Systematic studies of charmonium-, bottomonium-, and $B_c$-like tetraquark states

We study the mass splittings of $Q_1q_2\bar{Q}_3\bar{q}_4$ ($Q=c,b$, $q=u,d,s$) tetraquark states with chromomagnetic interactions between their quark components. Assuming that $X(4140)$ is the lowest $J^{PC}=1^{++}$ $cs\bar{c}\bar{s}$ tetraquark, we estimate the masses of the other tetraquark states. From the obtained masses and defined measure reflecting effective quark interactions, we find the following assignments for several exotic states: (1) both $X(3860)$ and the newly observed $Z_c(4100)$ seem to be $0^{++}$ $cn\bar{c}\bar{n}$ tetraquarks; (2) $Z_c(4200)$ is probably a $1^{+-}$ $cn\bar{c}\bar{n}$ tetraquark; (3) $Z_c(3900)$, $X(3940)$, and $X(4160)$ are unlikely compact tetraquarks; (4) $Z_c(4020)$ is unlikely a compact tetraquark, but seems the hidden-charm correspondence of $Z_b(10650)$ with $J^{PC}=1^{+-}$; and (5) $Z_c(4250)$ can be a tetraquark candidate but the quantum numbers cannot be assigned at present. We hope further studies may check the predictions and assignments given here.