Lund Jet Plane: Insights & Applications

• Lund Jet Plane is a two-dimensional map of a jet’s branching structure, constructed by reclustering and declustering techniques to expose QCD radiation patterns.
• It quantitatively connects analytic QCD predictions with experimental measurements across diverse collision environments including proton–proton and heavy-ion collisions.
• The LJP offers a versatile framework for machine learning, jet tagging, and generative modeling, enhancing studies in both standard model and beyond-standard-model physics.

Searching arXiv for recent and foundational work on the Lund jet plane. The Lund jet plane (LJP) is a two-dimensional representation of the internal branching structure of a jet, constructed by declustering a jet’s constituents and mapping each resolved emission into logarithmic coordinates that encode angular scale and hardness. In contemporary usage, the term usually denotes the primary Lund jet plane, obtained by reclustering with the Cambridge–Aachen (C/A) algorithm and following the hardest branch through the declustering tree, so that each step contributes one point to the plane (Dreyer et al., 2018). The LJP occupies a distinctive position in jet substructure: it is simultaneously close to analytic QCD, experimentally measurable, and well suited to statistical, machine-learning, and generative-model applications (Lifson et al., 2020, Collaboration, 2020). Measurements in proton–proton, heavy-flavor, heavy-ion, and heavy-particle-decay environments have established it as a high-granularity probe of radiation patterns, while recent work has extended it to tagging, fast simulation, multimodal transformers, and beyond-Standard-Model shower studies (Collaboration, 2023, collaboration et al., 29 May 2025, Collaboration, 9 Feb 2026, Gouskos et al., 26 May 2026).

1. Definition and kinematic construction

The standard construction begins from a reconstructed jet, often found with anti-$k_t$, after which its constituents are reclustered with the Cambridge–Aachen algorithm, whose distance measure is purely angular. One then declusters the resulting binary tree step by step. At each declustering, the parent pseudojet splits into two branches, which are ordered by transverse momentum so that the harder branch is retained as the continuing emitter and the softer branch is treated as the emitted radiation (Dreyer et al., 2018, Carrazza et al., 2019, Collaboration, 2020).

For a declustering $p \to p_a + p_b$, with $p_{t,a} > p_{t,b}$, the fundamental kinematic quantities are the angular separation

$$\Delta_{ab} = \sqrt{(y_a-y_b)^2 + (\phi_a-\phi_b)^2},$$

the softer-branch momentum fraction

$z = \frac{p_{t,b}}{p_{t,a}+p_{t,b}},$

and a transverse-momentum-like hardness variable. A common choice is

$k_t = p_{t,b}\,\Delta_{ab},$

while some works write the equivalent soft-collinear form $k_T \simeq z\,p_T\,\Delta$, or $k_T = z(1-z)p_T\Delta$ depending on convention and approximation (Carrazza et al., 2019, Cavallini et al., 2021, Collaboration, 2023). The most widely used primary-plane coordinates are then

$\left(\ln\frac{1}{\Delta},\,\ln k_t\right),$

or equivalently $(\ln(R/\Delta R), \ln k_T)$ when the original jet radius $p \to p_a + p_b$0 is kept explicit (Dreyer et al., 2018, Collaboration, 2023, Collaboration, 2024).

A closely related coordinate system replaces $p \to p_a + p_b$1 by $p \to p_a + p_b$2, yielding a $p \to p_a + p_b$3 primary plane density that was used in early ATLAS measurements (Collaboration, 2020). Other variants, including $p \to p_a + p_b$4, $p \to p_a + p_b$5, or tuples such as $p \to p_a + p_b$6, appear when the plane is adapted to specific observables or to token-based neural architectures (Carrazza et al., 2019, Gouskos et al., 26 May 2026).

The essential structural distinction is between the primary Lund plane and possible secondary planes. The primary plane follows only the hardest branch at each step and thus records one ordered sequence of emissions per jet. Secondary Lund planes are obtained by reclustering and declustering a softer branch itself, thereby probing radiation off a primary emission rather than off the main hard line (Dreyer et al., 2018, Baldenegro et al., 2024).

2. Analytic meaning and QCD phase space

The appeal of the LJP is rooted in the structure of soft-collinear QCD. In the leading soft approximation, emissions are approximately uniform in logarithmic angle and logarithmic momentum scale, so the average density in the perturbative region is roughly flat, up to the running of $p \to p_a + p_b$7, flavor factors, and kinematic boundaries (Dreyer et al., 2018, Lifson et al., 2020). In this sense the plane is not merely a visualization device: it is a direct map of emission phase space.

For the average primary Lund plane density,

$p \to p_a + p_b$8

or analogously in $p \to p_a + p_b$9, the soft-collinear expectation is approximately proportional to $p_{t,a} > p_{t,b}$0 times a color factor (collaboration et al., 29 May 2025, Collaboration, 2023). CMS states the leading expectation as

$p_{t,a} > p_{t,b}$1

while the all-order analytic treatment of the primary density in QCD derives a single-logarithmic resummation with running-coupling, hard-collinear, and soft-clustering effects included (Collaboration, 2023, Lifson et al., 2020).

This phase-space interpretation makes the physical regions of the plane transparent. Large $p_{t,a} > p_{t,b}$2 corresponds to collinear emissions; small $p_{t,a} > p_{t,b}$3 to wide-angle emissions. Large $p_{t,a} > p_{t,b}$4 corresponds to hard, perturbative branchings, while low $p_{t,a} > p_{t,b}$5 approaches the non-perturbative domain in which hadronization and underlying event become important (Carrazza et al., 2019, Havener, 2021). Measurements by ATLAS, CMS, and ALICE all identify a bulk perturbative region, an enhancement as $p_{t,a} > p_{t,b}$6 decreases toward the confinement scale because of the running coupling, and eventual distortion or turnover in the deeply non-perturbative regime (Collaboration, 2020, Collaboration, 2023, Havener, 2021).

The all-order calculation of the primary density established several further analytic features. For C/A declustering, the primary Lund plane density receives single-logarithmic higher-order corrections from running coupling, hard-collinear DGLAP evolution of the leading parton, soft non-global and clustering logarithms, and boundary effects near the jet edge; it was matched to exact NLO and compared successfully with ATLAS data across the perturbative domain down to about $p_{t,a} > p_{t,b}$7 GeV (Lifson et al., 2020). That analysis also identified a new source of boundary-induced clustering logarithms when anti-$p_{t,a} > p_{t,b}$8 defines the jet but C/A defines the declustering, especially near $p_{t,a} > p_{t,b}$9 (Lifson et al., 2020).

3. Experimental realization and measurements

The LJP has been measured in multiple collider environments and jet categories, each exploiting a different aspect of the observable.

In inclusive proton–proton collisions, ATLAS measured the primary LJP using charged particles in $$\Delta_{ab} = \sqrt{(y_a-y_b)^2 + (\phi_a-\phi_b)^2},$$0 TeV data for anti-$$\Delta_{ab} = \sqrt{(y_a-y_b)^2 + (\phi_a-\phi_b)^2},$$1 $$\Delta_{ab} = \sqrt{(y_a-y_b)^2 + (\phi_a-\phi_b)^2},$$2 jets with leading-jet $$\Delta_{ab} = \sqrt{(y_a-y_b)^2 + (\phi_a-\phi_b)^2},$$3 GeV, correcting the double-differential distribution in $$\Delta_{ab} = \sqrt{(y_a-y_b)^2 + (\phi_a-\phi_b)^2},$$4 to charged-particle level (Collaboration, 2020). The measurement showed the expected perturbative triangular structure, an enhancement toward the perturbative–non-perturbative boundary, and an average of

$$\Delta_{ab} = \sqrt{(y_a-y_b)^2 + (\phi_a-\phi_b)^2},$$5

in the fiducial region (Collaboration, 2020). No single generator described the full plane.

CMS later measured the primary LJP density in $$\Delta_{ab} = \sqrt{(y_a-y_b)^2 + (\phi_a-\phi_b)^2},$$6 of $$\Delta_{ab} = \sqrt{(y_a-y_b)^2 + (\phi_a-\phi_b)^2},$$7 TeV proton–proton data for jets with $$\Delta_{ab} = \sqrt{(y_a-y_b)^2 + (\phi_a-\phi_b)^2},$$8 or $$\Delta_{ab} = \sqrt{(y_a-y_b)^2 + (\phi_a-\phi_b)^2},$$9, $z = \frac{p_{t,b}}{p_{t,a}+p_{t,b}},$0 GeV, and $z = \frac{p_{t,b}}{p_{t,a}+p_{t,b}},$1, using charged-particle tracks and unfolding the distribution in $z = \frac{p_{t,b}}{p_{t,a}+p_{t,b}},$2 to stable-particle level (Collaboration, 2023). The observable was interpreted directly as the average number of emissions per jet per unit area in the log-plane, with total uncertainties typically at the $z = \frac{p_{t,b}}{p_{t,a}+p_{t,b}},$3–$z = \frac{p_{t,b}}{p_{t,a}+p_{t,b}},$4 level in the bulk and larger near the kinematic edge (Collaboration, 2023).

ALICE measured the primary Lund plane density in inclusive charged-particle jets with

$z = \frac{p_{t,b}}{p_{t,a}+p_{t,b}},$5

at $z = \frac{p_{t,b}}{p_{t,a}+p_{t,b}},$6 TeV, unfolding in jet $z = \frac{p_{t,b}}{p_{t,a}+p_{t,b}},$7, $z = \frac{p_{t,b}}{p_{t,a}+p_{t,b}},$8, and $z = \frac{p_{t,b}}{p_{t,a}+p_{t,b}},$9 (Havener, 2021). This lower-$k_t = p_{t,b}\,\Delta_{ab},$0 regime is especially sensitive to hadronization and underlying-event effects and therefore complements the high-$k_t = p_{t,b}\,\Delta_{ab},$1 ATLAS and CMS measurements (Havener, 2021).

ATLAS subsequently measured the LJP in hadronic top-quark and $k_t = p_{t,b}\,\Delta_{ab},$2-boson decays in semileptonic $k_t = p_{t,b}\,\Delta_{ab},$3 events using $k_t = p_{t,b}\,\Delta_{ab},$4 of $k_t = p_{t,b}\,\Delta_{ab},$5 TeV data (Collaboration, 2024). Here the observable was constructed from charged particles inside trimmed anti-$k_t = p_{t,b}\,\Delta_{ab},$6 $k_t = p_{t,b}\,\Delta_{ab},$7 jets with $k_t = p_{t,b}\,\Delta_{ab},$8 GeV. The measurement found a globally acceptable description for top jets from several generators, with Sherpa 2.2.10 performing best, but reported that all predictions are incompatible with the measured $k_t = p_{t,b}\,\Delta_{ab},$9-jet plane over the full fiducial region (Collaboration, 2024). The average emission multiplicities were reported as

$k_T \simeq z\,p_T\,\Delta$0

for top jets and

$k_T \simeq z\,p_T\,\Delta$1

for $k_T \simeq z\,p_T\,\Delta$2 jets (Collaboration, 2024).

Heavy-flavor measurements have used the LJP to isolate mass effects. LHCb presented the first measurement of the Lund plane for light-quark-enriched and beauty-initiated jets at $k_T \simeq z\,p_T\,\Delta$3 TeV, using both a $k_T \simeq z\,p_T\,\Delta$4-Lund plane and a $k_T \simeq z\,p_T\,\Delta$5-Lund plane (collaboration et al., 29 May 2025). The analysis directly observed a depletion of collinear radiation in beauty jets consistent with the dead-cone angle

$k_T \simeq z\,p_T\,\Delta$6

and identified this as the first direct observation of the beauty dead-cone effect on the LJP (collaboration et al., 29 May 2025).

The LJP has also entered heavy-ion studies. CMS measured the angular distribution of the hardest primary-Lund emission in fixed $k_T \simeq z\,p_T\,\Delta$7 slices for jets with $k_T \simeq z\,p_T\,\Delta$8 and $k_T \simeq z\,p_T\,\Delta$9 GeV in pp and PbPb collisions at $k_T = z(1-z)p_T\Delta$0 TeV (Collaboration, 9 Feb 2026). Using the formation-time estimate

$k_T = z(1-z)p_T\Delta$1

the measurement targeted early, high-$k_T = z(1-z)p_T\Delta$2 splittings and found no significant difference between the pp and PbPb angular distributions for the high-$k_T = z(1-z)p_T\Delta$3 region within uncertainties, consistent with those emissions occurring before substantial interaction with the QGP (Collaboration, 9 Feb 2026).

4. Observable relations, grooming, and phase-space diagnostics

The LJP is not an isolated construction: many standard jet-substructure observables are functionals of it. The original LJP paper emphasized that observables such as the groomed mass, $k_T = z(1-z)p_T\Delta$4, and the iterated soft-drop multiplicity correspond to selecting or integrating specific regions of the primary plane (Dreyer et al., 2018).

In the soft-drop procedure with condition

$k_T = z(1-z)p_T\Delta$5

the first primary declustering satisfying the cut defines the groomed splitting. The associated $k_T = z(1-z)p_T\Delta$6 and mass can therefore be read off from a point in the primary Lund plane above the soft-drop line (Dreyer et al., 2018). Similarly, the iterated soft-drop multiplicity counts the number of primary declusterings passing the same cut and is therefore an integral of the Lund density over the corresponding phase-space band (Dreyer et al., 2018).

The LJP also clarifies the action of grooming algorithms geometrically. In the ATLAS top and $k_T = z(1-z)p_T\Delta$7 study, trimming produced a visible depletion in the soft, wide-angle region of the plane, bounded approximately by $k_T = z(1-z)p_T\Delta$8 and $k_T = z(1-z)p_T\Delta$9, making the grooming-induced phase-space excision directly apparent (Collaboration, 2024).

Because the plane factorizes regions associated with perturbative showering, hadronization, and underlying event, it is an unusually sharp diagnostic for Monte Carlo generators. ATLAS, CMS, and ALICE all exploited this by comparing multiple generators and tunes, finding localized mismodeling that depends strongly on region: hard wide-angle regions probe matrix elements and shower ordering; low-$\left(\ln\frac{1}{\Delta},\,\ln k_t\right),$0 collinear regions probe hadronization; soft wide-angle regions probe UE and MPI (Collaboration, 2020, Havener, 2021, Collaboration, 2023). This localization is one reason the LJP has been proposed as a useful basis for generator tuning (Collaboration, 2020, Collaboration, 2024).

5. Jet tagging and machine learning on the Lund plane

The LJP has become an important low-level representation for classification tasks. The original paper demonstrated boosted electroweak-boson tagging using both machine learning and an explicitly interpretable log-likelihood based on approximately decorrelated plane regions (Dreyer et al., 2018). The central result was that much of the performance of sequence- or image-based models could be reproduced by a transparent additive likelihood built from the leading splitting and the density of non-leading emissions, implying that the key information used by ML is physically localizable in the plane (Dreyer et al., 2018).

Higgs tagging studies extended this approach. One line of work used $\left(\ln\frac{1}{\Delta},\,\ln k_t\right),$1 primary-Lund images as CNN inputs for $\left(\ln\frac{1}{\Delta},\,\ln k_t\right),$2 and $\left(\ln\frac{1}{\Delta},\,\ln k_t\right),$3 against QCD backgrounds in moderate- and high-boost regimes (Khosa et al., 2021, Khosa, 2021). In those analyses, the LJP-based CNN outperformed the jet color ring for $\left(\ln\frac{1}{\Delta},\,\ln k_t\right),$4, where the color-ring observable was nearly non-discriminating, and modestly improved on it for $\left(\ln\frac{1}{\Delta},\,\ln k_t\right),$5, especially at high boost (Khosa et al., 2021, Khosa, 2021). One of the papers further observed that restricting the input to the perturbative region $\left(\ln\frac{1}{\Delta},\,\ln k_t\right),$6 changed the performance measure by less than $\left(\ln\frac{1}{\Delta},\,\ln k_t\right),$7, suggesting that the classifier relied predominantly on perturbative features (Khosa et al., 2021).

A related analysis of $\left(\ln\frac{1}{\Delta},\,\ln k_t\right),$8 tagging combined a CNN on $\left(\ln\frac{1}{\Delta},\,\ln k_t\right),$9 LJP images with high-level color-sensitive observables such as pull, color ring, and $(\ln(R/\Delta R), \ln k_T)$0 (Cavallini et al., 2021). It reported the following AUC values:

Input set	AUC (truth)	AUC (reco)
CS observables	0.826	0.788
$(\ln(R/\Delta R), \ln k_T)$1+CR only	0.817	0.787
CNN (Lund only)	0.876	0.828
CS + CNN	0.893	0.846

These results established that the LJP alone is a powerful color-sensitive representation, and that the combination of a Lund-plane CNN score with theory-driven observables yields further gains (Cavallini et al., 2021). The same study also emphasized an important caveat: the CNN score was strongly correlated with the invariant mass of the $(\ln(R/\Delta R), \ln k_T)$2 system, similarly to $(\ln(R/\Delta R), \ln k_T)$3, which limits direct use in mass-agnostic $(\ln(R/\Delta R), \ln k_T)$4 searches unless decorrelation techniques are applied (Cavallini et al., 2021).

Recent work has tested whether explicit Lund information remains useful in the transformer era. The multimodal PLuM architecture projects both particle constituents and Lund-plane splittings into a shared latent space and processes them jointly with a unified transformer (Gouskos et al., 26 May 2026). Using up to $(\ln(R/\Delta R), \ln k_T)$5 Lund tokens per jet, each represented by

$(\ln(R/\Delta R), \ln k_T)$6

PLuM achieved systematic improvements over the particle-only ParT baseline for top and $(\ln(R/\Delta R), \ln k_T)$7 tagging (Gouskos et al., 26 May 2026). For $(\ln(R/\Delta R), \ln k_T)$8 vs QCD, the background rejection at $(\ln(R/\Delta R), \ln k_T)$9 signal efficiency improved from $p \to p_a + p_b$00 to $p \to p_a + p_b$01, while for top vs QCD it improved from $p \to p_a + p_b$02 to $p \to p_a + p_b$03 (Gouskos et al., 26 May 2026). The absence of comparable gains for $p \to p_a + p_b$04 and $p \to p_a + p_b$05 was interpreted as evidence that explicit hierarchical information remains complementary particularly for $p \to p_a + p_b$06-jet-rich topologies (Gouskos et al., 26 May 2026).

6. Generative modeling, domain transfer, and sample morphing

The LJP has also become a generative-modeling substrate because its pixelized or tokenized form is much more structured than raw detector images. A notable early example used $p \to p_a + p_b$07 binary or probabilistic Lund images derived from the primary plane, trained generative models on them, and compared several architectures (Carrazza et al., 2019).

In that work, the main model was a least-squares GAN called gLund, trained on averaged probabilistic Lund images with $p \to p_a + p_b$08, ZCA whitening, and pixel intensities mapped from $p \to p_a + p_b$09 to $p \to p_a + p_b$10 (Carrazza et al., 2019). The generated average Lund plane reproduced the reference density to 3–5% in the bulk region, while a VAE baseline deviated by up to $p \to p_a + p_b$11 (Carrazza et al., 2019). The study further showed that derived observables reconstructed from the images, including activated-pixel counts, soft-drop multiplicity, and an mMDT-based groomed mass proxy,

$p \to p_a + p_b$12

were well reproduced by the LSGAN and WGAN-GP models (Carrazza et al., 2019).

The same paper introduced CycleJet, a CycleGAN-based framework for unpaired mappings between different jet domains, including parton-level to detector-level and QCD to boosted-$p \to p_a + p_b$13 images (Carrazza et al., 2019). The standard cycle-consistency objective was used,

$p \to p_a + p_b$14

with $p \to p_a + p_b$15 and an identity-loss factor $p \to p_a + p_b$16 (Carrazza et al., 2019). The proposed use case was retroactive modification of existing MC samples: one could take stored parton-level jets, convert them to Lund images, and map them approximately to detector level without rerunning the full detector simulation (Carrazza et al., 2019).

More recent generative and classification studies in dark-sector showering have used the LJP because it factorizes perturbative and non-perturbative regions spatially. One paper on dark sector showers used the number of primary-Lund emissions above a $p \to p_a + p_b$17 cut as a robust observable and showed that LJP-based observables had better resilience to hadronization-model changes than jet mass, constituent multiplicity, or energy-sharing observables (Cohen et al., 2023). Another paper on dark gauge symmetries used primary-Lund declusterings as input to a Neural Sorter Mamba Network, with per-node features $p \to p_a + p_b$18 plus the emitted-branch four-momentum, and showed that the perturbative footprints of different gauge groups could be separated efficiently (Li et al., 24 Jun 2026).

7. Variants, secondary planes, and specialized physics applications

Although the primary plane is the canonical object, a range of specialized applications rely on modifications or extensions.

One recent proposal uses the secondary Lund plane to isolate a high-purity sample of gluon-initiated jets (Baldenegro et al., 2024). The logic is that if one identifies a soft branch of an asymmetric, near-collinear splitting, that branch is often a gluon because of the soft singularity in $p \to p_a + p_b$19 and $p \to p_a + p_b$20. In a practical dijet strategy with anti-$p \to p_a + p_b$21 $p \to p_a + p_b$22 jets, requiring

$p \to p_a + p_b$23

the primary Lund plane of the subleading jet acts as an effective secondary plane and yields a gluon fraction around 90% according to fixed-order and hadron-level studies (Baldenegro et al., 2024).

The LJP has also been used to probe mass-dependent radiation in heavy flavor. In LHCb’s beauty-jet measurement, the primary branch was followed differently depending on the sample: for light-quark jets, the harder branch was followed; for heavy-flavor jets, the branch containing the reconstructed heavy hadron was followed (collaboration et al., 29 May 2025). This made the primary sequence a direct probe of the heavy quark’s radiation history and exposed the suppression of hard-collinear branchings in beauty jets (collaboration et al., 29 May 2025).

In heavy-ion collisions, the primary plane has been used in a deliberately reduced form: CMS selected, for each jet, the single emission with highest $p \to p_a + p_b$24 and studied its angular distribution in fixed $p \to p_a + p_b$25 intervals (Collaboration, 9 Feb 2026). This is not a full density measurement, but it is directly motivated by the same primary-Lund construction and uses the plane as a formation-time-resolved basis (Collaboration, 9 Feb 2026).

Dark-sector shower studies likewise exploit the plane’s phase-space separation. One analysis of dark showers in the LJP showed that hadronization-induced differences are localized near the confinement scale $p \to p_a + p_b$26, while the perturbative region at higher $p \to p_a + p_b$27 is comparatively stable (Cohen et al., 2023). Another analysis of arbitrary dark gauge groups found that exact massive kinematics generate distinctive LJP boundaries, dead-cone thresholds, and even a bifurcation near $p \to p_a + p_b$28 for massive dark gauge bosons (Li et al., 24 Jun 2026).

8. Limitations, caveats, and open directions

Despite its versatility, the LJP has several well-defined limitations.

A first limitation is representation loss. Pixelized Lund images necessarily coarse-grain the declustering sequence. In the generative-model study, $p \to p_a + p_b$29 images recorded only binary occupancy, so multiple emissions falling in the same pixel were not distinguished in the baseline representation (Carrazza et al., 2019). Similar issues arise in $p \to p_a + p_b$30 image-based Higgs taggers (Cavallini et al., 2021, Khosa, 2021). This can affect observables sensitive to fine multi-emission correlations or dense regions near phase-space boundaries (Carrazza et al., 2019).

A second limitation is non-perturbative sensitivity. The low-$p \to p_a + p_b$31 region carries information but also depends strongly on hadronization, underlying event, and detector reconstruction. Analytic predictions currently degrade from $p \to p_a + p_b$32–$p \to p_a + p_b$33 precision at high transverse momenta to about $p \to p_a + p_b$34 near the lower edge of the perturbative region, around $p \to p_a + p_b$35 GeV (Lifson et al., 2020). Experimental measurements likewise find larger generator discrepancies in soft-wide-angle and soft-collinear regions (Collaboration, 2020, Havener, 2021, Collaboration, 2023).

A third issue is boundary and clustering effects. Near $p \to p_a + p_b$36, the interplay of anti-$p \to p_a + p_b$37 jet finding and C/A declustering produces boundary logarithms whose full resummation remains incomplete (Lifson et al., 2020). This has direct implications for the largest-angle bins in proton–proton measurements (Lifson et al., 2020).

A fourth concern is task-specific bias. In $p \to p_a + p_b$38 tagging, the Lund-plane CNN score was found to be notably correlated with the invariant mass of the $p \to p_a + p_b$39 system, which is problematic for generic $p \to p_a + p_b$40 tagging (Cavallini et al., 2021). This is not a pathology of the plane itself, but it shows that physically meaningful coordinates do not automatically imply decorrelated discriminants.

Finally, the primary plane does not encode the full jet tree. For tasks involving dense or intricate topologies, such as four-prong decays or secondary heavy-flavor structure, additional information in secondary planes or richer tree-based representations may be needed (Dreyer et al., 2018, Gouskos et al., 26 May 2026).

Several future directions are therefore recurrent across the literature: higher-resolution or multi-channel images; sparse or point-cloud architectures that avoid rasterization; explicit secondary-plane or full-tree representations; tighter integration with perturbative QCD constraints; and broader use of the plane for generator tuning, domain adaptation, and multimodal learning (Carrazza et al., 2019, Dreyer et al., 2018, Gouskos et al., 26 May 2026).

9. Summary

The Lund jet plane organizes the emissions inside a jet into a logarithmic map of angle and hardness, constructed experimentally by C/A reclustering and hardest-branch declustering (Dreyer et al., 2018). In its primary form, it provides a perturbatively meaningful emission density, a flexible basis for observables and grooming procedures, and a high-resolution probe of radiation patterns across perturbative and non-perturbative regimes (Lifson et al., 2020).

Its importance derives from a rare combination of properties: it is analytically calculable to high precision in QCD (Lifson et al., 2020), experimentally measurable in diverse environments (Collaboration, 2020, Collaboration, 2023, Havener, 2021, Collaboration, 2024, collaboration et al., 29 May 2025, Collaboration, 9 Feb 2026), interpretable enough to support likelihood-based tagging (Dreyer et al., 2018), and expressive enough to improve deep-learning systems even when constituent-level transformers are already strong (Gouskos et al., 26 May 2026). Applications now span boosted-object tagging, heavy-flavor physics, heavy-ion jet quenching, fast simulation, domain transfer, dark-sector phenomenology, and generator tuning (Carrazza et al., 2019, Cohen et al., 2023, Li et al., 24 Jun 2026).

In that sense, the LJP has become both a measurement observable and a representation language for jet substructure: one that links analytic QCD, experimental reconstruction, and modern machine learning in a common phase-space framework.