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Near-threshold dynamics of $χ_{c 1}$ (3872) and $T_{c c}^{+}$ (3875) states via effective range expansion and Monte Carlo uncertainty propagation

Department of Particle Physics, Faculty of Science, University of Geneva, P.O. Box 1205 Geneva, Switzerland and Laboratory of Nuclear, Atomic and Molecular Physics, University of Yaoundé I, P.O. Box 812 Yaoundé, Cameroon

Department of Particle Physics, Faculty of Science, University of Geneva, P.O. Box 1205 Geneva, Switzerland

Department of Physics, Bertoua Higher Teacher's Training College, University of Bertoua, P.O. Box 55 Bertoua, Cameroon

Department of Particle Physics, Faculty of Science, University of Geneva, P.O. Box 1205 Geneva, Switzerland

School of Mathematics, University of Manchester, Manchester M13 9PL, United Kingdom

Department of Particle Physics, Faculty of Science, University of Geneva, P.O. Box 1205 Geneva, Switzerland

Laboratory of Nuclear, Atomic and Molecular Physics, University of Yaoundé I, P.O. Box 812 Yaoundé, Cameroon

^*Contact author: andre.atangana@etu.unige.ch; aandreaime@yahoo.fr

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Phys. Rev. C 114, 045208 – Published 27 October, 2026

DOI: https://doi.org/10.1103/gck5-f5j5

Abstract

We investigate the internal structure of two exotic tetraquark candidates, $χ_{c 1} (3872)$ and $T_{c c}^{+} (3875)$ , whose masses lie within $1 MeV$ of the $D^{0} {\bar{D}}^{* 0}$ and $D^{0} D^{* +}$ thresholds, respectively. Using the effective-range expansion (ERE) supplemented by resonance compositeness relations on the second Riemann sheet, we extract scattering lengths $a$ and effective ranges $r$ directly from the experimental masses and widths and quantify the $D D^{*}$ molecular content through partial compositeness coefficients $X_{j}$ . A two-channel framework couples each state's inelastic detection channel ( $J / ψ π π$ ; $D^{0} D^{0} π^{+}$ ) to the near-threshold molecular channel. Full Monte Carlo uncertainty propagation with $N = 50 000$ Gaussian samples yields asymmetric $68 %$ confidence intervals for all observables. We find large negative scattering lengths $a = - 8 . 49_{- 1.05}^{+ 0.93}$ fm for $χ_{c 1} (3872)$ and $a = - 14 . 57_{- 1.66}^{+ 1.70}$ fm for $T_{c c}^{+}$ , both far exceeding the natural hadronic scale $\approx 1 / m_{π}$ and dominant molecular components $X_{2} > 0.95$ under full compositeness. The results are stable for total compositeness $X \in [0.5, 1.0]$ , where $1 - X$ represents an upper bound on any compact/CDD-pole contribution. In the present study, we go beyond the pole-position Castillejo-Dalitz-Dyson (CDD) criterion by explicitly constructing the inverse amplitude as an ERE background plus a scanned CDD pole term, finding no region of physically reasonable bare-state parameter space that reproduces the pole while leaving a natural background; we fully propagate the loop-function subtraction constant $a (μ_{g})$ as a Monte Carlo nuisance parameter jointly with the experimental mass and width, confirming it contributes a negligible ( $< 0.01 %$ ) additional uncertainty; and we test robustness against an assumed correlation between $M_{R}$ and $Γ_{R}$ , finding all results stable over a broad range of correlation coefficients. We demonstrate analytically and numerically the scheme-independence of the compositeness coefficients, provide explicit derivations of all ERE formulas from the pole condition, clarify the analytic continuation procedure for closed channels, and compare our results quantitatively with independent ERE, effective-field theory, and other relevant studies. The $Z_{c} (3900)$ case is treated separately to show the breakdown of the two-channel approximation for above-threshold states. Graphical illustrations of the Monte Carlo distributions are included.

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Phys. Rev. C 114, 045208– Published 27 October, 2026

Vol. 114, Iss. 4 — October 2026

Received 28 May 2026
Revised 3 August 2026
Accepted 1 September 2026

DOI: https://doi.org/10.1103/gck5-f5j5

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Funded by SCOAP³.

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Near-threshold dynamics of $χ_{c 1}$ (3872) and $T_{c c}^{+}$ (3875) states via effective range expansion and Monte Carlo uncertainty propagation

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Near-threshold dynamics of χc1(3872) and Tcc+(3875) states via effective range expansion and Monte Carlo uncertainty propagation

Abstract

Physics Subject Headings (PhySH)

Article Text

References (57)

Near-threshold dynamics of $χ_{c 1}$ (3872) and $T_{c c}^{+}$ (3875) states via effective range expansion and Monte Carlo uncertainty propagation