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Using proton-proton collisions at centre-of-mass energies $7$, $8$, and $13$TeV, with a total integrated luminosity of $9\,\text{fb}^{-1}$, the LHCbcollaboration has performed amplitude analyses of the $B^+\to D^+D^-K^+$,$B^+\to D^- D_s^+ \pi^+$ and $B^0\to \bar{D}^0 D_s^+ \pi^-$ decays, observingthat new $T_{cs}$ and $T_{c\bar s}$ resonances are required in order to explainthe experimental data. These signals could be the first observation oftetraquark candidates that do not contain a heavy quark-antiquark pair; infact, they consist of four different flavours of quarks, one of which is adoubly charged open-charm state. We present herein an analysis of the $T_{cs}$and $T_{c\bar s}$ states, which is an extension of our recently published studyof similar $T_{cc}^+$ exotic candidates. Our theoretical framework is aconstituent-quark-model-based coupled-channels calculation of $qq^\prime \bar s\bar c$ and $cq\bar s\bar q^{\prime}$ tetraquark sectors for $T_{cs}$ and$T_{c\bar s}$ structures, respectively. We explore the nature, and poleposition, of the singularities that appear in the scattering matrix withspin-parity quantum numbers: $J^P=0^\pm$, $1^\mp$, and $2^\pm$. The constituentquark model has been widely used in the heavy quark sector, and thus all modelparameters are already constrained from previous works. This makes ourpredictions robust and parameter-free. We find many singularities in thesolution of various scattering-matrix problems which are either virtual statesor resonances, but not bound states. Some of them fit well with theexperimental observations of the spin-parity, mass and width of $T_{cs}$ and$T_{c\bar s}$ candidates, and thus tentative assignments are made; however,with caution, because the experimental Breit-Wigner parameters are related tothe pole characteristics.

Unraveling the nature of the novel $\mathbf{T_{cs}}$ and $\mathbf{T_{c\bar s}}$ tetraquark candidates