# Guidance needed in finding scattering amplitude

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!

• "The" scattering amplitude does not exist. In general, one computes such amplitudes via Feynman diagrams. – ACuriousMind Nov 3 '14 at 16:09
• @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? – Fluctuations Nov 3 '14 at 16:11
• Aren't you missing kinetic terms for $\phi$ .? – Jerry Schirmer Nov 3 '14 at 20:33
• 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 $\phi$) – Jerry Schirmer Nov 3 '14 at 21:46
• Your professor probably just wanted there to be an implied $\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi$ term. – Jerry Schirmer Nov 3 '14 at 22:03

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.

• Thanks for your answer. Does this theory have a name in order to check for the rules on line? – Fluctuations Nov 3 '14 at 16:15
• 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. – Prof. Legolasov Nov 3 '14 at 16:16
• 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. – Fluctuations Nov 3 '14 at 17:22
• 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. – Prof. Legolasov Nov 3 '14 at 19:05