In a previous blog post, I introduced ν-Flows, a novel machine learning method that uses conditional normalizing flows to precisely reconstruct the kinematics of a single, elusive neutrino in particle collider experiments. ν-Flows marked a significant step forward by moving beyond single-point estimates, providing a full probabilistic understanding of the neutrino’s momentum and its associated uncertainties.
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
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arXiv:2307.02405.