Lecture 17 – Collision theory#
This notebook is an attempt to recreate the Mathematica notebook provided by Miguel Albaladejo. Another nice tutorial about the complex plane is this Julia notebook by Mikhail Mikhasenko.
Riemann sheets#
Square root example#
There are multiple solutions for \(x\) to the equation \(y^2 = x\) – the fact that we usually take \(y = \sqrt{x}\) to be the solution to this equation is just a matter of convention. It would be more complete to represent the solution as a set of points in the complex plane. In this case, we have the set \(S = \left\{\left(z, w\right)\in\mathbb{C}^2 | w^2=z\right\}\). This is set forms a Riemann surface in \(\mathbb{C}^2\) space.
plot_riemann_surfaces(
funcs=[
lambda z: -1 / np.sqrt(z),
lambda z: +1 / np.sqrt(z),
],
func_unicode="1/±√z",
mask=cut_t(10),
)
Note also that since \(y = e^{x + 2n \pi i}\) for \(\forall n \in \mathbb{Z}\), we have that \(x = \log(y) + 2n\pi i\):
Video explainers
Definition of the G(s) functions#
\[\begin{split}\displaystyle \begin{array}{rcl}
\sqrt[+]{z} &=& e^{\frac{i \arg^+\left(z\right)}{2}} \sqrt{\left|{z}\right|} \\
\arg^+\left(z\right) &=& \begin{cases} \arg{\left(z \right)} + 2 \pi & \text{for}\: \operatorname{im}{\left(z\right)} < 0 \\\arg{\left(z \right)} & \text{otherwise} \end{cases} \\
\end{array}\end{split}\]
\[\begin{split}\displaystyle \begin{array}{rcl}
G_{I}(s) &=& \frac{g_{0} - \log{\left(\frac{\sigma\left(s\right) - 1}{\sigma\left(s\right) + 1} \right)} \sigma\left(s\right)}{16 \pi^{2}} \\
G_{II}(s) &=& G_{I}(s) + \frac{i \sigma\left(s\right)}{8 \pi} \\
\sigma\left(s\right) &=& \sqrt[+]{- \frac{4 m^{2}}{s} + 1} \\
\end{array}\end{split}\]
\[\begin{split}\displaystyle \begin{array}{rcl}
m &=& 139 \\
g_{0} &=& 3.0 \\
\end{array}\end{split}\]
T-matrix definition#
\[\begin{split}\displaystyle \begin{array}{rcl}
S_{I}(s) &=& - \frac{i \sigma\left(s\right) T_{I}(s)}{8 \pi} + 1 \\
T_{I}(s) &=& \frac{1}{- G_{I}(s) + \frac{1}{V_1\left(s\right)}} \\
T_{II}(s) &=& \frac{1}{- G_{II}(s) + \frac{1}{V_1\left(s\right)}} \\
V_1\left(s\right) &=& - \frac{G_{V}^{2} h\left(\frac{m_{\rho}^{2}}{2 p^2\left(s\right)}\right) p^2\left(s\right)}{f_{\pi}^{4}} - \frac{2 \left(- \frac{2 G_{V}^{2} s}{f_{\pi}^{2} \left(- m_{\rho}^{2} + s\right)} + 1\right) p^2\left(s\right)}{3 f_{\pi}^{2}} \\
h\left(a\right) &=& a \left(a^{2} + 3 a + 2\right) \log{\left(1 + \frac{2}{a} \right)} - \frac{2 \left(3 a^{2} + 6 a + 1\right)}{3} \\
p^2\left(s\right) &=& - m^{2} + \frac{s}{4} \\
\end{array}\end{split}\]
\[\begin{split}\displaystyle \begin{array}{rcl}
f_{\pi} &=& 87.3 \\
G_{V} &=& \frac{\sqrt{f_{\pi}^{2} g_{v}^{2}}}{2} \\
g_{v} &=& 1 \\
m &=& 139 \\
m_{\rho} &=& 770 \\
g_{0} &=& -3 \\
\end{array}\end{split}\]