\(\Lambda_c^+ \to p \pi^+ K^-\)#
The decay chains follow the amplitude model that LHCb published for this decay. That model is distributed as a serialized JSON file, which Model serialization imports directly. Here, the qrules particle database is instead narrowed down to the twelve resonances of that model and their masses and widths are overwritten with the published values, so that the generated model stays as close to the published one as it can.
Decay definition#
Three of the published lineshapes are not relativistic Breit–Wigner functions. The \(\bar K^*_0(700)^0\) and \(\bar K^*_0(1430)^0\) use the Bugg parametrization, whose mass and width parameters are taken over as Breit–Wigner parameters here. The \(\Lambda(1405)\) is a two-channel Flatté, where the coupling of the \(\Sigma\pi\) channel reproduces a width of \(50.5\;\mathrm{MeV}\) at the pole, which is the value listed below.
flowchart LR
T0_N0["$$\Lambda_{c}^{+}$$"]
T0_1["$$1: p$$"]
T0_2["$$2: K^{-}$$"]
T0_3["$$3: \pi^{+}$$"]
T0_N1@{ shape: text, label: " " }
T0_4("$$\begin{gathered} \Lambda(1405) \\\ \Lambda(1520) \\\ \Lambda(1600) \\\ \Lambda(1670) \\\ \Lambda(1690) \\\ \Lambda(2000) \end{gathered}$$")
T0_N0 --- T0_4
T0_4 --- T0_N1
T0_N0 --- T0_3
T0_N1 --- T0_1
T0_N1 --- T0_2
T1_N0["$$\Lambda_{c}^{+}$$"]
T1_1["$$1: p$$"]
T1_2["$$2: K^{-}$$"]
T1_3["$$3: \pi^{+}$$"]
T1_N1@{ shape: text, label: " " }
T1_4("$$\begin{gathered} \Delta(1232)^{++} \\\ \Delta(1600)^{++} \\\ \Delta(1700)^{++} \end{gathered}$$")
T1_N0 --- T1_4
T1_4 --- T1_N1
T1_N0 --- T1_2
T1_N1 --- T1_1
T1_N1 --- T1_3
T2_N0["$$\Lambda_{c}^{+}$$"]
T2_1["$$1: p$$"]
T2_2["$$2: K^{-}$$"]
T2_3["$$3: \pi^{+}$$"]
T2_N1@{ shape: text, label: " " }
T2_4("$$\begin{gathered} \overline{K}_{0}^{*}(700)^{0} \\\ \overline{K}^{*}(892)^{0} \\\ \overline{K}_{0}^{*}(1430)^{0} \end{gathered}$$")
T2_N0 --- T2_4
T2_4 --- T2_N1
T2_N0 --- T2_1
T2_N1 --- T2_2
T2_N1 --- T2_3
index |
name |
LaTeX |
\(J^P\) |
mass (MeV) |
width (MeV) |
|---|---|---|---|---|---|
0 |
|
\(\Lambda_{c}^{+}\) |
\(\frac{1}{2}^+\) |
2,286 |
0 |
1 |
|
\(p\) |
\(\frac{1}{2}^+\) |
938 |
0 |
2 |
|
\(K^{-}\) |
\(0^-\) |
493 |
0 |
3 |
|
\(\pi^{+}\) |
\(0^-\) |
139 |
0 |
name |
LaTeX |
\(J^P\) |
mass (MeV) |
width (MeV) |
|---|---|---|---|---|
|
\(\Delta(1232)^{++}\) |
\(\frac{3}{2}^+\) |
1,232 |
117 |
|
\(\Delta(1600)^{++}\) |
\(\frac{3}{2}^+\) |
1,640 |
300 |
|
\(\Delta(1700)^{++}\) |
\(\frac{3}{2}^-\) |
1,690 |
380 |
|
\(\overline{K}_{0}^{*}(700)^{0}\) |
\(0^+\) |
824 |
478 |
|
\(\overline{K}^{*}(892)^{0}\) |
\(1^-\) |
895 |
47 |
|
\(\overline{K}_{0}^{*}(1430)^{0}\) |
\(0^+\) |
1,375 |
190 |
|
\(\Lambda(1405)\) |
\(\frac{1}{2}^-\) |
1,405 |
50 |
|
\(\Lambda(1520)\) |
\(\frac{3}{2}^-\) |
1,518 |
15 |
|
\(\Lambda(1600)\) |
\(\frac{1}{2}^+\) |
1,630 |
250 |
|
\(\Lambda(1670)\) |
\(\frac{1}{2}^-\) |
1,670 |
30 |
|
\(\Lambda(1690)\) |
\(\frac{3}{2}^-\) |
1,690 |
70 |
|
\(\Lambda(2000)\) |
\(\frac{1}{2}^-\) |
1,988 |
179 |
\[\begin{split}\begin{array}{c}
\Lambda_{c}^{+}\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=1} \left(\Delta(1232)^{++}\left[\frac{3}{2}^+\right] \xrightarrow[S=1/2]{L=1} p\left[\frac{1}{2}^+\right] \pi^{+}\left[0^-\right]\right) K^{-}\left[0^-\right] \\
\Lambda_{c}^{+}\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=1} \left(\Delta(1600)^{++}\left[\frac{3}{2}^+\right] \xrightarrow[S=1/2]{L=1} p\left[\frac{1}{2}^+\right] \pi^{+}\left[0^-\right]\right) K^{-}\left[0^-\right] \\
\Lambda_{c}^{+}\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=1} \left(\Delta(1700)^{++}\left[\frac{3}{2}^-\right] \xrightarrow[S=1/2]{L=2} p\left[\frac{1}{2}^+\right] \pi^{+}\left[0^-\right]\right) K^{-}\left[0^-\right] \\
\Lambda_{c}^{+}\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(\overline{K}_{0}^{*}(1430)^{0}\left[0^+\right] \xrightarrow[S=0]{L=0} K^{-}\left[0^-\right] \pi^{+}\left[0^-\right]\right) p\left[\frac{1}{2}^+\right] \\
\Lambda_{c}^{+}\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(\overline{K}_{0}^{*}(700)^{0}\left[0^+\right] \xrightarrow[S=0]{L=0} K^{-}\left[0^-\right] \pi^{+}\left[0^-\right]\right) p\left[\frac{1}{2}^+\right] \\
\Lambda_{c}^{+}\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(\overline{K}^{*}(892)^{0}\left[1^-\right] \xrightarrow[S=0]{L=1} K^{-}\left[0^-\right] \pi^{+}\left[0^-\right]\right) p\left[\frac{1}{2}^+\right] \\
\Lambda_{c}^{+}\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(\Lambda(1405)\left[\frac{1}{2}^-\right] \xrightarrow[S=1/2]{L=0} p\left[\frac{1}{2}^+\right] K^{-}\left[0^-\right]\right) \pi^{+}\left[0^-\right] \\
\Lambda_{c}^{+}\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=1} \left(\Lambda(1520)\left[\frac{3}{2}^-\right] \xrightarrow[S=1/2]{L=2} p\left[\frac{1}{2}^+\right] K^{-}\left[0^-\right]\right) \pi^{+}\left[0^-\right] \\
\Lambda_{c}^{+}\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(\Lambda(1600)\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=1} p\left[\frac{1}{2}^+\right] K^{-}\left[0^-\right]\right) \pi^{+}\left[0^-\right] \\
\Lambda_{c}^{+}\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(\Lambda(1670)\left[\frac{1}{2}^-\right] \xrightarrow[S=1/2]{L=0} p\left[\frac{1}{2}^+\right] K^{-}\left[0^-\right]\right) \pi^{+}\left[0^-\right] \\
\Lambda_{c}^{+}\left[\frac{1}{2}^+\right] \xrightarrow[S=3/2]{L=1} \left(\Lambda(1690)\left[\frac{3}{2}^-\right] \xrightarrow[S=1/2]{L=2} p\left[\frac{1}{2}^+\right] K^{-}\left[0^-\right]\right) \pi^{+}\left[0^-\right] \\
\Lambda_{c}^{+}\left[\frac{1}{2}^+\right] \xrightarrow[S=1/2]{L=0} \left(\Lambda(2000)\left[\frac{1}{2}^-\right] \xrightarrow[S=1/2]{L=0} p\left[\frac{1}{2}^+\right] K^{-}\left[0^-\right]\right) \pi^{+}\left[0^-\right] \\
\end{array}\end{split}\]
Lineshapes for dynamics#
\[\begin{split}\begin{aligned}
\mathcal{R}^\mathrm{BW}_{l_{R},l_{\Lambda_c}}\left(s\right) \;&=\; \frac{\frac{\mathcal{F}_{l_{R}}\left(s, m_{1}, m_{2}\right)}{\mathcal{F}_{l_{R}}\left(m_{0}^{2}, m_{1}, m_{2}\right)} \frac{\mathcal{F}_{l_{\Lambda_c}}\left(m_\mathrm{top}^{2}, \sqrt{s}, m_\mathrm{spectator}\right)}{\mathcal{F}_{l_{\Lambda_c}}\left(m_\mathrm{top}^{2}, m_{0}, m_\mathrm{spectator}\right)}}{m_{0}^{2} - i m_{0} \Gamma_{0}\left(s\right) - s} \\
\frac{\frac{\mathcal{F}_{l_{R}}\left(s, m_{1}, m_{2}\right)}{\mathcal{F}_{l_{R}}\left(m_{0}^{2}, m_{1}, m_{2}\right)} \frac{\mathcal{F}_{l_{\Lambda_c}}\left(m_\mathrm{top}^{2}, \sqrt{s}, m_\mathrm{spectator}\right)}{\mathcal{F}_{l_{\Lambda_c}}\left(m_\mathrm{top}^{2}, m_{0}, m_\mathrm{spectator}\right)}}{m_{0}^{2} - i m_{0} \Gamma_{0}\left(s\right) - s} \;&=\; \frac{\mathcal{F}_{l_{\Lambda_c}}\left(m_\mathrm{top}^{2}, \sqrt{s}, m_\mathrm{spectator}\right) \mathcal{F}_{l_{R}}\left(s, m_{1}, m_{2}\right)}{\left(m_{0}^{2} - i m_{0} \Gamma_{0}\left(s\right) - s\right) \mathcal{F}_{l_{R}}\left(m_{0}^{2}, m_{1}, m_{2}\right) \mathcal{F}_{l_{\Lambda_c}}\left(m_\mathrm{top}^{2}, m_{0}, m_\mathrm{spectator}\right)} \\
\mathcal{F}_{l_{\Lambda_c}}\left(s, m_{1}, m_\mathrm{spectator}\right) \;&=\; \sqrt{B_{l_{\Lambda_c}}^2\left(R_{\Lambda_c}^{2} q^2\left(s\right)\right)} \\
\Gamma_{0}\left(s\right) \;&=\; \frac{\Gamma_{0} \mathcal{F}_{l_{R}}\left(s, m_{1}, m_{2}\right)^{2} \rho\left(s\right)}{\mathcal{F}_{l_{R}}\left(m_{0}^{2}, m_{1}, m_{2}\right)^{2} \rho_{0}\left(m_{0}^{2}\right)} \\
B_{l_{\Lambda_c}}^2\left(R_{\Lambda_c}^{2} q^2\left(s\right)\right) \;&=\; \frac{\left|{h_{l_{\Lambda_c}}^{(1)}\left(1\right)}\right|^{2}}{R_{\Lambda_c}^{2} \left|{h_{l_{\Lambda_c}}^{(1)}\left(R_{\Lambda_c} \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_\mathrm{spectator}\right)^{2}\right) \left(s - \left(m_{1} + m_\mathrm{spectator}\right)^{2}\right)}{4 s} \\
\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} \\
h_{l_{\Lambda_c}}^{(1)}\left(z\right) \;&=\; \frac{\left(- i\right)^{l_{\Lambda_c} + 1} e^{i z} \sum_{k=0}^{l_{\Lambda_c}} \frac{\left(\frac{i}{2 z}\right)^{k} \left(k + l_{\Lambda_c}\right)!}{k! \left(- k + l_{\Lambda_c}\right)!}}{z} \\
\end{aligned}\end{split}\]
Model formulation#
model_builder = DalitzPlotDecompositionBuilder(DECAY, min_ls=(False, True))
for chain in model_builder.decay.chains:
model_builder.dynamics_choices.register_builder(chain, formulate_breit_wigner)
model = model_builder.formulate()
model.intensity
\[\displaystyle \sum_{\lambda_{0}=-1/2}^{1/2} \sum_{\lambda_{1}=-1/2}^{1/2} \sum_{\lambda_{2}=0} \sum_{\lambda_{3}=0}{\left|{\sum_{\lambda_0^{\prime}=-1/2}^{1/2} \sum_{\lambda_1^{\prime}=-1/2}^{1/2} \sum_{\lambda_2^{\prime}=0} \sum_{\lambda_3^{\prime}=0}{A^{1}_{\lambda_0^{\prime}, \lambda_1^{\prime}, \lambda_2^{\prime}, \lambda_3^{\prime}} d^{\frac{1}{2}}_{\lambda_1^{\prime},\lambda_{1}}\left(\zeta^1_{1(3)}\right) d^{\frac{1}{2}}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{1(3)}\right) + A^{2}_{\lambda_0^{\prime}, \lambda_1^{\prime}, \lambda_2^{\prime}, \lambda_3^{\prime}} d^{\frac{1}{2}}_{\lambda_1^{\prime},\lambda_{1}}\left(\zeta^1_{2(3)}\right) d^{\frac{1}{2}}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{2(3)}\right) + A^{3}_{\lambda_0^{\prime}, \lambda_1^{\prime}, \lambda_2^{\prime}, \lambda_3^{\prime}} d^{\frac{1}{2}}_{\lambda_1^{\prime},\lambda_{1}}\left(\zeta^1_{3(3)}\right) d^{\frac{1}{2}}_{\lambda_{0},\lambda_0^{\prime}}\left(\zeta^0_{3(3)}\right)}}\right|^{2}}\]
Notice that subsystem 3 (with the \(\varLambda^*\) resonances and recoil \(\pi^+\)) has been selected as reference subsystem for the Wigner‑\(d\) functions. This is computationally more efficient, because that subsystem contains most resonances.
Latex(aslatex(model.amplitudes, terms_per_line=1))
\[\begin{split}\begin{aligned}
A^{1}_{- \frac{1}{2}, - \frac{1}{2}, 0, 0} \;&=\; \sum_{\lambda_{R}=-1}^{1}{- \frac{\sqrt{2} \delta_{- \frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{1,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{1,\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}^{*}(892)^{0}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}^{*}(892)^{0}, 0, 0} d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=0}{- \frac{\sqrt{2} \delta_{- \frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}_{0}^{*}(1430)^{0}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}_{0}^{*}(1430)^{0}, 0, 0} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=0}{- \frac{\sqrt{2} \delta_{- \frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}_{0}^{*}(700)^{0}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}_{0}^{*}(700)^{0}, 0, 0} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\
A^{2}_{- \frac{1}{2}, - \frac{1}{2}, 0, 0} \;&=\; \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{6} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,1}\left(\sigma_{2}\right) C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Delta(1232)^{++}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Delta(1232)^{++}, 0, - \frac{1}{2}} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{6} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,1}\left(\sigma_{2}\right) C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Delta(1600)^{++}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Delta(1600)^{++}, 0, - \frac{1}{2}} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{6} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{2}\right) C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Delta(1700)^{++}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Delta(1700)^{++}, 0, - \frac{1}{2}} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} \\
A^{3}_{- \frac{1}{2}, - \frac{1}{2}, 0, 0} \;&=\; \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{3}\right) C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1405), - \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,0}\left(\sigma_{3}\right) C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1600), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1600), - \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{3}\right) C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1670), - \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{3}\right) C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Lambda(2000), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(2000), - \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{3}\right) C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1520), 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1520), - \frac{1}{2}, 0} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{3}\right) C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1690), 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1690), - \frac{1}{2}, 0} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} \\
A^{1}_{- \frac{1}{2}, \frac{1}{2}, 0, 0} \;&=\; \sum_{\lambda_{R}=-1}^{1}{\frac{\sqrt{2} \delta_{- \frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{1,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} - \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} - \frac{1}{2}}_{1,\lambda_{R},\frac{1}{2},- \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}^{*}(892)^{0}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}^{*}(892)^{0}, 0, 0} d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=0}{\frac{\sqrt{2} \delta_{- \frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} - \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} - \frac{1}{2}}_{0,\lambda_{R},\frac{1}{2},- \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}_{0}^{*}(1430)^{0}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}_{0}^{*}(1430)^{0}, 0, 0} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=0}{\frac{\sqrt{2} \delta_{- \frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} - \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} - \frac{1}{2}}_{0,\lambda_{R},\frac{1}{2},- \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}_{0}^{*}(700)^{0}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}_{0}^{*}(700)^{0}, 0, 0} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\
A^{2}_{- \frac{1}{2}, \frac{1}{2}, 0, 0} \;&=\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,1}\left(\sigma_{2}\right) C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Delta(1232)^{++}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Delta(1232)^{++}, 0, \frac{1}{2}} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,1}\left(\sigma_{2}\right) C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Delta(1600)^{++}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Delta(1600)^{++}, 0, \frac{1}{2}} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{2}\right) C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Delta(1700)^{++}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Delta(1700)^{++}, 0, \frac{1}{2}} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} \\
A^{3}_{- \frac{1}{2}, \frac{1}{2}, 0, 0} \;&=\; \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{3}\right) C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1405), \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,0}\left(\sigma_{3}\right) C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1600), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1600), \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{3}\right) C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1670), \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{3}\right) C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Lambda(2000), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(2000), \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{3}\right) C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1520), 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1520), \frac{1}{2}, 0} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{- \frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{3}\right) C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1690), 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1690), \frac{1}{2}, 0} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} \\
A^{1}_{\frac{1}{2}, - \frac{1}{2}, 0, 0} \;&=\; \sum_{\lambda_{R}=-1}^{1}{- \frac{\sqrt{2} \delta_{\frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{1,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{1,\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}^{*}(892)^{0}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}^{*}(892)^{0}, 0, 0} d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=0}{- \frac{\sqrt{2} \delta_{\frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}_{0}^{*}(1430)^{0}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}_{0}^{*}(1430)^{0}, 0, 0} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=0}{- \frac{\sqrt{2} \delta_{\frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}_{0}^{*}(700)^{0}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}_{0}^{*}(700)^{0}, 0, 0} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\
A^{2}_{\frac{1}{2}, - \frac{1}{2}, 0, 0} \;&=\; \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{6} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,1}\left(\sigma_{2}\right) C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Delta(1232)^{++}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Delta(1232)^{++}, 0, - \frac{1}{2}} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{6} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,1}\left(\sigma_{2}\right) C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Delta(1600)^{++}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Delta(1600)^{++}, 0, - \frac{1}{2}} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{- \frac{\sqrt{6} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{2}\right) C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Delta(1700)^{++}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Delta(1700)^{++}, 0, - \frac{1}{2}} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{31}\right)}{2}} \\
A^{3}_{\frac{1}{2}, - \frac{1}{2}, 0, 0} \;&=\; \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{3}\right) C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1405), - \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,0}\left(\sigma_{3}\right) C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1600), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1600), - \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{3}\right) C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1670), - \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{3}\right) C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Lambda(2000), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(2000), - \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{3}\right) C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1520), 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1520), - \frac{1}{2}, 0} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{3}\right) C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1690), 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1690), - \frac{1}{2}, 0} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{12}\right)}{2}} \\
A^{1}_{\frac{1}{2}, \frac{1}{2}, 0, 0} \;&=\; \sum_{\lambda_{R}=-1}^{1}{\frac{\sqrt{2} \delta_{\frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{1,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} - \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} - \frac{1}{2}}_{1,\lambda_{R},\frac{1}{2},- \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}^{*}(892)^{0}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}^{*}(892)^{0}, 0, 0} d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=0}{\frac{\sqrt{2} \delta_{\frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} - \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} - \frac{1}{2}}_{0,\lambda_{R},\frac{1}{2},- \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}_{0}^{*}(1430)^{0}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}_{0}^{*}(1430)^{0}, 0, 0} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=0}{\frac{\sqrt{2} \delta_{\frac{1}{2}, \lambda_{R} - \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} - \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} - \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} - \frac{1}{2}}_{0,\lambda_{R},\frac{1}{2},- \frac{1}{2}} \mathcal{H}^\mathrm{LS,production}_{\overline{K}_{0}^{*}(700)^{0}, 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\overline{K}_{0}^{*}(700)^{0}, 0, 0} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\
A^{2}_{\frac{1}{2}, \frac{1}{2}, 0, 0} \;&=\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,1}\left(\sigma_{2}\right) C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Delta(1232)^{++}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Delta(1232)^{++}, 0, \frac{1}{2}} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,1}\left(\sigma_{2}\right) C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Delta(1600)^{++}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Delta(1600)^{++}, 0, \frac{1}{2}} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{2}\right) C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Delta(1700)^{++}, 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Delta(1700)^{++}, 0, \frac{1}{2}} d^{\frac{3}{2}}_{\lambda_{R},- \frac{1}{2}}\left(\theta_{31}\right)}{2}} \\
A^{3}_{\frac{1}{2}, \frac{1}{2}, 0, 0} \;&=\; \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{3}\right) C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1405), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1405), \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{1,0}\left(\sigma_{3}\right) C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1600), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1600), \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{3}\right) C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1670), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1670), \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-1/2}^{1/2}{\frac{\sqrt{2} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{3}\right) C^{\frac{1}{2},\lambda_{R}}_{0,0,\frac{1}{2},\lambda_{R}} C^{\frac{1}{2},\lambda_{R}}_{\frac{1}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Lambda(2000), 0, \frac{1}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(2000), \frac{1}{2}, 0} d^{\frac{1}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{3}\right) C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1520), 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1520), \frac{1}{2}, 0} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=-3/2}^{3/2}{\frac{\sqrt{6} \delta_{\frac{1}{2} \lambda_{R}} \mathcal{R}^\mathrm{BW}_{2,1}\left(\sigma_{3}\right) C^{\frac{1}{2},\lambda_{R}}_{1,0,\frac{3}{2},\lambda_{R}} C^{\frac{3}{2},\lambda_{R}}_{\frac{3}{2},\lambda_{R},0,0} \mathcal{H}^\mathrm{LS,production}_{\Lambda(1690), 1, \frac{3}{2}} \mathcal{H}^\mathrm{decay}_{\Lambda(1690), \frac{1}{2}, 0} d^{\frac{3}{2}}_{\lambda_{R},\frac{1}{2}}\left(\theta_{12}\right)}{2}} \\
\end{aligned}\end{split}\]
By default, the aligned amplitudes are built up of Wigner-\(d\) functions, Clebsch–Gordan coefficients (\(C\)), a resonance parametrization (\(\mathcal{R}(\sigma)\)), and two coupling symbols \(\mathcal{H}^\text{prod}_{\dots}, \mathcal{H}^\text{dec}_{\dots}\). In some cases, you want to combine the couplings into one scaling factor. That can be done with the use_coefficients flag. The following example also explicitly specifies the reference subsystem (which is otherwise automatically chosen).
coefficient_model = model_builder.formulate(
reference_subsystem=1, use_coefficients=True
)
\[\begin{split}\begin{aligned}
A^{1}_{- \frac{1}{2}, - \frac{1}{2}, 0, 0} \;&=\; \sum_{\lambda_{R}=-1}^{1}{- \frac{\sqrt{2} \delta_{- \frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{1,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{1,\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{H}^\mathrm{\lambda,LS,\overline{K}^{*}(892)^{0}}_{0, \frac{1}{2}, 0, 0} d^{1}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=0}{- \frac{\sqrt{2} \delta_{- \frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{H}^\mathrm{\lambda,LS,\overline{K}_{0}^{*}(1430)^{0}}_{0, \frac{1}{2}, 0, 0} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\
\;&+\; \sum_{\lambda_{R}=0}{- \frac{\sqrt{2} \delta_{- \frac{1}{2}, \lambda_{R} + \frac{1}{2}} \mathcal{R}^\mathrm{BW}_{0,0}\left(\sigma_{1}\right) C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,0,\frac{1}{2},\lambda_{R} + \frac{1}{2}} C^{\frac{1}{2},\lambda_{R} + \frac{1}{2}}_{0,\lambda_{R},\frac{1}{2},\frac{1}{2}} \mathcal{H}^\mathrm{\lambda,LS,\overline{K}_{0}^{*}(700)^{0}}_{0, \frac{1}{2}, 0, 0} d^{0}_{\lambda_{R},0}\left(\theta_{23}\right)}{2}} \\
\end{aligned}\end{split}\]
Dalitz plot#
The intensity can now be rendered over the Dalitz plane, just as in Model serialization, with the difference that the model here was formulated from the ThreeBodyDecay above instead of imported from a serialized JSON file. There is therefore no compiler that infers the kinematics from serialized metadata: the amplitudes are written in terms of helicity angles, so the two Mandelstam variables that span the plane first have to be converted with a SympyDataTransformer.
Note that the couplings all carry their default value of \(1\), so the relative strength of the resonances is not the one that LHCb fitted. It is the lineshapes and the phase space that carry the published values.
The third Mandelstam variable is fixed by the other two.
\[\begin{split}\begin{aligned}
\sigma_{2} \;&=\; m_{0}^{2} + m_{1}^{2} + m_{2}^{2} + m_{3}^{2} - \sigma_{1} - \sigma_{3} \\
\end{aligned}\end{split}\]
intensity_expr = cached.xreplace(
cached.unfold(model),
{**model.parameter_defaults, **model.masses},
)
intensity_func = cached.lambdify(intensity_expr, backend="jax")
intensities = intensity_func(dalitz_data)
Tip
Compare this Dalitz plot, built directly from the reaction, with the one produced from the deserialized LHCb model in Model serialization.