Spin alignment#
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import logging
import graphviz
import qrules
import sympy as sp
from IPython.display import Math
import ampform
from ampform.io import improve_latex_rendering
LOGGER = logging.getLogger()
LOGGER.setLevel(logging.ERROR)
improve_latex_rendering()
As described in Spin alignment implementation, the âstandardâ helicity formalism is not suited for state transitions that have different decay topologies. For this reason, the HelicityAmplitudeBuilder can insert a number of Wigner-\(D\) function into the amplitude model in case there is more than one underlying Topology. It is easiest to see this by inspecting the resulting HelicityModel.intensity and its amplitudes:
reaction = qrules.generate_transitions(
initial_state=("J/psi(1S)", [-1, +1]),
final_state=["K0", "Sigma+", "p~"],
allowed_intermediate_particles=["Sigma(1660)", "N(1650)"],
allowed_interaction_types=["strong"],
formalism="helicity",
)
Show code cell source
src = qrules.io.asdot(
reaction,
collapse_graphs=True,
render_initial_state_id=True,
)
graphviz.Source(src)
âSpin alignmentâ can be now switched off or on by setting HelicityAmplitudeBuilder.align_spin:
builder = ampform.get_builder(reaction)
builder.align_spin = True
model = builder.formulate()
model.intensity
\[\displaystyle \sum_{m_{A}\in\left\{1,-1\right\}} \sum_{m_{0}=0} \sum_{m_{1}=-1/2}^{1/2} \sum_{m_{2}=-1/2}^{1/2}{\left|{\sum_{\lambda^{01}_{0}=0} \sum_{\mu^{01}_{0}=0} \sum_{\nu^{01}_{0}=0} \sum_{\lambda^{01}_{1}=-1/2}^{1/2} \sum_{\mu^{01}_{1}=-1/2}^{1/2} \sum_{\nu^{01}_{1}=-1/2}^{1/2} \sum_{\lambda^{01}_{2}=-1/2}^{1/2}{A^{01}_{m_{A}, \lambda^{01}_{0}, - \lambda^{01}_{1}, - \lambda^{01}_{2}} D^{0}_{m_{0},\nu^{01}_{0}}\left(\alpha^{01}_{0},\beta^{01}_{0},\gamma^{01}_{0}\right) D^{0}_{\mu^{01}_{0},\lambda^{01}_{0}}\left(\phi^{01}_{0},\theta^{01}_{0},0\right) D^{0}_{\nu^{01}_{0},\mu^{01}_{0}}\left(\phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{m_{1},\nu^{01}_{1}}\left(\alpha^{01}_{1},\beta^{01}_{1},\gamma^{01}_{1}\right) D^{\frac{1}{2}}_{m_{2},\lambda^{01}_{2}}\left(\phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{\mu^{01}_{1},\lambda^{01}_{1}}\left(\phi^{01}_{0},\theta^{01}_{0},0\right) D^{\frac{1}{2}}_{\nu^{01}_{1},\mu^{01}_{1}}\left(\phi_{01},\theta_{01},0\right)} + \sum_{\lambda^{02}_{0}=0} \sum_{\mu^{02}_{0}=0} \sum_{\nu^{02}_{0}=0} \sum_{\lambda^{02}_{1}=-1/2}^{1/2} \sum_{\lambda^{02}_{2}=-1/2}^{1/2} \sum_{\mu^{02}_{2}=-1/2}^{1/2} \sum_{\nu^{02}_{2}=-1/2}^{1/2}{A^{02}_{m_{A}, \lambda^{02}_{0}, - \lambda^{02}_{1}, - \lambda^{02}_{2}} D^{0}_{m_{0},\nu^{02}_{0}}\left(\alpha^{02}_{0},\beta^{02}_{0},\gamma^{02}_{0}\right) D^{0}_{\mu^{02}_{0},\lambda^{02}_{0}}\left(\phi^{02}_{0},\theta^{02}_{0},0\right) D^{0}_{\nu^{02}_{0},\mu^{02}_{0}}\left(\phi_{02},\theta_{02},0\right) D^{\frac{1}{2}}_{m_{1},\lambda^{02}_{1}}\left(\phi_{02},\theta_{02},0\right) D^{\frac{1}{2}}_{m_{2},\nu^{02}_{2}}\left(\alpha^{02}_{2},\beta^{02}_{2},\gamma^{02}_{2}\right) D^{\frac{1}{2}}_{\mu^{02}_{2},\lambda^{02}_{2}}\left(\phi^{02}_{0},\theta^{02}_{0},0\right) D^{\frac{1}{2}}_{\nu^{02}_{2},\mu^{02}_{2}}\left(\phi_{02},\theta_{02},0\right)}}\right|^{2}}\]
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latex = sp.multiline_latex(
sp.Symbol("I"),
model.intensity.evaluate(),
environment="eqnarray",
)
Math(latex)
\[\begin{split}\displaystyle \begin{eqnarray}
I & = & \left|{\sum_{\lambda^{01}_{0}=0} \sum_{\mu^{01}_{0}=0} \sum_{\nu^{01}_{0}=0} \sum_{\lambda^{01}_{1}=-1/2}^{1/2} \sum_{\mu^{01}_{1}=-1/2}^{1/2} \sum_{\nu^{01}_{1}=-1/2}^{1/2} \sum_{\lambda^{01}_{2}=-1/2}^{1/2}{A^{01}_{-1, \lambda^{01}_{0}, - \lambda^{01}_{1}, - \lambda^{01}_{2}} D^{0}_{0,\nu^{01}_{0}}\left(\alpha^{01}_{0},\beta^{01}_{0},\gamma^{01}_{0}\right) D^{0}_{\mu^{01}_{0},\lambda^{01}_{0}}\left(\phi^{01}_{0},\theta^{01}_{0},0\right) D^{0}_{\nu^{01}_{0},\mu^{01}_{0}}\left(\phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{- \frac{1}{2},\lambda^{01}_{2}}\left(\phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{- \frac{1}{2},\nu^{01}_{1}}\left(\alpha^{01}_{1},\beta^{01}_{1},\gamma^{01}_{1}\right) D^{\frac{1}{2}}_{\mu^{01}_{1},\lambda^{01}_{1}}\left(\phi^{01}_{0},\theta^{01}_{0},0\right) D^{\frac{1}{2}}_{\nu^{01}_{1},\mu^{01}_{1}}\left(\phi_{01},\theta_{01},0\right)} + \sum_{\lambda^{02}_{0}=0} \sum_{\mu^{02}_{0}=0} \sum_{\nu^{02}_{0}=0} \sum_{\lambda^{02}_{1}=-1/2}^{1/2} \sum_{\lambda^{02}_{2}=-1/2}^{1/2} \sum_{\mu^{02}_{2}=-1/2}^{1/2} \sum_{\nu^{02}_{2}=-1/2}^{1/2}{A^{02}_{-1, \lambda^{02}_{0}, - \lambda^{02}_{1}, - \lambda^{02}_{2}} D^{0}_{0,\nu^{02}_{0}}\left(\alpha^{02}_{0},\beta^{02}_{0},\gamma^{02}_{0}\right) D^{0}_{\mu^{02}_{0},\lambda^{02}_{0}}\left(\phi^{02}_{0},\theta^{02}_{0},0\right) D^{0}_{\nu^{02}_{0},\mu^{02}_{0}}\left(\phi_{02},\theta_{02},0\right) D^{\frac{1}{2}}_{- \frac{1}{2},\lambda^{02}_{1}}\left(\phi_{02},\theta_{02},0\right) D^{\frac{1}{2}}_{- \frac{1}{2},\nu^{02}_{2}}\left(\alpha^{02}_{2},\beta^{02}_{2},\gamma^{02}_{2}\right) D^{\frac{1}{2}}_{\mu^{02}_{2},\lambda^{02}_{2}}\left(\phi^{02}_{0},\theta^{02}_{0},0\right) D^{\frac{1}{2}}_{\nu^{02}_{2},\mu^{02}_{2}}\left(\phi_{02},\theta_{02},0\right)}}\right|^{2} \nonumber\\
& & + \left|{\sum_{\lambda^{01}_{0}=0} \sum_{\mu^{01}_{0}=0} \sum_{\nu^{01}_{0}=0} \sum_{\lambda^{01}_{1}=-1/2}^{1/2} \sum_{\mu^{01}_{1}=-1/2}^{1/2} \sum_{\nu^{01}_{1}=-1/2}^{1/2} \sum_{\lambda^{01}_{2}=-1/2}^{1/2}{A^{01}_{-1, \lambda^{01}_{0}, - \lambda^{01}_{1}, - \lambda^{01}_{2}} D^{0}_{0,\nu^{01}_{0}}\left(\alpha^{01}_{0},\beta^{01}_{0},\gamma^{01}_{0}\right) D^{0}_{\mu^{01}_{0},\lambda^{01}_{0}}\left(\phi^{01}_{0},\theta^{01}_{0},0\right) D^{0}_{\nu^{01}_{0},\mu^{01}_{0}}\left(\phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{- \frac{1}{2},\lambda^{01}_{2}}\left(\phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{\frac{1}{2},\nu^{01}_{1}}\left(\alpha^{01}_{1},\beta^{01}_{1},\gamma^{01}_{1}\right) D^{\frac{1}{2}}_{\mu^{01}_{1},\lambda^{01}_{1}}\left(\phi^{01}_{0},\theta^{01}_{0},0\right) D^{\frac{1}{2}}_{\nu^{01}_{1},\mu^{01}_{1}}\left(\phi_{01},\theta_{01},0\right)} + \sum_{\lambda^{02}_{0}=0} \sum_{\mu^{02}_{0}=0} \sum_{\nu^{02}_{0}=0} \sum_{\lambda^{02}_{1}=-1/2}^{1/2} \sum_{\lambda^{02}_{2}=-1/2}^{1/2} \sum_{\mu^{02}_{2}=-1/2}^{1/2} \sum_{\nu^{02}_{2}=-1/2}^{1/2}{A^{02}_{-1, \lambda^{02}_{0}, - \lambda^{02}_{1}, - \lambda^{02}_{2}} D^{0}_{0,\nu^{02}_{0}}\left(\alpha^{02}_{0},\beta^{02}_{0},\gamma^{02}_{0}\right) D^{0}_{\mu^{02}_{0},\lambda^{02}_{0}}\left(\phi^{02}_{0},\theta^{02}_{0},0\right) D^{0}_{\nu^{02}_{0},\mu^{02}_{0}}\left(\phi_{02},\theta_{02},0\right) D^{\frac{1}{2}}_{- \frac{1}{2},\nu^{02}_{2}}\left(\alpha^{02}_{2},\beta^{02}_{2},\gamma^{02}_{2}\right) D^{\frac{1}{2}}_{\frac{1}{2},\lambda^{02}_{1}}\left(\phi_{02},\theta_{02},0\right) D^{\frac{1}{2}}_{\mu^{02}_{2},\lambda^{02}_{2}}\left(\phi^{02}_{0},\theta^{02}_{0},0\right) D^{\frac{1}{2}}_{\nu^{02}_{2},\mu^{02}_{2}}\left(\phi_{02},\theta_{02},0\right)}}\right|^{2} \nonumber\\
& & + \left|{\sum_{\lambda^{01}_{0}=0} \sum_{\mu^{01}_{0}=0} \sum_{\nu^{01}_{0}=0} \sum_{\lambda^{01}_{1}=-1/2}^{1/2} \sum_{\mu^{01}_{1}=-1/2}^{1/2} \sum_{\nu^{01}_{1}=-1/2}^{1/2} \sum_{\lambda^{01}_{2}=-1/2}^{1/2}{A^{01}_{-1, \lambda^{01}_{0}, - \lambda^{01}_{1}, - \lambda^{01}_{2}} D^{0}_{0,\nu^{01}_{0}}\left(\alpha^{01}_{0},\beta^{01}_{0},\gamma^{01}_{0}\right) D^{0}_{\mu^{01}_{0},\lambda^{01}_{0}}\left(\phi^{01}_{0},\theta^{01}_{0},0\right) D^{0}_{\nu^{01}_{0},\mu^{01}_{0}}\left(\phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{- \frac{1}{2},\nu^{01}_{1}}\left(\alpha^{01}_{1},\beta^{01}_{1},\gamma^{01}_{1}\right) D^{\frac{1}{2}}_{\frac{1}{2},\lambda^{01}_{2}}\left(\phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{\mu^{01}_{1},\lambda^{01}_{1}}\left(\phi^{01}_{0},\theta^{01}_{0},0\right) D^{\frac{1}{2}}_{\nu^{01}_{1},\mu^{01}_{1}}\left(\phi_{01},\theta_{01},0\right)} + \sum_{\lambda^{02}_{0}=0} \sum_{\mu^{02}_{0}=0} \sum_{\nu^{02}_{0}=0} \sum_{\lambda^{02}_{1}=-1/2}^{1/2} \sum_{\lambda^{02}_{2}=-1/2}^{1/2} \sum_{\mu^{02}_{2}=-1/2}^{1/2} \sum_{\nu^{02}_{2}=-1/2}^{1/2}{A^{02}_{-1, \lambda^{02}_{0}, - \lambda^{02}_{1}, - \lambda^{02}_{2}} D^{0}_{0,\nu^{02}_{0}}\left(\alpha^{02}_{0},\beta^{02}_{0},\gamma^{02}_{0}\right) D^{0}_{\mu^{02}_{0},\lambda^{02}_{0}}\left(\phi^{02}_{0},\theta^{02}_{0},0\right) D^{0}_{\nu^{02}_{0},\mu^{02}_{0}}\left(\phi_{02},\theta_{02},0\right) D^{\frac{1}{2}}_{- \frac{1}{2},\lambda^{02}_{1}}\left(\phi_{02},\theta_{02},0\right) D^{\frac{1}{2}}_{\frac{1}{2},\nu^{02}_{2}}\left(\alpha^{02}_{2},\beta^{02}_{2},\gamma^{02}_{2}\right) D^{\frac{1}{2}}_{\mu^{02}_{2},\lambda^{02}_{2}}\left(\phi^{02}_{0},\theta^{02}_{0},0\right) D^{\frac{1}{2}}_{\nu^{02}_{2},\mu^{02}_{2}}\left(\phi_{02},\theta_{02},0\right)}}\right|^{2} \nonumber\\
& & + \left|{\sum_{\lambda^{01}_{0}=0} \sum_{\mu^{01}_{0}=0} \sum_{\nu^{01}_{0}=0} \sum_{\lambda^{01}_{1}=-1/2}^{1/2} \sum_{\mu^{01}_{1}=-1/2}^{1/2} \sum_{\nu^{01}_{1}=-1/2}^{1/2} \sum_{\lambda^{01}_{2}=-1/2}^{1/2}{A^{01}_{-1, \lambda^{01}_{0}, - \lambda^{01}_{1}, - \lambda^{01}_{2}} D^{0}_{0,\nu^{01}_{0}}\left(\alpha^{01}_{0},\beta^{01}_{0},\gamma^{01}_{0}\right) D^{0}_{\mu^{01}_{0},\lambda^{01}_{0}}\left(\phi^{01}_{0},\theta^{01}_{0},0\right) D^{0}_{\nu^{01}_{0},\mu^{01}_{0}}\left(\phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{\frac{1}{2},\lambda^{01}_{2}}\left(\phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{\frac{1}{2},\nu^{01}_{1}}\left(\alpha^{01}_{1},\beta^{01}_{1},\gamma^{01}_{1}\right) D^{\frac{1}{2}}_{\mu^{01}_{1},\lambda^{01}_{1}}\left(\phi^{01}_{0},\theta^{01}_{0},0\right) D^{\frac{1}{2}}_{\nu^{01}_{1},\mu^{01}_{1}}\left(\phi_{01},\theta_{01},0\right)} + \sum_{\lambda^{02}_{0}=0} \sum_{\mu^{02}_{0}=0} \sum_{\nu^{02}_{0}=0} \sum_{\lambda^{02}_{1}=-1/2}^{1/2} \sum_{\lambda^{02}_{2}=-1/2}^{1/2} \sum_{\mu^{02}_{2}=-1/2}^{1/2} \sum_{\nu^{02}_{2}=-1/2}^{1/2}{A^{02}_{-1, \lambda^{02}_{0}, - \lambda^{02}_{1}, - \lambda^{02}_{2}} D^{0}_{0,\nu^{02}_{0}}\left(\alpha^{02}_{0},\beta^{02}_{0},\gamma^{02}_{0}\right) D^{0}_{\mu^{02}_{0},\lambda^{02}_{0}}\left(\phi^{02}_{0},\theta^{02}_{0},0\right) D^{0}_{\nu^{02}_{0},\mu^{02}_{0}}\left(\phi_{02},\theta_{02},0\right) D^{\frac{1}{2}}_{\frac{1}{2},\lambda^{02}_{1}}\left(\phi_{02},\theta_{02},0\right) D^{\frac{1}{2}}_{\frac{1}{2},\nu^{02}_{2}}\left(\alpha^{02}_{2},\beta^{02}_{2},\gamma^{02}_{2}\right) D^{\frac{1}{2}}_{\mu^{02}_{2},\lambda^{02}_{2}}\left(\phi^{02}_{0},\theta^{02}_{0},0\right) D^{\frac{1}{2}}_{\nu^{02}_{2},\mu^{02}_{2}}\left(\phi_{02},\theta_{02},0\right)}}\right|^{2} \nonumber\\
& & + \left|{\sum_{\lambda^{01}_{0}=0} \sum_{\mu^{01}_{0}=0} \sum_{\nu^{01}_{0}=0} \sum_{\lambda^{01}_{1}=-1/2}^{1/2} \sum_{\mu^{01}_{1}=-1/2}^{1/2} \sum_{\nu^{01}_{1}=-1/2}^{1/2} \sum_{\lambda^{01}_{2}=-1/2}^{1/2}{A^{01}_{1, \lambda^{01}_{0}, - \lambda^{01}_{1}, - \lambda^{01}_{2}} D^{0}_{0,\nu^{01}_{0}}\left(\alpha^{01}_{0},\beta^{01}_{0},\gamma^{01}_{0}\right) D^{0}_{\mu^{01}_{0},\lambda^{01}_{0}}\left(\phi^{01}_{0},\theta^{01}_{0},0\right) D^{0}_{\nu^{01}_{0},\mu^{01}_{0}}\left(\phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{- \frac{1}{2},\lambda^{01}_{2}}\left(\phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{- \frac{1}{2},\nu^{01}_{1}}\left(\alpha^{01}_{1},\beta^{01}_{1},\gamma^{01}_{1}\right) D^{\frac{1}{2}}_{\mu^{01}_{1},\lambda^{01}_{1}}\left(\phi^{01}_{0},\theta^{01}_{0},0\right) D^{\frac{1}{2}}_{\nu^{01}_{1},\mu^{01}_{1}}\left(\phi_{01},\theta_{01},0\right)} + \sum_{\lambda^{02}_{0}=0} \sum_{\mu^{02}_{0}=0} \sum_{\nu^{02}_{0}=0} \sum_{\lambda^{02}_{1}=-1/2}^{1/2} \sum_{\lambda^{02}_{2}=-1/2}^{1/2} \sum_{\mu^{02}_{2}=-1/2}^{1/2} \sum_{\nu^{02}_{2}=-1/2}^{1/2}{A^{02}_{1, \lambda^{02}_{0}, - \lambda^{02}_{1}, - \lambda^{02}_{2}} D^{0}_{0,\nu^{02}_{0}}\left(\alpha^{02}_{0},\beta^{02}_{0},\gamma^{02}_{0}\right) D^{0}_{\mu^{02}_{0},\lambda^{02}_{0}}\left(\phi^{02}_{0},\theta^{02}_{0},0\right) D^{0}_{\nu^{02}_{0},\mu^{02}_{0}}\left(\phi_{02},\theta_{02},0\right) D^{\frac{1}{2}}_{- \frac{1}{2},\lambda^{02}_{1}}\left(\phi_{02},\theta_{02},0\right) D^{\frac{1}{2}}_{- \frac{1}{2},\nu^{02}_{2}}\left(\alpha^{02}_{2},\beta^{02}_{2},\gamma^{02}_{2}\right) D^{\frac{1}{2}}_{\mu^{02}_{2},\lambda^{02}_{2}}\left(\phi^{02}_{0},\theta^{02}_{0},0\right) D^{\frac{1}{2}}_{\nu^{02}_{2},\mu^{02}_{2}}\left(\phi_{02},\theta_{02},0\right)}}\right|^{2} \nonumber\\
& & + \left|{\sum_{\lambda^{01}_{0}=0} \sum_{\mu^{01}_{0}=0} \sum_{\nu^{01}_{0}=0} \sum_{\lambda^{01}_{1}=-1/2}^{1/2} \sum_{\mu^{01}_{1}=-1/2}^{1/2} \sum_{\nu^{01}_{1}=-1/2}^{1/2} \sum_{\lambda^{01}_{2}=-1/2}^{1/2}{A^{01}_{1, \lambda^{01}_{0}, - \lambda^{01}_{1}, - \lambda^{01}_{2}} D^{0}_{0,\nu^{01}_{0}}\left(\alpha^{01}_{0},\beta^{01}_{0},\gamma^{01}_{0}\right) D^{0}_{\mu^{01}_{0},\lambda^{01}_{0}}\left(\phi^{01}_{0},\theta^{01}_{0},0\right) D^{0}_{\nu^{01}_{0},\mu^{01}_{0}}\left(\phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{- \frac{1}{2},\lambda^{01}_{2}}\left(\phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{\frac{1}{2},\nu^{01}_{1}}\left(\alpha^{01}_{1},\beta^{01}_{1},\gamma^{01}_{1}\right) D^{\frac{1}{2}}_{\mu^{01}_{1},\lambda^{01}_{1}}\left(\phi^{01}_{0},\theta^{01}_{0},0\right) D^{\frac{1}{2}}_{\nu^{01}_{1},\mu^{01}_{1}}\left(\phi_{01},\theta_{01},0\right)} + \sum_{\lambda^{02}_{0}=0} \sum_{\mu^{02}_{0}=0} \sum_{\nu^{02}_{0}=0} \sum_{\lambda^{02}_{1}=-1/2}^{1/2} \sum_{\lambda^{02}_{2}=-1/2}^{1/2} \sum_{\mu^{02}_{2}=-1/2}^{1/2} \sum_{\nu^{02}_{2}=-1/2}^{1/2}{A^{02}_{1, \lambda^{02}_{0}, - \lambda^{02}_{1}, - \lambda^{02}_{2}} D^{0}_{0,\nu^{02}_{0}}\left(\alpha^{02}_{0},\beta^{02}_{0},\gamma^{02}_{0}\right) D^{0}_{\mu^{02}_{0},\lambda^{02}_{0}}\left(\phi^{02}_{0},\theta^{02}_{0},0\right) D^{0}_{\nu^{02}_{0},\mu^{02}_{0}}\left(\phi_{02},\theta_{02},0\right) D^{\frac{1}{2}}_{- \frac{1}{2},\nu^{02}_{2}}\left(\alpha^{02}_{2},\beta^{02}_{2},\gamma^{02}_{2}\right) D^{\frac{1}{2}}_{\frac{1}{2},\lambda^{02}_{1}}\left(\phi_{02},\theta_{02},0\right) D^{\frac{1}{2}}_{\mu^{02}_{2},\lambda^{02}_{2}}\left(\phi^{02}_{0},\theta^{02}_{0},0\right) D^{\frac{1}{2}}_{\nu^{02}_{2},\mu^{02}_{2}}\left(\phi_{02},\theta_{02},0\right)}}\right|^{2} \nonumber\\
& & + \left|{\sum_{\lambda^{01}_{0}=0} \sum_{\mu^{01}_{0}=0} \sum_{\nu^{01}_{0}=0} \sum_{\lambda^{01}_{1}=-1/2}^{1/2} \sum_{\mu^{01}_{1}=-1/2}^{1/2} \sum_{\nu^{01}_{1}=-1/2}^{1/2} \sum_{\lambda^{01}_{2}=-1/2}^{1/2}{A^{01}_{1, \lambda^{01}_{0}, - \lambda^{01}_{1}, - \lambda^{01}_{2}} D^{0}_{0,\nu^{01}_{0}}\left(\alpha^{01}_{0},\beta^{01}_{0},\gamma^{01}_{0}\right) D^{0}_{\mu^{01}_{0},\lambda^{01}_{0}}\left(\phi^{01}_{0},\theta^{01}_{0},0\right) D^{0}_{\nu^{01}_{0},\mu^{01}_{0}}\left(\phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{- \frac{1}{2},\nu^{01}_{1}}\left(\alpha^{01}_{1},\beta^{01}_{1},\gamma^{01}_{1}\right) D^{\frac{1}{2}}_{\frac{1}{2},\lambda^{01}_{2}}\left(\phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{\mu^{01}_{1},\lambda^{01}_{1}}\left(\phi^{01}_{0},\theta^{01}_{0},0\right) D^{\frac{1}{2}}_{\nu^{01}_{1},\mu^{01}_{1}}\left(\phi_{01},\theta_{01},0\right)} + \sum_{\lambda^{02}_{0}=0} \sum_{\mu^{02}_{0}=0} \sum_{\nu^{02}_{0}=0} \sum_{\lambda^{02}_{1}=-1/2}^{1/2} \sum_{\lambda^{02}_{2}=-1/2}^{1/2} \sum_{\mu^{02}_{2}=-1/2}^{1/2} \sum_{\nu^{02}_{2}=-1/2}^{1/2}{A^{02}_{1, \lambda^{02}_{0}, - \lambda^{02}_{1}, - \lambda^{02}_{2}} D^{0}_{0,\nu^{02}_{0}}\left(\alpha^{02}_{0},\beta^{02}_{0},\gamma^{02}_{0}\right) D^{0}_{\mu^{02}_{0},\lambda^{02}_{0}}\left(\phi^{02}_{0},\theta^{02}_{0},0\right) D^{0}_{\nu^{02}_{0},\mu^{02}_{0}}\left(\phi_{02},\theta_{02},0\right) D^{\frac{1}{2}}_{- \frac{1}{2},\lambda^{02}_{1}}\left(\phi_{02},\theta_{02},0\right) D^{\frac{1}{2}}_{\frac{1}{2},\nu^{02}_{2}}\left(\alpha^{02}_{2},\beta^{02}_{2},\gamma^{02}_{2}\right) D^{\frac{1}{2}}_{\mu^{02}_{2},\lambda^{02}_{2}}\left(\phi^{02}_{0},\theta^{02}_{0},0\right) D^{\frac{1}{2}}_{\nu^{02}_{2},\mu^{02}_{2}}\left(\phi_{02},\theta_{02},0\right)}}\right|^{2} \nonumber\\
& & + \left|{\sum_{\lambda^{01}_{0}=0} \sum_{\mu^{01}_{0}=0} \sum_{\nu^{01}_{0}=0} \sum_{\lambda^{01}_{1}=-1/2}^{1/2} \sum_{\mu^{01}_{1}=-1/2}^{1/2} \sum_{\nu^{01}_{1}=-1/2}^{1/2} \sum_{\lambda^{01}_{2}=-1/2}^{1/2}{A^{01}_{1, \lambda^{01}_{0}, - \lambda^{01}_{1}, - \lambda^{01}_{2}} D^{0}_{0,\nu^{01}_{0}}\left(\alpha^{01}_{0},\beta^{01}_{0},\gamma^{01}_{0}\right) D^{0}_{\mu^{01}_{0},\lambda^{01}_{0}}\left(\phi^{01}_{0},\theta^{01}_{0},0\right) D^{0}_{\nu^{01}_{0},\mu^{01}_{0}}\left(\phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{\frac{1}{2},\lambda^{01}_{2}}\left(\phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{\frac{1}{2},\nu^{01}_{1}}\left(\alpha^{01}_{1},\beta^{01}_{1},\gamma^{01}_{1}\right) D^{\frac{1}{2}}_{\mu^{01}_{1},\lambda^{01}_{1}}\left(\phi^{01}_{0},\theta^{01}_{0},0\right) D^{\frac{1}{2}}_{\nu^{01}_{1},\mu^{01}_{1}}\left(\phi_{01},\theta_{01},0\right)} + \sum_{\lambda^{02}_{0}=0} \sum_{\mu^{02}_{0}=0} \sum_{\nu^{02}_{0}=0} \sum_{\lambda^{02}_{1}=-1/2}^{1/2} \sum_{\lambda^{02}_{2}=-1/2}^{1/2} \sum_{\mu^{02}_{2}=-1/2}^{1/2} \sum_{\nu^{02}_{2}=-1/2}^{1/2}{A^{02}_{1, \lambda^{02}_{0}, - \lambda^{02}_{1}, - \lambda^{02}_{2}} D^{0}_{0,\nu^{02}_{0}}\left(\alpha^{02}_{0},\beta^{02}_{0},\gamma^{02}_{0}\right) D^{0}_{\mu^{02}_{0},\lambda^{02}_{0}}\left(\phi^{02}_{0},\theta^{02}_{0},0\right) D^{0}_{\nu^{02}_{0},\mu^{02}_{0}}\left(\phi_{02},\theta_{02},0\right) D^{\frac{1}{2}}_{\frac{1}{2},\lambda^{02}_{1}}\left(\phi_{02},\theta_{02},0\right) D^{\frac{1}{2}}_{\frac{1}{2},\nu^{02}_{2}}\left(\alpha^{02}_{2},\beta^{02}_{2},\gamma^{02}_{2}\right) D^{\frac{1}{2}}_{\mu^{02}_{2},\lambda^{02}_{2}}\left(\phi^{02}_{0},\theta^{02}_{0},0\right) D^{\frac{1}{2}}_{\nu^{02}_{2},\mu^{02}_{2}}\left(\phi_{02},\theta_{02},0\right)}}\right|^{2}
\end{eqnarray}\end{split}\]
This also introduces several Wigner rotation angles to the HelicityModel.kinematic_variables:
alpha = sp.Symbol("alpha_0^01", real=True)
model.kinematic_variables[alpha]
\[\displaystyle \operatorname{atan}_{2}{\left(\boldsymbol{B}\left(-\left(p_{0}\right)\right) \boldsymbol{B}\left({p}_{01}\right) \boldsymbol{B}\left(\boldsymbol{B}\left({p}_{01}\right) p_{0}\right)\left[:, 3, 2\right],\boldsymbol{B}\left(-\left(p_{0}\right)\right) \boldsymbol{B}\left({p}_{01}\right) \boldsymbol{B}\left(\boldsymbol{B}\left({p}_{01}\right) p_{0}\right)\left[:, 3, 1\right] \right)}\]
For more information about these angles, see Compute Wigner rotation angles in TR-015.
By default, align_spin is set to False and the total HelicityModel.intensity does not contain alignment Wigner-\(D\) functions:
builder.align_spin = False
non_aligned_model = builder.formulate()
non_aligned_model.intensity
\[\displaystyle \sum_{m_{A}\in\left\{1,-1\right\}} \sum_{m_{0}=0} \sum_{m_{1}=-1/2}^{1/2} \sum_{m_{2}=-1/2}^{1/2}{\left|{A^{01}_{m_{A}, m_{0}, m_{1}, m_{2}} + A^{02}_{m_{A}, m_{0}, m_{1}, m_{2}}}\right|^{2}}\]