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.

$p_1$ and $p_2$: momenta of incoming particles
$p_3$ and $p_4$: momenta of outgoing particles

In the center of mass frame the amplitude is [1] \begin{equation*} i \mathcal{M} = (ig)^2 \left( \frac{i}{s - m^2} +\frac{i}{t - m^2}+\frac{i}{u - m^2} \right) \end{equation*} where $s$, $t$, $u$ are the Mandelstam variables \begin{align*} s &= 4 E^2\\ t & = - \dfrac{1}{2} (s - 4 m^2) ( 1 - \cos\theta)\\ u & = - \dfrac{1}{2} (s - 4 m^2) ( 1 + \cos\theta) \end{align*} Here $E$ is the energy of incoming particle 1, which is equal to the energy of incoming particle 2. The amplitude $|\mathcal{M}|$ is a quite complicated function of the $s$ and $\theta$, but can be nicely visualised on a polar plot.

Polar plot of $|\mathcal{M}|$ for three values of energy.

In the above graph $|\mathcal{M}|$ is plotted for three values of $E$

1. The velocity of the incoming particles is $0.01 c$. This corresponds with $E = 1.00005 m$. ($c=1$ is the velocity of light)
2. The velocity of the incoming particles is $0.50 c$. This corresponds with $E = 1.1547 m$
3. The velocity of the incoming particles is $0.99 c$. This corresponds with $E = 7.08881 m$

In the non-relativistic case (1) the scattering is thus isotropic. In the ultra-relativistic case (3), the scattering is sharply peaked in the forward ($\theta = 0$) and backward ($\theta = \pi$) direction.

References
[1] Srednicki, Quantum Field Theory, chapter 11
[2] In Wikipedia, one can find polar plots of photon electron scattering for example.