Coupled-channel scattering of $DD, DD^*$ and $D^* D^*$ in isospin-$1$ from lattice QCD

TL;DR

This study delivers the first lattice QCD determination of coupled-channel scattering among $DD$, $DD^*$, and $D^*D^*$ in isospin $I=1$ for $J^P=0^+,1^+,2^+$, using three volumes at $m_\pi\approx391$ MeV where the $D^*$ is stable. Finite-volume spectra are computed via distillation and a variational analysis, and the Lüscher formalism is employed to constrain a multichannel $t$-matrix through several K-matrix and Chew–Mandelstam parametrizations; no unphysical poles are found within the explored energy region. The results show predominantly weak, repulsive $S$-wave interactions across all channels, with a weakly attractive $D^*D^*$ $S$-wave in $J^P=0^+$ and only small inter-channel couplings, yielding no bound, virtual, or resonant states in the studied region. These amplitudes complement prior isospin-$0$ lattice findings and provide crucial inputs for upcoming three-body analyses of $DD\pi$ and for addressing left-hand cut effects in $DD^*$ scattering, advancing the lattice QCD mapping of the doubly charmed sector and informing phenomenology of exotic hadrons like the $T_{cc}^+$ family. The work also underscores the consistency with heavy-quark spin symmetry and SU(3) perspectives while highlighting the need to treat left-hand cuts in future studies.

Abstract

The first coupled-channel determination of two-body $D$ and $D^*$ scattering amplitudes in isospin-$1$ from lattice quantum chromodynamics is presented. Using three lattice volumes at $m_π\approx 391~\mathrm{MeV}$, finite-volume energies relevant for the channels of interest are determined. Through the Lüscher formalism, these energies are used to constrain amplitudes of coupled $J^P=0^+$ $DD - DD^*$, $J^P=1^+$ $DD^*- D^*D^*$ and $J^P=2^+$ $DD - DD^* - D^* D^*$ scattering. All channels feature weakly repulsive interactions in $S$ wave, except for a weak $D^* D^*$ attraction in $J^P=0^+$. No amplitude singularities corresponding to physical states are found. Some of these amplitudes will be a necessary component of future lattice QCD analyses of $DDπ$ and $DD^*$ scattering taking into account three-body and left-hand cut effects.

Figures (10)

• Figure 1: Finite-volume spectrum for rest-frame irreps $[000]A_1^+$, $[000]T_1^+$ and $[000]A_2^+$. The points are the computed lattice energies and error bars are statistical with the additional systematic uncertainty added in quadrature, as described in the text, and colored based on the magnitude of their largest operator-state overlap. The translucent data points were not used in the following scattering analysis (see also Appendix \ref{['apx:operatortables']}). The inset histograms represent the magnitudes of the normalized operator-state overlaps associated with the energy level they are next to, as described in the text. The noninteracting $DD$ (blue), $DD^*$ (orange) and $D^*D^*$ (red) finite-volume energies are represented by solid lines. When the degeneracy of a noninteracting level is larger than one, this is indicated by a number in curly brackets, and the corresponding lattice levels are displaced horizontally to facilitate visualization. Following the same color scheme, the opening of kinematic thresholds is represented by horizontal dashed lines, where the $D_0^*D$ threshold ($a_t E_\mathrm{cm} \approx 0.734$) has been omitted due to its proximity to the $DD\pi$ threshold (see Table \ref{['tab:hadmasses']}).
• Figure 2: Same as Fig. \ref{['fig:fvspectra1']}, but for the $[000]T_2^+$ and $[000]E^+$ irreps.
• Figure 3: Phase shift (left) and $k \cot \delta_0$ (right) representations of the results for isovector $DD$ elastic scattering in $J^P=0^+$. The darker band represents the statistical error of the example parametrization quoted in Eq. \ref{['eq:0pelasticpars']}, while the lighter band is an envelope which accounts for the mass-anisotropy variation. The different curves correspond to the central value of all other reasonable fits listed in Table \ref{['tab:par_var_0pelastic']}. The data points are obtained by using the lattice energies in Eq. \ref{['eq:luscher']}. The relevant thresholds and left-hand cut positions are indicated on the horizontal axis. The intersection of the $k \cot \delta_0$ curves with the purple dotted line indicates the position of $t$-matrix singularities on the corresponding parametrizations.
• Figure 4: Left: Normalized squared amplitudes (upper) and lattice energies (lower) for coupled-channel $DD, D^*D^*$ scattering in $I=1$ and $J^P=0^+$. Other amplitudes that are consistent with zero are not shown for clarity. Right: Phase shift (upper) and inelasticity (lower) from the generalized Stapp parametrization applied to the same system, as described in the text. The inner and outer bands correspond to the statistical error and mass-anisotropy envelope of the example parametrization, as described in the text and quoted in Eq. \ref{['eq:0pcoupledpars']}. The curves represent the central value result of all parametrization variations yielding reasonable fits, as detailed in Table \ref{['tab:par_var_0pcoupled']}.
• Figure 5: Lattice energies (black points) and finite-volume spectrum as a function of $L/a_s$ (orange bands) on each irrep considered for analyzing $J^P=0^+,1^+$ and $2^+$ coupled-channel scattering. The spectrum bands are derived from the example parametrizations quoted in Eqs. \ref{['eq:0pcoupledpars']}, \ref{['eq:1p3pcoupledpars']}, \ref{['eq:2pcoupledpars']}. The bands only account for the statistical uncertainties on the quoted parameters. Non-interacting energies are represented by solid lines and threshold by dashed lines, following the same color scheme of Figs. \ref{['fig:fvspectra1']} and \ref{['fig:fvspectra2']}.