Spin alignment with data#
Phase space sample#
from tensorwaves.data import TFPhaseSpaceGenerator, TFUniformRealNumberGenerator
phsp_generator = TFPhaseSpaceGenerator(
initial_state_mass=PDG["Lambda(c)+"].mass,
final_state_masses={
0: PDG["p"].mass,
1: PDG["K-"].mass,
2: PDG["pi+"].mass,
},
)
rng = TFUniformRealNumberGenerator(seed=0)
phsp_momenta = phsp_generator.generate(1_000_000, rng)
Generate transitions#
reaction = qrules.generate_transitions(
initial_state=("Lambda(c)+", [-0.5, +0.5]),
final_state=["p", "K-", "pi+"],
formalism="helicity",
particle_db=particle_db,
)
Distribution without alignment#
Amplitude model formulated following Appendix C:
import ampform
from ampform.dynamics.builder import RelativisticBreitWignerBuilder
builder = ampform.get_builder(reaction)
builder.align_spin = False
builder.stable_final_state_ids = list(reaction.final_state)
builder.scalar_initial_state_mass = True
bw_builder = RelativisticBreitWignerBuilder()
for name in reaction.get_intermediate_particles().names:
builder.set_dynamics(name, bw_builder)
standard_model = builder.formulate()
standard_model.intensity
\[\displaystyle \sum_{m_{A}=-1/2}^{1/2} \sum_{m_{0}=-1/2}^{1/2} \sum_{m_{1}=0} \sum_{m_{2}=0}{\left|{{A^{01}}_{m_{A},m_{0},m_{1},m_{2}} + {A^{02}}_{m_{A},m_{0},m_{1},m_{2}} + {A^{12}}_{m_{A},m_{0},m_{1},m_{2}}}\right|^{2}}\]
Importing the parameter values given by Table 1:
\[\begin{split}\displaystyle \begin{array}{lc}
C_{\Lambda_{c}^{+} \to K^*_{0} p_{+1/2}; K^* \to K^{-}_{0} \pi^{+}_{0}} & 1 \\
C_{\Lambda_{c}^{+} \to K^*_{+1} p_{+1/2}; K^* \to K^{-}_{0} \pi^{+}_{0}} & 0.5+0.5i \\
C_{\Lambda_{c}^{+} \to K^*_{-1} p_{-1/2}; K^* \to K^{-}_{0} \pi^{+}_{0}} & 1i \\
C_{\Lambda_{c}^{+} \to K^*_{0} p_{-1/2}; K^* \to K^{-}_{0} \pi^{+}_{0}} & -0.5-0.5i \\
m_{K^*} & 0.9 \\
\Gamma_{K^*} & 0.2 \\
C_{\Lambda_{c}^{+} \to \Lambda^*_{-1/2} \pi^{+}_{0}; \Lambda^* \to K^{-}_{0} p_{+1/2}} & 1i \\
C_{\Lambda_{c}^{+} \to \Lambda^*_{+1/2} \pi^{+}_{0}; \Lambda^* \to K^{-}_{0} p_{+1/2}} & 0.8-0.4i \\
m_{\Lambda^*} & 1.6 \\
\Gamma_{\Lambda^*} & 0.2 \\
C_{\Lambda_{c}^{+} \to \Delta^*_{+1/2} K^{-}_{0}; \Delta^* \to p_{+1/2} \pi^{+}_{0}} & 0.6-0.4i \\
C_{\Lambda_{c}^{+} \to \Delta^*_{-1/2} K^{-}_{0}; \Delta^* \to p_{+1/2} \pi^{+}_{0}} & 0.1i \\
m_{\Delta^*} & 1.4 \\
\Gamma_{\Delta^*} & 0.2 \\
\end{array}\end{split}\]
Generate data#
Warning
It takes several minutes to lambdify the full expression and expressions for the Wigner rotation angles.
Spin alignment sum#
Now, with the spin alignment sum from ampform#245 inserted:
builder.align_spin = True
aligned_model = builder.formulate()
set_coefficients(aligned_model)
aligned_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)} + \sum_{\lambda^{12}_{0}=-1/2}^{1/2} \sum_{\lambda^{12}_{1}=0} \sum_{\mu^{12}_{1}=0} \sum_{\nu^{12}_{1}=0} \sum_{\lambda^{12}_{2}=0} \sum_{\mu^{12}_{2}=0} \sum_{\nu^{12}_{2}=0}{{A^{12}}_{m_{A},\lambda^{12}_{0},\lambda^{12}_{1},- \lambda^{12}_{2}} D^{0}_{m_{1},\nu^{12}_{1}}\left(\alpha^{12}_{1},\beta^{12}_{1},\gamma^{12}_{1}\right) D^{0}_{m_{2},\nu^{12}_{2}}\left(\alpha^{12}_{2},\beta^{12}_{2},\gamma^{12}_{2}\right) D^{0}_{\mu^{12}_{1},\lambda^{12}_{1}}\left(\phi^{12}_{1},\theta^{12}_{1},0\right) D^{0}_{\mu^{12}_{2},\lambda^{12}_{2}}\left(\phi^{12}_{1},\theta^{12}_{1},0\right) D^{0}_{\nu^{12}_{1},\mu^{12}_{1}}\left(\phi_{0},\theta_{0},0\right) D^{0}_{\nu^{12}_{2},\mu^{12}_{2}}\left(\phi_{0},\theta_{0},0\right) D^{\frac{1}{2}}_{m_{0},\lambda^{12}_{0}}\left(\phi_{0},\theta_{0},0\right)}}\right|^{2}}\]
Compare with Figure 2. Note that the distributions differ close to threshold, because the distributions in the paper are produced with form factors and an energy-dependent width.