Usage#

Overview#

Library of symbolic dynamics functions#

AmpForm offers a number of dynamics parametrization functions. The functions are expressed with sympy, so that they can easily be visualized, simplified, or modified:

import sympy as sp

from ampform.dynamics import relativistic_breit_wigner

m, m0, w0 = sp.symbols("m m0 Gamma0")
relativistic_breit_wigner(s=m**2, mass0=m0, gamma0=w0)
\[\displaystyle \frac{\Gamma_{0} m_{0}}{- i \Gamma_{0} m_{0} - m^{2} + m_{0}^{2}}\]
from ampform.dynamics import relativistic_breit_wigner_with_ff

m1, m2, L = sp.symbols("m1 m2 L")
relativistic_breit_wigner_with_ff(
    s=m**2,
    mass0=m0,
    gamma0=w0,
    m_a=m1,
    m_b=m2,
    angular_momentum=L,
    meson_radius=1,
)
\[\displaystyle \frac{\Gamma_{0} m_{0} \sqrt{B_{L}^2\left(q^2\left(m^{2}\right)\right)}}{- m^{2} + m_{0}^{2} - i m_{0} \Gamma_{0}\left(m^{2}\right)}\]
from ampform.dynamics.kmatrix import NonRelativisticKMatrix

n_poles = sp.Symbol("n_R", integer=True, positive=True)
NonRelativisticKMatrix.formulate(
    n_poles=n_poles,
    n_channels=1,
)[0, 0]
\[\displaystyle \frac{\sum_{R=1}^{n_{R}} \frac{{\Gamma}_{R,0} {\gamma}_{R,0}^{2} {m}_{R}}{- s + {m}_{R}^{2}}}{- i \sum_{R=1}^{n_{R}} \frac{{\Gamma}_{R,0} {\gamma}_{R,0}^{2} {m}_{R}}{- s + {m}_{R}^{2}} + 1}\]
matrix = NonRelativisticKMatrix.formulate(n_poles=1, n_channels=2)
matrix[0, 0].doit().simplify()
\[\displaystyle - \frac{{\Gamma}_{1,0} {\gamma}_{1,0}^{2} {m}_{1}}{s + i {\Gamma}_{1,0} {\gamma}_{1,0}^{2} {m}_{1} + i {\Gamma}_{1,1} {\gamma}_{1,1}^{2} {m}_{1} - {m}_{1}^{2}}\]

More dynamics functions can be found in the dynamics library, as well as on the Dynamics page!

Formulate amplitude models#

Together with QRules, AmpForm can automatically formulate amplitude models for generic, multi-body decays. These models can then be used as templates for faster computational back-ends with TensorWaves. Here’s an example:

import qrules

reaction = qrules.generate_transitions(
    initial_state=("psi(4160)", [-1, +1]),
    final_state=["D-", "D0", "pi+"],
    allowed_intermediate_particles=["D*(2007)0"],
    formalism="helicity",
)

import ampform
from ampform.dynamics.builder import create_relativistic_breit_wigner

builder = ampform.get_builder(reaction)
for particle in reaction.get_intermediate_particles():
    builder.set_dynamics(particle.name, create_relativistic_breit_wigner)
model = builder.formulate()
model.intensity
\[\displaystyle \sum_{m_{A}\in\left\{1,-1\right\}} \sum_{m_{0}=0} \sum_{m_{1}=0} \sum_{m_{2}=0}{\left|{{A^{02}}_{m_{A},m_{0},m_{1},m_{2}}}\right|^{2}}\]
symbol, expr = list(model.amplitudes.items())[0]
latex = sp.multiline_latex(symbol, expr)
Math(latex)
\[\begin{split}\displaystyle \begin{align*} {A^{02}}_{-1,0,0,0} = & \frac{C_{\psi(4160) \to D^{*}(2007)^{0}_{+1} D^{0}_{0}; D^{*}(2007)^{0} \to D^{-}_{0} \pi^{+}_{0}} \Gamma_{D^{*}(2007)^{0}} m_{D^{*}(2007)^{0}} D^{1}_{-1,1}\left(- \phi_{02},\theta_{02},0\right) D^{1}_{1,0}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right)}{- i \Gamma_{D^{*}(2007)^{0}} m_{D^{*}(2007)^{0}} - m_{02}^{2} + \left(m_{D^{*}(2007)^{0}}\right)^{2}} \\ & + \frac{C_{\psi(4160) \to D^{*}(2007)^{0}_{-1} D^{0}_{0}; D^{*}(2007)^{0} \to D^{-}_{0} \pi^{+}_{0}} \Gamma_{D^{*}(2007)^{0}} m_{D^{*}(2007)^{0}} D^{1}_{-1,-1}\left(- \phi_{02},\theta_{02},0\right) D^{1}_{-1,0}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right)}{- i \Gamma_{D^{*}(2007)^{0}} m_{D^{*}(2007)^{0}} - m_{02}^{2} + \left(m_{D^{*}(2007)^{0}}\right)^{2}} \\ & + \frac{C_{\psi(4160) \to D^{*}(2007)^{0}_{0} D^{0}_{0}; D^{*}(2007)^{0} \to D^{-}_{0} \pi^{+}_{0}} \Gamma_{D^{*}(2007)^{0}} m_{D^{*}(2007)^{0}} D^{1}_{-1,0}\left(- \phi_{02},\theta_{02},0\right) D^{1}_{0,0}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right)}{- i \Gamma_{D^{*}(2007)^{0}} m_{D^{*}(2007)^{0}} - m_{02}^{2} + \left(m_{D^{*}(2007)^{0}}\right)^{2}} \end{align*}\end{split}\]

In case of multiple decay topologies, AmpForm also takes care of spin alignment with [1]!

reaction = qrules.generate_transitions(
    initial_state="Lambda(c)+",
    final_state=["p", "K-", "pi+"],
    allowed_intermediate_particles=[
        "Lambda(1405)",
        "Delta(1232)++",
    ],
)

import graphviz

dot = qrules.io.asdot(reaction, collapse_graphs=True)
graphviz.Source(dot)
_images/usage_20_0.svg
builder = ampform.get_builder(reaction)
model = builder.formulate()
model.intensity
\[\displaystyle \sum_{m_{A}=-1/2}^{1/2} \sum_{m_{0}=-1/2}^{1/2} \sum_{m_{1}=0} \sum_{m_{2}=0}{\left|{\sum_{\lambda^{01}_{0}=-1/2}^{1/2} \sum_{\mu^{01}_{0}=-1/2}^{1/2} \sum_{\nu^{01}_{0}=-1/2}^{1/2} \sum_{\lambda^{01}_{1}=0} \sum_{\mu^{01}_{1}=0} \sum_{\nu^{01}_{1}=0} \sum_{\lambda^{01}_{2}=0}{{A^{01}}_{m_{A},\lambda^{01}_{0},- \lambda^{01}_{1},- \lambda^{01}_{2}} D^{0}_{m_{1},\nu^{01}_{1}}\left(\alpha^{01}_{1},\beta^{01}_{1},\gamma^{01}_{1}\right) D^{0}_{m_{2},\lambda^{01}_{2}}\left(\phi_{01},\theta_{01},0\right) D^{0}_{\mu^{01}_{1},\lambda^{01}_{1}}\left(\phi^{01}_{0},\theta^{01}_{0},0\right) D^{0}_{\nu^{01}_{1},\mu^{01}_{1}}\left(\phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{m_{0},\nu^{01}_{0}}\left(\alpha^{01}_{0},\beta^{01}_{0},\gamma^{01}_{0}\right) D^{\frac{1}{2}}_{\mu^{01}_{0},\lambda^{01}_{0}}\left(\phi^{01}_{0},\theta^{01}_{0},0\right) D^{\frac{1}{2}}_{\nu^{01}_{0},\mu^{01}_{0}}\left(\phi_{01},\theta_{01},0\right)} + \sum_{\lambda^{02}_{0}=-1/2}^{1/2} \sum_{\mu^{02}_{0}=-1/2}^{1/2} \sum_{\nu^{02}_{0}=-1/2}^{1/2} \sum_{\lambda^{02}_{1}=0} \sum_{\lambda^{02}_{2}=0} \sum_{\mu^{02}_{2}=0} \sum_{\nu^{02}_{2}=0}{{A^{02}}_{m_{A},\lambda^{02}_{0},- \lambda^{02}_{1},- \lambda^{02}_{2}} D^{0}_{m_{1},\lambda^{02}_{1}}\left(\phi_{02},\theta_{02},0\right) D^{0}_{m_{2},\nu^{02}_{2}}\left(\alpha^{02}_{2},\beta^{02}_{2},\gamma^{02}_{2}\right) D^{0}_{\mu^{02}_{2},\lambda^{02}_{2}}\left(\phi^{02}_{0},\theta^{02}_{0},0\right) D^{0}_{\nu^{02}_{2},\mu^{02}_{2}}\left(\phi_{02},\theta_{02},0\right) D^{\frac{1}{2}}_{m_{0},\nu^{02}_{0}}\left(\alpha^{02}_{0},\beta^{02}_{0},\gamma^{02}_{0}\right) D^{\frac{1}{2}}_{\mu^{02}_{0},\lambda^{02}_{0}}\left(\phi^{02}_{0},\theta^{02}_{0},0\right) D^{\frac{1}{2}}_{\nu^{02}_{0},\mu^{02}_{0}}\left(\phi_{02},\theta_{02},0\right)}}\right|^{2}}\]

Advanced examples#

The following pages provide more advanced examples of how to use AmpForm. You can run each of them as Jupyter notebooks with the launch button in the top-right corner.