Lorentz-Boosted Hadronic Top Quarks

• The paper demonstrates that Lorentz-boosted top quarks, with highly collimated decay products, can be robustly identified using fat-jet algorithms and tailored grooming techniques to preserve jet mass accuracy.
• Key methods include the use of substructure observables such as N-subjettiness, energy correlation functions, and advanced machine learning taggers to efficiently distinguish signal from background.
• Implications of these techniques extend to precision top quark mass measurements, differential cross-section studies, and searches for new physics in high-energy collider experiments.

A Lorentz-boosted hadronically decaying top quark is a top quark produced with transverse momentum $p_T$ substantially larger than its rest mass $m_t$ ($\approx$173 GeV), such that all three of its hadronic decay products ($t\rightarrow bW\rightarrow bq\bar{q}'$) are collimated into a single high-$p_T$ jet. Identification and measurement of such top quarks at hadron colliders are central for precision Standard Model studies and new-physics searches at high energy scales.

1. Kinematics and Topology of Lorentz-Boosted Hadronic Top Decays

The critical kinematic parameter for boosted tops is the Lorentz boost factor,

$$\gamma = \frac{E_t}{m_t} = \frac{\sqrt{p_T^2 + m_t^2}}{m_t}.$$

For $p_T\sim1$ TeV, $\gamma\gtrsim6$. The decay products are confined to a cone of

$\Delta R\sim\frac{2m_t}{p_T},$

which results in typical angular separations of $\Delta R\sim0.3$ for $p_T\sim1$ TeV (Auerbach et al., 2013). This forces all three decay prongs ($b$, $q$, $q'$) into a region much smaller than the usual jet radii ($R=0.4$), necessitating specialized "fat-jet" ($R=0.6$–$1.5$) reconstruction to capture the complete top decay.

The resulting jet’s invariant mass,

$m_j = \sqrt{E_j^2 - \lvert \vec{p}_j \rvert^2},$

peaks near $m_t\approx 180$ GeV (including detector resolution, missing energy from neutrinos or semileptonic $W$ decays, and associated radiation).

2. Experimental Identification and Tagging Strategies

The identification of boosted hadronic tops leverages:

• Jet reconstruction: Anti-$k_T$ or Cambridge–Aachen algorithms with large radius parameters (typically $R=0.8$–$1.5$) (Schätzel, 2014, Collaboration, 2015, Collaboration, 2022). Variable-$R$ schemes, where $R(p_T)\propto 1/p_T$, are used in certain algorithms (e.g., HOTVR) to optimize capture of the decay at all $p_T$ (Lapsien et al., 2016).
• Jet grooming: Pruning, trimming, and soft-drop/mMDT removal of soft, wide-angle radiation to stabilize jet mass and mitigate pileup/underlying event effects (Schätzel, 2014, Collaboration, 2022, Lapsien et al., 2016). Parameters (e.g. soft-drop $z_{\text{cut}}$, $\beta$) and grooming radius are tuned to maintain signal efficiency and mitigate mass sculpting.
• Substructure discrimination:
  • N-subjettiness ($\tau_N$): Measures degree to which a jet possesses $N$ subjets. The ratio $\tau_{32} = \tau_3/\tau_2$ is minimized for true three-prong top decays; typical selection is $\tau_{32} < 0.6$ (Kuutmann, 2014, Conway et al., 2016, Collaboration, 2016).
  • Energy correlation functions (ECFs): Ratios such as $N_3^{(\beta)}$ probe three-prong substructure and are highly effective in backgrounds with different radiation patterns (Collaboration, 2020, Collaboration, 2018).
  • HEPTopTagger variables: Jet mass windows, pairwise subjet masses consistent with $W$ and top kinematics, subjet pairings. A frequently used mass window is 140–210 GeV for the full jet and 65–95 GeV for the $W$ candidate (Schätzel, 2014, Collaboration, 2015).
  • Mass-drop and splitting scales: $k_T$ splitting scales ($\sqrt{d_{12}}$) and mass-drop criteria ($\mu$, $y_{\text{cut}}$) are used to test compatibility with boosted top topology (Schätzel, 2014).
• $b$-tagging in boosted environments: Performed either on subjets inside the fat-jet (using information such as CSVv2, DeepJet, or impact parameter significances) or via specialized algorithms in the jet center-of-mass frame. Subjet $b$-tagging gives critical background suppression, often by an order of magnitude, while maintaining $\sim$40% efficiency (Chen, 2013, Auerbach et al., 2013, Collaboration, 2016).
• Multivariate and Machine Learning (ML) Taggers:
  • Boosted-Decision Trees (BDTs): Combine substructure variables and $b$-tag to optimize separation (Collaboration, 2018, Collaboration, 2020).
  • DNN/CNN/LoLa architectures: Work on low-level four-vector (PF candidate) inputs or jet images, incorporating Lorentz symmetries (e.g., DeepAK8, LoLa) and achieving up to an order of magnitude background reduction relative to cut-based taggers at fixed efficiency (Butter et al., 2017, Collaboration, 2020).
  • Boosted Event Shape Tagger (BEST): Uses event shapes calculated in multiple Lorentz-boosted frames (for $t$, $W$, $Z$, $H$) as neural network input, further enhancing multi-class separation (Conway et al., 2016, Collaboration, 2020).

3. Performance Metrics and Systematics

Performance is quantified by tagging efficiency (signal acceptance) and background rejection (inverse mistag rate), both as functions of $p_T$ and jet characteristics.

Technique/Tagger	Signal Eff. ($\epsilon_t$)	Background Rejection ($1/\epsilon_{\text{bkg}}$)	Comments
Cut-based (e.g., $\tau_{32}+\text{mass}$)	$\sim$30–50%	$\sim 10$–$40$	$p_T > 400$ GeV typical
HOTVR	35–48%	12–25	Robust up to several TeV
ECF+BDT	30–50%	30–50	$N_3^{(\beta)}$, $b$-tag, BDT
CNN/DNN (ImageTop, DeepAK8)	up to 70%	$20\times$ higher than legacy	Multiclass (top, $W$, $Z$, $H$, QCD)

Systematic uncertainties are dominated by jet energy/mass calibration (JES/JMS), subjet $b$-tag scale factors, modeling of QCD radiation (ISR/FSR), pileup mitigation effectiveness, and MC generator parton-shower/tune choices. In data/MC comparisons, scale factors and uncertainties are derived in control regions (e.g., semileptonic $t\bar t$, $Z$+jets, multijets) for validation (Collaboration, 2020, Collaboration, 2015).

4. Applications: Precision Measurements and New Physics Searches

Lorentz-boosted hadronic top identification is pivotal for:

• Differential cross-section and top-mass measurements: The unfolded $m_{\text{jet}}$ and $p_T$ spectra of boosted top candidates are used to measure $d\sigma/dm_{\text{jet}}$, $d\sigma/dp_T$, and extract $m_t$ with precision. For example, (Collaboration, 2022) achieved $m_t=173.06\pm0.84$ GeV using XCone-reconstructed jets at $p_T>400$ GeV, exploiting in-situ $W$-mass calibration and substructure to constrain FSR uncertainties.
• BSM searches: High-mass resonances decaying to $t\bar t$ or monotop signatures necessitate efficient rejection of QCD backgrounds in the high-$p_T$ regime. Improved taggers—cut-based, BDT, or DNN—significantly enhance sensitivity and extend the reach to $M_{Z'}, M_{\phi},$ or $M_T$ in the TeV scale (Collaboration, 2018, Collaboration, 2015, Collaboration, 2016).
• SUSY and Exotics: In SUSY cascades and vector-like quark searches, Lorentz-boosted tops arise from heavy state decays with $p_T\sim300$–1000 GeV, making boosted top tagging essential for both background suppression and signal significance (Bandyopadhyay et al., 2010, Collaboration, 2016).
• Control of backgrounds and systematic effects: Event selection and tagging are designed to provide data-driven control over dominant backgrounds ($W/Z+$jets, $t\bar t$ lepton+jets, QCD multijets), and pileup effects are mitigated by grooming and PUPPI weighting.

5. Evolution of Tagging Algorithms

Historically, cut-based approaches using jet mass windows, splitting scales, and simple $\tau_{32}$ thresholds were standard (Schätzel, 2014, Kuutmann, 2014). Modern analyses rely increasingly on multivariate taggers:

• Energy Correlation Functions (ECF): E.g., $N_3^{(\beta)}$ provides robust QCD discrimination and, when decorrelated with jet mass, reduces classifier-induced sculpting.
• Variable-$R$ clustering and mass-jump veto: HOTVR achieves stable performance over a broad $p_T$ range by dynamically adjusting $R$ and freezing out genuine massive subjets (Lapsien et al., 2016).
• Deep Learning: Approaches using constituent four-vectors, convolutional filters (jet images), or Lorentz-invariant layers outperform legacy taggers, especially at the highest boosts. DeepAK8 and similar architectures gain up to a factor of 20 in background suppression at fixed efficiency, crucial for future high-luminosity scenarios (Collaboration, 2020, Butter et al., 2017).

6. Special Techniques: Subjet $b$-Tagging and Rest-Frame Methods

Standard $b$-tagging loses power at high $p_T$ due to collimation; associating tracks to subjets in the jet rest frame and using signed impact parameter significances yields an order of magnitude better background rejection at fixed signal efficiency, stable even in high pileup (Chen, 2013).

Additionally, methods leveraging rest-frame reclustering, energy asymmetry, and opening angle variables, combined in boosted decision trees, optimize discrimination between $t\rightarrow bW\rightarrow bqq'$ and QCD (Chen, 2013). These are complementary to substructure-cut and ML-based methods.

7. Outlook and Significance

Lorentz-boosted hadronic top quark reconstruction and tagging are now central in precision Standard Model and BSM analyses at the LHC. Tagging techniques—including advanced grooming, substructure observables, multivariate (BDT, DNN) architectures, and sophisticated $b$-tagging—decisively improve selection purity, signal sensitivity, and systematic control across a wide $p_T$ range. These developments are key enablers for analyses in high-luminosity runs and are broadly adopted or adapted by ATLAS, CMS, and theoretical groups (Auerbach et al., 2013, Kuutmann, 2014, Schätzel, 2014, Butter et al., 2017, Collaboration, 2022, Collaboration, 2020, Lapsien et al., 2016).