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Combinatorial Residual Network: a decay-chain-aware classifier without explicit jet assignment

The paper introduces the Combinatorial Residual Network (CoReNet), a highly efficient and parameter-light classifier that encodes decay-chain hierarchies directly into its architecture to solve jet assignment combinatorial problems without requiring explicit generator-level labels, achieving performance comparable to much larger transformer models across various heavy-particle final states.

Original authors: John Alison, Patrick Bryant, Alejandro Gomez Espinosa, Christian Guinto-Brody, Aniket Khanal, Chuyuan Liu, Sindhu Murthy, Guanglei Zhao

Published 2026-11-02

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CC BY 4.0

Original authors: John Alison, Patrick Bryant, Alejandro Gomez Espinosa, Christian Guinto-Brody, Aniket Khanal, Chuyuan Liu, Sindhu Murthy, Guanglei Zhao

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the bustling heart of the Large Hadron Collider, where protons collide at nearly the speed of light, physicists are hunting for the rarest events in the universe. When heavy particles decay, they often shatter into sprays of smaller particles called jets. The challenge is not just seeing these jets, but understanding which ones belong together. Imagine a room where a dozen people have just thrown their coats into a single pile; to figure out who wore which coat, you would have to try every possible combination of matching coats to owners. In particle physics, this matching problem grows explosively complex as more jets appear. If a heavy particle decays into four jets, there are dozens of ways to group them, and most of those groupings are wrong. For years, scientists have relied on complex computer programs to guess the correct groupings before they could even begin to analyze the data, a process that was slow and prone to error.

A team of researchers at Carnegie Mellon University has developed a new way to solve this puzzle, one that skips the guessing game entirely. They created a specialized computer program called the Combinatorial Residual Network, or CoReNet, which is designed to understand the structure of a particle decay without ever needing to decide which jet belongs to which parent particle first. Instead of forcing the computer to pick a single "correct" answer from a list of possibilities, the network looks at every possible grouping at the same time. It weighs each possibility, learns which combinations contain the most useful information, and combines them into a single, powerful judgment about whether an event is a signal of new physics or just background noise. This approach is remarkably efficient, requiring a fraction of the computing power of previous methods while delivering results that are just as accurate.

The researchers tested their new tool on a specific and difficult scenario: the production of pairs of Higgs bosons, which are known to decay into four jets of bottom quarks. This is a classic "needle in a haystack" problem, where the signal is buried under a mountain of ordinary particle collisions. In their simulations, the new network was able to distinguish the Higgs boson pairs from the background with a high degree of success, outperforming older methods that relied on hand-picked measurements of the jets. What makes this result particularly striking is the size of the network. The new tool has fewer than two thousand adjustable settings, whereas the previous state-of-the-art systems used for this task had hundreds of thousands. It is a bit like finding a way to solve a complex maze by understanding the shape of the walls, rather than trying every single path with a giant map.

The beauty of this design lies in how it mimics the physical reality of the decay. The network is built with a hierarchy that follows the steps of the particle's breakup, from the initial jets up to the final parent particles. It does not need to be told which jets are the "right" ones; it simply evaluates every logical grouping and learns which ones help it separate the signal from the noise. In their tests, the network learned to identify the correct groupings in about eighty-one percent of the cases where a correct answer existed, even though it was never explicitly taught to do so. This suggests that the network naturally discovers the underlying physics structure simply by trying to do its job of classification.

The researchers also showed that this method is flexible enough to handle more complicated scenarios, such as when a particle decays into a mix of jets and other particles like electrons or missing energy. In these cases, they were able to encode the decay patterns of both the signal and the background into the network's structure. This allowed the system to use the background's own structure against it, improving the accuracy of the search even further. The entire system trains in a matter of minutes on a single graphics card, a stark contrast to the hours required by older, much larger systems. This speed and efficiency mean that such tools could eventually be used in real-time during experiments, helping scientists make decisions about which collisions to keep and which to discard as they happen.

Ultimately, this work demonstrates that building specific knowledge about the physical world directly into the design of a computer program can be more powerful than simply throwing more data and more complex models at a problem. By respecting the symmetries and hierarchies of particle decay, the researchers created a tool that is not only faster and smaller but also more effective at finding the rare events that could reveal new laws of nature. The success of this approach suggests a new path forward for particle physics, where the architecture of the analysis tool itself becomes a guide to understanding the universe.

Technical Summary: Combinatorial Residual Network (CoReNet)

Problem Statement
In Large Hadron Collider (LHC) physics, analyzing final states with multiple jets from heavy particle decays (e.g., $t\bar{t}$ , $HH$ , or new resonances) presents a significant combinatorial challenge. To interpret an event, jets must be assigned to their parent particles. Since jets from different partons are difficult to distinguish beyond $b$ -tagging, the number of possible assignments grows factorially with the number of jets. Previous machine learning approaches typically treated jet assignment as a supervised intermediate step, using generator-level truth to train a model to assign jets, followed by a separate classification step. However, this assumes that learning the truth-matched pairing inherently improves signal-background separation, an assumption rarely tested directly. Furthermore, at the detector level, a "correct" pairing is often ill-defined or non-existent for background events, making the reliance on generator-level assignment targets suboptimal for the ultimate goal of classification.

Methodology: The Combinatorial Residual Network
The authors propose the Combinatorial Residual Network (CoReNet), a classifier that encodes the assumed decay chain of the final state directly into its architecture, eliminating the need for explicit jet assignment targets during training.

Architecture and Hierarchy: CoReNet builds event representations level-by-level following the decay hierarchy. For a process like $HH \to b\bar{b}b\bar{b}$ , four candidate jets are combined into all possible dijet pairs (six combinations), which are then grouped into three pairing hypotheses (ways to assign the four jets to two Higgs bosons).
Hypothesis Evaluation: Instead of selecting a single assignment, the network evaluates every assignment hypothesis simultaneously. It uses shared weights across objects at each level, meaning the network size depends on the internal representation width ( $d$ ) rather than the number of objects or hypotheses.
Symmetry Handling: The exchange symmetries of the decay (e.g., swapping the two jets in a dijet or the two Higgs candidates) are built into the construction using mean and absolute difference operations, rather than being learned.
Pooling and Output: Each hypothesis is scored by a shared linear layer. These scores are normalized via softmax to produce weights ( $q_{score}$ ), which are used to pool the hypothesis representations into a single event vector ( $e$ ). This vector is mapped to a binary classification output.
Training: The network is trained solely on the classification loss (binary cross-entropy) without any generator-level jet assignment targets. It requires no truth-level information for assignment.
Extensions: The architecture is adaptable to other topologies. For $HH \to b\bar{b}W^+W^-$ against semileptonic $t\bar{t}$ , the network can encode the decay chains of both the signal and the background simultaneously, reusing candidate objects and combining hypotheses with learned weights.

Key Contributions

Architecture Design: A novel network structure that integrates the decay hierarchy and combinatorial possibilities directly into the model, avoiding the "assignment-then-classify" pipeline.
Parameter Efficiency: By sharing weights across objects and hypotheses, CoReNet achieves high performance with a drastically reduced parameter count (approx. $10^3$ parameters) compared to state-of-the-art transformer-based models (approx. $10^5$ – $10^6$ parameters).
Training Efficiency: The model trains in minutes on a single GPU, compared to hours for larger reference models.
No Assignment Supervision: The method demonstrates that high-quality classification can be achieved without training on truth-matched jet assignments, addressing the issue of undefined pairings in background events.

Results
The authors evaluated CoReNet on two benchmark scenarios using fast-simulation samples:

$HH \to b\bar{b}b\bar{b}$ vs. QCD Multijet:
- CoReNet (with $d=8$ , 1508 parameters) achieved a ROC AUC of 0.8664.
- This significantly outperformed kinematic baselines (GBDT and MLP, AUC $\approx$ 0.829) and was comparable to or better than SPANet (a transformer-based reference with 559,846 parameters).
- The best SPANet configuration achieved an AUC of 0.8593, while configurations trained with strong assignment losses performed worse, falling below kinematic baselines.
- CoReNet's hypothesis weights selected the truth-matched pairing in 80.8% of signal events where it was defined, despite never being trained to do so.
$HH \to b\bar{b}W^+W^-$ vs. Semileptonic $t\bar{t}$ :
- CoReNet encoding only the signal decay chain ( $CoReNet_{HH}$ ) achieved an AUC of 0.8859.
- Encoding the background decay chain as well ( $CoReNet_{HH+t\bar{t}}$ ) improved the AUC to 0.9004 with only ~1282 additional parameters.
- This performance was comparable to the best SPANet configuration (AUC 0.9126), which used 155 times more parameters. Notably, SPANet's performance degraded when trained with assignment losses.

Significance and Claims
The paper claims that CoReNet demonstrates that encoding domain knowledge (the decay chain) into the network architecture is a highly efficient alternative to training large, general-purpose models on massive datasets. The results suggest that:

Explicitly learning jet assignment is not necessary for optimal classification; the network can learn to weight hypotheses effectively for discrimination without assignment targets.
Small, specialized networks can match the performance of large transformers while being orders of magnitude faster to train and deploy.
The construction is generalizable to any final state with a known decay chain, including cases where background topologies can be encoded alongside signal topologies to further improve discrimination.
The small parameter count makes the model suitable for resource-constrained environments, such as trigger-level selection or inference on FPGAs, where larger transformers are impractical.

The authors conclude that while symmetry-preserving attention networks like SPANet remain well-suited for tasks where reconstructing the specific jet assignment is the primary goal, CoReNet offers a superior approach for classification tasks where the separation of signal from background is the sole objective.

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