$DK$ and $BK$-like spectra from Laplace sum rule at NLO

Encouraged by the agreement, with the recent LHCb data on the $D^-K^+$ invariant mass from $B \rightarrow D^+D^-K^+$ decay, of our results for the masses of the $0^+$ and $1^-$ open charm $(\bar c\bar d)(u s)$ tetraquarks and molecules states from QCD spectral sum rules within stability criteria, which we review here, we extend our analysis to the $b$-quark channel. We find, in the $0^+$ case the lowest mass $M_{BK}=5195(15)~{\rm MeV}$ with $f_{BK}=8.3(2.4)~{\rm keV}$ and three (almost) degenerate states having respectively the masses $M_{SS}=5702(60)~{\rm MeV}$, $M_{AA}=5661(75)~{\rm MeV}$ and $M_{B^*K^*}=5720(71)~{\rm MeV}$ and couplings $f_{SS}=22.2(2.3)~{\rm keV}$, $f_{AA}=30.1(3.1)~{\rm keV}$ and $f_{B^*K^*}=26.5(2.8)~{\rm keV}$, from which we can associate a scalar tetramole with $M_{\mathcal T_{\mathcal M_0}}=5694(69)~{\rm MeV}~\text{and} ~ f_{\mathcal T_{\mathcal M_0}}=26.5(2.7)~{\rm keV}$. In the spin 1 case, we find four (almost) degenerate states associated with a tetramole having $M_{\mathcal T_{\mathcal M_1}}=5700(81)~{\rm MeV}~\text{and} ~ f_{\mathcal T_{\mathcal M_1}}=16.2(2.6)~{\rm keV}$. For the first radial excitation of the $BK$ molecule, we have $M_{(BK)_1} = 6265(146) ~{\rm MeV}$ and $f_{(BK)_1} = 22.8(3.2) ~{\rm keV}$ . For the remaining states, we associate a scalar and vector tetramoles having respectively $M_{\mathcal (T_{\mathcal M_0})_1}=7439(314)~{\rm MeV}, ~ f_{\mathcal (T_{\mathcal M_0})_1}=74.7(8.4)~{\rm keV}$ and $M_{\mathcal (T_{\mathcal M_1})_1}=7544(345)~{\rm MeV}, ~ f_{\mathcal (T_{\mathcal M_1})_1}=33.0(6.7)~{\rm keV}$.