How to derive the decay rate from the Lagrangian of an interaction? 
For instance, in Fermi theory of beta decay, the lagrangian is shown in the fig.
How do we derive the decay rate and the distribution of kinetic energy from that?

 A: You basically need to do perturbation theory in quantum field theory, i.e. calculate Feynman diagrams. In general, the decay rate $\Gamma(1\rightarrow1^{\prime} + 2^{\prime} + \dots + n^{\prime})$ of  a particle (call it particle 1) that decays into $n$ other particles (call them particles $1^{\prime}, 2^{\prime},\dots, n^{\prime}$) is calculated in terms of correlation functions or scattering amplitudes. More accurately, the partial decay rate is found by,
\begin{equation}
d\Gamma(1\rightarrow1^{\prime} + \dots + n^{\prime}) = \frac{1}{2E_1} \overline{|\mathcal{M}|^2} d\text{Lips}_{1n}
\end{equation}
with $E_1$ the energy of the initial particle, $\mathcal{M}$ the Feynman invariant amplitude calculated explicitly via Feynman diagrams and,
\begin{equation}
 d\text{Lips}_{mn} = \left(2\pi\right)^{d+1} \delta^{d+1}\left(\sum_{j=1}^{m}p_i - \sum_{j=1}^{n}p^{\prime}_{i}\right) \prod_{i=1}^{n} \frac{d^{d}\vec{p}^{\;\prime}_{i}}{\left(2\pi\right)^{d}2E^{\prime}_{i}}
\end{equation}
the Lorentz invariant phase space. An integral over $d\text{Lips}$ then gives you the decay rate. To be even more formal, $\overline{|\mathcal{M}|^2}$ is the Feynman invariant amplitude averaged over unmeasured particle spins.
The details of the theory tell you what each propagator and vertex contribute to the Feynman diagrams. For example, the Lagrangian density you wrote describes a 4-vertex interaction with a coupling constant $G_{F}/\sqrt{2}$. You also need a kinetic term in the Lagrangian density (for your case, you should have 4 kinetic terms for the proton, neutron, electron and neutrino) from which you will read the propagators. The rest is just a standard perturbation theory calculation, i.e. use the Feynman rules to compute the contribution of each diagram.
