D⁰ → K⁰ K⁺ K⁻#
The decay \(D^0 \to K^0K^+K^-\) has a spinless initial and final state, which means that there is no need to align spin with Dalitz-plot decomposition. This notebook shows that the model formulated by ampform is the same as that formulated by AmpForm-DPD. To simplify this comparison, we do not define any dynamics.
Decay definition#
index |
name |
LaTeX |
\(J^P\) |
mass (MeV) |
width (MeV) |
|---|---|---|---|---|---|
0 |
|
\(D^{0}\) |
\(0^-\) |
1,864 |
0 |
1 |
|
\(K^{0}\) |
\(0^-\) |
497 |
0 |
2 |
|
\(K^{-}\) |
\(0^-\) |
493 |
0 |
3 |
|
\(K^{+}\) |
\(0^-\) |
493 |
0 |
name |
LaTeX |
\(J^P\) |
mass (MeV) |
width (MeV) |
|---|---|---|---|---|
|
\(a_{0}(980)^{0}\) |
\(0^+\) |
980 |
75 |
|
\(a_{0}(980)^{-}\) |
\(0^+\) |
980 |
75 |
|
\(a_{0}(980)^{+}\) |
\(0^+\) |
980 |
75 |
|
\(f_{0}(980)\) |
\(0^+\) |
990 |
60 |
\[\begin{split}\begin{array}{c}
D^{0}\left[0^-\right] \to \left(a_{0}(980)^{+}\left[0^+\right] \to K^{0}\left[0^-\right] K^{+}\left[0^-\right]\right) K^{-}\left[0^-\right] \\
D^{0}\left[0^-\right] \to \left(a_{0}(980)^{-}\left[0^+\right] \to K^{0}\left[0^-\right] K^{-}\left[0^-\right]\right) K^{+}\left[0^-\right] \\
D^{0}\left[0^-\right] \to \left(a_{0}(980)^{0}\left[0^+\right] \to K^{-}\left[0^-\right] K^{+}\left[0^-\right]\right) K^{0}\left[0^-\right] \\
D^{0}\left[0^-\right] \to \left(f_{0}(980)\left[0^+\right] \to K^{-}\left[0^-\right] K^{+}\left[0^-\right]\right) K^{0}\left[0^-\right] \\
\end{array}\end{split}\]
Model formulation#
DPD model#
Note that, as opposed to Λc⁺ → pπ⁺K⁻ and J/ψ → K⁰Σ⁺p̅, there are no Wigner-\(d\) functions, because the final state is spinless.
\[\displaystyle \left|{\sum_{\lambda_0^{\prime}=0} \sum_{\lambda_1^{\prime}=0} \sum_{\lambda_2^{\prime}=0} \sum_{\lambda_3^{\prime}=0}{A^{1}_{\lambda_0^{\prime}, \lambda_1^{\prime}, \lambda_2^{\prime}, \lambda_3^{\prime}} + A^{2}_{\lambda_0^{\prime}, \lambda_1^{\prime}, \lambda_2^{\prime}, \lambda_3^{\prime}} + A^{3}_{\lambda_0^{\prime}, \lambda_1^{\prime}, \lambda_2^{\prime}, \lambda_3^{\prime}}}}\right|^{2}\]
\[\begin{split}\begin{array}{rcl}
A^{1}_{0, 0, 0, 0} &=& \sum_{\lambda_{R}=0}{\delta_{0 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{a_{0}(980)^{0}, 0, 0} \mathcal{H}^\mathrm{production}_{a_{0}(980)^{0}, \lambda_{R}, 0} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)} + \sum_{\lambda_{R}=0}{\delta_{0 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{f_{0}(980), 0, 0} \mathcal{H}^\mathrm{production}_{f_{0}(980), \lambda_{R}, 0} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)} \\
A^{2}_{0, 0, 0, 0} &=& \sum_{\lambda_{R}=0}{\delta_{0 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{a_{0}(980)^{+}, 0, 0} \mathcal{H}^\mathrm{production}_{a_{0}(980)^{+}, \lambda_{R}, 0} d^{0}_{\lambda_{R},0}\left(\theta_{31}\right)} \\
A^{3}_{0, 0, 0, 0} &=& \sum_{\lambda_{R}=0}{\delta_{0 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{a_{0}(980)^{-}, 0, 0} \mathcal{H}^\mathrm{production}_{a_{0}(980)^{-}, \lambda_{R}, 0} d^{0}_{\lambda_{R},0}\left(\theta_{12}\right)} \\
\end{array}\end{split}\]
There is an isobar Wigner-\(d\) function, which takes the following helicity angles as argument:
\[\begin{split}\begin{array}{rcl}
\theta_{23} &=& \operatorname{acos}{\left(\frac{2 \sigma_{1} \left(- m_{1}^{2} - m_{2}^{2} + \sigma_{3}\right) - \left(m_{0}^{2} - m_{1}^{2} - \sigma_{1}\right) \left(m_{2}^{2} - m_{3}^{2} + \sigma_{1}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{1}^{2}, \sigma_{1}\right)} \sqrt{\lambda\left(\sigma_{1}, m_{2}^{2}, m_{3}^{2}\right)}} \right)} \\
\theta_{31} &=& \operatorname{acos}{\left(\frac{2 \sigma_{2} \left(- m_{2}^{2} - m_{3}^{2} + \sigma_{1}\right) - \left(m_{0}^{2} - m_{2}^{2} - \sigma_{2}\right) \left(- m_{1}^{2} + m_{3}^{2} + \sigma_{2}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{2}^{2}, \sigma_{2}\right)} \sqrt{\lambda\left(\sigma_{2}, m_{3}^{2}, m_{1}^{2}\right)}} \right)} \\
\theta_{12} &=& \operatorname{acos}{\left(\frac{2 \sigma_{3} \left(- m_{1}^{2} - m_{3}^{2} + \sigma_{2}\right) - \left(m_{0}^{2} - m_{3}^{2} - \sigma_{3}\right) \left(m_{1}^{2} - m_{2}^{2} + \sigma_{3}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{3}^{2}, \sigma_{3}\right)} \sqrt{\lambda\left(\sigma_{3}, m_{1}^{2}, m_{2}^{2}\right)}} \right)} \\
\end{array}\end{split}\]
AmpForm model#
AmpForm does not formulate alignment Wigner-\(D\) functions. For the case of this spinless final state, this means the intensity is the same as that of the DPD model.
\[\displaystyle \sum_{m_{A}=0} \sum_{m_{0}=0} \sum_{m_{1}=0} \sum_{m_{2}=0}{\left|{A^{01}_{m_{A}, m_{0}, m_{1}, m_{2}} + A^{02}_{m_{A}, m_{0}, m_{1}, m_{2}} + A^{12}_{m_{A}, m_{0}, m_{1}, m_{2}}}\right|^{2}}\]
\[\begin{split}\begin{array}{rcl}
A^{01}_{0, 0, 0, 0} &=& C_{D^{0} \to K^{+}_{0} {a_{0}(980)^{-}}_{0}; a_{0}(980)^{-} \to K^{-}_{0} K^{0}_{0}} D^{0}_{0,0}\left(- \phi_{01},\theta_{01},0\right) D^{0}_{0,0}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) \\
A^{02}_{0, 0, 0, 0} &=& C_{D^{0} \to K^{-}_{0} {a_{0}(980)^{+}}_{0}; a_{0}(980)^{+} \to K^{+}_{0} K^{0}_{0}} D^{0}_{0,0}\left(- \phi_{02},\theta_{02},0\right) D^{0}_{0,0}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) \\
A^{12}_{0, 0, 0, 0} &=& C_{D^{0} \to K^{0}_{0} {a_{0}(980)^{0}}_{0}; a_{0}(980)^{0} \to K^{+}_{0} K^{-}_{0}} D^{0}_{0,0}\left(- \phi_{0},\theta_{0},0\right) D^{0}_{0,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) + C_{D^{0} \to K^{0}_{0} {f_{0}(980)}_{0}; f_{0}(980) \to K^{+}_{0} K^{-}_{0}} D^{0}_{0,0}\left(- \phi_{0},\theta_{0},0\right) D^{0}_{0,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) \\
\end{array}\end{split}\]
\[\begin{split}\begin{array}{rcl}
m_{01} &=& m_{{p}_{01}} \\
m_{02} &=& m_{{p}_{02}} \\
m_{12} &=& m_{{p}_{12}} \\
\phi_{0} &=& \phi\left({p}_{12}\right) \\
\phi^{01}_{0} &=& \phi\left(\boldsymbol{B_z}\left(\frac{\left|\vec{{p}_{01}}\right|}{E\left({p}_{01}\right)}\right) \boldsymbol{R_y}\left(- \theta\left({p}_{01}\right)\right) \boldsymbol{R_z}\left(- \phi\left({p}_{01}\right)\right) p_{0}\right) \\
\phi^{02}_{0} &=& \phi\left(\boldsymbol{B_z}\left(\frac{\left|\vec{{p}_{02}}\right|}{E\left({p}_{02}\right)}\right) \boldsymbol{R_y}\left(- \theta\left({p}_{02}\right)\right) \boldsymbol{R_z}\left(- \phi\left({p}_{02}\right)\right) p_{0}\right) \\
\phi_{01} &=& \phi\left({p}_{01}\right) \\
\phi^{12}_{1} &=& \phi\left(\boldsymbol{B_z}\left(\frac{\left|\vec{{p}_{12}}\right|}{E\left({p}_{12}\right)}\right) \boldsymbol{R_y}\left(- \theta\left({p}_{12}\right)\right) \boldsymbol{R_z}\left(- \phi\left({p}_{12}\right)\right) p_{1}\right) \\
\phi_{02} &=& \phi\left({p}_{02}\right) \\
\theta_{0} &=& \theta\left({p}_{12}\right) \\
\theta^{01}_{0} &=& \theta\left(\boldsymbol{B_z}\left(\frac{\left|\vec{{p}_{01}}\right|}{E\left({p}_{01}\right)}\right) \boldsymbol{R_y}\left(- \theta\left({p}_{01}\right)\right) \boldsymbol{R_z}\left(- \phi\left({p}_{01}\right)\right) p_{0}\right) \\
\theta^{02}_{0} &=& \theta\left(\boldsymbol{B_z}\left(\frac{\left|\vec{{p}_{02}}\right|}{E\left({p}_{02}\right)}\right) \boldsymbol{R_y}\left(- \theta\left({p}_{02}\right)\right) \boldsymbol{R_z}\left(- \phi\left({p}_{02}\right)\right) p_{0}\right) \\
\theta_{01} &=& \theta\left({p}_{01}\right) \\
\theta^{12}_{1} &=& \theta\left(\boldsymbol{B_z}\left(\frac{\left|\vec{{p}_{12}}\right|}{E\left({p}_{12}\right)}\right) \boldsymbol{R_y}\left(- \theta\left({p}_{12}\right)\right) \boldsymbol{R_z}\left(- \phi\left({p}_{12}\right)\right) p_{1}\right) \\
\theta_{02} &=& \theta\left({p}_{02}\right) \\
\end{array}\end{split}\]
Phase space sample#
def generate_phase_space(reaction: ReactionInfo, size: int) -> dict[str, jnp.ndarray]:
rng = TFUniformRealNumberGenerator(seed=0)
phsp_generator = TFPhaseSpaceGenerator(
initial_state_mass=reaction.initial_state[-1].mass,
final_state_masses={i: p.mass for i, p in reaction.final_state.items()},
)
return phsp_generator.generate(size, rng)
phsp = generate_phase_space(AMPFORM_MODEL.reaction_info, size=100_000)
ampform_phsp = ampform_transformer(phsp)
dpd_phsp = dpd_transformer(phsp)
WARNING: All log messages before absl::InitializeLog() is called are written to STDERR
E0000 00:00:1759918688.407197 3720 cuda_dnn.cc:8579] Unable to register cuDNN factory: Attempting to register factory for plugin cuDNN when one has already been registered
E0000 00:00:1759918688.411282 3720 cuda_blas.cc:1407] Unable to register cuBLAS factory: Attempting to register factory for plugin cuBLAS when one has already been registered
W0000 00:00:1759918688.422754 3720 computation_placer.cc:177] computation placer already registered. Please check linkage and avoid linking the same target more than once.
W0000 00:00:1759918688.422764 3720 computation_placer.cc:177] computation placer already registered. Please check linkage and avoid linking the same target more than once.
W0000 00:00:1759918688.422766 3720 computation_placer.cc:177] computation placer already registered. Please check linkage and avoid linking the same target more than once.
W0000 00:00:1759918688.422767 3720 computation_placer.cc:177] computation placer already registered. Please check linkage and avoid linking the same target more than once.
WARNING: All log messages before absl::InitializeLog() is called are written to STDERR
I0000 00:00:1759918696.312142 3795 service.cc:152] XLA service 0x7f945c18c820 initialized for platform Host (this does not guarantee that XLA will be used). Devices:
I0000 00:00:1759918696.312179 3795 service.cc:160] StreamExecutor device (0): Host, Default Version
I0000 00:00:1759918696.379613 3796 device_compiler.h:188] Compiled cluster using XLA! This line is logged at most once for the lifetime of the process.
Convert to numerical functions#
ampform_intensity_expr = cached.unfold(AMPFORM_MODEL)
dpd_intensity_expr = cached.unfold(DPD_MODEL)
ampform_func = cached.lambdify(
ampform_intensity_expr,
parameters=AMPFORM_MODEL.parameter_defaults,
)
dpd_func = cached.lambdify(
dpd_intensity_expr,
parameters=DPD_MODEL.parameter_defaults,
)