[Paper Review] States of $ρD^* \bar D^*$ with $J=3$ within the Fixed Center Approximation to the Faddeev equations

M. Bayar, Xiu-Lei Ren+1 more|arXiv (Cornell University)|Jan 13, 2015

Quantum Chromodynamics and Particle Interactions4 citations

TL;DR

This study investigates $J=3$ states in the $\rho D^{*}\bar{D}^{*}$ system using the Fixed Center Approximation (FCA) to the Faddeev equations, treating $D^{*}\bar{D}^{*}$ or $\rho D^{*}$ as pre-bound clusters. It predicts a narrow $I=1$ resonance at 4340 MeV with a width of about 50 MeV when the $\rho$ orbits the $D^{*}\bar{D}^{*}$ cluster, and less certain but strongly bound states around 4000 MeV ($I=0$) and 4200 MeV ($I=1$) when $\bar{D}^{*}$ orbits the $\rho D^{*}$ cluster.

ABSTRACT

We study the interaction of the a $ρ$ and $D^*$, $\bar D^*$ with spins aligned using the Fixed Center Approximation to the Faddeev equations. We select a cluster of $D^* \bar D^*$, which is found to be bound in $I=0$ and can be associated to the X(3915), and let the $ρ$ meson orbit around the $D^*$ and $\bar D^*$. In this case we find an $I=1$ state with mass around 4340 MeV and narrow width of about 50 MeV. We also investigate the case with a cluster of $ρD^*$ and let the $\bar D^*$ orbit around the system of the two states. The $ρD^*$ cluster is also found to bind and leads to the $D^*_2(2460)$ state. The addition of the extra $\bar D^*$ produces further binding and we find, with admitted uncertainties, a state of $I=0$ around 4000 MeV, and a less bound narrow state with $I=1$ around 4200 MeV.

Motivation & Objective

• To investigate the existence of three-body resonances with $J=3$ in the $\rho D^{*}\bar{D}^{*}$ system, a novel configuration in the hidden charm sector.
• To explore the role of strong vector-vector interactions in forming tightly bound three-body states with aligned spins.
• To apply the Fixed Center Approximation (FCA) to the Faddeev equations as a reliable method for systems where two particles form a pre-bound cluster.
• To provide theoretical support for experimental searches of high-spin charmonium-like states in the 4.0–4.4 GeV region.
• To extend previous studies on multi-vector states to include hidden charm systems, particularly $\rho D^{*}\bar{D}^{*}$.

Proposed method

• The FCA to the Faddeev equations is employed to model the three-body $\rho D^{*}\bar{D}^{*}$ system, assuming one pair forms a stable cluster.
• The $D^{*}\bar{D}^{*}$ cluster is treated as a bound state with $I=0$, identified with the $X(3915)$ resonance.
• The $\rho D^{*}$ cluster is assumed to bind in $I=1/2$, corresponding to the $D_{2}^{*}(2460)$ state.
• The scattering amplitude is calculated via coupled-channel Faddeev equations with unitary $t$-matrix elements derived from hidden gauge Lagrangians.
• The interaction is solved iteratively, with the third particle (e.g., $\rho$) scattering off the cluster, and the full amplitude is constructed from repeated rescattering diagrams.
• The $J=3$ quantum number is ensured by aligning the spins of all three vector particles, maximizing the vector-vector attraction.

Experimental results

Research questions

• RQ1 Can the $\rho D^{*}\bar{D}^{*}$ system form a bound state with $J=3$ when the $D^{*}\bar{D}^{*}$ pair is pre-bound?
• RQ2 What is the mass and width of the $\rho$-meson bound state orbiting the $D^{*}\bar{D}^{*}$ cluster in the $I=1$ channel?
• RQ3 How does the binding and width change when the $\bar{D}^{*}$ meson orbits the $\rho D^{*}$ cluster instead?
• RQ4 Is the Fixed Center Approximation reliable for systems where the orbiting particle is heavier than the cluster?
• RQ5 Can the $\rho D^{*}\bar{D}^{*}$ system produce a narrow resonance near 4.3 GeV, consistent with experimental searches?

Key findings

• A narrow $I=1$ resonance at 4340 MeV with a width of approximately 50 MeV is predicted when the $\rho$ meson orbits the $D^{*}\bar{D}^{*}$ cluster.
• The $D^{*}\bar{D}^{*}$ cluster is found to be bound in $I=0$, consistent with the $X(3915)$ state, and serves as a stable core for the three-body system.
• When $\bar{D}^{*}$ orbits the $\rho D^{*}$ cluster, the FCA predicts a bound $I=0$ state at around 4000 MeV and a less bound $I=1$ state at approximately 4200 MeV.
• The $I=1$ state with the $\rho$ orbiting the $X(3915)$ cluster is considered more reliable due to the lighter mass of the orbiting particle.
• The $I=0$ state with $\bar{D}^{*}$ orbiting the $\rho D^{*}$ cluster shows a large width of about 250 MeV, indicating possible inapplicability of FCA due to mass imbalance.
• The results suggest strong binding in the $J=3$ sector, supporting further investigation with full Faddeev or variational methods.