A new method for extracting poles from single-channel data based on Laurent expansion of T-matrices with Pietarinen power series representing the non-singular part

Abstract: We present a new approach to quantifying pole parameters of single-channel processes based on Laurent expansion of partial wave T-matrices. Instead of guessing the analytical form of non-singular part of Laurent expansion as it is usually done, we represent it by the convergent series of Pietarinen functions. As the analytic structure of non-singular term is usually very well known (physical cuts with branhcpoints at inelastic thresholds, and unphysical cuts in the negative energy plane), we show that we need one Pietarinen series per cut, and the number of terms in each Pietarinen series is automatically determined by the quality of the fit. The method is tested on a toy model constructed from two known poles, various background terms, and two physical cuts, and shown to be robust and confident up to three Pietarinen series. We also apply this method to Zagreb CMB amplitudes for the N(1535) 1/2- resonance, and confirm the full success of the method on realistic data. This formalism can also be used for fitting experimental data, and the procedure is very similar as when Breit-Wigner functions are used, but with one modification: Laurent expansion with Pietarinen series is replacing the standard Breit-Wigner T-matrix form.

• The paper introduces a Laurent+Pietarinen technique that decomposes T-matrices into simple pole terms and a regular part using adaptive power series.
• It demonstrates accurate resonance extraction by validating the method on both synthetic benchmarks and real hadron spectroscopy data.
• The approach minimizes model dependence by systematically encoding known analytic structures and physical thresholds in its expansion.

A Novel Laurent+Pietarinen Method for Single-Channel Pole Extraction

Motivation and Theoretical Foundations

Parametrization of hadronic resonances is crucial in scattering theory and hadron spectroscopy, with analytic structure of the $T$-matrix encoding fundamental properties such as resonance positions and couplings. Traditional approaches involve fitting $T$-matrices using Breit-Wigner forms or direct analytic continuations; however, significant model dependence arises from ambiguous choices for the analytic background contributions. This introduces uncertainties and complicates the pole-background separation, particularly problematic in single-channel analyses.

The presented work establishes a method for extracting pole parameters from single-channel amplitudes employing a Laurent expansion of partial wave $T$-matrices, with the non-singular part represented by a convergent Pietarinen power series. Crucially, the regular part's analytic structure is encoded with Pietarinen expansions, one per branch cut, thereby systematically incorporating known physical thresholds and negative energy cuts without ad hoc modeling. The number of terms per series is determined adaptively based on fit quality.

Formalism and Implementation

The method assumes all physical poles are first order. The Laurent expansion decomposes the $T$-matrix into a sum of simple pole terms and a regular part:

$$T(w) = \sum_{i=1}^k \frac{a_{-1}^{(i)}}{w_i - w} + B^L(w)$$

where $a_{-1}^{(i)}$ are residues and $w_i$ pole positions. The novelty is in expressing the regular background $B^L(w)$ as a sum of Pietarinen series:

$$B^L(w) = \sum_{n=0}^{M} c_n Z(w)^n + \sum_{n=0}^{N} d_n W(w)^n + \ldots$$

with each Pietarinen function (e.g. $Z(w)$) associated with a specific branch point (threshold):

$$Z(w) = \frac{\alpha - \sqrt{x_p - w}}{\alpha + \sqrt{x_p - w}}$$

where $x_p$ is the branchpoint, and the coefficients $\{c_n\}$ are fit parameters. Each cut in the analytic structure is thus represented with an explicit, rapidly convergent series, while negative energy cuts are efficiently approximated with one effective branch point.

Fitting is performed over real axis data, with branch points typically fixed to known physical threshold values, but in practice allowed to vary slightly to account for model reduction (i.e., effective representation of multiple cuts). The degree of the Pietarinen expansions is increased until a satisfactory fit, as gauged by reduced $\chi^2$ and visual quality, is achieved. Penalty functions (e.g. in the $\chi^2$) enforce minimal model complexity.

Robust Validation: Toy Model and Realistic Data

Synthetic Benchmarking

A comprehensive battery of toy model tests was performed, including combinations of two resonance poles, two positive energy cuts, and variable backgrounds. For each analytic feature present in the toy model, the method adaptively required and constructed the corresponding number of Pietarinen expansions. Pole positions and residues were always accurately recovered, with insensitivity to background strength or phase, demonstrating the robustness of the approach to analytic structure and parameter variations.

When only the modulus of the amplitude was available (reflecting limited experimental data), the method reliably reproduced pole positions, but residues remained underdetermined—consistent with theoretical expectations given the unobservability of relative phases.

Application to Hadron Spectroscopy Data

Applied to the Zagreb Carnegie-Mellon–Berkeley (CMB) $N(1535)\ 1/2^-$ amplitude, the method correctly identified two physical poles near $1.52$ and $1.65$ GeV (widths $0.09$ and $0.10$ GeV, respectively), and indicated a third, broad structure near $1.8$ GeV with greater uncertainty—consistent with the coupled-channel analytic continuation results in the literature. The automatically determined Pietarinen branch points clustered near the physical two-body thresholds ($\pi N$-elastic and $\eta N$-production), with an additional negative energy branch for background, matching physical expectations.

Key finding: Single-channel Laurent+Pietarinen (L+P) analysis accesses only those resonances strongly coupled to the observed channel; others are hinted but not well localized.

Direct Analysis of Experimental Single-Energy Solutions

For the GWU single-energy (SES) and energy-dependent (SP06) partial wave data sets, the L+P method demonstrated that direct pole extraction from experimental data is feasible without resorting to model-dependent analytic continuation. Notably, whereas the energy-dependent fit (SP06) did not require the $P_{11}(1710)$ state, the SES data analysis identified it as necessary, highlighting the importance of analysis methodology and providing a tool for revealing resonance content “hidden” in model-building procedures.

Implications and Perspectives

This formalism provides a procedural framework for pole extraction free from arbitrary modeling of the regular background. The method is numerically stable, rapidly converges, and robustly determines both pole positions and residues (when both real and imaginary amplitude data are available).

Contradictory to approaches relying solely on Breit-Wigner fits or specific background ansätze, L+P achieves minimal model dependence, encoding the entirety of known analytic structure, and allowing the fit to determine only what is justified by data and physics. This significantly increases the reliability and reproducibility of resonance determinations from partial wave data.

In practical terms, the L+P method delivers:

• Parameter-free modeling (apart from physical inputs such as the number of poles and known branch points).
• Direct applicability to both theoretical (model-generated) and experimental amplitudes.
• The ability to systematically refine the regular background with minimal required complexity.

From a theoretical standpoint, the Laurent+Pietarinen framework provides a systematic link between data and fundamental QCD-motivated amplitude structure, thus supporting resonance extractions as advocated by modern hadron spectroscopy.

Future developments should address:

• Extension to coupled-channel analyses, essential for comprehensive resonance identification.
• Incorporating unstable two-body threshold cuts (currently only real branch points are included).
• Systematic applications across full baryon and meson spectra with robust error propagation strategies.

Conclusion

The Laurent+Pietarinen method enables reliable extraction of scattering matrix pole parameters from single-channel data with unprecedented control over analytic model uncertainties. By encoding all known cut structures without speculative modeling, it systematically determines resonance parameters associated with observed channels, and sets a new standard for amplitude analyses in hadron spectroscopy and beyond. The method's success in reproduction of synthetic and real-world data underscores its value for precision hadron resonance determination and inspires future multi-channel generalizations.

Reference:

"A new method for extracting poles from single-channel data based on Laurent expansion of T-matrices with Pietarinen power series representing the non-singular part" (1212.1295)