Usage
Contents
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)
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.