Relativistic effects are important in the rigorous study of heavy quarks. In this paper, we study the relativistic corrections of semileptonic B c decays to charmonium with the instantaneous Bethe-Salpeter method. Within the Bethe-Salpeter framework, we use two methods to study the relativistic effects. One of them is to expand the transition amplitude in powers of q which is the relative momentum between the quark and antiquark, and the other is to expand the amplitude base on the wave functions. In the level of decay width, the results show that, for the transition of B c → η c , the relativistic correction is about 22%; for B c → J/ψ, it is about 19%; the relativistic effects of 1P final states are about 14 ∼ 46% larger than those of 1S final states; for 2S final states, they are about 19 ∼ 28% larger than those of 1S final states; for 3S final states, they are about 12 ∼ 13% larger than those of 2S final states; for 2P final states, they are about 10 ∼ 14% larger than those of 1P final states; for 3P final states, they are about 7 ∼ 12% larger than those of 2P final states. We conclude that the relativistic corrections of the B c decays to the orbitally or radially excited charmonium (2S, 3S, 1P, 2P, 3P) are quite large.

The heavy quark effective theory vastly reduces the weak-decay form factors of hadrons containing one heavy quark. Many works attempt to directly apply this theory to hadrons with multiple heavy quarks. In this paper, we examine this confusing application by the instantaneous Bethe-Salpeter method from a phenomenological perspective, and give the numerical results for decays to charmonium where the final states include , , , and . Our results indicate that the form factors parameterized by a single Isgur-Wise function deviate substantially from the full ones, especially when excited states are involved. The relativistic corrections ( corrections) require the introduction of more non-perturbative universal functions, similar to the Isgur-Wise function, which are the overlapping integrals of the wave functions with the relative momentum between the quark and antiquark.

The Isgur-Wise function vastly reduces the weak-decay form factors of ground heavy mesons.In this paper, we extract the Isgur-Wise functions from the instantaneous Bethe-Salpeter method, and give the numerical results for the B c decays to charmonium where the final states include 1S, 1P , 2S and 2P . The overlapping integral of the wave functions for the initial and final states is the leading order Isgur-Wise function, as the heavy quark effective theory does. In the case of accurate calculation, we need to introduce more high-order Isgur-Wise functions which are the overlapping integrals with the relative momentum between the quark and antiquark. The higher order Isgur-Wise functions provide greater relativistic corrections especially involving the excited state, and therefore are necessary to be adopted.

In this paper, we use the instantaneous Bethe-Salpeter method to calculate the semi-leptonic and nonleptonic production of the orbitally excited scalar D * 0 in B meson decays. When the final state is 1P state D * 0 (2400), our theoretical decay rate is consistent with experimental data. For D * J (3000) final state, which was observed by LHCb collaboration recently and here treated as the orbitally excited scalar D * 0 (2P), its rate is in the order of 10 −4 ∼ 10 −6. We find the special node structure of D * 0 (2P) wave function possibly results in the suppression of its branching ratio and the abnormal uncertainty. The 3P states production rate is in the order of 10 −5 .