## Thursday, June 30, 2016

### 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.
The one-loop amplitude is \begin{equation*} i \mathcal{M} = i V_4(s,t,u) + \left( \left( i V_3(s)\right) ^2 \Delta(s) + s \leftrightarrow t + t \leftrightarrow u \right) \end{equation*} Here, $V_3$ and $V_4$ are the one-loop expressions for the vertices, $\Delta$ is the one-loop expression for the propagator and $s$, $t$, $u$ are the Mandelstam variables. The propagator $\Delta$ is given by a one-dimensional integral, the vertex $V_3$ by a 2-dimensional integral and $V_4$ by a 3-dimensional integral. I do not write down explicit expressions as the formulas are quite long [1].

NIntegrate in Mathematica does not have problems calculating $\Delta$. However, I struggled a lot to calculate $V_3$ with NIntegrate, but at the end managed to calculate $V_3$ satisfactorily. When calculating $V_4$ with NIntegrate, I get many warnings and error messages in Mathematica. I spent a bit of time trying to resolve these, but did not succeed. I have ignored all warnings and error messages in Mathematica, the graphs below are thus possibly not accurate.

The coupling constant $g=10$ in the graphs below. This seems large, but I think that the perturbation series is essentially in $\alpha = g^2 / (4 \pi)^3 \sim 0.05$, which is sufficiently small. I also did not want to take $g$ too small because otherwise the loop corrections are barely visible. I plot $|\mathcal{M}|$ in the center of mass frame as function of the scattering angle $\theta$.

In figure 1, the velocity of the incoming particles is $0.10$. This corresponds with $E = 1.00504 m$

 Fig 1. $|\mathcal{M}|$ as function of $\theta$ black line: tree level, blue dots: one loop $v = 0.10$
In figure 2, the velocity of the incoming particles is $0.50$. This corresponds with $E = 1.1547 m$
 Fig 2. $|\mathcal{M}|$ as function of $\theta$ black line: tree level, blue dots: one loop $v = 0.50$
In figure 3, the velocity of the incoming particles is $0.90$. This corresponds with $E = 2.29416 m$
 Fig 3. $|\mathcal{M}|$ as function of $\theta$ black line: tree level, blue dots: one loop $v = 0.90$

References
[1] Explicit formulas can be found in Srednicki, equations (20.3 - 20.11)
[2] In a previous post I plotted $\mathcal{M}$ at tree level.
[3] Some Mathematica code can be found in a previous post.