Coupled-channel ${\mathrm{\Lambda }}_c{K}^{+}-p{D}_s$ interaction in the flavor SU(3) limit of lattice QCD

We study $S$-wave interactions in the $I(J^p)=1/2(1/2^{-}){\mathrm{\Lambda }}_c{K}^{+}-p{D}_s$ system on the basis of the coupled-channel Hadrons to Atomic nuclei from Lattice Quantum ChromoDynamics method. The potentials which are faithful to QCD S-matrix below the $p{D}^{*}$ threshold are extracted from Nambu-Bethe-Salpeter wave functions on the lattice in flavor SU(3) limit. For the simulation, we employ 3-flavor full QCD gauge configurations on a ${(1.93\mathrm{fm})}^3$ volume at $m_{\pi }\simeq 872\mathrm{MeV}$. We present our results of the S-wave coupled-channel potentials for the ${\mathrm{\Lambda }}_c{K}^{+}-p{D}_s$ system in the $1/2(1/2^{-})$ state as well as scattering observables obtained from the extracted potential matrix. We observe that the coupling between ${\mathrm{\Lambda }}_c{K}^{+}$ and $p{D}_s$ channels is weak. The phase shifts and scattering length obtained from the extracted potential matrix show that the ${\mathrm{\Lambda }}_c{K}^{+}$ interaction is attractive at low energy and stronger than the $p{D}_s$ interaction though no bound state at $m_{\pi }\ge 872\mathrm{MeV}$.

Figure 1

Effective mass plot of ${\mathrm{\Lambda }}_c,D_s,p$ and $K^{+}$ for (a) wall-wall (WW) and point-wall (PW), in addition, to have better comparison in the plot (b) only point-wall correlators are shown.

Figure 2

Time dependences of the diagonal potentials matrix, panel (a) for $V_{{\mathrm{\Lambda }}_c{K}^{+}}^{{\mathrm{\Lambda }}_c{K}^{+}}$ and panel (b) for $V_{p{D}_s}^{p{D}_s}$ as function of $r$ at $t-t_0=7$ (dark magenta), 8 (red), and 9 (blue).

Figure 3

The S-wave ${\mathrm{\Lambda }}_c{K}^{+}-p{D}_s$ coupled-channel potential matrix (red circle). The solid blue line is the corresponding fitted function. The time separation between sink and source is $t-t_0=8$. Panels show elements of the potential matrix, (a) $V_{{\mathrm{\Lambda }}_c{K}^{+}}^{{\mathrm{\Lambda }}_c{K}^{+}}$, (b) $V_{{\mathrm{\Lambda }}_c{K}^{+}}^{p{D}_s}$, (c) $V_{p{D}_s}^{{\mathrm{\Lambda }}_c{K}^{+}}$, and (d) $V_{p{D}_s}^{p{D}_s}$, respectively.

Figure 4

Hermitian (average) potential, ${\overline{V}}_{{\mathrm{\Lambda }}_c{K}^{+}}^{p{D}_s}$ in Eq. (15) and its fit function at $t-t_0=8$.

Figure 5

Panel (a) shows a comparison between the single channel ${\mathrm{\Lambda }}_c{K}^{+}$ potential (blue square) and a diagonal part of the coupled channel potential (red circle), and panel (b) presents the relevant phase shifts in the single and coupled channel at $t-t_0=8$. The mild difference between single and coupled channel phase shift is due to the effect of coupled channel. The ${\mathrm{\Lambda }}_c{K}^{+}$ phase shift are plotted against the energies from the ${\mathrm{\Lambda }}_c{K}^{+}$ threshold. In the case of single channel ${\mathrm{\Lambda }}_c{K}^{+}$ potential the corresponding fit parameters are given by Table 5 in Appendix pp2.

Figure 6

The phase shifts of S-wave (a) ${\mathrm{\Lambda }}_c{K}^{+}$, (b) $p{D}_s$ and (c) the inelasticity of the scattering defined as $\eta =|S_{{\mathrm{\Lambda }}_c{K}^{+}}^{{\mathrm{\Lambda }}_c{K}^{+}}|=|\cos 2\overline{\epsilon }|$ is plotted against the energies from the ${\mathrm{\Lambda }}_c{K}^{+}$ threshold for $t-t_0=7$ (blue), 8 (red) and 9 (green).

Figure 7

The scattering phase shifts of S-wave (a) ${\mathrm{\Lambda }}_c{K}^{+}$, (b) $p{D}_s$ and (c) the inelasticity parameter correspond to the phase shifts calculated with non-Hermitian (blue), ${\overline{V}}_{{\mathrm{\Lambda }}_c{K}^{+}}^{p{D}_s}$ (red), $V_{{\mathrm{\Lambda }}_c{K}^{+}}^{p{D}_s}$ (purple), and $V_{p{D}_s}^{{\mathrm{\Lambda }}_c{K}^{+}}$ (green) at $t-t_0=8$.