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How Much Information is in a Jet?

Kaustuv Datta, Andrew Larkoski

arXiv:1704.08249v2hep-phhep-ex

TL;DR

Jet-discrimination machine learning has shown strong performance, but its learned information is difficult to interpret. This paper tests minimal observable bases for boosted hadronic Z bosons versus QCD jets and finds that useful discrimination saturates at 4-body phase space, which is 8 dimensional.

  • Problem

    Machine-learning methods for jet identification have shown strong discrimination, but their physical interpretation and the additional information beyond standard observables remain unclear.

    Source: §1 · p. 2, §4 · p. 14

  • Method

    The authors measure IRC-safe observables that completely and minimally span M-body jet-substructure phase space, then use them to identify relevant discrimination information.

    Source: §1 · p. 3, §2 · p. 6, §4 · p. 14

  • Results

    Discrimination of boosted hadronic Z-boson jets from light-QCD-parton jets saturates with observables sensitive to 4-body phase space, which is 8 dimensional.

    Source: §1 · p. 3, §4 · p. 14

  • Takeaways & Limitations

    For this Z-versus-QCD jet task, the useful discrimination information can be captured by a relatively small, 8-dimensional observable description.

    Source: §1 · p. 3, §4 · p. 14

    The basis describes relative jet-substructure configuration, not the total jet (pT, η, φ) or global event properties.

    Source: §1 · p. 3

Abstract

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Machine learning techniques are increasingly being applied toward data analyses at the Large Hadron Collider, especially with applications for discrimination of jets with different originating particles. Previous studies of the power of machine learning to jet physics has typically employed image recognition, natural language processing, or other algorithms that have been extensively developed in computer science. While these studies have demonstrated impressive discrimination power, often exceeding that of widely-used observables, they have been formulated in a non-constructive manner and it is not clear what additional information the machines are learning. In this paper, we study machine learning for jet physics constructively, expressing all of the information in a jet onto sets of observables that completely and minimally span N-body phase space. For concreteness, we study the application of machine learning for discrimination of boosted, hadronic decays of Z bosons from jets initiated by QCD processes. Our results demonstrate that the information in a jet that is useful for discrimination power of QCD jets from Z bosons is saturated by only considering observables that are sensitive to 4-body (8 dimensional) phase space.

1 Introduction

The paper makes jet-discrimination machine learning more interpretable by testing minimal observable sets that resolve successive dimensions of jet-substructure phase space. For boosted hadronic Z bosons versus QCD jets, discrimination saturates at 4-body phase space, an 8-dimensional description.

1 Introduction

  • Boosted hadronic decays of heavy Standard Model particles can resemble light-parton QCD jets, making their identification important for LHC measurements and searches.Jet-identification observables and techniques have become standard tools in ATLAS and CMS.Source: §1 · p. 2
  • Prior machine-learning studies [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] showed strong jet discrimination, but their learned information was difficult to interpret and standard observables' omissions remained unclear.The paper also notes earlier efforts to quantify overcomplete bases of jet observables.Source: §1 · pp. 2–3
  • The authors construct a basis of 3M - 4 IRC-safe observables that spans M-body jet-substructure phase space, with measurements invertible to recover its coordinates.The basis describes relative emission configuration, not the total jet (pT, η, φ).Source: §1 · p. 3
  • A classifier built from a finite number of IRC-safe observables is generally Sudakov safe [22] [23].Source: §1 · p. 3
  • Discrimination power for boosted hadronic Z bosons versus QCD jets saturates when observables resolve 4-body phase space, which is 8 dimensional.Figure 1 plots Z-boson jet efficiency against QCD-jet rejection for mass plus observables defining different M-body phase spaces.Source: §1 · p. 3, Figure 1 · p. 4

2 Observable Basis

The section constructs a minimal set of IRC-safe N-subjettiness observables to span jet phase space, then generalizes the construction from two and three particles to M particles. Its observables span generic configurations, with a jet-mass measurement added to capture the overall energy scale.

2 Observable Basis

  • N-subjettiness measures radiation about N axes with angular exponent β>0, and the selected observables are intended to span phase space rather than represent a uniquely optimal basis.N-subjettiness is used for practical reasons because its evaluation is significantly faster than that of energy correlation functions; other observable choices could also span phase space.Source: §2 · p. 4
  • The phase-space expressions are illustrated in the narrow-jet, relatively collinear limit, which simplifies them without affecting their ability to span the variables.The authors state that the observable choice is not special and that optimizing the basis is beyond the paper’s scope.Source: §2 · pp. 4–5
  • For two particles, the two-dimensional phase space is specified by a momentum fraction and splitting angle, recoverable from two 1-subjettiness observables with different angular exponents.The particle momentum ordering is not assumed, so the expressions are symmetric under z ↔ 1−z.Source: §2 · pp. 5–6
  • For three particles, five N-subjettiness observables generically span the five-dimensional phase space, using three 1-subjettiness and two 2-subjettiness measurements.The displaced 1-subjettiness axis under E-scheme recombination is key; winner-take-all axes can leave non-degenerate configurations unspanned.Source: §2 · p. 6
  • For M-body phase space, coordinates comprise M−1 transverse-momentum fractions and 2M−3 pairwise angles, with the remaining angles fixed by planar geometry.The construction generalizes the lower-dimensional sets through successive declustering, adding three phase-space variables and three observables at each stage.Source: §2 · p. 7
  • For M particles, the construction uses 3M−4 N-subjettiness observables to span generic phase-space configurations, requiring nonzero energies and finite angles between particles.The count follows from adding three observables at each declustering stage; jet mass is also measured for sensitivity to the overall energy scale.Source: §2 · pp. 5, 7

3 Deep Learning Implementation

The study applies a neural network to jet mass and N-subjettiness observables, comparing minimal phase-space bases with standard discriminants and an overcomplete basis. Discrimination improves with higher-body observables until resolving 4-body phase space, with no improvement from 5- or 6-body information.

3 Deep Learning Implementation

  • The analysis uses simulated 13 TeV proton-proton events, anti-kT jets with R = 0.8 and pT ≥ 500 GeV, and particle-level observables without detector simulation.The signal is hadronically decaying Z bosons and the background is QCD jets; no mass cut is imposed on network inputs.Source: §3 · pp. 8–9
  • A single five-layer fully connected neural-network architecture is used across analyses, with 25 training runs per observable basis and the best model selected by area under the efficiency curve.Training uses dropout, early stopping, and Adam optimization.Source: §3 · p. 9
  • Discrimination improves with higher-body observables until 4-body phase space is resolved; 5- and 6-body observables add no improvement.Different network architectures and a boosted decision tree show the same saturation pattern.Source: §3 · pp. 10, 13
  • Standard N-subjettiness and energy-correlation ratios perform comparably to jet mass plus 3-body observables, while matching the 4-body basis requires additional 3- and 4-body information.The authors leave construction of these optimal observables for future work.Source: §3 · p. 10, Figure 5 · p. 12
  • The overcomplete 5-body basis does not improve discrimination over the minimal 4-body basis; its slight decrease is likely due to suboptimal training.The authors attribute the apparent decrease to the larger number of input observables.Source: §3 · p. 13, Figure 6 · p. 13

4 Conclusions

The paper uses a complete, minimal observable basis to make jet discrimination constructive, finding that useful Z-versus-QCD discrimination information is saturated at 4-body phase space. The conclusion also notes practical calculation challenges and possible extensions of the approach.

4 Conclusions

  • Useful discrimination information for boosted hadronic Z bosons versus light QCD jets is saturated by observables spanning 4-body, 8-dimensional phase space.The construction also permits observables that are non-zero for jets with four constituents to discriminate signal from background.Source: §4 · p. 14
  • The approach measures observables that completely and minimally span M-body phase space instead of feeding calorimeter energy deposits directly to the machine.Increasing the number of measured observables increases the effective resolution to emissions in the jet.Source: §4 · p. 14
  • An IRC-safe N-subjettiness basis is important for constructibility because its multi-observable cross sections can in principle be calculated in perturbative QCD.The paper notes that calculating these distributions in practice may be a significant challenge.Source: §4 · p. 14
  • Measuring relative angles and energy fractions preprocesses jet data, avoiding the rotation and rescaling often needed to compare image-based jet inputs.The observables are sensitive only to relative angles between particles and energy fractions.Source: §4 · pp. 14–15
  • Standard N-subjettiness ratios appear sensitive to most 3-body discrimination information, while studying how Z decays and QCD populate 4-body phase space could reveal remaining information.The paper anticipates applying the methods to quark-versus-gluon, QCD-versus-top, and multi-label jet classification.Source: §4 · p. 15

A Explicit Expressions for 3-Body Phase Space

The appendix presents explicit expressions for 1- and 2-subjettiness on a three-particle jet, treating 2-subjettiness before 1-subjettiness.

A Explicit Expressions for 3-Body Phase Space

  • The appendix gives explicit expressions for 1- and 2-subjettiness observables measured on a three-particle jet.It evaluates 2-subjettiness first, followed by 1-subjettiness.Source: A Explicit Expressions for 3-Body… · p. 15

A.1 2-subjettiness

For a three-particle jet, the 2-subjettiness construction uses two exclusive-kT axes and its observables determine a relative momentum fraction and pairwise angle.

A.1 2-subjettiness

  • With exclusive-kT axes and E-scheme recombination, one axis lies along particle 3, leaving particles 1 and 2 to contribute to 2-subjettiness around the other axis.The angles of particles 1 and 2 relative to that clustered axis are specified for the three-particle configuration.Source: §A.1 · p. 15, Figure 7 · p. 16
  • The 2-subjettiness values can be inverted to determine the relative momentum fraction z1z2/(z1+z2) and pairwise angle θ12.The displayed plane configuration labels the pairwise angles and momentum fractions of the three particles.Source: §A.1 · p. 16, Figure 7 · p. 16

A.2 1-subjettiness

The 1-subjettiness calculation locates the three-particle jet axis using momentum-weighted geometry, then expresses particle-to-axis angles and observables in terms of the configuration.

A.2 1-subjettiness

  • The calculation places particle 1 at the origin, particle 2 on the horizontal axis, and particle 3 at a generic point to determine angles to the net-momentum direction.The horizontal and vertical coordinates of particle 3 are calculated from the configuration.Source: §A.2 · p. 16
  • With E-scheme recombination, the jet axis is the momentum-weighted centroid of the three particles, using z3 = 1 − z1 − z2.The resulting angles from each particle to the jet axis are then specified.Source: §A.2 · p. 16
  • The three 1-subjettiness values are calculated from the particle-to-axis angles, with expressions simplified in terms of momentum fractions and pairwise angles.The setup uses the three pairwise angles θ12, θ13, and θ23 and the particle momentum fractions.Source: §A.2 · pp. 16–17

B Herwig Results

In the reduced Herwig sample, discrimination saturates with observables resolving 3- or 4-body phase space, though finite statistics are more evident than in Pythia.

B Herwig Results

  • The study uses Herwig 7.0.4 events generated in MadGraph, with 390,000 events each for pp → ZZ and pp → Z+ jet.The Herwig sample is about ten times smaller than the Pythia sample; 684,000 events were used for training, 76,000 for validation, and 20,000 for testing.Source: B Herwig Results · p. 17
  • Herwig jet-mass and one- versus two-prong validation distributions qualitatively agree with Pythia, but finite-statistics effects are more evident.The Herwig sample lacks the small shoulder of the Z-boson mass distribution seen in Pythia.Source: B Herwig Results · p. 17
  • Herwig discrimination saturates with observables sensitive to 3- or 4-body phase space, consistent with only limited useful information for distinguishing the jets.The authors note that the difference from Pythia's saturation point may reflect the smaller sample or differences between the parton showers.Source: B Herwig Results · p. 17

C Results with Other Architectures

The appendix presents discrimination results for a deeper neural network and a boosted decision tree, alongside Herwig jet-mass distributions.

C Results with Other Architectures

  • The appendix reports discrimination results for a neural network with one additional hidden layer and for a boosted decision tree.The passage introduces these results but does not state their outcomes.Source: C Results with Other Architectures · p. 17
  • Figure 8 compares Herwig jet-mass distributions for pp → Zj and hadronically decaying Z bosons from pp → ZZ.The plotted distributions use a minimum transverse momentum of 500 GeV and anti-kT jets with R = 0.8.Source: Figure 8 · p. 18
  • Figure 8 distinguishes the pp → Zj background in red dashed lines from the hadronic-Z signal in blue dotted lines.Source: Figure 8 · p. 18

C.1 A Deeper Neural Network

A deeper neural network confirms that discrimination is nearly saturated at 4-body phase space, with little additional gain from an overcomplete 5-body basis.

C.1 A Deeper Neural Network

  • The deeper network adds a 1000-node Dense layer immediately after the input layer, with dropout regularization of 0.2.It typically trained for about 50 epochs for each observable collection.Source: §C.1 · p. 18
  • The deeper network again finds saturation at 4-body phase space, while the overcomplete 5-body basis is only very slightly better than the 4-body basis.The authors interpret this as indicating that essentially all useful discrimination information has been extracted.Source: §C.1 · p. 18
  • Figures 11 and 12 show discrimination performance: adding observables to resolve higher-body phase space saturates at 4-body resolution, while the overcomplete 5-body basis is only very slightly better than the 4-body basis.Source: §C.1 · p. 18

C.2 Boosted Decision Tree

The boosted decision-tree results are consistent with discrimination saturating once 4-body phase space is resolved.

C.2 Boosted Decision Tree

  • The decision trees use ROOT's TMVA package and forests of 2500 trees; forests of 850 trees showed no significant improvement over 2500.Source: §C.2 · p. 18
  • Figures 10 and 11 plot signal efficiency against QCD rejection for neural-network observables resolving different phase-space orders; Figure 12 compares the overcomplete 5-body set with mass and minimal 3- and 4-body bases.The captions identify the comparisons but do not provide numerical values or establish a winner.Source: Figure 10 · p. 20, Figure 11 · p. 20, Figure 12 · p. 21
  • Boosted decision trees also show discrimination saturation once 4-body phase space is resolved.Source: §C.2 · p. 21
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