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In the same theme as my previous question, I have the diffusion process $$N+\pi \rightarrow N + \pi$$ where the Lagrangian for this theory is

$$L = \partial^\mu\psi\partial_\mu\psi^* - M²\psi\psi^*-\frac{1}{2}\partial^\mu\phi\partial_\mu\phi- m²\phi^2 + g\phi\psi\psi^*$$

where $M$ is the mass for Nucleons $N$ without spin and $m$ is the mass for the mesons $\pi$.

Am I wrong or the matrix element $\mathcal{M}$ is equal to

$$\mathcal{M} = (-ig)^2\left(\frac{-i}{s+M^2-i\epsilon}+\frac{-i}{u+M^2-i\epsilon}\right)$$

If I am correct, how can I do to get $|M|^2$ as a function of $\theta$ (and maybe something more), where $\theta$ is the scattering angle?

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  • $\begingroup$ why do you have two propagators? At tree level, this interaction should require two vertices, between which there is only one nucleon propagator. $\endgroup$
    – paulina
    Commented Jun 16 at 20:31
  • $\begingroup$ @paulina is what the exercise asks to calculate. :( $\endgroup$
    – LittleBlue
    Commented Jun 16 at 20:39
  • $\begingroup$ Is the formula you wrote given in the exercise? $\endgroup$
    – paulina
    Commented Jun 16 at 20:40
  • $\begingroup$ Yes. (more words to send the message xD) $\endgroup$
    – LittleBlue
    Commented Jun 16 at 20:42
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    $\begingroup$ 1. $N \pi \to N \pi$ is a scattering process and not a diffusion process. 2. Nucleons are fermions, they do have spin. 3. Your Lagrangian does not describe the interaction of a fermion with a scalar but of a complex scalar $\psi$ with a real scalar $\phi$. 4. Interpreted in this way, the relative sign of the kinetic terms of $\psi$ and $\phi$ is wrong. 5. If your problem is simply how to relate a Mandelstam variable to a scattering angle and how to compute the absolute square of a complex number, it is not "too much to ask" for, but a pure homework problem, being off-topic on this site. $\endgroup$
    – Hyperon
    Commented Jun 17 at 4:58

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