Lecture 17 – Collision theory

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.

Code for plotting Riemann sheets with plotly Hide code cell content

def plot_riemann_surfaces(
    funcs: list[Callable],
    func_unicode: str,
    boundaries: tuple[complex, float] | tuple[complex, complex] = (0, 1),
    resolution: int | tuple[int, int] = 50,
    colorize: bool = True,
    mask: Callable[[np.ndarray, np.ndarray], bool] | None = None,
) -> None:
    X, Y = create_meshgrid(boundaries, resolution)
    Z = X + Y * 1j
    T = [f(Z) for f in funcs]
    if mask is not None:
        the_mask = np.full(Z.shape, False)
        for t in T:
            the_mask |= mask(Z, t)
        if np.all(the_mask):
            raise ValueError("All points were masked away")
        X[the_mask] = np.nan
        Y[the_mask] = np.nan
        Z[the_mask] = np.nan
        for t in T:
            t[the_mask] = np.nan

    vmax = max(max(t.imag.max(), t.real.max()) for t in T)
    style = lambda i, t: dict(
        cmin=-vmax,
        cmax=+vmax,
        showscale=colorize,
        colorscale=(
            "RdBu_r"
            if colorize
            else [[0, "rgb(0, 0, 0)"], [1, DEFAULT_PLOTLY_COLORS[i - 1]]]
        ),
        surfacecolor=t.real if colorize else np.ones(shape=t.shape),
    )
    S_im = [
        go.Surface(x=X, y=Y, z=t.imag, **style(i, t), name=f"Sheet {i}")
        for i, t in enumerate(T, 1)
    ]
    S_re = [
        go.Surface(x=X, y=Y, z=t.real, **style(i, t), name=f"Sheet {i}")
        for i, t in enumerate(T, 1)
    ]
    fig = make_subplots(
        cols=2,
        specs=[[{"type": "surface"}, {"type": "surface"}]],
        subplot_titles=(f"Im {func_unicode}", f"Re {func_unicode}"),
    )
    for i in range(len(funcs)):
        fig.add_trace(S_im[i], col=1, row=1)
        fig.add_trace(S_re[i], col=2, row=1)
    fig.update_layout(height=550, width=1_000)
    fig.update_traces(colorbar=dict(title="Re/Im"))
    fig.show()


def create_meshgrid(
    boundaries: tuple[complex, float] | tuple[complex, complex] = (0, 1),
    resolution: int | tuple[int, int] = 50,
) -> tuple[np.ndarray, np.ndarray]:
    if isinstance(resolution, tuple):
        x_res, y_res = resolution
    else:
        x_res, y_res = resolution, resolution
    box_min, box_max = boundaries
    if isinstance(box_max, (float, int)):
        pos, r_max = box_min, box_max
        R, Θ = np.meshgrid(
            np.linspace(0, r_max, num=x_res),
            np.linspace(-np.pi, +np.pi, num=y_res),
        )
        X = R * np.cos(Θ) + pos
        Y = R * np.sin(Θ) + pos
        return X, Y
    x1 = complex(box_min).real
    x2 = complex(box_max).real
    y1 = complex(box_min).imag
    y2 = complex(box_max).imag
    return np.meshgrid(
        np.linspace(x1, x2, num=x_res),
        np.linspace(y1, y2, num=y_res),
    )


def cut_t(
    cutoff: float | tuple[float, float]
) -> Callable[[np.ndarray, np.ndarray], bool]:
    if isinstance(cutoff, tuple):
        re_cut, im_cut = cutoff
    else:
        re_cut, im_cut = cutoff, cutoff
    return lambda z, t: (np.abs(t.real) > re_cut) | (np.abs(t.imag) > im_cut)

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#

\[\begin{split}\displaystyle \begin{array}{rcl} S_{I}(s) &=& - \frac{i \sigma\left(s\right) T_{I}(s)}{8 \pi} + 1 \\ T_{I}(s) &=& \frac{1}{- G_{I}(s) + \frac{1}{V_1\left(s\right)}} \\ T_{II}(s) &=& \frac{1}{- G_{II}(s) + \frac{1}{V_1\left(s\right)}} \\ V_1\left(s\right) &=& - \frac{G_{V}^{2} h\left(\frac{m_{\rho}^{2}}{2 p^2\left(s\right)}\right) p^2\left(s\right)}{f_{\pi}^{4}} - \frac{2 \left(- \frac{2 G_{V}^{2} s}{f_{\pi}^{2} \left(- m_{\rho}^{2} + s\right)} + 1\right) p^2\left(s\right)}{3 f_{\pi}^{2}} \\ h\left(a\right) &=& a \left(a^{2} + 3 a + 2\right) \log{\left(1 + \frac{2}{a} \right)} - \frac{2 \left(3 a^{2} + 6 a + 1\right)}{3} \\ p^2\left(s\right) &=& - m^{2} + \frac{s}{4} \\ \end{array}\end{split}\]

\[\begin{split}\displaystyle \begin{array}{rcl} f_{\pi} &=& 87.3 \\ G_{V} &=& \frac{\sqrt{f_{\pi}^{2} g_{v}^{2}}}{2} \\ g_{v} &=& 1 \\ m &=& 139 \\ m_{\rho} &=& 770 \\ g_{0} &=& -3 \\ \end{array}\end{split}\]