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.

This post imported from StackExchange Physics at 2014-06-15 16:45 (UCT), posted by SE-user Melquíades