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#

load_workspace() loads the serialized definition and formulates its distributions, decays, and dynamics functions in one step. It also preserves kinematics, parameter points, and checksums for downstream backends and validation.

def to_latex(name: str) -> str:
    latex = {
        "Lc": R"\Lambda_c^+",
        "pi": R"\pi^+",
        "K": "K^-",
        "p": "p",
    }.get(name)
    if latex is not None:
        return latex
    mass_str = name[1:].strip("(").strip(")")
    subsystem_letter = name[0]
    subsystem = {"D": "D", "K": "K", "L": R"\Lambda"}.get(subsystem_letter)
    if subsystem is None:
        return name
    return f"{subsystem}({mass_str})"

Construct `ThreeBodyDecay`#

\[\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}\]

WORKSPACE.kinematics[INTENSITY_NAME]

mappingproxy({'initial_state': mappingproxy({'index': 0,
                            'name': 'Lc',
                            'spin': '1/2',
                            'mass': 2.28646}),
              'final_state': (mappingproxy({'index': 1,
                             'name': 'p',
                             'spin': '1/2',
                             'mass': 0.938272046}),
               mappingproxy({'index': 2,
                             'name': 'pi',
                             'spin': '0',
                             'mass': 0.13957018}),
               mappingproxy({'index': 3,
                             'name': 'K',
                             'spin': '0',
                             'mass': 0.493677}))})

Dynamics#

CHAIN_DEFS = get_decay_chains(MODEL_DEFINITION)

Vertices#

Blatt–Weisskopf form factor#

Propagators#

Breit–Wigner#

\[\begin{split}\displaystyle \begin{aligned} \mathcal{R}^\mathrm{BW}_{L}\left(s; m_{0}, \Gamma_{0}\right) \;&=\; \mathcal{R}^\mathrm{BW}\left(s; m_{0}, \Gamma_{0}\left(s\right)\right) \\ \mathcal{R}^\mathrm{BW}\left(s; m_{0}, \Gamma_{0}\left(s\right)\right) \;&=\; \frac{1}{m_{0}^{2} - i m_{0} \Gamma_{0}\left(s\right) - s} \\ \Gamma_{0}\left(s\right) \;&=\; \frac{\Gamma_{0} \mathcal{F}_{L}\left(s, m_{1}, m_{2}\right)^{2} \rho\left(s\right)}{\mathcal{F}_{L}\left(m_{0}^{2}, m_{1}, m_{2}\right)^{2} \rho_{0}\left(m_{0}^{2}\right)} \\ \mathcal{F}_{L}\left(s, m_{1}, m_{2}\right) \;&=\; \sqrt{B_{L}^2\left(R^{2} q^2\left(s\right)\right)} \\ \rho\left(s\right) \;&=\; \frac{\sqrt{\left(s - \left(m_{1} - m_{2}\right)^{2}\right) \left(s - \left(m_{1} + m_{2}\right)^{2}\right)}}{s} \\ B_{L}^2\left(R^{2} q^2\left(s\right)\right) \;&=\; \frac{\left|{h_{L}^{(1)}\left(1\right)}\right|^{2}}{R^{2} \left|{h_{L}^{(1)}\left(R \sqrt{q^2\left(s\right)}\right)}\right|^{2} q^2\left(s\right)} \\ q^2\left(s\right) \;&=\; \frac{\left(s - \left(m_{1} - m_{2}\right)^{2}\right) \left(s - \left(m_{1} + m_{2}\right)^{2}\right)}{4 s} \\ h_{L}^{(1)}\left(z\right) \;&=\; \frac{\left(- i\right)^{L + 1} e^{i z} \sum_{k=0}^{L} \frac{\left(\frac{i}{2 z}\right)^{k} \left(k + L\right)!}{k! \left(- k + L\right)!}}{z} \\ \end{aligned}\end{split}\]

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))

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

Multi-channel Breit–Wigner#

Each channel in the serialized model carries a gsq value, which is a coupling squared \(g_i^2\), not an energy-dependent width. It enters the lineshape through the channel term \(\Gamma^\text{ch}\) as follows.

Breit–Wigner with exponential#

The Bugg lineshapes are serialized as generic_function expression strings. The built-in dynamics builder parses these expressions and substitutes \(i\) and \(\sigma\) with the imaginary unit and the Mandelstam variable of the propagator node, respectively.

CHAIN_DEFS[18]

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

get_function_definition("K700_BuggBW", MODEL_DEFINITION)

mappingproxy({'name': 'K700_BuggBW',
              'type': 'generic_function',
              'expression': '1/(0.824^2 - σ - i * 0.824 * (σ - 0.23397706275638377) / (0.824^2 - 0.23397706275638377) * 0.478 * exp(-0.941060 * σ))'})

General propagator dynamics builder#

\[\begin{split}\displaystyle \begin{array}{c} \begin{array}{rcl} \mathcal{R}^\mathrm{BW}_\mathrm{multi}\left(\sigma_{2}; \Gamma_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}^\text{ch. 2}\right) &=& \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(1405)}, \Gamma^\text{ch}\left(\sigma_{2}; m_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}\right) + \Gamma^\text{ch}\left(\sigma_{2}; m_{\Lambda(1405)}, \Gamma_{\Lambda(1405)}^\text{ch. 2}\right)\right) \\ m_{\Lambda(1405)} &=& 1.4051 \\ \Gamma_{\Lambda(1405)} &=& 0.328725260215546 \\ m_{3} &=& 0.938272046 \\ m_{1} &=& 0.493677 \\ R_{\Lambda(1405)} &=& 0 \\ m_{a,2} &=& 1.18937 \\ m_{b,2} &=& 0.13957018 \\ \Gamma_{\Lambda(1405)}^\text{ch. 2} &=& 0.328725260215546 \\ \end{array} \\ \begin{array}{rcl} \frac{1}{- \sigma_{1} - 0.88510773180649964 i \left(\sigma_{1} - 0.23397706275638377\right) e^{- 0.94106 \sigma_{1}} + 0.678976} \\ \end{array} \\ \begin{array}{rcl} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{1}; m_{K(892)}, \Gamma_{K(892)}\right) &=& \mathcal{R}^\mathrm{BW}\left(\sigma_{1}; m_{K(892)}, \Gamma_{892}\left(\sigma_{1}\right)\right) \\ m_{K(892)} &=& 0.8955 \\ \Gamma_{K(892)} &=& 0.047299999999999995 \\ m_{2} &=& 0.13957018 \\ m_{3} &=& 0.493677 \\ R_\mathrm{res} &=& 1.5 \\ \end{array} \\ \end{array}\end{split}\]

Validation#

The propagators are compared with the reference values that the model file provides under misc.amplitude_model_checksums. The marks 🟢, 🟡, and 🔴 indicate an accuracy of \(<10^{-10}\), \(<10^{-2}\), and \(\geq10^{-2}\), respectively, on the difference between the reference and the computed value.

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()

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

Amplitude for the decay chain#

Helicity recouplings#

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

Recoupling deserialization#

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

Chain amplitudes#

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

\[\begin{split}\displaystyle \begin{aligned} A^{2}_{\lambda_{0}, \lambda_{1}, \lambda_{2}, \lambda_{3}} \;&=\; \sum_{\lambda_{R}=-1/2}^{1/2}{\left(-1\right)^{- \lambda_{2}} \left(-1\right)^{\frac{1}{2} - \lambda_{1}} \sqrt{2} c^{L1405[1/2]}_{\frac{1}{2}, 0, 0} \delta_{\lambda_{0}, \lambda_{R} - \lambda_{2}} \mathcal{H}^\text{helicity}\left(\lambda_{R},\lambda_{2}\middle|\frac{1}{2},0\right) \mathcal{R}^\mathrm{BW}_\mathrm{multi}\left(\sigma_{2}; \Gamma_{L1405}, \Gamma_{L1405}^\text{ch. 2}\right) \mathcal{H}^\text{parity}\left(\lambda_{3},\lambda_{1}\middle|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\lambda_{R},- \lambda_{1} + \lambda_{3}}\left(\theta_{31}\right)} \\ c^{L1405[1/2]}_{\frac{1}{2}, 0, 0} \;&=\; 7.38649400481717+1.971018433257411i \\ m_{L1405} \;&=\; 1.4051 \\ \Gamma_{L1405} \;&=\; 0.328725260215546 \\ m_{3} \;&=\; 0.938272046 \\ m_{1} \;&=\; 0.493677 \\ R_{L1405} \;&=\; 0 \\ m_{a,2} \;&=\; 1.18937 \\ m_{b,2} \;&=\; 0.13957018 \\ \Gamma_{L1405}^\text{ch. 2} \;&=\; 0.328725260215546 \\ \theta_{31} \;&=\; \operatorname{acos}{\left(\frac{2 \sigma_{2} \left(- m_{2}^{2} - m_{3}^{2} + \sigma_{1}\right) - \left(m_{0}^{2} - m_{2}^{2} - \sigma_{2}\right) \left(- m_{1}^{2} + m_{3}^{2} + \sigma_{2}\right)}{\sqrt{\lambda\left(m_{0}^{2}, m_{2}^{2}, \sigma_{2}\right)} \sqrt{\lambda\left(\sigma_{2}, m_{3}^{2}, m_{1}^{2}\right)}} \right)} \\ \end{aligned}\end{split}\]

Full amplitude model#

MODEL = WORKSPACE.distributions[INTENSITY_NAME]
MODEL.intensity

\[\displaystyle \sum_{\lambda_{0}=-1/2}^{1/2} \sum_{\lambda_{1}=-1/2}^{1/2} \sum_{\lambda_{2}=0} \sum_{\lambda_{3}=0}{\left|{\sum_{\lambda_0^{\prime}=-1/2}^{1/2} \sum_{\lambda_1^{\prime}=-1/2}^{1/2} \sum_{\lambda_2^{\prime}=0} \sum_{\lambda_3^{\prime}=0}{A^{1}_{\lambda_0^{\prime}, \lambda_1^{\prime}, \lambda_2^{\prime}, \lambda_3^{\prime}} d^{\frac{1}{2}}_{\lambda_1^{\prime},\lambda_{1}}\left(\zeta^1_{1(1)}\right) d^{\frac{1}{2}}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{1(1)}\right) + A^{2}_{\lambda_0^{\prime}, \lambda_1^{\prime}, \lambda_2^{\prime}, \lambda_3^{\prime}} d^{\frac{1}{2}}_{\lambda_1^{\prime},\lambda_{1}}\left(\zeta^1_{2(1)}\right) d^{\frac{1}{2}}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{2(1)}\right) + A^{3}_{\lambda_0^{\prime}, \lambda_1^{\prime}, \lambda_2^{\prime}, \lambda_3^{\prime}} d^{\frac{1}{2}}_{\lambda_1^{\prime},\lambda_{1}}\left(\zeta^1_{3(1)}\right) d^{\frac{1}{2}}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{3(1)}\right)}}\right|^{2}}\]

\[\begin{split}\displaystyle \begin{aligned} A^{2}_{- \frac{1}{2}, - \frac{1}{2}, 0, 0} \;&=\; \sum_{\lambda_{R}=-1/2}^{1/2}{- \sqrt{2} c^{\Lambda(1405)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \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}\middle|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{- \sqrt{2} c^{\Lambda(1405)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\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}\middle|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{- \sqrt{2} c^{\Lambda(1670)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}\middle|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{- \sqrt{2} c^{\Lambda(1670)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}\middle|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{- \sqrt{2} c^{\Lambda(2000)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(2000)}, \Gamma_{\Lambda(2000)}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}\middle|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{- \sqrt{2} c^{\Lambda(2000)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(2000)}, \Gamma_{\Lambda(2000)}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}\middle|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{- c^{\Lambda(1600)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \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(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}\middle|0,\frac{1}{2},-1\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{- c^{\Lambda(1600)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \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(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}\middle|0,\frac{1}{2},-1\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{26} c^{\Lambda(1520)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{2}}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}\middle|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{13}} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{26} c^{\Lambda(1520)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{2}}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}\middle|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{13}} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{26} c^{\Lambda(1690)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1690)}, \Gamma_{\Lambda(1690)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{2}}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}\middle|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{13}} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{26} c^{\Lambda(1690)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1690)}, \Gamma_{\Lambda(1690)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{2}}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}\middle|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{13}} \\ A^{3}_{- \frac{1}{2}, - \frac{1}{2}, 0, 0} \;&=\; \sum_{\lambda_{R}=-3/2}^{3/2}{c^{D(1232)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1232)}, \Gamma_{D(1232)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{3}}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(- \frac{1}{2},0\middle|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{c^{D(1232)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1232)}, \Gamma_{D(1232)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{3}}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(- \frac{1}{2},0\middle|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{c^{D(1600)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1600)}, \Gamma_{D(1600)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{3}}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(- \frac{1}{2},0\middle|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{c^{D(1600)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1600)}, \Gamma_{D(1600)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{3}}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(- \frac{1}{2},0\middle|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{26} c^{D(1700)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{3}; m_{D(1700)}, \Gamma_{D(1700)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{3}}, m_{3}\right) \mathcal{F}_{2}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(- \frac{1}{2},0\middle|\frac{1}{2},0,-1\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{13}} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{26} c^{D(1700)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{3}; m_{D(1700)}, \Gamma_{D(1700)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{3}}, m_{3}\right) \mathcal{F}_{2}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(- \frac{1}{2},0\middle|\frac{1}{2},0,-1\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{13}} \\ A^{1}_{- \frac{1}{2}, - \frac{1}{2}, 0, 0} \;&=\; \sum_{\lambda_{R}=0}{- \frac{c^{K(1430)[0]}_{- \frac{1}{2}, 0, 0} \delta_{- \frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{H}^\text{helicity}\left(\lambda_{R},- \frac{1}{2}\middle|0,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{- \sigma_{1} - 0.15769795991457068 i \left(\sigma_{1} - 0.23397706275638377\right) e^{- 0.020981 \sigma_{1}} + 1.890625}} \\ \;&+\; \sum_{\lambda_{R}=0}{- \frac{c^{K(1430)[0]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{H}^\text{helicity}\left(\lambda_{R},- \frac{1}{2}\middle|0,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{- \sigma_{1} - 0.15769795991457068 i \left(\sigma_{1} - 0.23397706275638377\right) e^{- 0.020981 \sigma_{1}} + 1.890625}} \\ \;&+\; \sum_{\lambda_{R}=0}{- \frac{c^{K(700)[0]}_{- \frac{1}{2}, 0, 0} \delta_{- \frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{H}^\text{helicity}\left(\lambda_{R},- \frac{1}{2}\middle|0,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{- \sigma_{1} - 0.88510773180649964 i \left(\sigma_{1} - 0.23397706275638377\right) e^{- 0.94106 \sigma_{1}} + 0.678976}} \\ \;&+\; \sum_{\lambda_{R}=0}{- \frac{c^{K(700)[0]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{H}^\text{helicity}\left(\lambda_{R},- \frac{1}{2}\middle|0,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{- \sigma_{1} - 0.88510773180649964 i \left(\sigma_{1} - 0.23397706275638377\right) e^{- 0.94106 \sigma_{1}} + 0.678976}} \\ \;&+\; \sum_{\lambda_{R}=-1}^{1}{- \frac{\sqrt{6} c^{K(892)[-1]}_{- \frac{1}{2}, 0, 0} \delta_{- \frac{1}{2}, \lambda_{R} + \frac{1}{2}} \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(\lambda_{R},- \frac{1}{2}\middle|-1,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\ \;&+\; \sum_{\lambda_{R}=-1}^{1}{- \frac{\sqrt{6} c^{K(892)[0]}_{- \frac{1}{2}, 0, 0} \delta_{- \frac{1}{2}, \lambda_{R} + \frac{1}{2}} \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(\lambda_{R},- \frac{1}{2}\middle|0,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\ \;&+\; \sum_{\lambda_{R}=-1}^{1}{- \frac{\sqrt{6} c^{K(892)[0]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2}, \lambda_{R} + \frac{1}{2}} \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(\lambda_{R},- \frac{1}{2}\middle|0,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\ \;&+\; \sum_{\lambda_{R}=-1}^{1}{- \frac{\sqrt{6} c^{K(892)[1]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2}, \lambda_{R} + \frac{1}{2}} \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(\lambda_{R},- \frac{1}{2}\middle|1,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\ A^{2}_{- \frac{1}{2}, \frac{1}{2}, 0, 0} \;&=\; \sum_{\lambda_{R}=-1/2}^{1/2}{\sqrt{2} c^{\Lambda(1405)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \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}\middle|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\sqrt{2} c^{\Lambda(1405)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\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}\middle|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\sqrt{2} c^{\Lambda(1670)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}\middle|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\sqrt{2} c^{\Lambda(1670)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}\middle|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\sqrt{2} c^{\Lambda(2000)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(2000)}, \Gamma_{\Lambda(2000)}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}\middle|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\sqrt{2} c^{\Lambda(2000)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(2000)}, \Gamma_{\Lambda(2000)}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}\middle|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{c^{\Lambda(1600)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \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(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}\middle|0,\frac{1}{2},-1\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{c^{\Lambda(1600)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \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(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}\middle|0,\frac{1}{2},-1\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{26} c^{\Lambda(1520)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{2}}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}\middle|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{13}} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{26} c^{\Lambda(1520)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{2}}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}\middle|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{13}} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{26} c^{\Lambda(1690)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1690)}, \Gamma_{\Lambda(1690)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{2}}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}\middle|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{13}} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{26} c^{\Lambda(1690)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1690)}, \Gamma_{\Lambda(1690)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{2}}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}\middle|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{13}} \\ A^{3}_{- \frac{1}{2}, \frac{1}{2}, 0, 0} \;&=\; \sum_{\lambda_{R}=-3/2}^{3/2}{c^{D(1232)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1232)}, \Gamma_{D(1232)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{3}}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(\frac{1}{2},0\middle|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{c^{D(1232)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1232)}, \Gamma_{D(1232)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{3}}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(\frac{1}{2},0\middle|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{c^{D(1600)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1600)}, \Gamma_{D(1600)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{3}}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(\frac{1}{2},0\middle|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{c^{D(1600)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1600)}, \Gamma_{D(1600)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{3}}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(\frac{1}{2},0\middle|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{26} c^{D(1700)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{3}; m_{D(1700)}, \Gamma_{D(1700)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{3}}, m_{3}\right) \mathcal{F}_{2}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(\frac{1}{2},0\middle|\frac{1}{2},0,-1\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{13}} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{26} c^{D(1700)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{3}; m_{D(1700)}, \Gamma_{D(1700)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{3}}, m_{3}\right) \mathcal{F}_{2}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(\frac{1}{2},0\middle|\frac{1}{2},0,-1\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{13}} \\ A^{1}_{- \frac{1}{2}, \frac{1}{2}, 0, 0} \;&=\; \sum_{\lambda_{R}=0}{\frac{c^{K(1430)[0]}_{- \frac{1}{2}, 0, 0} \delta_{- \frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{H}^\text{helicity}\left(\lambda_{R},\frac{1}{2}\middle|0,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{- \sigma_{1} - 0.15769795991457068 i \left(\sigma_{1} - 0.23397706275638377\right) e^{- 0.020981 \sigma_{1}} + 1.890625}} \\ \;&+\; \sum_{\lambda_{R}=0}{\frac{c^{K(1430)[0]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{H}^\text{helicity}\left(\lambda_{R},\frac{1}{2}\middle|0,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{- \sigma_{1} - 0.15769795991457068 i \left(\sigma_{1} - 0.23397706275638377\right) e^{- 0.020981 \sigma_{1}} + 1.890625}} \\ \;&+\; \sum_{\lambda_{R}=0}{\frac{c^{K(700)[0]}_{- \frac{1}{2}, 0, 0} \delta_{- \frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{H}^\text{helicity}\left(\lambda_{R},\frac{1}{2}\middle|0,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{- \sigma_{1} - 0.88510773180649964 i \left(\sigma_{1} - 0.23397706275638377\right) e^{- 0.94106 \sigma_{1}} + 0.678976}} \\ \;&+\; \sum_{\lambda_{R}=0}{\frac{c^{K(700)[0]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{H}^\text{helicity}\left(\lambda_{R},\frac{1}{2}\middle|0,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{- \sigma_{1} - 0.88510773180649964 i \left(\sigma_{1} - 0.23397706275638377\right) e^{- 0.94106 \sigma_{1}} + 0.678976}} \\ \;&+\; \sum_{\lambda_{R}=-1}^{1}{\frac{\sqrt{6} c^{K(892)[-1]}_{- \frac{1}{2}, 0, 0} \delta_{- \frac{1}{2}, \lambda_{R} - \frac{1}{2}} \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(\lambda_{R},\frac{1}{2}\middle|-1,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\ \;&+\; \sum_{\lambda_{R}=-1}^{1}{\frac{\sqrt{6} c^{K(892)[0]}_{- \frac{1}{2}, 0, 0} \delta_{- \frac{1}{2}, \lambda_{R} - \frac{1}{2}} \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(\lambda_{R},\frac{1}{2}\middle|0,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\ \;&+\; \sum_{\lambda_{R}=-1}^{1}{\frac{\sqrt{6} c^{K(892)[0]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2}, \lambda_{R} - \frac{1}{2}} \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(\lambda_{R},\frac{1}{2}\middle|0,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\ \;&+\; \sum_{\lambda_{R}=-1}^{1}{\frac{\sqrt{6} c^{K(892)[1]}_{\frac{1}{2}, 0, 0} \delta_{- \frac{1}{2}, \lambda_{R} - \frac{1}{2}} \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(\lambda_{R},\frac{1}{2}\middle|1,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\ A^{2}_{\frac{1}{2}, - \frac{1}{2}, 0, 0} \;&=\; \sum_{\lambda_{R}=-1/2}^{1/2}{- \sqrt{2} c^{\Lambda(1405)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \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}\middle|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{- \sqrt{2} c^{\Lambda(1405)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\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}\middle|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{- \sqrt{2} c^{\Lambda(1670)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}\middle|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{- \sqrt{2} c^{\Lambda(1670)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}\middle|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{- \sqrt{2} c^{\Lambda(2000)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(2000)}, \Gamma_{\Lambda(2000)}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}\middle|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{- \sqrt{2} c^{\Lambda(2000)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(2000)}, \Gamma_{\Lambda(2000)}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}\middle|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{- c^{\Lambda(1600)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \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(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}\middle|0,\frac{1}{2},-1\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{- c^{\Lambda(1600)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \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(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}\middle|0,\frac{1}{2},-1\right) d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{26} c^{\Lambda(1520)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{2}}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}\middle|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{13}} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{26} c^{\Lambda(1520)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{2}}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}\middle|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{13}} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{26} c^{\Lambda(1690)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1690)}, \Gamma_{\Lambda(1690)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{2}}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}\middle|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{13}} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{26} c^{\Lambda(1690)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1690)}, \Gamma_{\Lambda(1690)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{2}}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,- \frac{1}{2}\middle|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{13}} \\ A^{3}_{\frac{1}{2}, - \frac{1}{2}, 0, 0} \;&=\; \sum_{\lambda_{R}=-3/2}^{3/2}{c^{D(1232)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1232)}, \Gamma_{D(1232)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{3}}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(- \frac{1}{2},0\middle|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{c^{D(1232)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1232)}, \Gamma_{D(1232)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{3}}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(- \frac{1}{2},0\middle|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{c^{D(1600)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1600)}, \Gamma_{D(1600)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{3}}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(- \frac{1}{2},0\middle|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{c^{D(1600)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1600)}, \Gamma_{D(1600)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{3}}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(- \frac{1}{2},0\middle|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{26} c^{D(1700)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{3}; m_{D(1700)}, \Gamma_{D(1700)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{3}}, m_{3}\right) \mathcal{F}_{2}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(- \frac{1}{2},0\middle|\frac{1}{2},0,-1\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{13}} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{26} c^{D(1700)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{3}; m_{D(1700)}, \Gamma_{D(1700)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{3}}, m_{3}\right) \mathcal{F}_{2}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(- \frac{1}{2},0\middle|\frac{1}{2},0,-1\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{13}} \\ A^{1}_{\frac{1}{2}, - \frac{1}{2}, 0, 0} \;&=\; \sum_{\lambda_{R}=0}{- \frac{c^{K(1430)[0]}_{- \frac{1}{2}, 0, 0} \delta_{\frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{H}^\text{helicity}\left(\lambda_{R},- \frac{1}{2}\middle|0,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{- \sigma_{1} - 0.15769795991457068 i \left(\sigma_{1} - 0.23397706275638377\right) e^{- 0.020981 \sigma_{1}} + 1.890625}} \\ \;&+\; \sum_{\lambda_{R}=0}{- \frac{c^{K(1430)[0]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{H}^\text{helicity}\left(\lambda_{R},- \frac{1}{2}\middle|0,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{- \sigma_{1} - 0.15769795991457068 i \left(\sigma_{1} - 0.23397706275638377\right) e^{- 0.020981 \sigma_{1}} + 1.890625}} \\ \;&+\; \sum_{\lambda_{R}=0}{- \frac{c^{K(700)[0]}_{- \frac{1}{2}, 0, 0} \delta_{\frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{H}^\text{helicity}\left(\lambda_{R},- \frac{1}{2}\middle|0,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{- \sigma_{1} - 0.88510773180649964 i \left(\sigma_{1} - 0.23397706275638377\right) e^{- 0.94106 \sigma_{1}} + 0.678976}} \\ \;&+\; \sum_{\lambda_{R}=0}{- \frac{c^{K(700)[0]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{H}^\text{helicity}\left(\lambda_{R},- \frac{1}{2}\middle|0,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{- \sigma_{1} - 0.88510773180649964 i \left(\sigma_{1} - 0.23397706275638377\right) e^{- 0.94106 \sigma_{1}} + 0.678976}} \\ \;&+\; \sum_{\lambda_{R}=-1}^{1}{- \frac{\sqrt{6} c^{K(892)[-1]}_{- \frac{1}{2}, 0, 0} \delta_{\frac{1}{2}, \lambda_{R} + \frac{1}{2}} \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(\lambda_{R},- \frac{1}{2}\middle|-1,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\ \;&+\; \sum_{\lambda_{R}=-1}^{1}{- \frac{\sqrt{6} c^{K(892)[0]}_{- \frac{1}{2}, 0, 0} \delta_{\frac{1}{2}, \lambda_{R} + \frac{1}{2}} \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(\lambda_{R},- \frac{1}{2}\middle|0,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\ \;&+\; \sum_{\lambda_{R}=-1}^{1}{- \frac{\sqrt{6} c^{K(892)[0]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2}, \lambda_{R} + \frac{1}{2}} \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(\lambda_{R},- \frac{1}{2}\middle|0,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\ \;&+\; \sum_{\lambda_{R}=-1}^{1}{- \frac{\sqrt{6} c^{K(892)[1]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2}, \lambda_{R} + \frac{1}{2}} \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(\lambda_{R},- \frac{1}{2}\middle|1,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\ A^{2}_{\frac{1}{2}, \frac{1}{2}, 0, 0} \;&=\; \sum_{\lambda_{R}=-1/2}^{1/2}{\sqrt{2} c^{\Lambda(1405)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \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}\middle|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\sqrt{2} c^{\Lambda(1405)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\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}\middle|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\sqrt{2} c^{\Lambda(1670)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}\middle|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\sqrt{2} c^{\Lambda(1670)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(1670)}, \Gamma_{\Lambda(1670)}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}\middle|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\sqrt{2} c^{\Lambda(2000)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(2000)}, \Gamma_{\Lambda(2000)}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}\middle|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\sqrt{2} c^{\Lambda(2000)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=0}\left(\sigma_{2}; m_{\Lambda(2000)}, \Gamma_{\Lambda(2000)}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}\middle|0,\frac{1}{2},1\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{c^{\Lambda(1600)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \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(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}\middle|0,\frac{1}{2},-1\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{c^{\Lambda(1600)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \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(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}\middle|0,\frac{1}{2},-1\right) d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{26} c^{\Lambda(1520)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{2}}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}\middle|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{13}} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{26} c^{\Lambda(1520)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1520)}, \Gamma_{\Lambda(1520)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{2}}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}\middle|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{13}} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{26} c^{\Lambda(1690)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1690)}, \Gamma_{\Lambda(1690)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{2}}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}\middle|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{13}} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{26} c^{\Lambda(1690)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{2}; m_{\Lambda(1690)}, \Gamma_{\Lambda(1690)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{2}}, m_{2}\right) \mathcal{F}_{2}\left(\sigma_{2}, m_{3}, m_{1}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(0,\frac{1}{2}\middle|0,\frac{1}{2},-1\right) d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{13}} \\ A^{3}_{\frac{1}{2}, \frac{1}{2}, 0, 0} \;&=\; \sum_{\lambda_{R}=-3/2}^{3/2}{c^{D(1232)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1232)}, \Gamma_{D(1232)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{3}}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(\frac{1}{2},0\middle|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{c^{D(1232)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1232)}, \Gamma_{D(1232)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{3}}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(\frac{1}{2},0\middle|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{c^{D(1600)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1600)}, \Gamma_{D(1600)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{3}}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(\frac{1}{2},0\middle|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{c^{D(1600)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=1}\left(\sigma_{3}; m_{D(1600)}, \Gamma_{D(1600)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{3}}, m_{3}\right) \mathcal{F}_{1}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(\frac{1}{2},0\middle|\frac{1}{2},0,1\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{26} c^{D(1700)[-1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{3}; m_{D(1700)}, \Gamma_{D(1700)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{3}}, m_{3}\right) \mathcal{F}_{2}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|- \frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(\frac{1}{2},0\middle|\frac{1}{2},0,-1\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{13}} \\ \;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{26} c^{D(1700)[1/2]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{L=2}\left(\sigma_{3}; m_{D(1700)}, \Gamma_{D(1700)}\right) \mathcal{F}_{1}\left(m_{0}^{2}, \sqrt{\sigma_{3}}, m_{3}\right) \mathcal{F}_{2}\left(\sigma_{3}, m_{1}, m_{2}\right) \mathcal{H}^\text{helicity}\left(\lambda_{R},0\middle|\frac{1}{2},0\right) \mathcal{H}^\text{parity}\left(\frac{1}{2},0\middle|\frac{1}{2},0,-1\right) d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{13}} \\ A^{1}_{\frac{1}{2}, \frac{1}{2}, 0, 0} \;&=\; \sum_{\lambda_{R}=0}{\frac{c^{K(1430)[0]}_{- \frac{1}{2}, 0, 0} \delta_{\frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{H}^\text{helicity}\left(\lambda_{R},\frac{1}{2}\middle|0,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{- \sigma_{1} - 0.15769795991457068 i \left(\sigma_{1} - 0.23397706275638377\right) e^{- 0.020981 \sigma_{1}} + 1.890625}} \\ \;&+\; \sum_{\lambda_{R}=0}{\frac{c^{K(1430)[0]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{H}^\text{helicity}\left(\lambda_{R},\frac{1}{2}\middle|0,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{- \sigma_{1} - 0.15769795991457068 i \left(\sigma_{1} - 0.23397706275638377\right) e^{- 0.020981 \sigma_{1}} + 1.890625}} \\ \;&+\; \sum_{\lambda_{R}=0}{\frac{c^{K(700)[0]}_{- \frac{1}{2}, 0, 0} \delta_{\frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{H}^\text{helicity}\left(\lambda_{R},\frac{1}{2}\middle|0,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{- \sigma_{1} - 0.88510773180649964 i \left(\sigma_{1} - 0.23397706275638377\right) e^{- 0.94106 \sigma_{1}} + 0.678976}} \\ \;&+\; \sum_{\lambda_{R}=0}{\frac{c^{K(700)[0]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{H}^\text{helicity}\left(\lambda_{R},\frac{1}{2}\middle|0,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{- \sigma_{1} - 0.88510773180649964 i \left(\sigma_{1} - 0.23397706275638377\right) e^{- 0.94106 \sigma_{1}} + 0.678976}} \\ \;&+\; \sum_{\lambda_{R}=-1}^{1}{\frac{\sqrt{6} c^{K(892)[-1]}_{- \frac{1}{2}, 0, 0} \delta_{\frac{1}{2}, \lambda_{R} - \frac{1}{2}} \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(\lambda_{R},\frac{1}{2}\middle|-1,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\ \;&+\; \sum_{\lambda_{R}=-1}^{1}{\frac{\sqrt{6} c^{K(892)[0]}_{- \frac{1}{2}, 0, 0} \delta_{\frac{1}{2}, \lambda_{R} - \frac{1}{2}} \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(\lambda_{R},\frac{1}{2}\middle|0,- \frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\ \;&+\; \sum_{\lambda_{R}=-1}^{1}{\frac{\sqrt{6} c^{K(892)[0]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2}, \lambda_{R} - \frac{1}{2}} \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(\lambda_{R},\frac{1}{2}\middle|0,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\ \;&+\; \sum_{\lambda_{R}=-1}^{1}{\frac{\sqrt{6} c^{K(892)[1]}_{\frac{1}{2}, 0, 0} \delta_{\frac{1}{2}, \lambda_{R} - \frac{1}{2}} \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(\lambda_{R},\frac{1}{2}\middle|1,\frac{1}{2}\right) \mathcal{H}^\text{parity}\left(0,0\middle|0,0,1\right) d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\ \end{aligned}\end{split}\]

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

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

Numeric results#

intensity_expr = cached.xreplace(cached.unfold(MODEL), MODEL.variables)
intensity_expr = cached.xreplace(intensity_expr, MODEL.parameter_defaults)

Validation#

The following checks whether the full amplitude model has been deserialized correctly.

checksums = {
    checksum["point"]: checksum["value"]
    for checksum in WORKSPACE.checksums
    if checksum["distribution"] == INTENSITY_NAME
}
checksums

{'validation_point': 9345.853380852355}

checksum_points = {
    point["name"]: {par["name"]: par["value"] for par in point["parameters"]}
    for point in WORKSPACE.reference_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},
 'validation_point_m12sq': {'m_12_sq': 3.2},
 'validation_point_m23sq': {'m_23_sq': 1.4},
 'validation_point_m31sq': {'m_31_sq': 3.2}}

	Distribution	Point	Computed	Expected	Status
0	default_model	validation_point	9345.853380852332	9345.853381	🟢

Dalitz plot#

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

intensities = intensity_funcs[k](dalitz_data)

_images/b47c955081ad151ff131f12b075d47b50a107724c5eb4aae027b782c8bf19ab6.svg

	Propagator	Point	Difference	Status
0	L1405_Flatte	validation_point_m31sq	0.000000e+00	🟢
1	L1690_BW	validation_point_m31sq	1.404333e-15	🟢
2	D1232_BW	validation_point_m12sq	0.000000e+00	🟢
3	L1520_BW	validation_point_m31sq	1.570092e-16	🟢
4	L1600_BW	validation_point_m31sq	1.110223e-16	🟢
5	L2000_BW	validation_point_m31sq	0.000000e+00	🟢
6	D1600_BW	validation_point_m12sq	1.053250e-15	🟢
7	D1700_BW	validation_point_m12sq	1.241267e-16	🟢
8	K892_BW	validation_point_m23sq	3.554448e-16	🟢
9	L1670_BW	validation_point_m31sq	4.440892e-16	🟢

Model serialization

Contents

Model serialization#

Import model#

Construct ThreeBodyDecay#

Dynamics#

Vertices#

Blatt–Weisskopf form factor#

Propagators#

Breit–Wigner#

Multi-channel Breit–Wigner#

Breit–Wigner with exponential#

General propagator dynamics builder#

Validation#

Construct AmplitudeModel#

Unpolarized intensity#

Amplitude for the decay chain#

Helicity recouplings#

Recoupling deserialization#

Chain amplitudes#

Full amplitude model#

Numeric results#

Validation#

Dalitz plot#

Construct `ThreeBodyDecay`#

Construct `AmplitudeModel`#