J/ψ → π⁰ π⁺ π⁻

Contents

J/ψ → π⁰ π⁺ π⁻#

The decay \(J/\psi \to \pi^0 \pi^+ \pi^-\) has an initial state with spin, but has a spinless final state. This is a follow-up to D⁰ → K⁰ K⁺ K⁻, where there were no alignment Wigner-\(D\) functions.

Decay definition#

Model formulation#

DPD model#

There is an isobar Wigner-\(d\) function, which takes the following helicity angles as argument:

AmpForm model#

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)

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],
    ),
])