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arxiv: 2606.19040 · v1 · pith:4LPNB4ROnew · submitted 2026-06-17 · ✦ hep-ph · hep-ex· hep-lat· nucl-th

Three-body unitary determination of the f₁(1285) and f₁(1420) pole positions

Tao-Ran Hu , Hai-Long Fu , Feng-Kun Guo , Ulf-G. Mei{\ss}ner , Xu Zhang This is my paper

Pith reviewed 2026-06-26 20:20 UTC · model grok-4.3

classification ✦ hep-ph hep-exhep-latnucl-th
keywords three-body unitarityresonance polesf1(1285)f1(1420)K K-bar pi systemBESIII invariant masshadronic molecule
0
0 comments X

The pith

A three-body unitary amplitude fitted to BESIII data and continued to unphysical Riemann sheets places poles for the f1(1285) at 1277 - i12 MeV and the f1(1420) at 1435 - i40 MeV.

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

The paper constructs a unitary amplitude for the coupled pi a0 and K K-bar* channels in the 0+(1++) K K-bar pi system using the spectator-isobar representation. Short-range three-body forces are fixed by a fit to the relevant component of the BESIII invariant-mass distribution from J/psi to gamma K0S K0S pi0. Analytic continuation of the resulting amplitude locates two stable poles whose trajectories show the lower one arising from a dressed bare state and the higher one from an S-wave K K-bar* quasi-bound state. This matters because it supplies concrete complex masses and clarifies which resonance carries a molecular character. The work also isolates an extra pole generated purely by the P-wave pi a0 contact term that leaves little trace on the physical line shape.

Core claim

In the infinite-volume three-body unitary framework the fitted amplitude for the I^G(J^PC)=0^+(1^{++}) K K-bar pi system, when continued to the relevant unphysical Riemann sheets, produces two robust poles: sqrt(s_f1(1285)) = (1277±2±1) - i(12±1±0) MeV and sqrt(s_f1(1420)) = (1435±2±7) - i(40±2±1) MeV. Pole trajectories indicate the f1(1285) originates from dressing a bare state introduced in the potential, whereas the f1(1420) is predominantly dynamically generated from an S-wave K K-bar* quasi-bound state mixed with the nearby bare state, supporting its hadronic-molecule interpretation. An additional pole deeper in the complex plane appears on the same sheet as the f1(1285) and is generate

What carries the argument

The spectator-isobar representation of the coupled pi a0-K K-bar* amplitude that incorporates one-particle-exchange interactions required by three-body unitarity, plus short-range three-body contact terms constrained by data, followed by analytic continuation onto unphysical Riemann sheets.

If this is right

  • The f1(1285) is a dressed bare state while the f1(1420) is largely dynamically generated from a K K-bar* quasi-bound state.
  • The three-body framework automatically incorporates the triangle-singularity mechanism through the one-particle-exchange term.
  • An extra pole generated solely by the P-wave pi a0 contact interaction exists but produces negligible visible effect on the physical lineshape.
  • Detailed treatment of three-body cuts is required to solve the integral equation reliably.

Where Pith is reading between the lines

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

  • The extracted pole positions could be inserted into predictions for branching ratios in other three-body final states not yet measured.
  • The same unitary construction might be applied to related axial-vector resonances in the strange sector to test consistency of the molecular picture.
  • Production-rate differences between the two resonances could serve as an experimental handle to distinguish bare-state versus dynamically generated components.

Load-bearing premise

The short-range three-body interaction is fixed by fitting only the 0+(1++) component of the BESIII K0S K0S pi0 invariant-mass distribution in the J/psi to gamma K0S K0S pi0 decay.

What would settle it

A high-statistics measurement of the same invariant-mass distribution in an independent production channel whose shape deviates from the fitted amplitude on the physical sheet by more than the reported uncertainties.

read the original abstract

We study the $I^G(J^{PC})=0^+(1^{++})$ $K\bar K\pi$ system in an infinite-volume three-body unitary framework, focusing on the pole content of the region of the $f_1(1285)$ and $f_1(1420)$ resonances. The coupled $\pi a_0$-$K\bar K^*$ amplitude is constructed in the spectator-isobar representation, where the one-particle-exchange interaction required by three-body unitarity automatically incorporates the triangle-singularity mechanism. The short-range three-body interaction is constrained by fitting the $0^+(1^{++})$ component of the BESIII $K^0_SK^0_S\pi^0$ invariant-mass distribution in the $J/\psi\to\gamma(K^0_SK^0_S\pi^0)$ decay. Analytically continuing the fitted amplitude to the relevant unphysical Riemann sheets, we find two robust poles: \begin{align} \sqrt{s_{f_1(1285)}}&= \left(1277\pm2\pm1\right) -i\left(12\pm1\pm0\right)\text{MeV}\,,\notag\\ \sqrt{s_{f_1(1420)}}&= \left(1435\pm2\pm7\right) -i\left(40\pm2\pm1\right)\text{MeV}\,.\notag \end{align} The pole trajectories indicate that the $f_1(1285)$ originates from dressing a bare state introduced in the potential. In contrast, the $f_1(1420)$ is predominantly dynamically generated, and a single-channel analysis traces it to an $S$-wave $K\bar K^*$ quasi-bound state mixed with the nearby bare state, supporting its hadronic-molecule interpretation. We also find an additional pole deeper in the complex plane in the best-fit amplitude on the same Riemann sheet as the $f_1(1285)$. This additional pole is generated by the $P$-wave $\pi a_0$ contact interaction alone. It has a sizable cutoff and two-body-input dependence, and leaves little visible imprint on the physical lineshape. Finally, we provide a detailed and pedagogical appendix on how three-body cuts affect the solution of the integral equation.

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

1 major / 1 minor

Summary. The manuscript constructs a coupled-channel amplitude for the I^G(J^PC)=0^+(1++) K Kbar pi system in an infinite-volume three-body unitary spectator-isobar framework that includes one-particle exchange (automatically incorporating triangle singularities) plus a short-range three-body contact term. The single strength parameter of the contact term is fixed by a fit to the 0+(1++) projection of the BESIII K0S K0S pi0 invariant-mass distribution from J/psi -> gamma(K0S K0S pi0). Analytic continuation to the relevant unphysical Riemann sheets yields two poles whose positions are reported with statistical and systematic uncertainties; the f1(1285) is interpreted as a dressed bare state while the f1(1420) is predominantly dynamically generated from an S-wave K Kbar* quasi-bound state. An additional deeper pole generated by the P-wave pi a0 contact term is also found but has limited visible effect on the physical lineshape. A pedagogical appendix discusses three-body cuts in the integral equation.

Significance. If the extracted poles prove stable, the work supplies a unitary three-body determination of resonance parameters in the f1 sector that automatically respects three-body unitarity and triangle-singularity effects. The explicit separation of bare-state dressing versus dynamical generation, together with the detailed appendix on three-body cuts, constitutes a useful technical contribution to the study of light axial-vector mesons and hadronic molecules.

major comments (1)
  1. [Abstract and amplitude construction] Abstract (pole extraction paragraph) and amplitude-construction section: the short-range three-body interaction is fixed by a single-parameter fit to one invariant-mass distribution. The central claim that the two quoted poles are 'robust' (with the reported ±2±1 and ±2±7 uncertainties) is load-bearing, yet the manuscript provides no explicit demonstration that the pole locations remain stable under reasonable variations of the short-range functional form or under the inclusion of additional independent data sets; without such tests the quoted uncertainties may not fully capture model dependence on unphysical sheets.
minor comments (1)
  1. The notation sqrt(s_f1) for the complex pole position is conventional but would benefit from an explicit statement that it denotes the complex energy (rather than the square root of the Mandelstam variable) to avoid any ambiguity for readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract and amplitude construction] Abstract (pole extraction paragraph) and amplitude-construction section: the short-range three-body interaction is fixed by a single-parameter fit to one invariant-mass distribution. The central claim that the two quoted poles are 'robust' (with the reported ±2±1 and ±2±7 uncertainties) is load-bearing, yet the manuscript provides no explicit demonstration that the pole locations remain stable under reasonable variations of the short-range functional form or under the inclusion of additional independent data sets; without such tests the quoted uncertainties may not fully capture model dependence on unphysical sheets.

    Authors: We agree that explicit stability tests under variations of the short-range functional form would strengthen the robustness claim for the poles on unphysical sheets. The present work uses a minimal single-parameter contact term fitted to the single available BESIII invariant-mass distribution, with systematic uncertainties estimated from cutoff and two-body input variations. We did not perform additional fits with alternative momentum-dependent forms of the contact interaction. In the revised manuscript we will add such tests (e.g., a momentum-dependent contact term) and show the resulting pole trajectories to quantify residual model dependence. Regarding additional independent data sets, the analysis is deliberately scoped to the existing BESIII K0S K0S pi0 projection; a global fit incorporating further channels or experiments lies outside the present scope and is left for future work. The pole-trajectory analysis already provides supporting evidence for the quoted positions, but the planned tests will address the referee's concern directly. revision: partial

Circularity Check

0 steps flagged

No circularity: poles obtained via data fit plus analytic continuation in a unitary framework

full rationale

The derivation constructs a coupled-channel amplitude in the spectator-isobar representation using one-particle exchange (enforced by three-body unitarity) plus a single short-range contact term. The contact strength is fixed by a fit to the BESIII K0S K0S pi0 invariant-mass distribution; the pole positions are then located by analytic continuation of the resulting amplitude to unphysical Riemann sheets. Neither step reduces to the other by definition or by self-citation; the output poles are not inputs, and the framework does not invoke load-bearing self-citations or rename known results. The procedure is self-contained against external data and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the spectator-isobar representation of three-body unitarity and on the assumption that a fit to one experimental distribution sufficiently constrains the short-range interaction for reliable analytic continuation. Free parameters are the strengths of the short-range three-body contacts. The framework assumes the one-particle-exchange term automatically incorporates the triangle singularity without additional tuning.

free parameters (1)
  • short-range three-body interaction parameters
    Adjusted to reproduce the 0+(1++) component of the BESIII invariant-mass distribution
axioms (1)
  • domain assumption Spectator-isobar representation of the three-body amplitude with one-particle exchange
    Abstract states that this representation automatically incorporates the triangle-singularity mechanism required by three-body unitarity

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discussion (0)

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