Pairton: Iterative Reconstruction of Short-Lived Particles

Technical Summary: Pairton – Iterative Reconstruction of Short-Lived Particles

Problem Statement
In high-energy physics, particularly at the Large Hadron Collider (LHC), reconstructing short-lived particles (e.g., top quarks, Higgs bosons) from their decay products is a critical challenge. In complex events with high particle multiplicities, such as the fully hadronic decay of top quark pairs ($t\bar{t}$), the combinatorial background is significant. Traditional methods, such as likelihood-based algorithms (e.g., KLFitter), rely on evaluating all possible combinations of detected jets, which becomes computationally expensive and prone to errors as jet multiplicity increases. While recent machine learning approaches (SPANet, Topograph, HyPER, TIGER) have improved performance, there remains a need for frameworks that can effectively handle the hierarchical nature of decay chains and global event consistency without relying on rigid topological assumptions or fixed masking strategies.

Methodology
The authors propose Pairton, an iterative framework that reformulates particle reconstruction as a structured probabilistic inference problem over partially observed graphs.

1. Graph Representation:
   The event is modeled as a collection of graphs where nodes represent detected jets and edges represent decay relationships. Specifically, adjacency matrices ($A^\alpha$) are defined for particle types $\alpha \in \{W, t\}$. An edge $(i, j)$ exists if jets $i$ and $j$ are decay products of particle $\alpha$. Self-loops indicate jets not associated with a specific particle type. This representation allows structural assignments to be treated as dynamic pairwise edge features.

2. Factorized Probabilistic Formulation:
   The reconstruction task is defined as finding the most probable assignment of jets to $W$ bosons and $b$-quarks. The joint probability distribution is factorized into conditional probabilities:
   $$P(W_1, W_2, b_1, b_2 | x) = P(W_1 | x) \times P(W_2 | W_1, x) \times P(b_1, b_2 | W_1, W_2, x)$$
   This factorization mirrors the physical decay hierarchy, allowing the model to iteratively predict edges conditioned on previously inferred structures.

3. Discrete Diffusion and Masked Denoising:
   Inspired by absorbing state discrete diffusion models used in natural language processing, Pairton frames reconstruction as a reverse denoising process.

   • Forward Process: A hierarchical Markov chain masks the adjacency matrices in steps: first revealing both $W$ bosons, then one $W$, then none. This specific corruption scheme respects the physical topology of the $t\bar{t}$ decay, ensuring intermediate states have clear physical interpretations.
   • Training: The model $f_\theta$ is trained to predict masked entries in the adjacency matrices conditioned on the observed structure and jet features. The objective is a masked conditional likelihood (Binary Cross-Entropy loss).
   • Inference: Reconstruction proceeds iteratively from a fully masked state. The model predicts edges, updates the visible adjacency matrix, and uses this updated structure to condition subsequent predictions, effectively reversing the masking process.

4. Model Architecture:
   Pairton utilizes a Pairformer architecture, adapted from AlphaFold3. Unlike standard transformers or the Particle Transformer (ParT) which use fixed pairwise representations, the Pairformer dynamically updates pairwise features.

   • Inputs: Jet features (energy, $p_T$, $\eta$, $\phi$, $b$-tag status) and pairwise features (di-jet invariant mass, $\Delta\eta$, $\Delta\phi$, $\Delta R$).
   • Mechanism: The architecture augments attention scores with a learnable bias term derived from pairwise features ($S_{ij} = \frac{q_i \cdot k_j}{\sqrt{d}} + b_{ij}$). This allows the model to seamlessly incorporate the relational state between all jets without complex message-passing over higher-order tensors.

Key Contributions

• Iterative Graph Reconstruction: Introduces a paradigm where particle reconstruction is treated as an iterative masked prediction process over graph structures, leveraging the hierarchical nature of decay chains.
• Discrete Diffusion in HEP: Adapts absorbing state discrete diffusion (previously used in LLMs) to high-energy physics, moving beyond fixed-rate masking (BERT-style) or auto-regressive (GPT-style) approaches.
• Dynamic Pairwise Representations: Demonstrates the efficacy of the Pairformer architecture in this domain, where dynamically updated pairwise edge features significantly outperform static representations.
• General Framework: Provides a flexible approach that can be extended to other topologies beyond the $t\bar{t}$ system.

Results
The method was evaluated on a dataset of $6 \times 10^7$ simulated all-hadronic $t\bar{t}$ events at $\sqrt{s} = 13$ TeV. Performance was measured by reconstruction efficiency ($\epsilon$) for full events ($\epsilon_{t\bar{t}}$), top quarks ($\epsilon_t$), and $W$ bosons ($\epsilon_W$).

• State-of-the-Art Performance: Pairton outperforms existing methods (SPANet, Topograph, HyPER) across all jet multiplicities.
  • Inclusive Full-Event Efficiency: Pairton achieves 0.6944, compared to 0.670 for HyPER, 0.645 for Topograph, and 0.651 for SPANet.
  • High Multiplicity: The improvement is most pronounced in events with $\ge 8$ jets, where Pairton reaches 0.5459 full-event efficiency, significantly higher than HyPER (0.513) and SPANet (0.487).
• Mass Dependence: Pairton consistently outperforms competitors across the invariant mass spectrum ($m_{t\bar{t}}$), with the largest gains observed in the low-mass region where reconstruction is most difficult.
• Ablation Studies:
  • The Pairformer architecture provided a ~5% absolute increase in efficiency over a standard transformer.
  • The 3-step iterative generative process outperformed single-step prediction and hyperedge-based models.
  • Simply increasing model depth to match the computational cost of the iterative process did not yield similar gains, confirming the value of the iterative conditioning.

Significance and Claims
The paper claims that Pairton establishes a new, general, and flexible paradigm for particle reconstruction by bridging modern generative modeling (specifically discrete diffusion and iterative inference) with high-energy physics.

The authors emphasize that the significance lies in:

1. Global Consistency: By conditioning on previously inferred structures, the model naturally incorporates global event consistency constraints in a probabilistically sound manner.
2. Topological Flexibility: While the current implementation focuses on the hierarchical $t\bar{t}$ decay, the discrete diffusion formulation allows for arbitrary noising processes, making it adaptable to more complex topologies (e.g., $t\bar{t}H$) where the ordering of predictions might be interchangeable.
3. Performance: The method achieves state-of-the-art results on the fully hadronic $t\bar{t}$ benchmark, particularly in challenging high-multiplicity environments, demonstrating that iterative graph reconstruction is a powerful tool for event interpretation.

The work does not claim to solve all reconstruction problems universally but positions itself as a robust framework that can be readily extended to other final states involving short-lived particles.