Lund Jet Plane: QCD Radiation Insights

• Lund Jet Plane is a two-dimensional representation that maps jet radiation via iterative declustering, exposing soft-collinear QCD phase space.
• It distinguishes perturbative branchings from nonperturbative effects, facilitating precise measurements and calibration of jet substructure.
• Its applications span Monte Carlo validation, jet tagging, and machine learning, providing a unified framework for diverse QCD analyses.

Searching arXiv for recent and foundational papers on the Lund jet plane to ground the article in the literature. The Lund jet plane is a two-dimensional representation of the radiation pattern inside a reconstructed jet, obtained by iteratively declustering the jet and mapping each splitting into logarithmic coordinates that expose soft-collinear QCD phase space. Introduced as a per-jet realization of Lund diagrams, it provides a differential view of perturbative branchings, hadronization, and underlying-event contributions, and it has become a common framework for precision measurements, Monte Carlo validation, jet tagging, and theory-driven machine learning in jet substructure (Dreyer et al., 2018, Collaboration, 2020, Collaboration, 2023).

1. Definition and declustering procedure

The basic object is a jet reconstructed with a standard algorithm such as anti-$k_t$, whose constituents are then reclustered with the Cambridge–Aachen (C/A) algorithm. C/A is used because it orders mergings by angle, so undoing its clustering history exposes the angular hierarchy of the shower. At each declustering step, the parent splits into a harder branch and a softer branch. One records the splitting angle, the momentum-sharing fraction, and the relative transverse momentum of the softer branch,

$$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$

with the small-angle approximation used where appropriate (Dreyer et al., 2018, Havener, 2021).

The most common coordinates are

$$x=\ln\frac{1}{\Delta R}\quad \text{or}\quad \ln\frac{R}{\Delta R}, \qquad y=\ln k_T,$$

although some measurements and tagging studies use $\ln(1/z)$ as the vertical coordinate instead of $\ln k_T$ (Havener, 2021, Collaboration, 2024). The logarithmic mapping stretches the soft and collinear regions and makes the dominant QCD singularities visually explicit.

The primary Lund jet plane is obtained by following only the harder branch at each declustering step and recording the softer branch. This produces an ordered sequence of primary emissions. The full Lund tree retains every declustering node in the C/A tree, while secondary Lund planes are built by taking one of the softer primary branches and declustering it in the same way (Dreyer et al., 2018, Dreyer et al., 2021, Baldenegro et al., 2024).

2. Emission density, perturbative structure, and coordinate variants

The central observable is the average density of declusterings per jet,

$$\rho(\Delta,k_T) = \frac{1}{N_{\rm jets}} \frac{d^2N_{\rm emissions}}{d\ln(1/\Delta)\,d\ln k_T},$$

or, equivalently, with $d\ln(R/\Delta)$ in place of $d\ln(1/\Delta)$ (Collaboration, 2020, Collaboration, 2023). In the soft-collinear limit, the Lund plane is approximately uniformly populated: at leading logarithmic order the density is constant up to the running of $\alpha_s$ and the color factor of the initiating parton. In this sense, the plane is a direct map of the singular structure of QCD radiation (Dreyer et al., 2018, Cohen et al., 2023).

Beyond this leading picture, the distortions of the plane encode physically distinct effects. Running coupling introduces a $k_T$-dependence; hard-collinear evolution changes the flavor and momentum fraction of the leading branch; and soft large-angle radiation generates non-global and clustering logarithms. A key theoretical result of the original construction is that C/A declustering keeps these corrections at most single-logarithmic, whereas $$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$0- or anti-$$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$1-based declustering introduces double-logarithmic contamination in the plane (Dreyer et al., 2018, Lifson et al., 2020).

Because the axes are logarithmic, different sectors of the plane are associated with different dynamics. Low $$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$2 and large $$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$3 identify an underlying-event-dominated region; high $$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$4 and moderate $$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$5 correspond to perturbative branchings; and very low $$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$6 at small angle is where hadronization corrections are important (Havener, 2021). In high-$$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$7 inclusive CMS data, the unfolded density exhibits a perturbative plateau at large $$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$8, a rapid rise toward small $$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$9 from the running of $$x=\ln\frac{1}{\Delta R}\quad \text{or}\quad \ln\frac{R}{\Delta R}, \qquad y=\ln k_T,$$0, and a kinematical edge set by $$x=\ln\frac{1}{\Delta R}\quad \text{or}\quad \ln\frac{R}{\Delta R}, \qquad y=\ln k_T,$$1 (Collaboration, 2023).

The primary Lund sequence also subsumes several standard substructure observables. The soft-drop momentum fraction $$x=\ln\frac{1}{\Delta R}\quad \text{or}\quad \ln\frac{R}{\Delta R}, \qquad y=\ln k_T,$$2, groomed mass, and iterated soft-drop multiplicity can all be read off from the same primary declustering history, so the Lund plane is not merely a visualization but a unifying representation underlying multiple jet observables (Dreyer et al., 2018).

3. Experimental measurements and detector-level reconstruction

The Lund jet plane has moved from proposal to a broad experimental program spanning inclusive jets, heavy-particle decays, heavy-ion collisions, and heavy-flavor jets. Measurements generally reconstruct jets, recluster their constituents with C/A, extract primary declusterings, and correct detector effects with response matrices and iterative unfolding (Collaboration, 2020, Havener, 2021, Collaboration, 2023).

Measurement	Kinematic regime	Main result
ATLAS inclusive pp (Collaboration, 2020)	anti-$$x=\ln\frac{1}{\Delta R}\quad \text{or}\quad \ln\frac{R}{\Delta R}, \qquad y=\ln k_T,$$3, $$x=\ln\frac{1}{\Delta R}\quad \text{or}\quad \ln\frac{R}{\Delta R}, \qquad y=\ln k_T,$$4, $$x=\ln\frac{1}{\Delta R}\quad \text{or}\quad \ln\frac{R}{\Delta R}, \qquad y=\ln k_T,$$5 GeV	No single model is found to be in agreement with the measured data across the entire plane
ALICE inclusive pp (Havener, 2021)	charged-particle jets, $$x=\ln\frac{1}{\Delta R}\quad \text{or}\quad \ln\frac{R}{\Delta R}, \qquad y=\ln k_T,$$6–$$x=\ln\frac{1}{\Delta R}\quad \text{or}\quad \ln\frac{R}{\Delta R}, \qquad y=\ln k_T,$$7 GeV/$$x=\ln\frac{1}{\Delta R}\quad \text{or}\quad \ln\frac{R}{\Delta R}, \qquad y=\ln k_T,$$8, $$x=\ln\frac{1}{\Delta R}\quad \text{or}\quad \ln\frac{R}{\Delta R}, \qquad y=\ln k_T,$$9	Most generators agree within $\ln(1/z)$0; Herwig underestimates the very collinear, high-$\ln(1/z)$1 region by up to $\ln(1/z)$2–$\ln(1/z)$3
CMS inclusive pp (Collaboration, 2023)	$\ln(1/z)$4 or $\ln(1/z)$5, $\ln(1/z)$6 GeV	Pythia 8 CP5 underestimates the perturbative region by $\ln(1/z)$7; NLO + NLL pQCD reproduces shape and normalization within combined uncertainties
ATLAS top and $\ln(1/z)$8 jets (Collaboration, 2024)	$\ln(1/z)$9, large-$\ln k_T$0 jets with $\ln k_T$1 GeV	In $\ln k_T$2-initiated jets, all predictions are incompatible with the measurement
CMS PbPb versus pp (Collaboration, 9 Feb 2026)	$\ln k_T$3, $\ln k_T$4–$\ln k_T$5 GeV	No significant difference is observed for high-$\ln k_T$6 emission angular distributions
LHCb light- and beauty-jet LJP (collaboration et al., 29 May 2025)	anti-$\ln k_T$7, $\ln k_T$8, $\ln k_T$9 GeV	First direct observation of the dead-cone effect in beauty-quark jets

The first high-statistics inclusive pp measurement with ATLAS used charged particles in 13 TeV collisions and emphasized that the plane factorizes distinct physical effects more cleanly than many one-dimensional substructure observables; the comparison already showed that no parton shower model simultaneously described all regions (Collaboration, 2020). ALICE then extended the measurement to the intermediate $$\rho(\Delta,k_T) = \frac{1}{N_{\rm jets}} \frac{d^2N_{\rm emissions}}{d\ln(1/\Delta)\,d\ln k_T},$$0–$$\rho(\Delta,k_T) = \frac{1}{N_{\rm jets}} \frac{d^2N_{\rm emissions}}{d\ln(1/\Delta)\,d\ln k_T},$$1 GeV/$$\rho(\Delta,k_T) = \frac{1}{N_{\rm jets}} \frac{d^2N_{\rm emissions}}{d\ln(1/\Delta)\,d\ln k_T},$$2 range, where hadronization and underlying-event effects play a dominant role, and used a three-dimensional unfolding in jet $$\rho(\Delta,k_T) = \frac{1}{N_{\rm jets}} \frac{d^2N_{\rm emissions}}{d\ln(1/\Delta)\,d\ln k_T},$$3, $$\rho(\Delta,k_T) = \frac{1}{N_{\rm jets}} \frac{d^2N_{\rm emissions}}{d\ln(1/\Delta)\,d\ln k_T},$$4, and $$\rho(\Delta,k_T) = \frac{1}{N_{\rm jets}} \frac{d^2N_{\rm emissions}}{d\ln(1/\Delta)\,d\ln k_T},$$5 to correct the density (Havener, 2021).

CMS measured the inclusive primary Lund plane density for $$\rho(\Delta,k_T) = \frac{1}{N_{\rm jets}} \frac{d^2N_{\rm emissions}}{d\ln(1/\Delta)\,d\ln k_T},$$6 and $$\rho(\Delta,k_T) = \frac{1}{N_{\rm jets}} \frac{d^2N_{\rm emissions}}{d\ln(1/\Delta)\,d\ln k_T},$$7 jets with $$\rho(\Delta,k_T) = \frac{1}{N_{\rm jets}} \frac{d^2N_{\rm emissions}}{d\ln(1/\Delta)\,d\ln k_T},$$8 GeV using charged tracks, and compared the unfolded distribution to multiple showers and to an analytic NLO + NLL calculation with nonperturbative corrections. The measurement isolated shower-coupling, recoil-scheme, hadronization-cutoff, and underlying-event effects in distinct subregions of the plane (Collaboration, 2023).

The formalism has also been specialized to heavy-particle decays. In ATLAS $$\rho(\Delta,k_T) = \frac{1}{N_{\rm jets}} \frac{d^2N_{\rm emissions}}{d\ln(1/\Delta)\,d\ln k_T},$$9 events, both top- and $d\ln(R/\Delta)$0-jet Lund planes display a hard wide-angle decay region together with the usual collinear-soft shower structure, but the data revealed sizeable modeling failures, especially for $d\ln(R/\Delta)$1-initiated jets (Collaboration, 2024). In heavy-ion collisions, CMS measured the angular distributions of the largest-$d\ln(R/\Delta)$2 emissions in two $d\ln(R/\Delta)$3 slices, $d\ln(R/\Delta)$4 GeV and $d\ln(R/\Delta)$5 GeV, and found a flat PbPb/pp ratio within uncertainties, consistent with the interpretation that the earliest hard branchings occur before substantial interaction with the quark-gluon plasma (Collaboration, 9 Feb 2026).

4. Tagging, likelihood methods, and learning architectures

The Lund jet plane was proposed from the outset not only as a diagnostic observable but also as a discriminant. In the original study, boosted electroweak-boson tagging was formulated both as a transparent log-likelihood over primary declusterings and as a machine-learning problem on Lund-plane representations. The log-likelihood method exploited the approximate decorrelation of different regions of the plane, while sequence and image-based networks captured additional correlations and improved background rejection (Dreyer et al., 2018).

A particularly explicit realization is boosted Higgs tagging with Lund images. In one implementation, primary declusterings were pixelized on a $d\ln(R/\Delta)$6 grid and passed to a convolutional neural network with four 2D convolutional layers and a dense layer. For $d\ln(R/\Delta)$7, the resulting classifier achieved $d\ln(R/\Delta)$8 at the $d\ln(R/\Delta)$9 GeV benchmark and $d\ln(1/\Delta)$0 at the $d\ln(1/\Delta)$1 GeV benchmark, compared with $d\ln(1/\Delta)$2 and $d\ln(1/\Delta)$3 for the jet color-ring observable. For $d\ln(1/\Delta)$4, where the color ring was nearly non-discriminating, the Lund-plane CNN still achieved $d\ln(1/\Delta)$5 at $d\ln(1/\Delta)$6 GeV and $d\ln(1/\Delta)$7 at $d\ln(1/\Delta)$8 GeV (Khosa, 2021).

Quark-versus-gluon discrimination has been developed in parallel in analytic and neural forms. At single-logarithmic accuracy, the likelihood ratio can be written directly in terms of Lund declusterings, including full splitting kernels, Sudakov factors, flavor-changing transitions, and clustering-log corrections. On toy samples that exactly resum leading collinear single logarithms, the analytic likelihood ratio and the learned discriminant are equivalent; on full Monte Carlo samples, machine-learning models are usually superior, and the asymptotic analysis indicates that the gain comes from effects subleading to the analytic construction (Dreyer et al., 2021).

Recent architectures use the Lund representation as an explicit modality alongside raw particles. In PLuM, particle constituents and Lund-plane splittings are embedded into a shared latent space and processed by a unified transformer. The gains are selective: systematic improvements are seen for top-quark and $d\ln(1/\Delta)$9 tagging, but not for $\alpha_s$0 or $\alpha_s$1. For di-Higgs searches in the $\alpha_s$2 final state, at a $\alpha_s$3 di-Higgs efficiency working point, PLuM achieves $\alpha_s$4 higher background rejection than the particle-only baseline (Gouskos et al., 26 May 2026). This suggests that explicit hierarchical information about $\alpha_s$5-jet formation remains complementary to constituent-level representations even in highly expressive transformer models.

5. Heavy flavor, nuclear matter, and hidden-sector applications

Because the Lund plane isolates emissions by angle and hardness, it is especially well suited to mass effects. LHCb measured the plane for light-quark-enriched $\alpha_s$6jet events and for $\alpha_s$7-tagged beauty jets, and reported the first direct observation of the dead-cone effect in beauty-quark jets. The suppression appears in the collinear region of both the $\alpha_s$8-Lund and $\alpha_s$9-Lund projections, while the light- and beauty-jet planes agree at wide angle within uncertainties (collaboration et al., 29 May 2025).

This experimental result matches the theoretical motivation behind the Lund $k_T$0-jet plane. A single-logarithmic calculation for heavy-quark-initiated jets shows the familiar dead-cone suppression factor

$k_T$1

and includes running coupling, collinear evolution in a variable-flavor scheme, and numerically resummed soft clustering effects with full $k_T$2-mass dependence (Ghira et al., 19 Dec 2025). The heavy-quark case therefore turns the Lund plane into a direct probe of quasi-collinear factorization and mass-dependent QCD radiation.

In heavy-ion collisions, the Lund plane provides access to the time ordering of the shower. CMS designed a measurement specifically around the hypothesis that the earliest, highest-$k_T$3 emissions are formed before the quark-gluon plasma. The absence of angular reshaping for $k_T$4 GeV, together with comparisons to jet-quenching models, supports the interpretation of an approximately angle-independent quenching of already-formed hard subjets (Collaboration, 9 Feb 2026).

The same separation of perturbative and nonperturbative sectors has been exploited in dark-sector phenomenology. In hidden-valley and dark-QCD scenarios, the leading-log Lund density remains approximately flat in the perturbative region but becomes sensitive to unknown dark hadronization near $k_T$5. This motivates resilient observables such as the number of emissions above a perturbative $k_T$6 threshold or ratios of counts from perturbative and nonperturbative regions of the plane (Cohen et al., 2023). More recently, generalized parton showers for arbitrary gauge groups have been analyzed on the Lund plane and classified with deep sequential models, showing that perturbative emission topologies can be used to distinguish $k_T$7, $k_T$8, and $k_T$9 dark gauge structures, with robustness studies under infrared $$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$00 cuts and massive dark-gauge-boson thresholds (Li et al., 24 Jun 2026).

6. Secondary planes, derived observables, calibration, and computational extensions

The Lund formalism has broadened from the primary plane into a family of derived constructions. One example is the secondary Lund jet plane, obtained by declustering a selected soft branch. A recent proposal uses a dijet topology with a hard leading jet and an asymmetric, nearby subleading jet to obtain a gluon-enriched sample: fixed-order and hadron-level studies indicate gluon fractions of around $$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$01 for the subleading jet, with resilience to the overall color structure, the hard-scattering flavor channel, and the PDFs (Baldenegro et al., 2024). This turns the secondary plane into a tagger-free method for constraining gluon radiation patterns.

A more formal extension is the introduction of Lund-Tree Shapes, a class of declustering-tree observables built from the set of primary Lund declusterings in an event. At the differential level they interpolate between continuous event shapes and $$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$02 jet-resolution variables; at the cumulative level they define $$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$03-jet rates. Their all-order structure exponentiates in a simple way and, for the observables defined in that work, they are free of non-global logarithmic corrections. NNLL-resummed and NNLL+NNLO-matched predictions have been derived for two-leg processes at $$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$04, $$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$05, and $$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$06 colliders (Beekveld et al., 20 Nov 2025).

The Lund plane has also become a practical calibration object. CMS introduced a method for correcting the substructure of multiprong large-$$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$07 jets by reclustering them into exactly $$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$08 subjets, building a Lund plane for each prong, and reweighting simulated splittings with a data/MC ratio measured in a boosted $$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$09 control sample. The procedure improves agreement for observables such as $$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$10, $$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$11, $$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$12, $$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$13, and deep-tagger outputs, and establishes, for the first time, a robust calibration for jets with four or more prongs (Collaboration, 10 Jul 2025).

Computational uses have expanded in parallel. The gLund and CycleJet frameworks treat Lund images as sparse generative targets: averaged detector-level QCD jet images can be reproduced to within about $$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$14 over the bulk of the plane, and unpaired mappings between domains such as parton level and detector level or QCD and $$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$15 jets are achieved with similar few-percent accuracy in the bulk (Carrazza et al., 2019). Quantum-encoding work has pushed the representation further still: LP2B maps truncated Lund trees into qubit states, and a Quantum Tree-Topology Network built on this encoding matches large classical architectures on polarization tagging, remains competitive for $$\Delta R=\sqrt{(\Delta y)^2+(\Delta\phi)^2}, \qquad z=\frac{p_{T,s}}{p_{T,h}+p_{T,s}}, \qquad k_T \simeq p_{T,s}\,\Delta R \simeq z\,p_T\,\Delta R,$$16- and top-tagging, and was validated on a 3-qubit device (Napolitano et al., 15 Apr 2026).

Taken together, these developments position the Lund jet plane as both an observable and a representation. It is simultaneously a precision probe of parton-shower dynamics, a calibration target for detector-corrected substructure, a substrate for analytic resummation, and a hierarchical input space for classical, generative, multimodal, and quantum learning systems (Dreyer et al., 2018, Lifson et al., 2020).