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

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 $$\varphi$$?

• This might be helpful: en.wikipedia.org/wiki/S-matrix
– user87745
Commented May 18, 2020 at 9:57
• So the general formula for a scattering process is its scattering matrix, meaning the amplitude to go from the initial to the final state, and not the $\mathcal{M}$. Commented May 18, 2020 at 10:03

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.