The compact tetraquark states with fully heavy quark contents $QQ\overline{Q}\overline{Q}$ are studied as the bound states of the diquark-antidiquark within the Bethe-Salpeter framework. The (anti)diquark masses and form factors used are the same as we calculated the doubly heavy baryons in a previous work. Under the instantaneous approximation, the three-dimensional (Bethe-)Salpeter equation of the tetraquarks is derived and solved numerically to obtain the corresponding mass spectra and wave functions of the tetraquarks with $J^{PC}=0^{++}$, $1^{+-}$, and $2^{++}$. Our results show that the three ground states of $cc\overline{c}\overline{c}$ locate in the mass range of 6.4–6.5 GeV, and the $bb\overline{b}\overline{b}$ states in mass range of 19.2–19.3 GeV. The obtained relativistic wave functions naturally include the mixing effects from the possible $D$ (or $G$) partial waves, and then can be further used to do precise calculations of the tetraquark decays. Based on the obtained results, the LHCb’s observation $X(6900)$ is less likely to be the ground states of compact $cc\overline{c}\overline{c}$ tetraquarks but might be the first or second excited states. In addition, a widely used propagatorlike form factor is also investigated and discussed.

The spectrum and relativistic wave functions of B_c system are investigated via solving the complete Salpeter equation. Emphases are put on the study of the partial waves of each J^P state. Our study shows that there are three categories of J^P states. The first category contains 0⁻ and 0⁺ states, which are ¹S₀ dominant state with a small amount of P wave and ³P₀ dominant state with a small amount of S wave, respectively. The second category includes the natural parity states, such as 1⁻, 2⁺, 3⁻, etc. Taking the 1⁻ state as an example, we study it in two cases. One is the ³S₁ dominant state with a small amount of P and D waves, and the other is the ³D₁ dominant state but contains a large amount of S and P wave components. The third category includes the unnatural parity states, such as 1⁺, 2⁻, 3⁺, etc. For the 1⁺ spectrum, the states are grouped into pairs with different radial quantum numbers. Each pair contains two ¹P₁−³P₁ mixing states, and the corresponding mixing angles are calculated by using the relativistic wave functions.

We study the weak decays of $$\bar{B}_{(s)}$$ and $$B_c$$ into D-wave heavy–light mesons, including $$J^P=2^-$$ ( $$D_{(s)2},D'_{(s)2},B_{(s)2},

In this paper we study the weak decays of J / ψ and ϒ ( 1 S ) . The cases when the final mesons are pseudo-scalars or vectors are considered. Using the Bethe–Salpeter method, we calculate the hadronic transition amplitude and give the form factors. The energy spectra of leptons for the semi-leptonic channels are also presented for convenience. In the calculation of non-leptonic decays, the naive factorization is applied. And all types of such channels, namely, flavor-favored or suppressed and color-favored or suppressed, are calculated. Our results show that, for the semi-leptonic decay modes, the largest branching ratios are of the order of 10 ⁻¹⁰ both for J / ψ and ϒ ( 1 S ) decays, and the largest branching ratios of non-leptonic decays are of the order of 10 ⁻⁹ for J / ψ and 10 ⁻¹⁰ for ϒ ( 1 S ) .

The existence of heavy meson lepton number violating (LNV) processes shows the Majorana nature of the neutrino. Much of this theoretical and experimental researche focuses on this type of decay. Four-body epton-number violation (LNV) processes of the B meson may have sizable branching ratios as they share the same vertex and mixing parameters with the three-body case. Mixing parameters between the heavy Majorana neutrino and charged leptons extracted from the three-body case can be used to constrain the branching ratios of four-body decays of the B meson. So we can update the upper limits of these mixing parameters with new experimental data of the three-body LNV decays. We also analyze B 0 → D * − ℓ 1 + ℓ 2 + M 2 − using the updated parameters and estimate some channels’ reconstruction events using current experimental data from Belle.