Possible $\bar{D}^{(*)} \Xi_{cc}^{(*)}$ and $\Xi_{cc}^{(*)}\Xi_{cc}^{(*)}$ molecules as superflavor partners of $T_{cc}$

1. Problem Statement

The paper addresses the theoretical prediction of exotic hadrons that are "superflavor partners" of the recently discovered doubly charmed tetraquark, $T_{cc}^+$.

• Context: The LHCb experiment confirmed $T_{cc}^+$ (quark content $cc\bar{u}\bar{d}$) as a loosely bound $D^0 D^{*+}$ molecule with a very small binding energy ($\sim 360$ keV).
• Theoretical Gap: While $T_{cc}$ is well-studied, its partners involving doubly charmed baryons ($\Xi_{cc}$) remain unexplored. Specifically, the existence and properties of meson-baryon molecules ($\bar{D}^{(*)}\Xi_{cc}^{(*)}$) and baryon-baryon molecules ($\Xi_{cc}^{(*)}\Xi_{cc}^{(*)}$) are unknown.
• Key Challenge: The interaction strength in these systems depends heavily on the coupling constant of the $\sigma$ meson ($g_\sigma$), which is uncertain in current models. The authors aim to determine how the spectrum of these states changes under different assumptions for $g_\sigma$.

2. Methodology

The authors employ a One Boson Exchange Potential (OBEP) framework within the Heavy Quark Symmetry (HQS) and Superflavor Symmetry limits.

• Superflavor Symmetry: This symmetry relates a heavy antiquark ($\bar{Q}$) to a heavy diquark ($QQ$) because both transform as color antitriplets ($\bar{3}_c$).
  • By replacing the heavy antiquark in the $D^{(*)}$ meson with a heavy diquark, the authors map the $D^{(*)}D^{(*)}$ system (related to $T_{cc}$) to the $\bar{D}^{(*)}\Xi_{cc}^{(*)}$ and $\Xi_{cc}^{(*)}\Xi_{cc}^{(*)}$ systems.
  • This allows the use of the same coupling constants and cutoff parameters derived from the $T_{cc}$ analysis for these new systems.
• Interaction Lagrangians:
  • Effective Lagrangians are constructed for interactions between heavy mesons/baryons and light bosons ($\pi, \rho, \omega, \sigma$).
  • The fields include the anti-heavy meson superfield $H_a$ and the doubly heavy baryon superfield $\psi_\mu$.
• Potential Construction:
  • The potential $V(q)$ is derived from the scattering amplitude of $t$-channel boson exchange.
  • A dipole form factor $F(\vec{q})$ is applied at vertices to account for hadronic size, characterized by a cutoff parameter $\Lambda$.
  • The potential includes central, tensor, and spin-orbit forces arising from $\pi, \rho, \omega,$ and $\sigma$ exchanges.
• Numerical Solution:
  • The coupled-channel Schrödinger equation is solved using the Gaussian Expansion Method (GEM).
  • Resonant states are identified using the Complex Scaling Method (CSM).
• Parameter Sets:
  • The cutoff $\Lambda$ is tuned to reproduce the $T_{cc}$ binding energy ($340$ keV) for two distinct scenarios of the $\sigma$ coupling constant:
    1. $g_\sigma^L$ (Large): $g_\sigma = 3.4$ (derived from $g_{\sigma NN}/3$).
    2. $g_\sigma^S$ (Small): $g_\sigma = 0.76$ (derived from $g_\pi/2\sqrt{6}$).
  • This tuning results in $\Lambda_L \approx 1075$ MeV and $\Lambda_S \approx 1682$ MeV.

3. Key Contributions

• Systematic Prediction: The first comprehensive study predicting bound and resonant states for $\bar{D}^{(*)}\Xi_{cc}^{(*)}$ and $\Xi_{cc}^{(*)}\Xi_{cc}^{(*)}$ systems using the superflavor symmetry framework.
• Sensitivity Analysis: A detailed investigation into the dependence of the mass spectra on the uncertain $\sigma$ coupling constant ($g_\sigma$). The study demonstrates that while the existence of states is robust, their binding energies and dominant channel compositions vary significantly between the "Large" and "Small" $g_\sigma$ scenarios.
• Mechanism Elucidation: The paper clarifies the interplay between different boson exchanges. When $g_\sigma$ is small, the binding is driven by enhanced contributions from $\pi$ and $\rho$ exchanges (due to the larger cutoff $\Lambda$), whereas large $g_\sigma$ relies primarily on $\sigma$ exchange.

4. Results

A. $\bar{D}^{(*)}\Xi_{cc}^{(*)}$ System (Meson-Baryon)

• Bound States:
  • A bound state is found only for the isoscalar channel with $I(J^P) = 0(1/2^-)$.
  • Reason: Only this channel contains an S-wave $\bar{D}\Xi_{cc}$ component, which is necessary for binding given the strength of the forces. Higher spin states ($3/2^-, 5/2^-$) lack S-wave components and do not form bound states.
  • Binding Energy: The binding energy is significantly larger for $g_\sigma^S$ ($20.4$ MeV) than for $g_\sigma^L$ ($7.76$ MeV). This is because the smaller $g_\sigma$ requires a larger cutoff, which amplifies the attractive forces from $\pi$ and $\rho$ exchanges.
  • Composition: The $\bar{D}\Xi_{cc}(2S)$ channel is dominant in both cases, but mixing ratios with other channels (like $\bar{D}^*\Xi_{cc}$) increase in the $g_\sigma^S$ case due to enhanced tensor forces.
• Resonant States:
  • Resonances are found for $I(J^P) = 0(1/2^-), 0(3/2^-), 0(5/2^-)$ and $1(1/2^-)$.
  • These are identified as Feshbach resonances, arising from coupled-channel dynamics where S-wave bound states exist in lower channels.
  • For $I=1$, a resonance is found only for $g_\sigma^L$, as the isovector $\pi/\rho$ exchange is repulsive or too weak to bind without the strong $\sigma$ attraction.

B. $\Xi_{cc}^{(*)}\Xi_{cc}^{(*)}$ System (Baryon-Baryon)

• Bound States:
  • Isoscalar ($I=0$): Bound states are found for $0(1^+)$ and $0(2^+)$.
    • The $0(1^+)$ state is deeply bound ($24.7$ MeV for $g_\sigma^L$, $67.4$ MeV for $g_\sigma^S$). The dominant component is $\Xi_{cc}\Xi_{cc}(3S)$.
    • Similar to the meson-baryon case, the $g_\sigma^S$ scenario yields deeper binding due to the dominance of $\pi/\rho$ forces.
  • Isovector ($I=1$): Bound states are found for $1(0^+)$ and $1(1^+)$ only in the $g_\sigma^L$ case.
    • In the $g_\sigma^S$ case, the $\pi/\rho$ forces are insufficient to overcome the repulsion or lack of attraction in the isovector channel without the strong $\sigma$ contribution.
• Resonant States:
  • Multiple resonances are found for $I=0$ and $I=1$ (e.g., $0(2^+), 1(2^+)$).
  • Most are Feshbach resonances.
  • A notable difference: The quasi-bound state of $\Xi_{cc}^*\Xi_{cc}^*$ with $0(1^+)$ exists only for $g_\sigma^L$; it disappears for $g_\sigma^S$ due to insufficient attraction from non-$\sigma$ exchanges.

5. Significance

• Experimental Guidance: The paper provides specific mass predictions and quantum numbers ($J^P, I$) for future experimental searches at facilities like LHCb, Belle II, and PANDA. The distinct predictions for $g_\sigma^L$ vs. $g_\sigma^S$ offer a way to constrain the $\sigma$ coupling constant if these states are observed.
• Validation of Symmetry: The results reinforce the utility of superflavor symmetry in predicting the spectrum of heavy-hadron molecules, extending the understanding from meson-meson systems to meson-baryon and baryon-baryon systems.
• Insight into QCD Forces: The study highlights the delicate balance between short-range ($\sigma$) and medium-range ($\pi, \rho$) forces in exotic hadron formation. It suggests that even if the $\sigma$ coupling is small, the system can still be bound, but the nature of the binding mechanism shifts fundamentally.

In conclusion, the authors predict a rich spectrum of exotic hadrons that are superflavor partners of $T_{cc}$. While the existence of bound states is robust, their specific binding energies and the stability of isovector states are highly sensitive to the $\sigma$-meson coupling, providing a clear target for future theoretical and experimental verification.