The New Charm-Strange Resonances in the $D^{-}K^{+}$ Channel

In this work, based on the paper in Ref. [1], we attempt to estimate, from LSR, the masses and couplings of the $0^{+}$ and $1^{-}$ molecules and compact tetraquarks states for interpreting the recent LHCb data from $B\to D^{+}(D^{-}K^{+})$ decays [2, 3], where one finds two prominent peaks (units of MeV):

We have studied in Ref. [4] the masses and couplings of the $D^{0}K^{0}(0^{+})$ molecule and of the corresponding tetraquark states decaying into $D^{0}K^{0}$ but not into $D^{-}K^{+}$ and we found the lowest ground state masses:

We have used this result to interpret the nature of the $D^{*}_{s0}$(2317) compiled by PDG [5] where the existence of a $DK$ pole at this energy has been recently confirmed from lattice calculations of scattering amplitudes [6].

For the molecular state, we can interchange the $u$ and $d$ quarks in the interpolating current and deduce from SU(2) symmetry that the $D^{-}K^{+}(0^{+})$ molecule mass is degenerated with the $D^{0}K^{0}$ one. Compared with the LHCb data, one may invoke that this charged molecule can be responsible of the bump near the DK threshold around 2.4 GeV but is too light to explain the $X_{0,1}$ peaks.

For the tetraquark state, one may not use a simple SU(2) symmetry (rotation of $u$ and $d$ quarks) to deduce the ones decaying into $D^{-}K^{+}$ due to our present ignorance of the diquark dynamics (for some attempts see [7, 8]).

Therefore, recent analysis based on QSSR at lowest order (LO) of perturbation theory (PT) using some specific tetraquarks and / or molecules configurations appear in the literature [9, 10, 11, 12] (see also [13, 14]) to explain these new candidates for the exotic states.

We shall work with the Finite Energy version of the QCD Inverse Laplace sum rules (LSR) and their ratios [15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 24, 26, 27]:

where $M_{c}$ and $m_{s}$ are the on-shell / pole charm and running strange quark masses, $\tau$ is the LSR variable, $n=0,1$ is the degree of moments, $t_{c}$ is the threshold of the “QCD continuum” which parametrizes, from the discontinuity of the Feynman diagrams, the spectral function is evaluated by the calculation of the scalar correlator defined as:

where ${\cal O}^{J}_{\cal M,T}(x)$ are the interpolating currents for the tetraquarks ${\cal T}$ and molecules ${\cal M}$ states. The superscript $J$ refers to the spin of the particles.

The sum rule analysis for the $0^{+}$ and $1^{-}$ states present similar features for all currents in Table  1. Then, we show only explicitly the analysis of the $SS$ tetraquark channel for a better understanding on the extraction of our results.

We extend the analysis in Ref. [4] by using a “Two resonances” + $\theta(t-t_{c})$“QCD continuum” parametrization of the spectral function. To enhance the contribution of the 1st radial excitation, we shall also work with the ratio of moments ${\cal R}_{1}$ in addition to ${\cal R}_{0}$ for getting the masses. We use the same criteria involving the stability points in $(\tau,t_{c},\mu)$ and the optimal results are given in Table 2. We observe that the mass-splittings between the first radial excitation and the lowest ground state are in order of $\sim 1500$ MeV, which is much bigger than $\sim 500$ MeV typically used for ordinary mesons.

Our results indicate that the molecules and tetraquark states leading to the same final states are almost degenerated in masses. Therefore, we expect that the “physical state” is a combination of almost degenerated molecules and tetraquark states with the same quantum numbers $J^{PC}$ which we shall call: Tetramole $({\cal T_{M}}_{J})$.

$\bullet~{}$ Motivated by the recent LHCb data on the $D^{-}K^{+}$ invariant mass from $B\to D^{+}D^{-}K^{+}$ decay, we have systematically calculated the masses and couplings of some possible configurations of the molecules and tetraquarks states using QCD Laplace sum rules (LSR) within stability criteria where we have added to the LO perturbative term, the NLO radiative corrections, and the contributions from quark and gluon condensates up to dimension-6 in the OPE.

$\bullet~{}$ The peak around the $DK$ threshold can be due to $DK$ scattering amplitude $\oplus$ the $DK(2400)$ lowest mass molecule.

$\bullet~{}$ The ($0^{+}$) $X_{0}(2866)$ and $X_{J}(3150)$ (if it is a $0^{+}$ state) can e.g result from a mixing of the Tetramole (${\cal T_{M}}_{0}$) with the 1st radial excitation $(DK)_{1}$ of the molecule state $(DK)$ with a tiny mixing angle $\theta_{0}\simeq(5.2\pm 1.9)^{0}$.

$\bullet~{}$ The ($1^{-}$) $X_{1}(2904)$ and $X_{J}(3350)$ (if it is a $1^{-}$ state) can result from a mixing of the Tetramole (${\cal T_{M}}_{1}$) with its 1st radial excitation $({\cal T_{M}})_{1}$ with a tiny mixing angle $\theta_{1}\simeq(9.1\pm 0.6)^{0}$.

$\bullet~{}$ More data on the precise quantum numbers of the $X_{J}(3150)$ and $X_{J}(3350)$ states are needed for testing the previous two minimal mixing models proposal.