Spin alignment

Spin alignment

import logging

import graphviz
import qrules
import sympy as sp
from IPython.display import Math

import ampform

LOGGER = logging.getLogger()
LOGGER.setLevel(logging.ERROR)

As described in [TR-015] Spin alignment implementation, the ‘standard’ helicity formalism is not suited for state transitions that have different decay topologies. For this reason, the HelicityAmplitudeBuilder 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",
)
dot = qrules.io.asdot(reaction, collapse_graphs=True, size=4)
graphviz.Source(dot)
../../_images/spin-alignment_7_0.svg
\[\displaystyle \sum_{m_{A}\in\left\{1,-1\right\}} \sum_{m_{0}=0} \sum_{m_{1}=-1/2}^{1/2} \sum_{m_{2}=-1/2}^{1/2}{\left|{\sum_{\lambda^{01}_{0}=0} \sum_{\mu^{01}_{0}=0} \sum_{\nu^{01}_{0}=0} \sum_{\lambda^{01}_{1}=-1/2}^{1/2} \sum_{\mu^{01}_{1}=-1/2}^{1/2} \sum_{\nu^{01}_{1}=-1/2}^{1/2} \sum_{\lambda^{01}_{2}=-1/2}^{1/2}{{A^{01}}_{m_{A},\lambda^{01}_{0},- \lambda^{01}_{1},- \lambda^{01}_{2}} D^{0}_{m_{0},\nu^{01}_{0}}\left(\alpha^{01}_{0},\beta^{01}_{0},\gamma^{01}_{0}\right) D^{0}_{\mu^{01}_{0},\lambda^{01}_{0}}\left(\phi^{01}_{0},\theta^{01}_{0},0\right) D^{0}_{\nu^{01}_{0},\mu^{01}_{0}}\left(\phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{m_{1},\nu^{01}_{1}}\left(\alpha^{01}_{1},\beta^{01}_{1},\gamma^{01}_{1}\right) D^{\frac{1}{2}}_{m_{2},\lambda^{01}_{2}}\left(\phi_{01},\theta_{01},0\right) D^{\frac{1}{2}}_{\mu^{01}_{1},\lambda^{01}_{1}}\left(\phi^{01}_{0},\theta^{01}_{0},0\right) D^{\frac{1}{2}}_{\nu^{01}_{1},\mu^{01}_{1}}\left(\phi_{01},\theta_{01},0\right)} + \sum_{\lambda^{02}_{0}=0} \sum_{\mu^{02}_{0}=0} \sum_{\nu^{02}_{0}=0} \sum_{\lambda^{02}_{1}=-1/2}^{1/2} \sum_{\lambda^{02}_{2}=-1/2}^{1/2} \sum_{\mu^{02}_{2}=-1/2}^{1/2} \sum_{\nu^{02}_{2}=-1/2}^{1/2}{{A^{02}}_{m_{A},\lambda^{02}_{0},- \lambda^{02}_{1},- \lambda^{02}_{2}} D^{0}_{m_{0},\nu^{02}_{0}}\left(\alpha^{02}_{0},\beta^{02}_{0},\gamma^{02}_{0}\right) D^{0}_{\mu^{02}_{0},\lambda^{02}_{0}}\left(\phi^{02}_{0},\theta^{02}_{0},0\right) D^{0}_{\nu^{02}_{0},\mu^{02}_{0}}\left(\phi_{02},\theta_{02},0\right) D^{\frac{1}{2}}_{m_{1},\lambda^{02}_{1}}\left(\phi_{02},\theta_{02},0\right) D^{\frac{1}{2}}_{m_{2},\nu^{02}_{2}}\left(\alpha^{02}_{2},\beta^{02}_{2},\gamma^{02}_{2}\right) D^{\frac{1}{2}}_{\mu^{02}_{2},\lambda^{02}_{2}}\left(\phi^{02}_{0},\theta^{02}_{0},0\right) D^{\frac{1}{2}}_{\nu^{02}_{2},\mu^{02}_{2}}\left(\phi_{02},\theta_{02},0\right)}}\right|^{2}}\]
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.

Spin alignment can be switched off or on by setting HelicityAmplitudeBuilder.align_spin:

builder.align_spin = False
non_aligned_model = builder.formulate()
non_aligned_model.intensity
\[\displaystyle \sum_{m_{A}\in\left\{1,-1\right\}} \sum_{m_{0}=0} \sum_{m_{1}=-1/2}^{1/2} \sum_{m_{2}=-1/2}^{1/2}{\left|{{A^{01}}_{m_{A},m_{0},m_{1},m_{2}} + {A^{02}}_{m_{A},m_{0},m_{1},m_{2}}}\right|^{2}}\]