Model serialization

Model serialization#

This notebooks illustrates the use of the ampform_dpd.io.serialization module, which can be used to build amplitude models from the amplitude-serialization initiative.

Warning

The ampform_dpd.io.serialization module is a preview feature.

Import model#

with open("Lc2ppiK.json") as stream:
    MODEL_DEFINITION = json.load(stream)

Construct `ThreeBodyDecay`#

DECAY = to_decay(MODEL_DEFINITION, to_latex=to_latex)
Math(aslatex(DECAY, with_jp=True))

\[\begin{split}\displaystyle \begin{array}{c} \Lambda_c^+\left[J=\frac{1}{2}\right] \to \left(D(1232)\left[J=\frac{3}{2}\right] \to p\left[J=\frac{1}{2}\right] \pi^+\left[J=0\right]\right) K^-\left[J=0\right] \\ \Lambda_c^+\left[J=\frac{1}{2}\right] \to \left(D(1600)\left[J=\frac{3}{2}\right] \to p\left[J=\frac{1}{2}\right] \pi^+\left[J=0\right]\right) K^-\left[J=0\right] \\ \Lambda_c^+\left[J=\frac{1}{2}\right] \to \left(D(1700)\left[J=\frac{3}{2}\right] \to p\left[J=\frac{1}{2}\right] \pi^+\left[J=0\right]\right) K^-\left[J=0\right] \\ \Lambda_c^+\left[J=\frac{1}{2}\right] \to \left(K(1430)\left[J=0\right] \to \pi^+\left[J=0\right] K^-\left[J=0\right]\right) p\left[J=\frac{1}{2}\right] \\ \Lambda_c^+\left[J=\frac{1}{2}\right] \to \left(K(700)\left[J=0\right] \to \pi^+\left[J=0\right] K^-\left[J=0\right]\right) p\left[J=\frac{1}{2}\right] \\ \Lambda_c^+\left[J=\frac{1}{2}\right] \to \left(K(892)\left[J=1\right] \to \pi^+\left[J=0\right] K^-\left[J=0\right]\right) p\left[J=\frac{1}{2}\right] \\ \Lambda_c^+\left[J=\frac{1}{2}\right] \to \left(\Lambda(1405)\left[J=\frac{1}{2}\right] \to K^-\left[J=0\right] p\left[J=\frac{1}{2}\right]\right) \pi^+\left[J=0\right] \\ \Lambda_c^+\left[J=\frac{1}{2}\right] \to \left(\Lambda(1520)\left[J=\frac{3}{2}\right] \to K^-\left[J=0\right] p\left[J=\frac{1}{2}\right]\right) \pi^+\left[J=0\right] \\ \Lambda_c^+\left[J=\frac{1}{2}\right] \to \left(\Lambda(1600)\left[J=\frac{1}{2}\right] \to K^-\left[J=0\right] p\left[J=\frac{1}{2}\right]\right) \pi^+\left[J=0\right] \\ \Lambda_c^+\left[J=\frac{1}{2}\right] \to \left(\Lambda(1670)\left[J=\frac{1}{2}\right] \to K^-\left[J=0\right] p\left[J=\frac{1}{2}\right]\right) \pi^+\left[J=0\right] \\ \Lambda_c^+\left[J=\frac{1}{2}\right] \to \left(\Lambda(1690)\left[J=\frac{3}{2}\right] \to K^-\left[J=0\right] p\left[J=\frac{1}{2}\right]\right) \pi^+\left[J=0\right] \\ \Lambda_c^+\left[J=\frac{1}{2}\right] \to \left(\Lambda(2000)\left[J=\frac{1}{2}\right] \to K^-\left[J=0\right] p\left[J=\frac{1}{2}\right]\right) \pi^+\left[J=0\right] \\ \end{array}\end{split}\]

Dynamics#

Vertices#

Blatt-Weisskopf form factor#

\[\begin{split}\displaystyle \begin{array}{rcl} \mathcal{F}_{L}\left(s, m_{1}, m_{2}\right) &=& \mathcal{F}_{L}\left(s, m_{1}, m_{2}\right) \\ B_{L}^2\left(z\right) &=& \frac{\left|{h_{L}^{(1)}\left(1\right)}\right|^{2}}{z \left|{h_{L}^{(1)}\left(\sqrt{z}\right)}\right|^{2}} \\ q^2\left(s\right) &=& \frac{\left(s - \left(m_{1} - m_{2}\right)^{2}\right) \left(s - \left(m_{1} + m_{2}\right)^{2}\right)}{4 s} \\ \end{array}\end{split}\]

ff_L1520 = formulate_form_factor(
    vertex=CHAIN_DEFS[2]["vertices"][0],
    model=MODEL_DEFINITION,
)
Math(aslatex(ff_L1520))

\[\begin{split}\displaystyle \begin{array}{rcl} \mathcal{F}_{1}\left(m_{0}, \sigma_{2}, m_{2}\right) &=& \mathcal{F}_{1}\left(m_{0}, \sigma_{2}, m_{2}\right) \\ R_{Lc} &=& 5.0 \\ \end{array}\end{split}\]

Propagators#

Breit-Wigner#

\[\begin{split}\displaystyle \begin{array}{rcl} \mathcal{R}^\mathrm{BW}_{L}\left(s; m_{0}, \Gamma_{0}\right) &=& \mathcal{R}^\mathrm{BW}\left(s; m_{0}, \Gamma_{L}\left(s\right)\right) \\ \mathcal{R}^\mathrm{BW}\left(s; m_{0}, \Gamma_{0}\right) &=& \frac{1}{- 1.0 i \Gamma_{0} m_{0} + m_{0}^{2} - s} \\ \Gamma_{L}\left(s\right) &=& \Gamma_{0} \frac{m_{0}}{\sqrt{s}} \frac{F_{L}\left(R p_{_{m_{1},m_{2}}}\left(s\right)\right)^{2}}{F_{L}\left(R p_{_{m_{1},m_{2}}}\left(m_{0}^{2}\right)\right)^{2}} \left(\frac{p_{_{m_{1},m_{2}}}\left(s\right)}{p_{_{m_{1},m_{2}}}\left(m_{0}^{2}\right)}\right)^{2 L + 1} \\ \mathcal{F}_{L}\left(s, m_{1}, m_{2}\right) &=& \mathcal{F}_{L}\left(s, m_{1}, m_{2}\right) \\ p_{_{m_{1},m_{2}}}\left(s\right) &=& \frac{\sqrt{\lambda\left(s, m_{1}^{2}, m_{2}^{2}\right)}}{2 \sqrt{s}} \\ \lambda\left(x, y, z\right) &=& x^{2} - 2 x y - 2 x z + y^{2} - 2 y z + z^{2} \\ \end{array}\end{split}\]

K892_BW = formulate_breit_wigner(
    propagator=CHAIN_DEFS[20]["propagators"][0],
    resonance=to_latex(CHAIN_DEFS[20]["name"]),
    model=MODEL_DEFINITION,
)
Math(aslatex(K892_BW))

\[\begin{split}\displaystyle \begin{array}{rcl} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{1}; m_{K(892)}, \Gamma_{K(892)}\right) &=& \mathcal{R}^\mathrm{BW}\left(\sigma_{1}; m_{K(892)}, \Gamma_{1}\left(\sigma_{1}\right)\right) \\ m_{K(892)} &=& 0.8955 \\ \Gamma_{K(892)} &=& 0.047299999999999995 \\ m_{2} &=& 0.13957018 \\ m_{3} &=& 0.493677 \\ R_\mathrm{res} &=& 1.5 \\ \end{array}\end{split}\]

Multi-channel Breit-Wigner#

\[\begin{split}\displaystyle \begin{array}{rcl} \mathcal{R}^\mathrm{BW}_\mathrm{multi}\left(s; \Gamma_{1}, \Gamma_{2}\right) &=& \mathcal{R}^\mathrm{BW}_{L=0}\left(s; m_{0}, \frac{\Gamma_{1} m_{0} \mathcal{F}_{L_{1}}\left(s, m_{a,1}, m_{b,1}\right)^{2}}{\sqrt{s}} + \frac{\Gamma_{2} m_{0} \mathcal{F}_{L_{2}}\left(s, m_{a,2}, m_{b,2}\right)^{2}}{\sqrt{s}}\right) \\ \mathcal{R}^\mathrm{BW}_{L}\left(s; m_{0}, \Gamma_{0}\right) &=& \mathcal{R}^\mathrm{BW}\left(s; m_{0}, \Gamma_{L}\left(s\right)\right) \\ \mathcal{R}^\mathrm{BW}_{L=0}\left(s; m_{0}, \Gamma_{0}\right) &=& \frac{1}{- 1.0 i \Gamma_{0} m_{0} + m_{0}^{2} - s} \\ \Gamma_{L}\left(s\right) &=& \Gamma_{0} \frac{m_{0}}{\sqrt{s}} \frac{F_{L}\left(R p_{_{m_{1},m_{2}}}\left(s\right)\right)^{2}}{F_{L}\left(R p_{_{m_{1},m_{2}}}\left(m_{0}^{2}\right)\right)^{2}} \left(\frac{p_{_{m_{1},m_{2}}}\left(s\right)}{p_{_{m_{1},m_{2}}}\left(m_{0}^{2}\right)}\right)^{2 L + 1} \\ \mathcal{F}_{L}\left(s, m_{1}, m_{2}\right) &=& \mathcal{F}_{L}\left(s, m_{1}, m_{2}\right) \\ p_{_{m_{1},m_{2}}}\left(s\right) &=& \frac{\sqrt{\lambda\left(s, m_{1}^{2}, m_{2}^{2}\right)}}{2 \sqrt{s}} \\ \lambda\left(x, y, z\right) &=& x^{2} - 2 x y - 2 x z + y^{2} - 2 y z + z^{2} \\ \end{array}\end{split}\]

L1405_Flatte = formulate_multichannel_breit_wigner(
    propagator=CHAIN_DEFS[0]["propagators"][0],
    resonance=to_latex(CHAIN_DEFS[0]["name"]),
    model=MODEL_DEFINITION,
)
Math(aslatex(L1405_Flatte))

\[\begin{split}\displaystyle \begin{array}{rcl} \mathcal{R}^\mathrm{BW}_\mathrm{multi}\left(\sigma_{2}; \Gamma_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}^\text{ch. 2}\right) &=& \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(1405)}, \frac{\Gamma_{\Lambda(1405)} m_{\Lambda(1405)} \mathcal{F}_{0}\left(\sigma_{2}, m_{3}, m_{1}\right)^{2}}{\sqrt{\sigma_{2}}} + \frac{\Gamma_{\Lambda(1405)}^\text{ch. 2} m_{\Lambda(1405)} \mathcal{F}_{0}\left(\sigma_{2}, m_{a,2}, m_{b,2}\right)^{2}}{\sqrt{\sigma_{2}}}\right) \\ m_{\Lambda(1405)} &=& 1.4051 \\ \Gamma_{\Lambda(1405)} &=& 0.23395150538434703 \\ m_{3} &=& 0.938272046 \\ m_{1} &=& 0.493677 \\ R_{\Lambda(1405)} &=& 0 \\ m_{a,2} &=& 1.18937 \\ m_{b,2} &=& 0.13957018 \\ \Gamma_{\Lambda(1405)}^\text{ch. 2} &=& 0.23395150538434703 \\ \end{array}\end{split}\]

Breit-Wigner with exponential#

The model contains one lineshape function that is not standard, so we have to implement a custom propagator dynamics builder for this.

\[\begin{split}\displaystyle \begin{array}{rcl} \mathcal{R}^\mathrm{Bugg}\left(s\right) &=& \frac{1}{- \frac{i \Gamma_{0} m_{0} \left(- m_{1}^{2} + \frac{m_{2}^{2}}{2} + s\right) e^{- \gamma s}}{m_{0}^{2} - m_{1}^{2} + \frac{m_{2}^{2}}{2}} + m_{0}^{2} - s} \\ \end{array}\end{split}\]

CHAIN_DEFS[18]

{'propagators': [{'spin': '0',
   'node': [2, 3],
   'parametrization': 'K700_BuggBW'}],
 'weight': '0.068908 + 2.521444i',
 'vertices': [{'type': 'helicity',
   'helicities': ['0', '1/2'],
   'node': [[2, 3], 1],
   'formfactor': ''},
  {'type': 'parity',
   'helicities': ['0', '0'],
   'parity_factor': '+',
   'node': [2, 3],
   'formfactor': ''}],
 'topology': [[2, 3], 1],
 'name': 'K700'}

get_function_definition("K700_BuggBW", MODEL_DEFINITION)

{'name': 'K700_BuggBW',
 'slope': 0.94106,
 'type': 'BreitWignerWidthExpLikeBugg',
 'mass': 0.824,
 'width': 0.478}

def formulate_bugg_breit_wigner(
    propagator: Propagator, resonance: str, model: ModelDefinition
) -> DefinedExpression:
    function_definition = get_function_definition(propagator["parametrization"], model)
    node = propagator["node"]
    i, j = node
    s = to_mandelstam_symbol(node)
    mass = sp.Symbol(f"m_{{{resonance}}}", nonnegative=True)
    width = sp.Symbol(Rf"\Gamma_{{{resonance}}}", nonnegative=True)
    γ = sp.Symbol(Rf"\gamma_{{{resonance}}}", nonnegative=True)
    m1 = to_mass_symbol(i)
    m2 = to_mass_symbol(j)
    final_state = get_final_state(model)
    return DefinedExpression(
        expression=BuggBreitWigner(s, mass, width, m1, m2, γ),
        definitions={
            mass: function_definition["mass"],
            width: function_definition["width"],
            m1: final_state[i].mass,
            m2: final_state[j].mass,
            γ: function_definition["slope"],
        },
    )

CHAIN_18 = CHAIN_DEFS[18]
K700_BuggBW = formulate_bugg_breit_wigner(
    propagator=CHAIN_18["propagators"][0],
    resonance=to_latex(CHAIN_18["name"]),
    model=MODEL_DEFINITION,
)
Math(aslatex(K700_BuggBW))

\[\begin{split}\displaystyle \begin{array}{rcl} \mathcal{R}^\mathrm{Bugg}\left(\sigma_{1}\right) &=& \frac{1}{- \frac{i \Gamma_{K(700)} m_{K(700)} \left(- m_{2}^{2} + \frac{m_{3}^{2}}{2} + \sigma_{1}\right) e^{- \gamma_{K(700)} \sigma_{1}}}{- m_{2}^{2} + \frac{m_{3}^{2}}{2} + m_{K(700)}^{2}} + m_{K(700)}^{2} - \sigma_{1}} \\ m_{K(700)} &=& 0.824 \\ \Gamma_{K(700)} &=& 0.478 \\ m_{2} &=& 0.13957018 \\ m_{3} &=& 0.493677 \\ \gamma_{K(700)} &=& 0.94106 \\ \end{array}\end{split}\]

General propagator dynamics builder#

DYNAMICS_BUILDERS = {
    "BreitWignerWidthExpLikeBugg": formulate_bugg_breit_wigner,
}

\[\begin{split}\displaystyle \begin{array}{c} \begin{array}{rcl} \mathcal{R}^\mathrm{BW}_\mathrm{multi}\left(\sigma_{2}; \Gamma_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}^\text{ch. 2}\right) &=& \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(1405)}, \frac{\Gamma_{\Lambda(1405)} m_{\Lambda(1405)} \mathcal{F}_{0}\left(\sigma_{2}, m_{3}, m_{1}\right)^{2}}{\sqrt{\sigma_{2}}} + \frac{\Gamma_{\Lambda(1405)}^\text{ch. 2} m_{\Lambda(1405)} \mathcal{F}_{0}\left(\sigma_{2}, m_{a,2}, m_{b,2}\right)^{2}}{\sqrt{\sigma_{2}}}\right) \\ m_{\Lambda(1405)} &=& 1.4051 \\ \Gamma_{\Lambda(1405)} &=& 0.23395150538434703 \\ m_{3} &=& 0.938272046 \\ m_{1} &=& 0.493677 \\ R_{\Lambda(1405)} &=& 0 \\ m_{a,2} &=& 1.18937 \\ m_{b,2} &=& 0.13957018 \\ \Gamma_{\Lambda(1405)}^\text{ch. 2} &=& 0.23395150538434703 \\ \end{array} \\ \begin{array}{rcl} \mathcal{R}^\mathrm{Bugg}\left(\sigma_{1}\right) &=& \frac{1}{- \frac{i \Gamma_{K(700)} m_{K(700)} \left(- m_{2}^{2} + \frac{m_{3}^{2}}{2} + \sigma_{1}\right) e^{- \gamma_{K(700)} \sigma_{1}}}{- m_{2}^{2} + \frac{m_{3}^{2}}{2} + m_{K(700)}^{2}} + m_{K(700)}^{2} - \sigma_{1}} \\ m_{K(700)} &=& 0.824 \\ \Gamma_{K(700)} &=& 0.478 \\ m_{2} &=& 0.13957018 \\ m_{3} &=& 0.493677 \\ \gamma_{K(700)} &=& 0.94106 \\ \end{array} \\ \begin{array}{rcl} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{1}; m_{K(892)}, \Gamma_{K(892)}\right) &=& \mathcal{R}^\mathrm{BW}\left(\sigma_{1}; m_{K(892)}, \Gamma_{1}\left(\sigma_{1}\right)\right) \\ m_{K(892)} &=& 0.8955 \\ \Gamma_{K(892)} &=& 0.047299999999999995 \\ m_{2} &=& 0.13957018 \\ m_{3} &=& 0.493677 \\ R_\mathrm{res} &=& 1.5 \\ \end{array} \\ \end{array}\end{split}\]

Construct `AmplitudeModel`#

Unpolarized intensity#

λ0, λ1, λ2, λ3 = sp.symbols("lambda(:4)", rational=True)
amplitude_expr, _ = formulate_aligned_amplitude(MODEL_DEFINITION, λ0, λ1, λ2, λ3)
amplitude_expr.cleanup()

\[\displaystyle \sum_{\lambda_0^{\prime}=-1/2}^{1/2} \sum_{\lambda_1^{\prime}=-1/2}^{1/2}{A^{1}_{\lambda_0^{\prime}, \lambda_1^{\prime}, 0, 0} d^{\frac{1}{2}}_{\lambda_1^{\prime},\lambda_{1}}\left(\zeta^1_{1(2)}\right) d^{\frac{1}{2}}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{1(2)}\right) + A^{2}_{\lambda_0^{\prime}, \lambda_1^{\prime}, 0, 0} d^{\frac{1}{2}}_{\lambda_1^{\prime},\lambda_{1}}\left(\zeta^1_{2(2)}\right) d^{\frac{1}{2}}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{2(2)}\right) + A^{3}_{\lambda_0^{\prime}, \lambda_1^{\prime}, 0, 0} d^{\frac{1}{2}}_{\lambda_1^{\prime},\lambda_{1}}\left(\zeta^1_{3(2)}\right) d^{\frac{1}{2}}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{3(2)}\right)}\]

Amplitude for the decay chain#

Helicity recouplings#

\[\begin{split}\displaystyle \begin{array}{rcl} \mathcal{H}^\text{helicity}\left(\lambda_{a},\lambda_{b}|\lambda^{0}_{a},\lambda^{0}_{b}\right) &=& \delta_{\lambda_{a} \lambda^{0}_{a}} \delta_{\lambda_{b} \lambda^{0}_{b}} \\ \mathcal{H}^\text{parity}\left(\lambda_{a},\lambda_{b}|\lambda^{0}_{a},\lambda^{0}_{b},f\right) &=& f \delta_{\lambda_{a}, - \lambda^{0}_{a}} \delta_{\lambda_{b}, - \lambda^{0}_{b}} + \delta_{\lambda_{a} \lambda^{0}_{a}} \delta_{\lambda_{b} \lambda^{0}_{b}} \\ \mathcal{H}^\text{parity}\left(\lambda_{a},\lambda_{b}|l,s,j_{a},j_{b},j\right) &=& \frac{\sqrt{2 l + 1} C^{s,\lambda_{a} - \lambda_{b}}_{j_{a},\lambda_{a},j_{b},- \lambda_{b}} C^{j,\lambda_{a} - \lambda_{b}}_{l,0,s,\lambda_{a} - \lambda_{b}}}{\sqrt{2 j + 1}} \\ \end{array}\end{split}\]

Recoupling deserialization#

\[\begin{split}\displaystyle \begin{array}{rcl} \mathcal{H}^\text{helicity}\left(\lambda_{R},\lambda_{2}|\frac{1}{2},0\right) &=& \delta_{0 \lambda_{2}} \delta_{\frac{1}{2} \lambda_{R}} \\ \mathcal{H}^\text{parity}\left(\lambda_{3},\lambda_{1}|0,\frac{1}{2},1\right) &=& \delta_{- \frac{1}{2} \lambda_{1}} \delta_{0 \lambda_{3}} + \delta_{0 \lambda_{3}} \delta_{\frac{1}{2} \lambda_{1}} \\ \end{array}\end{split}\]

Chain amplitudes#

definitions = formulate_chain_amplitude(λ0, λ1, λ2, λ3, MODEL_DEFINITION, chain_idx=0)
Math(aslatex(definitions))

\[\begin{split}\displaystyle \begin{array}{rcl} A^{2}_{\lambda_{0}, \lambda_{1}, \lambda_{2}, \lambda_{3}} &=& c^{L1405[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{H}^\text{helicity}\left(\frac{1}{2},\lambda_{2}|\frac{1}{2},0\right) \mathcal{R}^\mathrm{BW}_\mathrm{multi}\left(\sigma_{2}; \Gamma_{L1405}, \Gamma_{L1405}^\text{ch. 2}\right) \mathcal{H}^\text{parity}\left(\lambda_{3},\lambda_{1}|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ c^{L1405[1/2]}_{\frac{1}{2}, 0, 0} &=& 7.38649400481717+1.971018433257411i \\ m_{L1405} &=& 1.4051 \\ \Gamma_{L1405} &=& 0.23395150538434703 \\ m_{3} &=& 0.938272046 \\ m_{1} &=& 0.493677 \\ R_{L1405} &=& 0 \\ m_{a,2} &=& 1.18937 \\ m_{b,2} &=& 0.13957018 \\ \Gamma_{L1405}^\text{ch. 2} &=& 0.23395150538434703 \\ \theta_{31} &=& \operatorname{acos}{\left(\frac{2 \sigma_{2} \left(- m_{2}^{2} - m_{3}^{2} + \sigma_{1}\right) - \left(m_{0}^{2} - m_{2}^{2} - \sigma_{2}\right) \left(- m_{1}^{2} + m_{3}^{2} + \sigma_{2}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{2}^{2}, \sigma_{2}\right)} \sqrt{\lambda\left(\sigma_{2}, m_{3}^{2}, m_{1}^{2}\right)}} \right)} \\ \end{array}\end{split}\]

Full amplitude model#

MODEL = formulate(
    MODEL_DEFINITION,
    additional_builders=DYNAMICS_BUILDERS,
    cleanup_summations=True,
    to_latex=to_latex,
)
MODEL.intensity

\[\displaystyle \sum_{\lambda_{0}=-1/2}^{1/2} \sum_{\lambda_{1}=-1/2}^{1/2}{\left|{\sum_{\lambda_0^{\prime}=-1/2}^{1/2} \sum_{\lambda_1^{\prime}=-1/2}^{1/2}{A^{1}_{\lambda_0^{\prime}, \lambda_1^{\prime}, 0, 0} d^{\frac{1}{2}}_{\lambda_1^{\prime},\lambda_{1}}\left(\zeta^1_{1(2)}\right) d^{\frac{1}{2}}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{1(2)}\right) + A^{2}_{\lambda_0^{\prime}, \lambda_1^{\prime}, 0, 0} d^{\frac{1}{2}}_{\lambda_1^{\prime},\lambda_{1}}\left(\zeta^1_{2(2)}\right) d^{\frac{1}{2}}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{2(2)}\right) + A^{3}_{\lambda_0^{\prime}, \lambda_1^{\prime}, 0, 0} d^{\frac{1}{2}}_{\lambda_1^{\prime},\lambda_{1}}\left(\zeta^1_{3(2)}\right) d^{\frac{1}{2}}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{3(2)}\right)}}\right|^{2}}\]

\[\begin{split}\displaystyle \begin{array}{rcl} A^{2}_{- \frac{1}{2}, - \frac{1}{2}, 0, 0} &=& c^{\Lambda(1405)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{R}^\mathrm{BW}_\mathrm{multi}\left(\sigma_{2}; \Gamma_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}^\text{ch. 2}\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1405)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{R}^\mathrm{BW}_\mathrm{multi}\left(\sigma_{2}; \Gamma_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}^\text{ch. 2}\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1520)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{2}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1520)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{2}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1600)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{2}; m_{\Lambda(1600)}, \Gamma_{\Lambda(1600)}\right) \mathcal{F}_{1}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}|0,\frac{1}{2},-1\right) d^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1600)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{2}; m_{\Lambda(1600)}, \Gamma_{\Lambda(1600)}\right) \mathcal{F}_{1}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}|0,\frac{1}{2},-1\right) d^{\frac{1}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1670)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1670)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1690)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1690)}, \Gamma_{\Lambda(1690)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{2}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1690)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1690)}, \Gamma_{\Lambda(1690)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{2}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(2000)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(2000)}, \Gamma_{\Lambda(2000)}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(2000)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(2000)}, \Gamma_{\Lambda(2000)}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ A^{3}_{- \frac{1}{2}, - \frac{1}{2}, 0, 0} &=& c^{D(1232)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1232)}, \Gamma_{D(1232)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{3}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(- \frac{1}{2},0|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{- \frac{1}{2},\frac{1}{2}}\left(\theta_{12}\right) \\ &+& c^{D(1232)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1232)}, \Gamma_{D(1232)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{3}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(- \frac{1}{2},0|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{\frac{1}{2},\frac{1}{2}}\left(\theta_{12}\right) \\ &+& c^{D(1600)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1600)}, \Gamma_{D(1600)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{3}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(- \frac{1}{2},0|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{- \frac{1}{2},\frac{1}{2}}\left(\theta_{12}\right) \\ &+& c^{D(1600)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1600)}, \Gamma_{D(1600)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{3}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(- \frac{1}{2},0|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{\frac{1}{2},\frac{1}{2}}\left(\theta_{12}\right) \\ &+& c^{D(1700)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{3}; m_{D(1700)}, \Gamma_{D(1700)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{3}, m_{3}\right) \mathcal{F}_{2}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(- \frac{1}{2},0|\frac{1}{2},0,-1\right) d^{\frac{3}{2}}_{- \frac{1}{2},\frac{1}{2}}\left(\theta_{12}\right) \\ &+& c^{D(1700)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{3}; m_{D(1700)}, \Gamma_{D(1700)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{3}, m_{3}\right) \mathcal{F}_{2}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(- \frac{1}{2},0|\frac{1}{2},0,-1\right) d^{\frac{3}{2}}_{\frac{1}{2},\frac{1}{2}}\left(\theta_{12}\right) \\ A^{1}_{- \frac{1}{2}, - \frac{1}{2}, 0, 0} &=& c^{K(1430)[0]}_{- \frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{Bugg}\left(\sigma_{1}\right) \mathcal{H}^\text{helicity}\left(0,- \frac{1}{2}|0,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{0}_{0,0}\left(\theta_{23}\right) \\ &+& c^{K(1430)[0]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{Bugg}\left(\sigma_{1}\right) \mathcal{H}^\text{helicity}\left(0,- \frac{1}{2}|0,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{0}_{0,0}\left(\theta_{23}\right) \\ &+& c^{K(700)[0]}_{- \frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{Bugg}\left(\sigma_{1}\right) \mathcal{H}^\text{helicity}\left(0,- \frac{1}{2}|0,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{0}_{0,0}\left(\theta_{23}\right) \\ &+& c^{K(700)[0]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{Bugg}\left(\sigma_{1}\right) \mathcal{H}^\text{helicity}\left(0,- \frac{1}{2}|0,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{0}_{0,0}\left(\theta_{23}\right) \\ &+& c^{K(892)[-1]}_{- \frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{1}; m_{K(892)}, \Gamma_{K(892)}\right) \mathcal{F}_{1}\left(\sigma_{1}, m_{2}, m_{3}\right) \mathcal{H}^\text{helicity}\left(-1,- \frac{1}{2}|-1,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{1}_{-1,0}\left(\theta_{23}\right) \\ &+& c^{K(892)[0]}_{- \frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{1}; m_{K(892)}, \Gamma_{K(892)}\right) \mathcal{F}_{1}\left(\sigma_{1}, m_{2}, m_{3}\right) \mathcal{H}^\text{helicity}\left(0,- \frac{1}{2}|0,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{1}_{0,0}\left(\theta_{23}\right) \\ &+& c^{K(892)[0]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{1}; m_{K(892)}, \Gamma_{K(892)}\right) \mathcal{F}_{1}\left(\sigma_{1}, m_{2}, m_{3}\right) \mathcal{H}^\text{helicity}\left(0,- \frac{1}{2}|0,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{1}_{0,0}\left(\theta_{23}\right) \\ &+& c^{K(892)[1]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{1}; m_{K(892)}, \Gamma_{K(892)}\right) \mathcal{F}_{1}\left(\sigma_{1}, m_{2}, m_{3}\right) \mathcal{H}^\text{helicity}\left(1,- \frac{1}{2}|1,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{1}_{1,0}\left(\theta_{23}\right) \\ A^{2}_{- \frac{1}{2}, \frac{1}{2}, 0, 0} &=& c^{\Lambda(1405)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{R}^\mathrm{BW}_\mathrm{multi}\left(\sigma_{2}; \Gamma_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}^\text{ch. 2}\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1405)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{R}^\mathrm{BW}_\mathrm{multi}\left(\sigma_{2}; \Gamma_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}^\text{ch. 2}\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1520)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{2}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1520)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{2}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1600)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{2}; m_{\Lambda(1600)}, \Gamma_{\Lambda(1600)}\right) \mathcal{F}_{1}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}|0,\frac{1}{2},-1\right) d^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1600)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{2}; m_{\Lambda(1600)}, \Gamma_{\Lambda(1600)}\right) \mathcal{F}_{1}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}|0,\frac{1}{2},-1\right) d^{\frac{1}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1670)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1670)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1690)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1690)}, \Gamma_{\Lambda(1690)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{2}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1690)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1690)}, \Gamma_{\Lambda(1690)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{2}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(2000)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(2000)}, \Gamma_{\Lambda(2000)}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(2000)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(2000)}, \Gamma_{\Lambda(2000)}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ A^{3}_{- \frac{1}{2}, \frac{1}{2}, 0, 0} &=& c^{D(1232)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1232)}, \Gamma_{D(1232)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{3}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(\frac{1}{2},0|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{- \frac{1}{2},\frac{1}{2}}\left(\theta_{12}\right) \\ &+& c^{D(1232)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1232)}, \Gamma_{D(1232)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{3}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(\frac{1}{2},0|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{\frac{1}{2},\frac{1}{2}}\left(\theta_{12}\right) \\ &+& c^{D(1600)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1600)}, \Gamma_{D(1600)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{3}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(\frac{1}{2},0|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{- \frac{1}{2},\frac{1}{2}}\left(\theta_{12}\right) \\ &+& c^{D(1600)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1600)}, \Gamma_{D(1600)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{3}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(\frac{1}{2},0|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{\frac{1}{2},\frac{1}{2}}\left(\theta_{12}\right) \\ &+& c^{D(1700)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{3}; m_{D(1700)}, \Gamma_{D(1700)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{3}, m_{3}\right) \mathcal{F}_{2}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(\frac{1}{2},0|\frac{1}{2},0,-1\right) d^{\frac{3}{2}}_{- \frac{1}{2},\frac{1}{2}}\left(\theta_{12}\right) \\ &+& c^{D(1700)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{3}; m_{D(1700)}, \Gamma_{D(1700)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{3}, m_{3}\right) \mathcal{F}_{2}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(\frac{1}{2},0|\frac{1}{2},0,-1\right) d^{\frac{3}{2}}_{\frac{1}{2},\frac{1}{2}}\left(\theta_{12}\right) \\ A^{1}_{- \frac{1}{2}, \frac{1}{2}, 0, 0} &=& c^{K(1430)[0]}_{- \frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{Bugg}\left(\sigma_{1}\right) \mathcal{H}^\text{helicity}\left(0,\frac{1}{2}|0,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{0}_{0,0}\left(\theta_{23}\right) \\ &+& c^{K(1430)[0]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{Bugg}\left(\sigma_{1}\right) \mathcal{H}^\text{helicity}\left(0,\frac{1}{2}|0,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{0}_{0,0}\left(\theta_{23}\right) \\ &+& c^{K(700)[0]}_{- \frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{Bugg}\left(\sigma_{1}\right) \mathcal{H}^\text{helicity}\left(0,\frac{1}{2}|0,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{0}_{0,0}\left(\theta_{23}\right) \\ &+& c^{K(700)[0]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{Bugg}\left(\sigma_{1}\right) \mathcal{H}^\text{helicity}\left(0,\frac{1}{2}|0,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{0}_{0,0}\left(\theta_{23}\right) \\ &+& c^{K(892)[-1]}_{- \frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{1}; m_{K(892)}, \Gamma_{K(892)}\right) \mathcal{F}_{1}\left(\sigma_{1}, m_{2}, m_{3}\right) \mathcal{H}^\text{helicity}\left(-1,\frac{1}{2}|-1,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{1}_{-1,0}\left(\theta_{23}\right) \\ &+& c^{K(892)[0]}_{- \frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{1}; m_{K(892)}, \Gamma_{K(892)}\right) \mathcal{F}_{1}\left(\sigma_{1}, m_{2}, m_{3}\right) \mathcal{H}^\text{helicity}\left(0,\frac{1}{2}|0,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{1}_{0,0}\left(\theta_{23}\right) \\ &+& c^{K(892)[0]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{1}; m_{K(892)}, \Gamma_{K(892)}\right) \mathcal{F}_{1}\left(\sigma_{1}, m_{2}, m_{3}\right) \mathcal{H}^\text{helicity}\left(0,\frac{1}{2}|0,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{1}_{0,0}\left(\theta_{23}\right) \\ &+& c^{K(892)[1]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{1}; m_{K(892)}, \Gamma_{K(892)}\right) \mathcal{F}_{1}\left(\sigma_{1}, m_{2}, m_{3}\right) \mathcal{H}^\text{helicity}\left(1,\frac{1}{2}|1,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{1}_{1,0}\left(\theta_{23}\right) \\ A^{2}_{\frac{1}{2}, - \frac{1}{2}, 0, 0} &=& c^{\Lambda(1405)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{R}^\mathrm{BW}_\mathrm{multi}\left(\sigma_{2}; \Gamma_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}^\text{ch. 2}\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1405)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{R}^\mathrm{BW}_\mathrm{multi}\left(\sigma_{2}; \Gamma_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}^\text{ch. 2}\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1520)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{2}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1520)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{2}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1600)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{2}; m_{\Lambda(1600)}, \Gamma_{\Lambda(1600)}\right) \mathcal{F}_{1}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}|0,\frac{1}{2},-1\right) d^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1600)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{2}; m_{\Lambda(1600)}, \Gamma_{\Lambda(1600)}\right) \mathcal{F}_{1}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}|0,\frac{1}{2},-1\right) d^{\frac{1}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1670)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1670)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1690)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1690)}, \Gamma_{\Lambda(1690)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{2}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1690)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1690)}, \Gamma_{\Lambda(1690)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{2}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(2000)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(2000)}, \Gamma_{\Lambda(2000)}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(2000)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(2000)}, \Gamma_{\Lambda(2000)}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ A^{3}_{\frac{1}{2}, - \frac{1}{2}, 0, 0} &=& c^{D(1232)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1232)}, \Gamma_{D(1232)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{3}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(- \frac{1}{2},0|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{- \frac{1}{2},\frac{1}{2}}\left(\theta_{12}\right) \\ &+& c^{D(1232)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1232)}, \Gamma_{D(1232)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{3}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(- \frac{1}{2},0|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{\frac{1}{2},\frac{1}{2}}\left(\theta_{12}\right) \\ &+& c^{D(1600)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1600)}, \Gamma_{D(1600)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{3}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(- \frac{1}{2},0|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{- \frac{1}{2},\frac{1}{2}}\left(\theta_{12}\right) \\ &+& c^{D(1600)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1600)}, \Gamma_{D(1600)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{3}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(- \frac{1}{2},0|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{\frac{1}{2},\frac{1}{2}}\left(\theta_{12}\right) \\ &+& c^{D(1700)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{3}; m_{D(1700)}, \Gamma_{D(1700)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{3}, m_{3}\right) \mathcal{F}_{2}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(- \frac{1}{2},0|\frac{1}{2},0,-1\right) d^{\frac{3}{2}}_{- \frac{1}{2},\frac{1}{2}}\left(\theta_{12}\right) \\ &+& c^{D(1700)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{3}; m_{D(1700)}, \Gamma_{D(1700)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{3}, m_{3}\right) \mathcal{F}_{2}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(- \frac{1}{2},0|\frac{1}{2},0,-1\right) d^{\frac{3}{2}}_{\frac{1}{2},\frac{1}{2}}\left(\theta_{12}\right) \\ A^{1}_{\frac{1}{2}, - \frac{1}{2}, 0, 0} &=& c^{K(1430)[0]}_{- \frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{Bugg}\left(\sigma_{1}\right) \mathcal{H}^\text{helicity}\left(0,- \frac{1}{2}|0,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{0}_{0,0}\left(\theta_{23}\right) \\ &+& c^{K(1430)[0]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{Bugg}\left(\sigma_{1}\right) \mathcal{H}^\text{helicity}\left(0,- \frac{1}{2}|0,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{0}_{0,0}\left(\theta_{23}\right) \\ &+& c^{K(700)[0]}_{- \frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{Bugg}\left(\sigma_{1}\right) \mathcal{H}^\text{helicity}\left(0,- \frac{1}{2}|0,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{0}_{0,0}\left(\theta_{23}\right) \\ &+& c^{K(700)[0]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{Bugg}\left(\sigma_{1}\right) \mathcal{H}^\text{helicity}\left(0,- \frac{1}{2}|0,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{0}_{0,0}\left(\theta_{23}\right) \\ &+& c^{K(892)[-1]}_{- \frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{1}; m_{K(892)}, \Gamma_{K(892)}\right) \mathcal{F}_{1}\left(\sigma_{1}, m_{2}, m_{3}\right) \mathcal{H}^\text{helicity}\left(-1,- \frac{1}{2}|-1,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{1}_{-1,0}\left(\theta_{23}\right) \\ &+& c^{K(892)[0]}_{- \frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{1}; m_{K(892)}, \Gamma_{K(892)}\right) \mathcal{F}_{1}\left(\sigma_{1}, m_{2}, m_{3}\right) \mathcal{H}^\text{helicity}\left(0,- \frac{1}{2}|0,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{1}_{0,0}\left(\theta_{23}\right) \\ &+& c^{K(892)[0]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{1}; m_{K(892)}, \Gamma_{K(892)}\right) \mathcal{F}_{1}\left(\sigma_{1}, m_{2}, m_{3}\right) \mathcal{H}^\text{helicity}\left(0,- \frac{1}{2}|0,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{1}_{0,0}\left(\theta_{23}\right) \\ &+& c^{K(892)[1]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{1}; m_{K(892)}, \Gamma_{K(892)}\right) \mathcal{F}_{1}\left(\sigma_{1}, m_{2}, m_{3}\right) \mathcal{H}^\text{helicity}\left(1,- \frac{1}{2}|1,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{1}_{1,0}\left(\theta_{23}\right) \\ A^{2}_{\frac{1}{2}, \frac{1}{2}, 0, 0} &=& c^{\Lambda(1405)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{R}^\mathrm{BW}_\mathrm{multi}\left(\sigma_{2}; \Gamma_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}^\text{ch. 2}\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1405)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{R}^\mathrm{BW}_\mathrm{multi}\left(\sigma_{2}; \Gamma_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}^\text{ch. 2}\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1520)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{2}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1520)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{2}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1600)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{2}; m_{\Lambda(1600)}, \Gamma_{\Lambda(1600)}\right) \mathcal{F}_{1}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}|0,\frac{1}{2},-1\right) d^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1600)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{2}; m_{\Lambda(1600)}, \Gamma_{\Lambda(1600)}\right) \mathcal{F}_{1}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}|0,\frac{1}{2},-1\right) d^{\frac{1}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1670)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1670)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1690)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1690)}, \Gamma_{\Lambda(1690)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{2}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(1690)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1690)}, \Gamma_{\Lambda(1690)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{2}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(2000)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(2000)}, \Gamma_{\Lambda(2000)}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{- \frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ &+& c^{\Lambda(2000)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(2000)}, \Gamma_{\Lambda(2000)}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\frac{1}{2},- \frac{1}{2}}\left(\theta_{31}\right) \\ A^{3}_{\frac{1}{2}, \frac{1}{2}, 0, 0} &=& c^{D(1232)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1232)}, \Gamma_{D(1232)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{3}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(\frac{1}{2},0|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{- \frac{1}{2},\frac{1}{2}}\left(\theta_{12}\right) \\ &+& c^{D(1232)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1232)}, \Gamma_{D(1232)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{3}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(\frac{1}{2},0|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{\frac{1}{2},\frac{1}{2}}\left(\theta_{12}\right) \\ &+& c^{D(1600)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1600)}, \Gamma_{D(1600)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{3}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(\frac{1}{2},0|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{- \frac{1}{2},\frac{1}{2}}\left(\theta_{12}\right) \\ &+& c^{D(1600)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1600)}, \Gamma_{D(1600)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{3}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(\frac{1}{2},0|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{\frac{1}{2},\frac{1}{2}}\left(\theta_{12}\right) \\ &+& c^{D(1700)[-1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{3}; m_{D(1700)}, \Gamma_{D(1700)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{3}, m_{3}\right) \mathcal{F}_{2}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(- \frac{1}{2},0|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(\frac{1}{2},0|\frac{1}{2},0,-1\right) d^{\frac{3}{2}}_{- \frac{1}{2},\frac{1}{2}}\left(\theta_{12}\right) \\ &+& c^{D(1700)[1/2]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{3}; m_{D(1700)}, \Gamma_{D(1700)}\right) \mathcal{F}_{1}\left(m_{0}, \sigma_{3}, m_{3}\right) \mathcal{F}_{2}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\frac{1}{2},0|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(\frac{1}{2},0|\frac{1}{2},0,-1\right) d^{\frac{3}{2}}_{\frac{1}{2},\frac{1}{2}}\left(\theta_{12}\right) \\ A^{1}_{\frac{1}{2}, \frac{1}{2}, 0, 0} &=& c^{K(1430)[0]}_{- \frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{Bugg}\left(\sigma_{1}\right) \mathcal{H}^\text{helicity}\left(0,\frac{1}{2}|0,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{0}_{0,0}\left(\theta_{23}\right) \\ &+& c^{K(1430)[0]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{Bugg}\left(\sigma_{1}\right) \mathcal{H}^\text{helicity}\left(0,\frac{1}{2}|0,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{0}_{0,0}\left(\theta_{23}\right) \\ &+& c^{K(700)[0]}_{- \frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{Bugg}\left(\sigma_{1}\right) \mathcal{H}^\text{helicity}\left(0,\frac{1}{2}|0,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{0}_{0,0}\left(\theta_{23}\right) \\ &+& c^{K(700)[0]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{Bugg}\left(\sigma_{1}\right) \mathcal{H}^\text{helicity}\left(0,\frac{1}{2}|0,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{0}_{0,0}\left(\theta_{23}\right) \\ &+& c^{K(892)[-1]}_{- \frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{1}; m_{K(892)}, \Gamma_{K(892)}\right) \mathcal{F}_{1}\left(\sigma_{1}, m_{2}, m_{3}\right) \mathcal{H}^\text{helicity}\left(-1,\frac{1}{2}|-1,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{1}_{-1,0}\left(\theta_{23}\right) \\ &+& c^{K(892)[0]}_{- \frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{1}; m_{K(892)}, \Gamma_{K(892)}\right) \mathcal{F}_{1}\left(\sigma_{1}, m_{2}, m_{3}\right) \mathcal{H}^\text{helicity}\left(0,\frac{1}{2}|0,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{1}_{0,0}\left(\theta_{23}\right) \\ &+& c^{K(892)[0]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{1}; m_{K(892)}, \Gamma_{K(892)}\right) \mathcal{F}_{1}\left(\sigma_{1}, m_{2}, m_{3}\right) \mathcal{H}^\text{helicity}\left(0,\frac{1}{2}|0,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{1}_{0,0}\left(\theta_{23}\right) \\ &+& c^{K(892)[1]}_{\frac{1}{2}, 0, 0} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{1}; m_{K(892)}, \Gamma_{K(892)}\right) \mathcal{F}_{1}\left(\sigma_{1}, m_{2}, m_{3}\right) \mathcal{H}^\text{helicity}\left(1,\frac{1}{2}|1,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0|0,0,1\right) d^{1}_{1,0}\left(\theta_{23}\right) \\ \end{array}\end{split}\]

\[\begin{split}\displaystyle \begin{array}{rcl} \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)} \\ \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)} \\ \zeta^0_{1(2)} &=& \operatorname{acos}{\left(\frac{- 2 m_{0}^{2} \left(- m_{1}^{2} - m_{2}^{2} + \sigma_{3}\right) + \left(m_{0}^{2} + m_{1}^{2} - \sigma_{1}\right) \left(m_{0}^{2} + m_{2}^{2} - \sigma_{2}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{2}^{2}, \sigma_{2}\right)} \sqrt{\lambda\left(m_{0}^{2}, \sigma_{1}, m_{1}^{2}\right)}} \right)} \\ \zeta^1_{1(2)} &=& - \operatorname{acos}{\left(\frac{2 m_{1}^{2} \left(- m_{0}^{2} - m_{3}^{2} + \sigma_{3}\right) + \left(m_{0}^{2} + m_{1}^{2} - \sigma_{1}\right) \left(- m_{1}^{2} - m_{3}^{2} + \sigma_{2}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{1}^{2}, \sigma_{1}\right)} \sqrt{\lambda\left(\sigma_{2}, m_{1}^{2}, m_{3}^{2}\right)}} \right)} \\ \zeta^0_{2(2)} &=& 0 \\ \zeta^1_{2(2)} &=& 0 \\ \zeta^0_{3(2)} &=& - \operatorname{acos}{\left(\frac{- 2 m_{0}^{2} \left(- m_{2}^{2} - m_{3}^{2} + \sigma_{1}\right) + \left(m_{0}^{2} + m_{2}^{2} - \sigma_{2}\right) \left(m_{0}^{2} + m_{3}^{2} - \sigma_{3}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{3}^{2}, \sigma_{3}\right)} \sqrt{\lambda\left(m_{0}^{2}, \sigma_{2}, m_{2}^{2}\right)}} \right)} \\ \zeta^1_{3(2)} &=& - \operatorname{acos}{\left(\frac{2 m_{1}^{2} \left(m_{2}^{2} + m_{3}^{2} - \sigma_{1}\right) + \left(- m_{1}^{2} - m_{2}^{2} + \sigma_{3}\right) \left(- m_{1}^{2} - m_{3}^{2} + \sigma_{2}\right)}{\sqrt{\lambda\left(\sigma_{2}, m_{3}^{2}, m_{1}^{2}\right)} \sqrt{\lambda\left(\sigma_{3}, m_{1}^{2}, m_{2}^{2}\right)}} \right)} \\ \end{array}\end{split}\]

\[\begin{split}\displaystyle \begin{array}{rcl} \sigma_{1} &=& m_{0}^{2} + m_{1}^{2} + m_{2}^{2} + m_{3}^{2} - \sigma_{2} - \sigma_{3} \\ \sigma_{2} &=& m_{0}^{2} + m_{1}^{2} + m_{2}^{2} + m_{3}^{2} - \sigma_{1} - \sigma_{3} \\ \sigma_{3} &=& m_{0}^{2} + m_{1}^{2} + m_{2}^{2} + m_{3}^{2} - \sigma_{1} - \sigma_{2} \\ m_{0} &=& 2.28646 \\ m_{1} &=& 0.938272046 \\ m_{2} &=& 0.13957018 \\ m_{3} &=& 0.493677 \\ \end{array}\end{split}\]

Numeric results#

intensity_expr = MODEL.full_expression.xreplace(MODEL.variables)
intensity_expr = intensity_expr.xreplace(MODEL.parameter_defaults)

Cached function file /home/runner/.cache/ampform_dpd/jax-v0.4.30/pythonhashseed-0+4560098846007579925.pkl not found, lambdifying...

Cached function file /home/runner/.cache/ampform_dpd/jax-v0.4.30/pythonhashseed-0-1452373578771090629.pkl not found, lambdifying...

Cached function file /home/runner/.cache/ampform_dpd/jax-v0.4.30/pythonhashseed-0+8385303269060950380.pkl not found, lambdifying...

Validation#

Error

The following serves as a numerical check on whether the amplitude model has been deserialized correctly. For now, this is not the case, see ComPWA/ampform-dpd#133 for updates.

checksums = {
    misc_key: {checksum["name"]: checksum["value"] for checksum in misc_value}
    for misc_key, misc_value in MODEL_DEFINITION["misc"].items()
    if "checksum" in misc_key
}
checksums

{'amplitude_model_checksums': {'validation_point': 9345.853380852355}}

checksum_points = {
    point["name"]: {par["name"]: par["value"] for par in point["parameters"]}
    for point in MODEL_DEFINITION["parameter_points"]
}
checksum_points

{'validation_point': {'cos_theta_31': -0.2309352648098208,
  'phi_31': 0.0,
  'm_31': 1.9101377207489973,
  'cos_theta_31_2': 0.0,
  'phi_31_2': 0.0,
  'm_31_2': 2.28646}}

	Distribution	Point	Computed	Expected	Status
0	amplitude_model_checksums	validation_point	nan	9345.853381	🔴

Dalitz plot#

\[\begin{split}\displaystyle \begin{array}{rcl} \sigma_{3} &=& m_{0}^{2} + m_{1}^{2} + m_{2}^{2} + m_{3}^{2} - \sigma_{1} - \sigma_{2} \\ \end{array}\end{split}\]

intensities = intensity_funcs[k](dalitz_data)