What is meant by the "general formula of a scattering process"?

Question

What is meant by the "general formula for the scattering process"?

In an old exercise my lecturer gave me, I am told to:

Give the Lagrangian for a scalar Yukawa Scattering:

$$\mathcal{L}= \frac{1}{2} \partial_\mu \phi \partial ^\mu \phi - \frac{1}{2} m^2 \phi^2 + \partial_\mu \varphi^* \partial^\mu \varphi - M^2 \varphi^* \varphi - g \varphi^* \varphi \phi - g\phi^3$$

where $\varphi,\bar{\varphi}$ are associated to a complex scalar field and $\phi$ is associated with a real scalar field.

I am told to write the general formula for the scattering process $a_1 + a_2 \to b_1 + b_2$ ,

I understand that the last two terms in the Lagrangian tell me that there are two interaction vertices in this Lagrangian, and I can draw them, but in this case, being it a "general formula", should I just not be leaving all my calculations in terms of $a_1$, $a_2$ , $b_1$ and $b_2$, instead of it being in terms of $\phi$ and of $\varphi$?

Answer 1

Since, fixed a process $a_1+a_2 \to b_1+b_2$, all the relevant values in the cross section, phase space, normalisation, conservation of momenta, are fixed, the only relevant term is then the scattering amplitude $|A|^2$, or simply the matrix element $A = \langle b_1,b_2 |S|a_1, a_2\rangle$.

Supposing that $a\equiv \varphi$ and $b\equiv \phi$, you have to evaluate the scattering matrix of the process $\varphi+\varphi \to \phi+\phi$ so a diagram of this kind

I leave to you to understand why the only possible propagator that you can use in this theory is a $\phi$ and not a $\varphi$. Notice that to construct this diagram you need two kind of vertices. All that you need is to look at the lagrangian.