Lattice study of scattering phase shifts for $DD^*$ and $BB^*$ systems using twisted boundary conditions: Search for bound state formation

TL;DR

The paper addresses the near-threshold scattering of DD^* and BB^* to search for doubly heavy tetraquarks, applying Lüscher's finite-volume method with partially twisted boundary conditions to access continuous momenta and disentangle S- and P-wave contributions. By combining three twisting schemes and RHQ-heavy-quark actions on 2+1 flavor PACS-CS lattices, the authors extract S- and P-wave phase shifts in I=0 and I=1 channels across multiple m_π, finding attractive I=0 interactions and a unitary BB^* limit at m_π = 295 MeV, consistent with a shallow BB^* bound state near threshold. They report a0^{BB^*} ≈ -7.6(5.8) fm at the physical point, suggesting a shallow bound state of order O(100) keV, while DD^* shows no bound state in this pion-mass range. The work also highlights operator-basis limitations in probing deeply bound T_{bb} states and suggests extending the operator basis and employing variational methods to resolve potential multiple bound states, including a possible second BB^* bound state. Overall, the study demonstrates a robust near-threshold analysis framework for heavy-heavy molecular systems with potential implications for doubly heavy tetraquark spectroscopy.

Abstract

We investigate the $S$- and $P$-wave phase shifts for the $DD^\ast$ and $BB^\ast$ scatterings using Lüscher's finite-size method under twisted boundary conditions to search for doubly charmed tetraquaks, $T_{cc}^+$, and doubly bottomed tetraquarks, $T_{bb}^-$ as the hadronic bound states. The $T_{cc}^+$ state was observed as a peak just bellow the $DD^*$ threshold by LHCb Collaboration, while the $T_{bb}^-$ state is a theoretically predicted tetraquark state having heavier quark flavors $bb\bar u \bar d$. Lüscher's finite-size method is one of the well established methods for calculating the scattering phase shifts between two hadrons in lattice QCD simulations. Several studies have used simulations under the periodic boundary condition to determine the scattering phase shifts at a few discrete momenta for the $DD^*$ system. However, the scattering phase shift has not been investigated for the $BB^*$ system. In this study, $S$- and $P$-wave scattering phase shifts for the $DD^*$ and $BB^*$ systems in both $I=0$ and $I=1$ channels under several types of partially twisted boundary conditions. The use of the partially twisted boundary conditions enables us to obtain the scattering phase shift at any momentum by continuously varying the twisting angle. It also allows us to easily access the $P$-wave scattering phase shifts through the mixing of $S$- and $P$-waves, which is induced by the imposed boundary conditions. The 2+1 flavor PACS-CS gauge ensembles at $m_π=295$, 411 and 569 MeV are used. For charm and bottom quarks, the relativistic heavy quark action is adopted to reduce the lattice discretization artifacts due to the heavy quark mass. We discuss the emergence of a shallow bound state with a binding energy of $O(100)$ keV at the physical pion mass in the $BB^*$ system, which has the quantum number $I(J^P)=0(1^+)$.

Figures (10)

• Figure 1: Effective mass plots of the $D$ and $D^*$ states (upper), and the $B$ and $B^*$ states (lower). Red lines indicate fit results and yellow bands display the fit ranges and 1 standard deviation estimated by the jackknife analysis.
• Figure 2: Check of the dispersion relations for the $D$ (upper) and $B$ (lower) states, which are calculated for ensemble A. For comparison, the continuum dispersion relation is denoted as the dotted line in each panel.
• Figure 3: The effective energy shift $\delta E_{\rm eff}$ as a function of $t$ for the $DD^{*}$ (upper) and $BB^{*}$ (lower) systems in the $I=0$ channel at $m_\pi=411$ MeV (ensemble B) with the trivial twisting angle, ${\bm \theta}=(0,0,0)$, corresponding the standard periodic boundary condition.
• Figure 4: Left: $k\cot\delta_0$ as a function of $q^2$ for the case of the $I=0$$DD^*$ scattering at $m_\pi=411~\mathrm{MeV}$ (Ensemble B). Right: $k^3\cot\delta_1$ as a function of $q^2$ for the case of the $I=0$$DD^*$ scattering at $m_\pi=411~\mathrm{MeV}$ (Ensemble B).
• Figure 5: $S$-wave phase shifts for the $DD^*$ and $BB^*$ scatterings in the $I=0$ channel: $m_\pi=295~\mathrm{MeV}$ (left), $m_\pi=411~\mathrm{MeV}$ (middle), and $m_\pi=569~\mathrm{MeV}$ (right). The upper panels illustrate the quantity $k\cot\delta_0$ as a function of $q^2$, while the phase shift $\delta_0$ is plotted as the two-hadron energy $E$, measured from the threshold, in the lower panels.