$D_{s0}^{*}\mathbf{(}2317\mathbf{)}$ Meson and $D$-Meson-Kaon Scattering from Lattice QCD

The scalar meson $D_{s0}^{*}(2317)$ is found $37(17)\mathrm{MeV}$ below the $DK$ threshold in a lattice simulation of the $J^P=0^{+}$ channel using, for the first time, both $DK$ as well as $\overline{s}c$ interpolating fields. The simulation is done on $N_f=2+1$ gauge configurations with $m_{\pi }\simeq 156\mathrm{MeV}$, and the resulting $M_{D_{s0}^{*}}-\frac{1}{4}(M_{D_s}+3M_{D_s^{*}})=266(16)\mathrm{MeV}$ is close to the experimental value $241.5(0.8)\mathrm{MeV}$. The energy level related to the scalar meson is accompanied by additional discrete levels due to $DK$ scattering states. The levels near threshold lead to the negative $DK$ scattering length $a_0=-1.33(20)\mathrm{fm}$ that indicates the presence of a state below threshold.

Figure 1

Energy levels from ensemble (1) (left panel) and ensemble (2) (right panel). For each case results with just quark-antiquark ($\overline{q}q$) and with a combined basis of $\overline{q}q$ and $DK$ interpolating fields are shown. The lower dashed lines indicate $M_D+M_K$ on both ensembles, while upper dashed lines show the energies of noninteracting $D(1)K(-1)$. The error bars include statistical and scale setting corrections.

Figure 2

Scattering length $a_0$ for $s$-wave $DK$ scattering with $I=0$. Our result is compared to the expectation from the indirect calculation in Ref. 42. The vertical line corresponds to the physical pion mass. For an explanation of the curves please refer to the main text.

Figure 3

The final result for the $D_{s0}^{*}(2317)$ mass is given by the crosses in the left and middle panels, while the experimental value is given in the right panel. Instead of the mass itself, we compare the values of $M_{L\to \infty }^{D_0^{*}(2317)}-M_{\overline{1S}}$, where $M_{\overline{1S}}^{\exp }=\frac{1}{4}(m_{D_s}+3m_{D_s^{*}})\simeq 2076\mathrm{MeV}$. The value of the bound state position in the infinite volume limit $M_{L\to \infty }^{D_0^{*}(2317)}$ is obtained from the pole condition $\cot \delta =i$. The two lowest energy levels from our simulation in the finite volume are given by the circles in the left and middle panels. Dashed lines represent the threshold for $DK$ in our simulation ($m_u=m_d$), and dotted lines the thresholds for $D^0{K}^{+}$, $K^0{D}^{+}$ in experiment.