$D\bar{D}$ interactions are weak near threshold in QCD

TL;DR

This study uses lattice QCD with three light-quark masses to examine near-threshold $D\bar{D}$ scattering in isospin-0, including the open hidden-charm channel $\eta_c\eta$ via a coupled-channel $S$-wave description. Employing the Lüscher framework with a $K$-matrix parameterization and Chew–Mandelstam phase-space, the authors find only weak, effectively decoupled interactions between $\eta_c\eta$ and $D\bar{D}$, across $m_\pi$ from 239 to 391 MeV. No bound-state or resonance poles near the $D\bar{D}$ threshold are required to describe the finite-volume spectra, in contrast to some earlier lattice claims, though deeply bound charmonia ($\chi_{c0}(1P)$, $\chi_{c2}(1P)$) are observed below threshold. The results imply that the next scalar resonance above $\chi_{c0}(1P)$ lies well above threshold (likely above 3900 MeV), highlighting a comparatively simple near-threshold dynamics for $D\bar{D}$ in this sector and suggesting that near-threshold phenomena are not generic features of QCD in this channel. Limitations include a single lattice spacing and a limited operator basis, underscoring the need for continuum extrapolation and broader interpolator sets in future work.

Abstract

We study near-threshold $D\bar{D}$ scattering in $S$ and $D$-wave to determine whether or not resonances or bound states are present. Working in the approximation where charm-annihilation is forbidden, with two degenerate light-quark flavors and a heavier strange quark, isospin is a good quantum number, and the only other other channel that is kinematically open is $η_cη$. Using lattice QCD we compute, as a function of varying light-quark mass, the $S$-matrix for coupled-channel $η_cη- D\bar{D}$ scattering and find only weakly interacting meson pairs. In contrast to several other studies, we find no evidence for any bound-state or resonance singularity in the energy region between the deeply-bound $χ_{c0}(1P)$ state and the $D_s\bar{D}_s$ threshold.

Figures (8)

• Figure 1: Finite-volume spectra in three irreps determined on the lattices described in Section \ref{['sec:lqcd']}. Energy levels are colored according to their dominant operator overlap determined in the diagonalization of the matrix of correlation functions: $\eta_c \eta$-like (red), $D\bar{D}$-like (green), $c\bar{c}$-like (black). Grey points in the hatched high-energy region are not used in the subsequent determination of scattering amplitudes. Curves indicate the non-interacting spectrum of meson pairs on the relevant volumes with the same color-coding, with the addition of $\psi \omega$ (purple) and $D_s \bar{D}_s$ (light green) appearing above the energy cutoff. Dashed lines show the relevant meson-meson kinematic thresholds.
• Figure 2: Coupled $\eta_c \eta - D\bar{D}$$S$-wave scattering amplitudes descriptions of finite-volume spectra. Top panels illustrate the magnitude with a normalization where the unitarity limit is 1 for $t_{\eta_c \eta, \eta_c \eta}$ (red) and $t_{D\bar{D}, D\bar{D}}$ (green). Center panels show channel phase-shifts, $\delta_{\eta_c \eta}$ (red), $\delta_{D\bar{D}}$ (green) and inelasticity, $\eta$, defined so that the diagonal elements of the $2\times 2$$t$-matrix are $t_{ii} = \tfrac{\eta e^{2 i \delta_i} - 1}{2i \rho_{i}}$ and the off-diagonal element is $t_{ij} = \tfrac{\sqrt{1-\eta^2} e^{i (\delta_i + \delta_j)}}{2 \sqrt{\rho_i \rho_j}}$. Bottom panels show the finite-volume spectrum given by the amplitude (orange curves) compared the spectrum constraining the amplitudes. (Left) Amplitudes of Eq. \ref{['eq:fit_856_no_poles']}. (Right) Amplitudes of Eq. \ref{['eq:fit_856_with_poles']}.
• Figure 3: $S$-wave coupled-channel phase-shifts, $\delta_{\eta_c \eta}$, $\delta_{D\bar{D}}$, and inelasticity $\eta$, as defined in the caption of Figure \ref{['fig:ref_amp_856']}, for a range of amplitude parameterizations capable of describing the finite-volume spectrum (Table \ref{['tab:amp_variations']}). For the amplitude given in Eq. \ref{['eq:fit_856_no_poles']}, the uncertainty bands are shown (inner band is the statistical uncertainty, outer band includes also the impact of varying the masses and anisotropy as described in the text). For other amplitude parameterizations, only the central value curves are plotted.
• Figure 4: $S$-wave scattering amplitudes for $m_\pi\sim 239$ MeV (left) and $330$ MeV (right). Same color coding as in top panels of Figure \ref{['fig:ref_amp_856']}.
• Figure 5: ${D\bar{D}}$ amplitudes plotted as $(m_D/k)\tan\delta$, with decoupled $S$-wave ${D\bar{D}}$ interactions. The large green circle shows the location of ${D\bar{D}}$ threshold. The energy level points selected have overlaps dominated by ${D\bar{D}}$ operators, and the $D$-wave has been fixed to zero in the quantization condition. The grey points are above the energy cutoff used in the fits for each lattice. The dashed grey curves on the left show $\pm|k|$ below threshold. We have removed 4 points from this plot for the $m_\pi\sim 391$ MeV case which have very large uncertainties. The grey bands and dot-dashed curves show amplitudes determined neglecting energy level correlations.