Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dynamical generation of charmonium-like tetraquarks in an off-shell coupled-channel formalism

Hee-Jin Kim, Hyun-Chul Kim

Abstract

We investigate the dynamical generation of charmonium-like ($I=0$) with spin-parity $J^{PC}=0^{++}, 1^{++}, 2^{++}$, and $3^{--}$ in the mass range of $3.6$ to $4.3$ GeV. We employ the off-shell coupled-channel formalism, constructing kernel amplitudes from effective Lagrangians that respect heavy-quark spin-flavor and chiral symmetries. To focus solely on dynamically generated states, we explicitly exclude $s$-channel pole diagrams and include only $t$- and $u$-channel meson exchanges. Solving the integral equations, we identify six poles in the complex energy plane. In the scalar ($0^{++}$) sector, we find a bound state below the $D\bar{D}$ threshold and a resonance at $\sqrt{s_R}=(3861-i\,23)\,\mathrm{MeV}$. For the axial-vector ($1^{++}$) sector, the experimentally observed $χ_{c1}(3872)$ is reproduced as a bound state near the $D\bar{D}^*$ threshold, alongside a broader resonance at $(3961-i\,32)\,\mathrm{MeV}$, which is a plausible candidate for the $X(3940)$. Furthermore, we find a narrow tensor ($2^{++}$) state at $4005\,\mathrm{MeV}$ and a vector ($3^{--}$) state at $4030\,\mathrm{MeV}$. The present results demonstrate that coupled-channel dynamics, particularly involving the $D^*\bar{D}^*$ channel, play a crucial role in the formation of these charmonium-like exotic states.

Dynamical generation of charmonium-like tetraquarks in an off-shell coupled-channel formalism

Abstract

We investigate the dynamical generation of charmonium-like () with spin-parity , and in the mass range of to GeV. We employ the off-shell coupled-channel formalism, constructing kernel amplitudes from effective Lagrangians that respect heavy-quark spin-flavor and chiral symmetries. To focus solely on dynamically generated states, we explicitly exclude -channel pole diagrams and include only - and -channel meson exchanges. Solving the integral equations, we identify six poles in the complex energy plane. In the scalar () sector, we find a bound state below the threshold and a resonance at . For the axial-vector () sector, the experimentally observed is reproduced as a bound state near the threshold, alongside a broader resonance at , which is a plausible candidate for the . Furthermore, we find a narrow tensor () state at and a vector () state at . The present results demonstrate that coupled-channel dynamics, particularly involving the channel, play a crucial role in the formation of these charmonium-like exotic states.
Paper Structure (10 sections, 27 equations, 7 figures, 6 tables)

This paper contains 10 sections, 27 equations, 7 figures, 6 tables.

Table of Contents

  1. Introduction
  2. General Formalism
  3. Results and Discussion
  4. $\chi_{cJ}$ states: $I^G(J^{PC})=0^+(J^{++})$
  5. $\chi_{c0}$ $J^{PC}=0^{++}$
  6. $\chi_{c1}$ $J^{PC}=1^{++}$
  7. $\chi_{c2}$ $J^{PC}=2^{++}$
  8. $\psi$ states: $I^G(J^{PC})=0^-(J^{--})$
  9. $\psi_3$ $J^{PC}=3^{--}$
  10. Summary and conclusions

Figures (7)

  • Figure 1: Mass spectrum of hidden-charm states in the range of $3.7$–$4.1$ GeV. The lines in the left column indicate the resonance states obtained in the present work, whereas those in the right column denote the experimental data for the states with $c\bar{c}$ content PDG:2024cfk.
  • Figure 2: Schematic diagram for the two-body coupled integral equations.
  • Figure 3: $t$- and $u$-channel Feynman diagrams for $12\to 34$.
  • Figure 4: Diagonal partial-wave transition amplitudes for the $I^G(J^{PC}) = 0^+(0^{++}$) channel as functions of $\sqrt{s}$.
  • Figure 5: Axial-vector transition amplitudes for diagonal elements as functions of $\sqrt{s}$ for $I^G(J^{PC}) = 0^+(1^{++}$) channel.
  • ...and 2 more figures