D⁰ → K⁰ K⁺ K⁻

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#

../_images/ffbb15ecd139f3904de55d1401a55060b0bd75593acdc167dfeac1d171f56321.svg

index	name	LaTeX	\(J^P\)	mass (MeV)
0	`D0`	\(D^{0}\)	\(0^-\)	1,864
1	`K0`	\(K^{0}\)	\(0^-\)	497
2	`K-`	\(K^{-}\)	\(0^-\)	493
3	`K+`	\(K^{+}\)	\(0^-\)	493

name	LaTeX	\(J^P\)	mass (MeV)	width (MeV)
`a(0)(980)+`	\(a_{0}(980)^{+}\)	\(0^+\)	980	75
`a(0)(980)-`	\(a_{0}(980)^{-}\)	\(0^+\)	980	75
`a(0)(980)0`	\(a_{0}(980)^{0}\)	\(0^+\)	980	75
`f(0)(980)`	\(f_{0}(980)\)	\(0^+\)	990	60
`pi(1)(1400)+`	\(\pi_{1}(1400)^{+}\)	\(1^-\)	1,354	330
`pi(1)(1400)-`	\(\pi_{1}(1400)^{-}\)	\(1^-\)	1,354	330
`pi(1)(1400)0`	\(\pi_{1}(1400)^{0}\)	\(1^-\)	1,354	330

\[\begin{split}\begin{array}{c} D^{0} \to \left(a_{0}(980)^{+} \to K^{0} K^{+}\right) K^{-} \\ D^{0} \to \left(a_{0}(980)^{-} \to K^{0} K^{-}\right) K^{+} \\ D^{0} \to \left(a_{0}(980)^{0} \to K^{-} K^{+}\right) K^{0} \\ D^{0} \to \left(f_{0}(980) \to K^{-} K^{+}\right) K^{0} \\ D^{0} \to \left(\pi_{1}(1400)^{+} \to K^{0} K^{+}\right) K^{-} \\ D^{0} \to \left(\pi_{1}(1400)^{-} \to K^{0} K^{-}\right) K^{+} \\ D^{0} \to \left(\pi_{1}(1400)^{0} \to K^{-} K^{+}\right) K^{0} \\ \end{array}\end{split}\]

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.

\[\displaystyle \left|{\sum_{\lambda_0^{\prime}=0} \sum_{\lambda_1^{\prime}=0} \sum_{\lambda_2^{\prime}=0} \sum_{\lambda_3^{\prime}=0}{A^{1}_{\lambda_0^{\prime}, \lambda_1^{\prime}, \lambda_2^{\prime}, \lambda_3^{\prime}} + A^{2}_{\lambda_0^{\prime}, \lambda_1^{\prime}, \lambda_2^{\prime}, \lambda_3^{\prime}} + A^{3}_{\lambda_0^{\prime}, \lambda_1^{\prime}, \lambda_2^{\prime}, \lambda_3^{\prime}}}}\right|^{2}\]

\[\begin{split}\begin{array}{rcl} A^{1}_{0, 0, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{\delta_{0 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{\pi_{1}(1400)^{0}, 0, 0} \mathcal{H}^\mathrm{production}_{\pi_{1}(1400)^{0}, \lambda_{R}, 0} d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)} + \sum_{\lambda_{R}=0}{\delta_{0 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{a_{0}(980)^{0}, 0, 0} \mathcal{H}^\mathrm{production}_{a_{0}(980)^{0}, \lambda_{R}, 0} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)} + \sum_{\lambda_{R}=0}{\delta_{0 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{f_{0}(980), 0, 0} \mathcal{H}^\mathrm{production}_{f_{0}(980), \lambda_{R}, 0} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)} \\ A^{2}_{0, 0, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{\delta_{0 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{\pi_{1}(1400)^{+}, 0, 0} \mathcal{H}^\mathrm{production}_{\pi_{1}(1400)^{+}, \lambda_{R}, 0} d^{1}_{\lambda_{R},0}\left(\theta_{31}\right)} + \sum_{\lambda_{R}=0}{\delta_{0 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{a_{0}(980)^{+}, 0, 0} \mathcal{H}^\mathrm{production}_{a_{0}(980)^{+}, \lambda_{R}, 0} d^{0}_{\lambda_{R},0}\left(\theta_{31}\right)} \\ A^{3}_{0, 0, 0, 0} &=& \sum_{\lambda_{R}=-1}^{1}{\delta_{0 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{\pi_{1}(1400)^{-}, 0, 0} \mathcal{H}^\mathrm{production}_{\pi_{1}(1400)^{-}, \lambda_{R}, 0} d^{1}_{\lambda_{R},0}\left(\theta_{12}\right)} + \sum_{\lambda_{R}=0}{\delta_{0 \lambda_{R}} \mathcal{H}^\mathrm{decay}_{a_{0}(980)^{-}, 0, 0} \mathcal{H}^\mathrm{production}_{a_{0}(980)^{-}, \lambda_{R}, 0} d^{0}_{\lambda_{R},0}\left(\theta_{12}\right)} \\ \end{array}\end{split}\]

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.

\[\displaystyle \sum_{m_{A}=0} \sum_{m_{0}=0} \sum_{m_{1}=0} \sum_{m_{2}=0}{\left|{A^{01}_{m_{A}, m_{0}, m_{1}, m_{2}} + A^{02}_{m_{A}, m_{0}, m_{1}, m_{2}} + A^{12}_{m_{A}, m_{0}, m_{1}, m_{2}}}\right|^{2}}\]

\[\begin{split}\begin{array}{rcl} A^{01}_{0, 0, 0, 0} &=& C_{D^{0} \to K^{+}_{0} {\pi_{1}(1400)^{-}}_{0}; \pi_{1}(1400)^{-} \to K^{-}_{0} K^{0}_{0}} D^{0}_{0,0}\left(- \phi_{01},\theta_{01},0\right) D^{1}_{0,0}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) + C_{D^{0} \to K^{+}_{0} {a_{0}(980)^{-}}_{0}; a_{0}(980)^{-} \to K^{-}_{0} K^{0}_{0}} D^{0}_{0,0}\left(- \phi_{01},\theta_{01},0\right) D^{0}_{0,0}\left(- \phi^{01}_{0},\theta^{01}_{0},0\right) \\ A^{02}_{0, 0, 0, 0} &=& C_{D^{0} \to K^{-}_{0} {\pi_{1}(1400)^{+}}_{0}; \pi_{1}(1400)^{+} \to K^{+}_{0} K^{0}_{0}} D^{0}_{0,0}\left(- \phi_{02},\theta_{02},0\right) D^{1}_{0,0}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) + C_{D^{0} \to K^{-}_{0} {a_{0}(980)^{+}}_{0}; a_{0}(980)^{+} \to K^{+}_{0} K^{0}_{0}} D^{0}_{0,0}\left(- \phi_{02},\theta_{02},0\right) D^{0}_{0,0}\left(- \phi^{02}_{0},\theta^{02}_{0},0\right) \\ A^{12}_{0, 0, 0, 0} &=& C_{D^{0} \to K^{0}_{0} {\pi_{1}(1400)^{0}}_{0}; \pi_{1}(1400)^{0} \to K^{+}_{0} K^{-}_{0}} D^{0}_{0,0}\left(- \phi_{0},\theta_{0},0\right) D^{1}_{0,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) + C_{D^{0} \to K^{0}_{0} {a_{0}(980)^{0}}_{0}; a_{0}(980)^{0} \to K^{+}_{0} K^{-}_{0}} D^{0}_{0,0}\left(- \phi_{0},\theta_{0},0\right) D^{0}_{0,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) + C_{D^{0} \to K^{0}_{0} {f_{0}(980)}_{0}; f_{0}(980) \to K^{+}_{0} K^{-}_{0}} D^{0}_{0,0}\left(- \phi_{0},\theta_{0},0\right) D^{0}_{0,0}\left(- \phi^{12}_{1},\theta^{12}_{1},0\right) \\ \end{array}\end{split}\]

\[\begin{split}\begin{array}{rcl} m_{01} &=& m_{{p}_{01}} \\ m_{02} &=& m_{{p}_{02}} \\ m_{12} &=& m_{{p}_{12}} \\ \phi_{0} &=& \phi\left({p}_{12}\right) \\ \phi^{01}_{0} &=& \phi\left(\boldsymbol{B_z}\left(\frac{\left|\vec{{p}_{01}}\right|}{E\left({p}_{01}\right)}\right) \boldsymbol{R_y}\left(- \theta\left({p}_{01}\right)\right) \boldsymbol{R_z}\left(- \phi\left({p}_{01}\right)\right) p_{0}\right) \\ \phi^{02}_{0} &=& \phi\left(\boldsymbol{B_z}\left(\frac{\left|\vec{{p}_{02}}\right|}{E\left({p}_{02}\right)}\right) \boldsymbol{R_y}\left(- \theta\left({p}_{02}\right)\right) \boldsymbol{R_z}\left(- \phi\left({p}_{02}\right)\right) p_{0}\right) \\ \phi_{01} &=& \phi\left({p}_{01}\right) \\ \phi^{12}_{1} &=& \phi\left(\boldsymbol{B_z}\left(\frac{\left|\vec{{p}_{12}}\right|}{E\left({p}_{12}\right)}\right) \boldsymbol{R_y}\left(- \theta\left({p}_{12}\right)\right) \boldsymbol{R_z}\left(- \phi\left({p}_{12}\right)\right) p_{1}\right) \\ \phi_{02} &=& \phi\left({p}_{02}\right) \\ \theta_{0} &=& \theta\left({p}_{12}\right) \\ \theta^{01}_{0} &=& \theta\left(\boldsymbol{B_z}\left(\frac{\left|\vec{{p}_{01}}\right|}{E\left({p}_{01}\right)}\right) \boldsymbol{R_y}\left(- \theta\left({p}_{01}\right)\right) \boldsymbol{R_z}\left(- \phi\left({p}_{01}\right)\right) p_{0}\right) \\ \theta^{02}_{0} &=& \theta\left(\boldsymbol{B_z}\left(\frac{\left|\vec{{p}_{02}}\right|}{E\left({p}_{02}\right)}\right) \boldsymbol{R_y}\left(- \theta\left({p}_{02}\right)\right) \boldsymbol{R_z}\left(- \phi\left({p}_{02}\right)\right) p_{0}\right) \\ \theta_{01} &=& \theta\left({p}_{01}\right) \\ \theta^{12}_{1} &=& \theta\left(\boldsymbol{B_z}\left(\frac{\left|\vec{{p}_{12}}\right|}{E\left({p}_{12}\right)}\right) \boldsymbol{R_y}\left(- \theta\left({p}_{12}\right)\right) \boldsymbol{R_z}\left(- \phi\left({p}_{12}\right)\right) p_{1}\right) \\ \theta_{02} &=& \theta\left({p}_{02}\right) \\ \end{array}\end{split}\]

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
I0000 00:00:1714472689.973854    3254 service.cc:145] XLA service 0x7f4db418cab0 initialized for platform Host (this does not guarantee that XLA will be used). Devices:
I0000 00:00:1714472689.973904    3254 service.cc:153]   StreamExecutor device (0): Host, Default Version
I0000 00:00:1714472689.997112    3254 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#

def unfold_intensity(model: HelicityModel) -> sp.Expr:
    unfolded_intensity = perform_cached_doit(model.intensity)
    unfolded_amplitudes = {
        symbol: perform_cached_doit(expr) for symbol, expr in model.amplitudes.items()
    }
    return unfolded_intensity.xreplace(unfolded_amplitudes)


ampform_intensity_expr = unfold_intensity(AMPFORM_MODEL)
dpd_intensity_expr = unfold_intensity(DPD_MODEL)

ampform_func = perform_cached_lambdify(
    ampform_intensity_expr,
    parameters=AMPFORM_MODEL.parameter_defaults,
)
dpd_func = perform_cached_lambdify(
    dpd_intensity_expr,
    parameters=DPD_MODEL.parameter_defaults,
)

Visualization#

Define widget UI Hide code cell source

def create_sliders() -> dict[str, ToggleButtons]:
    all_parameters = dict(AMPFORM_MODEL.parameter_defaults.items())
    all_parameters.update(DPD_MODEL.parameter_defaults)
    sliders = {}
    for symbol, value in all_parameters.items():
        value = "+1"
        if (
            symbol.name.startswith(R"\mathcal{H}^\mathrm{decay}") and "+" in symbol.name
        ) and any(s in symbol.name for s in ["{1}", "*", "rho"]):
            value = "-1"
        sliders[symbol.name] = ToggleButtons(
            description=Rf"\({sp.latex(symbol)}\)",
            options=["-1", "0", "+1"],
            value=value,
            continuous_update=False,
        )
    return sliders


def to_unicode(particle: Particle) -> str:
    unicode = particle.name
    unicode = unicode.replace("pi", "π")
    unicode = unicode.replace("rho", "p")
    unicode = unicode.replace("Sigma", "Σ")
    unicode = unicode.replace("~", "")
    unicode = unicode.replace("Σ", "~Σ")
    unicode = unicode.replace("+", "⁺")
    unicode = unicode.replace("-", "⁻")
    unicode = unicode.replace("(0)", "₀")
    unicode = unicode.replace("(1)", "₁")
    return unicode.replace(")0", ")⁰")


sliders = create_sliders()
resonance_selector = SelectMultiple(
    description="Resonance",
    options={to_unicode(p): p.latex for p in resonances},
    value=[resonances[0].latex, resonances[1].latex],
    layout=Layout(
        height=f"{14 * (len(resonances) + 1)}pt",
        width="auto",
    ),
)
hide_expressions = Checkbox(description="Hide expressions", value=True)
simplify_expressions = Checkbox(description="Simplify", value=True)
ipywidgets.link((hide_expressions, "value"), (simplify_expressions, "disabled"))

package_names = ("AmpForm", "AmpForm-DPD")
ui = HBox([
    VBox([resonance_selector, hide_expressions, simplify_expressions]),
    Tab(
        children=[
            Accordion(
                children=[
                    GridBox([
                        sliders[key]
                        for key in sorted(sliders)
                        if p.latex in key
                        if (
                            key[0] in {"C", "H"}
                            if package == "AmpForm"
                            else key.startswith(R"\mathcal{H}")
                        )
                    ])
                    for package in package_names
                ],
                selected_index=1,
                titles=package_names,
            )
            for p in resonances
        ],
        titles=[to_unicode(p) for p in resonances],
    ),
])