Steps behind Meson decay amplitudes

[Elmo]

TL;DR Summary

I have this paper on B meson decays but I cant seem to understand how certain expressions are obtained. My browsing of the references of this paper and similar google searches have revealed nothing new so I was hoping that someone can point me in the right direction.

I am a graduate student with a somewhat basic level of exposure to qft. As part of my degree I have been tasked with calculating meson decay amplitudes in a paper provided to me by my supervisor but unfortunately I seem to be lacking some prior knowledge to understand why certain expressions are written the way they are. In the following document I am supposed to calculate equation (1) but I have no idea how the expressions to be plugged in it (equations (4 and 6)) are obtained.
I am also confused about the diagrams used in Figure1.Are these different from the sort of diagrams for B decays as shown here in this link https://www.google.com.pk/url?sa=i&...ved=0CAIQjRxqFwoTCPCKwtfWxu4CFQAAAAAdAAAAABAD

Here is the picture of those eqtns and diagrams .I have also included that paper itself.

 

[LightHero]

The equations in the paper you have been provided with are derived using the techniques of quantum field theory (QFT). The goal of QFT is to calculate the probability of a certain particle transition (in this case meson decay) by analyzing the Feynman diagrams associated with that process. The diagrams in Figure 1 represent the interactions between the particles involved in the decay process and the arrows indicate the direction in which the particles are moving.By writing down the relevant interactions between the particles, it is possible to calculate the probability of the transition using perturbation theory. Perturbation theory is a mathematical technique which allows for the calculation of a physical process by expanding the interaction Hamiltonian into a series of terms. Each term in the expansion corresponds to a different Feynman diagram. In the case of meson decay, the Hamiltonian is expanded into two terms, one corresponding to the direct decay of the meson and the other to the exchange of virtual particles. The contribution from each of these terms can then be calculated separately and combined to give the total decay amplitude.The equations (4) and (6) in the paper are the amplitudes for the direct decay and virtual exchange processes, respectively. By plugging these expressions into equation (1), it is possible to calculate the total decay amplitude.I hope this helps to clarify the calculations you need to perform for your assignment. If you need more help or have any further questions about the techniques used in QFT, please do not hesitate to ask.