Understanding the Structure of Doubly-Heavy Tetraquarks based on the Diquark Model

Abstract

We investigate the $T_{cc}$ tetraquark, treating it as a bound state of a heavy diquark and a light antidiquark. Using the Silvestre-Brac potential and solving the Schrödinger equation via the Gaussian Expansion Method, we find that the excitation energy between the heavy diquark and light antidiquark is unexpectedly larger than that between the two light anti-quarks within the anti-diquark -- contrary to the naive expectation where the former is smaller than the latter. We trace this inversion of the mass hierarchy to the centrifugal force acting on the light degree of freedom. Applying the same framework to other systems ($T_{bb}, Λ_b, Λ_c$) yields qualitatively identical behavior, demonstrating the robustness of the mechanism. These results provide new insights into diquark dynamics and the mass structure of exotic hadrons.

Figures (9)

• Figure 1: Schematic picture of the $T_{cc}$ system (dashed gray ellipse) with a color-$\mathbf{3}$ light-antidiquark (dashed magenta ellipse) and a color-$\mathbf{\bar{3}}$ heavy-diquark (dashed cyan ellipse) subsystems, based on our $\overline{\text{LD}}$HD description. The subsystems consist either of two antiquarks (top), i.e., anti-up $\bar{u}$ and anti-down $\bar{d}$, or two charm-quarks $c$ (bottom). The interquark separations are $\rho$ (top and red solid line), $\rho_{cc}$ (bottom and blue solid line), and $\lambda$ (vertical and black solid line).
• Figure 2: Energy landscape of the $T_{cc}$ system for $n_\text{max}= 20$ as a function of the nonlinear range parameters $(r_1, r_\text{max})$. The red area corresponds to the most stable and lowest-energy configurations where the energy difference was less than $1$ MeV. This area contains $956$ possible combinations and has an energy value of $3.740\pm 0.001$ GeV, with $m_{\bar{u}\bar{d}}=0.666\pm0.001$ GeV and $m_{cc}=3.500\pm0.001$ GeV.
• Figure 3: Radial wave function of the $T_{cc}$ ground (blue) and excited state (orange) obtained with the optimized range parameters $\{n_\text{max}=20, r_1 = 0.1\, \text{fm}, r_\text{max}=6\, \text{fm}\}$. The smooth and localized shape confirms the stability and convergence of the basis. Note that although the excited wave function extends beyond $6$ fm and approaches zero around $10$ fm, the obtained energies remain unchanged under the choice of $r_\text{max}$. In addition, choosing $6$ fm helps to reduce computational time.
• Figure 4: Comparison of the predicted $T_{cc}$ spectrum obtained in the DQM (blue solid lines) with that from the chiral EFT framework PhysRevD.102.014004_Spectrum_of_singly_heavy_baryons__Kim (violet solid lines). The energy levels of corresponding states are shown together with their excitation energies relative to the ground state. While most states appear close in energy between the two approaches, the $\rho$-mode excitation lies approximately $200$ MeV lower in our model, indicating a significant sensitivity to the treatment of short-range diquark correlations.
• Figure 5: Heavy-quark mass ($m_c$) dependence of excited energies based on the HO model [see Eq. \ref{['eq:HO_Jacobian_coord']}] for the $T_{cc}$ system. The $\rho$-mode (blue solid line), $\lambda$-mode (orange solid line), and the $\rho_{cc}$-mode (green solid line) are calculated by Eq. \ref{['eq:oscillator_frequcies_HO_Jacobian_coord']}. The vertical gray line corresponds to the used charm mass $m_c=1.836$ GeV Silvestre-Brac. The initial parameters, $m_u$ and $K$, are given by the AL1 potential (see. Tab. \ref{['tab:AL1_parameters']}) and arbitrarily chosen, respectively.