Near to Pole Framework

Updated 1 December 2025

Near to Pole Framework is a set of analytic and computational methodologies that exploit complex-plane pole structures to model threshold and resonance phenomena.
It employs uniformization and pole-expansion techniques to extract positions and residues from experimental data, improving the accuracy of physical interpretations.
The framework is applied across diverse fields—from hadron spectroscopy to metamaterials and numerical analysis—and integrates machine learning for robust pole classification.

The Near to Pole Framework refers to a set of analytic, computational, and statistical methodologies across several disciplines—hadron spectroscopy, scattering theory, electromagnetic metamaterials, applied mathematics, and signal processing—that exploit the analytic structure of response functions, S-matrix elements, or Green's functions near singularities (poles) in the complex domain. In each context, physical observables in the vicinity of channel thresholds, geometric singularities, or spectral resonances are determined by the proximity and configuration of these poles on the relevant complexified parameter space. The framework formalizes the extraction, classification, and interpretation of such near-threshold or near-resonant pole phenomena, enabling precise modeling, inverse inference, and engineered design in both quantum and classical wave systems.

1. Analytic Structure and Physical Motivation

The central insight of the Near to Pole Framework is that observable enhancements or anomalies—such as resonance peaks in hadronic invariant-mass spectra, cusp-like singularities at scattering thresholds, sharp features in electromagnetic absorption/emission spectra, or slow convergence of Taylor series solutions—are governed by the analytic continuation and singularities (poles) of the underlying theoretical objects (S-matrix, scattering matrix, Green's function, permittivity function, etc.) in the complex domain of energy, frequency, or other spectral variables (Yamada et al., 2020, Molesky et al., 2012, Verschelde et al., 26 Apr 2024, Frohnert et al., 10 Jul 2025).

For example, in hadron spectroscopy, bound states, virtual states, and resonances correspond to poles of the S-matrix located on different sheets of the complex energy Riemann surface. When such a pole approaches an open-channel threshold, it generates pronounced enhancements in the cross-section, but the precise sheet location (physical/unphysical) may be experimentally ambiguous, and the width and position cease to map directly onto standard Breit–Wigner resonance parameters (Yamada et al., 2021).

Similarly, in electromagnetic metamaterials, poles of the effective permittivity function ( $\epsilon(\omega)$ ), such as epsilon-near-pole (ENP) resonances, underpin the emergence of narrow-band, polarization-insensitive high emissivity critical for thermophotovoltaic applications (Molesky et al., 2012).

In numerical analysis, the convergence and acceleration of series expansions for homotopy continuation methods are governed by the placement and proximity of singularities (poles) in the complex parameter plane; “near” poles can sharply limit convergence rates and the effectiveness of extrapolation techniques (Verschelde et al., 26 Apr 2024).

2. Formalism: Uniformization and Model Construction

The framework relies on transforming the multi-sheeted analytic structure of the response functions into a single-valued domain (uniformization), allowing a pole-expansion, often of Mittag–Leffler type, that systematically captures all relevant physical and “shadow” poles (Yamada et al., 2020, Yamada et al., 2021, 1804.01786, Frohnert et al., 10 Jul 2025). For two-channel S-matrix problems, the standard uniformizing variable $z$ or $\omega$ is chosen such that all Riemann sheets are mapped into disjoint sectors of the $z$ -plane:

$z = (1 + \sqrt{u})/(1 - \sqrt{u}), \quad \text{where} \quad u = (q_1 - \Delta)/(q_1 + \Delta), \quad \Delta = \sqrt{\epsilon_2^2 - \epsilon_1^2},$

or variant conformal maps (Yamada et al., 2020, Yamada et al., 2021, Frohnert et al., 10 Jul 2025).

Within this domain, observables (cross-sections, invariant mass distributions, emission spectra) are written as (truncated) sums over simple poles: $T(z) = \sum_{n=1}^M \frac{c_n}{z - z_n} - \frac{\overline{c_n}}{z + \overline{z_n}}$ or, for transmission/reflection matrices in optics,

$\mathcal{S}(k) = \mathcal{S}_{\text{BG}} + \sum_n \frac{\mathcal{R}_n}{k - k_n}$

where $k_n$ are complex resonance frequencies or wavenumbers, and the residues $\mathcal{R}_n$ are constructed from appropriately normalized quasi-normal modes or resonant states (1804.01786).

For machine-learning-based classification of line shapes, model-agnostic parametrizations of the S-matrix with independently placed poles (via Jost function constructions) are employed, allowing the synthetic generation of labeled training data spanning all relevant pole configurations (Frohnert et al., 10 Jul 2025).

3. Extraction of Resonance and Pole Information

A key aspect is the direct determination of the positions and residues of near-threshold or near-resonant poles from experimental observables, circumventing the limitations of Breit–Wigner fits and properly accounting for the branch point structure at thresholds. The near-to-pole methodology proceeds as follows (Yamada et al., 2020, Yamada et al., 2021, Frohnert et al., 10 Jul 2025):

Choose a suitable uniformizing variable to unfold the multi-sheeted energy/frequency plane.
Represent the observable (e.g., cross-section, $dN/dm$ ) as the imaginary part of the user-specified analytic function in the uniformized domain.
Fit experimental or synthetic data by varying pole positions and residues until the best agreement is reached, employing symmetry constraints (from unitarity) to ensure correct threshold behavior and sheet pairing.
Determine the physical significance of each pole by its distance from the physical line; enhancements are always governed by the closest poles, independent of which Riemann sheet they inhabit.

This pole-extraction framework is robust to multiple channels and can naturally incorporate statistical methods such as bootstrapping or classifier-chain ensembles for rigorous uncertainty estimation (Frohnert et al., 10 Jul 2025, Co et al., 27 Mar 2024).

4. Ambiguity, Model Discrimination, and Machine Learning

Near-threshold (or near-pole) features, especially in hadron spectroscopy, are inherently ambiguous: multiple pole arrangements on adjacent sheets yield similar or indistinguishable line shapes on the real energy axis (Zhou et al., 2015, Frohnert et al., 10 Jul 2025). As a consequence, model selection by conventional amplitude fitting is not unique.

The near-to-pole framework addresses this via uncertainty-aware machine learning approaches:

Construct balanced, synthetic datasets labeled by explicit pole configuration (e.g., number of poles in sheets $[bt]$ , $[bb]$ , $[tb]$ ).
Use feature engineering (time-series statistics, high-moment correlations) to map line shapes into low-dimensional feature spaces for robust classification.
Train ensemble classifiers (e.g., classifier chains, CatBoost, DNNs) to predict pole configuration with predictive uncertainty, rejecting high-entropy cases to achieve $\approx95\%$ validation accuracy while minimizing misclassification (Frohnert et al., 10 Jul 2025, Co et al., 27 Mar 2024).
Apply the framework to new data (e.g., $P_{c\bar{c}}(4312)^+$ in LHCb studies) to infer the most probable pole structure, detect the presence of genuine compact states versus virtual or molecular configurations, and provide quantified confidence levels.

This approach has been demonstrated to robustly disambiguate between dynamical (pole-driven) and kinematical (triangle/cusp) origins of spectral enhancements, provided the correct classes are represented in training (Co et al., 27 Mar 2024).

5. Applications Beyond Scattering Theory

The near-to-pole paradigm extends to electromagnetic metamaterials, numerical analysis, and geometric approximation theory.

Metamaterial Thermal Emitters: Engineering poles in the effective medium permittivity via subwavelength metal–dielectric geometries yields epsilon-near-pole metamaterials with tailored, narrowband, omnidirectional high emissivity. These ENP metamaterials deliver spectral, angular, and polarization control beyond the limits of blackbody or epsilon-near-zero designs and achieve thermophotovoltaic efficiencies exceeding the Shockley-Queisser limit at operation temperatures near $1500\,\mathrm{K}$ (Molesky et al., 2012).
Optics: Scattering matrices of open resonators can be uniformly expanded in resonant-state poles, providing analytic spectral dependence and bypassing the need for brute-force frequency sweeps, with applications to high-resolution sensing and chiral dichroism (1804.01786).
Numerical Homotopy Continuation: The effect of “near” poles in the complex parameter plane on the convergence rate of Taylor series (and the success of extrapolation algorithms) is quantified by the gap $\delta$ beyond the radius of convergence. If $\delta$ is small, acceleration fails beyond order $O(1/\delta)$ ; thus, robust extraction of polynomial roots or singular solutions must dynamically adapt step size and truncation order accordingly (Verschelde et al., 26 Apr 2024).
Isogeometric Analysis: In computational PDE, polar domain parameterizations with graded meshes towards a collapsed edge yield optimal approximation of solutions with corner singularities, provided the function space and projection operators are refined to account for reduced regularity near the pole (Apel et al., 15 May 2025).

6. Significance, Generalization, and Limitations

The Near to Pole Framework unifies the treatment of threshold, resonance, and cusp phenomena in a single analytic language dictated by the pole structure of the underlying theory. It provides a model-independent, symmetry-consistent, and data-driven route to physical interpretation, with broad applicability across quantum and classical fields.

Its principal limitations stem from ambiguities intrinsic to analytic continuation: distinct pole configurations may remain experimentally indistinguishable, particularly when only near-threshold line shapes are accessible. Machine-learning-based classification is limited by the representativeness of the synthetic training set and cannot, in the absence of external constraints (e.g., angular observables, polarization data), resolve all degeneracies.

Anticipated extensions include the expansion to higher channel number via suitably generalized uniformization, integration of additional observable modalities, and active-learning schemes to optimize the exploration of high-uncertainty regions of parameter space (Frohnert et al., 10 Jul 2025).

In summary, the Near to Pole Framework systematizes the analytic, numerical, and statistical machinery required to extract, classify, and interpret physical phenomena near spectral, geometrical, or threshold singularities, based upon the precise mapping and exploitation of complex-plane pole structure in the problem's core response function (Yamada et al., 2020, Yamada et al., 2021, Molesky et al., 2012, Frohnert et al., 10 Jul 2025, Verschelde et al., 26 Apr 2024).