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Jet Substructure at the Large Hadron Collider: A Review of Recent Advances in Theory and Machine Learning

Andrew J. Larkoski, Ian Moult, Benjamin Nachman

arXiv:1709.04464v2hep-phhep-exnucl-exnucl-th

TL;DR

Jet substructure supports Standard Model studies at the TeV scale and searches for new physics. This review synthesizes advances in theory and machine learning, including observables and calculations, and reports that combined measurements can improve parton-shower descriptions of jet radiation.

  • Problem

    Jet substructure is needed to study the Standard Model at the TeV scale and search for new physics.

  • Method

    The paper reviews advances in jet-substructure theory and machine learning, including calculations, new observables, and machine-learning techniques.

  • Results

    Combined measurements of perturbative fragmentation improve the parton-shower description of jet radiation.

  • Takeaways & Limitations

    The review provides a broad introduction and reference for developments in jet substructure and its applications.

  • Takeaways & Limitations

    Because the review covers a wide range of topics, some topics remain beyond its scope.

Abstract

from arXiv · show

Jet substructure has emerged to play a central role at the Large Hadron Collider (LHC), where it has provided numerous innovative new ways to search for new physics and to probe the Standard Model in extreme regions of phase space. In this article we provide a comprehensive review of state of the art theoretical and machine learning developments in jet substructure. This article is meant both as a pedagogical introduction, covering the key physical principles underlying the calculation of jet substructure observables, the development of new observables, and cutting edge machine learning techniques for jet substructure, as well as a comprehensive reference for experts. We hope that it will prove a useful introduction to the exciting and rapidly developing field of jet substructure at the LHC.

1. Introduction

Jet substructure has become central to LHC physics, supporting Standard Model studies and new-physics searches through increasingly sophisticated theoretical, experimental, and machine-learning techniques. This review offers a pedagogical and state-of-the-art overview while emphasizing representative examples, physical principles, and future challenges.

  • Motivation: Jet substructure exploits radiation patterns inside collimated particle sprays to study high-energy quarks and gluons.It can identify boosted hadronically decaying electroweak bosons and top quarks.
  • Motivation: The LHC's extended energy reach has driven new theoretical ideas and reconstruction techniques for previously unexplored regimes.These developments improve sensitivity to new physics and advance Standard Model probes.
  • Recent advances: Theoretical progress has produced precise predictions for many observables and enabled tailored observables already deployed in major experiments.Experimental and theoretical ideas have developed in a mutually reinforcing relationship.
  • Review scope: The review surveys theoretical developments, machine-learning applications, new observables, precision prospects, and connections to other areas of physics.It aims to serve both as a state-of-the-art reference and as a primer for newcomers.
  • Future directions: The review also outlines unresolved challenges and goals intended to guide future progress in jet substructure.Its machine-learning section introduces technical aspects while highlighting applications to current problems.
  • Review scope: The authors emphasize representative examples and underlying physical principles rather than exhaustive technical details.They provide references for calculational techniques beyond the review's scope.

2. Theory Developments

Jet-substructure theory develops calculable observables around infrared and collinear safety while extending predictive perturbative-QCD methods to broader classes of observables. The review surveys these calculations, precision improvements, measurements, and directions for data–theory comparison.

  • Foundations of jet-substructure calculations: Observable design balances sensitivity to the target physics against calculability from first principles of QCD.Modern observables probe multiple hard prongs, coherent soft radiation, and correlations between radiation inside and outside the jet.
  • Foundations of jet-substructure calculations: IRC safety makes jet observables insensitive to soft or exactly collinear emissions, preserving the real–virtual cancellation required for perturbative calculations.This property ensures that phase-space restrictions imposed by an observable do not disrupt the cancellation of divergences.
  • Theory developments: Perturbatively calculable jet observables can be systematically improved order by order, building on early predictions for jet mass and resummed substructure observables.The theoretical program expanded beyond jet mass to a broad range of observables, while retaining predictive power for some IRC-unsafe classes through new techniques.
  • Frontiers and applications: Fast, experimentally well-behaved IRC-safe jet algorithms and the LHC’s high collision energy and resolution helped make jet-substructure theory a mature field.The review also identifies calculation frontiers, broader QCD influences, and recommendations for data–theory comparison.
  • Theory developments: The review covers the physics, accuracy definitions, exclusive event generators, measured observables, and high-precision calculations used in jet substructure.Because observables are typically dominated by soft or collinear radiation, resummation to higher logarithmic accuracy is an important part of precision predictions.

2.1. Aspects of Jet Substructure Calculations

Reliable jet-substructure calculations combine resummation, fixed-order perturbative corrections, and non-perturbative effects. The review illustrates these ingredients using jet mass and discusses their theoretical accuracy, extensions, and limitations.

  • Scope: The review focuses on IRC-safe observables and does not cover the more general classes of observables whose calculations remain beyond the scope of this brief treatment.Many principles discussed remain similar for those more sophisticated calculations.
  • Perturbative structure: For jet mass with mJ ≪ pT J, fixed-order predictions diverge as mJ →0 because logarithmically enhanced terms are not resummed.Soft and collinear QCD radiation generates large logarithms of pT J/mJ.
  • Calculation ingredients: Jet-substructure predictions incorporate resummation, fixed-order corrections in αs, and non-perturbative corrections from hadronization and related effects.These ingredients are used to compare the theoretical sophistication of different calculations.
  • Resummation: Resummation makes the distribution vanish as the observable approaches zero and produces the characteristic Sudakov peak.The resummed prediction accounts for logarithmically enhanced contributions that invalidate the traditional fixed-order expansion in this region.
  • Non-perturbative corrections: Non-perturbative effects shift distributions at small observable values and can be modeled with a shape function convolved with the perturbative distribution.The shape function captures sensitivity to fluctuations at the scale ΛQCD but is not currently calculable from first principles.
  • Resummation: Higher-order resummation improves quantitative control: NLL calculations begin to describe distributional features, while calculations beyond NLL provide perturbative uncertainty estimates and convergence information.Several approaches are reviewed, including radiator calculations, generating functionals, factorization theorems, and effective field theory.
  • Parton showers and factorization: Parton showers factorize hard scattering, perturbative soft and collinear emissions, and hadronization, reproducing the LL Sudakov factor and extending to NLL through an appropriate coupling scheme.The review notes that all-orders factorization theorems have also been proven for several jet-substructure processes and are being extended to more differential observables.

2.2. Status of Jet Substructure Calculations

Jet substructure calculations are organized by the jet topology they probe, with analytic predictions and parton-shower studies covering one-prong, two-prong, and newer structures. These observables provide insight into dominant physics, while recoil and non-perturbative effects remain important considerations.

  • Organization: Calculations are organized by whether observables probe one-prong, two-prong, or other jet structures.The review primarily discusses IRC-safe observables, while also noting useful observables that are not IRC safe.
  • One-prong jets: Angularities describe radiation relative to a jet axis, while energy correlation functions use particle-pair angles and are recoil-free by construction.The angular exponent controls collinear sensitivity, with smaller α giving greater sensitivity; IRC safety requires α > 0.
  • One-prong jets: NLL calculations describe angularity distributions for quark and gluon jets, with Pythia comparisons shown separately at parton and hadron levels.The example uses angularity α = 3/2 in e+e− collisions.
  • One-prong jets: Ignoring recoil limits the observables that can be calculated, while recoil-sensitive broadening requires more complicated precision calculations.For angularities, recoil contributions are subleading when α > 1 but dominate in the soft limit for α ≤ 1.
  • One-prong jets: Recoil-free energy correlation functions are more powerful quark-versus-gluon discriminants at NLL accuracy than recoil-sensitive counterparts.This motivates designing observables that remain maximally sensitive to radiation while avoiding recoil effects.
  • Two-prong jets: Varying non-perturbative parameters produces large changes in predicted D(α)2 discrimination efficiencies across parton-shower and analytic calculations.Differences between analytic predictions and Pythia or Herwig++ support measuring and tuning non-perturbative parameters for multi-prong observables.
  • Two-prong jets: Grooming reduces overlap between one- and two-prong jets, improves discrimination, and places non-perturbative effects under theoretical control.Grooming strongly changes QCD-jet distributions but has a much smaller effect on Z-jet distributions; multi-prong observables remain more sensitive to hadronization corrections.
  • New structures and probes: The zg distribution approaches an ultraviolet fixed point described by Altarelli–Parisi splitting functions, while counting observables can reach the LL-optimal quark–gluon likelihood ratio.Herwig++ approaches the expected fixed-point distribution as jet transverse momentum increases.

2.3. Toward Higher-Order Resummation

The review surveys efforts to calculate jet-substructure observables at increasingly high theoretical precision, emphasizing resummation for challenging logarithmic structures and small jet radii. It discusses jet-mass calculations in e+e− and pp collisions, including complications from non-global logarithms and the hadronic environment.

  • Toward Higher-Order Resummation: Higher-precision calculations begin with simpler observables and extend toward mult prong-sensitive observables whose phase spaces are substantially more complicated.The review describes this progression as a strategy for developing theoretical control.
  • Toward Higher-Order Resummation: Non-global logarithms arise in jet-mass observables from correlations between in-jet and out-of-jet scales, making them difficult to compute and control.They are especially relevant when the jet mass is small relative to the jet energy or transverse momentum.
  • Toward Higher-Order Resummation: Small jet radii require resummation of logarithms of the jet radius to obtain controlled predictions.The review motivates this need by the granularity of LHC detectors and the high-luminosity environment.
  • Toward Higher-Order Resummation: Heavy jet-mass predictions in e+e− collisions reach NNNLL+NNLO accuracy.This is presented as an example of high-precision jet-substructure calculations.
  • Toward Higher-Order Resummation: Jet-mass calculations in pp collisions involve additional complications because measuring the largest-mass jet does not control all large logarithms present in the cross section.The hadronic environment also introduces effects associated with the underlying event and pile-up.
  • Toward Higher-Order Resummation: In e+e− collisions, a re-emission from the heavy hemisphere can set the light hemisphere mass, producing the leading non-global-emission configuration.The review illustrates this configuration with the light and heavy hemisphere masses.

HR HL

The review explains how non-global logarithms, underlying-event radiation, and pile-up complicate precision jet-mass predictions at hadron colliders. It then surveys grooming and related methods that suppress or eliminate these effects, enabling more precise measurements and theoretical comparisons.

  • HR HL: Non-global logarithms are a leading manifestation of correlations between in-jet and out-of-jet radiation scales.They arise in jet-mass measurements when the observable does not globally constrain radiation.
  • HR HL: Underlying-event contributions are subtle because they lack a field-theoretic definition and must be modeled in calculations.The underlying event is described as low-energy radiation from secondary parton scatterings, approximately uniform in rapidity.
  • HR HL: Pile-up can substantially bias jet measurements, but its lack of correlation with the hard scattering means it can in principle be removed.This distinguishes pile-up from the underlying event in comparisons between data and theory.
  • HR HL: NLL jet-mass predictions show that including non-global logarithms produces an especially large effect near the distribution peak.The comparison is made for pp →Z +j events using the ratio of jet mass to jet pT.
  • HR HL: Among the groomers studied, only the modified mass drop tagger successfully removed non-global logarithms from the mass distribution.The construction of mMDT was motivated by identifying a groomer that eliminates these logarithms.
  • HR HL: Soft drop was subsequently shown to eliminate non-global logarithms from the groomed jet mass.Grooming directly reduces contamination radiation within jets rather than suppressing radiation globally.

3. Machine Learning

The review connects jet substructure with machine learning through clustering, supervised tagging, regression, and generation. It emphasizes deep neural networks as flexible tools that can use high-dimensional information from the full jet radiation pattern.

  • Machine Learning: Jets are defined by clustering algorithms, linking jet physics to unsupervised machine learning while requiring infrared and collinear safety for physical meaning.This connection motivates the review’s emphasis on relating machine-learning algorithms to physical jet properties.
  • Machine Learning: Supervised learning underlies jet tagging because simulated and, to a limited extent, collision datasets can provide labeled examples of jet origin.Traditional approaches include boosted decision trees, random forests, and neural networks.
  • Machine Learning: The optimal tagger uses the likelihood ratio based on the full radiation pattern, but practical limits on data, computing, and simulation require approximations.These approximations include boosted decision trees, random forests, and neural networks.
  • Machine Learning: Deep neural networks can classify jets using all available information in their radiation patterns rather than only one-dimensional projections such as jet mass.Their flexibility and scalability have driven increasing use in jet physics.
  • Machine Learning: Machine-learning analysis encodes optimality through a loss function, making the choice of objective explicit for tasks such as jet tagging.For binary classification, the network output is trained toward 0 for one class and 1 for the other.
  • Machine Learning: Neural networks support classification, regression, and generation, with binary cross-entropy used for classification and mean squared error commonly used for regression.Generation learns a map from random numbers to structured data, such as parton-shower-like outputs.

3.1. Jet Representations and Preprocessing

Jet substructure can be represented as images, sequences, trees, graphs, unordered sets, or physics-inspired features, with each representation motivating different neural-network architectures. Preprocessing and representation choices trade detector granularity, physical structure, and information preservation.

  • Images: Jet images encode particle energy or transverse momentum on detector-like pixels, optionally adding channels for charge-separated energy.Convolutional neural networks exploit local filters and weight sharing, reducing trainable parameters.
  • Preprocessing and trade-offs: Images may be inefficient at finer detector granularity, while pixelating particles can lose relevant information; preprocessing such as rotating jets can reduce input complexity.The review emphasizes matching preprocessing to the chosen machine-learning architecture.
  • Sequences: Sequences process ordered jet constituents recurrently, combining each particle with an internal state whose final value supports classification.The ordering is often by jet pT, but no unique ordering is required.
  • Trees and graphs: Binary trees and generic graphs encode clustering histories or node connections, using recurrent or graph-convolutional networks respectively.Graphs may use adjacency matrices derived from standard jet clustering algorithms or more exotic constructions.
  • Unordered sets: Images and sequence/tree/graph approaches impose spatial, temporal, or relational structure even though final-state hadrons have no unique ordering or history.The unordered set of constituent four-vectors is therefore a natural representation, while point-cloud methods support variable-length sets.
  • Physics-inspired representations: Physics-inspired representations include energy flow polynomials, which provably span IRC-safe jet observables, and Lorentz networks that extract invariant features directly from four-vectors.Energy flow polynomials can support linear regression over a basis of functions.

3.2. Jet Tagging with Machine Learning

Machine learning jet tagging spans supervised, weakly supervised, and label-free strategies, motivated in part by ambiguous jet origins and unavailable per-jet labels. The section also emphasizes decorrelation and diagnostics needed for data-driven searches and validation.

  • Labeling challenges: Jet-origin categories are not always well-defined: containment can be ambiguous, and strongly charged jets cannot formally be treated independently of their environment.These issues motivate methods that do not rely on fixed labeling schemes.
  • Learning without per-jet labels: 118?

3.3. Regression Techniques

Regression extends machine learning beyond classification to continuous jet-physics targets such as energy calibration and pileup correction. The reviewed methods address spectrum-dependent bias, multidimensional detector information, and constituent-level corrections.

  • Pileup mitigation: End-to-end image regression predicts pileup-removed jet images from detector-level inputs and can use multidimensional information unavailable to more structured constructions.Graph networks provide another route by replacing PUPPI-like local kernels with learned constituent-level classification and weighting.
  • Energy calibration: A neural network trained to predict true energy from reconstructed observables approaches the conditional mean, making the calibration dependent on the training spectrum.The loss function determines the learned conditional statistic, such as the mean, median, or mode.
  • Energy calibration: Numerical inversion learns reconstructed observables from the true target and inverts that function, producing a nearly unbiased procedure under mild assumptions.Deep learning can generalize the inversion to multidimensional measured jet-radiation features.
  • Pileup mitigation: Pileup mitigation is a central calibration task, with constituent-based methods correcting jet inputs so that pileup dependence can be reduced across jet-substructure observables.The review contrasts established corrections with machine-learning approaches that automate or improve them.
  • Pileup mitigation: No direct comparison yet establishes whether structured PUPPI-like methods or direct image regression is superior, and future procedures may combine their strengths.This remains an open methodological boundary in the reviewed approaches.

3.4. Machine Learning for Jet Simulations

Machine learning is being applied to jet simulation through generative distributions, optimized phase-space integration, parton-shower modeling, and fast detector-response generation. The approaches promise speed and flexibility but remain limited by fidelity and extrapolation challenges.

  • Generative modeling: Generative models learn or sample probability distributions for applications ranging from parton-shower replacement or augmentation to accelerated detector-response simulation.The section treats generation as a natural extension of regression.
  • Phase-space integration: Deep learning can optimize phase-space grids for differential jet calculations, accommodating non-trivial region shapes and potentially reducing computation time.The target bottleneck is slow numerical integration needed to match fixed-order matrix elements.
  • Parton showers: QCD-structured generative models replace shower components with neural networks, using convolutional filters or recurrent histories to encode splitting and evolution structure.These physically inspired image- and tree-based approaches use far fewer parameters than typical deep networks.
  • Detector simulation: GANs and VAEs are promising for accurate generation in high-dimensional spaces, but substantial work remains to reach detector-simulation fidelity and establish suitable performance measures.Generative-model quality is not summarized adequately by the loss function alone.
  • Detector simulation: Generative models interpolate well but are likely unreliable for extrapolation because generated distribution tails are limited by what training data contain.This boundary still permits interpolation-focused applications.
  • Background estimation: GAN-based unbinned templates can estimate multijet backgrounds while removing binning effects and conditioning naturally on other jet observables.The method is presented for high-multiplicity jet searches.

3.5. Anomaly detection

Anomaly detection methods seek nonstandard jet substructure with limited reliance on predefined signal models. CWoLa and autoencoder approaches offer complementary ways to identify potentially uncovered signals, but each has important constraints.

  • CWoLa anomaly detection: CWoLa classifiers are asymptotically optimal for distinguishing examples from mixed samples when the underlying class proportions differ.This approach learns from mixed rather than pure samples, extending weak supervision toward anomaly detection.
  • CWoLa anomaly detection: CWoLa anomaly detection trains classifiers to distinguish a signal region from sidebands and enhance low-purity signals with nonstandard jet substructure.The method uses a known feature where a signal may be resonant and requires care to avoid sculpting artificial resonance structures.
  • CWoLa anomaly detection: The CWoLa approach requires sufficient signal because its mixed training samples must have different class proportions.This requirement limits the method when the signal contribution is too small to produce distinguishable mixtures.
  • Autoencoder anomaly detection: Autoencoders trained entirely on background use reconstruction differences for model-agnostic classification in regions that may contain signal.Jets not represented during training are expected to reconstruct poorly, providing an anomaly score based on compression and decompression.
  • Autoencoder anomaly detection: Autoencoder studies show sensitivity to jets with non-SM substructure and investigate reducing dependence on jet mass.Automatic or manual methods can reduce mass dependence, which matters because jet mass is important for background estimation in single-jet mass searches.
  • Complementary approaches: CWoLa and autoencoders are expected to provide complementary strengths needed to broadly cover BSM scenarios.The review presents both as model-agnostic approaches alongside conventional supervised methods.

3.6. The future

Jet-physics machine learning spans nearly physics-agnostic to strongly physics-inspired methods and continues to adapt state-of-the-art algorithms. Future work includes combining deep learning with QCD while considering approaches suited to complex hadronic final states.

  • The future: Deep learning for jet physics spans nearly physics-agnostic methods to strongly physics-inspired approaches.The field adapts state-of-the-art algorithms from industry to characterize jets.
  • The future: A dedicated workshop series has grown from increasing interest in deep learning for jet physics.The review describes the field as rapidly expanding and continuing to produce innovative methods.
  • The future: Combining deep learning with QCD is presented as a way to maximally exploit data at the LHC and beyond.The review connects future machine-learning developments with improving theoretical understanding.
  • The future: Model-agnostic anomaly-detection proposals are relevant because complex hadronic final states require data-driven background estimates.The review notes that approaches relying heavily on a background model may be poorly suited to this setting.

4. Conclusions

The review provides a comprehensive, pedagogical overview of advances in jet-substructure theory and machine learning, emphasizing the physical principles behind observables, calculations, and applications. It is intended as a useful starting point for further study in a rapidly developing field.

  • 4. Conclusions: The review surveys experimental and theoretical techniques developed to exploit jet substructure at the LHC.It frames jet substructure as playing a central role in current collider physics.
  • 4. Conclusions: The paper aims to provide a comprehensive yet pedagogical overview focused on advances in theory and machine learning.The stated scope combines introductory explanation with broad coverage of the field.
  • 4. Conclusions: The review emphasizes underlying physical principles guiding jet-substructure observables, calculations, and machine-learning applications.These principles are presented alongside developments in QCD, quantum field theory, and machine learning.
  • 4. Conclusions: The review is intended to remain useful as jet-substructure techniques and their theoretical and machine-learning foundations continue to evolve.The conclusion explicitly looks toward future developments and applications.
  • 4. Conclusions: The authors hope the review serves as a useful starting point for readers studying these topics in greater detail.The conclusion positions the paper as an entry point for contributing to the field.
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