Letter
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Cut structures and an observable singularity in the three-body threshold dynamics: The $T_{c c}^{+}$ case

Jun-Zhang Wang ^*, Zi-Yang Lin ^†, and Shi-Lin Zhu ^‡

School of Physics and Center of High Energy Physics, Peking University, Beijing 100871, China

^*wangjzh2022@pku.edu.cn
^†lzy_15@pku.edu.cn
^‡zhusl@pku.edu.cn

Phys. Rev. D 109, L071505 – Published 25 April, 2024

DOI: https://doi.org/10.1103/PhysRevD.109.L071505

Abstract

The three-body threshold effect, the distinctive and intriguing nonperturbative dynamics in the low-energy hadron-hadron scattering, has acquired compelling significance in the wake of the recent observation of the double-charm tetraquark $T_{c c}^{+}$ . This dynamics is characterized by the emergence of singular points and branch cuts within the interaction potential, occurring when the on-shell condition of the mediated particle is satisfied. The presence of these potential singularities indicates that the system is no longer Hermitian and also poses intractable challenges in obtaining exact solutions for dynamical scattering amplitudes. In this work, we develop a complex scaled Lippmann-Schwinger equation as an operation of analytical continuation of the $T$ matrix to resolve this problem. Through a practical application to the $D D^{*} \to D D^{*}$ process, we reveal complicated cut structures of the three-body threshold dynamics in the complex plane, primarily stemming from the one-pion exchange. Notably, our methodology succeeds in reproducing the $T_{c c}^{+}$ structure, in alignment with the quasibound pole derived from the complex scaling method within the Schrödinger equation framework. More remarkably, after solving the on-shell $T$ matrix on the positive real axis of momentum plane, we find an extra new structure in the $D D^{*}$ mass spectrum, which arises from a right-hand cut at a physical pion mass and should be observable in lattice QCD simulations and future high-energy experiments.

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References (46)

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See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevD.109.L071505 for more technical details about the one-pion exchange potential involving the three-body threshold dynamics, an alternative scheme to deal with the singularities of the potential function in the integral, and integral path schemes of solving the on-shell $T$ matrix with a positive real momentum, in addition to which the results of the predicted scattering phase shifts based on the existence of right-hand cut structure and the cut structures of the three-body threshold dynamics with involving the kinetic energy terms of the heavy mesons are presented.
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Phys. Rev. D 109, L071505– Published 25 April, 2024

Vol. 109, Iss. 7 — 1 April 2024

Received 18 January 2024
Accepted 28 March 2024

DOI: https://doi.org/10.1103/PhysRevD.109.L071505

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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Cut structures and an observable singularity in the three-body threshold dynamics: The $T_{c c}^{+}$ case

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Cut structures and an observable singularity in the three-body threshold dynamics: The Tcc+ case

Abstract

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (46)

Cut structures and an observable singularity in the three-body threshold dynamics: The $T_{c c}^{+}$ case