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If I have the Lagrangian $$\mathcal{L}=\bar{\psi}(i\gamma ^\mu \partial_\mu - m)\psi -g\bar{\psi}i\gamma^5\phi\psi,$$ where $g$ is a coupling constant.

How to find the scattering amplitude for $$ \phi \psi \to \phi \psi $$ What I only learned in class was electron-electron scattering and electron-proton scattering and can't seem to relate this case above to any of them. I ask only for guidance. Please and thank you!

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    $\begingroup$ "The" scattering amplitude does not exist. In general, one computes such amplitudes via Feynman diagrams. $\endgroup$
    – ACuriousMind
    Nov 3, 2014 at 16:09
  • $\begingroup$ @ACuriousMind I am aware of that, I just dont know how to deal with this problem since I can't relate this case to the e-e scattering nor to the e- p+ scattering. Just how to start? $\endgroup$ Nov 3, 2014 at 16:11
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    $\begingroup$ Aren't you missing kinetic terms for $\phi$ .? $\endgroup$ Nov 3, 2014 at 20:33
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    $\begingroup$ Well, without the kinetic term, then your interaction term becomes a constraint (if you vary with respect to $\phi$), or it has an undetermined external function (if you don't vary with respect to $\phi$) $\endgroup$ Nov 3, 2014 at 21:46
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    $\begingroup$ Your professor probably just wanted there to be an implied $\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi$ term. $\endgroup$ Nov 3, 2014 at 22:03

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Proceed as usual:

  1. Derive (or find somewhere) the Feynman rules for this theory.

  2. Draw the lowest-order diagrams contributing to the specific scattering process you are interested in

  3. Evaluate them

It should be even easier than in case of QED (I believe you studied electron-electron scatterings in QED)

UPD: this is called the pseudo-scalar Yukawa theory.

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  • $\begingroup$ Thanks for your answer. Does this theory have a name in order to check for the rules on line? $\endgroup$ Nov 3, 2014 at 16:15
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    $\begingroup$ It is called pseudo-scalar Yukawa theory. Be aware that there is also a scalar Yukawa theory without the $\gamma^5$ matrix and it differs from yours. $\endgroup$ Nov 3, 2014 at 16:16
  • $\begingroup$ Would it differ if I added to the above lagrangian $$1/2\partial_\mu \phi \partial^\mu \phi - 1/2 m^2 \phi^2 -1/4! \lambda \phi^4$$ I mean would it affect my Feynmann rules for this. $\endgroup$ Nov 3, 2014 at 17:22
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    $\begingroup$ Yes, a new type of propagating particle (in context of Yukawa interaction it is usually called a meson) would appear. The third term also adds the meson self-interaction vertex. $\endgroup$ Nov 3, 2014 at 19:05

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