Lattice study of $cc\bar u\bar s$ tetraquark channel in $D^{(*)}D^{(*)}_s$ scattering

TL;DR

This work targets the existence of near-threshold cc\bar{u}\bar{s} tetraquarks by computing elastic $DD_s$ scattering in the $J^{P}=0^{+}$ channel and the coupled $DD_s^*$–$D^*D_s$ scattering in the $J^{P}=1^{+}$ channel from lattice QCD using two CLS ensembles at $m_\pi\sim280$ MeV. The authors employ bilocal two-meson interpolators within the distillation framework and extract finite-volume spectra through a variational analysis, then convert these spectra into infinite-volume scattering amplitudes via both Lüscher’s formalism and a finite-volume Lippmann–Schwinger approach. Across both channels, they observe only small energy shifts relative to noninteracting levels, indicating weak interactions, and find no signatures of hadronic poles or near-threshold resonances in the physical region constrained by the lattice data. The results imply no near-threshold $cc\bar{u}\bar{s}$ tetraquark in the studied energy window, placing upper bounds on possible pole locations and motivating further multi-volume, multi-lattice-spacing studies to tighten the systematic uncertainties.

Abstract

We present the first lattice QCD determination of coupled $DD_s^*$ and $D^*D_s$ scattering amplitudes in the $J^{P}=1^{+}$ channel and elastic $DD_s$ scattering amplitude in the $J^{P}=0^{+}$ channel. The aim is to investigate whether tetraquarks with flavor $cc\bar u\bar s$ exist in the region near threshold. Lattice QCD ensembles from the CLS consortium with $m_π \sim 280$ MeV, $a\sim0.09$ fm and $L/a = 24, 32$ are utilized. Finite-volume spectra are determined via variational analysis of two-point correlation matrices, computed using large bases of operators resembling bilocal two-meson structures within the distillation framework. The scattering matrix for partial wave $l=0$ is determined using lattice eigenenergies from multiple inertial frames following Lüscher's formalism as well as following the solutions of Lippmann-Schwinger Equation in the finite-volume on a plane-wave basis. We observe small nonzero energy shifts in the simulated spectra from the noninteracting scenario in both the channels studied, which points to rather weak nontrivial interactions between the mesons involved. Despite the nonzero energy shifts, the lattice-extracted $S$-wave amplitudes do not carry signatures of any hadron pole features in the physical amplitudes in the energy region near the threshold.

Figures (17)

• Figure 1: Left: s-channel diagram. Right: u-channel diagram
• Figure 2: Effective energies as a function of the time interval for the GEVP eigenvalue correlators $\lambda^n(t)$. The plots are presented for the lowest six states in the $T_1^+$ irrep on the large volume ensemble. The bands represent the single-exponential fit estimates for the respective energies and their errors.
• Figure 3: Dependence of fitted energy estimates on the choice of $t_{min}$ for different levels in the $T_1^+$ irrep for the large volume ensemble. The red markers are determined from the ratio of correlators ($R^n(t)$ defined in the text) followed by adding the associated noninteracting level energy (indicated within each block) evaluated using continuum dispersion relation for single mesons. The black markers are based on fits to the $\lambda^{(n)}(t)$. The dashed horizontal black line in each subplot represents the reference noninteracting energy, calculated using the individual meson masses and continuum dispersion relation. The inset figures showcase the normalized operator state overlaps ($\tilde{Z}_n^i$), such that the largest value of $\tilde{Z}_n^i$ for a given operator $i$ across all states $\{n\}$ is unity. The $x$-axis in the inset figures denotes the operator index (only for the cases with single meson components having gamma structures $\gamma_5$ or $\gamma_i$) in the order provided in Table \ref{['tab:op_inelastic']}. The red, green and pink colored bars represent operators of type $DD_s^*$, $D^*D_s$, and $D^*D_s^*$. The operator ordering presented in the inset are listed in the legend appended above the figure.
• Figure 4: The FV energy eigenvalues for the coupled two-meson system involving channels $DD_s$(blue dashed), $DD_s^*$(red dot-dashed), $D^*D_s$(green solid), $D^*D_s^*$(purple solid). The $y$-axis represents the energies in the center-of-momentum frame in units of $DD_s$ (top panes) and $DD_s^*$ (bottom panes) threshold for varying spatial extent, of the boxes used, along the x-axis. The panes presented in the top are relevant for the study of scalar channel, whereas the ones in the bottom are relevant for the axialvector channel. Different panes correspond to FV irreps which are also labeled at the bottom with the corresponding $P^2$ value in brackets. The black markers are the lattice-extracted eigenenergies on the two ensembles utilized. The curves represent the energies of relevant low lying non-interacting levels as a function of volume. The two-meson thresholds are presented in gray solid lines, whereas the faded colored curves in the high energy regions within each irreps, correspond to the lowest shell of noninteracting two-meson levels that are ignored in the entire analysis. The lowest three particle threshold $DD_s\pi$ is the gray dotted horizontal line above the $D^*D_s^*$ threshold.
• Figure 5: FV eigenenergies in $T_1^+(0)$, $A_2(1)$, and $A_2(4)$ irreps determined with two operator bases, first excluding $D^*D_s$ and second excluding $DD_s^*$-type operators. We plot the energies in units of $DD_s^*$ threshold along the y-axis for different volumes of the box along the x-axis. The red circles / green diamonds are the lattice energy levels obtained with operator basis omitting $D^*D_s$ / $DD_s^*$ type operators. The magenta diamonds are levels determined by $D^*D_s^*$-like interpolators, and are unaffected by the other operators in the basis used. The curves represent the noninteracting two-meson energy levels as a function of volume.