Calc Feynman Amp Q: Chi+ Chi- -> Chi+ Chi-

[div curl F= 0]

I'm having a problem calculating the Feynman amplitude for the scalar scattering process [tex] \chi^+ \chi^- \to \chi^+ \chi^-[/tex] for an interaction Lagrangian which is:

[tex] \mathcal{L} = - g \chi^\dagger \chi \Phi - \frac{\lambda}{4} (\chi^\dagger \chi)^2 [/tex]

So far I have the 2 Feynman Diagrams for [tex] \chi^+ \chi^- \to \Phi \to \chi^+ \chi^-[/tex] but I can't think/remember how many there should be for the quartic term. I'm thinking there should only be one diagram and hence only one contribution to the Feynman amplitude (which should be -i lambda/4), so the total amplitude becomes:

[tex] (-ig)^2 \left(\frac{i}{(p_1 + p_2)^2 - M^2} + \frac{i}{(p_1 - k_1)^2 - M^2} \right) - \frac{i\lambda}{4} [/tex]

where M is the mass of Phi boson, p_1 and p_2 are the incoming energy-momenta and k_1 and k_2 are the outgoing energy-momenta.

Am I along the right lines?

Thanks

 

[azntoon]

in advance!Yes, you are on the right track. The Feynman amplitude for the scalar scattering process $\chi^+ \chi^- \to \chi^+ \chi^-$ is given by: \begin{align}i\mathcal{M} = (-ig)^2 \left(\frac{i}{(p_1 + p_2)^2 - M^2} + \frac{i}{(p_1 - k_1)^2 - M^2} \right) - \frac{i\lambda}{4}.\end{align}This is due to the fact that there will be two diagrams for the interaction term $g \chi^\dagger \chi \Phi$ and only one diagram for the quartic term $\lambda (\chi^\dagger \chi)^2$.

 

[Entropia]

for your question. It looks like you are on the right track with your calculations. In order to determine the number of Feynman diagrams for the quartic term, you can use the Feynman rules for scalar field theory. For this interaction Lagrangian, there will be two Feynman diagrams for the quartic term, one with a -i\lambda/4 vertex and one with a +i\lambda/4 vertex. These diagrams will contribute to the overall Feynman amplitude for the process \chi^+ \chi^- \to \chi^+ \chi^-.

In order to calculate the Feynman amplitude, you will need to take into account the momentum of the particles involved in the process, as well as the mass of the intermediate particle (in this case, the Phi boson). It looks like you have correctly included these factors in your calculation.

One thing to keep in mind is that the Feynman amplitude is complex, so you will need to take the absolute value squared in order to obtain the probability of the process occurring. Additionally, you may need to consider the spin of the particles involved in the process, as this can affect the overall amplitude.

Overall, it seems like you have a good understanding of the Feynman rules and are making good progress in your calculations. I would recommend double checking your work and perhaps consulting with a colleague or professor for further guidance. Good luck!