$D_s$ mesons with $DK$ and $D*K$ scattering near threshold

$D_s$ mesons are studied in three quantum channels ($J^P=0^{+}$, $1^{+}$ and $2^{+}$), where experiments have identified the very narrow $D_{s0}^{*}(2317)$, $D_{s1}(2460)$ and narrow $D_{s1}(2536)$, $D_{s2}^{*}(2573)$. We explore the effect of nearby $DK$ and $D^{*}K$ thresholds on the subthreshold states using lattice QCD. Our simulation is done on two very different ensembles of gauge configurations (2 or $2+1$ dynamical quarks, Pion mass of 266 or 156 MeV, lattice size $16^3\times 32$ or $32^3\times 64$). In addition to $\overline{q}q$ operators we also include meson-meson interpolators in the correlation functions. This clarifies the identification of the states above and below the scattering thresholds. The ensemble with $m_{\pi }\simeq 156\mathrm{MeV}$ renders the $D_{s1}(2460)$ as a strong interaction bound state 44(10) MeV below $D^{*}K$ threshold, which is in agreement with the experiment. The $D_{s0}^{*}(2317)$ is found 37(17) MeV below $DK$ threshold, close to the experiment value of 45 MeV. The narrow resonances $D_{s1}(2536)$ and $D_{s2}^{*}(2573)$ are also found close to the experimental masses.

Figure 1

Diagrammatic representation of the triangle contribution Eq. (26) for $D_s^{+}\to D^0{K}^{+}$.

Figure 2

Diagrammatic representation of the necessary Wick contractions. Source and sink are indicated by $t_i$ and $t_f$.

Figure 3

$A_1^{+}$: Effective energies as obtained for various subsets of operators for ensemble (2). The horizontal broken lines indicate the positions of $D(0)K(0)$ and $D(1)K(-1)$ in the noninteracting case. The boxes indicate the operators (listed in the Appendix) considered in each case [blue: $\overline{q}q$, red: $D(0)K(0)$, green: $D(1)K(-1)$].

Figure 4

Effective range fits for $A_1^{+}$, cf. Table 8. Ensemble (1) black dots, ensemble (2) red squares; the vertical arrows give the positions of the bound state for ensembles (1) and (2), see Table 8, the dashed line indicates the threshold.

Figure 5

Ensemble (1), $T_1^{+}$ with $t_0=3$: Effective energies of the lowest four energy levels. We compare the results obtained including the $D^{*}K$ operators (circles, $O_{1-11}$) with the results obtained without those (stars, $O_{1-8}$).The horizontal broken lines in the upper plot indicate the positions of $D^{*}(0)K(0)$ and $D^{*}(1)K(-1)$ in the noninteracting case. Note the “missing state” in the second case.

Figure 6

Ensemble (2), $T_1^{+}(O_{1,2,4,5,9,11})$: Effective energies of the lowest four energy levels; the fits are to the eigenvalues but here we show the result in the effective energy plots: 2-exponential fits (red) and 1-exponential fits (broken blue line) are consistent.

Figure 7

$T_1^{+}$: Effective energies as obtained for various subsets of operators for ensemble (1) (lhs) and (2) (rhs). The fit type and fit range is shown in Table 9. The horizontal broken lines indicate the positions of $D^{*}(0)K(0)$ and $D^{*}(1)K(-1)$ in the noninteracting case. The boxes indicate the operators considered in each case (blue: $\overline{q}q$, red: $D^{*}(0)K(0)$, green: $D^{*}(1)K(-1)$.

Figure 8

$T_1^{+}$: The upper figure shows the overlap factors $⟨n|O_i⟩$ for the lowest four eigenstates (ensemble (2), first choice, see Table 11). The abscissa gives the operator number $i$ according to the order in Tables 13 (1-8) and (a2) (9-11). Below the ratios $⟨n|O_i⟩/{\max }_m⟨m|O_i⟩$ for the same four eigenstates are shown.

Figure 9

Effective range fits for $T_1^{+}$, cf. Table 11. Ensemble (1): black dots, ensemble (2): red open/full (set 1/2) squares. The vertical arrows show the positions of the bound state, see Table 11, the dashed line indicates the threshold.

Figure 10

Resulting $D_s$ spectrum for all channels. The masses are presented with respect to spin-averaged mass $\frac{1}{4}(m_{D_s}+3m_{D_s^{*}})$. The diamonds and crosses display our lattice results, while black full lines correspond to experiment. The magenta diamonds show masses extracted via phase shift analysis and correspond to the pole position in the $T$ matrix. Masses extracted as energy levels in a finite box are displayed as blue crosses. The dotted lines correspond to $DK$ and $D^{*}K$ lattice thresholds, while dashed lines on the left of each pane are thresholds in experiment.