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arxiv: 2605.18075 · v1 · pith:2KLDIIU5new · submitted 2026-05-18 · ✦ hep-ph

Compact csbar{s}bar{s} Tetraquarks in the Charm--Strange Sector: Mass Spectra, Rearrangement Decays and Regge Trajectories with D_s Threshold Inputs

Chetan Lodha , Ajay Kumar Rai This is my paper

Pith reviewed 2026-05-20 10:01 UTC · model grok-4.3

classification ✦ hep-ph
keywords tetraquarksdiquark-antidiquarkCornell potentialrearrangement decaysRegge trajectoriesopen charmhidden strangenessDs mesons
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The pith

Compact cs s-bar s-bar tetraquarks form axial diquark-antidiquark states whose masses and decays are predicted by a Cornell potential tuned to the Ds spectrum.

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

This paper models the cs bar s bar s tetraquark as a compact bound state of a charm-strange diquark and a strange-antistrange antidiquark. It computes the mass spectrum in both semi-relativistic and non-relativistic versions of a Cornell potential whose parameters are fixed by reproducing the known Ds meson spectrum. Pauli symmetry forces the antidiquark to be axial-vector for the lowest states, and the calculation yields concrete predictions for rearrangement decays into Ds star eta, Ds star eta prime and Ds star phi pairs via spin-color recoupling. Orbital and radial Regge trajectories are built to organize the excited states. A sympathetic reader would care because the results supply specific mass and decay targets for experiments looking for exotic open-charm states that also carry hidden strangeness.

Core claim

The authors claim that the cs bar s bar s system forms compact tetraquarks in the bar 3-3 and 6-bar 6 color configurations, with the bar s bar s antidiquark restricted to an axial-vector state by Pauli symmetry, and that the same Cornell potential calibrated on conventional Ds mesons yields reliable masses, threshold structures, rearrangement decay widths to Ds star eta, eta prime and phi modes, and Regge trajectories for the J^P = 0^+, 1^+, 2^+ multiplet.

What carries the argument

Axial-vector diquark-antidiquark configuration [cs][bar s bar s] treated in the Cornell potential framework, with color and spin symmetries used to generate the spectrum and to obtain decay amplitudes through Fierz recoupling.

Load-bearing premise

The model assumes that the exotic four-quark cs bar s bar s state can be described by the identical Cornell potential parameters fitted to ordinary Ds mesons without extra corrections arising from its multi-quark nature.

What would settle it

A measured mass or decay branching ratio for a narrow resonance in the Ds eta or Ds phi invariant-mass spectrum that lies well outside the range predicted for the ground-state multiplet would directly test whether the potential calibrated on two-quark mesons applies to this configuration.

Figures

Figures reproduced from arXiv: 2605.18075 by Ajay Kumar Rai, Chetan Lodha.

Figure 1
Figure 1. Figure 1: Orbital Regge trajectories in the (L, M2 ) plane for the Ds meson family in the non-relativistic and semi-relativistic schemes for S = 0 and S = 1. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Radial Regge trajectories in the (n, M2 ) plane for the Ds meson family in the non-relativistic and semi-relativistic schemes for S = 0 and S = 1. (a) NR, S = 0 (b) NR, S = 1 (c) NR, S = 2 (d) SR, S = 0 (e) SR, S = 1 (f) SR, S = 2 [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Orbital Regge trajectories in the (L, M2 ) plane for the Tcss¯s¯ tetraquark in the 3¯−3 configuration. The first row shows the non-relativistic results and the second row shows the semi-relativistic results for S = 0, 1, 2. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Radial Regge trajectories in the (n, M2 ) plane for the Tcss¯s¯ tetraquark in the 3¯−3 configuration. The first row shows the non-relativistic results and the second row shows the semi-relativistic results for S = 0, 1, 2. (a) NR, S = 0 (b) NR, S = 1 (c) NR, S = 2 (d) SR, S = 0 (e) SR, S = 1 (f) SR, S = 2 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Orbital Regge trajectories in the (L, M2 ) plane for the Tcss¯s¯ tetraquark in the 6−6¯ configuration. The first row shows the non-relativistic results and the second row shows the semi-relativistic results for S = 0, 1, 2. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Radial Regge trajectories in the (n, M2 ) plane for the Tcss¯s¯ tetraquark in the 6−6¯ configuration. The first row shows the non-relativistic results and the second row shows the semi-relativistic results for S = 0, 1, 2. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Semi-relativistic form-factor comparison plots for the pseudoscalar, vector, scalar, axial-vector, [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Semi-relativistic differential decay-width distributions for electron and muon final states across [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
read the original abstract

This work presents a spectroscopy-focused study of the compact open-charm, multi-strange tetraquark configuration \(cs\bar{s}\bar{s}\), modeled as an axial diquark-antidiquark system \([cs][\bar{s}\bar{s}]\). The conventional \(D_s\) meson spectrum is retained as a calibration sector for the model parameters and as a reference for the dominant two-meson thresholds; however, the primary emphasis is placed on the mass spectrum, threshold structure, rearrangement decay mechanisms, and Regge systematics of the \(cs\bar{s}\bar{s}\) tetraquark. The spectrum is computed within a Cornell-potential framework using both semi-relativistic and non-relativistic treatments, with the \(\bar{\mathbf{3}}-\mathbf{3}\) and \(\mathbf{6}-\bar{\mathbf{6}}\) color configurations analyzed separately. Owing to the presence of two identical strange antiquarks, Pauli symmetry imposes restrictions on the \([\bar{s}\bar{s}]\) antidiquark, favoring axial-vector building blocks as the natural low-lying degrees of freedom for the \(J^P=0^+,1^+,2^+\) tetraquark multiplet. The strong-decay sector is formulated in terms of the rearrangement topology \(cs\bar{s}\bar{s}\rightarrow(c\bar{s})(s\bar{s})\), with the \(D_s^{(*)}\eta\), \(D_s^{(*)}\eta'\), and \(D_s^{(*)}\phi\) modes identified through spin-color Fierz recoupling and normalized using two-point and three-point QCD-sum-rule amplitudes. Orbital and radial Regge trajectories are constructed to characterize the excitation patterns of the compact tetraquark and to benchmark them against the calibrated \(D_s\) spectrum. This framework provides a dedicated phenomenological basis for identifying \(cs\bar{s}\bar{s}\) candidates in hidden-strangeness open-charm final states.

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. This paper presents a spectroscopy study of compact cs s-bar s-bar tetraquarks modeled as axial-vector diquark-antidiquark systems [cs][s-bar s-bar]. Mass spectra are computed in a Cornell-potential framework for both semi-relativistic and non-relativistic treatments, with separate analysis of 3-bar-3 and 6-6-bar color configurations. Rearrangement decays to Ds(*) eta, Ds(*) eta', and Ds(*) phi modes are formulated via spin-color Fierz recoupling and normalized with QCD sum-rule amplitudes. Orbital and radial Regge trajectories are constructed, with parameters calibrated on the conventional Ds meson spectrum to provide a phenomenological basis for identifying such states in hidden-strangeness open-charm final states.

Significance. If the results hold after addressing parameter transfer issues, the work supplies a dedicated framework for tetraquark candidates in open-charm hidden-strangeness channels, extending potential-model techniques to exotic configurations and offering testable mass and decay predictions benchmarked against Ds thresholds.

major comments (1)
  1. Model section / Cornell potential application: The same V(r) = -α/r + σ r calibrated on Ds mesons is applied directly to the [cs][s-bar s-bar] tetraquark without rescaling for the different color factors (1/2 for 3-bar-3 or -1/6 for 6-6-bar versus 4/3 for mesons). This unadjusted extrapolation is load-bearing for the quoted mass spectra, threshold structures, and rearrangement decay widths, as the string tension and Coulomb strength should be modified for the exotic configuration.
minor comments (1)
  1. Abstract and introduction: Clarify whether the Pauli symmetry restrictions on the [s-bar s-bar] antidiquark are enforced at the wave-function level or only at the level of allowed J^P multiplets.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and insightful comments on our manuscript. We address the major comment point by point below and are committed to improving the paper accordingly.

read point-by-point responses

  1. Referee: Model section / Cornell potential application: The same V(r) = -α/r + σ r calibrated on Ds mesons is applied directly to the [cs][s-bar s-bar] tetraquark without rescaling for the different color factors (1/2 for 3-bar-3 or -1/6 for 6-6-bar versus 4/3 for mesons). This unadjusted extrapolation is load-bearing for the quoted mass spectra, threshold structures, and rearrangement decay widths, as the string tension and Coulomb strength should be modified for the exotic configuration.

    Authors: We appreciate the referee highlighting this crucial point regarding the color structure in the potential. In our model, the Cornell potential parameters were fitted to the conventional Ds meson spectrum, which corresponds to a color factor of 4/3. For the tetraquark treated as a diquark-antidiquark system, the effective color factors are indeed different: approximately 1/2 for the 3-bar-3 configuration and -1/6 for the 6-6-bar configuration. We acknowledge that directly transferring the unrescaled potential represents an approximation that may affect the absolute mass values and derived quantities. However, since the parameters are phenomenological and the focus is on relative spectra and trajectories, this choice was made to maintain consistency with the threshold inputs. To strengthen the analysis, we will revise the manuscript to explicitly include the color-factor rescaling in the potential for each configuration. This will involve multiplying the Coulomb and linear terms by the appropriate factors, recomputing the mass spectra for both semi-relativistic and non-relativistic cases, and updating the Regge trajectories and decay width estimates. A new paragraph will be added in the model section to discuss the color factors and justify the rescaling procedure. revision: yes

Circularity Check

0 steps flagged

No significant circularity: Ds calibration used for extrapolation to distinct tetraquark system

full rationale

The derivation calibrates the Cornell potential on the conventional Ds meson spectrum as an input sector and then solves the Schrödinger or semi-relativistic equation for the separate [cs][s-bar s-bar] tetraquark configurations in 3-bar-3 and 6-6-bar color channels. The resulting mass spectra, rearrangement decay widths (normalized via independent QCD sum-rule amplitudes), and Regge trajectories are therefore predictions for the exotic system rather than quantities that reduce by construction to the Ds fit. No self-definitional loop, fitted-input-renamed-as-prediction, or load-bearing self-citation chain appears in the abstract or model description; the color-factor rescaling issue is a modeling assumption, not a circular reduction. The framework remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

The central results rest on standard quark-model assumptions plus parameters fitted to the conventional Ds spectrum; no machine-checked proofs or independent data releases are indicated.

free parameters (2)
  • Cornell potential parameters
    Fitted or adjusted using the Ds meson spectrum as calibration sector for both semi-relativistic and non-relativistic treatments.
  • Decay amplitude normalizations
    Normalized using two-point and three-point QCD-sum-rule amplitudes for the rearrangement modes.
axioms (2)
  • domain assumption Tetraquarks can be modeled as compact axial diquark-antidiquark systems with color configurations 3-bar-3 and 6-6-bar.
    Invoked in the modeling of the cs s-bar s-bar state and Pauli symmetry restrictions on the antidiquark.
  • domain assumption The Cornell potential framework calibrated on ordinary mesons applies to exotic tetraquarks without major additional corrections.
    Used to compute the spectrum and Regge trajectories.
invented entities (1)
  • cs s-bar s-bar tetraquark states with specific J^P quantum numbers no independent evidence
    purpose: To provide candidate resonances in hidden-strangeness open-charm channels.
    Postulated within the diquark-antidiquark picture; no independent falsifiable evidence supplied beyond the model itself.

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