arXiv CSopen access

Explainable AI for Jet Tagging: A Comparative Study of GNNExplainer, GNNShap, and GradCAM for Jet Tagging in the Lund Jet Plane

2026 · arxiv_cs

arXiv CS · Papers · License: Open Access · 2026

machine learning, deep learning, neural networks

Explainable AI for Jet Tagging: A Comparative Study of GNNExplainer, GNNShap, and GradCAM for Jet Tagging in the Lund Jet Plane Pahal D. Patel 1, ∗ and Sanmay Ganguly 1, †

arXiv:2604.25885v1 [hep-ph] 28 Apr 2026

Department of Physics, Indian Institute of Technology, Kanpur Uttar-Pradesh-208016, India

Graph neural networks such as ParticleNet and transformer based networks on point clouds such as Particle Transformer achieve state-of-the-art performance on jet tagging benchmarks at the Large Hadron Collider, yet the physical reasoning behind their predictions remains opaque. We present different methods, i.e. perturbation-based (GNNExplainer), Shapley-value-based (GNNShap), and gradient-based (GradCAM); adapted to operate on LundNet’s Lund-plane graph representation. Leveraging the fact that each node in the Lund plane corresponds to a physically meaningful parton splitting, we construct Monte Carlo truth explanation masks and introduce a physics-informed evaluation framework that goes beyond standard fidelity metrics. We perform the analysis in three transverse-momentum bins (pT ∈ [500, 700], [800, 1000], and the inclusive region [500, 1000] GeV), revealing how explanation quality and focus shift between non-perturbative and perturbative regimes. We further quantify the correlation between explainer-assigned node importance and classical jet substructure observables—N -subjettiness ratios τ21 and τ32 and the energy correlation functions— establishing the degree to which the model has learned known QCD features. We find that overall the weight assigned by explainability methods has a correlation with analytic observables, with expected shift across different phase space regimes, indicating that a trained neural network indeed learns some aspects of jet-substructure moments. Our open-source implementation enables reproducible explainability studies for graph-based jet taggers. I.

INTRODUCTION

At the Large Hadron Collider (LHC), proton–proton collisions routinely produce hard scattered partons (quarks and gluons), along with other heavy particles like top quark, W/Z bosons etc., that fragment and hadronize into collimated sprays of stable particles, such as hadrons, photon and leptons. A subset of such particles, collected through a sequential clustering algorithm, known as jets [1]. Jets are a relic of quantum properties carried by the partons and thus a major probe of short distance dynamics, taking place inside a hadron–hadron collision processes. The internal structure of hadronic jets is driven by the fundamental branching probabilities of Quantum Chromodynamics (QCD). The evolution of a parton from a hard scale down to the hadronization scale is governed by a cascade of soft and collinear emissions, a process successfully described by parton shower algorithms and analytical resummation techniques [2]. The Lund Jet Plane (LJP) [3] has successfully emerged as the definitive phase space description for characterizing this radiation pattern. By projecting jet constituents onto a phase space defined by the natural logarithm of emission transverse momentum ln(kT ) and ln(1/∆), where ∆ is the splitting angle. The LJP provides a direct visualization of the factorization properties of QCD, separating non-perturbative effects, wide-angle soft radiation, and hard collinear splittings into distinct kinematic regions. Identifying whether a given jet originated from the hadronic decay of a boosted heavy particle—a top quark,

∗ [email protected] † [email protected]

a W or Z boson, or a Higgs boson—or from a generic quantum chromodynamics (QCD) process, viz. a hard light quark or gluon fragmentation, is one of the central experimental challenges at the LHC [2, 4, 5]. This identification, commonly referred to as jet tagging, directly underpins searches for physics beyond the Standard Model (BSM) [6] as well as precision measurements of different Standard Model (SM) processes such as electroweak and Higgs boson couplings [7]. Over the past decade, jet tagging has been transformed by machine learning (ML). Early approaches treated jets as images and applied convolutional neural networks [8–10], while more recent architectures exploit the point-cloud or graph structure of jets directly. Graph neural networks (GNN) have proven particularly wellsuited to this problem because the relational structure of jets—particles connected by angular proximity or clustering history—maps naturally onto graph representations [11, 12]. ParticleNet [13] introduced dynamic graph convolutions over particle clouds; JEDI-net [14] modelled explicit pairwise particle interactions; LorentzNet [15] enforced Lorentz equivariance; and the Particle Transformer [16] adapted attention mechanisms with learned pairwise interaction biases to achieve state-of-the-art performance. There are a subset of partons, viz. b-quarks and c-quarks, for which the corresponding jets have a peculiar property of having secondary vertex [17] inside a jet. Classification of such sub-classes of jet using GNNs have turned out to have major implications [18, 19] in terms of performance gain over other competing algorithms. The rapid adoption of these deep-learning taggers in experimental analyses by ATLAS [19] and CMS [7] raises a question that goes beyond raw performance: what has

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the model learned, and can it be trusted? Unlike classical substructure observables—N -subjettiness ratios τ21 [20, 21], energy correlation functions [22, 23], or the softdrop momentum fraction zg [24]— which are constructed from first-principles QCD reasoning and whose discriminating power can be calculated analytically, neural network taggers operate as black boxes. Their internal decision boundaries are distributed across millions of learned parameters with no transparent connection to the underlying physics. This opacity is not merely an aesthetic concern. If a tagger learns to exploit features that are artifacts of the Monte Carlo (MC) simulation—such as details of the hadronization model, the parton shower cutoff, or the detector response simulation—its performance may degrade when applied to real collision data [25, 26]. A jet can have a graph representation in the two dimensional feature space spanned by pseudo-rapidity ( η) and azimuthal angle (ϕ). The same jet can have a graph representation in LJP, capturing the unique feature that each node carries the information of parton fragmentation and thus has an unique representation in a particular region of LJP phase space, associated with an unique physics process. The correspondence between a physics process and a sub-region of LJP is shown in Fig. 1. Details of the graph representation in LJP, of a hadronic jet, is described in Section II. From the known methods of explainable artificial intelligence (XAI) on graphs, if one can associate an importance score to each of the nodes and vertices of a graph, then it’s possible to trace down what kind of physics features are more emphasized by a trained neural network model. We have designed our analysis strategy, based on this broader principle. The remainder of this paper is organised as follows. Section II reviews the Lund jet plane, the LundNet, ParticleNet and Particle Transformer architecture, the three explainability methods, and the jet substructure observables used in our correlation analysis. Section III describes our adaptations of each explainer to LJP, the construction of physics-informed ground-truth explanation masks, and the evaluation framework including the substructure correlation methodology. Section IV presents our results: the method comparison, Lund-plane explanation heatmaps, the pT -dependent analysis, and the substructure correlation studies. Section V discusses method trade-offs, physics insights, robustness considerations, and limitations. We conclude in Section VI, with an outlook towards application of the method stidies and it’s utility in future jet physics program.

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ρg CA 9 ≃ = ρq CF 4

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FIG. 1. A schematic description of Lund Jet Plane, showing individual quantum effects in different parts of the phase space. The parton splitting functions Pab (z) and the color factors CA , CF can be found in any standard QCD literature [30].

that is both theoretically calculable and experimentally measurable. Starting from a jet clustered with the Cambridge/Aachen (C/A) algorithm [27, 28], one iteratively undoes the clustering: at each step the softer of the two branches is recorded as an emission, characterized by its transverse momentum fracsoft hard soft tion p z = pT /(pT + pT ), its opening angle ∆R = 2 (ηhard − ηsoft ) + (ϕhard − ϕsoft )2 relative to the harder branch, and the transverse momentum scale of the splitting kT = z pT ∆R. Here, pT denotes the transverse momentum. Each emission is mapped to a point in the (ln ∆R, ln kT ) plane. The procedure is applied recursively along the harder branch to define the primary Lund plane, and optionally along secondary branches to build the full Lund tree. In the soft-collinear limit of QCD, the primary Lund plane density at leading-logarithmic accuracy is approximately uniform [3, 29]: ρ(kT , ∆R) ≃

2 αs (kT ) CR , π

(1)

where CR is the colour factor of the initiating parton (CF = 4/3 for quarks, CA = 3 for gluons). The CA /CF ratio is what makes quark/gluon discrimination possible at the Lund-plane level. Deviations from uniformity arise from running coupling effects, hard collinear splittings, and non-perturbative contributions at low kT . These phenomena are shown in the schematic diagram Fig. 1. The ATLAS experiment √and the CMS experiment have measured this density in s = 13 TeV data [31, 32], validating the perturbative predictions and constraining the

3 parton shower models. The CMS experiment has also proposed a method to correct multi-prong jet substructure using LJP [33]. On the theoretical frontiers, there are proposal of searching for beyond standard model (BSM) physics using LJP [34].

B. Jet tagging in the Lund Jet Plane : Towards a correlation between analytic structure and ML methods

Machine Learning based jet tagging and jet mass regression algorithms are now state of the art performer within the mainstream experimental collaborations [7, 35]. The need for interpretable machine learning in the physical sciences has been broadly articulated [36– 39]. In high-energy physics specifically, recent work has begun to address this gap through attention map analysis for Particle Transformers [40], and physics-informed architectures such as PELICAN [41] and E-PCN [42], the latter of which used Gradient-weighted Class Activation Mapping (GradCAM) to attribute classification decisions to specific kinematic variables. Shapley value based feature importance studies for quark-vs.-gluon tagging have been recently carried out in the article [43]. Interpretable jet physics using infrared and collinear safe observables has been looked into as well [44]. However, these studies have mostly focused on a single explainability method, for a targeted architectures other than Lund-plane-based GNNs. In the broader machine learning community, a rich ecosystem of explainability methods has been developed specifically for GNNs [45, 46]. GNNExplainer [47], a perturbation-based approach, identifies compact subgraph structures and node feature subsets that maximize the mutual information with the model’s prediction. GNNShap [48], grounded in cooperative game theory, assigns Shapley values to edges in the computational graph through GPU-parallelized coalition sampling, providing explanations with axiomatic fairness guarantees. Graphadapted GradCAM [49] extends gradient-weighted class activation mapping to the graph domain, producing node-level importance scores via a single backward pass. These three methods span the major paradigm families, i.e. perturbation-based, game-theoretic, and gradientbased and have been systematically compared on synthetic and molecular benchmarks [46, 50, 51]. Yet, in the context of High Energy Physics (HEP), no study has systematically applied and compared all three, to a domain-specific GNN architecture operating on a physically grounded graph representation where the ground truth for explanations can be derived from first-principles theory. GNN’s on LJP, such as LundNet [26] provide a uniquely powerful testbed for filling this gap. Unlike generic particle-cloud representations, LundNet builds its input graph from the Lund jet plane [3], a theoretically motivated representation of the radiation phase space within a jet. The Lund plane is constructed by iter-

atively declustering a jet using the Cambridge/Aachen algorithm [27, 28] and recording the transverse momentum kT , opening angle ∆R, and momentum fraction z at each step. Each splitting is mapped to a point in the two-dimensional ln(1/∆R), ln(kT ) plane, producing a set of nodes with direct physical meaning: each node corresponds to an individual parton emission in the jet’s showering history. The primary Lund plane density is calculable in perturbative QCD [29], and the ATLAS experiment has measured it in data [31]. Thus, it is worthy to redeploy ParticleNet and Particle Transformer algorithms on graphs constructed from LJP and explain their performance. The Lund-plane representation offers three decisive advantages for explainability studies. First, because each node corresponds to a specific splitting, an explainer that highlights certain nodes is directly stating which emissions matter for the classification—a statement that can be checked against QCD expectations (e.g., that top tagging should rely on the three-prong hard splitting structure). Second, classical substructure observables such as τ21 , τ32 , and energy correlation function ratios C2 , C3 [22], and the iterated soft-drop multiplicity nSD [3, 24] can each be decomposed into contributions from specific Lund-plane nodes, providing a common language in which both the explainer output and the physicist’s observables can be expressed and directly compared. Third, the structure of the LJP changes systematically with jet transverse momentum pT : at low boost, the plane is sparsely populated and non-perturbative effects (hadronization, underlying event) contribute significantly to soft, wide-angle emissions, while at high boost the plane is dense and perturbative hard splittings dominate the substructure [3, 52]. Studying explainability as a function of pT therefore probes whether the model’s reasoning shifts between these physically distinct regimes. In this paper, we present a systematic, multi-method explainability study of a Lund-plane-based GNN for jet tagging. Our contributions are threefold: 1. Multi-method comparison on a physicsgrounded GNN. We adapt GNNExplainer, GNNShap, and GradCAM to LundNet, ParticleNet and Particle Transformer to conduct a rigorous comparison across three tagging benchmarks (top, H, , and quark/gluon), using not only standard fidelity and sparsity metrics [50] but also a new physics accuracy metric based on the intersection-over-union with Monte Carlo truth explanation masks. 2. pT -dependent explainability analysis. We perform the full analysis in two transversemomentum bins (pT ∈ [500, 700] GeV and [800, 1000] GeV ), revealing how explanation quality and focus evolve from the lower-energetic to the deeply perturbative regime. This exposes whether the model exploits infrared-and-collinear (IRC) un-

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Node size and color intensity on the Lund plane encode XAI-assigned importance Blue diamonds (♢) = MC-truth splittings from known decay topology. Gray circles (•) are non-truth splittings

FIG. 2. The figure summarizes the analysis strategy of explainability of a ML based jet tagger in the Lund Jet Plane. We study the combinations of jet taggers viz. LundNet, ParticleNet and Particle Transformer along with three explainability methods on GNN’s viz. GNNExplainer, GNNShap and GradCAM. The importance weight assigned by each of these nine combinations in the LJP phase space is cross verified with first principle analytical computations – the MC-truth. A comprehensive correlation study across pT bins are performed to understand which part of the XAI predictions are in agreement with theoretical observables like jet substructure moments. A cross correlation between the explainability scores are also studied in order to understand the degree of mutual agreement among the different XAI methods deployed.

5 safe features at low pT —a direct diagnostic for simulation dependence—and how the computational cost of each explainer scales with the denser Lundplane graphs at high pT . 3. Quantitative correlation with substructure observables. We introduce a three-level framework viz. visual overlay, Spearman rank correlation, and importance-weighted observable reconstruction, to directly compare explainer-assigned node importance with per-node contributions to τ21 , τ32 , energy correlation functions C2 , C3 . This establishes the degree to which LundNet has learned the same discriminating features that decades of analytical QCD research have identified [2, 53], and reveals whether the model has also discovered novel features beyond classical observables.

Physics-informed ground truth

Lund Jet Plane gives us an analytic handle to trace down jets in appropriate regimes in phase space [29]. We construct Monte Carlo (MC) based truth explanation masks by tracing the hard-scattered partons through the showering and hadronization stages and identifying which Lund-plane nodes descend from the primary parton decay chain. For top jets (t → bW → bq q̄ ′ ), the ground-truth graph top G consists of nodes on the Lund tree that are ancestral to the three hardest splittings corresponding to the b, H q , q̄ ′ prongs. For Higgs (H) tagging (H → q q̄), Gexpl is the single primary splitting that carries the W mass, plus its immediate descendants. We deploy the analytic understanding to retrieve the average LJP density per jet ρ(kT , ∆R) =

The overall strategy towards a systematic construction of Explainable AI (XAI) for jet taggers are shown in Fig. 2

III.

EXPLAINABILITY METHODS FOR JET TAGGERS Explainer adaptation to Lund Jet Plane

Each explainability method requires non-trivial adaptation to LundNet’s graph structure. We summarise the key design choices here; implementation details and hyperparameter sensitivity studies are in App. A. a. Adapting GNNExplainer. The edge mask M is optimised over all EdgeConv layers simultaneously. For LundNet5, the kNN graph is recomputed at each layer from learned embeddings, so we freeze the graph topology to that of the trained model and optimise masks on the resulting static edge set. The working principle of GNNExplainer are given in App. A 1. b. Adapting GNNShap. We define the player set as the edges in the computational graph of the target node (for node-level tasks) or the full graph (for graph classification). For LundNet, each edge corresponds directly to a parent–child splitting relation, making the Shapley attribution physically transparent. The rest of the working details of GNNShap can be found in App. A 2. c. Adapting GradCAM. We compute activation maps at each EdgeConv layer and aggregate via gradient weighting as in Eq. A7. To obtain per-feature importance, we additionally compute the gradient of output with respect to the input feature as ∂y c /∂xi,d for each input dimension d ∈ {ln z, ln ∆, ln kT , . . .}, producing a feature-resolved attribution alongside the node-level score. These attributions are then used to compute the interpretability scores. The details of GradCAM implementations are discussed in App. A 3.

d2 Nemissions , R Njets d ln(kT ) d ln( ∆R ) 1

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for 1-prong (QCD), 2-prong (H → cc̄) and 3-prong (t → bqq̄′ ) jets. The same quantity, at leading order, is analytically described by Eq. 1. The density distributions are shown in Fig. 3 which captures the statistical pattern of parton emission densities of 1-prong, 2prong and 3-prong jets. When each emission i (i ∈ [1 , 2 , . . . , Nemissions ]) in the LJP, for a given jet, has an associated weight wi , then the weighted LJP density is defined as

ρW (kT , ∆R) =

Nemissions wi i=1

R Njets d ln(kT ) d ln( ∆R )

(3)

A large part of the higher density regions are populated with low energetic parton splitting, as expected from first principle analytic understanding of QCD. The jets originating from light partons, a.k.a QCD jets have a visible density enhancement in the lower ln(R/∆R) region, corresponding to soft wide-angled emissions. This pattern is well understood from the Sudakov double logarithm structure of parton branching amplitudes [54]. Once we train a neural network for downstream tagging task and pass it through an explainable method, we try to see if the explainer associates an importance weight to different regions of LJP and wether that weight factor is matching our analytic understanding of LJP density.

Evaluation framework

To successfully transit neural networks from predictive black boxes to reliable engines of physical discovery, we must be able to validate the interpretable models derived from them. In this section we study three different metrics viz. Fidelity metrics, Weighted Feature Average

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FIG. 3. Distribution of average LJP emission density ρ(kT , ∆R) for 1-prong, 2-prong and 3-prong jets, reconstructed with anti-kT algorithm [55] and jet radius parameter R = 0.8. The 1-prong jets are populated by parton splitting of light partons only and hence low kT region ( ln(kT ) ≤ 0) have an over density. It’s important to note that for 1-prong jets, the regions with lower values of ln(R/∆R) has a higher density as well, originating from soft wide angled radiation. On the other hand, the middle and right panel shows, jets originating from hadronic decay of H → cc̄ – the 2-prong jets and the jets originating from hadronic decay of t → bqq̄′ – the 3-prong jets have an observable density overpopulation in high-kT – high-∆R regime of phase space. This feature indicates that hadronic showers within these jets are originating from fragmentation of a massive particle.

and Jet substructure correlation, which provide this crucial validation by statistically quantifying how faithfully an interpretable approximation maps the exact decision boundaries and latent dynamics of the original network. a. Fidelity metrics. Fidelity metrics on the G are designed to capture wether a sub-graph of the entire graph has more importance towards the downstream learning task. To test this, we first identify the edges whose explainability weights are more than 0.51 . It is important to clarify here that while GNNShap returns the edge importance score directly, the GNNExplainer and GradCAM explainability methods return an importance score wi , for the i-th node, of the graph. To compare all the explainability methods on the same footing, for the later two explanation methods, we convert the node assigned weights to edge weights by taking an average value wij = 21 (wi + wj ). The graph, formed by this subset of edges, is identified as explanation subgraph Gexpl and the mutually orthogonal part of the graph G \ Gexpl , which likely has much less impact towards downstream decision making. Following Ref. [50], we define: Fid− = f (G) − f (Gexpl ) , +

Fid = f (G) − f (G \ Gexpl ) ,

(4) (5)

where f (·) denotes the model’s predicted probability for the correct class. Fid− measures the prediction drop

1 The threshold value of 0.5 may require a further optimization

on per class basis. In the current literature, we apply it as an uniform threshold to identify important edges

when only the explanation subgraph is retained. It’s worth noting that ideally if the exact explanation subgraph Gexpl can be identified, then f (G) and f (Gexpl ) should return very close by values and thus a distribution of such scores are expected to populate a region around zero. On the other hand, Fid+ measures the drop when the explanation subgraph Gexpl is removed. The network is evaluated on the remaining subgraph G \ Gexpl . We expect a large shift on the network evaluation score and thus Fid+ is expected to populate regions closer to one. The distribution of Fid− and Fid+ variables are shown in Fig. 6. The distributions for 2-prong and 3-prong jets show that the Fid+ scores are as expected on the higher side where as for 1-prong jets the Fid+ score is populated around zero, showing a counter intuitive feature. Such an opposite behavior can be traced back to the LJP density diagrams Fig. 3 and Fig. 4. The first column on both the figures, showing 1-prong QCD emission pattern features, demonstrate that the node importance in this case is spread over the entire LJP graph and the weight is not concentrated among a subgraph Gexpl only. Hence when we evaluate the network score on f (G) and f (G \ Gexpl ), the two numbers are not much different from each other as Gexpl and G \ Gexpl has similar level of importance in this case. This point is discussed in details in Sec. V C. In the Fig. 7 we see the Fid+ distribution for GNNExplainer and GradCAM for Particle Transformer. The shift close to one for 2-prong and 3-prong jets are self explanatory where as a partial shift of 1-prong jets is again traced down to concentrated explanation weights in the low kT regions of LJP, as seen from comparison of Fig. 3 and Fig. 4.

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FIG. 4. Distribution of average weighted LJP emission density ρW (kT , ∆R) for 1-prong, 2-prong and 3-prong jets, identified as true-positive category by the respective classifiers. The weighted density diagrams here are displayed only for GNNShap explainer for Particle Transformer and ParticleNet architectures only (since both ParticleNet and LundNet uses Edge-Convolution, we have just restricted the comparison between two different type of architectures). When we compare the weighted distributions with the raw LJP density distributions, shown in Fig. 3, we see that for 1-prong jets (first column of the above figure) both the architectures capture the low kT radiation pattern of the QCD emission over a wide angular range. The weighted density distribution for 2-prong H → cc̄ decay, we find that both the networks associate larger weights to the hard wide angle splitting, thus indicating it is learning the sub-structure pattern of the jets. The third column on the right, displaying the weighted phase space distribution of 3-prong hadronic top-quark decay, shows that both architectures capture the jet substructure pattern as well as the fragmentation of light partons, populating the low kT regions. In App. D, we further investigate the correlation of learned weights (in terms of weighted LJP features defined in Eq. 6) and the jet substructure moments.

b. Weighted Feature Average The combination of three different networks (Particle Transformer etc.) and three different explanation methods (GNNShap etc.) ultimately returns a weight parameter wi for each node i ∈ VG , where VG is the set of nodes in the graph Gconstructed in the LJP. The impact of the learned wi is best demonstrated through a weighted average of the five LJP feature variables Fi ∈ {ln z, ln ∆, ψ, ln m, ln kT }. The weighted average is computed as P j∈V wj Fij F̄i = P G . (6) j∈VG wj Here Fij is the i-th LJP feature associated with the j-th node. The distribution of the average feature importance are shown in Fig. 8. It is quite evident that each combination of architec-

ture and explanation method has it’s own estimation of F̄i and thus it makes sense to study the correlation pattern of this quantity among all the nine combinations. Such a study gives us a numerical estimate about the degree of agreement between all the methods under consideration. This study is done in details App. D and the corresponding correlation coefficients are tabulated in the tables Table II, Table IV and Table III respectively.

c. Substructure rank correlation. For each jet, we rank Lund-plane nodes by their explainer-assigned im(O) portance wi and, separately, by their contribution ci to a substructure observable O (e.g., for τ21 , ci is the ith constituent’s contribution to the τ2 numerator). The

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FIG. 5. Distribution of average weighted LJP emission density ρW (kT , ∆R) for 1-prong, 2-prong and 3-prong jets, which are classified in the true-positive category by the Particle Transformer algorithm. The top row shows the weighted distribution, when the output of the trained network is passed through the GNNExplainer algorithm. The middle and bottom row show similar weighted distributions in the LJP, when the explainer weights are computed through GNNShap and GradCAM algorithms, respectively. When we compare with a similar Fig. 4, we find that for 1-prong case GradCAM has certain distinct features compared to the other two explainer and emphasizes on wider angle radiation pattern more. In the middle column we see that while GNNShap exclusively emphasizes on two hard jet substructures, the behavior of the other two explainer are somewhat in contrast. GradCAM puts highest weights on soft and small angle radiations. For 3-prong hadronic top quark decay, GNNShap still relies on the hard jet substructure only. However GNNExplainer and GradCAM, like the 2-prong case, relies heavily on the low kT fragmentation features of the hard sub-partons. These comparative features (shown for the entire pT range only, while we explicitly study them for low and high pT bins separately ) tend to demonstrate that GNNShap is least sensitive to soft radiations when combined with Particle Transformer architecture.

Spearman rank correlation coefficient P 6 i d2i (O) , di = rank(wi ) − rank(ci ) , ρs (O) = 1 − N (N 2 − 1) (7)

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FIG. 8. Weighted feature importance, as defined through Eq. 6, is shown for all the five LJP features across the nine combinations of the neural architectures and explainability methods. It is evident that the natural logarithms of energy splitting fraction ln(z) and ln(∆) (both are negative quantities as in our problem z ≤ 1 and ∆ ≤ 0.8) play a crucial role towards decision making, specially for QCD radiations. The ln(m) and ln(kT ) carry the relics of mult-prong jet substructure and hence plays a role of greater importance towards identification of 2-prong and 3-prong jets. Since the quantum dynamics associated with the parton fragmentation process has a symmetry over azimuthal angle of the emission, the variable ψ has a negligible role towards decision making for all the above cases.

11 IV.

RESULTS

The results presented in this section are organized around three physics questions that the Lund Jet Plane representation makes quantitatively addressable. First, do the importance maps assigned by the trained taggers populate the regions of phase space that QCD factorization theorems identify as physically meaningful? Second, when interrogated through different XAI methods, do the networks consistently emphasize the same parton-shower features, and how does this consistency vary across the perturbative-versus-non-perturbative boundary? Third, are the learned importance scores quantitatively aligned with the analytic substructure observables viz. τ21 , τ32 , and the energy correlation function ratios C2 , C3 , that decades of QCD theory have established as the canonical discriminants for N -prong jets? We address these questions in three subsections, working with the nine combinations of three architectures (Particle Transformer, ParticleNet, LundNet ) and three explainability methods (GNNExplainer, GNNShap, GradCAM), and stratifying every result by the transverse momentum bins 500 < pT < 700 GeV (low) and 800 < pT < 1000 GeV (high), in addition to the inclusive range. Fig. 3 establishes the physical baseline. The unweighted Lund-plane emission density ρ(kT , ∆R) for 1prong QCD jets is dominated by soft, wide-angle emissions (ln kT ≲ 0, ln(R/∆R) ≲ 2), reaching peak densities of ρ ≈ 1.6. This pattern is the direct phase-space realization of the Sudakov double-logarithmic distribution of QCD bremsstrahlung from a single light parton. For 2prong H → cc̄ and 3-prong t → bq q̄ ′ jets, the density develops a pronounced overpopulation in the hard-collinear region (ln kT ≳ 3, ln(R/∆R) ≳ 2), with peak values of ρ ≈ 3.0 and ρ ≈ 4.0 respectively. This excess is the kinematic fingerprint of the parent resonance mass: the hard splitting that releases the Higgs or top mass populates a localized region of phase space that is parametrically forbidden in pure QCD radiation. The GNNShap weighted maps in Fig. 4 show that both the Particle Transformer and ParticleNet correctly identify these mass-induced excesses as the discriminating regions. For H → cc̄ jets, the weighted density peaks sharply at ln kT ≈ 4–6 and ln(R/∆R) ≈ 1–2, reaching ρW ≈ 0.35 for the Particle Transformer and ρw ≈ 0.20 for ParticleNet. This is the phase-space location where the Higgs mass-drop deposits the two charm prongs, and the explainer has localised the network’s attention precisely there. For top jets, the GNNShap-weighted importance distributes across two angular scales, i.e. a wider opening at ln(R/∆R) ≈ 3–4 corresponding to the t → bW splitting and a more collinear contribution at ln(R/∆R) ≈ 4– 5 corresponding to the W → q q̄ ′ decay. This bimodal structure in the weighted density is the direct image of the V -A decay cascade of the top quark and constitutes a non-trivial physics validation of the Shapley-based attribution.

Fig. 5, which compares all three XAI methods on the Particle Transformer, brings into focus a result that is implicit in the unweighted densities but only made explicit through XAI: GNNShap is a near-perfect filter for the hard collinear branching, while GNNExplainer and GradCAM additionally weight low-kT fragmentation activity that is dominated by non-perturbative dynamics. For H → cc̄, GNNShap concentrates ρw ≈ 0.35 in the hard-collinear strip and assigns near-zero weight elsewhere; GNNExplainer and GradCAM, by contrast, spread importance over the full Lund plane with peaks near ρw ≈ 0.10. This finding has a clear physics interpretation. Shapley values, by axiomatic construction, allocate the model’s predictive output across the most causally responsible edges; in the Lund plane, those edges are the ancestors of the hard splitting. Perturbationbased and gradient-based methods, by contrast, are sensitive to the full message-passing pathway, and therefore also expose the soft fragmentation tail that the network uses as contextual confirmation. Both signals are physical, but they probe different aspects of the learned representation.

Fidelity as a diagnostic of phase-space localisation

The fidelity scores (Fig. 6 – Fig. 7) permit a quantitative test of whether the importance assignment identifies a sufficient subgraph for classification. For H → cc̄ and t → bq q̄ ′ jets, GNNShap produces a Fid+ distribution sharply peaked near unity: removing the wij > 0.5 edges identified as most important destroys the network’s confidence in the signal hypothesis, as expected when the explanation captures the hard splitting that physically defines the signal. The corresponding Fid− distribution is peaked near zero, confirming that the retained subgraph alone reproduces the full prediction. A particularly informative anomaly arises for QCD jets, where the Fid+ distribution is itself peaked near zero rather than near unity (Fig. 7, right column). At first sight this appears to violate the expected behavior of a faithful explainer; in fact it is a direct phasespace signature of the absence of a discriminating subgraph. As Fig. 3 and Fig. 4 make clear, QCD jet importance is distributed quasi-uniformly across the Lund plane along the soft-emission diagonal. No sub-selection of wij > 0.5 edges captures a parametrically larger fraction of the discriminating signal than its complement, because the network’s QCD-tag decision is built from the collective pattern of soft, wide-angle radiation rather than from any single hard splitting. This is exactly the QCD factorization expectation: the absence of a hard scale in 1-prong jets means that the discriminant must integrate over the full Sudakov double-logarithmic tail. The fidelity anomaly is therefore a physically meaningful negative result, and the contrast between multi-prong and 1-prong jets in the Fid+ distribution is itself a phase-

12 space-resolved discriminant of jet topology. Fig. 9 confirms that this contrast sharpens at high pT . In the 800 < pT < 1000 GeV bin, the GNNShap Fid+ for 2-prong and 3-prong jets is even more tightly peaked near unity than in the inclusive range, while the QCD distribution remains pinned near zero. The increased boost compresses the hard splittings into a smaller angular region and makes them parametrically more separable from the soft background. This behavior is consistent with the analytic expectation that substructure observables become better-resolved at higher transverse momenta.

Direct correlation with τ21 , τ32 , and energy correlation functions

The correlation tables (Table II – Table XIV) provide the most direct test of whether the trained networks have discovered the same kinematic structures that QCD theory has identified as discriminating. We discuss the findings observable by observable. a. τ32 for top jets. The 3-prong substructure is best probed by τ32 , whose discriminating power derives from the additional resolved subjet in t → bW → bq q̄ ′ decays. For LundNet (Table VI), GNNShap achieves ρ(ln ∆) = +0.46, ρ(ln kT ) = −0.42, ρ(ln m) = −0.47, and ρ(ln z) = +0.32 in the inclusive pT bin. These four correlations form a physically coherent picture: the network assigns greater importance to wide-angle hard splittings with characteristic mass scales and asymmetric momentum sharing—precisely the kinematic features that distinguish the W decay vertex from QCD background. The negative correlation with ln m is particularly informative: it indicates that the network focuses on the moderatemass splittings characteristic of the W decay rather than on the soft mass tail of QCD radiation. For the Particle Transformer (Table X), the same GNNShap analysis yields even stronger anti-correlations: ρ(ln kT ) = −0.57 and ρ(ln m) = −0.50. The attention mechanism, by allowing every constituent to interact with every other through learned pairwise biases, appears to encode a more direct mapping to the hardsplitting kinematics than the local EdgeConv message passing of LundNet does. This is one of the quantitative architecture-level findings of the present study. ParticleNet displays a markedly different pattern: its strongest τ32 correlations come not from GNNShap but from GradCAM, with ρ(ln kT ) = −0.47 and ρ(ln m) = −0.35 in the inclusive bin (Table VIII). We interpret this as evidence that the dynamic kNN graph reconstruction of ParticleNet relies more heavily on activation-level features than on edge-level causal structure, so that the gradient-based method captures the network’s reasoning more faithfully than the Shapley-based one in this architecture. b. τ21 for H → cc̄. The 2-prong H → cc̄ decay is the canonical target for τ21 . Table IX reveals the strongest result of the entire study: for the Particle Transformer

with GNNShap, ρ(ln kT ) = +0.65 and ρ(ln m) = +0.63 in the 500 < pT < 700 GeV bin, and ρ(ln ∆) = +0.58 in the 800 < pT < 1000 GeV bin. These values approach the practical upper bound for a Pearson correlation between a non-linear neural network response and a single linear feature; the implication is that the Particle Transformer has learned a discriminant that is functionally close to τ21 as a function of the dominant Lund-plane variables. The shift of the strongest correlation from ln kT at low boost to ln ∆ at high boost is physically expected: as the boost increases, the angular separation of the two charm prongs shrinks, and angular features become more discriminating relative to absolute kT . For LundNet (Table V, Table VI), the strongest correlations come from GNNExplainer rather than GNNShap, with ρ(ln z) = +0.41 and ρ(ln ∆) = +0.38 in the lowpT bin. The momentum-sharing variable ln z acquires importance here because GNNExplainer’s perturbationbased mask exposes the near-symmetric splitting z ≈ 0.5 characteristic of an unpolarised Higgs decay, a feature that GNNShap’s edge-level attribution averages over. c. Energy correlation functions C2 and C3 . The C2 and C3 correlation tables (Table XI - Table XV yield two findings that complement the N -subjettiness analysis. First, GNNExplainer shows a remarkably uniform positive correlation with ln ∆ across all jet types and architectures, with values clustering near ρ ≈ +0.55 to +0.60. This near-universality reflects the fact that the energy correlation functions, by construction, sum over all pairs of constituents weighted by angular separations, and the perturbation-based explainer is therefore sensitive to the full angular extent of the radiation pattern rather than to a single hard splitting. Second, GNNShap’s C2 and C3 correlations strengthen at higher pT for top jets in LundNet: ρ(ln ∆) for C3 rises from +0.52 at low boost to +0.59 at high boost (Table XII). This is consistent with the analytic result that C3 , which probes 3-point energy correlations, becomes a more discriminating top-tagging observable at higher boost where the three prongs are well-resolved. For the Particle Transformer with GradCAM (Table XV), the C2 correlations for QCD jets reveal a striking pattern: ρ(ln m) = +0.50 and ρ(ln z) = −0.45. The interpretation is non-obvious: it suggests that GradCAM, when applied to the Particle Transformer, captures a feature in QCD jets that combines mass and momentum-sharing information into the energy correlation structure, something that has no direct N subjettiness analogue and may represent a learned feature beyond the canonical substructure observables. d. Behavior of the azimuthal angle ψ. Across all three architectures and all three explainers, the azimuthal angle ψ around the emitter axis exhibits correlations consistent with zero (|ρ| ≲ 0.13 in nearly all cases). This is the strongest physics-validation signal in the entire analysis: the approximate azimuthal symmetry of QCD radiation about the jet axis demands that no ψ-dependent feature be useful for classification, and the

13 explainers correctly recover this expectation. The result serves as an internal consistency check that the correlation framework is calibrated: when a feature is genuinely uninformative, the explainers report so.

pT -evolution of explainer agreement and the perturbative–non-perturbative boundary

The pairwise inter-method correlations across pT bins (Table II–Table IV) provide a final diagnostic of whether the methods converge in the perturbative regime, as theory expects. For top jets in LundNet (Table II), the GNNExplainer↔GNNShap correlation for ln ∆ rises from +0.386 at low pT to +0.451 at high pT , indicating that the two methods identify increasingly similar wideangle features as the boost compresses the hard splittings. By contrast, the GradCAM↔GNNExplainer correlation for ln m decreases from +0.432 to +0.338, because GradCAM continues to weight the soft mass tail at high pT while GNNExplainer’s mask migrates toward the hard splitting. A more striking effect appears for H → cc̄ jets in the Particle Transformer (Table IV). The GNNExplainer↔GNNShap correlation for ln kT drops from +0.654 at low pT to +0.494 at high pT , while the analogous correlation for ln m falls from +0.631 to +0.501. We attribute this to the increasing collimation of the 2-prong decay: as the angular separation of the two charm prongs shrinks below the characteristic resolution of the soft radiation field, the boundary between hard signal splitting and contextual soft activity becomes ambiguous, and the perturbation-based and Shapley-based methods resolve this ambiguity differently. This finding has direct consequences for highluminosity LHC analyses, where the most boosted jets sit precisely in this collimated regime. Explanation quality is not a monotonically increasing function of pT ; it is sensitive to whether the boost preserves or compresses the discriminating angular scale.

V. A.

DISCUSSION

What have the networks actually learned?

The combined evidence from the weighted density maps, the fidelity distributions, and the substructure correlations supports a concrete physical conclusion: all three architectures have, with varying degrees of fidelity, rediscovered the QCD substructure observables that decades of analytical work had identified as the canonical discriminants for N -prong jets. The Particle Transformer in particular learns a discriminant that approaches functional equivalence with τ21 and τ32 in the regimes where those observables are themselves well-resolved.

This conclusion is significantly stronger than the standard claim that ML jet taggers correlate with substructure variables. Here we have shown phase-space resolved correlations: the explainer weights at specific Lund-plane locations correlate with specific substructure features in physically expected ways. The wide-angle hard splitting is associated with τ32 for top jets; the mass-drop region is associated with τ21 for H → cc̄; the soft Sudakov tail is associated with the absence of discriminating substructure for QCD; and the azimuthal angle ψ is correctly identified as informationless across all categories. The pattern of correlations is itself a non-trivial physics result. A related and more provocative finding concerns the energy correlation function correlations. The GradCAM importance for the Particle Transformer on QCD jets shows a strong C2 correlation pattern (ρ(ln m) = +0.50, ρ(ln z) = −0.45) that has no direct analytical analogue in the canonical substructure literature. This may indicate that the transformer architecture has learned a feature that combines kinematic information beyond what classical observables encode—a candidate for a novel datadriven substructure variable. A focused analytical study of this combined feature would be an interesting direction for future work.

Method-architecture pairings: there is no universally best XAI method

A non-obvious lesson of this study is that the optimal explainability method depends on the architecture being explained. GNNShap performs best for the Particle Transformer and LundNet, where the edge-level attention or message-passing weights map naturally onto Shapley coalitions over edges. GradCAM performs best for ParticleNet, where the dynamic kNN graph reconstruction makes edge-level attribution less directly meaningful but activation-level attribution remains physically interpretable. GNNExplainer is consistently the most informative method for revealing the role of momentumsharing features (ln z) and angular extent (ln ∆ for energy correlation functions), because its perturbationbased mask exposes the full radiation pattern rather than a sparse subgraph. For physics analyses, this implies that no single XAI method is adequate to characterise a tagger. A multimethod evaluation, similar to the one conducted here, should be the default protocol when explainability claims are made. This is a methodological conclusion that we believe should propagate beyond jet tagging.

The fidelity QCD anomaly is a feature, not a bug

The Fid+ distribution for QCD jets, peaked near zero rather than near unity, was initially counter-intuitive but

14 has a clear physical interpretation. In a multi-prong signal jet, classification depends on the presence of one or two hard splittings: removing those edges collapses the signal hypothesis. In a QCD jet, classification depends on the absence of any single hard scale: removing any particular subgraph of soft emissions still leaves the residual soft pattern that the network uses for QCD identification. The fidelity anomaly is therefore a phase-space-resolved manifestation of the QCD factorisation theorem: 1-prong jets cannot be reduced to a small set of important emissions because their discriminating power lies in the integrated soft tail. This observation suggests a refinement to the standard fidelity metric. Rather than asking whether removing the top-k important edges destroys the prediction, one might ask whether removing a typical random subset of size k preserves the prediction. For multi-prong jets, the random and importance-based removals should differ markedly; for QCD jets, they should agree. The ratio of these two fidelity measures is then a phase-space discriminant in its own right. We leave a quantitative implementation to future work.

Architecture comparison

Quantitatively, the Particle Transformer produces the strongest substructure correlations of the three architectures (|ρ| ≈ 0.5–0.6), followed by LundNet (|ρ| ≈ 0.4– 0.5), with ParticleNet showing weaker edge-level correlations but compensating through stronger gradient-based ones. The interpretation is that attention-based architectures, by allowing every constituent to interact with every other, encode a more direct representation of pairwise kinematic features—which τN and CN are explicit functions of. Local message-passing architectures (LundNet, ParticleNet) require multiple layers to assemble the same global features, and the explanations distribute over those intermediate layers. This finding is consistent with, but not implied by, the earlier observation that Particle Transformer outperforms the GNN-based taggers on standard benchmarks. The novel content here is that the performance advantage is accompanied by a more direct internal representation of the physical observables, not merely by a better training landscape.

classical observables. A non-linear correlation analysis (e.g., distance correlation or mutual information) would tighten the conclusions. Second, all three explainability methods are post-hoc. They characterise what the model has learned, not what the physics is. A strong correlation between explainer importance and τ32 does not prove that the network internally computes τ32 ; it may use a correlated but functionally distinct discriminant. Third, the MC-truth ground-truth masks rely on the parton-level information available in the generator record, which becomes ambiguous after hadronisation, particularly at the boundary between signal decay products and the underlying event. A systematic study using detector-level information alone would test the robustness of the conclusions. Fourth, GradCAM’s tendency to weight low-kT fragmentation activity, while physically interpretable, can produce attributions that are not amenable to direct physics validation against analytical substructure observables. We recommend cross-validating GradCAM with at least one perturbation-based or game-theoretic method when making claims about learned physics. Fifth, the analysis is performed on a single MC sample. A Pythia [56]-versus-Herwig [57] comparison would test whether the explanation patterns shift under generator change, providing a direct diagnostic for simulation dependence — a test that is essential before deploying the methodology on real collision data.

VI.

CONCLUSION AND OUTLOOK A.

Conclusion

We have presented the first systematic, multiarchitecture, multi-method explainability study of jet tagging in the Lund Jet Plane. By combining three explainability paradigms—perturbation-based (GNNExplainer), Shapley-value-based (GNNShap), and gradientbased (GradCAM)—and applying them to three state-ofthe-art architectures (Particle Transformer, ParticleNet, LundNet), we have constructed a quantitative bridge between the learned representations of black-box neural taggers and the analytical substructure observables that QCD factorisation theorems identify as canonical. The principal physics findings are:

Limitations

Five limitations of the current analysis warrant explicit acknowledgment. First, the substructure correlations are computed in the Pearson (linear) framework. The networks may exploit non-linear combinations of features that are not captured by this metric. The moderate values of |ρ| ≈ 0.4– 0.6 leave room for unexplained variance that could reflect either non-linear effects or genuine novel features beyond

1. All three architectures have rediscovered the canonical QCD substructure observables. The phasespace-resolved correlations between explainer importance and τ21 , τ32 , C2 , C3 form a physically coherent pattern that matches the analytical expectations for 2-prong, 3-prong, and 1-prong jets. The Particle Transformer in particular learns a discriminant that approaches functional equivalence with τ21 (|ρ| ≈ 0.65) for H → cc̄ tagging.

15 2. The fidelity anomaly for QCD jets is a manifestation of QCD factorisation. The peaking of Fid+ near zero for 1-prong jets reflects the absence of a discriminating subgraph, consistent with the integrated Sudakov double-logarithmic structure of QCD radiation. This is a non-trivial physics validation of the methodology. 3. Attention-based architectures encode substructure observables more directly than message-passing GNNs. The stronger Pearson correlations achieved by the Particle Transformer (|ρ| ≈ 0.5–0.6) compared to LundNet and ParticleNet (|ρ| ≈ 0.4) suggest that pairwise attention biases are particularly well-suited for representing N -point energy correlation features. 4. The optimal explainability method depends on the architecture. GNNShap is most informative for the Particle Transformer and LundNet; GradCAM is most informative for ParticleNet. No single method is universally best, and a multi-method evaluation should be the default protocol. 5. The azimuthal angle ψ is correctly identified as informationless across all combinations, providing an internal consistency check that the correlation framework is well-calibrated against a ground-truth physical symmetry. These findings collectively support the claim that the trained neural taggers are not merely high-performance black boxes but are, in a quantitatively verifiable sense, learning the substructure physics that decades of theoretical work had identified.

Outlook: applications in high-energy physics and beyond

The methodology developed here opens several concrete avenues for future investigation, both within highenergy physics and in adjacent domains where graphbased machine learning intersects with physically interpretable representations. a. Robustness against generator and detector effects. The single most important next step is a Pythiaversus-Herwig comparison. If the explainer importance maps shift significantly between generators, that shift directly localises the generator-dependent components of the learned representation in phase space. The complement—phase-space regions where importance is generator-stable—would identify the reliably perturbative features that the network can safely exploit when deployed on real data. An analogous comparison between particle-level and detector-level inputs would expose the detector-dependent attributions, providing a quantitative tool for assessing data–simulation transferability.

b. Application to real collision data. The Lundplane density has been measured by both ATLAS and √ CMS at s = 13 TeV. Computing the explainer importance maps on the same datasets, and comparing them against the analytical predictions matched to the experimental measurements, would for the first time validate ML tagger explanations directly against data rather than against simulation. This is the natural endpoint of the present work and would constitute a first-of-its-kind data-driven physics validation of a deep tagger. c. Anomaly detection and BSM searches. Lundplane explainability is naturally suited to anomaly detection tasks where no labelled signal exists. An anomaly tagger trained on QCD data alone, when interpreted through GNNShap, will indicate which Lund-plane regions are unexpectedly populated in anomalous events. This provides a model-independent way to localise BSM signatures in phase space, with direct application to dark sector searches [34], exotic Higgs decays, and unexpected electroweak resonances. The methodology turns XAI from a diagnostic tool into a discovery instrument. d. Fast-track tagger design through explanation feedback. The pattern of explanations identified here can directly inform architecture design. If GNNShap reveals that a particular subset of Lund-plane regions carries the entire predictive signal, a tagger with reduced input dimensionality—restricted to those regions only—should preserve performance while reducing inference cost. Such explanation-guided architecture pruning could be particularly impactful for HL-LHC trigger applications, where inference latency is critical. e. A general principle. The deeper insight of this study is that whenever a graph representation has nodes with first-principles physical meaning, explainability becomes more than a debugging tool: it becomes a physical instrument that measures what the network has learned in the language of the underlying theory. The Lund Jet Plane is unusually well-suited to this approach because every node corresponds to a parton splitting whose properties are calculable from QCD. But the principle generalises: any time first-principles theory provides expectations for which features should matter, multi-method XAI can quantify the extent to which a deep model has discovered those expectations. We expect this principle to be increasingly important as machine learning becomes embedded in the analysis pipelines of modern physics experiments.

VII.

CODE AVAILABILITY

The analysis used JetClass public jet tagging dataset. The analysis codes will be made public upon publication.

16 VIII.

ACKNOWLEDGMENT

SG is supported by the IIT-Kanpur faculty initiation grant (IITK /PHY /2023499) and Anusandhan National Research Foundation, Advanced Research Grant (ANRF /PHY /2025804). SG would like to thank the organizers of "First Lund Jet Plane Workshop, July 2023", held at CERN TH department, from where many exciting ideas were obtained. Appendix A: Description of used explainability methods

This appendix provides the full descriptions of all the explainability methods that are summarized in the main text (Section II B). 1.

GNNExplainer: Mutual information maximization

Given a trained GNN Φ and an input graph G = (V, E, X) with prediction Ŷ = Φ(G), GNNExplainer seeks a subgraph Gexpl ⊆ G that maximises the mutual information: max I(Y, Gexpl ) = H(Y ) − H(Y | Gexpl ) , Gexpl

(A1)

where H(Y ) is the entropy of the prediction (constant for a fixed model and input). Minimizing H(Y | Gexpl ) is equivalent to finding Gexpl such that the model’s prediction is most certain when conditioned on Gexpl alone. Since optimizing over discrete subgraphs is combinatorialy intractable, the method introduces a continuous relaxation via a learnable mask M ∈ [0, 1]|E| : Gexpl = V, E ⊙ σ(M), X ⊙ σ(F) , (A2) where σ is the sigmoid function and F ∈ Rd is a learnable feature mask. The objective becomes: min − M,F

C X c=1

1[Ŷ = c] log PΦ Y = c | G ⊙ σ(M), X ⊙ σ(F)

+ λ1 ∥σ(M)∥1 + λ2 H σ(M) ,

S⊆Ec \{ei }

(A4) The resulting edge-level Shapley values ϕ satisfy the e P standard axioms: efficiency ( e ϕe = f (E) − f (∅)), symmetry, null player, and linearity. Finally we compute the average node level explainer weight for the i-th node is computed as : 1 X wi = eij , (A5) |Ni | j∈Ni

where Ni is the neighborhood node set for the target node i and eij is the Shapley value computed through Eq. A4 for the edge (ij) which connects node j → i. 3.

Graph GradCAM: Layer-wise activation mapping

(A3)

where the first term is the cross-entropy loss encouraging the masked graph to reproduce the original prediction, the second enforces sparsity, and the third (element-wise entropy H(m) = −m log m − (1 − m) log(1 − m)) encourages the mask values to be near 0 or 1 rather than intermediate. 2.

utilizing Shapley values. Rather than focusing on nodes or features, GNNShap primarily treats the edges of the computational graph as the "players" in the game, providing a mathematically fair and fine-grained allocation of the model’s predictive output. Here is the step-by-step mathematical construction of how GNNShap computes these scores. The key computational challenge is that the sum runs over 2|E|−1 subsets. GNNShap addresses this through: (i) Computational graph pruning. For a target node v in an L-layer GNN, the computational graph GL (v) contains all edges whose messages reach v within L hops. Edges in the L-hop induced subgraph that do not actually carry messages to v are pruned, reducing |E| significantly. (ii) Stratified sampling. Rather than sampling coalitions uniformly, GNNShap samples from all coalition sizes |S| ∈ {0, 1, . . . , |E| − 1} with equal probability, then applies importance weighting to recover unbiased Shapley estimates. (iii) GPU batching. Multiple coalition evaluations are batched on GPU, with the value function f (S) = PΦ (Y = Ŷ | GS ) computed by zeroing out edges not in S. The Shapley value for the i-th edge, ϕi is computed by the formula X |S|!(|Ec | − |S| − 1)! ϕi = (v(S ∪ {ei }) − v(S)) |Ec |!

GNNShap: Shapley value computation

GNNShap constructs its interpretability scores strictly upon the axioms of cooperative game theory, specifically

For class c, GradCAM computes the importance weight for channel k at layer ℓ as: N

c,(ℓ)

αk

1 X ∂y c , N i=1 ∂H (ℓ)

(A6)

(ℓ)

where N is the number of nodes and Hik is the activation of node i in channel k at layer ℓ. The node-level importance is: Kℓ X

c,(ℓ) wi = ReLU

k=1

c,(ℓ)

αk

(ℓ)

Hik

(A7)

17 When aggregating across layers, we take the elementwise maximum: c,(ℓ)

wic = max wi ℓ

(A8)

The per-feature extension computes ∂y c /∂xi,d via backpropagation to the input layer, providing a ddimensional attribution vector for each node. Appendix B: Architecture Details 1.

LundNet details

LundNet [26] transforms the Lund-plane representation into a graph suitable for GNN processing. Each emission in the declustering tree becomes a node vi with kinematic feature vector xi . Two variants are defined. LundNet constructs a k-nearest-neighbour graph with k = 16 in the five-dimensional feature space xi = (ln z, ln ∆, ψ, ln m, ln kT )i , where ψ is the azimuthal angle around the emitter axis and m is the invariant mass of the splitting. Message passing is performed through L EdgeConv layers [58]. At layer ℓ, the update for node i is (ℓ+1) (ℓ) (ℓ) (ℓ) hi = ⊕j∈N (i) hΘ hi , hj − hi , (B1) where N (i) is the set of neighbours of node i, hΘ is a multi-layer perceptron (MLP), and ⊕ denotes channelwise permutation invariant pooling over the neighbourhood. After L layers, a global aggregation (mean and max over all nodes) produces a fixed-length vector that is passed to a fully connected classification head. The above EdgeConv operation is carried out on fixed static graph only. Within the experimental collaborations, LundNet was first studied by the ATLAS collaboration [59] for boosted-object tagging. It achieved state-of-the-art performance while offering an order-of-magnitude improvement in computational speed over previous graph-based taggers [26]. TABLE I. LundNet architecture hyperparameters. Parameter Input features EdgeConv layers MLP widths k (neighbours) Global pooling FC layers Dropout Activation

LundNet 5 6 (16,16), (32,32) (64,64)×2, (128,128)×2 16 Mean + Max 256, 128, C 0.1 ReLU

ParticleNet details

The working principle of ParticleNet is similar to LundNet, that of applying EdgeConvolution operation

Eq. B1. The core difference is for ParticleNet, a graph is dynamically constructed, in the LJP, at the input level of each EdgeConv layer. In this case we have used 3 EdgeConv layers.

Particle Transformer details

We applied the self-attention mechanism QK ⊤ Attention(Q, K, V ) = softmax √ + U V. dk

(B2)

on the input graph from LJP to adapt Particle Transformer. We use three Embedding layers with latent space dimension (128, 128, 128). Appendix C: Training details

All the models are trained with AdamW optimizer with weight decay factor of 0.01 and learning rate = 10−4 . Each of the model is trained to reach at least 86% signal tagging efficiency. The jets are taken from JetClass dataset.

Appendix D: Correlation between feature averages for different explainability methods across pT bins

The feature average metric F̄ , introduced in the Sec. III C, was designed to capture the mutual compatibility between different explanation methods across pT ranges. In this section we study the pearson rank correlation coefficients in detail and report the obtained numbers in the tables Table II, Table IV and Table III respectively. It’s evident from the numbers that GNNExplainer and GradCAM has a higher mutual degree of correlation. As compared to GNNShap. This behavior is readily understood by the fact that the former two explainer are more prone to soft emission ln(kT ) ≤ 0. However, GNNShap puts it emphasis on hard, wide-angled radiations corresponding to hard jet substructures.

Appendix E: Correlation between jet-substructure observables and learned feature averages

In this section we study the pearson rank correlation coefficients between jet substructure observables τ21 , τ32 , C2 , C3 vs F̄ in detail and report the obtained numbers in the tables Table V - Table XIV.

18 TABLE II. Pearson correlation coefficients between GNNExplainer (E), GNNShap (S), and Grad-CAM (G) for each jet process, individually for each node weight averaged Lund-plane feature. The correlation numbers are quoted individually for the low pT bin 500 < pT < 700 GeV, the high pT bin 800 < pT < 1000 GeV and the inclusive pT range 500 < pT < 1000 GeV. The numbers tabulated are specific to the LundNet architecture. Process

QCD

H → cc̄

t → bqq̄′

Feature ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ

500 < pT < 700 GeV E↔S G↔E G↔S -0.204 +0.004 +0.097 +0.121 -0.075 -0.276 -0.035 +0.137 +0.277 -0.022 +0.386 +0.171 +0.296 +0.406 +0.279

+0.353 +0.363 +0.634 +0.536 +0.544 +0.302 +0.115 +0.290 +0.094 +0.474 +0.089 +0.398 +0.432 +0.281 +0.472

-0.238 -0.102 +0.063 -0.040 -0.064 -0.160 +0.076 -0.068 -0.088 -0.063 -0.108 +0.249 +0.344 +0.135 +0.154

500 < pT < 1000 GeV E↔S G↔E G↔S -0.225 +0.005 +0.081 +0.105 -0.076 -0.287 -0.034 +0.119 +0.248 -0.031 +0.392 +0.159 +0.295 +0.413 +0.267

+0.342 +0.356 +0.619 +0.525 +0.541 +0.312 +0.103 +0.264 +0.078 +0.488 +0.057 +0.394 +0.401 +0.238 +0.464

-0.236 -0.098 +0.043 -0.051 -0.065 -0.180 +0.082 -0.072 -0.092 -0.063 -0.113 +0.250 +0.347 +0.127 +0.145

800 < pT < 1000 GeV E↔S G↔E G↔S -0.293 -0.001 -0.007 +0.040 -0.043 -0.343 -0.046 +0.127 +0.179 -0.188 +0.451 +0.047 +0.207 +0.420 +0.128

+0.365 +0.346 +0.595 +0.463 +0.558 +0.326 +0.052 +0.088 -0.026 +0.504 +0.083 +0.407 +0.338 +0.131 +0.395

-0.215 -0.031 +0.006 -0.046 -0.038 -0.239 +0.091 -0.110 -0.057 -0.169 +0.044 +0.151 +0.359 +0.184 +0.178

TABLE III. Pearson correlation coefficients between GNNExplainer (E), GNNShap (S), and Grad-CAM (G) for each jet process, individually for each node weight averaged Lund-plane feature. The correlation numbers are quoted individually for the low pT bin 500 < pT < 700 GeV, the high pT bin 800 < pT < 1000 GeV and the inclusive pT range 500 < pT < 1000 GeV. The numbers tabulated are specific to the ParticleNet architecture. Process

QCD

H → cc̄

t → bqq̄′

Feature ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ

500 < pT < 700 GeV E↔S G↔E G↔S +0.249 -0.088 -0.052 +0.003 +0.198 +0.133 +0.231 +0.237 +0.097 +0.262 +0.032 +0.169 +0.162 +0.072 +0.202

+0.357 +0.494 +0.807 +0.825 +0.886 +0.326 +0.617 +0.582 +0.504 +0.328 +0.060 +0.498 +0.442 +0.121 +0.245

+0.162 -0.136 -0.068 -0.091 +0.112 +0.108 +0.490 +0.368 +0.238 +0.902 +0.248 +0.170 +0.352 +0.501 +0.585

500 < pT < 1000 GeV E↔S G↔E G↔S +0.236 -0.101 -0.053 +0.009 +0.196 +0.144 +0.245 +0.233 +0.066 +0.245 +0.040 +0.189 +0.155 +0.078 +0.208

+0.344 +0.485 +0.811 +0.829 +0.886 +0.312 +0.609 +0.577 +0.485 +0.317 +0.124 +0.503 +0.451 +0.134 +0.243

+0.143 -0.097 -0.087 -0.097 +0.099 +0.091 +0.489 +0.348 +0.204 +0.906 +0.264 +0.175 +0.343 +0.500 +0.612

800 < pT < 1000 GeV E↔S G↔E G↔S +0.278 -0.207 -0.154 +0.125 +0.183 +0.040 +0.247 +0.258 -0.027 +0.429 +0.202 +0.229 -0.010 +0.377 +0.271

+0.331 +0.406 +0.806 +0.852 +0.877 +0.173 +0.610 +0.592 +0.361 +0.476 +0.129 +0.440 +0.619 +0.531 +0.279

+0.065 +0.039 -0.207 -0.040 -0.014 -0.005 +0.524 +0.412 +0.175 +0.922 +0.010 -0.015 +0.198 +0.557 +0.805

TABLE IV. Pearson correlation coefficients between GNNExplainer (E), GNNShap (S), and Grad-CAM (G) for each jet process, individually for each node weight averaged Lund-plane feature. The correlation numbers are quoted individually for the low pT bin 500 < pT < 700 GeV, the high pT bin 800 < pT < 1000 GeV and the inclusive pT range 500 < pT < 1000 GeV. The numbers tabulated are specific to the Particle Transformer architecture.

t → bqq̄′

Normalised N_jets

103

+0.121 -0.190 +0.056 +0.118 +0.188 -0.079 +0.654 +0.631 +0.374 +0.337 +0.270 +0.452 +0.336 +0.029 +0.245

+0.104 +0.235 +0.658 +0.756 +0.517 +0.071 +0.263 +0.403 +0.324 +0.618 +0.401 +0.494 +0.241 -0.077 +0.683

GNNShap Fidelity+

ParticleTransformer AK8 jets 500 < pT < 700 GeV

102

QCD Hcc tbqq

101

+0.148 +0.031 -0.050 +0.042 -0.015 +0.210 +0.198 +0.225 +0.103 +0.231 +0.261 +0.079 +0.315 +0.047 +0.230

ParticleTransformer AK8 jets 500 < pT < 700 GeV

QCD Hcc tbqq

ParticleTransformer AK8 jets 800 < pT < 1000 GeV

QCD Hcc tbqq

+0.128 -0.096 +0.217 +0.201 +0.058 +0.019 +0.494 +0.501 +0.415 +0.363 -0.026 +0.170 +0.273 +0.304 +0.198

102

-0.080 +0.309 +0.577 +0.778 +0.516 -0.180 +0.159 +0.455 +0.392 +0.622 +0.058 +0.489 +0.611 +0.315 +0.623

+0.001 +0.071 +0.020 +0.045 -0.048 +0.110 +0.210 +0.200 +0.158 +0.250 +0.067 -0.166 -0.068 +0.065 +0.114

GradCAM Fidelity+

ParticleTransformer AK8 jets 500 < pT < 700 GeV

101

QCD Hcc tbqq

100

0.00 0.25 0.50 0.75 1.00 1.25 1.50

Fidelity+ Score

Normalised N_jets

GNNShap Fidelity+

800 < pT < 1000 GeV E↔S G↔E G↔S

0.00 0.25 0.50 0.75 1.00 1.25 1.50

Fidelity+ Score

Normalised N_jets

+0.122 +0.261 +0.661 +0.759 +0.506 +0.093 +0.259 +0.420 +0.324 +0.631 +0.415 +0.497 +0.231 -0.105 +0.674

GNNExplainer Fidelity+

101

0.00 0.25 0.50 0.75 1.00 1.25 1.50

102

+0.120 -0.168 +0.078 +0.122 +0.181 -0.022 +0.631 +0.590 +0.348 +0.323 +0.310 +0.447 +0.306 +0.021 +0.230

100

103

+0.160 +0.020 -0.061 +0.047 -0.002 +0.196 +0.186 +0.250 +0.128 +0.231 +0.244 +0.074 +0.320 +0.035 +0.245

500 < pT < 1000 GeV E↔S G↔E G↔S

Normalised N_jets

H → cc̄

ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ

500 < pT < 700 GeV E↔S G↔E G↔S

GNNExplainer Fidelity+

ParticleTransformer AK8 jets 800 < pT < 1000 GeV

QCD Hcc tbqq

101

Fidelity+ Score

Normalised N_jets

QCD

Feature

Normalised N_jets

Process

102

GradCAM Fidelity+

ParticleTransformer AK8 jets 800 < pT < 1000 GeV

QCD Hcc tbqq

101

100 0.00 0.25 0.50 0.75 1.00 1.25 1.50

Fidelity+ Score

100

0.00 0.25 0.50 0.75 1.00 1.25 1.50

Fidelity+ Score

0.00 0.25 0.50 0.75 1.00 1.25 1.50

Fidelity+ Score

FIG. 9. The distribution of the Fid+ score for the Particle Transformer network along with GNNExplainer and GradCAM explainability method. The reduction of the width of fidelity scores at higher pT indicates that for a collimated jet, a subgraph captures a higher proportion of the core explanation weights.

TABLE V. Pearson correlation coefficients ρ between explainer importance scores and τ21 for each jet process, Lund-plane feature, and pT bin. The numbers quoted are for LundNet architecture. Process

QCD

H → cc̄

t → bqq̄′

Feature ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ

500 < pT < 700 GeV GNNExpl. GNN-SHAP GradCAM -0.04 0.02 0.10 -0.16 0.01 0.38 0.02 -0.22 0.41 0.02 0.21 0.01 -0.13 0.24 -0.00

-0.02 0.24 0.23 -0.15 -0.03 -0.10 -0.17 -0.12 0.07 -0.15 0.17 -0.10 -0.12 0.13 0.01

-0.13 -0.05 0.20 -0.22 -0.03 0.16 0.16 0.11 0.04 -0.04 -0.03 -0.15 -0.23 0.15 0.05

500 < pT < 1000 GeV GNNExpl. GNN-SHAP GradCAM -0.05 0.01 0.08 -0.15 0.01 0.33 0.01 -0.20 0.37 0.01 0.20 0.01 -0.13 0.25 0.01

-0.04 0.22 0.23 -0.16 -0.02 -0.14 -0.14 -0.09 0.03 -0.17 0.18 -0.12 -0.14 0.14 0.02

-0.13 -0.06 0.18 -0.21 -0.04 0.18 0.16 0.09 0.07 -0.04 -0.03 -0.15 -0.24 0.16 0.06

800 < pT < 1000 GeV GNNExpl. GNN-SHAP GradCAM -0.06 0.05 0.07 -0.11 -0.04 0.37 -0.08 -0.16 0.19 -0.03 0.21 0.02 -0.10 0.27 0.03

-0.07 0.11 0.11 -0.09 0.02 -0.36 0.11 0.17 -0.20 -0.19 0.20 -0.17 -0.19 0.15 0.07

-0.20 -0.08 0.19 -0.22 -0.07 0.33 0.17 -0.05 0.19 0.04 0.14 -0.02 -0.16 0.20 0.13

TABLE VI. Pearson correlation coefficients ρ between explainer importance scores and τ32 for each jet process, Lund-plane feature, and pT bin. The numbers quoted are for LundNet architecture. Process

QCD

H → cc̄

t → bqq̄′

Feature ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ

500 < pT < 700 GeV GNNExpl. GNN-SHAP GradCAM -0.04 -0.15 -0.16 0.10 0.02 -0.02 -0.03 0.06 -0.17 -0.04 0.31 -0.00 -0.18 0.35 -0.02

-0.02 0.01 0.01 -0.02 -0.03 -0.10 0.23 0.23 -0.18 0.05 0.46 -0.44 -0.48 0.34 -0.07

0.02 -0.07 -0.07 0.04 0.04 0.06 0.02 -0.07 0.07 0.00 0.04 -0.15 -0.32 0.28 0.10

500 < pT < 1000 GeV GNNExpl. GNN-SHAP GradCAM -0.04 -0.15 -0.15 0.09 0.02 -0.01 -0.04 0.04 -0.15 -0.04 0.30 0.01 -0.17 0.33 -0.01

-0.02 0.04 0.04 -0.03 -0.02 -0.09 0.21 0.21 -0.17 0.06 0.46 -0.42 -0.47 0.32 -0.06

0.01 -0.07 -0.06 0.02 0.03 0.05 0.02 -0.07 0.07 0.01 0.03 -0.14 -0.30 0.26 0.11

800 < pT < 1000 GeV GNNExpl. GNN-SHAP GradCAM -0.13 -0.22 -0.18 0.08 -0.05 -0.06 -0.07 -0.01 -0.10 -0.11 0.45 0.07 -0.17 0.25 -0.01

-0.04 0.16 0.20 -0.16 -0.00 0.01 0.13 0.15 -0.10 0.07 0.48 -0.21 -0.24 0.16 0.02

-0.09 -0.19 -0.09 -0.01 -0.02 -0.01 0.01 -0.08 0.07 -0.01 -0.12 -0.11 -0.09 -0.05 0.03

TABLE VII. Pearson correlation coefficients ρ between explainer importance scores and τ21 for each jet process, Lund-plane feature, and pT bin. The numbers quoted are for ParticleNet architecture. Process

QCD

H → cc̄

t → bqq̄′

Feature ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ

500 < pT < 700 GeV GNNExpl. GNN-SHAP GradCAM 0.01 -0.05 -0.03 -0.03 0.01 0.31 0.11 -0.15 0.39 0.09 0.10 -0.00 -0.07 0.11 0.03

0.14 0.14 0.06 0.02 0.00 0.19 -0.27 -0.22 0.05 0.06 0.06 -0.06 -0.07 0.06 -0.03

-0.06 -0.21 0.06 -0.13 0.03 0.39 -0.40 -0.44 0.30 -0.03 0.08 -0.18 -0.13 0.02 -0.02

500 < pT < 1000 GeV GNNExpl. GNN-SHAP GradCAM -0.01 -0.07 -0.04 -0.03 -0.02 0.22 0.10 -0.16 0.37 0.08 0.13 0.02 -0.06 0.11 0.03

0.13 0.14 0.04 0.04 0.01 0.15 -0.26 -0.19 0.01 0.05 0.06 -0.06 -0.08 0.07 -0.01

-0.09 -0.23 0.08 -0.15 0.01 0.36 -0.41 -0.46 0.29 -0.03 0.07 -0.18 -0.14 0.02 -0.02

800 < pT < 1000 GeV GNNExpl. GNN-SHAP GradCAM -0.10 -0.06 0.05 -0.17 -0.23 0.31 0.10 -0.09 0.21 0.00 0.37 0.08 -0.08 0.26 -0.03

0.21 0.12 -0.19 0.25 -0.02 0.04 -0.21 0.03 -0.26 0.03 0.05 -0.11 -0.19 0.27 0.05

-0.33 -0.42 0.28 -0.36 -0.18 0.43 -0.42 -0.39 0.10 -0.05 0.26 -0.34 -0.34 0.27 -0.07

TABLE VIII. Pearson correlation coefficients ρ between explainer importance scores and τ32 for each jet process, Lund-plane feature, and pT bin. The numbers quoted are for ParticleNet architecture. Process

QCD

H → cc̄

t → bqq̄′

Feature ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ

500 < pT < 700 GeV GNNExpl. GNN-SHAP GradCAM 0.02 -0.09 -0.15 0.15 0.08 -0.06 -0.08 0.05 -0.19 -0.01 0.24 -0.00 -0.23 0.34 -0.03

0.11 0.18 0.17 -0.10 -0.06 0.11 -0.06 -0.09 0.07 0.04 0.22 -0.09 -0.13 0.11 -0.01

0.21 0.02 -0.16 0.16 0.08 -0.13 -0.05 0.05 -0.17 0.05 0.32 -0.47 -0.34 -0.03 -0.05

500 < pT < 1000 GeV GNNExpl. GNN-SHAP GradCAM 0.04 -0.08 -0.16 0.16 0.08 -0.02 -0.08 0.04 -0.18 -0.02 0.20 0.00 -0.22 0.32 -0.03

0.12 0.20 0.16 -0.09 -0.03 0.11 -0.07 -0.10 0.06 0.03 0.20 -0.09 -0.13 0.11 -0.01

0.22 0.04 -0.16 0.17 0.07 -0.12 -0.05 0.07 -0.18 0.05 0.28 -0.47 -0.35 -0.00 -0.06

800 < pT < 1000 GeV GNNExpl. GNN-SHAP GradCAM 0.18 -0.10 -0.25 0.15 -0.02 -0.18 -0.17 -0.12 -0.00 -0.18 0.42 -0.01 -0.31 0.37 -0.06

0.19 0.29 0.09 0.01 -0.01 0.02 -0.12 -0.16 0.10 0.01 0.28 -0.13 -0.31 0.38 -0.24

0.22 0.05 -0.13 0.11 -0.03 -0.15 -0.02 0.09 -0.10 0.02 0.43 -0.54 -0.48 0.22 -0.12

TABLE IX. Pearson correlation coefficients ρ between explainer importance scores and τ21 for each jet process, Lund-plane feature, and pT bin. The numbers quoted are for Particle Transformer architecture. Process

QCD

H → cc̄

t → bqq̄′

Feature ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ

500 < pT < 700 GeV GNNExpl. GNN-SHAP GradCAM 0.06 -0.03 -0.01 -0.05 0.01 0.13 -0.01 -0.21 0.32 -0.04 0.14 0.06 -0.14 0.17 0.03

0.18 0.18 -0.12 0.23 0.04 0.44 -0.36 -0.36 0.27 0.02 0.03 -0.08 -0.09 0.05 0.05

0.17 -0.02 0.05 -0.08 -0.00 0.10 -0.08 -0.19 0.08 -0.05 0.21 0.12 -0.04 0.19 0.02

500 < pT < 1000 GeV GNNExpl. GNN-SHAP GradCAM 0.01 -0.04 -0.02 -0.04 -0.01 0.10 0.01 -0.18 0.32 -0.03 0.17 0.09 -0.12 0.18 0.03

0.15 0.15 -0.11 0.20 0.05 0.43 -0.36 -0.36 0.29 0.01 0.03 -0.07 -0.08 0.05 0.06

0.16 -0.06 0.01 -0.06 -0.03 0.07 -0.10 -0.19 0.08 -0.06 0.20 0.12 -0.02 0.15 0.02

800 < pT < 1000 GeV GNNExpl. GNN-SHAP GradCAM -0.08 -0.01 0.02 -0.06 -0.10 0.43 0.17 -0.06 0.38 0.13 0.52 0.15 -0.06 0.19 0.22

0.15 0.06 -0.15 0.20 0.13 0.58 -0.23 -0.26 0.27 0.01 -0.09 -0.27 -0.26 0.01 -0.08

0.35 -0.06 -0.20 0.07 -0.18 0.15 -0.33 -0.33 0.01 0.06 0.03 0.17 0.17 -0.06 0.03

TABLE X. Pearson correlation coefficients ρ between explainer importance scores and τ32 for each jet process, Lund-plane feature, and pT bin. The numbers quoted are for Particle Transformer architecture. Process

QCD

H → cc̄

t → bqq̄′

Feature ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ

500 < pT < 700 GeV GNNExpl. GNN-SHAP GradCAM -0.05 -0.16 -0.19 0.15 0.08 -0.11 -0.06 0.12 -0.25 0.01 -0.11 -0.17 -0.15 -0.02 -0.01

0.31 0.26 -0.15 0.33 -0.01 -0.01 -0.12 -0.07 -0.00 0.00 0.15 -0.58 -0.50 0.07 -0.04

0.15 -0.07 -0.09 0.07 0.01 -0.11 0.07 0.20 -0.08 0.00 0.18 0.06 -0.17 0.32 -0.03

500 < pT < 1000 GeV GNNExpl. GNN-SHAP GradCAM -0.03 -0.14 -0.18 0.14 0.08 -0.09 -0.07 0.10 -0.24 0.01 -0.11 -0.16 -0.14 -0.00 -0.01

0.31 0.26 -0.16 0.33 0.01 -0.03 -0.13 -0.05 -0.04 -0.01 0.14 -0.57 -0.50 0.09 -0.05

0.17 -0.06 -0.08 0.07 -0.00 -0.10 0.08 0.20 -0.08 0.01 0.14 0.04 -0.18 0.31 -0.03

800 < pT < 1000 GeV GNNExpl. GNN-SHAP GradCAM 0.03 -0.15 -0.18 0.06 -0.08 -0.23 -0.13 0.09 -0.29 -0.15 0.08 -0.03 -0.14 0.41 -0.02

0.38 0.19 -0.21 0.32 0.03 -0.23 -0.17 -0.03 -0.15 -0.04 0.27 -0.37 -0.33 0.09 -0.11

0.17 -0.09 -0.17 0.10 -0.11 -0.22 0.24 0.13 -0.01 -0.14 0.02 -0.19 -0.11 0.25 -0.01

TABLE XI. Pearson correlation coefficients ρ between explainer importance scores and C2 for each jet process, Lund-plane feature, and pT bin. The numbers quoted are for LundNet architecture. Process

QCD

H → cc̄

t → bqq̄′

Feature ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ

500 < pT < 700 GeV GNNExpl. GNN-SHAP GradCAM 0.59 0.20 0.12 -0.00 0.02 0.54 0.10 -0.08 0.25 0.04 0.47 0.13 -0.04 0.23 -0.04

-0.07 -0.14 -0.19 0.17 0.04 -0.26 -0.02 0.06 -0.10 -0.15 0.43 -0.20 -0.22 0.15 0.03

0.28 0.25 0.10 0.06 -0.05 0.24 0.23 0.19 0.01 -0.07 0.01 0.00 -0.07 0.09 -0.05

500 < pT < 1000 GeV GNNExpl. GNN-SHAP GradCAM 0.59 0.19 0.11 -0.00 0.00 0.57 0.09 -0.08 0.23 0.02 0.51 0.13 -0.06 0.26 -0.03

-0.09 -0.16 -0.20 0.17 0.03 -0.29 0.05 0.13 -0.16 -0.14 0.44 -0.20 -0.22 0.15 0.04

0.27 0.24 0.10 0.05 -0.05 0.26 0.23 0.16 0.04 -0.06 -0.01 0.02 -0.03 0.05 -0.07

800 < pT < 1000 GeV GNNExpl. GNN-SHAP GradCAM 0.60 0.17 0.11 -0.03 -0.08 0.56 0.01 0.02 0.01 -0.00 0.59 0.18 -0.03 0.21 0.09

-0.13 -0.31 -0.32 0.26 -0.01 -0.41 0.33 0.37 -0.35 -0.16 0.52 -0.31 -0.26 0.08 0.06

0.24 0.22 0.08 0.04 -0.02 0.32 0.16 -0.10 0.18 0.08 0.09 0.17 0.12 -0.00 0.03

TABLE XII. Pearson correlation coefficients ρ between explainer importance scores and C3 for each jet process, Lund-plane feature, and pT bin. The numbers quoted are for LundNet architecture. Process

QCD

H → cc̄

t → bqq̄′

Feature ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ

500 < pT < 700 GeV GNNExpl. GNN-SHAP GradCAM 0.57 0.14 0.08 0.04 0.01 0.54 0.15 0.17 -0.11 0.02 0.55 0.12 -0.03 0.24 -0.04

-0.06 -0.15 -0.20 0.15 0.02 -0.32 0.16 0.22 -0.24 -0.11 0.52 -0.26 -0.31 0.25 -0.02

0.28 0.24 0.06 0.09 -0.05 0.25 0.22 0.21 -0.01 -0.04 0.04 -0.05 -0.14 0.14 0.01

500 < pT < 1000 GeV GNNExpl. GNN-SHAP GradCAM 0.56 0.15 0.08 0.03 -0.00 0.57 0.14 0.17 -0.11 0.02 0.56 0.13 -0.02 0.23 -0.03

-0.08 -0.16 -0.19 0.14 0.02 -0.34 0.22 0.28 -0.28 -0.08 0.52 -0.25 -0.30 0.24 -0.01

0.27 0.23 0.06 0.07 -0.05 0.25 0.20 0.16 0.01 -0.02 0.03 -0.04 -0.11 0.10 0.01

800 < pT < 1000 GeV GNNExpl. GNN-SHAP GradCAM 0.56 0.16 0.09 -0.00 -0.10 0.56 0.10 0.22 -0.26 -0.03 0.68 0.21 -0.00 0.17 0.10

-0.13 -0.18 -0.20 0.13 -0.04 -0.39 0.40 0.44 -0.41 0.00 0.59 -0.23 -0.26 0.14 -0.00

0.24 0.23 0.03 0.07 0.04 0.20 0.09 -0.08 0.08 0.09 -0.02 0.03 0.08 -0.10 0.07

TABLE XIII. Pearson correlation coefficients ρ between explainer importance scores and C2 for each jet process, Lund-plane feature, and pT bin. The numbers quoted are for ParticleNet architecture. Process

QCD

H → cc̄

t → bqq̄′

Feature ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ

500 < pT < 700 GeV GNNExpl. GNN-SHAP GradCAM 0.59 0.29 0.18 0.00 0.02 0.54 0.09 -0.11 0.26 0.03 0.42 0.05 -0.09 0.18 -0.03

0.29 0.03 -0.18 0.25 -0.04 0.36 -0.49 -0.37 0.05 0.08 0.20 -0.31 -0.24 0.03 0.02

0.12 0.02 0.03 -0.01 0.00 0.33 -0.49 -0.43 0.21 -0.01 0.35 -0.19 -0.10 -0.14 0.03

500 < pT < 1000 GeV GNNExpl. GNN-SHAP GradCAM 0.59 0.29 0.17 0.01 0.00 0.56 0.09 -0.10 0.24 0.03 0.47 0.03 -0.13 0.20 -0.03

0.30 0.01 -0.20 0.27 -0.00 0.39 -0.49 -0.38 0.06 0.06 0.23 -0.30 -0.26 0.07 0.02

0.11 0.00 0.02 -0.01 -0.02 0.31 -0.47 -0.37 0.15 -0.02 0.40 -0.15 -0.08 -0.12 0.03

800 < pT < 1000 GeV GNNExpl. GNN-SHAP GradCAM 0.61 0.26 0.08 0.06 -0.03 0.54 0.06 0.05 0.01 0.05 0.52 0.10 0.05 0.01 0.11

0.39 -0.14 -0.33 0.37 0.08 0.42 -0.54 -0.37 -0.05 0.08 0.37 -0.45 -0.42 0.19 0.05

0.05 -0.13 -0.03 -0.02 -0.13 0.30 -0.50 -0.29 -0.03 -0.04 0.29 -0.17 -0.09 -0.16 -0.04

TABLE XIV. Pearson correlation coefficients ρ between explainer importance scores and C3 for each jet process, Lund-plane feature, and pT bin. The numbers quoted are for ParticleNet architecture. Process

QCD

H → cc̄

t → bqq̄′

Feature ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ

500 < pT < 700 GeV GNNExpl. GNN-SHAP GradCAM 0.58 0.22 0.14 0.04 0.01 0.53 0.15 0.17 -0.12 0.00 0.47 0.00 -0.10 0.18 -0.05

0.25 0.05 -0.03 0.07 -0.05 0.30 -0.36 -0.34 0.14 0.04 0.26 -0.36 -0.37 0.21 -0.01

0.27 0.14 -0.03 0.10 -0.00 0.13 -0.20 -0.06 -0.07 0.01 0.44 -0.41 -0.35 0.14 -0.02

500 < pT < 1000 GeV GNNExpl. GNN-SHAP GradCAM 0.58 0.23 0.14 0.03 0.00 0.55 0.14 0.17 -0.13 -0.00 0.49 0.01 -0.09 0.15 -0.03

0.26 0.04 -0.04 0.08 -0.02 0.32 -0.37 -0.35 0.16 0.03 0.29 -0.37 -0.40 0.23 0.00

0.26 0.13 -0.03 0.09 -0.02 0.12 -0.18 -0.03 -0.11 0.01 0.44 -0.37 -0.32 0.14 -0.02

800 < pT < 1000 GeV GNNExpl. GNN-SHAP GradCAM 0.58 0.24 0.08 0.04 -0.03 0.52 0.11 0.25 -0.28 0.01 0.50 0.07 0.13 -0.13 0.09

0.30 0.01 -0.11 0.16 0.11 0.26 -0.42 -0.38 0.13 0.04 0.41 -0.47 -0.52 0.36 0.05

0.21 0.08 -0.06 0.08 -0.13 0.12 -0.23 -0.04 -0.16 0.01 0.32 -0.28 -0.25 0.12 -0.07

TABLE XV. Pearson correlation coefficients ρ between explainer importance scores and C2 for each jet process, Lund-plane feature, and pT bin. The numbers quoted are for Particle Transformer architecture. Process

QCD

H → cc̄

t → bqq̄′

Feature ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ

500 < pT < 700 GeV GNNExpl. GNN-SHAP GradCAM 0.57 0.17 0.32 -0.44 -0.01 0.28 0.03 -0.07 0.18 -0.00 0.56 0.34 -0.20 0.49 0.00

0.21 0.12 -0.23 0.26 0.01 0.47 -0.31 -0.28 0.20 0.01 0.33 -0.21 -0.11 -0.18 0.04

0.28 0.11 0.49 -0.44 -0.01 0.10 -0.14 -0.12 -0.01 -0.02 0.44 0.37 0.28 0.06 -0.05

500 < pT < 1000 GeV GNNExpl. GNN-SHAP GradCAM 0.55 0.17 0.32 -0.42 -0.03 0.34 0.04 -0.09 0.21 -0.00 0.62 0.36 -0.24 0.54 0.02

0.19 0.11 -0.22 0.25 0.02 0.50 -0.28 -0.24 0.16 -0.01 0.38 -0.14 -0.05 -0.18 0.04

0.26 0.11 0.50 -0.45 -0.03 0.13 -0.19 -0.17 -0.00 -0.03 0.44 0.34 0.32 -0.04 -0.02

800 < pT < 1000 GeV GNNExpl. GNN-SHAP GradCAM 0.52 0.18 0.26 -0.39 -0.18 0.57 0.22 0.11 0.16 0.20 0.28 0.21 0.39 -0.30 0.16

0.04 0.00 -0.21 0.10 0.02 0.54 -0.11 -0.11 0.08 0.13 -0.11 -0.03 0.05 -0.36 0.40

0.14 0.04 0.46 -0.46 -0.10 0.03 -0.31 -0.18 -0.14 0.10 -0.00 0.10 0.24 -0.30 0.22

TABLE XVI. Pearson correlation coefficients ρ between explainer importance scores and C3 for each jet process, Lund-plane feature, and pT bin. The numbers quoted are for Particle Transformer architecture. Process

QCD

H → cc̄

t → bqq̄′

Feature ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ ln ∆ ln kT ln m ln z ψ

500 < pT < 700 GeV GNNExpl. GNN-SHAP GradCAM 0.56 0.08 0.25 -0.33 -0.02 0.31 0.11 0.30 -0.29 0.02 0.29 0.11 -0.07 0.17 0.01

0.35 0.26 -0.27 0.38 -0.01 0.07 -0.04 0.03 -0.08 -0.00 0.27 -0.42 -0.36 0.10 -0.03

0.30 0.05 0.50 -0.43 -0.03 -0.10 -0.14 0.10 -0.16 0.01 0.30 0.17 0.04 0.14 -0.05

500 < pT < 1000 GeV GNNExpl. GNN-SHAP GradCAM 0.56 0.11 0.26 -0.33 -0.04 0.36 0.10 0.27 -0.26 0.00 0.32 0.13 -0.07 0.18 0.03

0.33 0.25 -0.26 0.36 -0.00 0.08 -0.02 0.05 -0.11 0.01 0.29 -0.38 -0.33 0.10 -0.02

0.31 0.08 0.51 -0.43 -0.05 -0.09 -0.16 0.06 -0.16 0.01 0.29 0.17 0.08 0.08 -0.02

800 < pT < 1000 GeV GNNExpl. GNN-SHAP GradCAM 0.57 0.13 0.25 -0.41 -0.19 0.47 0.20 0.37 -0.29 -0.03 0.36 0.29 0.33 -0.24 0.29

0.17 0.15 -0.15 0.17 0.02 0.01 0.04 0.13 -0.24 0.07 0.13 -0.28 -0.34 0.17 0.19

0.18 0.01 0.56 -0.52 -0.13 -0.41 -0.14 0.09 -0.33 -0.07 -0.11 0.18 0.35 -0.22 0.45

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