Three-body study of the $T_{cc}(3875)^+$ from lattice QCD

Abstract

We discuss an ongoing first lattice study of the doubly-charmed tetraquark $T_{cc}^+$(3875) via a three-body approach. We investigate the $DDπ$ system in the $I=0$, $C=2$ sector, where the $T_{cc}^+$ appears as a pole in the $J^P = 1^+$ $DDπ$ elastic scattering amplitude. The approach automatically incorporates two-body $DD^*$ and three-body $DDπ$ effects and treats left-hand cuts due to single $π$ exchanges. Two CLS ensembles, X252 and X253, with pion mass $M_π\approx 280$ MeV, are used, and an operator set comprised of two- and three-hadron and tetraquark operators is employed to extract finite-volume energies. Additional inputs are required for the three-body finite-volume analysis, in the form of amplitudes for the $I=1$ $DD$ and $I=1/2$ $Dπ$ two-body subsystems. We present preliminary results for these subchannels and perform exploratory three-body spectra determinations for simple choices of the three-particle K-matrix $\mathcal{K}_{\text{df}, 3}$, allowing a first comparison to the lattice spectrum.

Figures (3)

• Figure 1: Plot of the $s$-wave phase shift for the $I=1$$DD$ subsystem. The solid blue line is the best ERE3 fit described in the text and the red points show the data. Two levels included in the fit (ground states of the $(L\boldsymbol P/2\pi)^2=2,3$ frames) are not shown, as their error bars cross free-energy singularities of the zeta function. The dashed black line shows the bound state condition and the shaded gray region corresponds to the left-hand cut due to two-pion exchange.
• Figure 2: The $D\pi$ spectrum on each ensemble is shown on the left. On the right, the $s$- and $p$-wave phase shift plots show the best fits described in the main text. The rest frame $A_{1g}$ and $T_{1u}$ levels included in the fit are displayed in red. The $D^*_0$ appears as a virtual bound state in $s$-wave, while the $D^*$ is a bound state in the $p$-wave, as expected. The inset zooms in on the intersection of the bound state condition (dashed line).
• Figure 3: Comparison of the selected $DD^* + DD\pi$ levels extracted on X252 (black) with the QC3 predictions for vanishing $\mathcal{K}_{\rm df, 3}$ (red), and the two fits described in the main text (blue and orange). The relevant two- and three-body thresholds are shown as dashed lines.