Possible explanation of Hoehler's clustering: effective partial-wave mixing induced by truncation

Abstract: Hoehler noted that resonance poles obtained from different partial waves in $πN$ scattering appear to bunch together near a small set of common complex energies, and suggested that this could indicate mixing between angular momenta. Here, we examine whether at least part of this pattern could arise effectively from the extraction procedure itself. Exact partial-wave unitarity preserves the separation of angular momenta in the infinite problem, whereas practical pole extraction from bilinear observables requires truncation of the partial-wave series. Combined with the truncation-induced mixing mechanism established in Ref.~\cite{Svarc2026}, this provides a natural source by which fitted partial-wave coefficients can inherit overlapping pole-bearing content, thereby offering a plausible contribution to Hoehler-type clustering.

• The paper proposes that truncation in bilinear observable fits induces effective mixing between partial waves, leading to Hӧhler's clustering.
• It rigorously distinguishes between exact unitarity-preserving angular momentum separation and apparent mixing arising from truncated data fits.
• The findings emphasize the need for improved fitting procedures to accurately isolate resonance poles in hadron spectroscopy.

Effective Partial-Wave Mixing Induced by Truncation as a Possible Explanation for H\"ohler's Clustering

Background and Motivation

The longstanding issue of "H\"ohler's clustering" in $\pi N$ scattering refers to the observed near-degeneracy of complex pole positions across partial waves with different angular momenta, as initially identified by H\"ohler [H\"ohler, Landolt-B\"ornstein (1983)]. These resonance poles, extracted from partial-wave analyses, appear to bunch together near a limited set of complex energies, suggesting either a fundamental dynamical relationship between distinct angular-momentum channels or an artifact of the extraction procedure. The direct association of resonances with poles of partial-wave amplitudes underpins baryon spectroscopy and S-matrix analysis, making the elucidation of such clustering central for both phenomenological models and the interpretation of experimental data.

This work rigorously analyzes whether H\"ohler's clustering can be explained, at least in part, as a consequence of effective partial-wave mixing induced by the truncation inherent to practical pole extraction from bilinear observables, as opposed to being a reflection of the underlying analytic structure of the scattering amplitude. This explanation leverages the mechanism recently formalized in [Svarc, (Švarc, 13 Apr 2026)], focusing on the nonlinearities and cross-talk introduced by truncation and nonlinear fitting strategies in empirical analysis.

Analysis of Angular-Momentum Mixing Mechanisms

The paper delineates two distinct sources of apparent partial-wave mixing:

1. Dynamical Mixing at the Level of the Exact Amplitude: The study demonstrates that, within the exact, infinite partial-wave framework, S-matrix unitarity rigorously ensures separation of angular momenta. Unitarity is represented as an identity in the angular-momentum basis, which enforces independent conservation of each partial wave. The analysis employs formal operator arguments for the $2\to2$ scattering problem, highlighting that an isolated pole in the invariant amplitude cannot simultaneously populate several distinct angular-momentum sectors without violating exact unitarity. Therefore, genuine dynamical mixing of stable, isolated poles across partial waves is strongly constrained and, in practice, forbidden in the absence of explicit symmetry breaking or nontrivial dynamical couplings.
2. Effective Mixing in Practical Extraction Procedures: The extraction of resonance properties relies on fitting observables (such as cross sections) that are bilinear in the partial-wave amplitudes, and these fits are necessarily performed with truncated partial-wave expansions due to finite data and computational resources. The paper expounds on the algebraic structure of truncated Legendre expansions, establishing that while the exact amplitude expansion coefficients are unmixed, the fitted coefficients obtained from truncated bilinear fits inherit contributions from both included and excluded angular-momentum components. The system of equations linking observables to coefficients is nonlinear and coupled: the extracted lower-order partial-wave coefficients are, in general, nonlinear functionals of a mixture of lower and higher true partial-wave amplitudes. These findings are formalized and proved in [Svarc, (Švarc, 13 Apr 2026)].

Connection to H\"ohler’s Clustering and Phenomenological Implications

The key implication is that, under practical conditions, resonance poles extracted in different nominal partial waves may share overlapping analytic content due to truncation-induced effective mixing. The extracted lower-order coefficients do not correspond to isolated physical angular-momentum states but rather encode admixtures resulting from the nonlinear least-squares minimization of truncated bilinear expansions. Consequently, pole positions associated with different partial waves can cluster artificially, reflecting effective cross-wave correlations rather than physical degeneracies or Lorentz multiplet structure.

This framework naturally explains the occurrence of H\"ohler-type clustering in older phenomenological analyses and its diminished visibility in modern Particle Data Group (PDG) listings, where resonance claims are subjected to stricter scrutiny regarding truncation effects and completeness.

Theoretical and Practical Consequences

This outcome has several important theoretical and methodological consequences:

• Interpretation of Partial-Wave Analysis Results: The effective nature of fitted partial-wave coefficients necessitates caution when ascribing physical meaning to resonance pole positions obtained from truncated analyses—especially when comparing across partial waves or between different truncated expansions.
• Guidance for Future Analyses: The results indicate that increasing the range and completeness of included partial waves and developing improved nontruncated fitting procedures will reduce effective mixing and clarify the dynamical content of resonance poles.
• Context for Existing Phenomenology: The mechanism provides a robust physical and mathematical rationale for why H\"ohler’s clusters appear in historical analyses and cautions against overinterpreting such clustering as evidence of deep dynamical symmetries without considering extraction artifacts.
• Generalization to Other Processes: Although focused on $\pi N$ scattering, the analysis is applicable to any process where partial-wave analysis is performed on bilinear (or higher-order) observables with finite truncation, including many hadronic and electromagnetic reactions.

Conclusion

The clustering of resonance pole positions across different partial waves in phenomenological analyses, as first noted by H\"ohler, can be naturally attributed in large part to effective partial-wave mixing induced by the truncation of bilinear observables in practical fitting procedures. This effect arises not from fundamental dynamical mixing, which is forbidden by exact unitarity, but from the nonlinear, coupled structure of the coefficient extraction problem when the observable basis is truncated. This clarification has strong implications for the interpretation of extracted resonance properties and motivates refinement of analysis techniques to better isolate genuine dynamical features of the hadron spectrum. The proposed mechanism represents an unavoidable and physically natural explanation for at least a significant component of observed cross-wave clustering patterns.

Reference:

A. \v{S}varc, "Possible explanation of Hoehler's clustering: effective partial-wave mixing induced by truncation" (2604.26652).