 #1
MT777
 4
 0
 Homework Statement:
 Find the Feynman rules in the momentum space for the following interaction lagrangian [itex] \mathcal{L}_{\text{I}} = (1/2!) \lambda \phi_1^2 \phi_2 + \text{h.c.} [/itex], where [itex] \phi_1 [/itex] has charge [itex] e [/itex] and [itex] \phi_2 [/itex] is neutral. Calculate at tree level the efficient cross section from the center of mass of the elastic process [itex] \phi_1 + \phi_2 \rightarrow \phi_1 + \phi_2 [/itex].
 Relevant Equations:

The tree level cross section from the center of mass from two incoming particles [itex] A [/itex] and [itex] B [/itex] is
[tex] \Bigg(\frac{\text{d} \sigma}{\text{d} \Omega} \Bigg)_{\text{CM}} = \frac{1}{2 E_A 2 E_B} \frac{1}{v_A  v_B} \frac{\textbf{p}}{4 (2 \pi)^4 \sqrt{S}} \mathcal{M}^2 [/tex]
Hi there. I'm trying to solve the problem mentioned above, the thing is I'm truly lost and I don't know how to start solving this problem. Sorry if I don't have a concrete attempt at a solution. How do I derive the Feynman rules for this Lagrangian? What I think happens is that in momentum space, for each line there is a Feynman propagator while for each vertex there is a coupling term with a Dirac delta due to conservation of momentum, but how do I draw the corresponding Feynman diagram from the interaction Lagrangian?
For the cross section, who is [itex] \mathcal{M} [/itex] in this case?
For the cross section, who is [itex] \mathcal{M} [/itex] in this case?