Question

(4) A famous “toy” model in quantum field theory is the scalar φ^3
theory. The only field
in the theory is a neutral scalar (spin-zero) particle, which is represented by a line
without an arrow, and the only vertex in the theory is where three scalar lines meet.
(a) Draw all the leading-order Feynman diagrams for the elastic scattering of two
such scalars. As before, label the external momenta to help you make sure that
you obtain all of the diagrams.
(b) Now do the same for the process of one scalar decaying into three scalars. Which
of these two processes [(a) or (b)] has more independent diagrams?

Answer 1

(a) The elastic scattering of the elastic scattering of two sclalars are given by three different leading order Feynman diagrams. The initial momenta are and the final momenta are  . Note that in the following diagrams, time runs from bottom to top (increases to the top) (note that only the black blob means a vertex, in the third diagram, the to lines in the upper part do not actually intesect):

(b) You should note that that the one scalar cannot decay into three identical scalars unless it is massless, since it is kinematically forbidden. The diagrams will be as follows, in the tree level.

We thus see that the number of independent diagrams are same in each case.