Pairton: Iterative Reconstruction of Short-Lived Particles

Andreas Hermansen; Chris Scheulen; Tobias Golling

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Pairton: Iterative Reconstruction of Short-Lived Particles

Andreas Hermansen, Chris Scheulen, Tobias Golling

arXiv:2608.14278v1Aug 14, 2026hep-phcs.LGhep-ex

TL;DR

Reconstructing short-lived particles from their decay products is crucial for studying their properties and interactions. Pairton treats reconstruction as iterative probabilistic inference over partially observed graphs and achieves state-of-the-art performance on fully hadronic t̄t reconstruction.

Problem
Reconstructing short-lived particles from detected decay products is crucial for studying their properties and interactions.
Method
Pairton iteratively predicts missing graph edges in a masked probabilistic reconstruction process conditioned on previously inferred structure.
Results
Pairton achieves state-of-the-art performance across full-event, top-quark, and W-boson reconstruction efficiencies on the fully hadronic t̄t benchmark.
Takeaways & Limitations
Iterative graph reconstruction provides a flexible particle-reconstruction paradigm that can extend beyond the all-hadronic t̄t system to more complex topologies.

Abstract

from arXiv · show

We present Pairton, an iterative framework for reconstructing short-lived particles in high-energy collision events. By formulating particle reconstruction as a masked prediction process over graph structures, Pairton learns conditional distributions consistent with a factorised decomposition of decay products and iteratively predicts edges in the adjacency matrix representing particle decay relationships. Leveraging a pairformer-based architecture with dynamically updated pairwise representations, our method incorporates global event consistency. We demonstrate state-of-the-art performance on fully hadronic $t\bar{t}$ decays. Pairton provides a general, flexible paradigm for particle reconstruction and can be readily extended to other topologies, bridging ideas from modern generative modelling and high-energy physics.

I. INTRODUCTION · II. CURRENT APPROACHES · III. METHOD

Pairton addresses the combinatorial challenge of reconstructing short-lived particles by treating decay assignments as iterative probabilistic inference over partially observed graphs. The paper situates this approach among existing reconstruction methods and develops it for fully hadronic t̄t events, with extension to other final states.

I. INTRODUCTION: Short-lived particles produced at the LHC decay almost instantaneously into detectable leptons and hadronic jets, making reconstruction crucial for studying their properties.The decay products are measured by sub-detectors surrounding the collision point.
I. INTRODUCTION: High particle multiplicities create substantial combinatorial backgrounds, complicating accurate identification of short-lived particles’ decay products.The fully hadronic decay of a top quark pair is given as a representative difficult process.
I. INTRODUCTION: Pairton formulates particle reconstruction as structured probabilistic inference over partially observed graphs and iteratively predicts missing decay edges conditioned on inferred structure.Nodes represent final-state objects, while edges encode decay relationships.
II. CURRENT APPROACHES: Existing approaches include combinatorial likelihood methods, transformer-based SPANet, graph-based Topograph, hypergraph-based HyPER, and topology-agnostic hierarchical TIGER.These methods use different representations or architectures for assigning decay products.
II. CURRENT APPROACHES: Machine learning methods achieve enhanced performance compared to traditional reconstruction methods.The comparison is stated for the methods discussed in the current-approaches section.
I. INTRODUCTION: The all-hadronic t̄t benchmark requires assigning two W-decay jets and one corresponding b-jet to each top-quark decay, while additional radiation and pile-up complicate reconstruction.Methods developed and tested on t̄t can be extended to other final states involving short-lived particles.
III. METHOD: The optimal reconstruction is defined as the most probable assignment of the two W bosons and two b-jets conditioned on event data, allowing relevant jet permutations.Each W_i is an unordered pair of jet indices, while each b_i is a single index.
III. METHOD: Pairton factorises the joint conditional distribution into sequential reconstruction steps and represents them with iteratively updated graphs for masked denoising.The updated graphs are fed back into the model during subsequent particle assignments.

Graph Representation

Pairton represents each event as multiple graphs over a shared set of detected-jet nodes, with graph edges encoding decay-product relationships for W bosons and top quarks. Partially observed adjacency matrices become dynamic pairwise features that condition subsequent predictions on relational structure.

Graph construction: Each event is represented as a collection of simple graphs sharing detected jets as nodes, with one graph for each particle type α ∈ {W, t}.This formulation targets all-hadronic t¯t decays.
Graph construction: Edges connect jets that are decay products of the corresponding particle type, while self-loops indicate that a jet is unassociated with that type.The graph structure therefore encodes particle assignments and non-associations directly.
Adjacency matrices: Particle assignments are encoded as adjacency matrices Aα for each particle type.The adjacency-matrix formulation provides the structural representation used by the model.
Architectural advantage: Partially observed adjacency matrices are embedded as dynamic pairwise edge features that condition predictions on relational states between all jets.This lets the graph neural network use explicit structural assignments without complex message passing over higher-order tensors.

Masked Denoising Formulation

Pairton formulates particle reconstruction as discrete denoising over adjacency matrices, using a hierarchy-aware masking process that mirrors the t̄t decay topology. It trains masked conditional predictions and reverses the process iteratively during inference.

Formulation: Reconstruction is formulated as a discrete denoising problem over adjacency matrices using a partially deterministic Markov masking process.The process produces partially observed adjacency matrices from ground-truth adjacency matrices.
Masking process: The masking level determines which particles are visible, with configurations revealing both W bosons, one W boson, or no particles.Logically inferable entries retain true values, while other entries receive a mask token M.
Masking process: The hierarchical corruption scheme mirrors the t̄t decay topology and avoids physically uninterpretable intermediate configurations caused by randomly masking individual edges.It also lets reverse-process predictions condition unambiguously on previously predicted particles.
Training objective: The model predicts masked adjacency entries conditioned on observed structure and jet features using masked conditional likelihood training.The objective averages over training data and masking levels, with δM selecting masked entries.
Novelty: The method introduces discrete diffusion to high-energy physics by learning to reverse masking over adjacency matrices, contrasting with fixed-rate masked and autoregressive approaches.Prior examples cited include Masked Particle Modeling on Sets and OmniJet-α.
Inference: At inference, reconstruction reverses masking by iteratively predicting particle assignments and refining predictions with previously selected structures.The method starts from a fully masked adjacency matrix and implements the factorised reconstruction process.

Model Architecture

Pairton models conditional adjacency predictions with a pairformer architecture that augments attention using pairwise features and dynamically updates those representations. It combines jet-level and pairwise jet features with embedded visible adjacency matrices for iterative reconstruction.

Pairformer architecture: Pairton uses a pairformer-based architecture, a transformer modification that augments attention scores with learnable pairwise-feature biases.The architecture was initially developed as a primary component of AlphaFold3.
Pairformer architecture: Unlike Particle Transformer, which uses fixed pairwise representations, Pairton’s pairformer dynamically updates pairwise features.The passage identifies this dynamic updating as the architectural distinction from ParT.
Input features: Inputs include jet energy, transverse momentum, pseudo-rapidity, azimuthal angle, b-tag status, and pairwise dijet mass and angular-separation features.Pairwise inputs include mij, Δηij, Δφij, and ΔRij; selected features undergo preprocessing transformations.
Adjacency conditioning: Visible adjacency matrices are embedded and added to embedded pairwise jet features before the features enter the pairformer stack.During training, visible matrices derive from true matrices and masking; during inference, they derive iteratively from previous inference steps.

IV. RESULTS · Dataset

The dataset comprises simulated all-hadronic t¯t events at √s = 13 TeV, processed through parton-shower and ATLAS-detector simulations. After selection, events were split into training, validation, and testing subsets, with fully reconstructible W-boson and event samples retained for evaluation.

Dataset: The study uses 6 × 10^7 all-hadronic t¯t events simulated at √s = 13 TeV.MadGraph5 aMC@Nlo 2.9.16 generated the matrix elements at next-to-leading order in QCD.
Dataset: Pythia 8.306 modeled the parton shower, while Delphes 3.5.0 simulated the ATLAS detector response.
Dataset: Jets were reconstructed with anti-kt using FastJet, with radius R = 0.4, pT = 25 GeV, and |η| < 2.5 requirements.
Dataset: b-jet tagging was simulated using a pT-dependent efficiency applied to jets containing b-hadrons.
Dataset: Selected events were randomly partitioned into training, validation, and testing subsets containing 90 %, 5 %, and 5 % of events, respectively.
Dataset: Requiring at least one fully reconstructible W boson left roughly 8.8×10^6 events for training and validation.
Dataset: Roughly 2.4 × 10^6 of the retained events were fully reconstructible.The paper refers to Reference for exact numbers and the complete event-generation setup.
IV. RESULTS: The results compare SPANet, Topograph, HyPER, and Pairton reconstruction efficiencies for full-event, top-quark, and W-boson reconstruction.Pairton uncertainties were estimated by varying initial random initialization and shuffling training batches five times; the other models report central values only.

Reconstruction Performance and Statistical Stability · Ablation Studies · V. CONCLUSION

Pairton achieves strong and stable reconstruction performance, including the largest gains in difficult regions, while ablations support the pairformer architecture. The conclusion frames iterative graph reconstruction as a general framework that can extend beyond fully hadronic t̄t events.

Reconstruction Performance and Statistical Stability: Reconstruction performance is measured by full-event, top-quark, and W-boson efficiencies, or true positive rates.A W boson is correctly reconstructed when its two associated jets are identified, regardless of permutation.
Reconstruction Performance and Statistical Stability: Pairton significantly outperforms SPANet, Topograph, and HyPER, with the largest improvements at high jet multiplicities.High multiplicities make full t̄t reconstruction difficult because of large jet combinatorics and a crowded detector environment.
Reconstruction Performance and Statistical Stability: Direct efficiency comparison with TIGER is not fair because the models use different training statistics and predict different reconstruction types.TIGER can predict partial reconstructions, whereas Pairton requires and predicts complete assignments; TIGER efficiencies are nevertheless comparable to HyPER.
Reconstruction Performance and Statistical Stability: Pairton’s uncertainties were estimated from five retrainings and showed small variations, indicating stable optimisation and low sensitivity to initialisation.Comparison-method uncertainties were unavailable under equivalent retraining statistics and therefore were not included in Table II.
Reconstruction Performance and Statistical Stability: Pairton outperforms earlier methods across the t̄t-mass spectrum, with the largest gains at low m_t̄t where efficiencies are lowest.The differential efficiencies are calculated using truth-matched reconstructed jets in fully reconstructible events.
Ablation Studies: The pairformer architecture improves full-event reconstruction efficiency by roughly 5 % absolute over a standard transformer.The ablation studies compare pairformer with both a standard transformer and a bias-augmented transformer baseline.
V. CONCLUSION: Pairton reformulates particle reconstruction as a discrete diffusion process over graphs, using factorised distributions and a masked edge predictor.The model learns conditional distributions for reconstructed particles through probabilistic event reconstruction.
V. CONCLUSION: Iterative conditioning on partially reconstructed structures enables global consistency, while dynamic pairwise representations support state-of-the-art fully hadronic t̄t reconstruction.The framework is presented as general and extensible to more complex topologies, including t̄tH reconstruction, and as a bridge between generative modelling and event interpretation.

Appendix A: Connection between Masked Denoising and Auto-regressive Factorization

The masked binary-cross-entropy objective learns conditional distributions for adjacency entries given observed structure and input features. Structured masking stages align these predictions with an auto-regressive factorization, which inference recovers by reversing the masking chain.

Masked denoising objective: Binary cross-entropy minimization yields the conditional distribution of each adjacency entry given observed structure and input features.The result is obtained by applying the Bernoulli cross-entropy minimizer entry-wise.
Structured masking: The masking process defines which adjacency variables are observed or predicted through partially observed matrices.The process can reveal no particles, one W boson with its implied entries, or both W bosons with their implied entries.
Conditional factorization: Successive predictions correspond to P(W | x), P(W2 = (i, j) | W1, x), and P({b1 = k} | {W1 = (i, j)}, W2, x).The first W-boson edges combine mutually exclusive assignments, while later stages condition on already revealed particles.
Conditional factorization: Reversing the masking chain makes inference exactly mirror the chain-rule factorization of the joint distribution over particle assignments.The training procedure is therefore consistent with learning the conditionals in the auto-regressive factorization.

Appendix B: Model Hyperparameters

Pairton generally follows the original Pairformer hyperparameters while using a smaller architecture with RMSNorm replacing LayerNorm. The resulting implementation has approximately 1.2 × 10^6 learnable parameters and achieves roughly 2000-event-per-second inference throughput on a single RTX 3080 unless otherwise specified.

Pairton generally follows the original Pairformer hyperparameters from Reference.
The model uses a much smaller architecture and replaces LayerNorm with RMSNorm.The selected configuration results in approximately 1.2 × 10^6 learnable parameters.

Appendix C: Model Ablation Studies

The ablations show that Pairton’s pairformer architecture outperforms transformer-based alternatives, while conditioning on partially masked adjacency matrices and iterative prediction further improves reconstruction. A three-step iterative design was selected for the final model because four steps offered similar performance, though recovery from incorrect assignments remains weaker.

Neural network architecture: The pairformer architecture achieves superior reconstruction performance to standard transformer architectures and transformers with attention-matrix bias.The architecture comparison used matched embedding dimensions, adjusted layer counts, and otherwise fixed hyperparameters; partially masked adjacency matrices were withheld from these models.
Neural network architecture: Conditioning the iterative pairformer on partially masked adjacency matrices produces a further performance improvement over the unconditioned pairformer.The iterative pairformer was trained specifically to measure the gain from access to these matrices.
Reconstruction method: The iterative predictive model outperforms the noniterative baseline, hyperedge-based reconstruction, and deeper pairformer alternatives, showing the gain is not solely due to increased test-time compute.The comparisons include a three-step iterative model, a HyPER-style hyperedge method with a pairformer embedding graph neural network, and a twelve-block pairformer reference.
Reconstruction method: The iterative model is poorer at recovering from incorrect assignments, possibly because training uses only correct partial adjacency matrices for later assignments.The authors suggest training with both correct and incorrect previous assignments as a possible improvement.
Reconstruction method: Because three-step and four-step models perform similarly, the final model uses three-step iterative reconstruction to increase event throughput.The three steps reconstruct the first W boson, second W boson, and two b quarks simultaneously.

Appendix D: Algorithm for Graph Reconstruction

The reconstruction algorithm greedily selects edges using model logits, accounting for edge and self-edge contributions in W assignments and updating choices iteratively until all particles are selected. The authors note that this local maximization is tractable but may not be optimal generally.

Greedy decoding: The decoder uses a greedy approximation, selecting the edge that locally maximizes the objective at each prediction step.This is motivated by the tractable optimal assignment obtained under independent predictions.
Limitation: Local maximization may not yield the best reconstruction algorithm in general, and other reconstruction algorithms could be explored.The limitation concerns the greedy approximation rather than the feasibility of the procedure.
W-boson assignment: W-boson assignment maximizes contributions from the candidate inter-node edge together with each node’s possible self-edge.The procedure selects the edge indices that locally maximize the combined quantity.
b-quark assignment: For already selected W bosons, the algorithm chooses the b-quark pair maximizing the sum of two connecting-edge contributions minus the new node’s self-contribution.The pair is selected jointly for both b quarks using the top adjacency logits and chosen W-boson assignments.
Overall reconstruction: The full reconstruction alternates edge-logit prediction with the appropriate selection algorithm until all particles are selected.Algorithm 3 combines model predictions from data and partial adjacency matrices with W- and b-assignment procedures.