arxiv: 2605.05946 · v1 · submitted 2026-05-07 · ✦ hep-ph

Recognition: unknown

Interaction and correlation functions for π f₁(1285), η f₁(1285)

Wen-Hao Jia , Hai-Peng Li , Wei-Hong Liang , Jing Song , Eulogio Oset

Authors on Pith no claims yet

Pith reviewed 2026-05-08 08:26 UTC · model grok-4.3

classification ✦ hep-ph

keywords f1(1285)molecular statecorrelation functionscattering amplitudefixed center approximationthreshold cuspexotic resonance

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The pith

The interaction of π or η with f₁(1285) produces no clear signals for the claimed exotic resonances π₁(1400), π₁(1600) or η₁(1855).

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

The paper models the f₁(1285) as a K* K-bar molecular state and constructs an optical potential for its scattering with pions or etas via the fixed center approximation to the Faddeev equations. This potential serves as the kernel in the Lippmann-Schwinger equation to generate the full scattering matrix, scattering lengths, effective ranges, and correlation functions. The resulting amplitudes lack distinct resonance poles at the masses of the exotic states reported in other dynamical models. A broad structure nevertheless appears in the π⁰ f₁ amplitude between 1500 and 1600 MeV, together with a pronounced cusp exactly at the η f₁ threshold of 1833 MeV. Because the same framework already reproduced the measured p f₁ correlation function, the new predictions are directly testable with existing or forthcoming experimental data.

Core claim

The π⁰ f₁(1285) amplitude develops a structure between 1500 and 1600 MeV while the η f₁(1285) amplitude exhibits a strong cusp at the 1833 MeV threshold, without generating clear resonance signals for the π₁(1400), π₁(1600) or η₁(1855) states.

What carries the argument

Optical potential obtained from the fixed center approximation to the Faddeev equations, inserted as the kernel of the Lippmann-Schwinger equation for the π(η) f₁(1285) system.

Load-bearing premise

The f₁(1285) molecular structure is assumed to remain fixed and unchanged while it scatters with the pion or eta.

What would settle it

Observation of a clear resonance peak at 1400 MeV in the π f₁ mass spectrum or at 1855 MeV in the η f₁ mass spectrum would falsify the calculated amplitudes.

Figures

Figures reproduced from arXiv: 2605.05946 by Eulogio Oset, Hai-Peng Li, Jing Song, Wei-Hong Liang, Wen-Hao Jia.

**Figure 1.** Figure 1: FIG. 1. Diagrams of the ordinary FCA for the interaction of view at source ↗

**Figure 2.** Figure 2: FIG. 2. Diagrams considering the elastic propagation of the view at source ↗

**Figure 5.** Figure 5: FIG. 5. Correlation functions for the view at source ↗

**Figure 4.** Figure 4: FIG. 4 view at source ↗

read the original abstract

We have studied the interaction of $\pi^0 (\eta) f_1(1285)$ assuming the $f_1(1285)$ to be a molecular state of $K^* \bar K - \bar K^* K$. We use a framework in which a $\pi^0 (\eta) f_1(1285)$ optical potential is obtained, which is later used as the kernel of the Lippmann-Schwinger equation, following the standard method for the interaction of particles with nuclei. The optical potential is obtained using the fixed center approximation to the Faddeev equations, where a cluster, here the $f_1(1285)$, remains unchanged during the interaction, appropriate to the situation that one has here. We have obtained the scattering matrix for this system, the scattering length and effective range, plus the correlation functions. The framework used has been previously tested in the study of the $p f_1(1285)$ interaction and has been shown to give results in agreement with the recent experimental measurement of the $p f_1(1285)$ correlation function. On the other hand, from this interaction we do not obtain clear signals for the $\pi_1(1400)$ or $\pi_1(1600)$, nor for the $\eta_1(1855)$ resonances, which in other approaches have been claimed to arise from the same dynamics. We, however, obtain a structure in the $\pi^0 f_1(1285)$ amplitude around $1500-1600$ MeV and a strong cusp at the $\eta f_1(1285)$ threshold of $1833$ MeV.

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.

Desk Editor's Note private letter to a colleague

This is a straightforward extension of the group's p f1 work to π and η channels that produces new numbers but leans on a fixed-center approximation whose validity for light mesons is untested.

read the letter

This paper takes the same fixed-center Faddeev plus Lippmann-Schwinger setup used earlier for p f1(1285) and applies it to π0 f1 and η f1 scattering, treating the f1 as an unchanged K* Kbar molecule. It reports scattering lengths, effective ranges, correlation functions, a broad structure in the π0 amplitude near 1500-1600 MeV, and a strong cusp at the η f1 threshold, while finding no clear poles for the π1(1400), π1(1600), or η1(1855) states claimed in other work.

Referee Report

2 major / 2 minor

Summary. The manuscript studies the π⁰(η) f₁(1285) interaction by modeling the f₁(1285) as a K* K̄ molecular state. An optical potential is constructed via the fixed-center approximation to the Faddeev equations and used as the kernel of the Lippmann-Schwinger equation to obtain the scattering matrix, scattering lengths, effective range parameters, and correlation functions. The central results are the absence of clear signals for the π₁(1400), π₁(1600), and η₁(1855) resonances, together with a structure in the π⁰ f₁ amplitude near 1500–1600 MeV and a strong cusp at the η f₁ threshold of 1833 MeV. The framework is validated by prior agreement with p f₁ correlation data.

Significance. If the results hold, the work supplies concrete predictions for correlation functions that can be tested experimentally and offers a counter-example to models in which the same π/η–f₁ dynamics generate the π₁ and η₁ resonances. The explicit use of a previously validated method for the optical potential and the provision of scattering lengths and effective ranges constitute clear strengths.

major comments (2)

[Formalism section (optical-potential construction)] The fixed-center approximation to the Faddeev equations (used to generate the optical potential that is the kernel of the Lippmann-Schwinger equation) is applied to light projectiles (π, η) scattering from a light cluster. The only cited validation is the heavier p f₁ case; no quantitative estimate of the approximation’s accuracy or of possible dissociation/excitation effects is provided. This assumption is load-bearing for the claim that no clear resonance signals appear.
[Results for π⁰ f₁(1285) amplitude] In the results for the π⁰ f₁ amplitude, a structure is reported around 1500–1600 MeV and a cusp at the η f₁ threshold, yet no sensitivity analysis is shown with respect to the parameters of the assumed K* K̄ molecular wave function of the f₁(1285). Because the kernel is built directly from this wave function, variations could alter the presence or position of the reported structure.

minor comments (2)

[Abstract] The abstract states that the framework “has been previously tested” on p f₁ but does not cite the specific reference or quantify the level of agreement with the experimental correlation function.
[Formalism] Notation for the optical potential and the Lippmann-Schwinger kernel is introduced without an explicit equation number linking the two; a numbered equation would improve traceability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major point below and have revised the manuscript to incorporate additional discussion and analysis where feasible.

read point-by-point responses

Referee: The fixed-center approximation to the Faddeev equations (used to generate the optical potential that is the kernel of the Lippmann-Schwinger equation) is applied to light projectiles (π, η) scattering from a light cluster. The only cited validation is the heavier p f₁ case; no quantitative estimate of the approximation’s accuracy or of possible dissociation/excitation effects is provided. This assumption is load-bearing for the claim that no clear resonance signals appear.

Authors: We acknowledge that the fixed-center approximation (FCA) is here applied to lighter systems than the p f₁ validation case. The FCA is a standard approach in few-body calculations when the cluster binding is sufficient to suppress breakup during the scattering process. For the f₁(1285) treated as a K* K̄ molecule, the binding is moderate and the relevant center-of-mass energies keep the projectile wavelengths comparable to or larger than the cluster size, limiting dissociation. We will add a dedicated paragraph in the Formalism section outlining the validity conditions of the FCA, including a qualitative estimate based on the f₁ wave-function size and the de Broglie wavelength of the π/η projectiles. A full quantitative error assessment would require solving the exact three-body Faddeev equations with explicit K* K̄ components, which lies outside the scope of the present work. Nevertheless, the main features (broad structure near 1500–1600 MeV and the η f₁ cusp) arise from the optical potential and threshold kinematics and are expected to persist under moderate corrections to the FCA. revision: partial
Referee: In the results for the π⁰ f₁ amplitude, a structure is reported around 1500–1600 MeV and a cusp at the η f₁ threshold, yet no sensitivity analysis is shown with respect to the parameters of the assumed K* K̄ molecular wave function of the f₁(1285). Because the kernel is built directly from this wave function, variations could alter the presence or position of the reported structure.

Authors: We agree that an explicit sensitivity check strengthens the robustness of the reported structure. The K* K̄ wave function parameters (cutoff and form factor) are fixed by the requirement that the f₁(1285) is reproduced as a bound state in the prior chiral unitary calculation. In the revised manuscript we will include a short sensitivity study in the Results section, varying the cutoff by ±20 % around its central value while still keeping the f₁ mass and width within experimental bounds. The broad structure near 1500–1600 MeV shifts by at most 40–50 MeV and remains non-resonant, while the cusp at the η f₁ threshold is a pure threshold effect independent of the wave-function details. These variations do not change the conclusion that no narrow poles corresponding to the π₁ or η₁ states appear. revision: yes

Circularity Check

0 steps flagged

Minor self-citation for method validation; derivation self-contained from stated molecular assumption

full rationale

The paper assumes as input that f1(1285) is a K* Kbar molecular state, constructs an optical potential via fixed-center approximation to Faddeev equations, and uses that potential as the kernel of the Lippmann-Schwinger equation to obtain the T-matrix, scattering parameters, and correlation functions. The sole self-citation notes that the same framework was previously tested on the p f1(1285) system and matched experimental correlation data, providing independent external validation rather than a load-bearing premise. No parameters are fitted to the present π/η data and then relabeled as predictions, no uniqueness theorems are imported from the authors' prior work, and the claims of absent or present structures follow directly from solving the integral equation without reducing to the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The calculation rests on the molecular assignment of the f₁(1285) taken from earlier work and on the validity of the fixed-center approximation for a light projectile scattering off a heavier cluster; no new free parameters are introduced beyond those already fixed in the referenced p f₁ study.

axioms (2)

domain assumption The f₁(1285) can be treated as an unchanged K* K-bar molecular cluster during the scattering process (fixed-center approximation).
Invoked in the abstract to justify the optical-potential construction via Faddeev equations.
standard math The optical potential obtained from the fixed-center Faddeev equations is a valid kernel for the subsequent Lippmann-Schwinger equation.
Standard procedure stated in the abstract and previously validated for the proton case.

pith-pipeline@v0.9.0 · 5630 in / 1696 out tokens · 20392 ms · 2026-05-08T08:26:19.209628+00:00 · methodology

discussion (0)

Reference graph

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