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arxiv: 2601.20816 · v2 · submitted 2026-01-28 · ✦ hep-ph

Elastic phase shift analysis reveals the geometric origin of the residue phase

S. Ceci , R. Omerovi\'c , H. Osmanovi\'c , M. Uroi\'c , M. Vuk\v{s}i\'c , B. Zauner This is my paper

Pith reviewed 2026-05-16 10:31 UTC · model grok-4.3

classification ✦ hep-ph
keywords residue phasegeometric phasehadron resonancesphase shift analysisAdler zerosππ scatteringπK scatteringπN scattering
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The pith

The residue phase of light hadron resonances is set mostly by the geometric angle from threshold to pole.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that in elastic scattering of light hadrons the phase of the residue at a resonance pole is fixed primarily by a simple geometric quantity. This quantity is the angle δ₀ formed at the nearest threshold between the real axis and the line connecting to the complex pole. Analysis of phase-shift data for ππ, πK and πN channels confirms that vector resonances lie close to the pure geometric prediction, while scalar resonances are shifted by 10° to 15°. The shifts are interpreted as the effect of nearby Adler zeros. A reader would care because the result offers a baseline that separates kinematic geometry from dynamical content when extracting resonance parameters.

Core claim

The complex-plane structure of light hadron resonances is governed by a unified geometric framework in which the threshold position plays a decisive role. The residue phase θ is primarily determined by the geometric phase δ₀, defined as the angle between the resonance pole and the real axis as seen from the threshold. Vector resonances align closely with this geometric baseline. Scalar resonances exhibit systematic deviations of 10°–15°, which are identified as the dynamical imprint of Adler zeros.

What carries the argument

The geometric phase δ₀, the angle at the threshold between the real axis and the line to the resonance pole, which directly sets the residue phase θ.

If this is right

  • Vector resonances in ππ, πK and πN channels follow the geometric prediction to high accuracy.
  • Scalar resonances show a consistent 10°–15° offset attributed to Adler zeros.
  • The framework supplies a baseline for interpreting residue phases directly from elastic phase-shift data.
  • Threshold location becomes the dominant kinematic ingredient for the complex-plane location of the residue.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Resonance parameters taken from experiment could be corrected for this geometric contribution before comparison with dynamical models.
  • The same geometric relation may be tested in other near-threshold systems such as heavy-meson resonances.
  • Improved phase-shift data near thresholds would allow a quantitative separation of the Adler-zero contribution.
  • The approach links to dispersion relations that already enforce threshold constraints on scattering phases.

Load-bearing premise

The threshold position plays a decisive role in fixing the residue phase, and observed deviations for scalar resonances are due to Adler zeros rather than fitting artifacts or other effects.

What would settle it

A high-precision extraction of the residue phase for a well-measured vector resonance such as the ρ(770) that deviates by more than a few degrees from the geometric angle δ₀ calculated from its pole and the ππ threshold.

Figures

Figures reproduced from arXiv: 2601.20816 by B. Zauner, H. Osmanovi\'c, M. Uroi\'c, M. Vuk\v{s}i\'c, R. Omerovi\'c, S. Ceci.

Figure 1
Figure 1. Figure 1: FIG. 1: Partial fits of our model to the meson phase shift [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Partial fits of our model to the data. The phase of the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The self-consistency plot. Each “*” is at the position [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

We show that the complex-plane structure of light hadron resonances is governed by a unified geometric framework where the threshold position plays a decisive role. By applying this framework to $\pi\pi$, $\pi K$, and $\pi N$ phase shifts, we show that the residue phase $\theta$ is primarily determined by the geometric phase $\delta_0$ (the angle between pole and real axis seen from the threshold). While vector resonances exhibit excellent alignment with this geometric baseline, scalar resonances show systematic deviations of $10^\circ$--$15^\circ$, which we identify as the dynamical imprint of Adler zeros.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that the complex-plane structure of light hadron resonances is governed by a unified geometric framework in which the threshold position plays a decisive role. By analyzing ππ, πK, and πN phase shifts, it asserts that the residue phase θ is primarily determined by the geometric phase δ₀ (the angle between the pole and the real axis viewed from the threshold). Vector resonances align closely with this baseline, while scalar resonances exhibit systematic 10°–15° deviations identified as the dynamical imprint of Adler zeros.

Significance. If substantiated with quantitative support, the result would provide a compact geometric account of residue phases across channels, underscoring the role of thresholds and chiral features such as Adler zeros. The multi-channel phase-shift application is a constructive element. The absence of an explicit derivation linking Adler zeros to the observed deviation magnitude, however, leaves the central attribution under-supported.

major comments (2)
  1. [Abstract] Abstract: the assertion that scalar deviations of 10°–15° constitute the 'dynamical imprint of Adler zeros' is not backed by an explicit analytic or numerical calculation demonstrating how an Adler zero at its chiral location shifts the residue phase by that amount once the geometric baseline δ₀ is subtracted. Without this link, alternative sources (parametrization artifacts, residual inelasticity) cannot be excluded.
  2. [Phase-shift analysis] The geometric phase δ₀ is constructed directly from the fitted pole position, which is obtained from the same phase-shift data used to extract the residue phase θ. This raises a circularity concern: the reported alignment may partly reflect the shared fitting procedure rather than an independent geometric prediction. A test that isolates the determination of δ₀ from the residue extraction is needed to establish the claim.
minor comments (1)
  1. [Notation] The precise mathematical definitions of δ₀ and θ should be stated explicitly with equations to remove any ambiguity in how the angles are measured from the threshold and the residue.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address each major comment below and will incorporate revisions to strengthen the quantitative support for our claims.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion that scalar deviations of 10°–15° constitute the 'dynamical imprint of Adler zeros' is not backed by an explicit analytic or numerical calculation demonstrating how an Adler zero at its chiral location shifts the residue phase by that amount once the geometric baseline δ₀ is subtracted. Without this link, alternative sources (parametrization artifacts, residual inelasticity) cannot be excluded.

    Authors: We agree that the manuscript would benefit from an explicit demonstration. In the revised version we will add a new subsection containing a minimal analytic model (a chiral amplitude with an Adler zero fixed at its expected location) together with a numerical scan that isolates the residue-phase shift after subtracting δ₀. This will quantify the expected 10°–15° deviation and help exclude parametrization artifacts. revision: yes

  2. Referee: [Phase-shift analysis] The geometric phase δ₀ is constructed directly from the fitted pole position, which is obtained from the same phase-shift data used to extract the residue phase θ. This raises a circularity concern: the reported alignment may partly reflect the shared fitting procedure rather than an independent geometric prediction. A test that isolates the determination of δ₀ from the residue extraction is needed to establish the claim.

    Authors: The concern is valid. Although δ₀ is a purely geometric angle fixed once the pole coordinates and threshold are known, both quantities are extracted from the same data set. To remove any shared-procedure bias we will add a supplementary analysis that takes pole positions from independent literature determinations (dispersion-relation extractions and multi-channel fits that do not use our parametrization) and recomputes δ₀ for direct comparison with the corresponding θ values. This will test the geometric relation on an independent footing. revision: yes

Circularity Check

1 steps flagged

Residue phase θ claimed as determined by geometric δ₀ reduces to re-expression of parameters from the same phase-shift fits

specific steps
  1. fitted input called prediction [Abstract]
    "we show that the residue phase θ is primarily determined by the geometric phase δ₀ (the angle between pole and real axis seen from the threshold). While vector resonances exhibit excellent alignment with this geometric baseline, scalar resonances show systematic deviations of 10°--15°, which we identify as the dynamical imprint of Adler zeros."

    δ₀ is computed from the complex pole location, which is extracted by fitting the same phase-shift data that also yields the residue and therefore θ. The assertion that θ is 'determined by' δ₀ is thus a geometric re-labeling of two parameters obtained from one fit, not an independent prediction or first-principles result.

full rationale

The paper's core result states that θ is primarily fixed by δ₀, where δ₀ is the angle subtended by the pole position relative to threshold. Both the pole coordinates (hence δ₀) and the residue (hence θ) are obtained by fitting the identical elastic phase-shift data sets for ππ, πK and πN. The reported alignment for vectors and the 10–15° scalar deviations are therefore internal relations among fitted quantities rather than an independent geometric prediction. No separate analytic derivation or external constraint is supplied that would make δ₀ determine θ outside the fit parametrization itself. The attribution of scalar deviations to Adler zeros is noted observationally but lacks the quantitative mapping required to elevate it beyond a post-hoc interpretation of the same data.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; ledger cannot be populated with specific free parameters or axioms from the manuscript. The framework implicitly relies on standard S-matrix analyticity and the existence of Adler zeros in scalar channels, but these are not audited here.

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