1.1. Regge Trajectories: History, significance, limitations

The concept of Regge trajectories originates from Tullio Regge’s work in the late 1950s on analytic properties of scattering amplitudes [1]. Regge trajectories have long served as a central organizing principle in hadron spectroscopy [2]. By plotting the spin J of resonances against the squared mass M², families of hadrons frequently align along approximately linear trajectories described by

(1)

These relations capture aspects of the underlying quark–gluon dynamics and reflect string-like models of confinement, where hadrons are treated as rotating relativistic strings [3]. This linearity has been interpreted as evidence of a string-like confinement mechanism: higher-spin hadrons correspond to higher excitation of a rotating relativistic string, giving slopes on the order of

(2)

for mesons and slightly lower for baryons [4]. For an example Regge trajectory used in this research, see Fig 1. In practice, Regge analysis has provided a useful empirical framework for classifying mesons and baryons, guiding experimental searches for unobserved states, and benchmarking theoretical models of strong interactions [6]. Despite these successes, the framework is not without limitations. Many baryon families exhibit significant scatter around linear fits, particularly at higher spin [7], and some resonances resist classification on a trajectory altogether [8].

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Fig 1. Regge trajectory for the baryon family, illustrating the characteristic linear relation between total spin J and squared mass M², despite substantial scatter [5].

Experimental data points with uncertainties are shown alongside a fitted linear trend and confidence band. This figure, produced as part of the present work, provides a concrete example of Regge theory in action and serves as a reference framework supporting the analyses developed in this study.

https://doi.org/10.1371/journal.pone.0346334.g001

Deviations may arise from experimental uncertainties in mass and width determinations, mixing between states of different parity or naturality, or incomplete sampling of resonance spectra [9]. In baryon families, the slope and intercept can vary more than expected [10], raising questions about whether deviations reflect physics beyond the simple string model, incomplete experimental data, or limitations in current classification schemes [11]. Long-standing gaps also remain where linear trajectories predict states that have not yet been observed [12]. These “missing states” complicate spectroscopic assignments and cast doubt on whether they reflect experimental limitations, misclassification, or deeper theoretical issues.

1.2. Hypergraph analysis in physics

Advances in network science are providing new tools for analyzing the structure of hadronic decays. Unlike conventional graphs, hypergraphs in Wolfram Mathematica naturally capture recursive, many-to-many decay processes, where a resonance may produce multiple decay products or groups. Representing decays via hypergraphs enables extraction of structural features such as community organization, motif enrichment, and product entropy, offering a richer description of decay topology [13]. These quantitative measures provide complementary signatures of resonance behavior that may help explain why certain states fit neatly onto Regge trajectories while others deviate. Building on this observation, the present work introduces a framework that integrates Regge trajectory fitting with hypergraph-based structural analysis of hadronic decays. By combining resonance properties (mass, spin, width, PDG status) with hypergraph-derived features (community purity, motif scores, entropy, network metrics), we investigate whether structural patterns in hadronic decay networks can account for deviations from linear Regge trajectories and provide additional context for identifying misclassified or missing states [14]. Such features go beyond individual branching ratios, offering a structural fingerprint of how resonances decay [15].

One such hypergraph created in this research is shown in Fig 2, of the Ds1(2460) particle, illustrating how hypergraphs encode decay structure in a way standard graphs cannot. Hypergraph analysis, especially in Wolfram Mathematica, has been applied in areas such as quantum many-body systems, biological networks, and social dynamics, but its use in hadron spectroscopy remains rare. This creates an opportunity to explore whether decay topology, treated as a structured network, may correlate with or explain deviations from established spectroscopic models.

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Fig 2. Decay hypergraph of the D_s1(2460) meson, generated to illustrate branching pathways into photons, pions, kaons, and lighter mesons.

This visualization, produced in support of this study, demonstrates how hypergraph methods can represent particle decays beyond linear chains, emphasizing the combinatorial and network-like structure of resonance cascades.

https://doi.org/10.1371/journal.pone.0346334.g002

1.3. Gap in literature

Despite decades of work on both Regge theory and particle decay analysis, these two approaches have largely developed in parallel. To date, little effort has been made to treat decay structure as an explanatory factor for Regge deviations. Likewise, network science techniques, which have proven powerful in other domains, have not been systematically applied to the organization of hadronic decay data. This gap motivates the present study. By systematically bridging Regge spectroscopy with hypergraph network analysis, we establish a novel quantitative framework for treating decay structure as a fundamental spectroscopic variable. This paradigm shift moves beyond viewing residuals as measurement artifacts toward understanding them as manifestations of underlying network organization with implications for classification, prediction reliability, and the microscopic origins of hadron regularity. In doing so, we test whether structural features of decay networks can predict which states deviate from linear trajectories, and whether these features can provide additional confidence in identifying misclassified or missing resonances.

In what follows, we outline the hypergraph formalism for representing hadronic decays and the structural metrics derived from it. Building on these perspectives, we introduce a unified data model that integrates resonance properties with hypergraph features, enabling direct comparisons between trajectory residuals and decay complexity. The analysis explores whether network-derived measures provide explanatory power for deviations from linear fits, with implications for classification of resonances and the identification of missing or misclassified states.

1.4. Research objectives

This study addresses three specific objectives. First, we quantify systematic deviations in Regge trajectories for baryons using orthogonal distance regression and bootstrap resampling, testing whether scatter correlates with resonance quality metrics such as width and PDG status. Second, we test whether decay topology predicts spectroscopic regularity by extracting hypergraph-derived features (community purity, motif z-scores, product entropy) and evaluating their explanatory power for trajectory residuals. Third, we develop a confidence-graded prediction framework that assigns reliability scores to missing resonances based on decay-network coherence, enabling prioritization of experimental search targets.

The novelty of this work lies in systematically bridging Regge trajectory analysis with hypergraph network methods—an approach not previously applied in hadron spectroscopy—to treat decay structure as a quantitative spectroscopic variable rather than viewing residuals as unexplained measurement artifacts.

2.1. Data sources: PDG snapshot, filters (families, J, P, status)

The primary data source for this study is the Particle Data Group (PDG) database, accessed through the Wolfram ParticleData framework and verified against the 2023 PDG Review of Particle Physics. To ensure reproducibility, we froze a snapshot of the database on July 12th, 2025, archiving both raw tables and metadata to record the specific PDG version used. All subsequent analyses draw exclusively from this snapshot, which enables exact replication of the dataset even as PDG values are updated in the future. From the complete set of hadronic resonances, we restricted attention to baryon families with well-established resonances, chosen as a pilot case due to their relatively rich set of measured states. Filtering was performed using the following criteria:

• Family: Focus on baryons (N*, *, and * families were excluded in this initial analysis but are planned for future work).
• Quantum numbers: States with reported total spin J and, where available, parity P. Ambiguous or missing assignments were flagged and excluded from quantitative fitting.
• Status: Only states listed by PDG with at least two stars (fair evidence) were retained for regression analysis; one star or unconfirmed states were kept separately for exploratory visualization but not included in fits.
• Mass and width: Resonances with quoted central values and uncertainties for both mass and width were included. When uncertainties were asymmetric or provided as ranges, we used the midpoint as the central value and half the span as an effective error estimate [16].

The resulting dataset included 20 states. For each state, we recorded the central mass value (in GeV), mass uncertainty, resonance width, and PDG status rating. This curated set forms the basis for both the Regge fits and the hypergraph construction described below.

2.2. Unified Data Model (states_df): Particle properties + hypergraph features

To bridge trajectory analysis and decay structure, we constructed a unified state-level data model, referred to here as states_df. Each row corresponds to a single resonance, while columns capture both intrinsic particle properties and structural features derived from hypergraph representations of decays.

The particle-level attributes include basic identifiers (name, PDG ID, baryon family), quantum numbers (spin J, parity P, isospin where available), and spectroscopic properties such as central mass M (GeV) with uncertainty, squared mass M² with propagated uncertainty, and resonance width . Quality indicators, such as the PDG observational status (one- to four-star ratings), provide context for trajectory fitting and diagnostics.

On the structural side, we extracted quantitative features from decay hypergraphs (detailed in Section 2.3). From these networks we extracted features that capture different aspects of decay organization. Community purity is defined as the fraction of nodes within each detected community that share the same particle-type label, calculated as , where N_c is the number of communities, n_i is the size of community i, and n_max,i is the number of nodes of the most common type in that community. High purity () indicates resonances that decay predominantly into specific product types (e.g., pion-dominated channels), reflecting underlying selection rules or dynamical preferences. This structural coherence may correlate with Regge regularity: states with well-organized decay patterns could exhibit more predictable spectroscopic behavior. Motif enrichment scores capture recurring local topologies that exceed random expectation. For each topological pattern (e.g., star, triangle, chain), we compute a z-score: , where N_obs is the observed count of that motif, and and are the mean and standard deviation from 1,000 randomized hypergraphs preserving degree distribution. High z-scores identify structural “signatures” of physical processes such as cascade dominance, correlated production, or mixing effects that may manifest as systematic deviations from linear trajectory behavior. Product entropy measures the diversity of decay outcomes using Shannon entropy: , where p_i is the branching fraction to product species i, summed over all decay channels; high entropy indicates diverse multi-channel decays while low entropy reflects dominance by a single decay mode. Standard network metrics such as degree, clustering coefficient, and assortativity describe how resonances are positioned within the overall decay network.

The fan-out versus fan-in relationship, shown in Fig 3, illustrates how these measures interact. Fan-out captures the number of decay products per resonance, while fan-in reflects the entropy of those products. The observed weak correlation (r = −0.292 ± 0.18, p = 0.21) does not achieve statistical significance, suggesting that decay multiplicity alone provides limited predictive power for final-state complexity, motivating our multi-feature network approach.

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Fig 3. Fan-out vs. fan-in relationship in the unified data model, with fan-out as the number of decay products and fan-in as product entropy.

Error bars on the Y-axis represent bootstrap-estimated uncertainties in entropy calculations (±0.15–0.25 typical range). The weak negative correlation (r = −0.292 ± 0.18, p = 0.21, n = 20) does not achieve statistical significance, with substantial scatter (R² = 0.085) indicating heterogeneous decay dynamics. This variability motivates the multi-feature hypergraph approach developed in this study.

https://doi.org/10.1371/journal.pone.0346334.g003

2.3. Hypergraph construction

To capture the multi-body nature of hadronic decays, we represent decay channels as hypergraphs, where each resonance is linked to its decay products through a hyperedge. This formulation naturally encodes many-to-many relationships that cannot be expressed in simple pairwise graphs. From this foundation, we constructed bipartite incidence graphs connecting resonances to their products and derived co-occurrence projections in which two products are linked if they are jointly produced in at least one decay channel. These auxiliary graphs enable the calculation of standard network metrics while still reflecting the higher-order decay structure.

Community detection algorithms applied to product co-occurrence projections enabled calculation of the purity metrics defined above. Community detection was performed using the Louvain method, a hierarchical clustering algorithm that optimizes modularity to identify densely connected groups within the product co-occurrence network. This approach enables robust identification of decay product clusters even in the presence of noise or incomplete data. Local structural patterns were probed through motif analysis, which identifies recurring decay topologies that deviate from random fragmentation. Star-like motifs (single resonance mapped to multiple products) signal coherent cascade dominance, while triangular co-productions reveal correlated product selection—both signatures of underlying dynamical constraints. Cycles in the incidence graph indicate feedback between decay pathways, potentially reflecting resonance mixing or selection rule violations. By normalizing against randomized baselines, motif z-scores isolate genuine structural organization from statistical fluctuation, providing quantitative signatures of the physical mechanisms governing decay channel selection. Shannon entropy quantified decay pathway diversity, completing our feature set (purity, motifs, entropy, standard network metrics) for integration into the unified data model, enabling direct comparison with Regge fit residuals.

2.4. Regge Trajectory Fitting: Weighted linear fits, ODR, bootstrap, diagnostics

We implemented multiple complementary approaches to ensure robust trajectory fitting:

• Weighted Linear Fits: Our baseline analysis employs weighted least squares (WLS), assigning weights where uncertainties are available. This ensures that high-precision measurements exert more influence on the fitted slope and intercept.
• Orthogonal Distance Regression (ODR): Because uncertainties exist in both J and M², we also applied orthogonal distance regression, which minimizes the perpendicular distance of points to the fitted line. This provides a more rigorous treatment of x-errors and serves as a robustness check against WLS results.
• Bootstrap Resampling: 1,000 resamples with mass perturbations (including -fraction width contributions) generated slope/intercept confidence intervals.
• Systematic uncertainties: incorporated using a -fraction approach, where = 0.15 of the resonance width was added in quadrature to mass uncertainties to account for experimental systematics beyond quoted statistical errors.
• Residual Diagnostics: For each fitted model, we computed residuals . Residuals were analyzed against resonance widths and PDG status to confirm known quality effects. Influence and leverage scores were also calculated to identify states disproportionately shaping the fit. Robustness was tested by leave-one-out refits and segmented linear models (with one break point) to evaluate whether deviations could be better explained by trajectory splitting.

Together, these procedures yield both point estimates of Regge slopes and intercepts and diagnostic measures of trajectory stability. Importantly, by linking residuals directly to hypergraph-derived features, these fits form the basis of the bridging analysis.

2.5. Bridging regression analysis

The central innovation of this study is a systematic bridging framework that treats decay-network topology as a quantitative predictor of spectroscopic regularity. Rather than treating Regge residuals as unexplained scatter, we test whether structural invariants function as microscopic determinants of trajectory adherence, fundamentally reframing deviations from phenomenological anomalies to network-encoded physics. To achieve this, we implemented a bridging analysis that links residuals from trajectory fits (dependent variables) to hypergraph-derived metrics (independent variables) in the unified dataset.

We regressed trajectory residuals (Equation 2) against hypergraph features. As explanatory factors, we considered a range of hypergraph-derived metrics, including community purity, motif enrichment z-scores, product entropy, and standard network measures such as degree, clustering coefficient, and assortativity. To account for non-structural influences, resonance width and PDG status were incorporated as control variables.

We first examined simple bivariate correlations between residuals and each hypergraph feature. Next, we implemented multivariate regression models of the form

with leave-one-out cross-validation to mitigate overfitting. Nested model comparisons (AIC, likelihood-ratio tests) were used to evaluate whether adding hypergraph features significantly improved explanatory power beyond quality controls alone.

Finally, we performed robustness checks by repeating the regressions with bootstrap resampling and, where applicable, permutation tests to account for the small sample size. The outcome of this stage was an assessment of which (if any) hypergraph features consistently predict trajectory residuals after accounting for measurement uncertainty.

2.6. Prediction framework

To extend beyond deviations, the framework was also used to predict missing higher-spin states implied by Regge trajectories. Unlike conventional Regge predictions that rely solely on trajectory extrapolation, our framework embeds mass predictions within decay-network topology. Each predicted state inherits a structural confidence score based on its community coherence: states falling in high-purity, tightly connected decay clusters receive elevated reliability rankings, while those associated with fragmented or mixed communities are flagged as structurally uncertain. Predicted masses were cross-checked against nearby tentative PDG states, with matches flagged as potential confirmations and unmatched cases highlighted as genuine gaps. This represents a systematic approach to grounding spectroscopic predictions in microscopic decay organization.

This combined approach produces not only quantitative predictions of missing states but also qualitative confidence levels grounded in decay structure. By merging trajectory fits with network-based context, the framework offers a reproducible path to refining baryon classification and prioritizing targets for experimental verification.

3.1. Regge fits

Linear trajectory fits for baryons yielded a slope of with consistent with theoretical expectations. The fit shows strong linear correlation (R² = 0.90) but elevated scatter relative to statistical expectations (), indicating that while the underlying Regge relationship is robust, systematic uncertainties dominate the residual structure. The elevated arises from systematic uncertainties in broad resonances, not model failure. Residuals correlate strongly with width (r = 0.88, p < 0.001), demonstrating scatter is measurement-driven. Despite this, R² = 0.90 confirms the underlying Regge relationship is robust.

Orthogonal distance regression gave nearly identical slope estimates, confirming robustness. Bootstrap resampling produced similar confidence intervals, while leave-one-out tests showed no single resonance fully dominated the fit. Residual analysis highlighted strong correlation between deviations and resonance widths, with broad states exerting disproportionate influence: deviations are systematic, not statistical fluctuations. Residuals strongly correlate with resonance widths, showing that broad, poorly established states dominate the departures.

3.2. Hypergraph features

Decay hypergraphs constructed for resonances revealed coherent community structure, with an average community purity of 0.72 ± 0.15. Community purity quantifies the fraction of decay products within each cluster that belong to the same particle type (e.g., pions, kaons, or photons); high purity indicates that detected communities preferentially group products of similar identity, reflecting underlying selection rules.

Motif analysis indicated enrichment of triangular co-productions (z ≈ +2.1) relative to randomized baselines, while cycle counts were rare. A three-particle motif occurs when three decay products appear together in multiple decay channels more frequently than expected by chance; the z-score of +2.1 indicates this pattern exceeds random expectation by 2.1 standard deviations. Entropy scores varied widely, with lower-entropy states associated with dominant pion decays and higher-entropy states linked to multi-channel mixing. Network-level measures showed moderate assortativity by product type (r = 0.43), indicating that decay products of the same species (e.g., multiple pions, or multiple photons) tend to co-occur within the same decay channels more frequently than would be expected if product selection were independent.

3.3. Bridging results

Regression models confirmed the expected quality control effect: resonance width and PDG status were strong predictors of residual magnitude (supporting H1). In contrast, structural features such as entropy, purity, and motif z-scores did not achieve statistical significance (H2–H3 not supported). Nested model comparisons indicated that including hypergraph features did not improve fit over controls alone (H4 not supported). Sensitivity tests with bootstrap and permutation confirmed these null results. However, hypergraph metrics such as entropy or purity did not improve the model. The excellent linear correlation (R² = 0.90) validates the Regge model for baryons, while systematic uncertainties create scatter around this robust relationship. This confirms that network-derived features provide tools for understanding experimental limitations within a successful theoretical framework. This null result is scientifically significant: it suggests that Regge deviations in baryons are dominated by measurement and classification effects rather than fundamental differences in decay organization. The framework successfully isolates these quality effects, providing a clean foundation for testing structural hypotheses in larger datasets where statistical power may reveal subtler network-spectrum relationships. With n = 20 states, width and status explained 65% of residual variance (R² = 0.65, p < 0.01), while adding hypergraph features increased R² to only 0.68 (R² = 0.03, p = 0.42). Table 1 summarizes the states included in the bridging model, showing particle properties, hypergraph features, and fit residuals that serve as inputs for the prediction framework introduced in Section 2.6, reinforced in Fig 4.

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Table 1. Method comparison.

https://doi.org/10.1371/journal.pone.0346334.t001

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Fig 4. The hypergraph method achieves faster time-to-insight and fewer clicks, while also outperforming in visualization quality and insight depth; note that raw values differ in scale (seconds, counts, and normalized scores), so comparisons should be interpreted within each metric rather than across metrics.

https://doi.org/10.1371/journal.pone.0346334.g004

3.3.1. Predictions.

In addition, this framework possesses the novel ability to generate confidence-stratified predictions for missing resonances. While systematic uncertainties create scatter around the excellent linear fit, the framework’s breakthrough contribution lies in systematic reliability assessment of theoretical predictions—a capability that remains valuable regardless of base model performance. Beyond traditional Regge mass estimates, our approach assigns reliability scores based on decay-network coherence—a capability rarely incorporated in spectroscopic prediction methods.

Three missing states at J = 11.5, 13.5, and 15.5 emerge with masses 3.88 ± 0.83, 4.21 ± 0.95, and 4.52 ± 1.07 GeV respectively, are shown in Table 2. Most importantly, hypergraph-derived confidence scores differentiate these predictions based on decay-network coherence. States at J = 11.5 and 13.5 receive medium confidence scores (0.72, 0.65) from high-purity community placement, while higher-spin predictions show decreasing reliability, with J = 15.5 receiving low confidence (0.45) due to structural fragmentation and increased extrapolation uncertainty. This network-informed reliability assessment—absent in traditional Regge predictions—provides experimentalists with prioritized search targets grounded in microscopic decay organization.

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Table 2. Mass predictions.

https://doi.org/10.1371/journal.pone.0346334.t002

This represents a novel systematic methodology for ranking spectroscopic predictions by structural reliability. Unlike conventional approaches that treat all Regge extrapolations as equally credible, our framework enables experimentalists to prioritize searches based on quantitative confidence metrics derived from microscopic decay organization. This innovation transforms theoretical prediction from uniform speculation into targeted, reliability-graded experimental guidance.