Semileptonic decays of the $B_c$ meson I am struggling with calculating the exclusive semileptonic $B_c^+\rightarrow J/\psi l^+\nu_l$ decay. I learnt that the amplitude is given by a product of the leptonic current $L^{\mu}$ and the hadronic current $H^{\mu}$
$$
 \mathcal{M}(B_c\rightarrow J/\psi l^+\nu_l)=\frac{G_F}{\sqrt{2}}V_{cb}L^{\mu}H_{\mu}
$$
where $V_{cb}$ is the CKM parameter, $L^{\mu}$ and $H^{\mu}$ are expressed as
$$
 L^{\mu}=\bar{u}_l\gamma^{\mu}(1-\gamma^5)v_{\nu},\quad
 H^{\mu}=\langle J/\psi|J^{\mu}(0)|B_c\rangle
$$
where $J^{\mu}$ is the $V$-$A$ weak current. However, I did not know how this result can be derived. Could anyone provide some help?
There is a second problem. On the tree-level, we have the following Feynman diagram

If we calculate $\bar{b}\rightarrow\bar{c}l^+\nu_l$ as a three-body decay in the electroweak theory (not the four-fermion approximation adopted above), how does it relate to $B_c^+\rightarrow J/\psi l^+\nu_l$?
 A: For this process, the interaction Hamiltonian is given by:
$$\mathcal{H}_{\rm int}=-\frac{g}{\sqrt 2}\left(V_{cb}\bar{b}_L\gamma^\mu c_L W^-_\mu+\bar{\nu}_L\gamma^\mu\ell_L W^+_\mu\right).$$
After integrating-out the heavy bosons, we obtain the following Hamiltonian
$$\mathcal{H}_{\rm eff}=-\dfrac{G_F}{\sqrt{2}}V_{cb}[\bar{b}\gamma^\mu(1-\gamma_5)c][\bar{\nu}\gamma^\mu(1-\gamma_5)\ell],$$
where $G_F/\sqrt{2}=g^2/(8 m_W^2)$ is the Fermi constant.
To obtain the tree-level amplitude for the process $B_c\to J/\psi \ell^+ \nu$, we consider the following matrix element
$$\mathcal{A}(B_c\to J/\psi \ell^+ \nu)=-i\langle J/\psi \,\ell^+\, \nu_\ell |\mathcal{H}_{\rm eff} | B_c\rangle. $$
If you write explicitly the leptonic fields in terms of creation and annihilation operators, then you will notice that
$$\mathcal{A}(B_c\to J/\psi \ell^+ \nu)=i \dfrac{G_F}{\sqrt{2}}V_{cb}\bar{u}_\nu \gamma^\mu (1-\gamma_5) v_\ell \langle J/\psi| \bar{b}\gamma^\mu(1-\gamma_5)c| B_c\rangle.$$
Note that we have isolated the hadronix matrix element from the rest. Now, if we are able to find this element by using Lattice QCD methods or experimental results, then we will be able to compute the decay rate and other observables. [However, I don't think this is possible for this particular transition at present.]
For your second question, if you consider only valence quarks in the mesons, then you are using a tree-level approximation to describe hadronic states. This is a crude approximation, because QCD is non-perturbative at low energies. You can improve it by computing high-order QCD corrections, but you will never have a reliable result.
