$D$-wave heavy-light mesons from QCD sum rules

We study the $D$-wave $\overline{c}s$ heavy meson doublets $(1^{-},2^{-})$ and $(2^{-},3^{-})$ using the method of QCD sum rule in the framework of heavy quark effective theory. Choosing the same threshold values ${\omega }_c$ around 2.7 Gev, we calculate the masses of the $1^{-}$ and $3^{-}$ states. They are $m_{D_{s1}^{*}}=2.81\pm 0.10\mathrm{GeV}$ and $m_{D_{s3}^{*}}=2.85\pm 0.08\mathrm{GeV}$, consistent with the newly observed $D_{s1}^{*}(2860)$ and $D_{s3}^{*}(2860)$ states by LHCb. The masses of their $2^{-}$ partners are calculated to be $2.82\pm 0.10$ and $2.81\pm 0.08\mathrm{GeV}$. The mass splittings within the same doublet are calculated to be $m_{D_{s2}}-m_{D_{s1}^{*}}=0.016\pm 0.007\mathrm{GeV}$ and $m_{D_{s3}^{*}}-m_{D_{s2}^{\prime }}=0.039\pm 0.014\mathrm{GeV}$.

Figure 1

Various contributions to Eq. (20), as functions of the Borel mass $M_B$ in units of ${\mathrm{GeV}}^7$ at ${\omega }_c=2.7\mathrm{GeV}$. The labels indicate the dimension up to which the OPE terms are included.

Figure 2

The solid curve shows the pole contribution and the dashed curve shows the continuum contribution ($=1-\text{pole}$ contribution), when $J_{3,-,5/2}$ is used and ${\omega }_c$ is fixed to be 2.7 GeV.

Figure 3

The variation of ${\mathrm{\Lambda }}_{-,3/2}$ and $f_{-,3/2}$ with respect to the Borel mass $T$ and the threshold value ${\omega }_c$. The short-dashed, solid and long-dashed curves are obtained by fixing ${\omega }_c=2.5$, 2.7, and 2.9 GeV, respectively. Our working region is $0.35\mathrm{GeV}<T<0.48\mathrm{GeV}$.

Figure 4

The variation of ${\mathrm{\Lambda }}_{-,5/2}$ and $f_{-,5/2}$ with respect to the Borel mass $T$ and the threshold value ${\omega }_c$. The short-dashed, solid and long-dashed curves are obtained by fixing ${\omega }_c=2.5$, 2.7, and 2.9 GeV, respectively. Our working region is $0.39\mathrm{GeV}<T<0.47\mathrm{GeV}$.

Figure 5

The variation of $K_{-,3/2}$ and ${\mathrm{\Sigma }}_{-,3/2}$ with respect to the Borel mass $T$ and the threshold value ${\omega }_c$. The short-dashed, solid and long-dashed curves are obtained by fixing ${\omega }_c=2.5$, 2.7, and 2.9 GeV, respectively. Our working region is $0.35\mathrm{GeV}<T<0.48\mathrm{GeV}$.

Figure 6

The variation of $K_{-,5/2}$ and ${\mathrm{\Sigma }}_{-,5/2}$ with respect to the Borel mass $T$ and the threshold value ${\omega }_c$. The short-dashed, solid and long-dashed curves are obtained by fixing ${\omega }_c=2.5$, 2.7 and 2.9 GeV, respectively. Our working region is $0.39\mathrm{GeV}<T<0.47\mathrm{GeV}$.