In this work, we perform a systematical investigation about the possible hidden and doubly heavy molecular states with open and hidden strangeness from interactions of $$D^{(*)}{{\bar{D}}}^{(*)}_{s}$$ D ( ∗ ) D ¯ s ( ∗ ) /$$B^{(*)}{{\bar{B}}}^{(*)}_{s}$$ B ( ∗ ) B ¯ s ( ∗ ) , $${D}^{(*)}_{s}{{\bar{D}}}^{(*)}_{s}$$ D s ( ∗ ) D ¯ s ( ∗ ) /$${{B}}^{(*)}_{s}{{\bar{B}}}^{(*)}_{s}$$ B s ( ∗ ) B ¯ s ( ∗ ) , $${D}^{(*)}D_{s}^{(*)}$$ D ( ∗ ) D s ( ∗ ) /$${B}^{(*)}B_{s}^{(*)}$$ B ( ∗ ) B s ( ∗ ) , and $$D_{s}^{(*)}D_{s}^{(*)}$$ D s ( ∗ ) D s ( ∗ ) /$$B_{s}^{(*)}B_{s}^{(*)}$$ B s ( ∗ ) B s ( ∗ ) in a quasipotential Bethe-Salpeter equation approach. The interactions of the systems considered are described within the one-boson-exchange model, which includes exchanges of light mesons and $$J/\psi /\varUpsilon $$ J / ψ / Υ meson. Possible molecular states are searched for as poles of scattering amplitudes of the interactions considered. The results suggest that recently observed $$Z_{cs}(3985)$$ Z cs ( 3985 ) can be assigned as a molecular state of $$D^*{\bar{D}}_s+D{\bar{D}}^*_s$$ D ∗ D ¯ s + D D ¯ s ∗ , which is a partner of $$Z_c(3900)$$ Z c ( 3900 ) state as a $$D{\bar{D}}^*$$ D D ¯ ∗ molecular state. The calculation also favors the existence of hidden heavy states $$D_s{\bar{D}}_s/B_s{\bar{B}}_s$$ D s D ¯ s / B s B ¯ s with spin parity $$J^P=0^+$$ J P = 0 + , $$D_s{\bar{D}}^*_s/B_s{\bar{B}}^*_s$$ D s D ¯ s ∗ / B s B ¯ s ∗ with $$1^{+}$$ 1 + , and $$D^*_s{\bar{D}}^*_s/B^*_s{\bar{B}}^*_s$$ D s ∗ D ¯ s ∗ / B s ∗ B ¯ s ∗ with $$0^+$$ 0 + , $$1^+$$ 1 + , and $$2^+$$ 2 + . In the doubly heavy sector, the bound states can be found from the interactions $$(D^*D_s+DD^*_s)/(B^*B_s+BB^*_s)$$ ( D ∗ D s + D D s ∗ ) / ( B ∗ B s + B B s ∗ ) with $$1^+$$ 1 + , $$D_s{\bar{D}}_s^*/B_s{\bar{B}}_s^*$$ D s D ¯ s ∗ / B s B ¯ s ∗ with $$1^+$$ 1 + , $$D^*D^*_s/B^*B^*_s$$ D ∗ D s ∗ / B ∗ B s ∗ with $$1^+$$ 1 + and $$2^+$$ 2 + , and $$D^*_sD^*_s/B^*_sB^*_s$$ D s ∗ D s ∗ / B s ∗ B s ∗ with $$1^+$$ 1 + and $$2^+$$ 2 + . Some other interactions are also found attractive, but may be not strong enough to produce a bound state. The results in this work are helpful for understanding the $$Z_{cs}(3985)$$ Z cs ( 3985 ) , and future experimental search for the new molecular states.

In this work, we preform a systematic investigation about hidden heavy and doubly heavy molecular states from the $$D^{(*)}{\bar{D}}^{(*)}/B^{(*)}{\bar{B}}^{(*)}$$ D ( ∗ ) D ¯ ( ∗ ) / B ( ∗ ) B ¯ ( ∗ ) and $$D^{(*)}D^{(*)}/{\bar{B}}^{(*)}{\bar{B}}^{(*)}$$ D ( ∗ ) D ( ∗ ) / B ¯ ( ∗ ) B ¯ ( ∗ ) interactions in the quasipotential Bethe–Salpeter equation (qBSE) approach. With the help of Lagrangians with heavy quark and chiral symmetries, interaction potentials are constructed within the one-boson-exchange model in which we include the $$\pi $$ π , $$\eta $$ η , $$\rho $$ ρ , $$\omega $$ ω and $$\sigma $$ σ exchanges, as well as $$J/\psi $$ J / ψ or $$\varUpsilon $$ Υ exchange. Possible bound states from the interactions considered are searched for as the pole of scattering amplitude. The results suggest that experimentally observed states, $$Z_c(3900)$$ Z c ( 3900 ) , $$Z_c(4020)$$ Z c ( 4020 ) , $$Z_b(10610)$$ Z b ( 10610 ) , and $$Z_b(10650)$$ Z b ( 10650 ) , can be related to the $$D{\bar{D}}^{*}$$ D D ¯ ∗ , $$D^*{\bar{D}}^{*}$$ D ∗ D ¯ ∗ , $$B{\bar{B}}^{*}$$ B B ¯ ∗ , and $$B^*{\bar{B}}^{*}$$ B ∗ B ¯ ∗ interactions with quantum numbers $$I^G(J^P)=1^+(1^{+})$$ I G ( J P ) = 1 + ( 1 + ) , respectively. The $$D{\bar{D}}^{*}$$ D D ¯ ∗ interaction is also attractive enough to produce a pole with $$0^+(0^+)$$ 0 + ( 0 + ) which is related to the X(3872). Within the same theoretical frame, the existence of $$D{\bar{D}}$$ D D ¯ and $$B{\bar{B}}$$ B B ¯ molecular states with $$0(0^+)$$ 0 ( 0 + ) are predicted. The possible $$D^*{\bar{D}}^*$$ D ∗ D ¯ ∗ molecular states with $$0(0^+, 1^+, 2^+)$$ 0 ( 0 + , 1 + , 2 + ) and $$1(0^+)$$ 1 ( 0 + ) and their bottom partners are also suggested by the calculation. In the doubly heavy sector, no bound state is produced from the $$DD/{\bar{B}}{\bar{B}}$$ D D / B ¯ B ¯ interaction while a bound state is found with $$0(1^+)$$ 0 ( 1 + ) from $$DD^*/{\bar{B}}{\bar{B}}^*$$ D D ∗ / B ¯ B ¯ ∗ interaction. The $$D^*D^*/{\bar{B}}^*{\bar{B}}^*$$ D ∗ D ∗ / B ¯ ∗ B ¯ ∗ interaction produces three molecular states with $$0(1^+)$$ 0 ( 1 + ) , $$0(2^+)$$ 0 ( 2 + ) and $$1(2^+)$$ 1 ( 2 + ) .

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