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$ |

Fig 2. $|\mathcal{M}|$ as function of $\theta$ black line: tree level, blue dots: one loop $ v = 0.50$ |

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.

## No comments:

## Post a Comment