We calculate the masses of the $QQbar{q}bar{q}$ ($Q=c,b$; $q=u,d,s$) tetraquark states with the aid of heavy diquark-antiquark symmetry (HDAS) and the chromomagnetic interaction (CMI) model. The masses of the highest-spin ($J=2$) tetraquarks that hav
e only the $(QQ)_{bar{3}_c}(bar{q}bar{q})_{3_c}$ color structure are related with those of conventional hadrons using HDAS. Thereafter, the masses of their partner states are determined with the mass splittings in the CMI model. Our numerical results reveal that: (i) the lightest $ccbar{n}bar{n}$ ($n=u,d$) is an $I(J^P)=0(1^+)$ state around 3929 MeV (53 MeV above the $DD^*$ threshold) and none of the double-charm tetraquarks are stable; (ii) the stable double-bottom tetraquarks are the lowest $0(1^+)$ $bbbar{n}bar{n}$ around 10488 MeV ($approx116$ MeV below the $BB^*$ threshold) and the lowest $1/2(1^+)$ $bbbar{n}bar{s}$ around 10671 MeV ($approx20$ MeV below the $BB_s^*/B_sB^*$ threshold); and (iii) the two lowest $bcbar{n}bar{n}$ tetraquarks, namely the lowest $0(0^+)$ around 7167 MeV and the lowest $0(1^+)$ around 7223 MeV, are near-threshold states. Moreover, we discuss the constraints on the masses of double-heavy hadrons. Specifically, for the lowest nonstrange tetraquarks, we obtain $T_{cc}<3965$ MeV, $T_{bb}<10627$ MeV, and $T_{bc}<7199$ MeV.
We have systematically investigated the mass spectrum and rearrangement decay properties of the exotic tetraquark states with four different flavors using a color-magnetic interaction model. Their masses are estimated by assuming that the $X(4140)$ i
s a $csbar{c}bar{s}$ tetraquark state and their decay widths are obtained by assuming that the Hamiltonian for decay is a constant. According to the adopted method, we find that the most stable states are probably the isoscalar $bsbar{u}bar{d}$ and $csbar{u}bar{d}$ with $J^P=0^+$ and $1^+$. The width for most unstable tetraquarks is about tens of MeVs, but that for unstable $cubar{s}bar{d}$ and $csbar{u}bar{d}$ can be around 100 MeV. For the $X(5568)$, our method cannot give consistent mass and width if it is a $bubar{s}bar{d}$ tetraquark state. For the $I(J^P)=0(0^+),0(1^+)$ double-heavy $T_{bc}=bcbar{u}bar{d}$ states, their widths can be several MeVs.
In a chromomagnetic model, we analyse the properties of the newly observed $P_c(4457)^+$, $P_c(4440)^+$, and $P_c(4312)^+$ states. We estimate the masses of the $(uud)_{8_c}(cbar{c})_{8_c}$ and $(uds)_{8_c}(cbar{c})_{8_c}$ pentaquark states by consid
ering the isospin breaking effects. Their values are determined by calculating mass distances from the $Sigma_c^{++}D^-$ and $Xi_c^{prime+}D^-$ thresholds, respectively. It is found that the isospin breaking effects on the spectrum are small. From the uncertainty consideration and the rearrangement decay properties in a simple model, we find that it is possible to assign the $P_c(4457)^+$, $P_c(4440)^+$, and $P_c(4312)^+$ as $J^P=3/2^-$, $1/2^-$, and $3/2^-$ pentaquark states, respectively. The assignment in the molecule picture can be different, in particular for the $P_c(4312)^+$. The information from open-charm channels, e.g. ${cal B}[P_ctoSigma_c^{++}D^-]/{cal B}[P_cto J/psi p]$, will play an important role in distinguishing the inner structures of the $P_c$ states. Discussions and predictions based on the calculations are also given.
