Two-particle scattering at one loop

In his book on quantum field theory, Srednicki performs many calculations in the $\phi^3$ theory in six dimensions, which has Lagrangian \begin{equation*} \mathcal{L} = \frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi - \frac{1}{2} m^2 \phi^2 + \frac{g}{6} \phi^3 \end{equation*} For example, in chapter 20 Srednicki calculates the two particle scattering amplitude, including all one-loop corrections. In this post I illustrate the result.

Loop correction to the 3-point vertex

In chapter 18 in Srednicki, the loop correction to the 3-point vertex in $\phi^3$ theory in six dimensions is calculated. In this post, I give comments on its numerical calculation in Mathematica.

Loop correction to the propagator in $\phi^3$ theory

The loop correction to the propagator in the $\phi^3$ theory in six dimensions is given by the Feynman diagrams

Cross section in $\phi^3$ theory

In his book on quantum field theory, Srednicki performs many calculations in the $\phi^3$ theory in six dimensions, which has Lagrangian \begin{equation*} \mathcal{L} = \frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi - \frac{1}{2} m^2 \phi^2 + \frac{g}{6} \phi^3 \end{equation*} The $\phi^3$ theory in six dimensions is a nice theory to explain many aspects of quantum field theory, because it is a renormalizable theory with only scalar fields. Of course the theory is not realistic because it has six dimensions and the vacuum is not stable, but it is instructive to see some aspects of quantum field theory explained without the extra complications coming from spinors or gauge fields. I also find the $\phi^3$ less cumbersome to calculate with than the more familiar $\phi^4$ theory in four dimensions.

One of the first calculations one can do is to calculate the amplitude of the scattering $\phi\phi \to \phi\phi$ at tree level.