$X_{0,1}$(2900) and $(D^-K^+)$ invariant mass from QCD Laplace sum rules at NLO

Abstract

We revisit, improve and complete some recent estimates of the $0^{+}$ and $1^-$ open charm $(\bar c \bar d)(us)$ tetraquarks and the corresponding molecules masses and decay constants from QCD spectral sum rules (QSSR) by using QCD Laplace sum rule (LSR) within stability criteria where the factorised perturbative NLO corrections and the contributions of quark and gluon condensates up to dimension-6 in the OPE are included. We confront our results with the $D^-K^+$ invariant mass recently reported by LHCb from $B^+\to D^+(D^-K^+)$ decays. We expect that the bump near the $D^-K^+$ threshold can be originated from the $0^{++}(D^-K^+)$ molecule and/or $D^-K^+$ scattering. The prominent $X_{0}$(2900) scalar peak and the bump $X_J(3150)$ (if $J=0$) can emerge from a {\it minimal mixing model}, with a tiny mixing angle $θ_0\simeq (5.2\pm 1.9)^0$, between a scalar {\it Tetramole} (${\cal T_M}_0$) (superposition of nearly degenerated hypothetical molecules and compact tetraquarks states with the same quantum numbers) having a mass $M_{{\cal T_M}_0}$=2743(18) MeV and the first radial excitation of the $D^-K^+$ molecule with mass $M_{(DK)_1}=3678(310)$ MeV. In an analogous way, the $X_1$(2900) and the $X_J(3350)$ (if $J=1$) could be a mixture between the vector {\it Tetramole} $({\cal T_M}_1)$ with a mass $M_{{\cal T_M}_1}=2656(20)$ MeV and its first radial excitation having a mass $M_{({\cal T_M}_1)_1}=4592(141)$ MeV with an angle $θ_1\simeq (9.1\pm 0.6)^0$. A (non)-confirmation of the previous {\it minimal mixing models} requires an experimental identification of the quantum numbers of the bumps at 3150 and 3350 MeV.