence of our selection variables. For instance, the probability of an event having a “fake” vertex is largely uncorrelated with whether it also contains a “fake” muon. By carefully measuring these independent probabilities, we can construct a robust prediction of the background, ensuring that any observed excess would be statistically significant.

Figure 2: Schematic of the background estimation strategy. By measuring the rates in validation regions, we can predict the expected yield in the signal region using the transfer factor approach. (Image: ATLAS Collaboration/CERN)

Innovation in Run 3

The foundation of this search is the 164 fb⁻¹ of data collected at a center-of-mass energy of 13.6 TeV between 2022 and 2024. This represents not just a larger sample, but a higher-quality one due to Run 3 upgrades.

The new displaced-muon trigger allowed us to lower the transverse momentum thresholds for muons to just 20 GeV. In previous operational years, we often had to rely on much higher energy thresholds or additional signatures like large missing transverse momentum. This improvement has significantly expanded our sensitivity to lower-mass LLPs, which may have decay products with lower energy but are equally critical for a complete understanding of new physics that goes beyond our current theories.

Reading the Results

Upon analysis of the initial Run 3 dataset, the observed data were found to be remarkably consistent with the Standard Model background predictions. We observed only three events in our “far” signal region and one in our “near” signal region – results that align well with our expectations from instrumental fakes and rare background processes.

While these results do not constitute a discovery, they allow us to set the most stringent limits to date on these specific R-parity-violating models.

Figure 3: The 95% confidence level exclusion limits for long-lived Top Squarks. The new results significantly extend our reach in both mass (up to 1.85 TeV) and lifetime compared to previous searches. (Image: ATLAS Collaboration/CERN)

The primary highlights of the exclusion limits include:

• Higgsinos: We have excluded Higgsino masses up to 1.6 TeV for proper lifetimes near 0.1 nanoseconds. For these specific models, this is an improvement in sensitivity of nearly two orders of magnitude compared to results from LHC Run 1.
• Top Squarks: The exclusion limit was pushed to 1.85 TeV for intermediate lifetimes, representing a significant step forward in our hunt for supersymmetric partners.

A Stepping Stone for the Future

This result stands as one of the first major “Exotics” publications from the Run 3 era. It demonstrates the efficacy of the ATLAS detector upgrades and the sophisticated new trigger and reconstruction strategies developed by our teams.

By targeting the displaced and the invisible, we are systematically closing the gaps in our search for the fundamental laws of nature. Having coordinated this analysis and drafted the resulting publication, it was rewarding to see how our results fully exploit the improved technical capabilities of the ATLAS detector in Run 3. As the LHC continues to provide high-quality collision data, we will continue to refine these techniques. The remaining years of Run 3 will allow us to probe even deeper into the mass–lifetime plane, bringing us closer to discovering the particles that have, until now, remained hidden from view.

Publication Details:

Author: ATLAS Collaboration
Title: Search for massive, long-lived particles in events with displaced vertices and displaced muons in pp collisions at an energy of 13.6 TeV with the ATLAS experiment
Reference: Submitted to Phys. Lett. B. [arXiv:2603.01991].

The answer is complex. Unlike its lighter cousins, the top quark is famously ephemeral. It has a lifetime of roughly 0.5 yoctoseconds (10^-24 seconds) – so short that it typically decays before it can form a hadron. In the time it takes for a bound state to organize, the top quark is already gone. For this reason, a true, stable toponium meson cannot exist. However, quantum mechanics offers a loophole. When top quarks are produced very slowly, close to the energy threshold for creating a pair, they “feel” each other’s presence through the strong force just before they decay. This fleeting interaction creates a quasi-bound state – a spectral “ghost” of toponium that enhances the production rate of top-quark pairs at specific energies.

In a new result from the ATLAS Collaboration, we have found significant evidence of this phenomenon.

The Hunt for a Spectral Signature

Detecting this effect requires a perfect storm of experimental precision and theoretical advancement. The signal manifests as a subtle distortion in the production rate of top-quark pairs (tt̄) near the threshold. To see it, we analyzed the full LHC Run 2 dataset collected between 2015 and 2018, amounting to 140 fb⁻¹ of proton-proton collision data at 13 TeV collision energy.

We focused on the dilepton decay channel, where the top and antitop quarks decay into an electron and a muon, or pairs of electrons or muons, accompanied by neutrinos and jets. This channel is the “gold standard” for cleanliness, allowing us to reconstruct the invariant mass of the top-quark system (m_tt̄) with high precision. This observable is our primary window into the threshold region.

Observation requires knowing exactly what to look for. Standard perturbative QCD (pQCD) calculations, which work beautifully at high energies, break down when top quarks move slowly relative to one another. To describe this non-relativistic regime, we need a specialized framework known as Non-Relativistic QCD (NRQCD).

Recent breakthroughs by theorists Benjamin Fuks, Kaoru Hagiwara, Kai Ma, and Ya-Juan Zheng [1, 2] provided the map we needed. They developed sophisticated simulations that reweight standard predictions using an NRQCD Green’s function. This accounts for the complex resummation of interactions between the slowly moving quarks, predicting a broader, distinct enhancement in the cross-section just below the 2m_t threshold.

Unveiling the Signal

In our analysis, we pitted the data against two hypotheses:

1. The Standard Picture: A baseline pQCD model that assumes independent top quark production.
2. The Toponium Picture: An extended model incorporating the NRQCD quasi-bound state effects.

To distinguish between them, we didn’t just look at the mass. We also examined the angular separation of the leptons in the laboratory frame. Since the quasi-bound state forms in a specific quantum spin state (a spin-singlet), it leaves a unique imprint on the angular distribution of the decay products. By combining the mass spectrum with these angular variables, we maximized our sensitivity to the signal.

The results were striking. When we compared our data to the baseline model, we saw a clear excess of events right where the toponium signal was predicted to be.

The statistical significance of this excess is above 8 standard deviations, far beyond the 5-standard-deviation threshold traditionally required to claim a discovery. We measured the production cross-section of this toponium component to be 9.3 +1.4/-1.3 pb, which aligns remarkably well with the theoretical predictions from NRQCD.

Figure: The data (black points) clearly overshoot the standard prediction (standard histogram) in the low mass regions, aligning instead with the toponium prediction (orange). The middle panel shows this excess explicitly. (Image: ATLAS Collaboration/CERN)

A New Window into the Strong Force

This observation is more than just finding a new particle state; it is a validation of our understanding of the strong force in a regime that is notoriously difficult to calculate. It confirms that even the fleeting top quark can participate in spectroscopic binding, however briefly.

This result aligns with recent findings from the CMS Collaboration [3] and opens a new era of threshold top physics at the LHC. As we collect more data, we can use this quasi-bound state as a laboratory to measure the top quark’s mass and other properties with unprecedented precision, potentially revealing subtle cracks in the Standard Model where new physics might be hiding.

With the High-Luminosity LHC on the horizon, we will soon be able to study the shape of this resonance in even greater detail. For now, we celebrate the capture of this long-sought spectral state – a testament to the power of combining cutting-edge experiment with advanced theoretical insights.

Publication:

Author: ATLAS Collaboration.
Title: Observation of a cross-section enhancement near the tt̅ production threshold in √s = 13 TeV pp collisions with the ATLAS detector.
Reference: Submitted to Rep. Prog. Phys. [arXiv:2601.11780].

Further Reading:

[1] B. Fuks, K. Hagiwara, K. Ma, and Y.-J. Zheng, “Signatures of toponium formation in LHC run 2 data,” Phys. Rev. D 104 (2021) 034023, arXiv:2102.11281, DOI:10.1103/PhysRevD.104.034023.

[2] B. Fuks, K. Hagiwara, K. Ma, and Y.-J. Zheng, “Simulating toponium formation signals at the LHC,” Eur. Phys. J. C 85 (2025) 157, arXiv:2411.18962, DOI:10.1140/epjc/s10052-025-13853-3.

[3] CMS Collaboration, “Observation of a pseudoscalar excess at the top quark pair production threshold,” Rep. Prog. Phys. 88 (2025) 087801, arXiv:2503.22382, DOI:10.1088/1361-6633/adf7d3.

This new measurement from the ATLAS Collaboration, published in Physics Letters B, marks a significant milestone: the first dedicated ATLAS measurement of top-quark pair production in association with charm quarks (tt̄ + charm). I had the pleasure of presenting this result first at the TOP 2024 conference. This process is of paramount importance because it forms a substantial background to searches for rare Standard Model processes, such as the production of top-quark pairs with Higgs bosons decaying to bottom quarks (tt̄H(bb̄)), and even the elusive four-top-quark production (tt̄tt̄). Understanding and precisely measuring tt̄ + charm production is therefore essential for unlocking the full discovery potential of the LHC.

Bottom (b) quarks have been extensively studied in association with top quarks (tt̄ + ≥1 b), and charm quarks present an equally unique challenge. Their production mechanisms are complex, involving gluon radiation that splits into charm-anticharm pairs, or charm quarks originating directly from the initial state. These processes can lead to final states with one or more charm jets, making them difficult to distinguish from other phenomena.

We addressed this complexity by separately determining the production rates of tt̄ + two or more charm jets (tt+≥2c) and tt̄ + one charm jet (tt+1c). This distinction is vital as these processes are sensitive to different underlying production mechanisms. We analyzed the full LHC Run 2 dataset, corresponding to an integrated luminosity of 140 fb⁻¹, considering events with one or two charged leptons in the final state. These leptonic decay channels of the top quarks provide cleaner signatures for analysis.

Figure: Feynman diagrams illustrating representative production modes of top-quark pairs with additional charm quarks. (Image: ATLAS Collaboration/CERN)

A Novel Approach to Charm Tagging

To accurately identify and categorize events with charm quarks, we developed a custom flavour-tagging algorithm called the b/c-tagger. This specialized tool is designed for the simultaneous identification of both charm and bottom jets, allowing us to define analysis regions that are highly sensitive to tt+1c and tt+≥2c production. This innovative approach is key to disentangling the complex heavy-flavour landscape in top-quark events.

The b/c-tagger leverages advanced techniques to distinguish jets originating from charm quarks from those produced by bottom quarks or lighter quarks and gluons. This capability is crucial for reducing backgrounds and isolating the tt̄ + charm signal. By applying this tagger, we were able to define fiducial phase spaces – specific regions of the detector’s acceptance – where we could precisely measure the production rates of tt+1c and tt+≥2c.

Beyond absolute production rates, we also extracted the ratios of tt+1c, tt+≥2c, and tt̄ + b-jets to the overall tt̄ + jets production. These ratios provide valuable insights into the relative contributions of different heavy-flavour production mechanisms and allow for more detailed comparisons with theoretical predictions from next-to-leading order + parton-shower simulations.

Results: Unveiling the Charm Signatures

Our analysis involved meticulously selecting events with one or two leptons and then applying the b/c-tagger to categorize jets based on their heavy-flavour content. This allowed us to define distinct analysis regions sensitive to tt+1c and tt+≥2c production. The measured production rates were then compared with predictions from various Monte Carlo simulations.

The results provide valuable insights into the complex interplay of top quarks and charm quarks. We found that the measured production rates are systematically higher than theoretical predictions from all tested Monte Carlo simulations. While the measurements remain broadly consistent within uncertainties, their precision enables a more refined comparison and places tighter constraints on theoretical modelling. This is particularly important for processes where charm quarks are produced through gluon splitting, a challenging aspect to model accurately in Quantum Chromodynamics.

Figure: Measured fiducial cross-sections for top-quark pair production with additional charm jets compared to various simulations. (Image: ATLAS Collaboration/CERN)

Conclusion: A New Era for Top-Quark Physics

This first dedicated measurement of top-quark pair production in association with charm quarks by the ATLAS Collaboration is a significant achievement. By developing and deploying the b/c-tagger, we have opened new avenues for exploring complex final states involving charm quarks and provided high-quality experimental data that will directly influence both theory and experiment.

The implications are far-reaching:

• Enhanced precision for rare processes: A more accurate understanding of tt̄ + charm production improves the sensitivity of searches for rare Standard Model processes such as tt̄H(bb̄) and four-top-quark production, where it constitutes a major background.
• Refined theoretical models: Our measurements serve as stringent tests for calculations of tt̄ + heavy-flavour production, reducing uncertainties and improving the reliability of predictions.
• Path to future discoveries: With a better handle on tt̄ + charm production, we are paving the way for more sensitive investigations into processes like tt̄H(cc̄), as well as more precise studies at the High-Luminosity LHC.

In summary, this work represents a vital step forward in understanding heavy-flavour production in top-quark events and sets the stage for exciting developments in the years to come.

For more details, you can refer to the ATLAS Physics Briefing on this result.

Publication:

Author: ATLAS Collaboration.
Title: Measurement of top-quark pair production in association with charm quarks in proton-proton collisions at √s = 13 TeV with the ATLAS detector.
Reference: Phys. Lett. B 860 (2025) 139177.
DOI:10.1016/j.physletb.2024.139177 | arXiv:2409.11305.

However, many crucial processes in particle physics involve not just one, but multiple neutrinos. A prime example is the production of top-quark pairs where both top quarks decay leptonically (dileptonic tt̄ events). In such events, two neutrinos are produced, and their combined presence makes full event reconstruction even more challenging. While the missing transverse momentum (pTmiss) still provides a handle on the total momentum carried away by all invisible particles, it doesn’t tell us how this momentum is shared between individual neutrinos, nor does it provide information about their longitudinal momenta.

Traditional analytical methods for handling multiple neutrinos, such as Neutrino Weighting or the Ellipse method, often rely on strong kinematic assumptions (e.g., fixed W boson and top quark masses) and can struggle to find solutions for all events. These methods can also be computationally intensive and may introduce biases. This is where ν²-Flows comes in: an extension of our original ν-Flows framework specifically designed to tackle the complexities of multi-neutrino final states.

Our new work, published in Physical Review D, demonstrates how ν²-Flows can accurately reconstruct the momenta of multiple neutrinos and the correlations between them, even in challenging dileptonic tt̄ events. This advancement not only provides more accurate event reconstruction but also significantly improves the statistical precision of downstream physics analyses.

How ν²-Flows Extends the ν-Flows Concept

The core idea behind ν²-Flows remains the same as its predecessor: leveraging conditional normalizing flows to learn the probability distribution of neutrino momenta given the observed particles in an event. However, to handle multiple neutrinos and more complex event topologies, we’ve introduced several key enhancements:

1. Scaling to Multiple Neutrinos

The most direct extension is the ability to predict the momenta of multiple neutrinos simultaneously. For dileptonic tt̄ events, where two neutrinos are present, we double the dimensionality of the conditional normalizing flow (from three to six, representing the three momentum components for each of the two neutrinos). The architecture is designed to scale natively for any desired neutrino multiplicity, requiring only a predefined ordering for the neutrinos.

2. Enhanced Feature Extraction with Attention Transformers

While ν-Flows used an attention-pooled deep set to process jets, ν²-Flows introduces a more sophisticated feature extraction network based on attention transformers. This is crucial because dileptonic events can have varying numbers of both jets and leptons, and we need a method that is permutation invariant (meaning the order of input particles doesn’t affect the output) and can accommodate these variable multiplicities. The transformer encoder, with its multi-headed attention layers, allows for a more comprehensive and robust representation of the entire event.

Here’s a schematic overview of the ν²-Flows architecture:

Figure: A schematic of the ν²-Flows network for learning the conditional likelihood of multiple neutrinos in the event. It uses a transformer encoder with cross-attention and a learnable class token to embed an event representation for any multiplicity of physics objects. (Image: Phys. Rev. D 109, 092008 (2024))

In this architecture:

• Object-Specific Embeddings: Jets, leptons, and missing transverse momentum are independently embedded into a higher-dimensional space using dedicated multi-layer perceptrons (MLPs).
• Transformer Encoder: The embedded objects interact through several layers of multi-headed attention within a transformer encoder. This allows the network to learn complex relationships between all particles in the event.
• Contextual Conditioning: Additional event information, such as object multiplicities, is injected as conditional information into the transformer encoder blocks.
• Learnable Class Token: A learnable class token is used with repeated cross-attention to obtain a single global vector from the transformer. This vector serves as the conditioning input for the normalizing flow.

This enhanced feature extraction allows ν²-Flows to learn a rich, contextual representation of the event, which is then used to guide the transformations within the normalizing flow, enabling accurate prediction of multiple neutrino momenta.

Performance and Impact: Beyond Single Neutrinos

We applied ν²-Flows to simulated dileptonic tt̄ events and compared its performance against traditional analytical methods like Neutrino Weighting and the Ellipse method. Our results demonstrate significant improvements across several key metrics:

1. Superior Neutrino Kinematics Reconstruction

ν²-Flows accurately reproduces the true kinematics of both neutrinos, including their transverse momenta, pseudorapidities, and energies. In contrast, traditional methods often overestimate transverse momenta and energy, and tend to bias neutrino pseudorapidities towards zero. Furthermore, ν²-Flows correctly captures the angular separation between the two neutrinos, which is often poorly modeled by other approaches.

2. More Accurate W Boson and Top Quark Reconstruction

Unlike traditional methods that impose fixed W boson and top quark masses, ν²-Flows learns these distributions directly from the training data. This results in a more accurate reconstruction of the W boson and top quark invariant masses, with ν²-Flows closely following the full underlying target distribution. While traditional methods force these masses to nominal values, ν²-Flows provides a more realistic and less biased representation.

3. Improved tt̄ System Kinematics

ν²-Flows significantly improves the reconstruction of the full tt̄ system’s kinematics, including its invariant mass, transverse momentum, and rapidity. While some traditional methods might perform adequately for certain observables, ν²-Flows consistently provides a much better reproduction of the true distributions across the entire kinematic phase space.

4. Enhanced Statistical Precision in Unfolded Distributions

One of the most impactful results of ν²-Flows is its direct benefit to the statistical precision of unfolded distributions in physics analyses. We demonstrated this by performing a simplified double differential tt̄ dilepton analysis, measuring observables like the invariant mass of the tt̄ system and the angular separation of the two leptons. Compared to Neutrino Weighting, ν²-Flows leads to an improvement in statistical precision for each bin by a factor of 1.5 to 2, and up to a factor of four compared to the Ellipse method. This means that analyses using ν²-Flows can achieve more precise results with the same amount of data, or achieve similar precision with less data.

5. Robustness and Efficiency

ν²-Flows is robust to variations in the training sample and performs well even when applied to events simulated with different generators or with additional initial state radiation. Furthermore, unlike computationally intensive traditional methods, ν²-Flows offers fast inference times, making it practical for large-scale data analysis. A single event inference takes around 70 milliseconds on a CPU, and this can be reduced to just 0.03 milliseconds per event when processed in parallel on a GPU.

The Future of Neutrino Reconstruction

ν²-Flows represents a significant leap forward in our ability to reconstruct complex multi-neutrino events in particle physics. By extending the powerful framework of conditional normalizing flows and incorporating advanced deep learning architectures like attention transformers, we can now more accurately and efficiently unravel the kinematics of invisible particles.

This work has profound implications for a wide range of physics analyses at the LHC and beyond:

• Unlocking New Discoveries: More precise neutrino reconstruction can lead to enhanced sensitivity in searches for rare Standard Model processes and new physics phenomena, where multi-neutrino final states are often crucial.
• Refining Precision Measurements: Improved reconstruction of W bosons, top quarks, and the full tt̄ system will enable more accurate measurements of fundamental particle properties, contributing to a deeper understanding of the Standard Model.
• Broader Applicability: The generalized architecture of ν²-Flows can be easily extended to other final states with varying neutrino multiplicities and reconstructed object types, making it a versatile tool for the particle physics community.

In essence, ν²-Flows empowers physicists to see the invisible with unprecedented clarity, pushing the boundaries of what’s possible in high-energy physics research.

Publication:

Authors: John Andrew Raine, Matthew Leigh, Knut Zoch, and Tobias Golling.
Title: ν²-Flows: Fast and improved neutrino reconstruction in multi-neutrino final states with conditional normalizing flows.
Reference: Phys. Rev. D 109 (2024) 012005.
DOI:10.1103/PhysRevD.109.012005 | arXiv:2307.02405.

However, event reconstruction faces a significant hurdle: the combinatorial challenge. When collisions produce many detectable particles, determining which of these came from which original particle becomes a complex task. The number of possible assignments can grow exponentially with the number of detected particles, making it computationally intractable for traditional methods.

Traditional methods, such as the χ² method and Kinematic Likelihood Fitters, attempt to assign detected particles to their hypothesized parent particles based on kinematic constraints. While successful, these methods can suffer from biases by forcing combinations to match exact particle masses, potentially leading to inaccurate measurements. More modern machine learning approaches, like those using attention transformers such as SPA-Net, have shown improvements in reducing these biases and handling the combinatorial complexity. However, even these methods often focus on identifying groups of particles without explicitly reconstructing the full decay chain or predicting the properties of the intermediate particles.

A Case Study: Top Quark Pair Production

To illustrate the combinatorial challenge, consider the case of top quark pair production in the all-hadronic decay channel, a process we focused on in our publication. In this process, both top quarks decay hadronically: each top quark decays to a W boson and a bottom quark, and each W boson subsequently decays to two quarks. This results in six jets in the detector. For such an event, there are 720 possible ways to assign these jets to the original particles. Even after accounting for symmetries, this number is reduced to 90. As the number of jets increases, this combinatorial complexity rapidly escalates; for seven jets, the combinations jump to 630, and for eight jets, it becomes 2520. This exponential growth highlights why traditional methods struggle to test every possible combination, especially in the high-multiplicity environments of LHC collisions.

The Topograph Solution: Physics-Inspired Graph Neural Networks

Topographs are inspired by Feynman diagrams, which are visual representations of particle interactions. In a Feynman diagram, particles and their interactions form a network, or a graph. Topographs build upon this concept by representing the particle decay process as a graph where:

• Nodes represent particles, both the observed final-state particles (like jets) and the unobserved intermediate particles (like W bosons and top quarks).
• Edges represent the connections between these particles, indicating a parent-daughter relationship in a decay.

Unlike previous graph neural network (GNN) approaches that often create fully connected graphs (where every particle is potentially connected to every other particle), Topographs inject the intermediate particles directly into the graph as their own nodes. This allows the network to explicitly learn the properties of these intermediate particles. Furthermore, instead of connecting all objects to one another, Topographs connect observed particles to their potential mother particles. This significantly reduces the complexity of the graph. For example, for a top quark decay, a fully connected GNN would have a quadratic number of edges, while a Topograph has a linear number of edges with respect to the number of reconstructed objects. This linear scaling is a crucial advantage, making Topographs computationally much more efficient for complex events.

Figure: Two solutions to the combinatorics problem: a fully connected graph (left) and a topograph (right). The topograph captures the decay structure while avoiding unnecessary connections. (Image: Phys. Rev. D 107 (2023) 116019)

How Topographs Work

The key innovation of Topographs lies in how they make predictions for the most likely jet assignment to the top quarks. The process works through several interconnected components:

1. Explicit Modeling of Intermediate Particles: Topographs inject nodes for intermediate particles – such as W bosons and top quarks – directly into the graph structure. These nodes are initialized using attention-weighted pooling from the input jets, where different networks generate attention weights for each intermediate particle type. This allows the network not only to assign jets to these particles but also to predict their kinematic properties through dedicated regression components.

2. Physics-Inspired Connections: The edges in the Topograph reflect known decay patterns, connecting daughter particles (jets) only to plausible parent particles. This physics prior constrains the graph in a meaningful way, reducing complexity and guiding the learning process toward physically valid solutions. For instance, jets are connected to both W boson nodes and top quark nodes – the connections follow the expected decay topology.

3. Message Passing for Information Exchange: Information flows along the graph through multiple message passing layers, enabling each node to update its understanding based on its connections. In our implementation, we use four message passing steps where jets, edges, and injected W and top nodes are iteratively updated. This iterative exchange helps the model refine both the particle assignments and the predictions of their properties.

4. Edge Scoring and Assignment: Every edge receives a learned score indicating how likely it is that the connected particles are related by a decay. These scores are calculated by passing the edge properties through a classification network. To resolve the combinatorics, we use an iterative assignment procedure: first, the edge with the highest score is labeled as a true edge, with the jet assigned to this parton. Next, all edges connected to the corresponding jet and parton are removed, and the next highest edge is chosen. This process continues until all six partons are assigned, ensuring each jet and each parton is used exactly once.

5. Regression for Intermediate Particle Properties: Beyond just solving the assignment problem, Topographs predict the kinematic properties (three-momentum components) of the intermediate W bosons and top quarks through dedicated regression networks. These predictions are trained using the truth-level properties of the particles, providing more accurate and less biased estimates than simply reconstructing these quantities from the assigned jets.

By incorporating these physics-inspired structures and direct predictions, Topographs not only solve the combinatorial assignment problem but also provide a more complete and accurate reconstruction of the particle physics process. The approach scales linearly with the number of reconstructed objects, making it computationally feasible even for high-multiplicity events.

Performance and Significance: Outperforming the State-of-the-Art

We applied Topographs to the challenging case of top quark pair production in the all-hadronic decay channel, training and evaluating on a dataset of 20 million simulated events. Our results demonstrate that Topographs:

• Outperform traditional methods: Topographs significantly outperform the χ² method in event reconstruction efficiency, showing an improvement of around 10 percentage points, especially at higher jet multiplicities. For events with exactly six jets and at least two b-tagged jets, Topographs achieve 81.7% efficiency compared to 72.7% for the χ² method. This means Topographs are much better at correctly identifying which jets came from which parent particles.

• Match state-of-the-art machine learning techniques: Topographs achieve performance comparable to SPANet, a leading machine learning technique in this field, in terms of overall reconstruction efficiency. Both methods show similar performance across all jet multiplicities, with sub-percent differences in reconstruction efficiencies, indicating that Topographs are a competitive and powerful alternative to existing advanced methods.

• Provide richer information: Beyond just assigning jets, Topographs directly predict the kinematic properties (momentum, energy) of the intermediate W bosons and top quarks. For correctly assigned events, these predictions show improved resolution compared to those obtained by simply reconstructing these quantities from the assigned jets. The regression predictions demonstrate narrower peaks and reduced bias, providing more accurate estimates of the intermediate particle properties.

• Scalability: The linear scaling of Topographs with the number of reconstructed objects is a major breakthrough. While our implementation includes fully connected message passing layers for information exchange (which introduces quadratic scaling), the core Topograph structure scales linearly. This means that as experiments at the LHC become even more complex, producing events with even higher particle multiplicities, the fundamental Topograph approach will remain computationally feasible, unlike traditional methods that quickly become intractable.

• Interpretability: The individual edge scores from Topographs can be aggregated to provide a confidence measure for the combinatorial assignment. This allows for event filtering based on assignment confidence and helps identify events where complete reconstruction may not be possible due to missing partons.

Impact on Particle Physics

The development of Topographs represents a significant step forward in particle physics event reconstruction. By combining the power of graph neural networks with a deep understanding of particle physics principles, Topographs offer a robust and efficient solution to a long-standing challenge. This innovation has several key implications:

• Enhanced Discovery Potential: More accurate and efficient event reconstruction allows physicists to better analyze complex collision events, potentially leading to the discovery of new particles or phenomena that were previously hidden in the noise.

• Improved Precision Measurements: The ability to precisely predict the properties of intermediate particles will enable more accurate measurements of fundamental particle properties, helping to refine our understanding of the Standard Model of particle physics and search for deviations that could hint at new physics. The regression capabilities of Topographs provide less biased estimates of particle kinematics, crucial for precision measurements.

• Future-Proofing Analyses: As the LHC undergoes upgrades and future colliders are designed to produce even more complex events, the linear scaling of Topographs ensures that event reconstruction remains a tractable problem, paving the way for future discoveries. The modular nature of Topographs also allows them to be adapted to different physics processes beyond top quark pair production.

• Broader Applications: The Topograph approach is not limited to top quark physics. Due to their modular design based on particle blocks, Topographs can be generalized to almost any particle physics process. Applications could extend to jet identification, reconstructing displaced vertices from heavy flavor hadron decays, or analyzing constituents in large radius jets in boosted topologies.

In essence, Topographs are not just a new algorithm; they represent a new way of thinking about event reconstruction in particle physics, offering a powerful tool to unlock the universe’s deepest secrets through more accurate and efficient analysis of high-energy collision data.

Publication:

Authors: Lukas Ehrke, John Andrew Raine, Knut Zoch, Manuel Guth, and Tobias Golling.
Title: Topological Reconstruction of Particle Physics Processes using Graph Neural Networks.
Reference: Phys. Rev. D 107 (2023) 116019.
DOI:10.1103/PhysRevD.107.116019 | arXiv:2303.13937.

However, there’s a elusive particle that poses a unique challenge: the neutrino. Neutrinos are famously shy; they interact only through the weak nuclear force and typically pass through detector material without leaving any measurable signal. Their presence is inferred indirectly, by measuring the momentum imbalance of all visible particles in the plane perpendicular to the beam pipe. This imbalance is called missing transverse momentum, and it serves as our experimental proxy for the net transverse momentum of all undetected particles. Crucially, there’s no such direct measurement for the neutrino’s momentum in the longitudinal direction, leaving its full kinematics largely unknown.

Many analyses in collider physics investigate processes that involve neutrino production. A prime example is the study of the top quark, the most massive elementary particle. Top quarks decay almost instantaneously, and about a third of the time, their decay products include a W boson that subsequently decays leptonically, producing a neutrino. To fully reconstruct the top quark system and precisely measure its properties, knowing the neutrino’s full momentum is vital. Traditional methods for estimating neutrino momentum, such as those based on kinematic constraints (e.g., assuming the W boson’s mass), often have drawbacks: they can introduce biases, may not yield real solutions, or can result in ambiguous multiple solutions without a clear way to choose the correct one.

This is where ν-Flows comes in. We introduce a novel machine learning approach designed to fully reconstruct the neutrinos produced in collisions from the missing transverse momentum and observed event kinematics. Our method moves beyond single-point estimates, instead providing a probabilistic approach that can give the likelihood over a range of viable solutions, offering a more complete and accurate picture of the neutrino’s momentum.

ν-Flows: A Probabilistic Approach with Conditional Normalizing Flows

At its core, ν-Flows utilizes conditional normalizing flows, which are powerful deep invertible neural networks (INNs). Normalizing flows are essentially sophisticated mathematical functions that can map a complex probability distribution (like the unknown neutrino momentum) into a simpler, well-understood distribution (like a standard normal distribution). The ‘conditional’ aspect means that this mapping is influenced by other observed information from the event, such as the kinematics of other particles.

Unlike traditional methods that provide a single, often ambiguous, solution for the neutrino’s momentum, ν-Flows learns the full conditional likelihood over the neutrino’s kinematics. This means it can tell us not just one possible value, but the probability of a range of values, along with interpretable uncertainties. This is a significant advantage, as it allows us to recover degrees of freedom that were previously unconstrained.

Figure: A schematic overview of the conditional invertible neural network (INN) used in ν-Flows. It shows how observed event variables are used to condition the prediction of the neutrino momentum vector. (Image: SciPost Phys. 14 (2023) 159)

How ν-Flows Works

1. Learning from Data: ν-Flows is trained on simulated particle collision events where the true neutrino kinematics are known. This allows the model to learn the complex probabilistic relationship between the observed particles (leptons, jets, missing transverse momentum) and the unobserved neutrino’s momentum.

2. Conditional Inputs: The network takes various observed event variables as conditioning inputs. These include the components of the missing transverse momentum, the kinematics of the signal lepton, the kinematics and b-tagging information of the reconstructed jets, and event-level information like jet multiplicities. A Deep Set architecture is used to efficiently process the variable number of jets in an event.

3. Invertible Neural Networks: The core of ν-Flows is built using conditional invertible neural networks (INNs). These networks are designed to be bijective (meaning they have a unique inverse), efficiently invertible, and have a tractable Jacobian. This allows for efficient density estimation and the generation of new data by sampling from the learned probability distribution.

4. Probabilistic Output: Instead of a single value, ν-Flows outputs a full probability distribution for the neutrino’s momentum. This distribution can be multi-modal, reflecting the inherent ambiguities in neutrino reconstruction, such as the two solutions from the W boson mass constraint. The model learns to reproduce these physical relationships directly from the data.

5. Sampling and Mode Estimation: From the learned conditional probability density, we can either sample individual neutrino momentum values, denoted ν-Flows(sample), or identify the most probable solution, ν-Flows(mode). While sampling provides a less biased estimate, the mode estimation offers a more precise point estimate by selecting the peak of the distribution.

By leveraging these techniques, ν-Flows provides a powerful and flexible framework for neutrino reconstruction, moving beyond deterministic solutions to offer a comprehensive probabilistic understanding of the invisible particle’s kinematics.

Performance and Impact: Outperforming Traditional Methods

We demonstrated the applicability of ν-Flows in a case study involving semileptonic top-quark pair decays, where a single neutrino is present in the final state. Our results show significant improvements over traditional methods:

Figure: Comparison of neutrino pseudorapidity (η) reconstruction for two events. The true values are shown in black. Traditional methods (green, blue) often provide ambiguous or biased point estimates, while ν-Flows (orange) provides full probability distributions, revealing the inherent uncertainties and multiple possible solutions. (Image: SciPost Phys. 14 (2023) 159)

• Improved Momentum Reconstruction: For all components of the neutrino’s four-momentum, the distribution generated by ν-Flows(sample) closely matches the true momentum distribution. Traditional methods and even a standard feed-forward neural network (ν-FF) tend to introduce biases, particularly for the longitudinal momentum, often overestimating the fraction of events with longitudinal momentum close to zero. ν-Flows, especially in its mode estimation configuration, shows excellent correlation with the true values and no obvious bias.

• Accurate Invariant Mass Reconstruction: ν-Flows(sample) accurately reproduces the invariant mass distribution of the lepton-neutrino system, closely matching the true distribution. While the W boson mass constraint method forces this distribution to a single value, ν-Flows learns the natural width of the W boson. For the leptonic top quark’s invariant mass, ν-Flows(mode) significantly reduces the variance and produces a distribution most similar to the true one, outperforming other methods that introduce biases or larger spreads.

• Enhanced Jet-Parton Assignment: A crucial downstream task in many top-quark analyses is assigning reconstructed jets to their original partons. We assessed the impact of ν-Flows on this task using the χ² jet association method. Our results show that using neutrino estimates from ν-Flows (both sample and mode) leads to an improved matching efficiency compared to the standard kinematic approach. For events with nine jets, the accuracy of correctly identifying the b-jet from the leptonically decaying top quark increased by a factor of 1.41 when using ν-Flows(mode) compared to the traditional W boson mass constraint method. This improvement is particularly pronounced in events with higher jet multiplicities, where the neutrino’s kinematics play a more significant role in the assignment.

Figure: The fraction of events where the correct b-jet from the leptonically decaying top quark was identified using various neutrino estimation methods, binned by the number of reconstructed jets. ν-Flows significantly improves the accuracy, especially for events with higher jet multiplicities. (Data: SciPost Phys. 14 (2023) 159)

• Interpretable Uncertainties: A key advantage of ν-Flows is its ability to provide interpretable uncertainties. For events where reconstruction is inherently difficult (e.g., due to poor detector resolution or misidentified particles), the likelihood distribution produced by ν-Flows becomes broader, indicating higher uncertainty. This allows physicists to identify and potentially filter out poorly reconstructed events from downstream analyses, leading to more robust results.

Broader Impact and Future Directions

ν-Flows represents a significant advancement in neutrino reconstruction, offering a more accurate, less biased, and probabilistically rich approach. While our initial study focused on semileptonic top-quark pair decays, the method is highly adaptable and can be extended to a wide variety of processes involving any number of invisible particles. This includes other top-quark decay channels (like dileptonic tt̄), Higgs boson studies, and even searches for physics beyond the Standard Model that predict new weakly interacting massive particles.

This work lays the groundwork for:

1. Improved Precision Measurements: By providing more accurate neutrino kinematics, ν-Flows can directly enhance the precision of measurements involving neutrinos, leading to a deeper understanding of fundamental particle properties and interactions.
2. Enhanced Sensitivity in Searches for New Physics: Better neutrino reconstruction means reduced uncertainties and backgrounds in searches for rare processes or new particles, potentially increasing the sensitivity of these crucial investigations.
3. New Avenues for Analysis: The probabilistic nature of ν-Flows opens up new possibilities for event selection and analysis strategies, allowing physicists to leverage the full information content of neutrino kinematics.

In essence, ν-Flows provides a powerful new tool for particle physicists, enabling us to more effectively probe the universe’s deepest secrets, even those hidden by invisible particles.

Publication:

Authors: Matthew Leigh, John Andrew Raine, Knut Zoch, and Tobias Golling.
Title: ν-Flows: conditional neutrino regression.
Reference: SciPost Phys. 14 (2023) 159.
DOI:10.21468/SciPostPhys.14.6.159 | arXiv:2207.00664.

Our work from the ATLAS Collaboration, published in 2020 in the Journal of High Energy Physics (JHEP), presents comprehensive measurements of the combined production of top-quark pairs with a photon (ttγ) and a single top quark with a W boson and a photon (tWγ). These processes are particularly interesting because they allow us to probe the electroweak coupling of the top quark and search for potential deviations from Standard Model predictions that could signal new physics.

This analysis builds upon previous efforts by using the full dataset recorded by the ATLAS detector between 2015 and 2018, corresponding to an integrated luminosity of 139 fb⁻¹. We focused on events in the electron-muon (eμ) channel, which offers a clean final state with relatively small background contributions. This allowed us to perform precise measurements without relying on complex multivariate analysis techniques for signal-background separation.

Crucially, this measurement is performed at the parton level, enabling direct comparison with state-of-the-art theoretical calculations (G. Bevilacqua et al., JHEP 10 (2018) 158, JHEP 01 (2019) 188). We measured both the inclusive cross-section and differential cross-sections as functions of various kinematic variables of the photon and leptons. These detailed measurements provide a stringent test of theoretical models and contribute to our understanding of these fundamental interactions.

Here are example Feynman diagrams illustrating these processes:

Figure: Example Feynman diagrams at leading order for tt̄γ (left) and tWγ production (right) in the eμ channel. The top-quark mass resonances are marked with double-lined arrows, while W bosons are shown in red. (Image: ATLAS Collaboration/CERN)

The Experimental Approach: Sifting Through Collisions

To perform this measurement, we meticulously selected events recorded by the ATLAS detector that matched the signature of ttγ or tWγ production in the eμ channel. This involved identifying exactly one electron and one muon with opposite charges, at least two jets (with at least one b-tagged jet), and precisely one photon. All selected particles had to meet strict quality and kinematic criteria to ensure their reliable reconstruction.

Signal and Background Modelling

Understanding both the signal (ttγ and tWγ production) and various background processes is paramount for an accurate measurement. We relied on sophisticated Monte Carlo simulations to model these processes. These simulations account for the complex interactions within the LHC and the response of the ATLAS detector. A key aspect of our modelling involved carefully handling photons: some simulations intrinsically produce photons (dedicated simulations), while others produce them stochastically through radiation (inclusive simulations). A rigorous overlap-removal procedure was applied to avoid double-counting and ensure consistency.

Background processes were categorized based on the origin of the reconstructed photon: hadronic fakes (where a hadron mimics a photon), electron fakes (where an electron is misidentified as a photon), and genuine prompt photons from other Standard Model processes. Each of these backgrounds was carefully estimated and accounted for in the analysis.

Measuring the Inclusive Cross-Section

To determine the overall rate of ttγ and tWγ production, we performed a profile likelihood fit to the distribution of a variable called S_T. S_T is defined as the scalar sum of the transverse momenta of all reconstructed objects in the event (leptons, photons, jets, and missing transverse momentum). This variable proved to be effective in separating signal from background and was less sensitive to systematic uncertainties compared to other kinematic distributions.

The fit allowed us to extract the fiducial inclusive cross-section, which represents the production rate within a specific, well-defined region of the detector’s acceptance. This fiducial region is chosen to closely mimic the conditions of theoretical calculations, allowing for direct comparison. The measured cross-section was found to be in good agreement with the latest theoretical predictions at next-to-leading order (NLO) in Quantum Chromodynamics.

Here’s a look at the post-fit distribution of the S_T variable:

Figure: Post-fit distribution of the S_T variable. The uncertainty band represents the post-fit uncertainties. Underflow and overflow events are included in the first and last bins. The lower panel shows the ratio of the data to the prediction. (Image: ATLAS Collaboration/CERN)

Unveiling Differential Cross-Sections

Beyond the overall production rate, we also measured differential cross-sections. These measurements provide more granular information by showing how the production rate varies as a function of specific kinematic variables. We looked at variables such as the photon’s transverse momentum and pseudorapidity, as well as angular separations between the photon and leptons. These differential measurements are particularly sensitive to the underlying dynamics of the interaction and can reveal subtle effects not visible in the inclusive measurement.

To obtain these differential cross-sections, we corrected the observed data for detector effects (like resolution and acceptance) using an iterative unfolding procedure. This process effectively reconstructs the true parton-level distributions from the detector-level observations. The measured differential cross-sections were then compared with both NLO theoretical calculations and various Monte Carlo simulations.

Our findings show that the shapes of the measured differential distributions are generally well described by both the leading-order Monte Carlo predictions and, even better, by the NLO theory prediction. This agreement provides strong validation of our understanding of these complex processes.

One particularly interesting observable is the minimum angular separation, ΔR, between the photon and a lepton. This variable is sensitive to the angle between the top quark and the radiated photon, offering insights into the structure of the tγ coupling. Here’s how the data compares to predictions for this variable:

Figure: Normalised differential cross-section as a function of the minimum ΔR between the photon and a lepton. Data are compared with an NLO Quantum Chromodynamics calculation. The lower panel shows the ratio of prediction to data. (Image: ATLAS Collaboration/CERN)

Systematic Uncertainties: Quantifying Our Confidence

In any precision measurement in particle physics, a thorough understanding and quantification of systematic uncertainties are crucial. These uncertainties arise from various sources, including experimental factors (like detector calibration, object reconstruction efficiencies, and energy scales) and theoretical modelling uncertainties (related to the choice of Quantum Chromodynamics scales, parton shower models, and parton distribution functions).

We evaluated the impact of each systematic uncertainty on both the inclusive and differential cross-section measurements. For the inclusive cross-section, these uncertainties were incorporated into a profile likelihood fit. For the differential measurements, each uncertainty was determined individually for each bin of the measured distribution. The dominant sources of systematic uncertainty were found to be related to the modelling of the signal and background processes.

Conclusion: A Step Forward in Understanding Top-Quark Physics

This comprehensive analysis by the ATLAS Collaboration provides precise measurements of the combined ttγ and tWγ production cross-sections in the eμ decay channel at 13 TeV. Our results, based on the full Run 2 dataset, are in excellent agreement with state-of-the-art NLO theoretical predictions. This agreement not only validates our understanding of these fundamental processes but also demonstrates the remarkable precision achieved by the ATLAS experiment.

These measurements contribute significantly to the ongoing program of top-quark physics at the LHC. By probing the electroweak coupling of the top quark and providing detailed differential distributions, we lay the groundwork for future searches for new physics and further refinements of the Standard Model. This work exemplifies the collaborative effort within the ATLAS Collaboration to push the boundaries of high-energy physics research.

Publication:

Author: ATLAS Collaboration.
Title: Measurements of inclusive and differential cross-sections of combined tt̄γ and tWγ production in the eμ channel.
Reference: JHEP 09 (2020) 049.
DOI:10.1007/JHEP09(2020)049 | arXiv:2007.06946.